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Q: Firebase logs "undefined" on console when it should not I am using firebase 10.9.2 (Latest version)
I use the onValue function to go through the data in my live database. For some reason, it always logs (and is set to) undefined.
Due to how fast firebase evolves, I can not find a fix for this current issue.
Please tell me what can I do with the code so that it stops returning undefined.
// Import the functions you need from the SDKs you need
import { initializeApp } from "https://www.gstatic.com/firebasejs/9.6.8/firebase-app.js";
// TODO: Add SDKs for Firebase products that you want to use
// https://firebase.google.com/docs/web/setup#available-libraries
// Your web app's Firebase configuration
const firebaseConfig = {
apiKey: "AIzaSyBU21edukO9sxUR13wYy6yYW9ZYK5ax4tM",
authDomain: "team-seas-demo.firebaseapp.com",
databaseURL: "https://team-seas-demo-default-rtdb.asia-southeast1.firebasedatabase.app",
projectId: "team-seas-demo",
storageBucket: "team-seas-demo.appspot.com",
messagingSenderId: "864989587728",
appId: "1:864989587728:web:7eda88906cd9a7fe659d95"
};
// Initialize Firebase
const app = initializeApp(firebaseConfig);
import {getDatabase, ref, set, child, update, remove, onValue}
from "https://www.gstatic.com/firebasejs/9.6.8/firebase-database.js";
var db = getDatabase(app);
onValue(ref(db, "Groups/"), function (snapshot){
const data = snapshot.val();
console.log(data); //This part works, but returns the whole object
console.log(data.GroupName) //This part returns undefined...
});
P.S: Yes, my rules on read are set to true
This is how my database looks:
Image of database
A: I think it's better to use the onSnapshot() function
See an example here https://firebase.google.com/docs/firestore/query-data/listen
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,491
|
Arthroleptella är ett släkte av groddjur. Arthroleptella ingår i familjen Pyxicephalidae.
Kladogram enligt Catalogue of Life:
Källor
Externa länkar
Stjärtlösa groddjur
Arthroleptella
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,785
|
package cz.metacentrum.perun.webgui.model;
import com.google.gwt.core.client.JavaScriptObject;
/**
* Overlay type for MemberCandidate object.
*
* @author Pavel Zlámal <zlamal@cesnet.cz>
*/
public class MemberCandidate extends JavaScriptObject {
protected MemberCandidate() {}
/**
* Returns associated RichUser
* @return RichUser
*/
public final native User getRichUser() /*-{
return this.richUser;
}-*/;
/**
* Returns associated Candidate
* @return Candidate
*/
public final native Candidate getCandidate() /*-{
return this.candidate;
}-*/;
/**
* Returns associated Member
* @return Member
*/
public final native Member getMember() /*-{
return this.member;
}-*/;
/**
* Returns Perun specific type of object
*
* @return type of object
*/
public final native String getObjectType() /*-{
if (!this.beanName) {
return "JavaScriptObject"
}
return this.objecttype;
}-*/;
/**
* Sets Perun specific type of object
*
* @param type type of object
*/
public final native void setObjectType(String type) /*-{
this.objecttype = type;
}-*/;
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,708
|
Q: Image extension JPG or PNG I wrote a simple Python program and its working fine.
Basically what I wrote that enter image URL and its save to the computer but I want to add one more extra thing that couldn't fix it. I want to avoid error if image URL don't have Jpg or PNG format. How Should i do that?
Code is below:
import requests
def download():
url = input("Enter Image URL >> ")
response = requests.get(url)
file_name = url.split("/")[-1]
if file_name == "jpg":
with open(file_name, "wb") as saveImg:
if saveImg.write(response.content):
print("[+] Download Complete")
else:
print("[-] Download Failed.")
else:
print("Failed")
download()
A: You wrote:
if file_name == "jpg":
You want:
import os
_, ext = os.path.splitext(file_name)
if ext == "jpg":
or perhaps:
if ext in ("jpg", "png"):
or
if ext.lower() in ("jpg", "png"):
If you don't import os, you might still be able to get away with something like if file_name.endswith(".jpg").
A: updated code >>>
import requests
import os.path
def download():
url = input("Enter Image URL >> ")
response = requests.get(url)
file_name = url.split("/")[-1]
_, ext = os.path.splitext(file_name)
if ext in (".jpg", ".png"):
with open(file_name, "wb") as saveImg:
if saveImg.write(response.content):
print("[+] Download Complete")
else:
print("[-] Download Failed.")
else:
print("[-] Please Enter Valid Image URL.")
download()
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,877
|
package org.kaleidofoundry.core.store.module;
import org.junit.Before;
import org.junit.Test;
import org.kaleidofoundry.core.context.Context;
import org.kaleidofoundry.core.context.RuntimeContext;
import org.kaleidofoundry.core.store.ClasspathFileStore;
import org.kaleidofoundry.core.store.FileStore;
import org.kaleidofoundry.core.store.FileStoreConstants;
import org.kaleidofoundry.core.store.FileSystemStore;
import org.kaleidofoundry.core.store.FtpStore;
import org.kaleidofoundry.core.store.HttpFileStore;
import org.kaleidofoundry.core.store.annotation.Classpath;
import org.kaleidofoundry.core.store.annotation.File;
import org.kaleidofoundry.core.store.annotation.Ftp;
import org.kaleidofoundry.core.store.annotation.Http;
import com.google.inject.Guice;
import com.google.inject.Inject;
import com.google.inject.Injector;
import com.google.inject.name.Named;
import static org.junit.Assert.*;
/**
* @author jraduget
*/
public class RawResourceModuleTest {
private Injector injector;
private Sample sample;
@Before
public void setup() {
// guice injector
injector = Guice.createInjector(new org.kaleidofoundry.core.store.module.FileStoreModule());
// test guice instance
sample = injector.getInstance(Sample.class);
}
@Test
public void defaultFileStore() {
assertNotNull(sample);
assertNotNull(sample.defaultStore);
assertTrue(sample.defaultStore instanceof FileSystemStore);
}
// ** find fileStore by binding annotation **********************************************************************
@Test
public void fileStoreByAnnotation() {
assertNotNull(sample);
assertNotNull(sample.fileSystemStore);
assertTrue(sample.fileSystemStore instanceof FileSystemStore);
}
@Test
public void ftpStoreByAnnotation() {
assertNotNull(sample);
assertNotNull(sample.ftpStore);
assertTrue(sample.ftpStore instanceof FtpStore);
}
@Test
public void httpFileStoreByAnnotation() {
assertNotNull(sample);
assertNotNull(sample.httpStore);
assertTrue(sample.httpStore instanceof HttpFileStore);
}
@Test
public void classpathFileStoreByAnnotation() {
assertNotNull(sample);
assertNotNull(sample.classpathStore);
assertTrue(sample.classpathStore instanceof ClasspathFileStore);
}
// ** find fileStore by qualifier name **************************************************************************
@Test
public void fileFileStoreByName() {
assertNotNull(sample);
assertNotNull(sample.fileSystemStore);
assertTrue(sample.fileSystemStore instanceof FileSystemStore);
}
@Test
public void ftpStoreByName() {
assertNotNull(sample);
assertNotNull(sample.ftpNamedStore);
assertTrue(sample.ftpNamedStore instanceof FtpStore);
}
@Test
public void httpFileStoreByName() {
assertNotNull(sample);
assertNotNull(sample.httpNamedStore);
assertTrue(sample.httpNamedStore instanceof HttpFileStore);
}
@Test
public void classpathFileStoreByName() {
assertNotNull(sample);
assertNotNull(sample.classpathNamedStore);
assertTrue(sample.classpathNamedStore instanceof ClasspathFileStore);
}
}
/**
* sample class to test injection without any {@link RuntimeContext} / {@link Context}
*/
class Sample {
// default file store
@Inject
FileStore defaultStore;
// injection using custom binding guice annotation
@Inject
@Ftp
FileStore ftpStore;
@Inject
@Http
FileStore httpStore;
@Inject
@File
FileStore fileSystemStore;
@Inject
@Classpath
FileStore classpathStore;
// injection using custom binding guice named
@Inject
@Named(FileStoreConstants.FtpStorePluginName)
FileStore ftpNamedStore;
@Inject
@Named(FileStoreConstants.HttpStorePluginName)
FileStore httpNamedStore;
@Inject
@Named(FileStoreConstants.FileSystemStorePluginName)
FileStore fileNamedSystemStore;
@Inject
@Named(FileStoreConstants.ClasspathStorePluginName)
FileStore classpathNamedStore;
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,731
|
\section*{\abstractname}%
\else
\begin{center}%
{\bfseries \normalsize\abstractname\vspace{\z@}
\end{center}%
\quotation
\fi}
{\if@twocolumn\else\endquotation\fi}
\makeatother
\crefname{figure}{Figure}{Figures}
\labelcrefformat{subequation}{#2(#1)#3}
\labelcrefrangeformat{subequation}{#3(#1)#4 to #5(#2)#6}
\let\originalleft\left
\let\originalright\right
\renewcommand*{\left}{\mathopen{}\mathclose\bgroup\originalleft}
\renewcommand*{\right}{\aftergroup\egroup\originalright}
\newcommand*{\seccoord}[1]{\ensuremath\hat{#1}}
\newcommand*{\secx}{\ensuremath\seccoord{x}}
\newcommand*{\secy}{\ensuremath\seccoord{y}}
\newcommand*{\secz}{\ensuremath\seccoord{z}}
\newcommand*{\ratsec}[1]{\ensuremath\hat{#1}}
\newcommand*{\ensuremath\ratsec{o}}{\ensuremath\ratsec{o}}
\newcommand*{\sechomol}[1]{\ensuremath\mathcal{#1}}
\newcommand*{\ensuremath\sechomol{Z}}{\ensuremath\sechomol{Z}}
\newcommand*{\ensuremath V_{q = 1}}{\ensuremath V_{q = 1}}
\newcommand*{\ensuremath\Delta_{(a)}}{\ensuremath\Delta_{(a)}}
\DeclareMathOperator{\Adj}{Adj}
\DeclareMathOperator{\SO}{SO}
\DeclareMathOperator{\Sp}{Sp}
\DeclareMathOperator{\SU}{SU}
\DeclareMathOperator{\U}{U}
\DeclareMathOperator{\gE}{E}
\DeclareMathOperator{\gF}{F}
\DeclareMathOperator{\gG}{G}
\newcommand*{\ensuremath\mathfrak{su}}{\ensuremath\mathfrak{su}}
\newcommand*{\ensuremath\mathfrak{u}}{\ensuremath\mathfrak{u}}
\newcommand*{\mathcal{N}}{\mathcal{N}}
\newcommand*{\mathcal{O}}{\mathcal{O}}
\newcommand*{\mathcal{G}}{\mathcal{G}}
\newcommand*{\mathcal{S}}{\mathcal{S}}
\newcommand*{\mathbb{P}}{\mathbb{P}}
\newcommand*{\mathbb{F}}{\mathbb{F}}
\newcommand*{\mathbb{Z}}{\mathbb{Z}}
\newcommand*{\mathbb{Q}}{\mathbb{Q}}
\newcommand*{\mathbb{C}}{\mathbb{C}}
\newcommand*{\mathbb{R}}{\mathbb{R}}
\newcommand*{\mathsf{G}}{\mathsf{G}}
\newcommand*{\mathsf{r}}{\mathsf{r}}
\newcommand*{\mathsf{R}}{\mathsf{R}}
\newcommand*{\mathscr{B}}{\mathscr{B}}
\newcommand*{\mathscr{I}}{\mathscr{I}}
\newcommand*{\mathscr{L}}{\mathscr{L}}
\newcommand*{\mathscr{O}}{\mathscr{O}}
\newcommand*{\mathscr{V}}{\mathscr{V}}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\Res}{Res}
\newcommand*{\ensuremath\SU(3) \times \SU(2) \times \U(1)}{\ensuremath\SU(3) \times \SU(2) \times \U(1)}
\newcommand*{\ensuremath(\SU(3) \times \SU(2) \times \U(1)) / \Z_6}{\ensuremath(\SU(3) \times \SU(2) \times \U(1)) / \mathbb{Z}_6}
\newcommand*{\ensuremath\asu(3) \oplus \asu(2) \oplus \au(1)}{\ensuremath\ensuremath\mathfrak{su}(3) \oplus \ensuremath\mathfrak{su}(2) \oplus \ensuremath\mathfrak{u}(1)}
\newcommand*{\ensuremath\SU(4) \times \SU(3) \times \SU(2)}{\ensuremath\SU(4) \times \SU(3) \times \SU(2)}
\newcommand*{\ensuremath(\SU(4) \times \SU(2) \times \SU(2)) / \Z_2}{\ensuremath(\SU(4) \times \SU(2) \times \SU(2)) / \mathbb{Z}_2}
\newcommand*{\dzerotilde}[0]{\ensuremath\tilde{d}_0}
\newcommand*{\canonclass}[0]{\ensuremath K}
\newcommand*{\mat}[2][b]{\ensuremath\begin{#1matrix}#2\end{#1matrix}}
\newcommand*{\locus}[1]{\ensuremath\{#1\}}
\newcommand*{\tuning}[2]{\ensuremath#1 \to #2}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\Id}{Id}
\DeclareMathOperator{\PD}{PD}
\DeclareMathOperator{\rk}{rk}
\DeclareMathOperator{\sign}{sign}
\DeclareMathOperator{\Span}{span}
\DeclareMathOperator{\trace}{tr}
\newcommand*{\ensuremath\mathrm{t}}{\ensuremath\mathrm{t}}
\newcommand*{\mathcal{F}}{\mathcal{F}}
\newcommand*{\mathfrak{g}}{\mathfrak{g}}
\newcommand*{\mult}[1]{\ensuremath n_{#1}}
\newcommand*{\nv}[1]{\ensuremath \nu_{\langle #1 \rangle}}
\newcommand{\multDim}[2][]{\ensuremath x^{#1}_{#2}}
\newcommand*{\chIndex}[1]{\ensuremath\chi_{#1}}
\newcommand*{\httv}{\ensuremath H^{2,2}_\text{vert}}
\newcommand*{\hvtt}{\ensuremath H^\text{vert}_{2,2}}
\newcommand*{\mathop{}\!\mathrm{d}}{\mathop{}\!\mathrm{d}}
\newcommand*{\mathop{}\!\mathrm{D}}{\mathop{}\!\mathrm{D}}
\newcommand{\patrick}[1]{\footnote{\textcolor{orange}{\textbf{PJ:\ #1}}}}
\newcommand{\wati}[1]{\footnote{\textcolor{blue}{\textbf{WT:\ #1}}}}
\newcommand{\andrew}[1]{\footnote{\textcolor{red}{\textbf{AT:\ #1}}}}
\newcommand{\clean}{
\renewcommand{\patrick}[1]{}
\renewcommand{\wati}[1]{}
\renewcommand{\andrew}[1]{}
}
\newcommand{\remove}[1]{}
\clean
\title{
\Huge Chiral matter multiplicities
and resolution-independent structure
in 4D F-theory models}
\author[$\dag$]{\Large Patrick Jefferson}
\author[$\dag$]{\Large Washington Taylor}
\author[$\dag\dag$]{\Large Andrew P. Turner}
\affil[$\dag$]{\normalsize \emph{Center for Theoretical Physics, Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA}}
\affil[$\dag\dag$]{\normalsize \emph{Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA}}
\date{\texttt{pjeffers@mit.edu}~~ \texttt{wati@mit.edu} ~~\texttt{turnerap@sas.upenn.edu}}
\begin{document}
\maketitle
\begin{tikzpicture}[remember picture,overlay]
\node[anchor=north east,inner sep=0pt] at (current page.north east)
{$\begin{array}{ccc}&&\\ \\ \text{MIT-CTP-5293}&&\end{array}$};
\end{tikzpicture}
\thispagestyle{empty}
\begin{abstract}
\noindent Motivated by questions related to the landscape of flux compactifications, we combine new and existing techniques into a
systematic, streamlined approach for computing vertical fluxes and chiral matter multiplicities in 4D F-theory models. A central feature of our
approach is the conjecturally resolution-independent intersection pairing of the vertical part
of the integer middle cohomology of smooth elliptic CY fourfolds, relevant for computing chiral indices and related aspects of 4D F-theory flux vacua.
We illustrate our approach by analyzing vertical flux backgrounds for F-theory
models with simple, simply-laced gauge groups and generic matter content, as well as models with $\U(1)$ gauge factors.
We explicitly analyze resolutions of these F-theory models in which
the elliptic fiber is realized as a cubic in $\mathbb{P}^2$ over an
arbitrary (e.g., not necessarily toric) smooth base, and confirm the
resolution-independence of the intersection pairing of the vertical part of the middle
cohomology. In
each model we study, we find that vertical flux
backgrounds can produce nonzero multiplicities for all anomaly-free
chiral matter field combinations, suggesting that F-theory geometry
imposes no additional linear constraints beyond those implied by
anomaly cancellation.
\end{abstract}
\flushbottom
\newpage
\tableofcontents
\addtocontents{toc}{\protect\thispagestyle{empty}}
\setcounter{page}{1}
\section{Introduction}
\label{sec:intro}
F-theory \cite{VafaF-theory, MorrisonVafaI, MorrisonVafaII} provides a
powerful geometric framework for describing a large class of
supersymmetric string theory vacua. In particular, F-theory can be
used to describe a vast number of 4D $\mathcal{N} = 1$ supergravity
theories with gauge symmetries.
Because F-theory provides a uniquely broad
nonperturbative perspective on the set of supersymmetric string vacuum solutions,
there are several questions that can be addressed fruitfully in this
context. First is the question of the extent to which F-theory, or
string theory more generally, can provide a UV description of any
low energy field theory without a known obstruction to coupling to
gravity; this question has been usefully framed as the problem of
delineating the \emph{swampland} \cite{VafaSwamp, OoguriVafaSwamp} of
apparently consistent low energy effective theories of gravity not realized in string
theory. Second is the question of how the gauge group, chiral matter
content, and other physical features of the observed Standard Model of
particle physics can be realized in string theory, and the extent to
which this physics is typical or requires extensive fine tuning.
There has been a great deal of work on each of these questions in the
context of F-theory over the last two decades. However, neither question has been answered
definitively, and progress on these questions appears to be hindered by a combination of unresolved conceptual issues and
limitations of available computational techniques for analyzing
flux vacua. In this paper, we
investigate some aspects of F-theory flux backgrounds that are relevant for both
of these questions. As part of our investigation, we bring together a variety of methods (some known and some new) to
frame a systematic approach for characterizing chiral matter in broad classes of 4D F-theory models
Many of the known methods we employ in our approach have been explored
in different threads of the literature, as there has been extensive research on understanding how chiral matter arises from
fluxes in 4D F-theory models. Chiral matter in F-theory GUT models
was described locally in \cite{Donagi:2008ca, BeasleyHeckmanVafaI,
BeasleyHeckmanVafaII}, and a more systematic description in terms of
fluxes and global geometry was developed in
\patrick{Added reference here. (AT)}\cite{Marsano:2010ix,Grimm:2011tb,Braun_2012,Marsano_2011,KRAUSE20121,Grimm:2011fx}, among others. Much of this work is reviewed in
\cite{WeigandTASI}; in many of these papers the approach focuses on
geometric ``matter surfaces,'' while in \cite{Grimm:2011fx} and some
related works \cite{Grimm:2011sk,Cvetic:2012xn}
an alternative approach is used that relates fluxes
to Chern--Simons terms in 3D. We use the latter approach for explicit
computations here, though the resulting insights may shed
light on some subtle aspects of the geometry of matter
surfaces.
Our approach for studying 4D F-theory vacua offers computational and
conceptual simplifications relevant for the questions posed at the
beginning of the paper. The computational advantage of our approach
is that it combines the results of the earlier work just mentioned with the
techniques of \cite{Esole:2017kyr} (used for computing intersection
numbers) into a streamlined algorithm for analyzing chiral matter and
vertical fluxes, which allows us to easily survey large families of
F-theory flux vacua. We demonstrate the utility of our
approach by analyzing numerous examples, some not previously studied
in the literature, of flux vacua in models with fixed gauge group
$\mathsf{G}$ over arbitrary smooth threefold base. Conceptually, our
approach is novel in that while previous work on chiral matter in 4D
F-theory models has been based on specific choices of resolution of
the singularities in the Weierstrass model defining the F-theory
compactification, in this paper we focus on analyzing the chiral
multiplicities, as well as the linear constraints they satisfy, in
terms of resolution-independent geometric structures.
The main resolution-independent structure that we make use of here is
related to the intersection pairing on a particular subgroup of
the middle cohomology $H^4(X,\mathbb{Z})$ of a smooth elliptic Calabi--Yau
(CY) fourfold $X$ resolving the singular Weierstrass
model. Specifically, we study the \emph{nondegenerate} intersection
pairing $M_\text{red}$ acting on the (``vertical'') cohomology
subgroup $\httv(X,\mathbb{Z}) \subset H^4(X,\mathbb{Z})$ generated by products of divisors in $X$. The
intersection pairing $M_\text{red}$ can be obtained by assembling the
quadruple intersection numbers of $X$ into a matrix $M$ and
removing its nullspace. While the quadruple intersection numbers of
divisors are generically dependent on the choice of resolution $X$ of
the singular Weierstrass model, we find that in all models we study
$M_\text{red}$ (and hence implicitly $M$ as well) is independent of
the choice of $X$, up to an
integral change of basis. Since $M_\text{red}$ encodes fluxes relevant for
computing chiral matter multiplicities, we highlight the importance of
$M_\text{red}$ as the primary geometric object of interest for
analyzing chiral matter and
vertical flux backgrounds in a manifestly
resolution-independent manner.
The apparent resolution-independence of $M_\text{red}$ and $M$
suggests that
this intersection structure is in some sense an intrinsic mathematical
feature of the singular Weierstrass model used for F-theory and may
have a direct interpretation in this geometric language as well as in type
IIB string theory, without any need for explicit resolution, although
to our knowledge this statement has not been proven in the
mathematical literature.
Just as the resolution-independent Dynkin diagram associated with a Kodaira singularity type
encodes the resolution-invariant physics of the nonabelian gauge
algebra of an F-theory compactification, this (conjecturally) resolution-independent
part of the intersection structure encodes the resolution-invariant physics connecting
vertical fluxes and chiral matter.\patrick{Added footnote with additional references to resolution-independent structures.}\andrew{Changed redundant wording of last sentence of footnote, good with me.}\footnote{Other resolution-independent structures encoded in the intersection numbers of CY resolutions have been identified in the context of F-theory and M-theory compactifications. For example, the combined fiber diagrams (CFDs) of \cite{Apruzzi:2019kgb} appearing in non-flat resolutions of singular elliptic CY threefolds were shown to be manifestly flop-invariant. Furthermore,
the intersection pairing between curves and divisors in smooth CY threefolds was shown to have invariant Smith normal form in \cite{Morrison:2020ool}.}
The set of tools that this analysis provides for exploring the
landscape of 4D F-theory flux vacua positions us to clarify aspects of
the first question raised at the beginning of the paper. While 4D anomaly cancellation is satisfied by all
F-theory constructions that have been studied\footnote{For example,
general analyses of such conditions were carried out in
\cite{LinWeigandG4, Bies_2017, Corvilain:2017luj,Cheng:2021zjh}.} and is expected to hold in all
4D $\mathcal{N}=1$ supergravity theories that can be constructed in
F-theory,
it is unknown whether or not all anomaly-free families of chiral
matter can be realized in F-theory. Interestingly, it turns out that
in all such cases we study this is indeed true, hinting that (at least
at the level of the number of independent linear combinations of chiral matter allowed for
the most generic matter representations associated with a given gauge
group) every
anomaly-free configuration of 4D chiral multiplets may have a UV
completion in F-theory.
Our approach offers further insight into this question, as we also find that (with one
exception)
for all cases free of codimension-three $(4, 6)$ singularities, the number of independent
parameters for vertical
flux backgrounds in $H_{2,2}^{\text{vert}}(X,\mathbb{Z})$ that lift to consistent F-theory
flux backgrounds with an unbroken gauge group---equivalently, the rank of $M_\text{red}$ minus the
number of constraints required to preserve 4D local Lorentz and gauge
symmetry---is equal to the number of allowed independent
families of anomaly-free chiral matter.
Part of this result is to be expected: since the physics of any F-theory model (in particular, the number of independent families of chiral matter multiplets) should be
resolution-invariant, given the relationship between chiral multiplicities and vertical flux backgrounds, it should follow that the
number of independent vertical flux backgrounds is also a resolution-invariant property of the theory; for models free of
$(4, 6)$ points, our results confirm this expectation and show that
these quantities are in fact equal to one another. What is perhaps unexpected is that our results also show (again in models free of $(4,6)$ points) that the number of independent vertical flux backgrounds is in fact equal to the total
number of linearly-independent families allowed by 4D anomaly cancellation.\footnote{In a future publication \cite{46}, we explore
models containing $(4, 6)$ loci, in which the rank of $M_\text{red}$
is increased and may include extra flux backgrounds describing
additional, strongly coupled chiral degrees of freedom localized at
the $(4, 6)$ points.} Resolution-independence of $H_{2,2}^{\text{vert}}(X,\mathbb{Z})$ thus implies that the
possible set of independent vertical flux backgrounds in $H_{2,
2}^\text{vert}(X,\mathbb{Z})$ that exist for any choice of resolution $X$
should be characterizable, like the chiral spectrum, in a resolution-invariant
fashion. Furthermore, if $M_\text{red}$ and $M$ are indeed
resolution-independent, it may be possible to characterize the
nullspace of $M$ in a resolution-independent manner, e.g., in some
canonical form. Since the
nullspace of $M$ restricted to the subspace of 4D symmetry-preserving
fluxes can be identified with the set of linear constraints (of which
the 4D anomaly cancellation conditions must necessarily be a subset),
this potentially points to a more systematic method for exploring
possible swampland-like conditions obstructing the F-theory
realization of certain families of chiral matter multiplets, or
showing that no such additional linear conditions can exist, as we
essentially conjecture here.
Regarding the second question raised at the beginning of this paper, one of the initial motivations was to analyze
chiral matter in the family of $\ensuremath(\SU(3) \times \SU(2) \times \U(1)) / \Z_6$ models found in
\cite{Raghuram:2019efb}. This model has three
independent families of generic chiral matter fields that satisfy 4D
anomaly cancellation, one of which corresponds to the
matter content of the Minimal Supersymmetric Standard Model (MSSM).
This seems to be the broadest class of F-theory models that have a
tuned Standard Model--like gauge group\footnote{That is, a gauge group
that is directly tuned in the Weierstrass model, as opposed to one
that arises from breaking a larger GUT group or that is imposed as a
generic feature of the F-theory base geometry.}, and which naturally
includes Standard Model--like matter. One subclass of these models
arises naturally through a toric fiber (``$F_{11}$'') construction
\cite{KleversEtAlToric}, and has only the single family of chiral
matter fields associated with the MSSM; chiral matter in some Standard
Model--like $F_{11}$ constructions was recently intensively
investigated in \cite{CveticEtAlQuadrillion}. The approach developed
here gives us a means to check whether F-theory models of the more
general tuned $\ensuremath(\SU(3) \times \SU(2) \times \U(1)) / \Z_6$ type naturally contain chiral matter in the other
two allowed families, or whether these are forbidden by string
geometry for some reason and hence belong to the swampland. We find
that indeed all three of the allowed chiral matter types are allowed;
we briefly summarize these results here and report further on the
details of this analysis in a forthcoming publication \cite{321-fluxes}.
The structure of this paper is as follows: in \cref{sec:overview}, we
give an overview of the main ideas, technical contributions, and
results of the paper; in
\cref{sec:4dflux,sec:constraints-homology}, we
explore two complementary approaches to analyzing the set of
topologically distinct fluxes
that preserve 4D local Lorentz and gauge symmetry, which differ by the order in which the symmetry constraints and equivalence
relations in homology are imposed;
\cref{3Dcompare} reviews the strategy we use
for
determining the precise relationship between the chiral indices
$\chi_{\mathsf{r}}$ and vertical fluxes by exploiting their connection
to one-loop Chern--Simons couplings appearing in the 3D low energy
effective action describing the F-theory Coulomb branch; in
\cref{sec:exampleADE}, we use our approach to study various models with
simple gauge group $\mathsf{G}_\text{na}$; in \cref{sec:abelianmodel}, we
discuss the generalization of our analysis to models with gauge group
$\mathsf{G} = (\mathsf{G}_\text{na} \times \U(1))/\Gamma$, using the
$(\SU(2) \times \U(1)) / \mathbb{Z}_2$ model to illustrate various
properties and summarizing the results for $\ensuremath(\SU(3) \times \SU(2) \times \U(1)) / \Z_6$; finally, \cref{sec:conclusions} contains concluding
remarks and future directions. A number of technical results related
to e.g. anomaly cancellation, intersection theory, and resolutions are
collected in the appendices.
\section{Overview}
\label{sec:overview}
In this section, we give an overview of the steps needed to
systematically describe a class of 4D F-theory models with a given
gauge group and ultimately to compute the chiral matter content from vertical fluxes using our approach. In
particular, we try to make clear how various techniques in the
existing literature are integrated into our approach, and where this paper
makes novel contributions.
The current state of knowledge for
many parts of this analysis
are reviewed in more detail in Weigand's excellent review article
\cite{WeigandTASI}.
We are interested in finding a general formulation of the
chiral matter multiplicities for a variety of F-theory constructions
with different gauge groups, in a way that can be expressed succinctly in terms
of the geometry of the base of the F-theory compactification and a
choice of fluxes. In particular, for a given choice of gauge group
and generic\footnote{See \cref{sec:Weierstrass} for a precise definition of the notion of ``generic matter'' in F-theory compactifications.} matter representations, we are interested in identifying closed
form expressions for the chiral matter multiplicities in a
base-independent (and resolution-independent) fashion.
General expressions of this type have been found previously in
the literature using related but distinct combinations of techniques
for various\patrick{Replaced ``certain''. (AT)} gauge groups, such as $\SU(5)$ \cite{Marsano_2011}, $\text{E}_6$ \cite{Kuntzler:2012bu}, $\U(1) \times \U(1)$,
$(\SU(5) \times \U(1)\times\U(1)) / \mathbb{Z}_5$ \cite{Cveti__2014}, and
$(\SU(3) \times \SU(2) \times \U(1)^2) / \mathbb{Z}_6$
\cite{LinWeigandG4} (see also \cite{Marsano:2009gv}).\patrick{Added references to Sakura's papers. (AT)}
At a very heuristic level, the analysis can be described as follows:
for any specific choice of gauge group, it should be possible to
identify a multi-parameter family of Weierstrass models that describes
F-theory models over an arbitrary base with that gauge group and
generic matter. A resolution $X$ of any of the corresponding CY
fourfolds
gives rise to a well-defined set of intersection numbers, which can be
organized into a matrix $M$ containing the intersection pairing $M_{\text{red}}$ acting on the set of homology
classes $H_{2, 2}^\text{vert}(X,\mathbb{Z})$.
The intersection pairing is relevant for computing fluxes through certain homology classes dubbed ``matter surfaces''
that encode the multiplicities of chiral matter fields.
The chiral matter multiplicities that are fixed by the choice of gauge
flux and the intersection numbers can also be related directly to the
3D physics arising from a circle reduction of the F-theory
model.
While the choice of
resolution and its associated intersection numbers are not unique,
it should be possible in general to describe
the multiplicities of chiral matter fields in the 4D limit
in a resolution-independent fashion that depends
only on the intersection structure of the compactification base and a
choice of fluxes in an appropriate basis.
One of the key ingredients in this paper is the identification of a
resolution-independent piece of the intersection structure of $X$, namely $M_{\text{red}}$, that
is relevant for understanding the chiral multiplicities.
We now describe each of the
steps in this procedure in a bit more detail, framing the analysis of
the remainder of the paper.
\subsection{Selection of the base}
The first step in choosing an F-theory compactification is the choice
of complex threefold base $B$.
From the IIB string theory point of view, the 10D IIB
theory is compactified on $B$, which we take here to be a
compact K\"ahler threefold. Note that $B$ need not be a CY
manifold, i.e., the canonical class $K$ of $B$ need not be trivial,
though $-K$ must be an effective class. The F-theory model
\cite{VafaF-theory, MorrisonVafaI, MorrisonVafaII} is described by a
Weierstrass model
\begin{equation}
y^2 = x^3 + f x + g\,,
\end{equation}
defining an elliptic CY fourfold $X_0$
with base $B$, where $f, g$ are sections of the line bundles $\mathscr{O}(-4K),\mathscr{O}(-6K)$, respectively.
In general, the CY fourfold $X_0$ has singularities associated
with loci in the base where the elliptic fiber degenerates.
Degenerations over codimension-one loci in the base are associated
with the gauge group of the F-theory model and degenerations over
codimension-two loci are associated with matter.
F-theory is frequently analyzed as a limit of M-theory on a smooth
resolution $X$ of $X_0$, but the physics should in principle be
independent of resolution as discussed further in \cref{sec:resolution}.
The number of possible bases $B$ is quite large. The primary constraint is that $B$ cannot contain a divisor $\Sigma$ (codimension-one
algebraic surface) that has a normal bundle that is so negative that
$(f, g)$ need to vanish to orders $(4, 6)$ on $\Sigma$. When such a
divisor exists, the singularity structure of the total space of the
elliptic fibration goes beyond the classification of Kodaira
\cite{Kodaira} and N\'{e}ron \cite{Neron}; there is no smooth CY
resolution and the resulting geometry lies at infinite distance in the
moduli space of compactifications.\footnote{Note that there can be higher-codimension $(4, 6)$
singularities without a crepant (CY) resolution (see,
e.g., \cite{Klemm:1996ts}); these geometries, however, lie at finite
distance in moduli space and seem physically relevant as F-theory
compactifications.} A large
range of elliptic CY fourfolds have been studied in the
literature, see, e.g., \cite{Klemm:1996ts,Kreuzer:1997zg,Gray:2013mja}. Restricting to the simple case of toric
$B$, the number of possible bases has been shown by explicit
construction to be at least $10^{755}$ \cite{HalversonLongSungAlg} and
is estimated through Monte Carlo analysis to be of order closer to
$10^{3000}$ \cite{TaylorWangLandscape}.
Many of these bases have codimension-two loci where $(f, g)$ vanish to
orders $(4, 6)$. These codimension-two loci are generally associated with nonperturbative massless excitations
in the low energy 4D theory, see,
e.g., \cite{Candelas:2000nc,Achmed-Zade:2018idx,Hayashi:2009ge}; in 6D,
such excitations are generally associated with a superconformal sector
in the theory \cite{SeibergSCFT, HeckmanMorrisonVafa}, and while there
are some parallel aspects of 4D F-theory models
\cite{Apruzzi:2018oge} the structure of
these sectors in four space-time dimensions is less well
understood.
Much of the detailed analysis of chiral matter in 4D F-theory models
has been done in the context of toric geometry. One advantage of
toric bases is that there are many powerful and simple tools for
computing resolutions, intersection numbers, and other relevant
features of toric varieties that extend to many elliptic CY
fourfolds over toric bases that can be described as hypersurfaces in
toric varieties. At least
in the case of elliptic CY threefolds with relatively large
Hodge numbers over complex surface bases, toric constructions seem to
give a good representative sample of the set of possibilities
\cite{TaylorWangNon-toric}, although for 4D F-theory models with
chiral matter, some features such as GUT breaking are not easily seen
in purely toric contexts (see, e.g., \cite{Marsano:2009wr,
Braun:2014xka}). Toric geometry has been used with great efficacy in many
examples in the literature, e.g.,
\cite{KleversEtAlToric,Knapp:2011wk,Braun:2011ux,Buchmuller:2017wpe}. By
contrast, our analysis employs resolution techniques developed to study
Weierstrass models defined over a generic (toric or non-toric) base
for certain gauge group and matter structures---see,
e.g., \cite{Esole:2011sm,Esole:2014hya}.
Resolutions of general classes of
elliptic fourfolds including non-toric constructions have also been
considered in, e.g., \cite{Cveti__2014, LinWeigandG4}, using somewhat
different approaches.
\subsection{Non-degeneracy of the intersection pairing on the base}
\label{sec:nondegeneracy}
For any threefold base $B$, there is a triple intersection form
$D_\alpha \cdot D_\beta \cdot D_\gamma$
on the space $ H_{2, 2} (B,\mathbb{Z}) \cong H^{1, 1} (B,\mathbb{Z})$ of divisors on $B$. One feature of a
general F-theory threefold base that we will use in various places is
the observation that for any such smooth $B$, the triple
intersection form is nondegenerate, in the sense that for any
divisor $A = A^\alpha D_\alpha$,
there exists some $D', D''$ for which
$A \cdot D' \cdot D'' \ne 0$, so that there
exists a curve $C$ whose class is of the form $C = D' \cdot D''$ with $C \cdot A \ne
0$.
For a toric base, this follows from the standard result that the ring
in intersection theory generated by the divisors (i.e., the Chow
ring) generates the full linear space of homology classes $H_{i, i}
(B,\mathbb{Z})$, combined with Poincar\'{e} duality, which states that the
space of curves $H_{1, 1} (B,\mathbb{Z})$ is dual to the space of divisors
under the intersection product. More generally, the stated
result follows from the hard Lefschetz theorem (see, e.g., \cite{Griffiths:433962}), which asserts that
$J:H^{1,1} (B,\mathbb{Q}) \to H^{2, 2}(B,\mathbb{Q})$ is an isomorphism
over $\mathbb{Q}$ for any compact K\"ahler manifold $B$, where $J$ is a
K\"ahler class (equivalently, a
cohomology class Poincar\'e dual to the pullback of the hyperplane section in a projective realization of
$B$ when $B$ is a smooth complex projective variety.) This nondegeneracy plays a useful role in our analysis of the
structure of fluxes and the intersection numbers of CY
fourfolds that can be realized as elliptic fibrations over $B$.
\subsection{Weierstrass model: gauge group and matter content}
\label{sec:Weierstrass}
A central feature of a 4D F-theory model is the gauge group $\mathsf{G}$ realized in
the effective 4D theory constructed by compactifying F-theory on a
Weierstrass model defined over a given threefold base $B$. In general, $\mathsf{G}$ is encoded in the
Kodaira type of the singularities in the elliptic fibration over various
divisors in $B$.
The gauge group $\mathsf{G}$ can arise either because it is forced from the geometry
of $B$ or through explicit tuning of the Weierstrass model. In
the first case, geometrically ``non-Higgsable'' gauge group factors
can arise when certain divisors in $B$ have normal bundles that
are sufficiently negative that $(f, g)$ are forced to vanish to orders
at least $(1, 2)$ over those divisors \cite{MorrisonTaylorClusters, MorrisonTaylor4DClusters}.
Virtually all of the large number of threefold bases that support
elliptic CY fourfolds have multiple non-Higgsable gauge group
factors \cite{TaylorWangMC, HalversonLongSungAlg,
TaylorWangLandscape}. The gauge group can also be tuned by choosing
a Weierstrass model where $f, g$, and the discriminant $\Delta =
4 f^3 +27 g^2$ vanish to the appropriate orders over a given divisor in $B$
necessary to guarantee a desired nonabelian gauge factor. $\U(1)$ gauge factors
can also be non-Higgsable \cite{MartiniTaylorSemitoric,
morrison2016nonhiggsable, WangU1s} or tuned, and are subtler, as
they rely on the global structure of the Mordell-Weil group of
rational sections.
The allowed matter content in a given theory depends on the more
detailed structure of singularities in the elliptic fibration over
codimension-two loci in $B$. There is a natural distinction in
F-theory between ``generic'' matter content for given $\mathsf{G}$,
associated with the simplest codimension-two singularity types, and
more exotic matter representations that can be realized through more
complicated singularities. This notion of genericity can be made precise
in 6D, where generic matter content is associated with the
branch of moduli space of the largest dimension for a fixed $\mathsf{G}$ and
anomaly coefficients that are not ``too large'' \cite{TaylorTurnerGeneric}. For given $\mathsf{G}$, in general we expect that there is a
universal construction of a multi-parameter family of Weierstrass
models that realize the full geometric moduli space of elliptic
CY varieties over an arbitrary base that realize $\mathsf{G}$ and have generic matter content for that gauge
group.
Such ``universal'' $\mathsf{G}$ models were studied in
\cite{Raghuram:2019efb}\footnote{In that paper these universal
Weierstrass model constructions were referred to as ``generic'';
here we change terminology to ``universal'' to avoid confusion with
other uses of the term generic.}, where the universal $\ensuremath(\SU(3) \times \SU(2) \times \U(1)) / \Z_6$
Weierstrass model with generic matter was constructed, and a
moduli-counting argument was introduced to check that a universal
$\mathsf{G}$ model is fully parameterized; other universal $\mathsf{G}$
models with generic matter representations include the Tate-tuned
models with various nonabelian gauge factors (see,
e.g., \cite{BershadskyEtAlSingularities}), and the Morrison--Park
universal $\U(1)$ model \cite{MorrisonParkU1}.
In general, the parameters of the universal Weierstrass construction
for a given $\mathsf{G}$
include discrete parameters associated with the divisor classes
supporting the gauge factors and continuous parameters associated with
complex structure moduli of the associated $X_0$. These discrete parameters, along with the canonical class of the base, form what for us will be the characteristic data of the F-theory model.
While the definition
of ``generic'' matter representations is most clear in 6D theories, the same representations are naturally generic
for 4D F-theory constructions in terms of the dimension of the
geometric moduli space and the complexity of the singularities;
universal F-theory models with fixed $\mathsf{G}$ and these generic
matter representations such as the Tate-tuned and Morrison--Park
models take the same parameterized form in 6D and in 4D theories.
In this paper, we work with various universal $\mathsf{G}$ models
with generic chiral matter content in 4D, meaning
that we consider multi-parameter Weierstrass models with $\mathsf{G} = \SU(N),
\SO(4k+2), \text{E}_6,(\SU(2) \times \U(1))/\mathbb{Z}_2$ over
arbitrary threefold bases $B$ that need not be toric. Note that even over
toric $B$, only some universal $\mathsf{G}$ models have known toric
constructions with fibers that can be constructed torically as
elliptic curves within toric 2D fibers. For example, some $\SU(N)$
models can be constructed in this way torically, and a subset
of the universal $\ensuremath(\SU(3) \times \SU(2) \times \U(1)) / \Z_6$ models can be so constructed torically, but not all.
In this paper we focus on the degrees of freedom of 4D F-theory models
encoded in the Weierstrass model through the axiodilaton of type IIB
theory and the fluxes that come from the 3-form field $C_3$ in the
M-theory picture. There can be additional degrees of freedom such as
``T-branes'' \cite{Cecotti:2010bp} encoded in the world-volume
dynamics of the 7-branes of the IIB theory; in the analysis here we do
not consider the matter or other structures that these degrees of
freedom may produce in the effective 4D $\mathcal N=1$ supergravity theory.
\subsection{Resolution and intersection numbers}
\label{sec:resolution}
As described above, we are interested in general families of Weierstrass models with particular
structures of codimension-one and codimension-two singularities.
Given a Weierstrass model in such
a family, the standard
approach taken for understanding F-theory models is to resolve the
singular Weierstrass geometry into a smooth elliptic
CY manifold and analyze the theory as a limit of M-theory,
see, e.g., \cite{Grimm:2010ks,WeigandTASI}.
While this approach gives the best understood way of analyzing the
physics of the resulting 4D F-theory model, the physics
should be independent of the specific resolution; indeed, from the
nonperturbative type IIB point of view, the physics should be
well-defined directly in the context of the singular Weierstrass model.
Note further that there can be terminal
singularities at higher (i.e., $\geq 2$) codimension that do not admit a CY
resolution at all; in many cases, these can be present without any apparent
significant effects on the resulting physical model
\cite{arras2016terminal,Grassi:2018rva}.
One of the general motivations for the methods we explore in this
paper is to find ways of characterizing the resolution-independent
aspects of the physics of 4D chiral matter.
Given a Weierstrass
model defined by a singular elliptic CY fourfold,
there are, in general, multiple distinct resolutions $X$ that preserve
the CY structure of the singular fourfold. For most of the analysis here we do not
concern ourselves with terminal singularities or higher-codimension
$(4, 6)$ loci where there is no ``flat'' resolution $X$
respecting the elliptic structure (although, see \cref{sec:46} for
some further comments on codimension-three $(4, 6)$ loci).\patrick{Added a reference to the box graph paper in the footnote about (partial resolutions). Also made some changes to the same footnote, please have a look. (AT)}\footnote{\label{partial} In
this paper, when we refer to a CY fourfold $X$ as a ``resolution''
of a singular Weierstrass model $X_0$
obtained by a sequence of blowups, we generally mean that $X$ is a partial resolution that is smooth through generic codimension-three loci in $B$, but may nonetheless contain singularities over special codimension-three loci (i.e., points) in $B$. Apart from the special case of $(4, 6)$ singularities, these codimension-three singularities do not affect the results of our analysis, hence we ignore them and permit this abuse of terminology. Note that when $B$ is restricted to be a twofold $B^{(2)}$, these codimension-three singularities are absent; in the models we consider here there are also no codimension-two terminal singularities, hence the resulting CY spaces $X$ are in general genuine resolutions over $B^{(2)}$. A comprehensive analysis of the network of genuine CY fourfold resolutions (i.e., through codimension-three in $B$) using the physics of the low-energy effective 3D $\mathcal{N} = 2$ description of the F-theory Coulomb branch is presented in \cite{Hayashi:2014kca} (see also \cite{Lawrie:2012gg}); in \cite{Hayashi:2014kca}, particular attention is given to the geometry of the singular elliptic fibers over codimension-two and codimension-three loci.} Since simple and manifestly
resolution-independent methods are currently lacking for a complete
analysis of physics like chiral matter, we use
specific resolutions for explicit calculations and try to extract and
identify the
resolution-independent parts of the results.
One of the key features of a resolved CY fourfold $X$ is the
set of quadruple intersection numbers of divisors $\hat{D}_I$.\footnote{We use hats to denote divisors in the fourfold $X$, as
opposed to divisors in the base $B$; a glossary of notation commonly used throughout the paper is
given in \cref{sec:notation}.}
Expanding an arbitrary set of divisors $\hat C, \hat D, \hat E, \hat F$ in an
appropriate basis $\hat D_I$, we may write
\begin{equation}
\hat C \cdot \hat D \cdot \hat E \cdot \hat F = C^I D^J E^K F^L(\hat D_I \cdot \hat D_J \cdot \hat D_K \cdot \hat D_L)\,.
\end{equation}
These intersection numbers can also describe
aspects of the dual pairing associated with Poincar\'{e} duality
between divisors
(codimension-one algebraic surfaces) and curves
(codimension three) in the fourfold, e.g., when a curve is realized as
an intersection of three divisors.
As we discuss in further detail below, the quadruple intersection
numbers of $X$ are not in general resolution-independent, although some are resolution-independent.
A natural basis for the divisors in a fourfold with an elliptic
fibration structure is suggested through the Shioda--Tate--Wazir
\cite{Wazir} formula
\begin{equation}
h^{1, 1}(X) = 1 + h^{1, 1} (B) + \rk\mathsf{G}\,,
\label{eq:Wazir}
\end{equation}
where the 1 comes from the zero section of the elliptic fibration, the
second term comes from pullbacks of divisors in the base to the total
space of the elliptic fibration, and the last term contains Cartan
divisors of nonabelian gauge factors and additional sections for the
free abelian part of $\mathsf{G}$. In view of this decomposition, and
following standard notation in the literature (e.g., \cite{Grimm:2011sk}), we
use the following conventions for indices $I$:
\begin{itemize}
\item $I=0$ denotes the zero section
\item $I=a$ denotes a generating section associated to a non-Cartan $\U(1)$ gauge factor
\item $I=\alpha$ denotes a divisor $\hat D_\alpha = \pi^*(D_\alpha)$
realized as a pullback of a divisor in the base
\item $I=i_s$ denotes a Cartan divisor of a nonabelian factor $\mathsf{G}_s \subset \mathsf{G}$.
\end{itemize}
To make contact with the low energy gauge theoretic description of the Coulomb branch physics in the M-theory duality frame, we sometimes convert to the ``physical'' basis $\hat D_{\bar I} = \sigma_{\bar I}^I \hat D_I$ (see \labelcref{physbasis} for the definition of $\sigma_{\bar I}^I$), where
\begin{itemize}
\item $\bar I = \bar 0$ denotes the $\U(1)_{\text{KK}}$ divisor
\item $\bar I = i=(a,i_s)$ collectively denotes all other $\U(1)$ divisors.
\end{itemize}
Many of the intersection numbers are independent of resolution and are
known for a general elliptic fibration. For example
\cite{Grimm:2011sk}, when $\hat D_0$ is a holomorphic section and
there are no abelian factors, the quadruple intersection numbers of
$X$ can be pushed to the base $B$\footnote{Strictly speaking, the intersection products $\hat D_I \cdot \hat D_J \cdot \hat D_K \cdot \hat D_L$ live in the Chow ring of the variety $X$ (the Chow ring encodes the intersection structure in a smooth
algebraic variety; see \cref{fluxesint} for further discussion). However, since the pushforward is
computed with respect to the canonical projection $\pi\colon X
\to B$, the resulting intersection product $\pi_*(\hat D_I \cdot \hat D_J \cdot \hat D_K \cdot \hat D_L)$ lives in the Chow ring of
$B$. For simplicity of notation we are often sloppy and omit explicit pushforward maps such as $\pi_*$ when the appropriate Chow ring is otherwise clear from the context.}, so that
\begin{equation}
\begin{aligned}
\pi_*(\hat D_0^4 ) = K^3\,, ~~~~\pi_*(\hat D_0 \cdot \hat D_\alpha \cdot \hat D_\beta \cdot \hat D_\gamma) =
D_\alpha \cdot D_\beta \cdot D_\gamma\,.
\end{aligned}
\end{equation}
Other intersection numbers, particularly those with 3 or 4 indices of
type $i_s$ or $a$, depend not only on the matter content of the theory
but also on the particular resolution; see \cref{intersection} for a
more comprehensive discussion
of the general structure of these intersection numbers.
One issue that arises in certain situations, for instance when $\mathsf{G} = (\mathsf{G}_\text{na}\times\U(1)^k)/\Gamma$, is
that there may not be a holomorphic section of the Weierstrass model;
this occurs, in particular, when the section associated with the
identity element of the Mordell--Weil group intersects with one or more
sections associated with generators of $\U(1)$ factors over the
discriminant locus. In some of these cases, the procedure of
resolving the singular CY geometry and analyzing various physical
properties in the dual M-theory frame on $X$ is more easily
accomplished in models in which the elliptic fiber is realized as a
general cubic in $\mathbb{P}^2$ rather than the usual Weierstrass model
(e.g., the general cubic is used to define the resolutions studied in
\cite{KleversEtAlToric}.)
In this paper, we generalize the mathematical techniques of
\cite{Esole:2017kyr} to compute intersection numbers of resolutions of
models in which the elliptic fiber is presented as a general cubic in
$\mathbb{P}^2$. These mathematical techniques enable us to evaluate the
quadruple intersection numbers in terms of linear combinations of the
triple intersection numbers of an arbitrary base $B$, much in the same manner
as described above:
\begin{equation}
\label{firstpush}
\pi_*(\hat D_I \cdot \hat D_J \cdot \hat D_K \cdot \hat D_L)=W_{IJKL}= W^{\alpha \beta \gamma}_{IJKL} D_\alpha \cdot D_\beta \cdot D_\gamma\,.
\end{equation}
With the aid of a symbolic computing tool, the action of the above map
$\pi_*$ can easily be used to compute intersection numbers (and other
characteristic numbers\footnote{For example, the same methods have
been used to compute the generating function of the Euler
characteristics of smooth (up to codimension two) elliptic $n$-folds
resolving singular Weierstrass models with gauge symmetry $\mathsf{G}$; see
\cite{Esole:2017kyr} for further details.}) of $X$ in terms of
rational expressions involving divisor classes of the
ambient fivefold in which $X$ is realized as a hypersurface.\footnote{It should be
possible to straightforwardly adapt these techniques to models in
which the elliptic fiber is realized as a complete intersection in
$\mathbb{P}^n$.} The techniques used to evaluate the pushforward map
$\pi_*$ are described in detail in \cref{pushapp}.
Note that the
physical results obtained from a given resolutions $X$ should, and do in the
cases we analyze, match with the expected physics from any other
resolution, including the general structure of the tensors $W_{IJKL}^{\alpha \beta \gamma}$ (see \cref{intersection}) appearing on the right hand side of \cref{firstpush}.
\subsection{Fluxes, consistency conditions, and linear algebra}
\label{sec:fluxes-algebra}
In order to obtain a chiral matter spectrum in 4D, it is necessary to
turn on a nontrivial flux background, which in the M-theory duality
frame corresponds to a nontrivial profile for the 4-form field
strength $\mathop{}\!\mathrm{d} C_3$ whose key properties we now summarize.
\subsubsection{Flux conditions}
It was argued in \cite{Witten:1996md} that the cohomology class $G =\mathop{}\!\mathrm{d}
C_3 \in H^4(X,\mathbb{R})$ satisfies a shifted quantization
condition\footnote{It turns out that $c_2$ belongs to
$H^{2,2}_{\text{vert}}(X,\mathbb{Z})$ as defined in
\cref{eq:verticalcohomology}.
}
\begin{equation}
\label{eq:c2quant}
G_\mathbb{Z} = G - \frac{c_2(X)}{2} \in H^{4}(X,\mathbb{Z})\,.
\end{equation}
To preserve supersymmetry, $G$ must satisfy
\begin{equation}
G \in H^{2,2}(X, \mathbb{R}) \cap H^4(X, \mathbb{Z}/2)
\end{equation}
along with the primitivity condition
\begin{equation}
J \wedge G = 0\,,
\end{equation}
where $J$ is the K\"ahler form of $X$ \cite{Becker:1996gj,Gukov:1999ya}. Finally, there is a tadpole condition
\cite{Sethi:1996es,Dasgupta:1996yh,Dasgupta:1999ss} requiring that the net number of M2-branes (which are dual to D3-branes in the F-theory frame) is non-negative to ensure a stable vacuum,
\begin{equation}
\label{D3tadpole}
N_\text{M2} = \frac{\chi}{24} - \frac{1}{2} \int_X G \wedge G \in \mathbb{Z}_{\ge 0}\,,
\end{equation}
where $\chi = \int_X c_4$ is the Euler characteristic of
$X$; the integrality of $N_{\text{M2}}$ follows from \cref{eq:c2quant}, as explained in \cite{Witten:1996md}.
Additional conditions must be imposed to ensure that $G$ dualizes to a
suitable F-theory flux background. To preserve Poincar\'{e} symmetry, we
require that the following fluxes vanish:
\cite{Dasgupta:1999ss},
\begin{equation}
\label{eq:Poincare}
\int_{S_{0\alpha}} G =0\,, \quad \int_{S_{\alpha \beta} } G = 0\,.
\end{equation}
Furthermore, to ensure that the flux background does not break the gauge symmetry $\mathsf{G}$ in the 4D limit, it is necessary to impose the conditions (see, e.g., \cite{Donagi:2008ca})
\begin{equation}
\label{eq:gauge}
\int_{S_{i \alpha} } G = 0\,,
\end{equation}
where we emphasize that $i$ collectively indexes all divisors dual to gauge $\U(1)$s on the F-theory Coulomb branch.
Note that since $G$ may be a half-integral
cohomology class, in principle it seems there could be circumstances under which
no flux satisfies these conditions; in all cases we have considered,
however, the integrals \cref{eq:Poincare} and \cref{eq:gauge} take
integer values and the constraints can be satisfied, and this is
likely always true---in particular, the results of
\cite{Collinucci:2010gz} show that the fluxes appearing in \cref{eq:Poincare}
are always integer-valued for any $G$ on a smooth elliptic fourfold.
We describe these conditions explicitly in some general
classes of models with gauge groups $\SU(2), \SU(5)$ in \cref{sec:su2}
and \cref{sec:su5}.
\subsubsection{Vertical fluxes and intersection pairing}
\label{sec:vertical-pairing}
In addition to the usual Hodge decomposition, the cohomology group $H^4(X)$ admits the finer orthogonal decomposition \cite{Greene:1993vm,Braun:2014xka}
\begin{equation}
\label{ortho}
H^{4}(X,\mathbb{C}) = \httv(X,\mathbb{C}) \oplus H^{2,2}_\text{rem}(X,\mathbb{C}) \oplus H^{4}_\text{hor}(X,\mathbb{C})\,.
\end{equation}
As in much of the previous literature, for the most part
in this paper we focus on
integral ``vertical'' fluxes, i.e., flux backgrounds $G$
belonging to the subgroup $H^{2,2}_{\text{vert}}(X, \mathbb{R}) \cap
H^4(X,\mathbb{Z})$ spanned by wedge products of cohomology classes in
$H^{1,1}(X)$, where $H^{1,1}(X)$ has a basis $\PD(\hat D_I)$ of
harmonic $(1,1)$-forms on $X$ dual to divisors $\hat D_I$.\footnote{Note that when $c_2 (X)$ is not even, $G$ is a
half-integer class; we neglect this refinement in our notation in
various places, essentially restricting to the simplified cases
where $c_2 (x)$ is even,
except when it is directly relevant to the discussion.} More precisely, for the
purposes of this paper, we focus on fluxes belonging to the sublattice $H_{\text{vert}}^{2,2}(X,\mathbb{Z}) \subset H^{2,2}(X, \mathbb{R}) \cap
H^4(X,\mathbb{Z})$, which we define as follows:
\begin{equation}
\label{eq:verticalcohomology}
\httv(X,\mathbb{Z}) := \Span_{\mathbb{Z}}(H^{1,1}(X,\mathbb{Z}) \wedge H^{1,1}(X,\mathbb{Z}))\,,
\end{equation}
which is to say that a ``vertical'' class for us means a class belonging to
the integer linear span of forms $\PD(\hat D_I) \wedge \PD(\hat
D_J)$. Note that it is in principle possible for there to exist integral
vertical cohomology classes that do not lie in
$H^{2,2}_{\text{vert}}(X,\mathbb{Z})$ as given by this definition
and therefore it is possible that
our definition excludes some consistent vertical flux backgrounds that
could be included by permitting non-integer coefficients.
As reviewed in \cite{WeigandTASI}, vertical fluxes play a primary role
in determining the chiral matter content of a 4D F-theory
compactification, and for the most part we
ignore components in
$H^{4}_\text{hor}(X,\mathbb{C}) \oplus H^{2,2}_\text{rem}(X,\mathbb{C})$
for flux backgrounds.\footnote{The possibility that Poincar\'{e}
duality and the inclusion of fluxes in
$H^{4}_\text{hor}(X,\mathbb{C})\oplus H^{2,2}_\text{rem}(X,\mathbb{C})$ may give a broader class of possible
matter multiplicities is explored in \cref{sec:quantization-1},
and more specifically in the case of the $\SU(5)$ model with generic
matter in \cref{sec:su5}.} Denoting by $\Lambda_S$ the $[h^{1, 1}(X) (h^{1, 1}(X) + 1)/2]$-dimensional integral lattice
spanned by the
surfaces
$S_{IJ} = \hat D_I \cap \hat
D_J$ (here treated as formally independent objects for each $IJ$ pair),
we can
conveniently encode
the
flux integrals over vertical surfaces via the intersection pairing
matrix
\begin{equation}
\label{eq:m-o}
\begin{aligned}
M &\colon \Lambda_S \times \Lambda_S \rightarrow \mathbb{Z}\,, \\
M_{(IJ) (KL)} &=
S_{IJ} \cdot S_{KL} =
\int_X \PD(S_{IJ}) \wedge \PD(S_{KL})\,.
\end{aligned}
\end{equation}
In the second line above, $\cdot$ indicates the intersection pairing
on homology. As we explain in more detail in \cref{fluxesint}, $M$ can
thus
be viewed as an integral bilinear form
on
vectors $\phi = \phi^{IJ} S_{IJ} \in \Lambda_S$, giving
$\Lambda_S$ the structure of an integral lattice.\footnote{While sometimes
physicists refer to any subgroup of $\mathbb{R}^n$ that is isomorphic to
$\mathbb{Z}^n$ as a lattice without reference to
any associated bilinear form, throughout this paper we reserve the
term lattice for a free abelian group of finite rank with a symmetric
bilinear form.} Following \cite{Grimm:2011fx}, we define the fluxes
\begin{equation}
\Theta_{IJ} = \int_{S_{IJ}} G = \int_X G \wedge \PD(S_{IJ})
= M_{(IJ) (KL)} \phi^{KL}\,,
\label{eq:constraints-theta-m}
\end{equation}
where $\phi \in \Lambda_S$ represents the components of the Poincar\'{e} dual of the
flux background $G$ expanded in a collection of classes
$S_{IJ}$. Throughout the paper, we
refer to $\phi$ as a ``flux background''.
(Note that when $c_2 (X)$ is not even, the possible values of
$\phi$ are shifted appropriately
by a half-integer lattice element $\PD(c_2 (X)/2) \in
\Lambda_S/2$.)
In terms of the above notation, the symmetry constraints \labelcref{eq:Poincare,eq:gauge} can be expressed as
\begin{equation}
\label{eq:general-constraints}
\Theta_{I\alpha} = 0\,,
\end{equation}
where we note that for vertical fluxes the above conditions are both
necessary and sufficient to preserve 4D gauge symmetry and local
Lorentz symmetry.
The fluxes $\Theta_{IJ}$ can be written as linear functions of the flux backgrounds
$\phi^{IJ}$, with coefficients given by
the pushforwards of the intersection
numbers of divisors of $X$.
In explicit computations we can for certain resolutions of $\mathsf G$ models (defined over arbitrary $B$) formally solve the equations
\cref{eq:general-constraints} for a subset of the $\phi$s, so that
the nonzero $\Theta$s that encode the chiral matter multiplicities
are again linear functions of the remaining $\phi$s with coefficients
that are polynomial functions of the intersection numbers.
Imposing the symmetry constraints is equivalent to restricting the
flux background $\phi^{IJ}$ to lie in a sublattice $\Lambda_C \subset
\Lambda_S$. For
a given resolution $X$, the sublattice
$\Lambda_C$ can be viewed as the lattice of $\phi^{IJ}$ whose image
under $M$ is the sublattice of $\Theta_{IJ}$ satisfying
\labelcref{eq:general-constraints},
which in turn encodes the multiplicities of 4D chiral matter, as we
review in more detail in the following subsection. We emphasize that while the
intersection numbers entering the matrix $M$ are generically
resolution-dependent, we expect that
the allowed chiral multiplicities must be
resolution-independent, consistent with the expectation that every set
of M-theory vacua defined by a set of distinct resolutions $X$ lifts to a common set of
F-theory vacua on a singular elliptic CY fourfold $X_0$.
What we have described above is essentially the standard perspective
on analyzing chiral matter in 4D F-theory flux vacua. We now discuss a
complementary perspective that illuminates additional aspects of the
analysis. We begin with the observation that not all the cycles
$S_{IJ}$ are independent in homology \cite{LinWeigandG4,
Bies_2017}. This implies
that $M$ generically has a nontrivial nullspace, where the elements
of the nullspace represent equivalence relations in homology, and hence
the rank of the matrix $M$ is equal to the dimension $h_{2,
2}^\text{vert} (X)$ of $H_{2,2}^{\text{vert}}(X,\mathbb{Z})$. We denote by
$M_\text{red}$ the \emph{nondegenerate} intersection pairing
\begin{equation}
M_\text{red}\colon \hvtt(X,\mathbb{Z}) \times \hvtt(X,\mathbb{Z}) \to \mathbb{Z}\,,
\end{equation}
where we describe $M_\text{red}$ explicitly as a matrix by restricting
the action of the matrix $M$
to the quotient of $\Lambda_S$ by the nullspace, which
projects
$\Lambda_S$
to the
quotient lattice $\hvtt(X,\mathbb{Z})$.
While the reduced matrix $M_\text{red}$ produces the same results for
the multiplicities of chiral matter as $M$ does in the procedure described above,
$M_\text{red}$ is a simple and useful tool for analyzing various
aspects of fluxes and chiral matter (e.g. the number of independent families of chiral matter combinations can in principle be inferred from the rank of $M_{\text{red}}$, without having to explicitly compute chiral indices.)
Furthermore, we provide evidence suggesting that while the full set of quadruple
intersection numbers are not
in general resolution-independent, $M_\text{red}$ is independent of the choice of
$X$ up to a change of basis. Among other things, this implies
that $M_\text{red}$ makes the resolution-invariance of the chiral
matter multiplicities manifest in terms of a canonical subspace of
homology classes that parametrize the space of vertical fluxes lifting
to consistent F-theory flux backgrounds.
Since the dimension of the null space of $M$ is just $(h^{1, 1}(X) (h^{1, 1}(X) + 1)/2)
- h_{2,
2}^\text{vert} (X)$, resolution-independence of $M_\text{red}$ also
implies resolution-independence (up to an integral change of basis) of
$M$; this argument is spelled out more explicitly in \cref{invariance,sec:lattice-reduce}.
Previous work
\cite{LinWeigandG4} has implemented the quotient by the nullspace taking
$\Lambda_S$ to $H_{2,2}^{\text{vert}}(X)$ in explicit resolutions by using methods
related to the Stanley-Reisner ideal; here we carry out this quotient using a general form of $M$ for various $\mathsf G$ models defined over arbitrary
$B$. To our knowledge, the observation that $M_\text{red}$
and $M$ are resolution-invariant has not been made previously either in the
mathematics or physics literature.
Both the symmetry constraints and the
projection
into nontrivial homology classes
have a simple geometric interpretation, and it is clear that
the composition of these two operations in either order leads to the same sublattice equipped with a
nondegenerate bilinear form.
Given the original lattice
$\Lambda_S$ with bilinear form $M$,
the constraints \labelcref{eq:Poincare,eq:gauge}
restrict $\phi$ to the sublattice $\Lambda_C$.
If $\dim(\Lambda_S)= m$ and there are $k$ (non-null) constraints, then $\dim(\Lambda_C) = m-k$.
Imposing the homological equivalence
relation $\phi \sim \psi \Leftrightarrow M (\phi-\psi) = 0$ (i.e.,
quotienting out $\Lambda_C$ by the nullspace $V$ of $M$, which
satisfies $V\subset \Lambda_C$) gives us the
lattice of independent vertical flux backgrounds
$\Lambda_\text{phys} = \Lambda_{C}\mathclose{}/\mathopen{}\sim$, with the nondegenerate
bilinear form $M_\text{phys}$ that is the restriction of $M_C$ to
$\Lambda_C \mathclose{}/\mathopen{}\sim$.
We can describe this procedure explicitly in a given basis for $\Lambda_S$.
In particular, if we define an
$m \times(m-k)$ matrix $C$ to have columns given by a set of
generators of the lattice $\Lambda_C\subset\Lambda_S$, then
$C:\mathbb{Z}^{m-k} \rightarrow\Lambda_S$
describes a lattice embedding of $\mathbb{Z}^{m-k}$ into $\Lambda_C \subset\Lambda_S$, and
$M_C :=C^\ensuremath\mathrm{t} M C$
is the restriction of $M$ to $\Lambda_C$ that results from imposing
the symmetry constraints, expressed in a natural basis for
$\Lambda_C$. The resulting form of $M_C$ plays an important role in
our analysis, although with the simplest choice of coordinates there
are some subtleties with integrality conditions that we discuss in
more detail in \cref{sec:quantization-1}
and \cref{constraintsolutions}.
Alternatively, we can
first impose the
homological equivalence relation on $M$,
leading to the reduced intersection pairing matrix
$M_\text{red}$, and then impose the
symmetry constraints.
The preferred order in which to perform these two operations depends on the circumstances. Nevertheless, these two operations lead to the same result when both are performed either
over $\mathbb{Z}$ (more generally, over $\mathbb{R}$),
so the analysis can be carried out in either
order---see \cref{fig:commute}.
\Cref{sec:constraints-homology,sec:4dflux} essentially describe different perspectives on
our analysis that arise from performing these two different orders of
operation.
Each of these two approaches has value for understanding the structure
of chiral matter multiplicities; explicit computation of $M_C$ in many
cases gives us general expressions for the chiral matter
multiplicities in a base-independent fashion, while the structure of
$M_\text{red}$ gives us insight into resolution-independence and the
discrete sets of allowed chiral matter fields.
To maintain clear control of the
discrete quantization of allowed fluxes $\Theta$, however, some care is needed.
While every integer quantized choice of flux background $\phi \in
\Lambda_S$ must correspond to an integer vertical flux background
$G$ by Poincar\'{e} duality, in some circumstances (i.e., when
non-vertical fluxes are included) there may exist
quantized flux backgrounds $G$ that give rise to more general
fractional choices of $\phi$. We restrict attention in our analysis
primarily to fluxes corresponding to integrally quantized $\phi \in
\Lambda_S$ (except for the possible half-integer shift from $c_2 (X)/2$), though we discuss in some places the possibility of more
general fluxes, which may in turn lead to a larger set of possible
chiral matter multiplicities.
These issues are discussed further in \cref{sec:quantization-1}.
\begin{figure}
\begin{center}
\begin{tikzpicture}
\node(1) at (-2,2) {$\Lambda_{S}$};
\node(2) at (2,2) {$\Lambda_C$};
\node(3) at (-2,-2) {$H_{2,2}^{\text{vert}}(X,\mathbb{Z})$};
\node(4) at (2,-2) {$\Lambda_\text{phys}$};
\draw[big arrow] (1) -- node[above,midway]{$$} (2);
\draw[big arrow] (1) -- node[left,midway]{$$} (3);
\draw[big arrow] (2) -- node[right,midway]{$$}(4);
\draw[big arrow] (3) -- node[below,midway]{$$}(4);
\end{tikzpicture}
\end{center}
\caption{Our approach to analyzing vertical fluxes and chiral
matter involves the interplay of two commuting operations on
the lattice of vertical flux backgrounds $\Lambda_S$
spanned by the vertical cycles $S_{IJ} = \hat D_I \cap \hat
D_J$. One operation is the restriction of the $\Lambda_S$ to
the sublattice $\Lambda_C$ of backgrounds satisfying the
symmetry constraints \labelcref{eq:Poincare,eq:gauge} necessary to
preserve 4D local Lorentz and gauge symmetry. The other
operation is the restriction of $\Lambda_S$ to the
vertical homology $\hvtt(X,\mathbb{Z})$ by quotienting $\Lambda_S$ by homologically trivial cycles. Performed in either order, the composition of these two operations
lead to the same sublattice $\Lambda_\text{phys}$ of
consistent F-theory flux backgrounds that preserve gauge symmetries in the low energy effective 4D $\mathcal N=1$ description of the F-theory compactification. We present evidence suggesting that
$\hvtt(X,\mathbb{Z})$ equipped with its symmetric bilinear form $M_\text{red}$ is resolution-independent up to an integer change of basis.}
\label{fig:commute}
\end{figure}
\subsection{Chiral matter multiplicities}
\label{sec:matter-multiplicities}
The by now standard result in the F-theory literature \cite{DonagiWijnholtModelBuilding,Braun_2012,Marsano_2011,KRAUSE20121,Grimm:2011fx}
is that for any complex
representation $\mathsf{r}$ of the gauge group $\mathsf{G}$, the chiral
index is
\begin{equation}
\chIndex{\mathsf{r}} = n_{\mathsf{r}} - n_{\mathsf{r}^*} = \int_{S_{\mathsf{r}}} G \,,
\label{eq:chiral-index}
\end{equation}
where the homology class $S_{\mathsf{r}} \in H_{4}(X,\mathbb{Z})$ is a
``matter surface''. For local F-theory matter, it
is expected \cite{Marsano_2011,Borchmann:2013hta,Bies:2017fam} that any cycle belonging to the class $S_{\mathsf{r}}$ is topologically the fibration of
an irreducible component $C_w$ of the elliptic fibers (where $w \in \mathsf{r}$ is any weight) over an irreducible codimension-two component (i.e., a ``matter curve'', not to be confused with a matter surface) $C_{\mathsf{r}} \subset \{\Delta^{(2)} = 0\}$ of the discriminant locus
$\{\Delta = 0\} \subset B$, associated to the local matter transforming in the quaternionic representation $\mathsf{r} =\mathsf{r} \oplus \mathsf{r}^*$. In practice the flux of $G$ through $S_{\mathsf{r}}$ is computed by way of Poincar\'e duality, i.e.
\begin{equation}
\int_{S_{\mathsf{r}}} G = \int_X G \wedge \text{PD}(S_{\mathsf{r}})\,,
\end{equation}
and hence the analysis of vertical flux backgrounds we describe depends crucially on being able to identify an appropriate cohomology class $\text{PD}(S_{\mathsf{r}})$ dual to $S_{\mathsf{r}}$ (note that the choice of $\text{PD}(S_{\mathsf{r}})$ in general may depend on the choice of resolution $X$). With one exception \cite{Braun_2012}, in all known examples $\text{PD}(S_{\mathsf{r}})$ can be characterized as an element of $\httv(X)$ \cite{WeigandTASI} (or equivalently $S_{\mathsf{r}} \in H_{2,2}^{\text{vert}}(X)$.) However, the precise
definition is subtle and it is unclear that
$S_{\mathsf{r}}$ always has non-trivial components in
$H^{2,2}_{\text{vert}}(X)$; see \cref{sec:puzzle} for a discussion
about this subtlety in the context of certain resolutions of the
$\SU(5)$ model. Our default assumption in this paper is that
$S_{\mathsf{r}}$ always contains a non-trivial component in
$H_{2,2}^{\text{vert}}(X,\mathbb{Z})$. Note that in cases we study where $c_2 (X)$ is not an even class and hence (because of \cref{eq:c2quant}) the flux $G$ is a half-integer class in cohomology, the chiral indices \cref{eq:chiral-index} nonetheless take integer values. This is presumably guaranteed for all physically allowed configurations though we do not know of a complete proof.
As discussed above, the fiber of $S_{\mathsf{r}}$ is a curve $C_w$ that an M2-brane wraps leading to 3D
matter characterized by BPS central charges
$C \cdot \hat D_i = w_i$ (here $\hat D_i$ are Cartan divisors associated to $\U(1)$ gauge factors characterizing the low energy physics and $w_i$ are the Dynkin coefficients of the weight $w$). Thus by Poincar\'{e} duality, we can
construct in homology the class $C$ associated
with a particle labeled by any weight of any representation; note
that to utilize Poincar\'{e} duality in this context one must project
out, e.g., the fiber and zero section, as described in
\cite{Morrison:2021wuv}. We describe an explicit example of a matter
curve and some related quantization subtleties in the simple case of
SU(2) in \cref{sec:su2}.
Unfortunately, however, there is no universal
approach known to explicitly construct $S_{\mathsf{r}}$ in homology
simply from topological and representation theoretic considerations, without using
a specific resolution. The issue is that, as reviewed in
\cite{WeigandTASI}, the image of $S_{\mathsf{r}}$ in $\Lambda_C$ does not simply contain
components of the form $S_{i \alpha} =\hat D_i \cap \hat D_\alpha$; indeed, these must be
projected out to preserve gauge invariance. Rather, the image of $S_{\mathsf{r}}$ in $\Lambda_C$ must also include components in the $S_{ij}$ directions (as described explicitly in this paper by \cref{eqn:SC}), and since the intersection properties of the classes $S_{ij}$ in general may depend on the choice of $X$, it follows that the
precise form of $S_{\mathsf{r}}$ is not known from first principles in a
resolution-independent fashion.
While the approach described in this paper does not rely on explicit computation of the
matter surfaces, we remark that despite the apparent resolution-dependence of $S_{\mathsf{r}}$ the resolution independence of $M_\text{red}$
suggests
that there exists a natural description of $H_{2,2}^{\text{vert}}(X,\mathbb{Z})$ in terms of which
the vertical components of the matter surfaces for any given
anomaly-free combination of representations can be characterized in a
resolution-independent fashion.
Before addressing the explicit computation of chiral matter indices
we recall that, as described above, after both imposing the symmetry
conditions and quotienting by the homology relations encoded in the
nullspace of $M$, we are left with
a set of
independent flux backgrounds $\phi$, associated with a nontrivial
$[\rk M_\text{phys}]$-dimensional sublattice $M_{\text{phys}} \Lambda_{\text{phys}} \subset M \Lambda_{S}$ that for
a given $X$ encodes the
4D chiral matter multiplicities $\chi_{\mathsf{r}}$. Thus, even without explicitly computing $S_{\mathsf{r}}$, we can expect in such cases for there to be
a $[\rk M_\text{phys}]$-dimensional space of $\chi_{\mathsf{r}}$ that
can be realized by turning on different combinations of
$\phi^{IJ}$. Since
F-theory constructions are expected to always be consistent at low energies, these
combinations of $\chi_{\mathsf{r}}$ should always satisfy 4D anomaly
cancellation.
Therefore, we expect that the rank of $M_\text{phys}$, or equivalently the rank of $M_\text{red}$
minus the number of independent constraints in
(\ref{eq:general-constraints}), encodes
the number of linearly independent combinations of chiral matter
fields available in the theory.
As is evident from the above discussion, to explicitly compute $\chi_{\mathsf{r}}$
one must either identify $S_{\mathsf{r}}$, or proceed by
more indirect means. Here we proceed in the latter fashion and
follow a strategy similar to that of \cite{Cvetic:2012xn} (see also \cite{Cveti__2014,Cvetic:2015txa}), which exploits
the following relationship between the set of $\Theta_{IJ}$ satisfying the symmetry constraints and linear combinations of $\chi_{\mathsf{r}}$ given by 3D one-loop Chern--Simons couplings appearing on the Coulomb branch of the 4D F-theory vacuum compactified on a circle:
\begin{equation}
\Theta_{\bar{I}\bar{J}} = -\Theta_{\bar{I}\bar{J}}^{\text{3D}},~~~~ \bar{I} = \bar{0}, i
\end{equation}
where
\begin{equation}
\Theta_{\bar I \bar J} := \sigma_{\bar I}^I \sigma_{\bar J}^J \Theta_{IJ}\,.
\end{equation}
(Recall that the index $\bar{0}$ denotes the abelian Kaluza--Klein gauge field associated to the compact circle and we use the index $i$ to collectively denote all other $\U(1)$ gauge fields; see \labelcref{physbasis} for the precise definition of $\sigma^I_{\bar I}$.) In the above equation the couplings $\Theta_{\bar{I}\bar{J}}^{\text{3D}}$, which receive contributions from integrating out all massive fermions on the Coulomb branch, can be expressed as linear combinations
\begin{equation}
\label{eq:thetaconvert}
\Theta_{ij} = x^{\mathsf{r}}_{ij} \chi_{\mathsf{r}},~~~~ x^{\mathsf{r}}_{ij} \in \mathbb{Q}\,.
\end{equation}
For every resolution that satisfies our default assumption that each
matter surface has a vertical component,
the above linear system can
be inverted, which allows us to then write an explicit formula
\begin{equation}
\label{eq:chiconvert}
\chi_{\mathsf{r}} = x^{ij}_{\mathsf{r}} \Theta_{ij}\,.
\end{equation}
We expect the set of allowed chiral multiplicities $\chi_{\mathsf{r}}$
that can be realized for integer flux backgrounds $\phi^{ij}$
to be independent of the choice of
resolution $X$, up to a choice of basis for $\phi^{ij}$.
\subsection{Linear constraints and anomaly cancellations}
\label{sec:linear-anomaly}
The anomaly conditions on any 4D theory are linear relations on the
$\chi_{\mathsf{r}}$ (these conditions are reviewed in
\cref{4Danomalyreview}). There are also linear relations that automatically
hold on $\Theta_{IJ}$ by virtue of the nullspace of $M$. Connections
between the anomaly relations and these geometric conditions were
identified in \cite{Bies_2017} (see also \cite{Grimm:2015zea, Corvilain:2017luj}). Our general finding here is that in
all cases we consider the linear relations on $\Theta_{ij}$ imposed by
the nullspace conditions and symmetry constraints are precisely the
same as the anomaly conditions, so that not only does geometry encode
the anomaly conditions, but there are also no
further linear constraints coming from F-theory on the set of allowed
chiral multiplicities, and thus fluxes exist that can turn on all
anomaly-allowed combinations of chiral matter fields in all the cases
we explore here.
In general, the linear constraints that hold on the fluxes
$\Theta_{IJ}$ for an F-theory background where the 4D gauge
group is unbroken (and hence equal to the geometric gauge group $\mathsf{G}$) are the union
of those that come from the nullspace of $M$ and the constraints
\labelcref{eq:general-constraints}. It is helpful to consider how this set
of constraints arise in the two approaches characterized in
\cref{fig:commute}. When the nullspace of $M$ is quotiented
out first, giving $M_\text{red}$, and then the constraints are imposed,
it is clear that the constraints listed above are precisely the constraints on the resulting
$\Theta$s that can arise. The situation is slightly subtler,
however, when the constraints are imposed first. In particular, the
signature of the inner product matrix $M$ is not generally
semi-definite, so in principle
there can be vectors of vanishing norm that are not null vectors of
$M$. If one of the constraints \labelcref{eq:general-constraints} can
be described as $w M \phi$ for a vector $w$ of this
type, then when the constraints are imposed first the matrix $M_C$
could have additional null vectors beyond those associated with
homological equivalence in $\Lambda_S$; this would
occur when the vector
$w$ also lies in $\Lambda_C$. The subsequent quotient by
homologically trivial cycles (null vectors of $M$) does
not remove such null vectors from $M_C$. Nonetheless, any such vector
would still correspond to a linear combination of the nullspace and symmetry
constraints.\footnote{A simple proof of this statement can be made as
follows: assume without loss of generality that $M$ has no
nullspace, and the constraints are of the form $w \Theta =
w M \phi= 0$ for $w\in W$, and $\Lambda_C$ is the
orthocomplement $W^\perp$ of the set of constraints $W$. Then any
null vector $u \in\Lambda_C$ of $M_C$
satisfies $u M_C \phi = u M \phi = 0$
for any $\phi \in W^{\perp}$.
But then $u\in (W^{\perp})^{\perp}= W$, so $u$ is a constraint
vector. A similar proof follows when $M$ has nontrivial nullspace,
though $u$ can also have a component then in this nullspace. }
While this situation does not arise in practice in any of the models
we analyze here, we do not have any way of strictly ruling it out,
particularly for models with one or more $\U(1)$ factors, so this
possibility must be kept in mind throughout the analysis.
\subsection{Quantization of fluxes and matter multiplicities}
\label{sec:quantization-1}
One thorny issue, which has not been fully resolved to our knowledge
anywhere in the literature, is the precise quantization condition on
the fluxes and the consequent constraints on the multiplicities of
chiral matter. Even in the most well-understood $\SU(5)$ F-theory GUT
constructions, this question is left open in analyses of which we are
aware.
Note that this question is independent of issues related to the
shifted quantization condition \cref{eq:c2quant}.
We do not fully resolve this quantization issue here but we do shed
some light on the question and provide a set of sufficient conditions
for matter with certain multiplicities to exist.
In the basis for $\Lambda_S$ given by the surfaces $S_{IJ}$, the
coefficients $\phi^{IJ}$ are always
integers for purely vertical fluxes (or in some cases half-integers, when $c_2 (X)$ is not even)\footnote{For the most part we frame the discussion in terms of cases where $c_2 (X)$ is an even class, so that the quantization issue of \cref{eq:c2quant} leaves $G$ as an integer cohomology class; it should be kept in mind however that when $c_2 (X)$ is not an even class, some of the flux background parameters must be half-integer, i.e. $\phi^{IJ} \in \mathbb{Z} + \tfrac{1}{2}$. We consider explicit examples of this in section \cref{sec:su5}.}, so that
the lattice vectors $\phi$ live in the lattice $H_{2,2}^{\text{vert}}(X,\mathbb{Z}) =
\Lambda_S/ \sim$ obtained by quotienting out homologically trivial
$\phi$. However,
a basic observation is that the matrix $M_\text{red}$ that gives an inner
product on this space (and which maps $\phi$ to some corresponding
$\Theta$) does not in general have determinant equal to $\pm1$, so
that the possible values of $\Theta$ that can be realized
generically imply a nontrivial
quantization on possible chiral matter multiplicities induced by
vertical fluxes. Furthermore, the symmetry constraints
\labelcref{eq:Poincare,eq:gauge} impose further constraints on the
allowed values of $\phi$ and hence the resulting nonzero
$\Theta$
and associated chiral multiplicities may be subject to additional quantization
constraints.
More explicitly, in many situations such as that of a purely
nonabelian gauge group,
the condition that certain $\Theta$s
must vanish, needed to preserve local Lorentz and gauge symmetry of the
4D theory, can be written schematically in the form
\begin{equation}
\left(\begin{array}{c}
0\\\Theta''
\end{array} \right)
= \left(\begin{array}{cc}
M' & Q\\
Q^T & M''
\end{array} \right)
\left(\begin{array}{c}
\phi'\\
\phi''
\end{array} \right) \,,
\label{eq:constraint-ordered}
\end{equation}
so we have
\begin{equation}
M' \phi' + Q \phi'' = 0\,.
\label{eq:mq}
\end{equation}
In the basis for $\Lambda_S$ given by the surfaces $S_{IJ}$, the
coefficients $\phi^{IJ}$ comprising $\phi', \phi''$ are always
integers. When the matrix $M'$ has a non-unit determinant, we
can think of the image of $M'$ acting on vectors $\phi'$ in $\mathbb{Z}^k$ as a
$k$-dimensional lattice $\Lambda'$. We can solve the equation
(\ref{eq:mq}) for integer values of $\phi'$ if and only if $Q \phi''
\in \Lambda'$. This gives a quantization condition on the flux
coefficients $\phi''$ that is both necessary and sufficient to have an
integer solution for $\phi'$. Thus, we can determine a condition on
$\phi$, and hence on the nonzero $\Theta$s that parameterize the
chiral matter, which is sufficient to guarantee the existence of an
allowed flux background in $H_{2,2}^{\text{vert}}(X,\mathbb{Z})$. As we see in the more explicit analyses of
\cref{sec:exampleADE}, in cases of a simple gauge group like $\mathsf{G} = \SU(N)$,
this kind of analysis leads to a natural understanding of the
quantization condition on the fluxes from the appearance of the Cartan
matrix of $\mathsf{G}$ in the role of at least a block of the matrix $M'$.
The story is somewhat more complicated in the presence of $\U(1)$ factors.
The analysis just summarized focuses only on vertical fluxes. From
Poincar\'{e} duality of $H_{2,2}(X,\mathbb{C})$, we know that there must exist a
flux so that $\int_SG = 1$ for any primitive element $S$ in $H_{2,2}
(X, \mathbb{Z})$. As mentioned above, however, the intersection form is not
in general unimodular on
$H^{2, 2}_\text{vert} (X,\mathbb{Z})$. Thus, a complete analysis of the set of
possible chiral matter multiplicities available
may require including flux backgrounds with components in
$H^{2, 2}_\text{rem} (X,\mathbb{Z}) \oplus
H^{4}_\text{hor} (X,\mathbb{Z})$ and/or fractional coefficients in terms of the basis $\PD(S_{IJ})$ for $H^{2, 2}_\text{vert}(X)$---see \cref{fig:half} for an illustration of this point.
\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=2.5]
\node[] at (2.8,0) {\scalebox{1}{$H_{2,2}^{\text{vert}}(X)$}};
\node[] at (0,2.6) {$H_{4}^{\text{hor}}(X) \oplus H_{2,2}^{\text{rem}}(X)$};
\draw[dashed,thick,color=gray] (2*.5,-2*.1) -- (2*.5,2*1.1);
\draw[dashed,thick,color=gray] (2*1,-2*.1) -- (2*1,2*1.1);
\draw[dashed,thick,color=gray] (2*0,-2*.1) -- (2*0,2*1.1);
\draw[dashed,thick,color=gray] (-2*.1,2*.5) -- (2*1.1,2*.5);
\draw[dashed,thick,color=gray] (-2*.1,2*1) -- (2*1.1,2*1);
\draw[dashed,thick,color=gray] (-2*.1,2*0) -- (2*1.1,2*0);
\draw[big arrow,scale=2,very thick] (-2*.05,0) -- (1.2,0);
\draw[big arrow,scale=2,very thick] (0,-2*.05) -- (0,1.2);
\node[] at (2*.65,0.8) {$(\tfrac{1}{2},\tfrac{1}{2})$};
\node[] at (2*1,-2*.15) {$(1,0)$};
\node[] at (2*0,-2*.15) {$(0,0)$};
\node[draw,circle,fill=blue,color=blue,scale=.8] at (2,0) {};
\node[draw,circle,fill=blue,color=blue,scale=.8] at (0,0) {};
\node[draw,circle,fill=red,color=red,scale=.8] at (1,1) {};
\end{tikzpicture}
\end{center}
\caption{Toy example of a possible realization of $ H_4(X,\mathbb{Z})$ as an integral unimodular lattice (note that we implicitly include the shift by $c_2/2$), where we take the bilinear pairing to be $M_{\text{red}} \oplus M_{\text{red}}^{\perp} = \text{diag}(2,2)$. In this example we denote lattice vectors by $ (\phi,\psi)$, with $\phi \in H_{2,2}^{\text{vert}}(X), \psi \in H_{4}^{\text{hor}}(X) \oplus H_{2,2}^{\text{rem}}(X)$. As can be seen by requiring the inner product $2 \phi^2 + 2 \psi^2$ to take integer values, the restriction of $(\phi,\psi)$ to $(\phi,0)$ (represented by blue dots in the above graph) requires $\phi \in H_{2,2}^{\text{vert}}(X,\mathbb{Z}) \cong \mathbb{Z}$ to preserve the integrality of the lattice, i.e. $\phi \in \tfrac{1}{2} \mathbb{Z}$ is forbidden. However, there exist lattice vectors with $\psi \ne 0$ for which $\phi \in \tfrac{1}{2} \mathbb{Z}$, for instance the vector $(\tfrac{1}{2},\tfrac{1}{2})$ represented by the red dot in the above graph. This example shows that vertical flux backgrounds $\phi$ with rational coefficients $\phi^{IJ} \in \mathbb{Q}$, which preserve the integrality of both the lattice and chiral indices, could in principle exist.}
\label{fig:half}
\end{figure}
This is discussed in more detail in the
case of $\SU(5)$ in \cref{sec:su5}.
Flux backgrounds with such fractional coefficients have been analyzed previously
in, e.g., \cite{CveticEtAlQuadrillion}.
In that
context, in the notation of this paper, fractional values of $\phi$
are considered and the necessary constraints that $M \phi$
gives integer values (i.e. that $G$ integrated over any surface in
$H^\text{vert}_{2, 2}(X)$ is integral) are imposed. However, not all such fractional values of
$\phi$ necessarily correspond to allowed fluxes. As an example of this point, consider the self-dual lattice defined by the symmetric bilinear form
$\diag(2, 2)$. This lattice consists of all vectors $(x, y)$ with $x, y \in \mathbb{Z}/2,x
+ y \in\mathbb{Z}$. The vector $(1/2, 0)$ has integer inner product with the
elements of a non-unimodular basis $(1, 0), (0, 1)$, but it is not an
element of the given self-dual lattice. For similar reasons,
the conditions that $M \phi$ is integral are not by themselves sufficient to
guarantee that $\phi$ is an integer homology class.
This question is further complicated by the lack of understanding of
the components of $\phi$ in
$H_{2, 2}^\text{rem} (X,\mathbb{Z}) \oplus
H_{4}^\text{hor} (X,\mathbb{Z})$. Thus,
it is
difficult to ascertain exactly which fractional values of $\phi$
correspond to vectors in the unimodular lattice $H_4 (X,\mathbb{Z})$.
We discuss this further in some specific examples in \cref{sec:su2}
and \cref{sec:su5}.
When $H_{2, 2}^\text{vert}(X,\mathbb{Z})= H_4 (X,\mathbb{Z}) \cap H_{2, 2} (X,\mathbb{R})$,
and $\phi$ is allowed to be a general element of $H_4 (X,\mathbb{Z})$,
then the unimodularity of $H_4 (X,\mathbb{Z})$ implies that the proper conditions for the vertical component $\phi_\text{vert}$ are that it should lie in the dual lattice to $H_{2, 2}^\text{vert}(X,\mathbb{Z})$ and also in the constrained lattice $\Lambda_C$, but is not subject to any further apparent constraints. This does not, however, imply that even in this case any chiral multiplicity is possible, without further information about whether the matter surface has components in $H^{2, 2}_\text{rem} (X,\mathbb{Z}) \oplus
H^{4}_\text{hor} (X,\mathbb{Z})$; we leave a more detailed investigation of these integrality conditions to future work.
\subsection{Codimension-three $(4, 6)$ loci}
\label{sec:46}
Many F-theory geometries contain $(4, 6)$ (or
higher) singularities in the elliptic fibration over codimension-three loci in the base \cite{Candelas:2000nc,TaylorWangMC}; these are
often associated with non-flat fibers in the resolution
\cite{Lawrie:2012gg}.
In this paper, we focus on geometries without codimension-three $(4, 6)$
singular loci in the elliptic fibration. We have also analyzed a
variety of situations, such as universal models with gauge groups $\mathsf{G}
= \SU(N\geq 7), \SO(N \geq 12), \gE_7$ and other cases that do
have codimension-three $(4, 6)$ loci, where we find that
there is an additional allowed flux background parameter and the rank of $M_\text{red}$
is larger than expected from the 4D anomaly cancellation
conditions. A similar extra flux background parameter and increased rank arises
in the single exceptional case $\SO(11)$ without codimension-three $(4,
6)$ points where we find a mismatch between the available anomaly-free
families of charged matter and the rank of $M_\text{red}$. Analysis of these
models will be described in a separate publication \cite{46}.
\subsection{Summary of new results}
\label{summary}
We summarize here the main results of the paper:
\begin{itemize}
\item{} We show, by way of example, that the pushforward technology of \cite{Esole:2017kyr}
can be used to easily compute the vertical fluxes of resolutions of
singular Weierstrass models with any nonabelian gauge symmetry
subgroup over an arbitrary smooth base. We also show that $\U(1)$ gauge
factors can be incorporated into the analysis in a manner that
depends explicitly on the intersections of the associated height
pairing divisors with the curve classes of the base. We present an
explicit expression for the vertical fluxes in terms of the
pushforwards of the intersection numbers of the resolved elliptic CY
fourfold to the base; in the special case of a purely nonabelian
gauge group, these intersection numbers only depend on the
intersections of the canonical class of the base and the classes of
the gauge divisors supporting the simple factors of the gauge group.
\item{} We find that the reduced intersection pairing $M_\text{red}$ on the
vertical middle cohomology $H^{2, 2}_\text{vert} (X,\mathbb{Z})$ is
independent of resolution (up to a change of
basis)
in all cases
we consider explicitly, although the full set of quadruple intersection numbers
are not generally resolution-independent.
We furthermore show that this
resolution-independence holds
for all
F-theory models with nonabelian gauge symmetry and generic matter,
when the physically relevant $M_\text{phys}$ is resolution invariant
and obeys certain compatibility conditions related to the weight lattice of
the gauge algebra.
We conjecture that the resolution-independence of $M_\text{red}$
(and hence also of the full matrix $M$ including the nullspace built
from formal product surfaces)
holds
more generally for F-theory models with arbitrary gauge groups,
including those with $\U(1)$ factors, and give some explicit examples supporting this conjecture.
\item{} Exploiting M-theory/F-theory duality, we match 3D Chern--Simons couplings with the vertical fluxes to obtain the chiral matter multiplicities associated to various examples of universal $\mathsf{G}$ models, some not previously studied in the literature. In particular we study low rank examples of models with simple, simply laced gauge group and generic matter (see \cref{tab:fluxtable}), as well as the universal $(\SU(2) \times \U(1)) / \mathbb{Z}_2$ model.
\item{} With one exception ($\mathsf{G} =\SO(11)$), we find in all cases we
study
that the
number of independent vertical fluxes remaining after imposing
constraints necessary to preserve local Lorentz and gauge symmetry
in 4D---equivalently, the rank of the nondegenerate intersection
pairing of the vertical cohomology of the resolved elliptic CY
fourfold with integer coefficients, minus the number of
independent symmetry
constraints---is equal to the number of allowed
independent families of 4D
chiral matter multiplicities plus the number of non-minimal
codimension-three singularities in the F-theory base.\footnote{The
resolutions we study of models with codimension-three $(4, 6)$ loci
are non-flat fibrations in which the fibers over the $(4, 6)$ loci
contain a K\"ahler surface as an irreducible component.}
This means that
the number of independent allowed fluxes precisely matches the number
of families of chiral matter allowed through anomaly cancellation,
except for the special case $\mathsf{G} = \SO(11)$
and theories with codimension-three $(4, 6)$ points, which we will treat in
more detail in \cite{46}.
The
$\SO(11)$ model, despite not admitting chiral $\SO(11)$
representations nonetheless has nontrivial vertical flux and thus
is an apparent exception to this general characterization.
(See \cite{DelZotto:2018tcj} for a
discussion of additional puzzles related to the $\SO(11)$
F-theory model; see also the comments in \cref{sec:46}.)
\end{itemize}
\section{Fluxes preserving 4D local Lorentz and gauge symmetry}
\label{sec:4dflux}
\Cref{sec:4dflux,sec:constraints-homology} present two different
perspectives on the relation between flux backgrounds $\phi^{IJ}$ and
fluxes $\Theta_{IJ}$ corresponding to the two paths from the upper
left to the lower right of the
commuting diagram in \cref{fig:commute}. In this section, we describe the
sublattice of flux backgrounds $\Lambda_C \subset \Lambda_S$, which is the
preimage of the lattice of fluxes $M \Lambda_S$ preserving 4D
local Lorentz and gauge symmetry. The matrix elements of the inner
product matrix $M_C$ on the constrained space depend on the
pushforwards $W_{IJKL}$ of quadruple intersection numbers of a smooth
elliptic CY fourfold $X$ resolving a Weierstrass model with gauge
symmetry $\mathsf{G} = (\mathsf{G}_\text{na} \times \U(1)) / \Gamma$ (cases with
additional $\U(1)$ factors are a straightforward generalization of the
results presented here.) Note that in this section and the next, we do not concern ourselves with the
shifted quantization condition \labelcref{eq:c2quant} but simply treat
$\Lambda_S$ as an integral lattice, with the understanding that
sometimes this quantization condition gives an overall half-integer
shift that must be incorporated in specific contexts.
In \cref{fluxesint} we review the general relationship between
vertical fluxes $\Theta_{IJ}$ and intersection numbers of the types of
smooth elliptic CY fourfolds $X$ with which we concern
ourselves. \cref{constraintsolutions} presents an explicit expression
for $M_C$ that is valid in most of the cases we consider. In \cref{sec:homologyrel} we discuss further the
relationship between the nullspace of $M_C$ and linear constraints on
$\Theta_{IJ}$, along with the relationship of these constraints to 4D
anomaly cancellation.
\subsection{Computing vertical fluxes with intersection theory}
\label{fluxesint}
Given a smooth CY fourfold $X$ and a basis of divisors $\hat D_I$
where $I = 0,1,\alpha, i_s$ (we take $I=a= 1$ to be the only index, if
there is one,
denoting a U(1) section---see the discussion immediately below
\labelcref{eq:Wazir} for more details about the index structure), we
may expand a vertical flux background $G \in
H^{2,2}_{\text{vert}}(X,\mathbb{Z})$ in a basis of wedge products of
$(1,1)$-forms dual to divisors, $\PD(\hat D_I)$, where `PD' denotes the Poincar\'{e} dual.\footnote{Lefshetz's theorem on
(1,1)-classes applied to projective varieties such as $X$ guarantees that given a basis of divisors $\hat D_I$ there always corresponds a Poincar\'e dual basis of harmonic $(1,1)$ forms $\PD(\hat D_I)$---see
e.g. \cite{Bizet:2014uua} for a related discussion.}
In our analysis here we formally work in the Chow ring, which exhibits the intersection properties of elements of (co)homology that have a description in terms of
algebraic subvarieties. The reason for this is that the pushforward technology
that we use, which is described in more detail in
\cref{intersection}, is defined with respect to the Chow ring. For
the purposes of the analysis here, however, the only elements of the
Chow ring that concern us are the classes of divisors $\hat{D}_I$ and
their intersections
$S_{IJ} = \hat D_I \cap \hat D_J \in \Lambda_S$, which can be
understood directly as elements of the homology groups $H_{3, 3}
(X,\mathbb{Z})$ and $H_{2, 2} (X,\mathbb{Z})$ respectively.
As described in \labelcref{eq:constraints-theta-m},
the integrals
of a flux background $G$ over the cycles of vertical
surfaces can be
evaluated in terms of intersection products of $\hat D_I$
\begin{equation}
\Theta_{IJ} = \int_{S_{IJ}} G = \int_{X} G \wedge \PD(S_{IJ}) =\phi^{KL} S_{KL} \cdot S_{IJ}= \phi^{KL} \hat D_K \cdot \hat D_L \cdot \hat D_I \cdot \hat D_J\,,
\end{equation}
and so we may parametrize a candidate
vertical flux $G$ in terms of its Poincar\'{e} dual class $\phi$ in the
Chow ring of $X$ as
\phi = \phi^{IJ} S_{IJ}
$,
leading to the more succinct expression
\begin{equation}
\Theta_{IJ} = \phi \cdot S_{IJ}\,.
\end{equation}
This correspondence between integrals over cycles and
intersection products implies that the intersection pairing matrix $M
: \Lambda_S \times \Lambda_S \rightarrow \mathbb{Z}$ can be described as a
matrix with indices given by pairs $IJ$, where the matrix elements
are expressed in terms
of quadruple intersection numbers,
\begin{equation}
M_{(IJ)(KL)} =S_{IJ} \cdot S_{KL}\,.
\end{equation}
Thus, essentially every computation relevant for determining the
multiplicities of chiral matter can be characterized in terms of
linear algebra and performed using intersection
theory.
In what follows, we assume that the smooth fourfold $X$ is a resolution of a singular Weierstrass model belonging to a family defined by the characteristic data $(K, \Sigma_s, W_{01})$, where $K$ is the canonical class of $B$, $\Sigma_s$ is the class of the gauge divisor in $B$ associated to the nonabelian gauge subalgebra $\mathfrak{g}_{s}$ and $W_{01}$ is the class of the pushforward $\pi_*(\hat D_0\cdot \hat D_1)$ of the intersection of the zero and generating sections. We moreover assume that the resolved elliptic CY fourfold $\pi : X \rightarrow B$ can be realized as a hypersurface inside an ambient fivefold that is the blowup of a $\mathbb{P}^2$ bundle. These assumptions allow us to evaluate the quadruple intersection numbers explicitly by computing their pushforward to the Chow ring of the base $B$,
\begin{equation}
\pi_*( \hat D_I \cdot \hat D_J \cdot \hat D_K \cdot D_L ) =W_{IJKL}= W_{IJKL}^{\alpha \beta \gamma} D_{\alpha} \cdot D_{ \beta} \cdot D_{ \gamma}\,,
\end{equation}
where the right side of the above equation can be expressed as a
linear combination of triple intersection products of the classes of
the characteristic data $(K, \Sigma_s, W_{01})$. Furthermore, since
$X$ is an elliptic fibration, for certain multi-indices $IJKL$ the
pushforwards $W_{IJKL}$ have additional structure that remains
applicable for all known crepant resolutions. For convenience we
suppress the explicit pushforward map $\pi_*$ when the appropriate
ring is otherwise clear from the context. See \cref{intersection} for
additional mathematical details about the pushforward map $\pi_*$ and
the structure of the intersection numbers.
\subsection{Explicit solutions of the symmetry constraints}
\label{constraintsolutions}
The main result of this subsection is an explicit expression for the
matrix $M_C$ and the resulting possible
fluxes $\Theta_{IJ}$ that are the integrals of flux backgrounds
$\phi^{IJ}$ restricted to live in the sublattice $\Lambda_C$
satisfying the symmetry constraints \labelcref{eq:Poincare,eq:gauge}.
We now sketch the essential features of the computation; the details of this derivation can be found in
\cref{fluxderivation}. As we have seen, the symmetry constraints
\labelcref{eq:Poincare} and \labelcref{eq:gauge} imply $\Theta_{I\alpha} = 0$.
By ordering $\Theta_{IJ}$
so that those that $\Theta_{I\alpha}$ are listed first and likewise for $\phi^{I\alpha}$, as a matrix equation the symmetry constraints take the schematic form \labelcref{eq:constraint-ordered}, which we reproduce here for convenience:
\begin{equation}
\left(\begin{array}{c}
0\\\Theta''
\end{array} \right)
= \left(\begin{array}{cc}
M' & Q\\
Q^T & M''
\end{array} \right)
\left(\begin{array}{c}
\phi'\\
\phi''
\end{array} \right) \,.
\end{equation}
In solving the symmetry constraints \labelcref{eq:Poincare} and
\labelcref{eq:gauge} it is often convenient to eliminate (when possible)
the parameters $\phi^{I\alpha}$, whose indices match those of the
$\Theta_{I\alpha}$, which we require to vanish. We sometimes refer to
objects carrying indices $\hat I \hat J$ (i.e., pairs $IJ$ for which
$I, J \ne \alpha$) as ``distinctive'', and all others choices of
index $I\alpha$ as ``non-distinctive''. For example, in the above matrix
equation, $\phi''$ and $\Theta''$ have distinctive indices. However,
even when we cannot solve for all non-distinctive $\phi$ parameters
explicitly, we nevertheless sometimes denote by $\Theta''$ the
set of fluxes obeying the symmetry constraints.
Note also that even when we can solve for all the
non-distinctive $\phi$ parameters it is sometimes useful to solve for
a different set of $\phi$s; see, e.g., \cref{eq:21-indirect}.
In cases for which the matrix block $M'$ is
nondegenerate
we can solve the equation \labelcref{eq:constraint-ordered} for the non-distinctive $\phi'$ parameters in terms
of the distinctive $\phi''$ parameters, giving
\begin{equation}
\phi' = -(M')^{-1} Q \phi'' \,.
\label{eq:solution-abstract}
\end{equation}
When $|\det M'| = 1$, there is an integer solution in $\phi'$ for
any $\phi''$. When $|\det M'| > 1$, however,
the integrality condition imposed on $\phi''$ by requiring that the symmetry constraints be solved over $\mathbb{Z}$ is subtler, as discussed in
\cref{sec:quantization-1}.
In some cases, such as imposing the constraints on $M_\text{red}$
after removing null vectors for a purely nonabelian group, the
corresponding matrix $M'$ in the non-distinctive directions is
non-degenerate and invertible, and this procedure of solving for the
$\phi'$ flux backgrounds can be performed
exactly as in \cref{eq:solution-abstract}. In other cases, in
particular when we consider the constraints directly on $M$ and null
vectors still are included among the $\phi$s, the matrix $M'$ is
degenerate and cannot be inverted. In many such cases we can impose
the constraints by simply using the pseudoinverse of $M'$ for
$(M')^{-1}$, which for a symmetric matrix basically means taking the
inverse on the orthocomplement of the null space and the zero matrix
on the null space. This is equivalent to simply removing the null
space and then taking the inverse. Note that this works when the null vectors of $M'$ are also null vectors of $M$.
We can use this more general sense of \cref{eq:solution-abstract} to write
an expression for the restriction of $M$ to $\Lambda_{C} \subset
\Lambda_S$ when $M'$ is nondegenerate and thus invertible on the
orthocomplement of the null space.
In the following we denote by $(M')^{-1}$ the pseudoinverse of $M'$.
In
particular, as outlined in \cref{sec:vertical-pairing} we can define
the $m \times (m - k)$ matrix
\begin{equation}
\label{eq:c}
C = \left(\begin{array}{c}
-(M')^{-1} Q\\
\Id_{m-k}
\end{array}\right) \,,
\end{equation}
which defines an embedding of $\mathbb{Z}^{m-k}$ associated with the
$m-k$
distinctive directions into a rational extension of
the full lattice $\Lambda_S$, $C:\mathbb{Z}^{m-k} \rightarrow
\Lambda_C (\mathbb{Q})
\subset \Lambda_S (\mathbb{Q})$.
Note that null directions in $M'$ are associated with constraints that
are automatically satisfied, so the corresponding combinations of
$\phi'$ vanish, in accord with the definition of the pseudoinverse.
As discussed in \cref{sec:quantization-1}, when $\det M' = \pm 1$
(for the non-null part of $M'$)
this map gives a one-to-one correspondence between $\mathbb{Z}^{m-k}$ and
$\Lambda_C$; otherwise, the domain of $C$ must be taken to be the
subset
$\operatorname{dom} C = C^{-1} (\Lambda_C)$.
In general, given such a mapping $C$, we can give an explicit description of the
the inner product form
\begin{equation}
M_C =C^\ensuremath\mathrm{t} M C= M'' -Q^\ensuremath\mathrm{t} (M')^{-1} Q\,,
\label{eq:mc}
\end{equation}
which
gives the intersection pairing of flux backgrounds in the constrained space
$\Lambda_{C}$ as parameterized by $\phi''$, recalling that in some
situations there may be additional discrete constraints on the
$\phi''$ values allowed for a valid flux background.
We analyze these integrality conditions in more
detail for
general nonabelian gauge groups in \cref{integrality}, and for specific examples in \cref{sec:exampleADE,F6model}.
In much of the discussion, however, we elide this subtlety.
Carrying this description slightly further,
we can also define an $m \times m$ matrix
\begin{equation}
P =\left( 0_{m\times k} \hspace*{0.1in} C \right) \,,
\end{equation}
which is idempotent ($P^2 = P$) and gives by right multiplication of
the matrix $M$
\begin{equation}
MP = P^\ensuremath\mathrm{t} MP =
\left(\begin{array}{cc}
0 & 0\\
0 & M_C
\end{array} \right).
\label{eq:MP}
\end{equation}
This extends the embedding map $C$ to be defined on all of
$\Lambda_S$, where the extra (non-distinctive) parameters are
essentially thrown out in the map, which becomes a projector; this
form of $M_C$ will be useful in some places. In particular note that
the $\Theta$s that result from the action of $MP$ on a given set of
$\phi$s satisfying the constraint equations span the set of possible
vertical fluxes. Recalling that $M_\text{phys}$ can be defined as the
inner product on $\Lambda_C\mathclose{}/\mathopen{}\sim$ after taking
the quotient by homologically trivial cycles, we have
\begin{equation}
\rk M_\text{phys} = \rk M_C \,.
\end{equation}
This rank corresponds to the number of linearly independent
families of allowed fluxes.
We present now
a formal expression for the matrix elements of $M_C$ in the case of a
gauge group $\mathsf{G} = (\mathsf{G}_\text{na} \times \U(1))/\Gamma$ for generic
characteristic data.
This set of expressions is valid whenever $M'$ is nondegenerate and
invertible (or pseudo-invertible, using null vectors of $M'$ that are also null vectors of $M$).
This condition always holds when $\mathsf{G}$ is purely
nonabelian and the F-theory geometry admits a holomorphic zero section, and is true in most situations we have considered with
simple bases and/or generic characteristic data
when the gauge group contains $\U(1)$ factors.
As shown in the example in \cref{sec:21-exception}, however, there are some cases
where $M'$ is degenerate even after removing null vectors in the non-distinctive directions; in such situations we can still analyze the
spectrum by solving for a different set of $\phi$s, but the
formulae given here do not apply in this form.
The fluxes satisfying the symmetry constraints take the form\footnote{Hatted indices are of type
$\hat I = 0,1,i_s$, i.e., a restriction of the usual indices to the
case $I\ne \alpha$.}
\begin{equation}
\label{eq:thetaprime}
\Theta_{ \hat I \hat J} = {M_{C}}_{(\hat I \hat
J)(\hat K \hat L)} {\phi}^{ \hat K \hat L} \,.
\end{equation}
In the above equation, the matrix elements of $M_C$ can be expressed as
\begin{equation}
\label{eq:geotheta1}
{M_C}_{( \hat I \hat J) (\hat K \hat L) } = { M_{C_\text{na}}}_{( \hat I \hat J) (\hat K \hat L) } - { M_{C_\text{na}}}_{( \hat I \hat J)(1\alpha)} M_{C_\text{na}}^{+(1\alpha)(1\beta)} { M_{C_\text{na}}}_{(1\beta)( \hat K \hat L)}\,,
\end{equation}
where $ { M_{C_\text{na}}}= C_\text{na}^\ensuremath\mathrm{t} M C_\text{na}$ is
the restriction of $M$ to the sublattice $\Lambda_{ C_\text{na}}$ of
backgrounds only satisfying the purely nonabelian constraints
$\Theta_{i_s \alpha} =0$. The components of $ M_{C_\text{na}}$ are
\begin{equation}
\begin{aligned}
\label{eq:omegabar}
{M_{C_\text{na}}}_{ ({I} {J}) (K L)} &= W_{ I J K L}- W_{ I J|i_{s}} \cdot W^{i_{s}| j_{s'}} W_{ K Lj_{s'}} - W_{0 I J} \cdot W_{ K L} - W_{ I J} \cdot W_{0 K L} \\
&\quad+ W_{00}\cdot W_{ I J} \cdot W_{ K L}
\end{aligned}
\end{equation}
where in particular
\begin{align}
\begin{split}
\label{eq:omegabar2}
{ M_{C_\text{na}}}_{ (1 \alpha) (KL)} &= D_\alpha \cdot W_{\bar 1 KL}\\
&=D_\alpha \cdot (-W_{1|k_{s''}} W^{k_{s''}| i_s} W_{i_s KL} + W_{1 IJ}-W_{0 KL}+(W_{00}-W_{01}) \cdot W_{ KL})
\end{split} \\
\begin{split}
\label{eq:omegabar3}
{ M_{C_\text{na}}}_{(1\alpha)(1\beta)}&= D_\alpha \cdot D_\beta \cdot W_{\bar 1 \bar 1}\\
&= D_\alpha \cdot D_\beta \cdot (- W_{1|k_{s''}} W^{k_{s''}|i_s} W_{1 i_s }+2(W_{00} - W_{01}) )
\end{split}
\end{align}
and $M_{C_\text{na}}^{+(1\alpha)(1\beta)}$ is the inverse of ${
M_{C_\text{na}}}_{(1\alpha)(1\beta)}$.
The structure of the various pushforwards $W_{IJ}$ is explained in more detail in
\cref{ellipticintersection}; for example in \cref{eq:omegabar3},
$W_{\bar 1 \bar 1}$ is equal to (minus) the height pairing
divisor associated to the $\U(1)$. For a purely nonabelian gauge group, there are no indices of the form
$(1 \alpha)$, the second term in \cref{eq:geotheta1} can be dropped, and
$M_C = M_{C_\text{na}}$ from \cref{eq:omegabar}.
The fact that the restriction of $M'$ (see \cref{eq:constraint-ordered})
to the nonabelian part of the theory (i.e., taking all
indices $I \alpha$ except $1 \alpha$) contains a non-trivial invertible submatrix for generic characteristic data over arbitrary $B$ can be
deduced from the explicit form of the components of $M'$, which are all
resolution-independent, as discussed in more detail in \cref{sec:nonabelian}.
The presence of a $\U(1)$ factor introduces additional complications,
as we now describe in more detail. The submatrix
$M_{C_\text{na}}^{+(1\alpha)(1\beta)}$ is generically the inverse of
the matrix ${ M_{C_\text{na}}}_{(1\alpha)(1\beta)} = [[W_{\bar 1
\bar 1} \cdot D_\alpha \cdot D_\beta]]$.\footnote{Barred
(``physical'') indices are of type $\bar{I} = \bar{0},
\bar{1}, \alpha, i_s$. In the basis $\hat D_{\bar{I}}$,
$\hat D_{\bar{0}}$ is the KK $\U(1)$ divisor and $\hat
D_{\bar{1}}$ is the abelian $\U(1)$ divisor (i.e., the image of
the generating section under the Shioda map), whereas in the basis
$\hat D_{I}$, $\hat D_0$ is simply the zero section and $\hat D_1$
is the generating section. The matrices $\sigma^{I}_{\hat I}$ in
\labelcref{physbasis} and their inverses can be used to convert
between these two bases.}
For bases with $h^{1,1}(B)$ not too large relative to $h^{1,1}(W_{\bar 1 \bar 1}$\footnote{Since $W_{\bar 1 \bar 1})$ is the class of a surface in $B$, whenever $h^{1,1}(W_{\bar 1 \bar 1})< h^{1,1}(B)$ the matrix $[[W_{\bar 1 \bar 1} \cdot D_\alpha \cdot D_\beta]]$ will be singular.} and generic characteristic data, this matrix is invertible.
When $M_{C_\text{na}(1\alpha)(1\beta)}$
is not invertible, however, the expression \labelcref{eq:geotheta1} is no longer valid; in
such cases a further analysis must be done, which often involves
solving for a different set
of $\phi$ components, though the essentially the same procedure (i.e. solving the symmetry constraints $\Theta_{I\alpha} = 0$ by eliminating certain components of $\phi$) still works.
An explicit example of this type of situation is illustrated in \cref{sec:21-exception}.
In general, we expect that while some null vectors of
$M_{C_\text{na}(1\alpha)(1\beta)}$ can be dealt with by solving some
of the constraints $\Theta_{1 \alpha} = 0$ for other $\phi$s, this
can be done for at most the total number of
parameters $\phi^{\hat{I}\hat{J}}$, $ \frac{1}{2}
(\rk\mathsf{G} + 2)(\rk\mathsf{G} + 1) $. Null vectors of
$M_{C_\text{na}(1\alpha)(1\beta)}$ that cannot be treated in this manner, i.e., by solving
for other $\phi^{\hat I \hat J}$, should correspond to extra null vectors of
$M$. It is also possible that even in the cases where null vectors of
$M_{C_\text{na}(1\alpha)(1\beta)}$ can be treated by solving for
parameters $\phi^{\hat I \hat J}$, this may increase
the number of null vectors of $M$ and decrease the number of
independent possible fluxes $\Theta$ (since $ \rk M = \dim M - \operatorname{nullity} M $ is equal to the number of independent flux backgrounds plus the number of independent constraints, so that
an increase in the number of null vectors corresponds either to a
decrease in the number of independent constraints or a decrease in the
number of independent fluxes); we have not encountered any
explicit examples where this behavior occurs, though we have not attempted to systematically construct such examples. We
explore the detailed structure of null vectors of $M$ in more detail
in \cref{sec:constraints-homology}.
Note that the analysis here can in general lose information about the
integer quantization on the $\phi$s, since in principle the inverse
matrices $W^{i_{s}| j_{s'}}$ and
$M_{C_\text{na}}^{+(1\alpha)(1\beta)}$ may be rational and not
integer valued. We address these issues more explicitly in \cref{sec:constraints-homology} in the context of the analysis where the nullspace
is removed first to give the reduced matrix $M_\text{red}$.
\subsection{Homology relations and anomaly cancellation}
\label{sec:homologyrel}
As discussed in \cref{sec:linear-anomaly}, the null vectors of $M_C$,
considered as elements of $\Lambda_C \subset\Lambda_S$, encode the full
set of F-theory constraints on the possible vertical fluxes
$\Theta_{IJ}$, which must include at least the anomaly cancellation
conditions but in principle may impose stronger constraints. (See
\cite{Bies_2017} for a closely related discussion about anomalies in
F-theory.) When we can explicitly solve for a subset of the $\phi$
variables and write an expression for $M_C$ in terms of the remaining
variables, such as is done in terms of the distinctive parameters
$\phi''$ in the preceding section, we can gain explicit information
that is relevant for understanding 4D chiral matter
multiplicities---in particular, the nullspace of such an $M_C$
contains complete information about the linear constraints satisfied
by the F-theory fluxes, as we now explain in more detail. This
approach to understanding the number of independent families of chiral
matter available in universal F-theory models for a given $\mathsf{G}$
complements the related analysis of this question using $M_\text{red}$
as discussed in the following section. In the remainder of this
discussion we assume that we have an explicit description of $M_C$ in
terms of a subset of the flux degrees of freedom, as realized
concretely in the preceding subsection in cases where $M'$ is
(pseudo-)invertible, so that in this subspace $\Theta'' = M_C
\phi''$ and the remaining $\Theta$s vanish.
Notice that since $M_C$ is symmetric, any null vector $\nu$ satisfying
$ M_C \nu = \nu^\ensuremath\mathrm{t} M_C =0$ must also satisfy $\nu^\ensuremath\mathrm{t}
\Theta'' = \nu^\ensuremath\mathrm{t} M_C \phi'' = 0$. Thus, identifying the
nullspace of $M_C$ is equivalent to identifying the linear constraints
that must be satisfied by the fluxes $\Theta''$. This can be
accomplished in all purely nonabelian models admitting a resolution with a holomorphic zero section by
using the explicit expression for the nontrivial matrix
elements of $M_C$ given in \labelcref{eq:omegabar}.
The physical significance of the nullspace equations $ \nu^\ensuremath\mathrm{t}
\Theta'' =0$ is that they are the complete set of linear conditions
that must be obeyed by the symmetry constrained fluxes; provided it is possible to express the chiral
multiplicities as rational linear combinations of fluxes as in
\labelcref{eq:chiconvert}, this further implies that the nullspace
equations lead to the full set of linear constraints that must be
obeyed by the chiral matter multiplicities. Since all allowed
F-theory models are by assumption consistent with 4D anomaly
cancellation, the nullspace equations include as a subset the linear
4D anomaly constraints.
This observation has immediate applications to the question of whether
or not F-theory geometry imposes additional linear constraints on
chiral matter multiplicities beyond those associated with 4D anomaly
cancellation, as the nullspace equations can easily be recovered from
\labelcref{eq:omegabar}. When $\mathsf{G}$ is purely nonabelian and the corresponding resolution admits a holomorphic zero section, the fact that
\labelcref{eq:omegabar} is true for arbitrary base $B$ implies that the
linear constraints on the chiral multiplicities can in principle be
read off for all $\mathsf{G}$ models in full generality, provided a
resolution $X$ can be identified such that the chiral indices can be
expressed in terms of the vertical fluxes $\Theta''$. In
\cref{sec:exampleADE} we make extensive use of this structure to
confirm that for all universal $\mathsf{G}$ models of this type that we study,
F-theory geometry imposes no additional linear constraints on
the chiral multiplicities of matter charged under $\mathsf{G}$ beyond the 4D anomaly cancellation
constraints; we also find this to be true for all models we study with
$\U(1)$ gauge factors, as discussed in \cref{sec:abelianmodel}.
For models with a $\U(1)$ gauge factor, some additional care
is needed since, as explained towards the end of
\cref{constraintsolutions},
it does
not seem possible to easily compute a fully general form for
the nullspace of $M_C$ for a model
with $\U(1)$ gauge factors over an arbitrary base. Nevertheless, in
many circumstances it does appear possible to first solve for
$\Lambda_{C_\text{na}}$, then further restrict
$\Lambda_{C_\text{na}}$ to the sublattice $\Lambda_{C_\text{na}}
\cap \{\phi^{1\alpha} = 0\}$, for which the remaining symmetry
constraints $\Theta_{1\alpha} =0$ can be solved over arbitrary $B$
without modifying the nullspace equations $\nu^\ensuremath\mathrm{t} \Theta''
=0$.
The basic idea here is that as long as there exists a linearly independent subset of null vectors of $M$ that span the $S_{1\alpha}$ directions,
setting $\phi^{1 \alpha} = 0$ for all $\alpha$ will not reduce the rank of the set of
$\Theta$s that are realized by acting with $M$ on
$\Lambda_{C}$, and hence will not change the nullspace
equations
$\nu^\ensuremath\mathrm{t} \Theta''
=0$ that encode linear constraints on the matter multiplicities. We
expect that generically the null vectors should have this property,
and while we cannot prove that this is always the case we have not
encountered any instances where this does not hold. Thus, we can
often simplify the analysis of the linear constraints from null vectors by
restricting to background fluxes satisfying $\phi^{1\alpha} = 0$.
(Note, however, that even though we do not generally expect this to modify the
number of linear constraints, this strategy will not keep track of
the precise lattice of allowed fluxes, for reasons similar to the
analysis following \cref{eq:9664}.)
With the restriction to $\Lambda_{C_\text{na}}
\cap \{\phi^{1\alpha} = 0\}$, the $\U(1)$ symmetry constraints take the
form
\begin{align}
M_{C_\text{na}(1\alpha)(\hat I \hat J) } \phi^{\hat I \hat J} = 0\,.
\end{align}
In this case, the expressions for the symmetry constrained fluxes
induced by flux backgrounds restricted to the sublattice
$\phi^{1\alpha}=0$ only depend polynomially on triple intersections of the
characteristic data since setting $\phi^{1\alpha}=0$ eliminates
dependence on the matrix $W_{\bar 1 \bar 1 } \cdot D_\alpha
\cdot D_\beta$; therefore we can again compute the symmetry-preserving fluxes
in terms of the
characteristic data without committing to a specific choice of
$B$. Provided there are
null vectors with components spanning the $S_{1\alpha}$ directions as
described above, we can
then easily determine the linear constraints in this simpler setting
with the understanding that the same constraints apply to the
unrestricted fluxes as well, at least for generic characteristic
data. We do not attempt to specify the precise conditions under which
this is true; rather, we simply note that we have yet to identify any
counterexamples, i.e., any specific models with more restrictive
linear constraints among the fluxes (when $\phi^{1\alpha} \ne 0$) than
those implied by anomaly cancellation. We give an explicit example of
this type of analysis in \cref{F6model}.
\section{Reduced intersection pairing}
\label{sec:constraints-homology}
In the previous section we explained how to restrict the lattice of
vertical flux backgrounds $\Lambda_S$ to the sublattice $\Lambda_{C}$
of vertical M-theory flux backgrounds that lift to consistent F-theory
flux backgrounds compatible with unbroken 4D local Lorentz and gauge
symmetry $\mathsf{G}$. Specifically, we showed how to compute the symmetric bilinear form matrix $M_C$ on $\Lambda_C$ so that
the symmetry constrained fluxes $\Theta''$ can be realized explicitly
as elements of the lattice $M_C \Lambda_{C}$.
In this section, we present a complementary approach, namely first
quotienting out the nullspace of $M$ to get the reduced inner product
matrix $M_\text{red}$,
and then imposing
$\Theta_{I\alpha} = 0$. The conceptual advantage of this approach
centers on the observation that $M_\text{red}$ (equivalently, the lattice
$H_{2,2}^{\text{vert}}(X,\mathbb{Z})$, equipped with the intersection pairing $M_{\text{red}}$) appears to be independent of the choice
of resolution $X$.
It is also slightly easier in this approach to keep track of the
integer quantization on the fluxes.
Furthermore, $M_\text{red}$ may be
used to consider F-theory models with flux backgrounds that break part
of the gauge symmetry, though we do not explore such configurations
here.
In \cref{sec:methodology}, we briefly describe how to obtain the vertical
cohomology as a lattice quotient, $H_{2,2}^{\text{vert}}(X,\mathbb{Z}) =
\Lambda_{S}\mathclose{}/\mathopen{}\sim$, with some details of this analysis
relegated to \cref{sec:lattice-reduce}.
In
\cref{invariance}, we summarize the evidence suggesting that
$M_\text{red}$ is independent of the choice of resolution up to an
integral change of basis. Although we are unable to produce a
completely general expression for $M_\text{red}$, in
\cref{sec:nonabelian,abelianfactor} we describe the nullspace of the
intersection pairing $M$ in as much detail as we are able for various $\mathsf{G}$ models, and we defer specific examples to
\cref{sec:exampleADE,sec:abelianmodel}. \cref{rank} presents an
immediate physical application of the invariance of $M_\text{red}$.
\subsection{Nullspace quotient and integrality structure}
\label{sec:methodology}
Considered as an abstract lattice
quotient, the integrality
structure of $\Lambda_{\text{red}}: =\Lambda_S\mathclose{}/\mathopen{}\sim$ is automatically respected and the
quantization condition on flux backgrounds $\phi \in \Lambda_{\text{red}}$ is clear---it is simply
the condition that $\Lambda_{\text{red}}$ only contains integral elements. It is not
always completely straightforward, however, starting from a given
matrix $M$ and associated nullspace, to compute an integer
basis for $\Lambda_\text{red} = H_{2, 2}^\text{vert} (X,\mathbb{Z})$ and the
associated symmetric bilinear form $M_\text{red}$ explicitly. For
example, when the nullspace of an integer matrix describing the
bilinear form on a lattice is determined, any null vector that
contains a unit entry in some coordinate $IJ$ can be modded out by
simply removing that vector. If there are no obvious unit entries,
however, the projection to integer homology is less transparent, and
in general one must identify an appropriate basis for the quotient
lattice.
A general methodology for performing this quotient and determining the
resulting inner product matrix $M_\text{red}$ is described in
\cref{sec:lattice-reduce}.
In general, this will require a choice of basis vectors for
$\Lambda_\text{red}$ that have multiple nonzero components in the
original basis for $\Lambda_S$.
In all the cases we have studied explicitly, it is possible to
identify a subset of the basis vectors of $\Lambda_S$ that form a good
basis for $\Lambda_\text{red}$; while we have not tried to prove that
this is always possible it simplifies the analysis in the cases where
this works.
\subsection{Resolution independence}
\label{invariance}
The quotient of the lattice $\Lambda_S$ by the nullspace of
$M$ gives the lattice of vertical classes
$H_{2,2}^{\text{vert}}(X,\mathbb{Z})$. The restriction of the intersection
pairing on $M$ gives a nondegenerate symmetric bilinear form $M_\text{red}$
that maps pairs of elements of $H_{2,2}^{\text{vert}}(X,\mathbb{Z})$ to
$\mathbb{Z}$. An intriguing feature of $M_\text{red}$ is that in all examples
we study, $M_\text{red}$ appears to be independent of the choice of
resolution $X$ up to an integer change of basis. It is thus tempting
to conjecture that given any two resolutions $X, \tilde{X}$ of a singular
Weierstrass model with corresponding nondegenerate intersection
pairing matrices (resp.) $M_\text{red}, \tilde{M}_\text{red}$, there exists
a matrix $U$ such that
\begin{align}
\tilde{M}_\text{red} = U^\ensuremath\mathrm{t} M_\text{red} U \,, \quad U \in
\GL(h^{2,2}_{\text{vert}}(X), \mathbb{Z})\,,
\label{eq:mu-conditions}
\end{align}
where $h^{2,2}_{\text{vert}}(X) =
h^{2,2}_{\text{vert}}(\tilde{X})$. Note that since $\GL(n, \mathbb{Z})$ is a group, every $U$ must be invertible and therefore $\det U = \pm{}1$. While we have not checked the
resolution-independence of $M_\text{red}$ for every possible resolution
of every $\mathsf{G}$ model we study, nor for all choices of characteristic
data $(K,\Sigma_s, W_{01})$, for all cases in which we compute the
matrices $M_\text{red}, \tilde{M}_\text{red}$ explicitly, we find that there is indeed
an invertible integer matrix $U$ satisfying
\cref{eq:mu-conditions}. Note that it is somewhat easier to check that
the determinant, rank, and signature are equal for any pair
$M_\text{red}, \tilde{M}_\text{red}$. Since $M_\text{red}$ is symmetric,
Sylvester's law of inertia implies that any two intersection pairing
matrices with these common features are congruent to one another via
an invertible real (not necessarily integer) matrix $U$. However, this
is not enough to show that $U$ is an integral matrix, so to show that
$M_\text{red}$ and $\tilde{M}_\text{red}$ are equivalent it appears necessary to
explicitly compute a matrix $U$ satisfying $\tilde{M}_\text{red} = U^\ensuremath\mathrm{t}
M_\text{red} U$ and confirm that it is a unimodular matrix.
When $M_\text{red}, \tilde{M}_\text{red}$ are related by an integer change of basis $U$, it furthermore follows that the associated degenerate
matrices $M, \tilde{M}$ are also related by an integer change of basis. This can be seen by first putting each of
the
$M$ matrices in
the canonical form \labelcref{eq:pmp} with $M_\text{red}$ in the upper left
block, as described in
\cref{sec:lattice-reduce}, and then using a linear transformation with $U$ in the upper
left block and the identity in the remaining part of the matrix to
relate the two canonical forms of $M, \tilde{M}$.
In purely nonabelian cases $\mathsf{G} = \mathsf{G}_\text{na} = \prod_{s}
\mathsf{G}_s$, a general form for a
matrix $U$ relating two different versions of $M_\text{red}$ can be
constructed explicitly provided that we make the physically natural
assumption that $M_\text{phys}$ is the same for both resolutions; we
carry out this analysis in \cref{proof}.
As discussed in more detail in
\cref{nonabwithchiral},
the resulting $U$ is only constrained to be rational and not integral
from these considerations, and a certain compatibility condition is
required for $U$ to be integral. In all cases we have considered,
however, we have found an integral $U$ of this form.
In more general models
with $\U(1)$ gauge factors and rational zero sections, we do not know
of such an explicit construction of $U$; nevertheless, a similar
general structure should hold in those cases,
and for specific choices
of characteristic data it still appears to be true that
$M_\text{red}$ is independent of the specific choice of $X$ as we
illustrate in the context of the $(\SU(2) \times \U(1))/\mathbb{Z}_2$
model in \cref{sec:abelianmodel}.
If $M_\text{red}$ is indeed resolution independent, this further
suggests that the vertical cohomology $H^{2,2}_{\text{vert}}(X,\mathbb{Z})$ of
any elliptic CY fourfold $X$ resolving a singular Weierstrass model
with gauge symmetry $\mathsf{G}$ is in a sense a birational invariant
characterizing properties of the singular locus of
$X_0$.
\subsection{Purely nonabelian gauge groups}
\label{sec:nonabelian}
Before discussing the more general case including a $\U(1)$ gauge factor,
we study some general properties of the nullspace of the intersection pairing $M$
in the situation that the gauge group is purely nonabelian,
\begin{equation}
\mathsf{G} = \mathsf{G}_\text{na} = \prod_s \mathsf{G}_s\,,
\end{equation}
and the zero section is holomorphic.
\subsubsection{Null space structure of $M$
with purely nonabelian gauge group}
\label{sec:null-nonabelian}
When the group $\mathsf{G}$
is purely nonabelian,
the intersection pairing $M$ between pairs of vertical cycles $S_{00}$, $S_{0 \alpha}$, $S_{0 i_s}$,
$S_{\alpha \beta}$, $S_{\alpha i_s}$, $S_{i_s j_t}$ can be expressed as
\begin{equation}
M=
\left( \begin{array}{c c c c c c}
K^3 & [K^2 \cdot D_\alpha] & 0 & [K \cdot D_\alpha \cdot D_\beta] & 0 & 0\\
\hspace*{0.0in}
[K^2 \cdot D_{\alpha'}] & %
[[K \cdot D_\alpha \cdot D_{\alpha'}]] & 0 & %
[[D_\alpha \cdot D_{\alpha'} \cdot D_{\beta}]] & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 \\
%
\hspace*{0.0in}
[K \cdot D_{\alpha'}\cdot D_{\beta'}] & [[D_\alpha \cdot
D_{\alpha'} \cdot D_{\beta'}]] & 0 & 0 & 0 & [[W_{\alpha' \beta' i_s j_{t}}]]\\
0 & 0 & 0 & 0 & [[W_{\alpha' \alpha i'_{s'} i_{s}}]] & [[W_{\alpha' i'_{s'} i_s j_t}]\\
0 & 0 & 0 & [[W_{\alpha \beta i'_{s'} j'_{t'}}]] & [[W_{\alpha i_s i'_{s'} j'_{t'}}]] & [[W_{i'_{s'} j'_{t'} i_s j_t}]]
\end{array} \right).
\label{fullmat}
\end{equation}
(Above, single brackets $[\cdot]$ denote a sub-vector and double
brackets $[[ \cdot ]]$ denote a submatrix; moreover, unprimed free indices correspond to rows while primed free indices correspond to columns.)
Note that, as described in more detail in
\cref{intersection}, the only intersection numbers in \cref{fullmat} that are
resolution-dependent are those that contain at least three indices of
type $I =i_s$; the values in the upper left are all included explicitly,
and we have
\begin{equation}
W_{\alpha \beta i_sj_t} = D_\alpha \cdot D_\beta \cdot W_{i_sj_t} =W_{i_s|j_t} D_\alpha \cdot D_\beta \cdot \Sigma_s= -\delta_{st} D_\alpha \cdot D_\beta
\cdot \Sigma_s \kappa_{ij}^{(s)}\,,
\label{eq:omegakappa}
\end{equation}
where
\begin{align}
\kappa_{ij}^{(s)} =-W_{i_s | j_s}
\end{align}
is the inverse Killing metric
of the gauge factor $s= t$ (which is equal to the
Cartan matrix for ADE groups)
and
$\Sigma_s$ is the divisor supporting that gauge factor.
The nullspace of $M$ is the
set of solutions to the equation
\begin{align}
M_{(IJ)(KL)} \nu^{KL} =0\,.
\end{align}
Some elements of the nullspace correspond to linear combinations of
intersections
$S_{\alpha \beta}$ that are trivial in the base
homology.\footnote{For example, when $B = (\mathbb{P}^1)^{\times 3}$ with
classes $H_i, i = 1, 2, 3$ corresponding to points in the three
factors crossed with $\mathbb{F}_0 \cong\mathbb{P}^1\times\mathbb{P}^1$ from the other two factors, the only nontrivial
intersection is $H_1 \cdot H_2\cdot H_3 = 1$, and the curves $H_i \cap
H_i$ are trivial in homology.}
From the conditions of Poincar\'{e} duality on the base and
nondegeneracy of the triple intersection product as discussed in
\cref{sec:nondegeneracy}, the number of independent homology classes
represented by $S_{\alpha \beta}$ and $S_{0 \alpha}$ are both equal to
$h^{2,2} (B) =h^{1, 1} (B)$; null directions associated
with trivial homology classes in the linear space of $S_{\alpha
\beta}$ can thus be removed, though for notational simplicity we
continue to use the same symbol $S_{\alpha \beta}$ for the reduced
basis.
Similarly, there are (linear combinations of) intersections $
S_{\alpha i_s}$ that correspond to trivial classes $D_\alpha \cdot
\Sigma_s$ in the base. In general, the number of independent nontrivial
classes $S_{\alpha i_s}$ is at most $h^{1, 1}(\Sigma_s)$, but may be
smaller.
All these null
vectors depend only on the geometry of the base.
After removing the nullspace elements associated with the base
geometry, which are independent of resolution, we can proceed further
by solving explicitly for more general nullspace elements; we find
that additional elements of the nullspace are generated by the
vectors
\begin{align}
\begin{split}
\label{linrel}
\nu_{\langle 0 \rangle} &= (1 , - [K^\alpha], 0 ,0,0,0) \\
\nu_{\langle i'_{s'}\rangle} &= (0,0,[\delta_{i'_{s'}}^{j_t}],0,0,0) \\
\nu_{C\langle a \rangle}&=\nu_{C\langle a\rangle}^{i'_{s'} j'_{t'}} (0,[W_{i'_{s'} j'_{t'}}^\alpha ], 0,-[W_{i'_{s'} j'_{t'}}^\alpha K^\beta], -[W^{k_u|k'_{u'}} W_{i'_{s'} j'_{t'}|k'_{u'}}^\alpha] ,[\delta_{i'_{s'} j'_{t'}}^{k_u l_v}] )\,,
\end{split}
\end{align}
where the expression in parentheses in the third line above may be
viewed as the components of a basis of symmetry-constrained 4-cycles
$S_{C i_s j_t} = CS_{i_sj_t}\in \Lambda_C \subset\Lambda_S$ given by
\begin{equation}
\label{eqn:SC}
S_{Ci_sj_t} = W_{i_s j_t}^{\alpha} (S_{0\alpha} -
K^\beta S_{\alpha \beta} )-W^{k_v|l_u} W^\alpha_{i_s
j_t |l_u} S_{\alpha k_v} + S_{i_s j_t} \,,
\end{equation}
which satisfy
\begin{equation}
S_{C i_s j_t} \cdot S_{C k_u l_v} = S_{i_s j_t} \cdot
S_{Ck_u l_v} = M_{C (i_s j_t)(k_u l_v)} \,,
\end{equation}
and $\nu^{i_s j_t}_{C\langle a \rangle}$ are the coefficients of null
vectors $M_C$, i.e.
\begin{align}
\label{4Cid}
0 &= \nu_{C\langle a\rangle}^{i'_{s'} j'_{t'}} ( W_{k_u l_v }^\alpha K^\beta
W_{i'_{s'}|j'_{t'}} - W^{m_w|k'_{u'}}
W_{k_ul_v|k'_{u'}}^\alpha W_{m_w i'_{s'}|j'_{t'}}^\beta+
W_{k_u l_v i'_{s'}| j'_{t'}}^{\alpha \beta} )
\end{align}
or equivalently
\begin{align}
\begin{split}
\label{MCnull}
0
&=\nu_{C\langle a\rangle}^{i'_{s'} j'_{t'} } M_{C(k_u l_v ) (i'_{s'} j'_{t'})}\,.
\end{split}
\end{align}
In the above expression we have used the fact that the expression in parentheses
in \cref{4Cid} is $M_C$ in the special case of a purely nonabelian
gauge group and holomorphic zero section---see \labelcref{eq:omegabar}
and note here $W_{00} = K$.
The above computation shows the null vectors of $M_C$ are, in these cases, in
one-to-one correspondence with the null vectors of $M$ (note that in
this situation the subtlety of zero-norm null vectors of $M_C$
described in the second paragraph of \cref{sec:linear-anomaly} does
not arise since in the notation of \cref{eq:c}, the sub-matrix $M'$
restricted to the non-distinctive parameters $\phi^{I \alpha}$ is invertible
after removing the null vectors that depend on the base
geometry, as well as those from the first two rows of \cref{linrel}).
In principle the appearance of the inverse matrix $W^{k |
k'}$ in the third set of vectors (\ref{linrel}) may mean that
even when (\ref{4Cid}) is satisfied for all $\nu_{C\langle a\rangle}^{ij}$, these
null vectors are rational with integer $\nu_{C\langle a\rangle}^{ij}$, so that the $ij$ fluxes cannot simply be
projected out, as discussed in
\cref{sec:methodology}. In all cases we have examined explicitly,
however, the entries are integer despite the presence of the inverse
matrix; we suspect that this occurs generally, though we have not
tried to prove this statement.
As discussed in \cref{sec:homologyrel}, the structure of the nullspace
elements corresponds with constraints on the fluxes $\Theta_{IJ}$. In particular,
the property $M=
M^\ensuremath\mathrm{t}$ implies that the above nullspace equations must also be
satisfied by the fluxes:
\begin{align}
\label{homrel}
S_{KL} \cdot S_{IJ} \nu^{IJ} =0~~\implies ~~\Theta_{IJ} \nu^{IJ} = \nu^{IJ} S_{IJ} \cdot S_{KL} \phi^{KL} =0\,.
\end{align}
The linear relations on fluxes coming from the first two classes of
null vectors in \cref{linrel}, namely
\begin{align}
\Theta_{IJ} \nu_{\langle 0\rangle}^{IJ} =\Theta_{00} - K^\alpha \Theta_{0\alpha} = 0,~~~~\Theta_{IJ} \nu_{\langle i'_{s'}\rangle}^{IJ}= \Theta_{0 i'_{s'}} =0
\end{align}
are generally true in the special case of a holomorphic zero section; see, e.g.,
\cite{Grimm:2011sk}.
The possible coefficients $\nu_{C\langle a\rangle}^{i'_{s'} j'_{t'}}$ appearing in the third
linear condition
\begin{align}
\label{eq:thirdcondition}
\Theta_{i'_{s'}j'_{t'}} \nu_{C\langle a\rangle}^{i'_{s'} j'_{t'}} =\phi^{k_u l_v} M_{C(k_u l_v)(i'_{s'} j'_{t'})} \nu_{C\langle a\rangle}^{i'_{s'} j'_{t'}} = 0
\end{align}
can be determined
in any given situation by explicitly identifying the nullspace vectors
of the form in the last line in \cref{linrel}.
In cases where there are no constraints on these coefficients, these
conditions force all fluxes $\Theta_{ij}$ to vanish and there is no
chiral matter.
While in principle for any base and characteristic data the nullspace
of the intersection matrix $M$ is straightforward to compute directly,
because of the relation \labelcref{MCnull}, the structure of the
constrained
matrix $M_C$ studied in the previous section can be used to analyze
the nullspace of $M$ in general classes of situations. In some cases, for the purposes of practical
computation, this analysis can be simplified
when rows/columns of $M_{C}$
vanish identically.
The subset of indices $i'_{s'}j'_{t'}$ for which
$M_{C(k_u l_v)(i'_{s'} j'_{t'})} = 0$ vanishes for all $k_u l_v$ are
indices for which the coefficients $\nu_{C\langle a\rangle}^{i'_{s'} j'_{t'}}$ can be set
equal to unity. In these cases, the appearance of the Kronecker delta
function in explicit coefficients of $S_{Ci'_{s'} j'_{t'}}$ given in
\cref{linrel} indicates that the basis elements $S_{i'_{s'}
j'_{t'}}$ are redundant and may be removed from the generating
set. On the other hand, the subset of indices $i'_{s'} j'_{t'}$ for
which $M_{C(k_u l_v)(i'_{s'} j'_{t'})}$ does not vanish for all
$k_ul_v$ are those for which nontrivial elements can be found spanning
the nullspace of $M_C$ by taking appropriate linear combinations of
$S_{C i'_{s'} j'_{t'}}$. Removing these primitive directions from the
lattice $\Lambda_C$ completes the nullspace quotient and leaves behind
a basis of homologically nontrivial cycles spanning $\Lambda_C/
\sim$. Since $M_C$ can be used to indirectly define these remaining
elements via \cref{MCnull}, it follows that explicitly computing
$M_C$ automatically determines $M_\text{red}$; we elaborate on this
point in \cref{nonabwithchiral}.
We next examine the structure of $M_\text{red}$ in the context of specific models with purely nonabelian gauge symmetry.
\subsubsection{Nonabelian groups without chiral matter}
Let us first assume that
all of the vectors $S_{C i'_{s'} j'_{t'}}$ appearing in
the third class of vectors listed in \cref{linrel} are null vectors; i.e.,
the expression in parentheses in \labelcref{4Cid}
vanishes identically for all coefficients $\nu^{i'_{s'}j'_{t'}}_C$, so that $S_{Ci'_{s'}j'_{t'}} =0$ in homology.
We can then use the null vectors to eliminate the $ (\rk\mathsf
G_\text{na})^2 + \rk\mathsf{G}_\text{na} + 1$ redundant elements
$S_{00}, S_{0 i_s}, S_{i_sj_t}$ to form a basis consisting of the at most $h^{1,1}(B)(
h^{1,1}(B)+ \rk\mathsf{G}_\text{na} +1)$ classes $S_{0\alpha},
S_{\alpha \beta}, S_{\alpha i_s}$. (Since there may exist additional null vectors, in addition to $S_{0\alpha}$ we keep only homologically
nontrivial basis elements among $S_{\alpha \beta}, S_{\alpha i_s}$, and hence the total number basis elements may be less than $h^{1,1}(B)(
h^{1,1}(B)+ \rk\mathsf{G}_\text{na} +1)$.) This reduces the intersection matrix
$M$ to the intersection pairing
\begin{align}
\label{resindependenthatM}
M_\text{red} = [[D_\alpha \cdot D_{\alpha'}]] \otimes \begin{pmatrix} K & [D_\beta] & 0 \\ [D_{\beta' }] & 0 &0 \\
0 & 0 &[[W_{i'_{s'}i_s}]] \end{pmatrix}\,,
\end{align}
where the Kronecker product $\otimes$ in the above expression is understood to imply the intersection product $\cdot$ component-wise. Note that the integrality condition on flux backgrounds in
$H^{2,2}_\text{vert}(X,\mathbb{Z})$ is preserved through the projection to this
basis when all the components in the vectors appearing in
\cref{linrel} are integer, which as discussed above occurs in all the
cases we have studied.
The pairing $M_\text{red}$ is
manifestly independent of any choice of resolution $X$ with a holomorphic zero section, since in such cases $W_{i'_{s'} i_s}= \delta_{s'
s}\kappa^{(s)}_{i' i} \Sigma_s$ and the characteristic data $K, \Sigma_s$ remains unchanged.
Since the intersection pairing on $B$ is
nondegenerate, as discussed in \cref{sec:nondegeneracy},
and we have explicitly removed null combinations of $S_{\alpha i_s}$,
$M_\text{red}$ is manifestly nondegenerate and resolution-independent.
It follows immediately that the
symmetry constraints \labelcref{eq:Poincare,eq:gauge} $\Theta_{0\alpha} = \Theta_{\alpha \beta} =
\Theta_{\alpha i_s} =0$ force all independent fluxes to vanish.
Stated different, the symmetry constraints together with
\cref{eq:thirdcondition}
imply $\Theta_{i_s j_t} = 0$. Hence there are no nontrivial fluxes in
these cases, and consequently no chiral matter in the resulting 4D F-theory models, as assumed.
We give explicit examples of systems of this kind in
\cref{sec:exampleADE}, in particular for the groups $\mathsf{G} =
\SU(N<5)$.\footnote{Another way to verify that a given model does not contain chiral matter is to check that the rank of
$M_\text{red}$ is equal to the number of symmetry constraints
$\Theta_{I\alpha}=0$, provided there are no $(4, 6)$ singularities. There appears to be one puzzling exception to this general characterization,
namely the case $\mathsf{G} = \SO(11)$;
we will address the $\SO(11)$ model along with the physics of $(4,6)$ points
models further in a separate publication \cite{46}.}
\subsubsection{Nonabelian groups with chiral matter}
\label{nonabwithchiral}
Next we consider the case that the expression in parentheses
in \cref{4Cid} is non-vanishing for some non-empty subset of indices
$i'_{s'} j'_{t'}$. This implies that there are allowed nontrivial
flux backgrounds $\phi$ and corresponding fluxes $\Theta$; however,
not all of the nonzero fluxes $\Theta_{i_s j_t}$ are
independent in $M_C \Lambda_C$. We can remove all null vectors and project $\phi = \phi^{IJ}S_{IJ}$ onto a
basis
of surfaces $S_{0\alpha}, S_{\alpha \beta}, S_{\alpha i_s}, S_{j_t k_u}$ (again keeping only homologically non-trivial $S_{\alpha \beta}, S_{\alpha i_s}$), leading to the intersection pairing \begin{align}
\label{Mrednonabelian}
M_\text{red} = \begin{pmatrix}
[[ D_{\alpha'} \cdot K \cdot D_\alpha ]] & [[D_{\alpha'} \cdot D_{\alpha} \cdot D_\beta ]] & 0 & 0 \\
[[ D_{\alpha' } \cdot D_{\beta'} \cdot D_{\alpha } ]] &0 &0 & [[ W_{\alpha' \beta' j_t k_u}]] \\
0 & 0& [[ W_{\alpha' i'_{s'}\alpha i_s} ]] & [[ W_{\alpha' i'_{s'} j_t k_u}]] \\
0 & [[ W_{j'_{t'} k'_{u'}\alpha \beta} ]] & [[W_{j'_{t'} k'_{u'}\alpha i_s}]] &[[ W_{j'_{t'} k'_{u'}j_t k_u }]]
\end{pmatrix}\,,
\end{align}
where we keep in mind that only a linearly independent subset of the
$(\rk\mathsf{G}_\text{na})(\rk\mathsf{G}_\text{na} +1)/2$ possible 4-cycles $S_{j_t k_u}$ is
represented in the above expression for $M_\text{red}$.
In principle this choice of a subset of the
basis elements may not be compatible with the integral lattice
structure through the projection, but as mentioned above this kind of
issue does not occur for any of the cases we have considered
explicitly and we can always choose such a basis in these cases. The
specific set of
independent fluxes of the form $\Theta_{i_s j_t}$ (equivalently,
the set of independent 4-cycles of the form $S_{i_s j_t}$) depends
on the characteristic data of the resolution $X$, and hence we cannot
be more precise at this point without specifying the characteristic
data of the elliptic fibration, although we expect that for every
F-theory model the number of independent fluxes is independent of
resolution. Nevertheless, we can see clearly that
imposing the symmetry conditions on the reduced
intersection pairing $M_\text{red}$ leaves behind a subset of
independent fluxes in the ``pure Cartan'' (i.e. $S_{i_sj_t}$) directions that
parametrize the combinations of 4D chiral indices realized by the F-theory compactification.
We give more explicit examples of systems of this kind in
\cref{sec:exampleADE}, see in particular \cref{tab:fluxtable}.
While it is not obvious that \labelcref{Mrednonabelian} is resolution
independent,
as shown in \cref{proof}, with some natural physical assumptions
(essentially that $M_\text{phys}$ is the same for the two resolutions)
we can
determine an explicit form for
a change of basis matrix $U$
that converts between two different presentations of
$M_\text{red}$ associated to any pair of resolutions $X,\tilde X$
for which $M_\text{phys}, \tilde M_{\text{phys}}$ are related by an integral linear
transformation
$\tilde{M}_\text{phys}= U_p^\ensuremath\mathrm{t} M_\text{phys} U_p$. The
transformation $U$ has the general form
\begin{align}
\label{Umatrix}
U =
\begin{pmatrix}
1 & u\\
0 & U_p
\end{pmatrix} \,,
\end{align}
where $u$ may contain rational parts with a denominator of
$\det \kappa$.
As discussed in \cref{proof}, $U$ is an integral matrix when a certain
compatibility condition is satisfied between the off-diagonal blocks
on the lowest row and rightmost column of the two presenations of
$M_\text{red}$, for an allowed choice of equivalence $U_p$ (which has
an ambiguity up to automorphisms of $M_\text{phys}$). In all cases we
have analyzed this compatibility condition is satisfied for some
$U_p$, and the resulting $U$ is an integer change of basis, but we do
not have a completely general proof that this is always the case.
In a related fashion, there is a transformation of
the form \labelcref{Umatrix} that takes $M_\text{red}$ to a canonical product
form
\begin{align}
\label{Mredcanonical}
M_\text{red}^\text{cp} =
U^{\text t} M_\text{red} U =
\begin{pmatrix}
[[ D_{\alpha'} \cdot K \cdot D_\alpha ]] & [[D_{\alpha'} \cdot D_{\alpha} \cdot D_\beta ]] & 0 & 0 \\
[[ D_{\alpha' } \cdot D_{\beta'} \cdot D_{\alpha } ]] &0
&0 & 0 \\
0 & 0& [[ W_{\alpha' i'_{s'}\alpha i_s} ]] & 0 \\
0 & 0 & 0 & M_\text{phys}/(\det \kappa)^2
\end{pmatrix}\,,
\end{align}
where we simply use the upper right components of $U$ to transform away the
off-diagonal bottom row and right column of \cref{Mrednonabelian}.
This inner product matrix must be treated with respect to the lattice
$\Lambda^\text{cp} =U^{-1} \mathbb{Z}^n$, which is not in general an integer lattice in this case. This form
is however useful since the symmetry constraints can be solved
trivially by setting all components except the last of the flux
background $\phi \in\Lambda_\text{cp}$ to vanish; an explicit example
of this is illustrated in \cref{sec:5-independence}.
The appearance of $\det \kappa$ in the bottom right component comes
from the fact that in general the off-diagonal values of $U$, associated with this transformation to the canonical product form in \cref{Mredcanonical}, are
rational with denominator $\det \kappa$, and $\Lambda_{\rm phys} =
((\det \kappa)\mathbb{Z})^m$,
as discussed further in
\cref{proof}.
\subsection{Gauge groups with a $\U(1)$ factor}
\label{abelianfactor}
For the more general case of Weierstrass models with gauge group $\mathsf{G} = ( \mathsf{G}_\text{na} \times \U(1) )/ \Gamma$, we find in practice it is typically easier to compute resolutions of physically-equivalent singular models in which the elliptic fiber is realized as a general cubic in $\mathbb{P}^2$, see, e.g., \cite{KleversEtAlToric}. These models generically admit rational (as opposed to holomorphic) sections associated to $\U(1)_\text{KK}, \U(1)$ and consequently the structure of the pushforwards of quadruple intersection numbers involving the divisors $\hat D_0, \hat D_1$ are not known in full generality as is the case in models with a single holomorphic zero section. For example, in these cases
\begin{align}
W_{000 \gamma} \ne K^2 \cdot D_\gamma,~~~~ W_{0000} \ne K^3
\end{align}
and so on. This unfortunately complicates the computation of
$M_\text{red}$ as our incomplete understanding of intersection
products involving $\hat D_0, \hat D_1$ makes the solutions to the nullspace
equations unclear, and thus at present we are unable to present even a
formal general expression for $M_\text{red}$ for models with $\U(1)$
factors over arbitrary $B$ with general characteristic
data. Nevertheless, we can follow the general procedure to
construct $M_\text{red}$ outlined in
\cref{sec:methodology} for any specific $B$ and $\mathsf{G}$, and we find in all examples we have
considered that $M_\text{red}$ is also resolution independent for
models with a $\U(1)$ gauge factor---see \cref{sec:abelianmodel} for some
examples. It is natural to conjecture that this is always the case
although a more general proof is clearly desirable.
\subsection{Dimension of $\Lambda_\text{phys}$}
\label{rank}
One immediate application of the conjectural resolution invariance of
$M_\text{red}$ is for understanding the number of independent F-theory
vertical flux backgrounds and fluxes that can arise in a given
model. After computing $M_\text{red}$, one can impose the symmetry
constraints in order to further restrict the lattice
$H_{2,2}^{\text{vert}}(X,\mathbb{Z})$ to the sublattice $\Lambda_\text{phys}$
of independent F-theory flux backgrounds; the restriction of the
action of $M_\text{red}$ to $\Lambda_\text{phys}$ can be expressed as
a matrix $M_\text{phys}$.
The number of independent fluxes $\Theta$ subject to the symmetry constraints is equal to $\rk
M_\text{phys}$.
While in principle, as discussed in \cref{sec:linear-anomaly},
$M_\text{phys}$ can have null vectors associated with constraints
characterized by zero-norm non-null elements of $\Lambda_S$, we have
not encountered any situations where this occurs. Indeed, this is
impossible in purely nonabelian theories since all non-distinctive
flux background parameters $\phi'$ are determined by the constraints as linear
functions of the $\phi''$ as in \cref{eq:solution-abstract}. We
suspect, but have not proven, that this also does not happen in
theories with $\U(1)$ factors. When there are no such null vectors of
$M_\text{phys}$, then we have
\begin{align}
\rk M_\text{phys}
= \dim \Lambda_\text{phys} &=
\text{\# independent fluxes}\\
\nonumber & =
\rk M_\text{red} -
\text{\# independent constraints} \,.
\end{align}
The number of independent constraints is at most the number of basis
elements
$S_{0\alpha}, S_{\alpha \beta}, S_{\alpha
i}$, i.e., $h^{1,1}(B) + \frac{1}{2} h^{1,1}(B) (h^{1,1}(B) +1 +
2\rk\mathsf{G})$, but in general can be smaller when there are homologically
trivial cycles $S_{\alpha \beta}, S_{\alpha i}$ as discussed in
\cref{sec:null-nonabelian}. This number must be resolution-independent
and can be identified directly from the structure of $M_\text{red}$
and the geometry of $B$.
Aside from the $\SO(11)$ model (which will be
treated further in
\cite{46}), all 4D F-theory models
with generic matter that we have studied in which
codimension-three $(4, 6)$ points are absent have the property that the rank of
$M_\text{phys}$ is equal to the number of independent realized families of
chiral matter multiplicities:
\begin{equation}
\label{eq:conjecture}
\rk M_\text{phys} = \text{\# of families of 4D chiral matter multiplets realized in F-theory.}
\end{equation}
The fact that $\text{rk}\, M_{\text{phys}}$ is at least equal to the number of independent chiral matter multiplicities seems to follow from the assumption that all matter surfaces $S_{\mathsf{r}}$ have a non-trivial vertical component. On the other hand, the fact that $\text{rk} \, M_{\text{phys}}$ is no larger than the number of independent chiral matter multiplicities suggests that every independent vertical flux $\Theta_{ij}$ corresponds to some family of charged chiral matter. We have furthermore found in all of these cases that the number of independent families of chiral matter multiplicities realized in F-theory matches the number of independent families satisfying 4D anomaly cancellation. We
know of no natural geometric reason why this should be true; the observation that
in all cases considered this holds can be thought of as a statement
regarding the absence of swampland type models in which entire
families of anomaly-free 4D supergravity theories would lack an
F-theory realization.
If it is indeed true that the number of chiral multiplets can be read
off from the properties of $M_\text{red}$, computing $M_\text{red}$
may serve as an efficient strategy for scanning the F-theory landscape
for vacua that impose stronger constraints than 4D anomaly
cancellation without requiring the additional step of computing the
matter surfaces $S_{\mathsf{r}}$.
\section{Computing chiral indices}
\label{3Dcompare}
In \cref{sec:4dflux} and \cref{sec:constraints-homology} we gave a
general prescription for computing the lattice of vertical F-theory
flux backgrounds for $\mathsf{G}$ models with gauge group $\mathsf{G} = (
\mathsf{G}_\text{na} \times \U(1))/\Gamma$ and chiral matter transforming in
representation $\oplus \mathsf{r}^{\oplus n_{\mathsf{r}}}$. Here, we review a
method to compute the multiplicities
\begin{align}
\chi_{\mathsf{r}} =\int_{S_{\mathsf{r}}} G = n_{\mathsf{r}} - n_{\mathsf{r}^*}
\end{align}
of the 4D chiral matter representations $\mathsf{r}$ in terms of the fluxes $\Theta''$, without explicit knowledge of the matter surface $S_{\mathsf{r}}$.
\cref{3DCS} reviews the general relationship \cite{Grimm:2011fx} (see also \cite{Grimm:2011sk,Cvetic:2012xn}) between Chern--Simons couplings appearing in the low energy effective 3D $\mathcal{N}=2$ supergravity action describing M-theory compactified on a CY fourfold in a nontrivial flux background $G$, and the vertical fluxes $\int_{S_{IJ}} G =\Theta_{IJ}$. In \cref{sec:CS}, we explain how to compute the chiral indices by solving the linear system obtained by matching the vertical fluxes to one loop exact field theoretic expressions for CS couplings appearing in the 3D $\mathcal{N}=2$ supergravity action, using a similar strategy to that used in \cite{Cveti__2014}.
\subsection{3D Chern--Simons terms and M-theory fluxes}
\label{3DCS}
The key step in our analysis that enables us to determine the chiral indices $\chi_{\mathsf{r}}$ in terms of vertical fluxes without explicit knowledge of the matter surfaces $S_{\mathsf{r}}$ is the identification \cite{Grimm:2011sk,Grimm:2011fx}
\begin{align}
\label{identification}
\Theta_{\bar{I}\bar{J}} = -\Theta_{\bar{I}\bar{J}}^{\text{3D}}\,, \quad \bar{I} = \bar{0}, i\,.
\end{align}
On the right hand side of the above equation, $\Theta_{\bar{I}\bar{J}}^{\text{3D}}$ are Chern--Simons (CS) couplings that characterize the 3D effective action describing M-theory compactified on $X$ at low energies (recall that the index $\bar I =\bar{0}$ denotes the KK $U(1)$, see \cref{physbasis}).
The identification \labelcref{identification} holds for all M-theory
compactifications on CY fourfolds $X$ with nontrivial flux backgrounds
$G$, and follows from the dimensional reduction of 11D supergravity on
$X$. In the special case that $X$ is a resolution of a
singular elliptic CY fourfold, M-theory/F-theory duality implies that
the low energy effective 3D theory is a Kaluza--Klein (KK) theory equivalent
to a circle compactification of the 4D $\mathcal{N}=1$ supergravity
theory describing a flux compactification of F-theory on the singular
fourfold. Because of this duality, the one-loop exact quantum dynamics
on the F-theory Coulomb branch gets mapped to the classical dynamics of
M-theory; in particular, this means that the contributions of massive
fermions on the F-theory Coulomb branch are captured by classical CS
couplings $\Theta_{\bar I \bar J}^{\text{3D}}$.
Concretely, given a collection of real Coulomb branch moduli $\varphi$ corresponding to the holonomies of Cartan $\U(1)$ gauge fields around the KK circle, the F-theory Coulomb branch is characterized by a collection of massive BPS hyperinos, with masses given by
\begin{equation}
\label{eq:BPSmass}
m_\text{hyp} = n m_\text{KK} + \varphi \cdot w\,, \quad n \in \mathbb{Z}\,, \quad \varphi \cdot w = \varphi^i w_i\,, \quad i = 1, \dots, \rk \mathsf{G}\,,
\end{equation}
where $w_i$ may be regarded as the Dynkin coefficients of a weight in a basis of fundamental weights, associated with the charges (under $\U(1)^{\text{rk} \,\mathsf G}$) of each hyperino on the Coulomb branch. In terms of the Cartan charges $(n, w_i)$ above, the one-loop exact CS couplings are given by \cite{Cvetic:2012xn}
\begin{equation}
\label{eq:qIJ1loop}
\begin{aligned}
\Theta_{i j}^\text{3D} &= \sum_{w} ( \tfrac{1}{2} + \floor{|r_\text{KK}\varphi \cdot w| }) \,\sign(\varphi \cdot w)w_{i} w_{j}\,, \\
\Theta_{ \bar{0}i}^\text{3D} &=\sum_{w} ( \tfrac{1}{12} +\tfrac{1}{2} \floor{|r_\text{KK} \varphi \cdot w|} ( \floor{|r_\text{KK}\varphi \cdot w| } + 1) ) w_{i}\,, \\
\Theta_{ \bar{0} \bar 0}^\text{3D} &=\sum_{w} \tfrac{1}{6} \floor{|r_\text{KK}\varphi \cdot w| } (\floor{|r_\text{KK}\varphi \cdot w|} + 1) (2 \floor{|r_\text{KK}\varphi \cdot w|} + 1)\,,
\end{aligned}
\end{equation}
where $r_\text{KK} := 1/m_\text{KK}$ is the KK radius.
The sign and floor functions in the above expressions encode the dependence of the CS couplings on the phase of the vector multiplet moduli space parametrized by the Coulomb branch moduli $\varphi_i$ and KK modulus $m_\text{KK}$; we return to the issue of explicitly evaluating these functions shortly. For now, we point out that the CS couplings can be expressed as linear combinations of the chiral indices $\chi_{\mathsf{r}}$ by making the replacement $\Sigma_w \rightarrow \sum_{\mathsf{r}}\mult{\mathsf{r}} \sum_{w \in \mathsf{r}}$ (where $n_{\mathsf{r}}$ is the multiplicity of each type of representation $\mathsf{r}$ appearing in the 4D spectrum and we only sum over each distinct representation $\mathsf{r}$ once) and using the fact that the summands are odd under $\mathsf{r} \to \mathsf{r}^*$. Combining \cref{identification} and \cref{eq:qIJ1loop}, we may thus write
\begin{equation}
\Theta_{\bar{0} \bar{0}} = x_{\bar{0} \bar{0}}^{\mathsf{r}} \chi_{\mathsf{r}}\,, \quad \Theta_{\bar{0}i} = x_{\bar{0}i}^{\mathsf{r}} \chi_{\mathsf{r}}\,, \quad \Theta_{ij} = x_{ij}^{\mathsf{r}} \chi_{\mathsf{r}}\,,
\end{equation}
and under our assumption that all matter surfaces have components in $S_{IJ}$\footnote{See \cref{sec:puzzle} for a possible counterexample to this assumption.} it is possible to invert the
coefficients $x^{\mathsf{r}}_{ij}$ so that
\begin{equation}
\chi_{\mathsf{r}} =x_{\mathsf{r}}^{ij} \Theta_{ij} =x_{\mathsf{r}}^{ij} S_{Cij} \cdot S_{kl} \phi^{kl} = S_{\mathsf{r} } \cdot \phi\,,
\end{equation}
where on the right hand side of the above equation we have used the fact that the matter surfaces are given by
\begin{equation}
S_{\mathsf{r}} = x_{\mathsf{r}}^{ij} S_{Cij}
\end{equation}
and $S_{Cij}$ are defined in \cref{eqn:SC}.
\subsection{Computing 3D Chern--Simons terms using triple intersection numbers}
\label{sec:CS}
The explicit expressions for $\Theta_{\bar I \bar J}^{\text{3D}}$ given in the previous subsection depend on the values of the moduli-dependent functions $\text{sign}(\varphi \cdot w)$ and $\floor{|r_\text{KK} \varphi \cdot w |}$ as input. These functions partially characterize the field theoretic regime (i.e., the ``phase'') of the F-theory Coulomb branch described by the 3D KK theory, or in geometric terms the regime of the K\"ahler moduli space to which the resolution $X$ corresponds. The Coulomb branch phase can in principle be computed geometrically by studying the fibers of $X$ over the codimension-two components of the discriminant locus in the base $B$, which carry local matter transforming in the representation $\mathsf{r} = \mathsf{r} \oplus \mathsf{r}^* $.
Unfortunately, this procedure is often delicate and sometimes
difficult to carry out systematically, so we instead use an
alternative approach that relies on the assumption that the
hypermultiplet representations characterizing the gauge sector of a 6D
supergravity theory can be recovered from a 5D KK theory, at least for
representations $\mathsf{r}$ that correspond to local matter in the
F-theory geometry. In particular, we exploit the fact that the matter
representations are encoded in codimension-two components of the
discriminant locus in the base $B^{(2)}$ of an elliptic CY threefold
to extract the phase of the Coulomb branch from the triple
intersections of Cartan divisors $\hat D_i$. Closely following the strategy described in \cite{Cveti__2014}, we now explain in detail
how to use this trick to compute the sign and floor functions
appearing in the field theoretic expressions for the 3D CS couplings
in the previous subsection.
Recall that in the case of M-theory compactified on an elliptic CY
threefold $X^{(3)}$, M-theory/F-theory duality (similar to the case of
a CY fourfold) identifies the triple intersection numbers with
one-loop quantum corrected CS couplings in 5D,
\begin{align}
\label{5Dmatch}
\hat D_{\bar I} \cdot \hat D_{\bar J} \cdot \hat D_{\bar K} = k^{\text{5D}}_{\bar I \bar J \bar K},~~~~ \bar I =\bar{0}, i\,,
\end{align}
where field theoretic expressions analogous to \cref{eq:qIJ1loop} have also been worked out for the 5D one-loop CS couplings \cite{Grimm:2013oga} (see also \cite{Witten:1996qb,Intriligator:1997pq,Bonetti:2011mw}):
\begin{align}
\begin{split}
\label{5DCS}
k_{ijk}^\text{5D} &= -\sum_{w} ( \floor{|r_\text{KK} \varphi \cdot w |} +\tfrac{1}{2}) \, \sign(\varphi \cdot w) w_i w_j w_k \\
k_{\bar{0}ij}^\text{5D} &=-\sum_{w} ( \tfrac{1}{12} +\tfrac{1}{2} \floor{|r_\text{KK} \varphi \cdot w|} ( \floor{|r\varphi \cdot w| } + 1) ) w_{i}w_j \\
k_{\bar{0} \bar{0}i}^\text{5D}&=-\sum_{w} \tfrac{1}{6} \floor{|r_\text{KK}\varphi \cdot w| } (\floor{|r_\text{KK}\varphi \cdot w|} + 1) (2 \floor{|r_\text{KK}\varphi \cdot w|} + 1) \,\sign(\varphi \cdot w) w_i\,.
\end{split}
\end{align}
Importantly, the sign and floor functions appearing in \labelcref{5DCS} are the same as those in \labelcref{eq:qIJ1loop}, which means they can equally well be determined from the 5D CS couplings provided the 5D and 3D CS couplings correspond to the same Coulomb branch phase in an appropriate sense.
It turns out to be possible to determine the 5D CS terms from the types of (partial) resolutions $X$ we consider, as the sequences of blowups we use to obtain $X$ for a given $\mathsf G$ model defined over a threefold base $B$ can also be used to obtain resolutions $X^{(3)}$ of the same $\mathsf G$ model defined over a twofold base $B^{(2)}$.\footnote{Recall that in our case we only consider resolutions of singularities through codimension-two sub-loci of the discriminant locus in $B$. This is actually true for bases of arbitrary dimension, $B^{(d)}$, so long as the sequence of blowups used to obtain a (partial) resolution is formally identical through codimension two. In the case of a twofold (i.e. $d=2$), this implies that the resulting threefold $X^{(3)}$ is a genuine resolution.} Consequently, for a given $\mathsf G$ model and a common sequence of blowups resolving singularities through codimension two, the triple intersection numbers of $X^{(3)}$ are closely related to the quadruple
intersection numbers $W_{\bar I \bar J \bar K \alpha}$
of $X$. More precisely, the pushforwards $W_{\bar I \bar J \bar K}$ are
formally identical to the pushforwards of the triple intersection
numbers of $X^{(3)}$ to the base, with the key difference
that the pushforwards $W_{\bar I \bar J \bar K}$ are ``promoted'' from numbers to classes
of curves in the threefold base $B$. In the fourfold case, one then
computes quadruple intersection numbers involving three divisors
carrying nonabelian Cartan indices by computing the intersections of
these classes with other divisors in the base, i.e., $W_{\bar I
\bar J \bar K} \cdot D_\alpha$. One can use this fact to
make the formal identification
\begin{align}
\label{5Didentification}
k^{\text{5D}}_{\bar I \bar J \bar K} \rightarrow W_{\bar I \bar J \bar K}
\end{align}
provided we replace specific coefficients in the sums \labelcref{5DCS}
with the intersection products of classes of certain curves in $B$.
If, as in the 4D case, we organize the expressions for the CS
couplings in \labelcref{5DCS} into sums over representations by making
the replacement $\sum_{w} \to \sum_{\mathsf{R}} n_{\mathsf{R}} \sum_{w
\in \mathsf{R}}$ (where $n_{\mathsf{R}}$ is the multiplicity of
hypermultiplets in the 6D spectrum and we
only sum over each distinct quaternionic representation $\mathsf{r}$ once), then we simply need to promote
$n_{\mathsf{r}}$ to the classes of matter curves
$C_{\mathsf{r}}$ (matter curves are discussed in
\cref{sec:matter-multiplicities}; see also \cref{intersection} for an explicit description of how $C_{\mathsf{r}}$\footnote{Note that $C_{\mathsf{r}}$ are known for large classes of singular F-theory models \cite{Grassi:2011hq} and (in contrast to $S_{\mathsf{r}}$) are manifestly resolution-independent.} appear in the expressions for $W_{i_sj_tk_u}$.)\footnote{The fact that the pushforward technology used to
evaluate the intersection numbers does not rely explicitly on the
dimension of $B$ is a key part of what makes it such an efficient computational tool for this purpose.}
Alternatively, we could rephrase this discussion as indicating that
the formal expressions $W_{\bar I \bar J \bar K}$
should match the triple intersection numbers that arise when the threefold base
$B$ is instead ``demoted'' to a twofold $B^{(2)}$. Either way, the
upshot is that the sign and floor functions are captured by the terms
$W_{\bar I \bar J \bar K}$, as is made clear by the
matching \labelcref{5Didentification}. Since the linear system
\labelcref{5Didentification} does not involve any undetermined
parameters, the system can be solved explicitly for the values of
$\text{sign}(\varphi \cdot w)$ and $\floor{|r_\text{KK} \varphi \cdot
w|}$. Thus, we find that matching triple intersections with the low
energy effective 5D physics of M-theory compactified on an elliptic CY
threefold $X^{(3)}$ allows us to circumvent the task of computing the
sign and floor functions directly from geometry, and we may
subsequently use these values as input for the 3D case.
We illustrate this procedure for the $\SU(2)$ model in \cref{sec:su2}.
\section{Models with simple gauge group}
\label{sec:exampleADE}
We apply our systematic approach for analyzing flux backgrounds
described in the previous sections in several examples of models with
simple nonabelian gauge groups, $\mathsf{G} = \mathsf{G}_\text{na}$. In
\cref{simpleADE}, we explain why the only simple $\mathsf{G}$ models with
generic matter admitting nontrivial chiral multiplicities are the
simply-laced groups $\mathsf{G}_\text{na} = \SU(N), \SO(4 k + 2), \gE_6$,
with $N \ge 5, k \ge 2$. \cref{simplefluxes} describes the general
features of the F-theory fluxes $\Theta''$ for these models; the full
set of results can be found in \cref{tab:fluxtable}. We turn our attention
to specific examples in \cref{sec:su2,sec:su5,su6,so10,e6}.
\subsection{Chiral matter for simply-laced gauge groups}
\label{simpleADE}
The groups $\SU(N), N \ge 5$, $\SO(4 k + 2), k
\ge 2$, and $\gE_6$ are precisely the compact simple Lie groups for
which we expect a one-dimensional family of anomaly-consistent chiral matter
spectra with generic matter in 4D; all other compact simple Lie groups have no
chiral solutions to the anomaly cancellation conditions with only generic
matter representations. For reference, generic matter in these models includes the following complex representations:
\begin{itemize}
\item{} $\SU(N)$: fundamental and two-index antisymmetric;
\item{} $\SO(4k+2)$: spinor;
\item{} $\text{E}_6$: fundamental.
\end{itemize}
To see that these are the only gauge groups admitting chiral generic matter, note first that the set of generic matter
for a simple gauge group comprises three representations if the group has an
independent quartic Casimir and two representations otherwise (this can be
seen in the 6D context as coming from the fact that the anomaly cancellation
conditions depend on quadratic and quartic invariants of the gauge group). One
of these representations is always the adjoint, which is self-conjugate, and
thus there are at most two representations that can contribute chirally to the
spectrum in any case. For groups with an independent cubic Casimir, the number
of independent chiralities is further reduced by one by the 4D anomaly
cancellation equations. One can then carry out a case-by-case analysis of the
compact simple Lie groups to determine the number of independent chiralities
in each case.
For $\SU(N), N \ge 5$, there is an independent quartic Casimir,
giving two complex generic representations, and an independent cubic Casimir,
reducing the number of independent chiral families to one. For $\SU(2)$, every
representation is self-conjugate; for $\SU(3)$, there is an independent cubic
Casimir but no independent quartic Casimir; and for $\SU(4)$, there is an
independent cubic Casimir and the two-index antisymmetric representation is
self-conjugate; thus, in all these cases, there are no chiral solutions. The
group $\SO(N)$ only has complex representations for $N = 4 k + 2$, with only
the spinor being complex among generic matter representations, and has no
independent cubic Casimir, thus having a one-dimensional family of generic
chiral spectra for these $N$. None of the exceptional groups
has an independent quartic Casimir, giving only one generic representation
other than the adjoint, and of these, only $E_6$ has complex representations;
the $E_6$ fundamental is complex, and $E_6$ has no independent cubic Casimir,
leaving a one-dimensional family of generic chiral spectra.
Thus, we expect a
one-dimensional family of chiral solutions for the simple gauge groups
$\SU(N), N \ge 5$, $\SO(4 k + 2), k \ge 2$, and $\gE_6$, and no chiral
solutions for all other compact simple Lie groups.
\subsection{Summary of F-theory fluxes for simple nonabelian models}
\label{simplefluxes}
\subsubsection{Fluxes in universal (simple) $\mathsf{G}$ models}
\label{sec:fluxes-universal}
Universal $\mathsf{G}$ models with simple nonabelian gauge symmetry can be described in
F-theory using Tate models, i.e., Weierstrass models presented in Tate
form \cite{BershadskyEtAlSingularities}
\begin{align}
y^2 z + a_1 xyz + a_3 y z^2 - (x^3 + a_2 x^2 z + a_4 x z^2 + a_6 z^3 ) = 0
\end{align}
with a choice of tuning
\begin{align}
a_n = a_{n,m_n} \sigma^{m_n}\,,
\end{align}
which characterizes the sections $a_n$ of
the $n$th tensor power of the anticanonical bundle of
the base $B$ in the vicinity of the gauge divisor $\sigma = 0$. Note
that the divisor class of $\sigma = 0$ is $[\sigma] = \Sigma$ and
hence the divisor classes of the tuned sections are
\begin{align}
[a_{n,m_n}] = n(-K) -m_n \Sigma\,.
\end{align}
In all nontrivial cases that we study without codimension-three $(4, 6)$ singularities, and excluding the case $\mathsf{G} = \SO(11)$, we use the methods of \cref{pushapp,sec:resolutions} to show there is a one-dimensional family of independent F-theory fluxes preserving the 4D gauge group $\mathsf{G}_\text{na}$ that take the form
\begin{align}
\Theta_\text{phys} = \phi \Sigma \cdot [p(a_{n,s_{n}})] \cdot [p'(a_{n,s_{n}})], ~~~~ \phi \in \mathbb{Z}\,,
\end{align}
where in the above formula the bracketed expressions are the classes of polynomials $p$ of the sections $a_{n,s_{n}}$ that depend on the choice of gauge group.
See \cref{tab:fluxtable} for results for various groups $\mathsf{G}_\text{na}$.
The physical significance and interpretation of the above results is perhaps more transparent in the lattice $M_C\Lambda_C$ of symmetry-constrained fluxes. For example, consistent with the
argument in the previous subsection, that each model admits at most a one-parameter family of chiral
multiplicities, we find in each case we study that the non-trivial fluxes $\Theta_{ij} \in M_C \Lambda_C$ can be expressed as
\begin{equation}
\label{eq:general-indices}
\Theta_{ij}=M_{C(ij)(kl)}\phi^{kl} \propto \frac{\ell(\phi^{kl})}{\det\kappa} \Sigma \cdot [p(a_{n,s_{n}})] \cdot [p'(a_{n,s_{n}})] =
\chi_{\mathsf{r}}\,,
\end{equation}
for some complex representation $\mathsf{r}$ and where
$\ell(\phi^{kl})$
is a linear combination of the parameters $\phi^{kl}$ whose precise
form depends on $\mathsf{G}$;
here, since $\mathsf{G}$ is simply-laced, $\kappa_{ij} = - W_{i|j}$ is the Cartan
matrix for $\mathsf{G}$. Moreover, since $\text{rk}\, M_C =1$ and $M_C^{\text{t}} = M_C$, the coefficients of the parameters $\phi^{ij}$ in the linear expressions $\ell(\phi^{ij})$ are identical to the proportionality constants relating
different nontrivial $\Theta_{ij}$, it follows that straightforwardly that
\begin{equation}
\label{selfdualG}
\Theta_{kl} \phi^{kl} = \frac{\ell(\phi^{ij})^2}{\det\kappa} \Sigma \cdot [p(a_{n,s_{n}})] \cdot [p'(a_{n,s_{n}})]= \int_X G \wedge G\,,
\end{equation}
which is non-negative provided $\Sigma \cdot [p] \cdot [p'] \geq 0$
\footnote{In order
for Tate models to be consistent and not give rise to a larger gauge
algebra, it is necessary that the line bundles to which the
$a_{n,s_n}$ appearing in the table
are associated have non-empty spaces of sections. Mathematically, this
can be expressed as the requirement that the divisor classes
$[a_{n,s_n}]$ are effective; the resulting intersection products are
negative as long as these divisors are non-rigid.}; this is consistent with the assertion that $G$ is self-dual \cite{Gukov:1999ya}, which in turn implies $\int G \wedge G = \int G \wedge *G = \int |G|^2 \geq 0$; see e.g. \cref{SU5square} for a specific example of \cref{selfdualG} in the context of the $\SU(5)$ model. As we explain in \cref{integrality}, $\ell(\phi^{ij}) / \det\, \kappa$ is an integer and hence $\Theta_{ij}$ are manifestly integer-valued.
With the exception of the $\SO(11)$ model, in models that do not admit nontrivial chiral matter multiplicities
and that are also free of codimension-three $(4, 6)$ singularities we
expect that $M_C = 0$, i.e., that $\Lambda_C$ is completely spanned by null
vectors of $M$; this would imply that
$\Theta_{ij} = 0$ for all $ij$. This expectation is validated in a
number of examples: for instance, we have confirmed that the
$\SU(N<5), \SU(6)^\circ, \SO(12), \text{Sp}(6), E_7, E_8, \text{F}_4$, and $\text{G}_2$ models do not admit any nontrivial F-theory fluxes
preserving the gauge symmetry. This property seems to be true
generally, consistent with the expectation that the sublattice
$\Lambda_\text{phys}$ should be
empty for 4D models not admitting chiral matter.
\subsubsection{Integrality conditions for symmetry preserving flux backgrounds}
\label{integrality}
Before proceeding to examples, let us justify the integrality
condition $\ell(\phi^{ij})/ \det\kappa \in \mathbb{Z}$, which ensures
that our expression \labelcref{eq:omegabar} for $M_{C(IJ)(KL)}$ leads to
integer fluxes $\Theta_{ij}$ in \labelcref{eq:general-indices}. This
integrality condition
is of course guaranteed, provided that the symmetry constraints
$\Theta_{I\alpha} =0$ are solved over $\mathbb{Z}$ (assuming $\phi^{IJ} \in \mathbb{Z}$), since $M$
is an integer matrix, but nevertheless for the
sake of clarity we spell out explicitly how the integrality condition
propagates through to the final expressions in the case where the
constraints are imposed first and there is an additional quantization
condition on the domain of the mapping $C$ defined in \cref{eq:c}.
This integrality condition is an explicit example of the type of
quantization constraint discussed earlier in \cref{sec:quantization-1}
and \cref{constraintsolutions}.
To see how this works, we use the symmetry constraints to derive a condition on the combination of intersection numbers and parameters in the numerator of \labelcref{eq:general-indices}. First notice that the local Lorentz
symmetry constraint $\Theta_{\alpha \beta} =0$ implies (see
\labelcref{newconstraints1,newconstraints2})
$\phi^{0\gamma} =- \phi^{00} K^\gamma + \phi^{ij} \kappa_{ij}
\Sigma^\gamma$ and $\phi^{\beta \gamma} = - \phi^{ij} \kappa_{ij}
K^\beta\Sigma^\gamma $. Since these expressions are polynomial in the
distinctive parameters, it is evident that solving the
constraints does not impose any conditions on
$\phi^{ij}$. We thus turn our attention to the gauge symmetry
constraints $\Theta_{\alpha i} = 0$, which imply $\phi^{\beta j}
\kappa_{ij} \Sigma^\gamma = - \phi^{jk} \Delta^{\beta}_{\mathsf{R}}
\rho^{\mathsf{R}}_{ijk} \Sigma^\gamma$ and for nonzero
$\Sigma^\gamma$ further imply
\begin{align}
\phi^{\beta j}\kappa_{ij}= - \phi^{jk} \Delta_{\mathsf{R}}^\beta \rho_{ijk}^{\mathsf{R}}\,,
\end{align}
where we note that $\rho_{ijk}^{\mathsf{r}}$ in the above equation is
defined by the intersection numbers $W_{ijk} \cdot D_\alpha =
\rho_{ijk}^{\mathsf{r}} C_{\mathsf{r}} \cdot D_\alpha$; see
\cref{3Cnumbers}. Since $\phi^{\beta j}$ is assumed to be an
integral lattice vector for every $\beta$, the right-hand side of the
above equation must lie in the root lattice of the group $\mathsf
G$. Comparing the above equation to the list of necessary
and sufficient conditions in \cref{tab:root-lattice} of \cref{sec:root-lattice} for a lattice vector to lie in the root lattice of a simple Lie group,
we obtain for each universal (simple and simply-laced) $\mathsf G$ model an integrality condition of the form
\begin{align}
\frac{\Delta_{\mathsf{R}}^\beta \rho_{ijk}^{\mathsf{R}} c^i\phi^{jk} }{\det\kappa } \in \mathbb{Z}\,,
\end{align}
where the choice of coefficients $c^i$ depends on $\mathsf{G}$. In all cases we study, we find that $\Delta_{\mathsf{R}}^\beta \rho_{ijk}^{\mathsf{R}} c^i\phi^{jk} \equiv D^\beta \ell(\phi^{ij}) \mod \det\kappa$, and hence for generic coefficients $\Delta_{\mathsf{R}}^{\beta} \rho^{\mathsf{R}}_{ijk}$ the above condition reduces to
\begin{align}
\frac{\ell(\phi^{ij})}{\det\kappa} \in
\, \mathbb{Z} \,.
\end{align}
The above integrality condition must be satisfied for any allowed
set of integer fluxes $\phi^{ij}$ that preserve 4D local Lorentz and gauge
symmetry, guaranteeing that all chiral matter indices
(\ref{eq:general-indices}) automatically take integer values. This is demonstrated explicitly in the case of $\mathsf{G} = \SU(5)$ in \cref{sec:su5}.
This analysis in general gives a sufficient condition, for each of the
$\mathsf G$ models studied here, for the chiral matter spectrum to have certain
multiplicities. As discussed in \cref{sec:quantization-1}, however,
inclusion of fluxes in $H_4^{\text{hor}}(X,\mathbb{Z}) \oplus H_{2,2}^\text{rem} (X,\mathbb{Z})$ may permit a
broader set of possible chiral multiplicities.
\begin{table}
\centering
\scalebox{.94}{$
\begin{array}{|c|c|c|c|c|c|c|}
\hline
\substack{\text{Kodaira}\\\text{fiber}}& \mathsf{G} & \Delta^{(2)} &\Theta_\text{phys}\, (\textcolor{red}{\Theta_{(4,6)}}) &\substack{\text{codim-3}\\\text{enhancement}} & \substack{\text{codim-3}\\\text{$(4, 6)$ loci}} & \substack{\text{geometric}\\\text{constraints}}\\\hline
\textcolor{blue}{\text{I}_{5}^{\text{s}}}& \textcolor{blue}{\SU(5)} & \begin{array}{l}a_1^4(a_{6,5} a_1^2\\-a_{3,2} a_{4,3} a_1
\\+a_{2,1} a_{3,2}^2)\end{array}&\phi \Sigma \cdot [a_1] \cdot [a_{6,5}] &\text{none} &\text{none}& \chi_{\textbf{5}} + \chi_{\textbf{10}} =0 \\\hline
\textcolor{blue}{\text{I}_{6}^{\text{s}}}& \textcolor{blue}{\SU(6)}& \begin{array}{l} a_1^4 (a_1^2 a_{6,6}\\-a_1 a_{3,3} a_{4,3}\\-a_{4,3}^2) \end{array} &\phi \Sigma \cdot [a_1] \cdot [a_{4,3}^2] & (2,3,9)&\text{none}& \chi_{\textbf{6}} + 2 \chi_{\textbf{15}} =0 \\\hline
\textcolor{red}{\text{I}_{6}^{\text{s}}}& \textcolor{red}{\SU(6)^\circ} &\begin{array}{l} a_1^3 (a_{6,6} a_1^3\\-a_{3,2} a_{4,4} a_1^2\\+a_{2,2} a_{3,2}^2 a_1\\-a_{3,2}^3)\end{array}& (\textcolor{red}{\phi \Sigma \cdot [a_1] \cdot [a_{3,2}^3]}) & (4,6,12)&a_1 = a_{3,2}=0 & \chi_{\textbf{6}} =0\\\hline
\textcolor{blue}{\text{I}_{7}^{\text{s}}}& \textcolor{blue}{\SU(7)} & \begin{array}{l} a_1^4 (a_1^2 a_{6,7}\\-a_1 a_{3,3} a_{4,4}\\+a_{2,1} a_{3,3}^2)\end{array}& \begin{array}{c}\phi \Sigma \cdot [a_1] \cdot [a_{6,7}] \\ (\textcolor{red}{\phi' \Sigma \cdot [a_1] \cdot [a_{2,1}]} )\end{array}&\text{none}&a_1 = a_{2,1} = 0& \chi_{\textbf{7}} + 3 \chi_{\textbf{21}} = 0\\\hline
\text{I}_{6}^{\text{ns}} & \text{Sp}(6) &\begin{array}{l} a_2^2 (a_2 a_{3,3}^2\\-a_{4,3}^2\\+4 a_2 a_{6,6})\end{array} &0&\text{---}& a_2 = a_{4,3}=0 &\text{---}\\\hline
\textcolor{blue}{\text{I}_1^{* \text{s}}} &\textcolor{blue}{ \SO(10)}&a_{2,1}^3 a_{3,2}^2 & \phi \Sigma \cdot [a_{2,1}] \cdot [a_{6,5}] & \text{none}&\text{none} &\text{any $\chi_{\textbf{16}}$} \\\hline
\text{I}_2^{* \text{ns}} & \SO(11) &\begin{array}{l} a_{2,1}^2 (4 a_{2,1} a_{6,5}\\-a_{4,3}^2)\end{array} &\phi \Sigma \cdot [a_{2,1}] \cdot [a_{6,5}] & \text{none} &\text{none}&\text{---}\\\hline
\textcolor{red}{\text{I}_2^{* \text{s}}} & \textcolor{red}{\SO(12)} &\begin{array}{l} a_{2,1}^2 (4 a_{2,1} a_{6,5}\\-a_{4,3}^2)\end{array}& (\textcolor{red}{\phi \Sigma \cdot [a_{2,1}] \cdot [a_{4,3}^2]}) & (4,6,12) &a_{2,1} = a_{4,3} = 0&\text{---}\\\hline
\textcolor{blue}{\text{IV}^{* \text{s}}} & \textcolor{blue}{\text{E}_6} &a_{3,2}^4 &\phi \Sigma \cdot [a_{3,2} ] \cdot [a_{6,5}] & \text{none} &\text{none}&\text{any $\chi_{\textbf{27}}$}\\\hline
\textcolor{red}{\text{III}^{*}} & \textcolor{red}{\text{E}_7} &a_{3,3}^4& (\textcolor{red}{\phi \Sigma \cdot [a_{4,3}] \cdot [a_{6,5}]}) & (4,6,12) &a_{4,3}=a_{6,5}=0 &\text{---} \\\hline
\text{IV}^{* \text{ns}} & \text{F}_4 &a_{6,4}^2&0&\text{---}& a_{4,3} = a_{6,4} = 0&\text{---} \\\hline
\text{I}_{0}^{* \text{ns}} & \text{G}_2 &4 a_{4,2}^3+27 a_{6,3}^2&0&\text{---}&\text{none} &\text{---}\\\hline
\end{array}
$}
\caption{F-theory fluxes for universal $\mathsf{G}$ models with arbitrary
characteristic data. The
final column matches the linear 4D anomaly conditions in all known
examples. $\Delta^{(2)}$ is the codimension-two component of the
discriminant locus restricted to the gauge divisor $\sigma
=0$. The fifth
column indicates what kind of singularity enhancement of the
elliptic fiber (if any) occurs at these points. Models that
admit 4D chiral matter are indicated in blue and satisfy
$\Theta_{\text{phys}} = \chi_{\mathsf{r}_*}$ (where
$\chi_{\mathsf{r}_*}$ is the minimal chiral index), while
models whose fluxes are proportional to some number of
codimension-three $(4, 6)$ loci are indicated in
red. The $\SO$-type groups listed above
range from those of smallest rank that admit nontrivial flux to
those of largest rank for which the corresponding model does not
have $(4, 6)$ loci in codimension two; the same is true of the
$\Sp$-type groups, with the caveat that none have been
identified that admit nontrivial flux (note that the bases of
divisors and resolutions for the $\text{I}_{2n}^{\text{ns}}$ and
$\text{I}_{2n+1}^\text{ns}$ models appear to be identical.) The
$\text{E}_8$ model is suppressed because it has codimension-two
$(4, 6)$ singularities for generic characteristic data. $\SU(N)$
models with $N>6$ contain codimension-three $(4, 6)$ points;
however for $\SU(6)^\circ, \SU(7)$ the geometric constraints are stated under the restriction that the characteristic data are chosen to ensure
that these points are absent, noting that under analogous conditions a
similar pattern of fluxes may persist for $\SU(N>7)$. The extra fluxes associated to $(4,6)$ points (in red) and the $\SO(11)$ model flux will be treated in a forthcoming publication \cite{46}.}
\label{tab:fluxtable}
\end{table}
\subsection{$\SU(2)$ model}
\label{sec:su2}
We now discuss explicit examples.
We begin with a very simple example that has been well studied in the
literature, but which nevertheless illustrates the issue of unimodularity of the
intersection pairing $M_\text{red}$, namely the universal $\SU(2)$
model. (For additional background about $\SU(N)$ models and their
resolutions, see \cref{sec:appendix-sun}.)
For $\mathsf G = \SU(2)$, we find that the reduced
intersection pairing $M_\text{red}$ is resolution-invariant and the
constraints that local Lorentz and $\SU(2)$ symmetry are unbroken in 4D forces
all the flux backgrounds $\phi$ to vanish, so there are no nontrivial
vertical fluxes and no chiral matter. Identical conclusions follow for
$\mathsf G=\SU(3)$ and $\SU(4)$.
\subsubsection{Absence of chiral matter}
The SU(2) model is characterized by an I$_2$ Kodaira singularity over
the divisor $\Sigma$ and has matter in the representations $\textbf{2} ,
\textbf{3}$ with weights
\begin{align}
\begin{split}
w^{\textbf{2}}_{+} &= (1),~~~~ w^{\textbf{2}}_- = (-1)\\
w^{\textbf{3}}_{+}&= (2),~~~~w^{\textbf{3}}_0 =(0),~~~~w^{\textbf{3}}_{-}=(-2)\,.
\end{split}
\end{align}
The unique resolution $X_1 \rightarrow X_0$ admitting a holomorphic zero section consists of a single blowup. In this case, there is a single $\SU(2)$ Cartan divisor $\hat D_i$ whose nonzero quadruple intersection numbers are (see \cref{pushsu2} and above for details on how to evaluate the pushforwards explicitly)
\begin{align}
\begin{split}
\label{SU2int}
W_{\alpha \beta ii} &= W_{ii} \cdot D_{\alpha} \cdot D_{\beta} = -2 \Sigma \cdot D_{\alpha} \cdot D_{\beta} \\
W_{\alpha iii} &=W_{iii} \cdot D_\alpha = 2 \Sigma \cdot (2 K-\Sigma) \cdot D_\alpha\\
W_{iiii}&=W_{iiii} = 2 \Sigma \cdot ( -4 K^2 + 2 K \cdot \Sigma - \Sigma^2)\,,
\end{split}
\end{align}
with the remaining intersection numbers involving $\hat D_i$ vanishing.
The $\SU(2)$ model provides a simple illustration of the procedure,
discussed towards the end of \cref{3Dcompare}, for using low energy
effective 5D physics as a shortcut to determine the values of the sign
and floor functions appearing in the field theoretic expressions for
the 3D CS couplings. In this case, the floor functions vanish because
the zero section is holomorphic. Furthermore, matching the
pushforwards of the above intersection numbers with the 5D CS
couplings (where the multiplicities $n_{\mathsf{R}}$ are replaced by
the matter curves $C_{\mathsf{R}}$) fixes the values of the sign
functions to be $\text{sign}(\varphi \cdot w_{\pm{}}^{\mathsf{R}}) =
\pm{}1$. In detail,
\begin{align}
\begin{split}
k_{\bar{0}ii} &= - \frac{1}{12} \sum_{w\in \mathsf{r}} C_{\mathsf{r}} \sum_{i=\pm{}} (w^{\mathsf{r}}_i)^2\\
&=- \frac{1}{12} \left[\frac{1}{2} \Sigma \cdot (\Sigma + K ) \left( 2^2 + (-2)^2 \right) + \Sigma \cdot (-8 K - 2 \Sigma ) \left( 1^2 +(-1)^2 \right) \right]\\
&=(-2 \Sigma) \cdot (-\frac{1}{2} K)\\
&=W_{\bar{0}ii}
\end{split}\\
\begin{split}
k_{iii} &=- \frac{1}{2} \sum_{w\in \mathsf{R}} C_{\mathsf{R}} \sum_{i=\pm{}} (w^{\mathsf{R}}_i)^3 \text{sign}(\varphi \cdot w_i) \\
&= - \frac{1}{2}\left[ \frac{1}{2} \Sigma \cdot (\Sigma + K ) \left( 2^3 - (-2)^3 \right) + \Sigma \cdot (-8 K - 2 \Sigma ) \left( 1^3 - (-1)^3 \right) \right]\\
&=2 \Sigma \cdot (2 K-\Sigma)\\
&=W_{iii}\,.
\end{split}
\end{align}
The matrix $M$ from this resolution takes
the form \cref{fullmat}, where the $W$ entries with two or more $i$
indices in the bottom right blocks are all even. Note also that the second Chern
class in the basis of \cref{fullmat} is given by $c_2 (X) =(27, [-39K^\alpha], 0, [(c_2
(B))^{\alpha \beta} + 11K^\alpha K^\beta], [7K^\alpha],0)$; since for any any
F-theory base the class $c_2 (B) + K^2$ is even \cite{Collinucci:2010gz}, it follows that the constrained fluxes $\Theta_{I \alpha}$ are
integral even when $c_2 (X)$ is not even. Thus, we can always remove
the null vectors and impose the constraints by setting $\Theta_{I
\alpha} = 0$ without worrying about half-integer shifts.
It is straightforward to verify that there is one null vector of the
form $\nu_{C \langle a \rangle}$, so $M_\text{red}$ takes the resolution-independent form
\cref{resindependenthatM} and $M_C = M_\text{phys} = 0$. It follows
that 4D SU(2) models exist but do not admit chiral matter. This
conclusion is well known for F-theory models with a tuned SU(2) gauge
invariance and is reviewed in \cite{WeigandTASI}. This result is
consistent with expectations from 4D anomaly cancellation since the
$\textbf{2}$ of $\SU(2)$ is a self-conjugate representation.
\subsubsection{Unimodularity and integrality of the intersection pairing}
\label{sec:2-unimodularity}
We pause briefly to illustrate issues of unimodularity and integrality of the reduced intersection pairing $M_\text{red}$ in the context of the $\SU(2)$ model. Although the lattice $H^{4}(X,\mathbb{Z})$ is unimodular, the simple example of $\SU(2)$ illustrates that fact the intersection pairing acting on $H^{2,2}_{\text{vert}}(X_1,\mathbb{Z})$ is generically not unimodular since
the absolute value of the determinant of $M_\text{red}$ is generically greater than one. As an example, for a gauge divisor $\Sigma = H \subset B = \mathbb{P}^3$ where $H$ is the hyperplane class, we find
\begin{align}
\label{SU2P3mat}
M_\text{red} = \begin{pmatrix} -4 & 1 &0 \\ 1 & 0 & 0 \\ 0 & 0 &-2 \end{pmatrix},~~~~\det M_\text{red}= 2\,.
\end{align}
While one might imagine that we have
simply chosen the wrong basis for $H^{2, 2}_\text{vert}
(X_1,\mathbb{Z})$, the story is slightly subtler.
To further analyze the situation, we briefly digress to
a related situation in the case of 6D F-theory compactifications,
focusing in particular on the parallel case where we have a tuned
$\SU(2)$ F-theory model over the base $B = \mathbb{P}^2$. In this case, the
triple intersection numbers of the Cartan divisors $\hat D_i$
have the related value $W_{iii}= 2$. Thus, the intersection
number of the Cartan divisor $\hat D_i$ with the curve $C_{ii}= \hat D_i \cap
\hat D_i$ is 2. Unlike in the 4D case, the dimension of $H^4
(X^{(3)})$ is equal to that of $H^{2}(X^{(3)})$ by Poincar\'{e}
duality, and again by Poincar\'{e} duality we know that there must be a curve $C$ satisfying $C
\cdot \hat D_i = 1, C \cdot \hat D_{I \ne i} =0$ corresponding to a
(possibly massive) state in the fundamental representation of
$\SU(2)$ (see
\cite{Morrison:2021wuv} for a related discussion). Thus, in
this situation the curve $C_{ii}$ is not a primitive curve in $H_2
(X^{(3)},\mathbb{Z})$, but rather $C= C_{ii}/2$ is such a primitive curve
and is Poincar\'{e} dual to $\hat D_i$.
This same story cannot hold, however, in the 4D
SU(2) model. We do expect that there is a matter surface $S$
associated with matter in the fundamental representation of
$\SU(2)$. This surface cannot simply be identified with $S_{ii}/2$,
however, since the intersection of that surface with itself under the
matrix \labelcref{SU2P3mat} is $(1/2)\times (-2) \times (1/2) = -1/2$. Thus,
the Poincar\'{e} dual of the surface $S_{ii}\in H_{2, 2}^\text{vert}
(X_1,\mathbb{Z})$ is not itself contained entirely in $H_{2, 2}^\text{vert} (X_1,\mathbb{Z})$
and we see that the orthogonal decomposition \labelcref{ortho} of $H_{4}(X_1,\mathbb{Z})$
with respect to the intersection pairing does not hold over $\mathbb{Z}$.
As
we discuss below in the context of $\SU(5)$ models with chiral matter, this point
indicates that the assumption $\phi^{IJ} \in \mathbb{Z}$ may be
too restrictive for our analysis to explore all possible chiralities.
Rather, it appears to be necessary to extend
the analysis to account for contributions from the orthogonal
complement of $H_{2,2}^{\text{vert}}(X)$ in $H_{4}(X)$, which to our
knowledge has yet to be completely understood.
\subsection{$\SU(5)$ model}
\label{sec:su5}
The $\SU(5)$ model (see \cref{sec:appendix-sun}) is the simplest example of a universal SU($N$) model with chiral
matter. The full set of resolutions of the SU(5) model admitting a
holomorphic zero section were worked out in \cite{Esole:2014hya} (see
also \cite{Esole:2011sm,Hayashi:2013lra}) and the chiral indices were
computed for a subset of these resolutions in \cite{Grimm:2011fx} using similar methods to those described in this paper, as well as by other methods in e.g. \cite{Blumenhagen:2009yv,Grimm:2009yu}.
The SU(5) model describes chiral matter in the fundamental
($\textbf{5}$) and two-index antisymmetric (\textbf{10})
representations.
The 4D chiral anomaly cancellation condition requires that
\begin{align}
\label{SU5anomaly}
\chIndex{\bm{5}} +\chIndex{\bm{10}} = 0\,.
\end{align}
We begin by analyzing the model with a specific resolution, $X_4
\rightarrow X_0$, which was
described as the toric `phase I' resolution in \cite{Grimm:2011fx} and as
the resolution `$\mathscr{B}_{1,3}$' in \cite{Esole:2014hya}. The signs
associated to the central charges of BPS particles transforming in the (complex) representations $\textbf{5}, \textbf{10}$
can be found in \cref{SUNsigns}.
In \cref{sec:5-independence} we consider several other
resolutions and show explicitly that $M_\text{red}$ is the same up to
an integral choice of basis for each of these resolutions.
\subsubsection{Chiral matter multiplicities}
Plugging intersection numbers into \labelcref{eq:omegabar} for the
specific resolution just mentioned, we learn that there are four nontrivial
constrained fluxes $\Theta_{ij}$ (Cartan divisors take indices
$i=2,\dots,5$) that satisfy three linear relations, in agreement with
the solution described in \cite{Grimm:2011fx}:
\begin{align}
\label{SU5fluxrelations}
\Theta_{33} = - \Theta_{35}= -\Theta_{44} =\Theta_{45}\,.
\end{align}
In particular (compare also to \cite{Marsano_2011})\patrick{Added this. (AT)},
\begin{align}
\label{SU5fluxintegral}
\Theta_{33} = \frac{\ell(\phi^{ij})}{5} \Sigma \cdot [a_1] \cdot [a_{6,5}]= \frac{1}{5} (\phi^{33} - \phi^{35}-\phi^{44} + \phi^{45} ) \Sigma \cdot K \cdot (6 K + 5 \Sigma) \,.
\end{align}
Note that the proportionalities in \cref{SU5fluxrelations} imply
\begin{align}
\label{SU5square}
\phi^{33} \Theta_{33} + \phi^{35} \Theta_{35} + \phi^{44} \Theta_{44} + \phi^{45} \Theta_{45} = ( \phi^{33} - \phi^{35} - \phi^{44} + \phi^{45} )\Theta_{33} = \ell(\phi^{ij} ) \Theta_{33}\,.
\end{align}
Comparing \cref{SU5fluxrelations} to the one-loop 3D CS couplings,
we learn that
\begin{equation}
\chIndex{\bm{5}} =-\Theta_{33}, \hspace*{0.1in} \chIndex{\bm{10}} = -\Theta_{44} \,,
\end{equation}
and thus we recover the 4D anomaly cancellation equation
\labelcref{SU5anomaly}.
As explained in \cref{integrality}, solving the the gauge symmetry
constraints $\Theta_{i\alpha}=0$ over $\mathbb{Z}$ ensures that the flux \labelcref{SU5fluxintegral}
is integer-valued despite the factor of 5 in the denominator; we go through this analysis in some detail here to
illustrate this point. First note that a necessary and
sufficient condition for an
integral vector $v_i$ to lie in the $\mathfrak{su}(5)$ root lattice is (see \cref{sec:root-lattice}) $v_2 + 2 v_3 + 3 v_4 + 4 v_5 \equiv 0 \, (\text{mod}\, 5 )$, which is equivalent to the condition
\begin{equation}
-2
v_2+ v_3+4v_4+2v_5 \in 5 \mathbb{Z}\,.
\label{eq:5-condition}
\end{equation}
We can use this condition
and the logic following (\ref{eq:mq}) to determine a further constraint on the
parameters $\phi^{ij}$ by noting that from (\ref{fluxconstraints4}) and
(\ref{CartanW}), the gauge symmetry constraints imply
\begin{equation}
\Theta_{\alpha i}^{\text{d}} = \phi^{\beta j} (D_{\beta} \cdot D_{\alpha} \cdot
\Sigma) \kappa_{ij} \,,
\end{equation}
where the superscript `d' indicates that $\Theta^{\text{d}}_{\alpha
i}$ is the part of $\Theta_{\alpha i}$ that only depends explicitly
on the distinctive parameters $\phi^{\hat I \hat J}$. It follows that
$\Theta^{\text{d}}_{\alpha i}$ lies in the tensor product of the
$\mathfrak{su}(5)$ root lattice (for the $i$ index) and the sublattice
of $H_{1, 1}$ spanned by $\Sigma \cap D_\alpha$ (for the $\alpha$
index). Applying the linear combination from the condition
(\ref{eq:5-condition}) to the fluxes $\Theta_{\alpha i }^{\text{d}}$
we find
\begin{align}
\begin{split}
-2\Theta^{\text{d}}_{\alpha 2} + \Theta^{\text{d}}_{\alpha 3} + 4 \Theta^{\text{d}}_{\alpha 4} + 2 \Theta^{\text{d}}_{\alpha 5}
&=-(\Sigma \cdot D_\alpha) \cdot K (\phi^{33} -\phi^{35} - \phi^{44}+ \phi^{45} ) \,.
\end{split}
\end{align}
and thus $K (\phi^{33} - \phi^{35} - \phi^{44} +\phi^{45})$
must lie in $5 H^{1, 1} (B,\mathbb{Z})$. This condition, which is necessary
to ensure that the full SU(5) gauge symmetry is preserved, is
sufficient to guarantee that the chiral matter multiplicities
determined by the flux \labelcref{SU5fluxintegral} are
integer-valued. Note that for a generic base $B$, $K$ is not 5 times an
integral divisor, so the parameters must typically satisfy the condition
\begin{align}
\label{SU5quantcond}
(\phi^{33} - \phi^{35} - \phi^{44} +\phi^{45}) \in 5 \mathbb{Z}\,.
\end{align}
\subsubsection{Flux quantization}
\label{sec:flux-quantization-5}
We illustrate the non-unimodularity of $M_\text{red}$ and the
quantization of the parameters $\phi^{ij}$ in some further
detail with a concrete one-parameter family of $\SU(5)$ examples. Consider the case
$B = \mathbb{P}^3, K =- 4 H, \Sigma = nH$ where $H$ is the hyperplane
class of $\mathbb{P}^3$.
Using the parametrization of the nullspace of $M_C$ given in \labelcref{linrel} along with the explicit results \labelcref{SU5fluxrelations}, we find that a suitable basis for $H_{2,2}^{\text{vert}}(X,\mathbb{Z})$ is $S_{0 \alpha},
S_{\alpha \alpha}, S_{\alpha i}, S_{35}$ (with the index $\alpha$ for the only base divisor)
in terms of which the intersection pairing is
given by
\begin{equation}
M_\text{red} = \left(
\begin{array}{ccccccc}
-4 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & -2 n & n & 0 & 0 & 4 n \\
0 & 0 & n & -2 n & n & 0 & -4 n \\
0 & 0 & 0 & n & -2 n & n & 4 n \\
0 & 0 & 0 & 0 & n & -2 n & -4 n \\
0 & 0 & 4 n & -4 n & 4 n & -4 n & -4 n^2 \\
\end{array}
\right)
\label{eq:mr-2}
\end{equation}
In this example, $K$ is not 5 times an integer divisor, and the
integrality condition \labelcref{SU5quantcond} follows from assuming
that $\phi^{IJ} \in \mathbb{Z}$. This can be seen explicitly as follows:
imposing the local Lorentz and gauge symmetry constraints, one finds that
the flux backgrounds $\phi^{\alpha i}$ can be solved for in terms of the flux backgrounds
$\phi^{ij}$
\begin{align}
\begin{split}
\phi^{\alpha 2} = \frac{8}{5} \phi^{35},~~~~ \phi^{\alpha 3} = - \frac{4}{5} \phi^{35},~~~~ \phi^{\alpha 4} = \frac{4}{5} \phi^{35},~~~~ \phi^{\alpha 5} = - \frac{8}{5} \phi^{35}\,.
\end{split}
\end{align}
The second Chern class for this class of models over $\mathbb{P}^3$ in the
reduced basis
for the resolution $\mathscr{B}_{1,3}$ is
\begin{equation}
c_2(X_4) = 2 (n-22) S_{H3}+2 (3 n-34) S_{H4}+2 (n-20) S_{H5}+48 S_{0H}+182 S_{HH}-16 S_{H2}+S_{35}\,.
\end{equation}
Since all coefficients except that of $S_{35}$ are even integers, we
see that the proper shifted lattice for allowed values of $\phi$ keeps
all $\phi^{IJ}$ integral except $\phi^{35}$, which must take a
half-integral value.
The above relations imply that for any nontrivial integer solution to
these conditions we have\footnote{Note that the quantization condition is insensitive to the fact that we eliminated the redundant homology classes before imposing the Poincar\'{e} and gauge symmetry conditions. In particular, the homology relations can be used to show that $\phi^{35}, \phi^{44}, \phi^{45}$ are each proportional to $\phi^{33}$, and these proportionality factors can be used to convert \labelcref{SU5quantcond} into a relation of the form \labelcref{P3SU5quantcond}.}
\begin{align}
\label{P3SU5quantcond}
\phi^{35} = \frac{5}{2} (2k + 1) \in \frac{5}{2} (2 \mathbb{Z} + 1)\,.
\end{align}
The single flux spanning $M \Lambda_\text{phys}$ is then given
by
\begin{align}
\Theta_{35} = \chIndex{\bm{5}} = \frac{4}{5} n (24-5n)
\phi^{35}
= 2n (24 -5n) (2k + 1) \,,
\label{eq:33-n}
\end{align}
so the chiral matter multiplicities are necessarily integral.
On the other hand,
clearly unimodularity of $H_{2,2}^{\text{vert}}(X_4,\mathbb{Z})$ is not satisfied as for any $n \ge 1$
\begin{align}
\det M_\text{red} =n^5 (20 n-96) \ne \pm{}1\,.
\end{align}
For example, for $n = 1$ the determinant is $-76$. This means that
Poincar\'{e} duality guarantees that there are flux backgrounds that are not of
the simple form characterized by (half-)integer $\phi^{ij}$.
This is parallel to the situation discussed for $\SU(2)$ models above
in \cref{sec:2-unimodularity}. Taking for example
the $n = 1$ case, as for SU(2) the elements of the dual lattice
to the
lattice $H_{2,2}^{\text{vert}}(X_4,\mathbb{Z})$ with inner product
(\ref{eq:mr-2}) do not have integer norms. Thus,
the Poincar\'{e}
dual to e.g.\ the surface $S_{33}$ must project to a fractional
vector in $H_{2,2}^{\text{vert}}(X_4)$ and therefore must also contain a component of
$H_{2,2}^\text{rem}(X_4) \oplus H_{4}^\text{hor}(X_4)$.
An interesting
question, which to our knowledge is not addressed anywhere in the
literature and to which the answer seems unknown, is whether or not including
such flux backgrounds can produce chiral matter multiplicities that are more
general than those given by, e.g., \cref{eq:33-n}.\footnote{As discussed in \cref{sec:quantization-1}, some necessary conditions on fractional coefficients for flux backgrounds in $H_{2,2}^{\text{vert}}(X,\mathbb{Z})$ have been considered in \cite{CveticEtAlQuadrillion}, but not all flux backgrounds satisfying these conditions need be permissible.}
For example, for $n = 1$ the allowed chiral multiplicities from this
analysis should be $38, 114, \ldots$.
Naively it might
seem that Poincar\'{e} duality would suggest that arbitrary integer
matter multiplicities should be possible since there is always an
integral flux background in $H_{4}(X_4,\mathbb{Z})$
that gives $\Theta_{33} = 1, \Theta_{\alpha i} = 0$.
It may be,
however, that the components of $H_4(X_4,\mathbb{Z})$ that must be
turned on for this flux background Poincar\'{e} dual to $S_{33}$ (recall the discussion of Poincar\'e duality and its relation to flux quantization at the of \cref{sec:quantization-1}) would break gauge invariance
(as discussed, e.g., in \cite{Braun:2014xka})
or some other necessary feature of the F-theory vacuum so that such
further chiral multiplicities would be ruled out. Appealing to
a heterotic dual description also does not immediately clarify this
question, since (as demonstrated in e.g. \cite{Grimm:2011fx}) the chiral multiplicities achieved through the spectral cover construction
match the F-theory chiral multiplicities coming from purely vertical flux backgrounds, though
it is possible that additional chiral multiplicities could be realized through
more general bundle constructions. We leave further investigation of
these questions for future work.
\subsubsection{Resolution-independence of the reduced intersection pairing}
\label{sec:5-independence}
In this section we demonstrate the resolution independence of $M_\text{red}$ for the three resolutions $\mathscr{B}_{1,3}, \mathscr{B}_{1,2}, \mathscr{B}_{2,1}$ described in \cite{Esole:2014hya}. In order to compute $M_\text{red}$ for these three cases, we first write down the symmetry constrained fluxes:
\begin{align}
\begin{split}
\mathscr{B}_{1,3} ~&:~ \Theta_{33} =- \Theta_{44} =- \Theta_{35} = \Theta_{45}\\
\mathscr{B}_{1,2} ~&:~ \Theta_{34} = -2 \Theta_{44} = -\Theta_{35} = \Theta_{45} \\
\mathscr{B}_{2,1} ~&: ~ \Theta_{23} = -2 \Theta_{33} = - \Theta_{24} = \Theta_{34}\,.
\end{split}
\end{align}
The indices $jk$ of the above fluxes determine the basis elements $S_{0\alpha}, S_{\alpha \beta}, S_{\alpha i}, S_{jk}$ spanning $H_{2,2}^{\text{vert}}(X,\mathbb{Z})$ in each of these three resolutions. We illustrate this specifically in the case $B = \mathbb{P}^3, \Sigma= nH$, where $H =D_\alpha$ is the hyperplane class of $\mathbb{P}^3$. First, we compare the resolutions $\mathscr{B}_{1,3}$ and $\mathscr{B}_{1,2}$, for which a common basis is $S_{0\alpha}, S_{\alpha \beta}, S_{\alpha i}, S_{35}$. We find (see \labelcref{eq:mr-2})
\begin{align}
\begin{split}
\label{Mred13}
M_\text{red}(\mathscr{B}_{1,3}) &= M_\text{red}(\mathscr{B}_{1,2}) = \left(
\begin{array}{ccccccc}
-4 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & -2 n & n & 0 & 0 & 4 n \\
0 & 0 & n & -2 n & n & 0 & -4 n \\
0 & 0 & 0 & n & -2 n & n & 4 n \\
0 & 0 & 0 & 0 & n & -2 n & -4 n \\
0 & 0 & 4 n & -4 n & 4 n & -4 n & -4 n^2 \\
\end{array}
\right), \end{split}
\end{align}
i.e., the two intersection pairing matrices are identical for these two resolutions. On the other hand, to compare the two resolutions $\mathscr{B}_{1,2}$ and $\mathscr{B}_{2,1}$, we must use a different basis. A suitable basis in which to compare $M_{\text{red}}(\mathscr{B}_{1,2}), M_{\text{red}}(\mathscr{B}_{2,1})$ is $S_{0\alpha}, S_{\alpha \beta}, S_{\alpha i}, S_{34}$, for which we find
\begin{align}
\label{Mred21}
M_\text{red}(\mathscr{B}_{1,2}) =M_\text{red}(\mathscr{B}_{2,1}) =\scalebox{.88}{$ \left(
\begin{array}{ccccccc}
-4 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & n \\
0 & 0 & -2 n & n & 0 & 0 & 0 \\
0 & 0 & n & -2 n & n & 0 & 16 n-2 n^2 \\
0 & 0 & 0 & n & -2 n & n & 3 n^2-20 n \\
0 & 0 & 0 & 0 & n & -2 n & 4 n \\
0 & n & 0 & 16 n-2 n^2 & 3 n^2-20 n & 4 n & -6 n^3+72 n^2-224 n \\
\end{array}
\right)$}\,.
\end{align}
Again, we find that for an appropriate choice of basis the
intersection pairing matrices are identical. This implies that were we
to identify $M_\text{red}$ for the resolutions $\mathscr{B}_{1,3}$ and
$\mathscr{B}_{2,1}$, we would be forced to identify a change of
basis, from \labelcref{Mred13} to \labelcref{Mred21}; the explicit
matrix $U$ presented in \labelcref{Umatrix} does the job for a particular choice of sign in $U_p = (\pm{} 1)$:
solving for the undetermined coefficients in $U$ we find that
they take integer values compatible with the congruence
\begin{align}
M_\text{red}(\mathscr{B}_{2,1}) = U^\ensuremath\mathrm{t} M_\text{red}(\mathscr{B}_{1,3}) U\,.
\end{align}
A related change of basis illuminates further
the question of flux quantization discussed in \cref{sec:flux-quantization-5}.
As discussed in general in \cref{nonabwithchiral} and
\cref{proof}, there is a (non-integral)
change of basis $U$ of the form \labelcref{Umatrix} from both of the forms
\labelcref{Mred13} and \labelcref{Mred21} to a canonical product form
\labelcref{Mredcanonical}, given here by
\begin{align}
M_\text{red}^\text{cp}
=U^{\text{t}} M_\text{red} U =\scalebox{.88}{$ \left(
\begin{array}{ccccccc}
-4 & 1 & 0 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & -2 n & n & 0 & 0 & 0 \\
0 & 0 & n & -2 n & n & 0 & 0 \\
0 & 0 & 0 & n & -2 n & n & 0 \\
0 & 0 & 0 & 0 & n & -2 n & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & n (96-20n)/5\\
\end{array}
\right)$}.
\label{eq:mcr5}
\end{align}
Because $U$ is non-integral in these cases, the lattice $\Lambda^\text{cp}$ of flux
backgrounds on which \cref{eq:mcr5} acts is not $\mathbb{Z}^n$. On the other
hand, the constraint equations simply set all but the last of the flux
background parameters to vanish. The non-integer elements of $U$
involve terms of the form $m/5, m \in\mathbb{Z}$ in the lowest row, and the
set of physical flux background parameters $(0, \ldots, 0, \phi)
\in\Lambda_\text{cp}$ are thus constrained so $\phi \in 5
(2\mathbb{Z} + 1)/2$, analogous
to the constraint \labelcref{SU5quantcond}.
From this
analysis we see that the chiral multiplicity given by,
e.g., $\chIndex{\bm{5}}=\Theta_{35}$ is $\chIndex{\bm{5}}=2n (24 -5n)
(2k + 1)$, with $ k \in\mathbb{Z}$, in agreement with
\cref{eq:33-n}. We can relate the canonical form \labelcref{eq:mcr5}
to the lattice $\Lambda_\text{phys} =\mathbb{Z}$, under which the intersection
form becomes $M_\text{phys} = (5n (96-20n))$.
We expect on physical grounds that any valid F-theory resolutions
should give rise to the same physics and
the same $M_\text{phys}$. Note that while any two such
resolutions would admit non-integer transformations $U, V$ taking
$M_\text{red}$ to the canonical form \cref{eq:mcr5}, this is not quite
sufficient to prove that $M_\text{red}$ are the same in those two
resolutions since there is no guarantee from what we have said here
that $UV^{-1}$ is an integer matrix, as discussed further in
\cref{nonabwithchiral} and \cref{proof}.
It is also worth noting that \cref{eq:mcr5} gives an example of the
self-duality condition as discussed in more general terms in
\cref{sec:fluxes-universal}. Namely, the $\SU(5)$ Weierstrass model
on $ \mathbb{P}^3$ is only consistently defined without enhancement
when $n \leq 4$, in which case $M_\text{phys}$ has a positive matrix
entry and the flux background $\phi$ is self-dual.
\subsection{$\SU(6)$ model}
\label{su6}
The $\SU(6)$ model describes chiral matter in the fundamental
(\textbf{6}) and two-index antisymmetric (\textbf{15})
representations.
The 4D anomaly conditions give
\begin{align}
\chIndex{\bm{6}} + 2\chIndex{\bm{15}} =0.
\label{eq:su6-anomaly}
\end{align}
The signs of the BPS central charges associated to the fundamental and two-index antisymmetric representations can be found in \cref{SUNsigns}. Plugging the intersection numbers into \labelcref{eq:omegabar}, we find that the only nonzero fluxes $\Theta_{ij}$ (Cartan indices are $i= 2,\dots, 6$) satisfy the linear relations
\begin{align}
\label{SU6fluxes}
\Theta_{33} = - \Theta_{34} = \Theta_{35} = \Theta_{45} = - \frac{1}{3} \Theta_{55} = - \Theta_{36} = \Theta_{56}\,,
\end{align}
where
\begin{align}
\begin{split}
\label{SU6fluxintegral}
\Theta_{33} &= \frac{\ell(\phi^{ij})}{6} \Sigma \cdot [a_1] \cdot [a_{4,3}^2]\\
&= \frac{1}{6} ( \phi^{33} - \phi^{34} + \phi^{35} - \phi^{36} + \phi^{45} - 3 \phi^{55} + \phi^{56} )\Sigma \cdot K \cdot (8K + 6\Sigma)\,.
\end{split}
\end{align}
Matching fluxes with the corresponding one-loop CS couplings implies $0=\Theta_{22} = - \chIndex{\bm{6}} - 2 \chIndex{\bm{15}}$, which, using \cref{SU6fluxes}, reproduces the 4D anomaly cancellation condition
\labelcref{eq:su6-anomaly}.
Inverting the linear system arising from matching the fluxes with 3D one-loop CS couplings, we find that the chiral indices can be expressed geometrically as
\begin{align}
-\frac{\chIndex{\bm{6}}}{2} = \chIndex{\bm{15}} = \Theta_{33}\,.
\end{align}
Note that the gauge symmetry condition \labelcref{eq:gauge} implies, in an analogous fashion to the $\mathsf{G} = \SU(5)$ case,
\begin{align}
2K^\alpha (\phi ^{33}-\phi ^{34}+\phi ^{35}-\phi ^{36}+\phi ^{45}- \phi ^{56}) \in 6 \mathbb{Z}\,,
\end{align}
ensuring that $\Theta_{33} \in \mathbb{Z}$.
For comparison, we also comment on the flux for an alternative Tate tuning \cite{HuangTaylorLargeHodge} (see also \cite{MorrisonTaylorMaS,AndersonGrayRaghuramTaylorMiT}) of the $\SU(6)$ model, denoted the $\SU(6)^\circ$ model, that has matter in the exotic three-index antisymmetric representation rather than the usual two-index antisymmetric representation. Because the three-index antisymmetric representation is self-conjugate, one would naively expect the space of vertical F-theory fluxes to be empty. However, it turns out the $\SU(6)^\circ$ model contains codimension-three $(4, 6)$ singularities, leading to a nontrivial flux presumably given by the integral of the flux background over the surface component of the non-flat fiber visible at the $(4, 6)$ point in the resolution $X_5$. See \cref{tab:fluxtable} for additional details.
\subsection{$\SO(10)$ model}
\label{so10}
The SO(10) Tate model is characterized by a I$_1^{* \text{split}}$
singularity over a gauge divisor $\Sigma$ and contains chiral matter
in the spinor representation ($\textbf{16}$); the multiplicity of
matter in this representation is unconstrained by anomalies. We label
Cartan divisors with indices $i =2,\dots, 6$. The signs of the BPS
central charges associated to the spinor can be found in
\cref{SO10signs}.
Using \labelcref{eq:omegabar}, we find that the F-theory fluxes satisfy the homology relations
\begin{align}
\Theta_{22} = -\Theta_{24} = \Theta_{25} = \Theta_{44} = - \Theta_{46}=- \Theta_{55} =\frac{1}{2} \Theta_{66}
\end{align}
where
\begin{align}
\begin{split}
\label{SO10fluxintegral}
\Theta_{22} &=\frac{\ell(\phi^{ij})}{4} \cdot [a_{2,1}] \cdot [a_{6,5}] \\
&=\frac{1}{4} (\phi ^{22}-\phi ^{24}+\phi ^{25}+\phi ^{44}-\phi ^{46}-\phi ^{55}+2 \phi ^{66}) \Sigma \cdot (2 K+\Sigma )\cdot (6 K+5 \Sigma )\,.
\end{split}
\end{align}
Matching with 3D one-loop CS terms, we find
\begin{align}
\chi_{\textbf{16}} = - \Theta_{22}\,.
\end{align}
The gauge symmetry condition \labelcref{eq:gauge} implies
\begin{align}
(2 K + \Sigma)^\alpha (\phi ^{22}-\phi ^{24}+\phi ^{25}+\phi ^{44}-\phi ^{46}-\phi ^{55}+2 \phi ^{66}) \in 4 \mathbb{Z}\,,
\end{align}
hence $\Theta_{22}$ is integer-valued. We again find no linear constraints on the $\SO(10)$ chiral spectrum other than those implied by anomaly cancellation.
\subsection{$\text{E}_6$ model}
\label{e6}
Our final purely nonabelian example is $\mathsf G = \text{E}_6$, which is the only
exceptional group with complex representations and hence the only
exceptional group admitting chiral matter preserving the full gauge
symmetry. The E$_6$ Tate model is characterized by a IV$^{*
\text{split}}$ singularity over gauge divisor $\Sigma$ and contains
chiral matter in the fundamental ($\textbf{27}$) representation, with
a multiplicity unconstrained by local anomalies. Additional details
about the resolution and corresponding signs of BPS central charges
can be found in \cref{E6appendix}.
Computing intersection numbers and substituting their values into the
expression in \labelcref{eq:omegabar}, we find that the nontrivial constrained fluxes satisfy the homology relations
\begin{align}
\frac{1}{2} \Theta_{22} = -\Theta_{25} = - \Theta_{33}= \Theta_{35}=-\Theta_{55}=\Theta_{56}=- \frac{1}{2} \Theta_{66}
\end{align}
where
\begin{align}
\begin{split}
\label{E6fluxintegral}
\Theta_{35}& =\frac{\ell(\phi^{ij})}{3} \Sigma \cdot [a_{3,2} ] \cdot [a_{6,5}] \\
&=\frac{1}{3} ( 2 \phi^{22} - \phi^{25} -\phi^{33} + \phi^{35} - \phi^{55} + \phi^{56} - 2 \phi^{66} ) \Sigma\cdot (3 K + 2\Sigma) \cdot( 6 K + 5 \Sigma)\,.
\end{split}
\end{align}
Comparing with the corresponding 3D one-loop CS couplings, we find
\begin{align}
\chIndex{\bm{27}} = \Theta_{35}\,,
\end{align}
in agreement with, e.g., Eq.~(4.12) in \cite{Kuntzler:2012bu}.\patrick{Added this. (AT)} Note that the gauge symmetry conditions \labelcref{eq:gauge} imply
\begin{align}
\Sigma^\alpha (2 \phi ^{22}-\phi ^{25}-\phi ^{33}+\phi ^{35}-\phi ^{55}+\phi ^{56}-2 \phi ^{66} ) \in 3 \mathbb{Z}\,,
\end{align}
which ensures that $\Theta_{35}$ is integer-valued.\footnote{Curiously, in the special case that $\Sigma^\alpha \in 3 \mathbb{Z}$ the gauge symmetry condition does not appear to place any special conditions on the parameters $\phi^{ij}$.}
\section{Models with a $\U(1)$ gauge factor}
\label{sec:abelianmodel}
We now turn to the more general case of models with gauge symmetry
$\mathsf{G} = ( \mathsf{G}_\text{na} \times \U(1) )/ \Gamma$. As discussed in
previous sections, these models are complicated by the fact that the
fluxes do not simply depend on the mutual triple intersections of the
characteristic data $(K,\Sigma_s, W_{01})$, but rather also depend on
the intersection products of all divisors $D_\alpha \in B$ with the
height pairing divisor $W_{\bar 1 \bar 1}$ associated to the
$\U(1)$ factor---this is a reflection of the global geometric nature
of $\U(1)$ gauge factors in F-theory, in contrast to the local nature
of nonabelian gauge factors $\mathsf{G}_s \subset \mathsf{G}_\text{na}$. One notable
consequence is that the nullspace of $M_C$ is not obviously computable
for such models in a very general way without explicitly specifying $B$.
A possible workaround to this complication, as
discussed at the end of \cref{sec:homologyrel}, is to further restrict
to the sublattice $\Lambda_S \cap \{ \phi^{1\alpha} =0\}$; we describe
an example of
this analysis in \cref{sec:21-rcc}. In the rest of
this section we focus attention on specific bases $B$, where we can
explicitly carry out the full flux analysis.
In \cref{F6model} we analyze the $(\SU(2) \times \U(1) )/\mathbb{Z}_2$ model
in detail. In \cref{321model}, we briefly summarize the results of
the forthcoming paper \cite{321-fluxes} in which we use the methods of
this paper to analyze the universal $\ensuremath(\SU(3) \times \SU(2) \times \U(1)) / \Z_6$ model from
\cite{Raghuram:2019efb}.
\subsection{$(\SU(2) \times \U(1) )/\mathbb{Z}_2$ model}
\label{F6model}
Perhaps the simplest example of a model with gauge group $\mathsf{G}$
containing a $\U(1)$ gauge factor is the $F_{6}$ model studied in
\cite{KleversEtAlToric}, with $\mathsf{G}= (\SU(2) \times \U(1))/\mathbb{Z}_2$ and
matter transforming in the representations $\textbf{1}_{1},
\textbf{1}_2, \textbf{2}_{\frac{1}{2}},
\textbf{2}_{-\frac{3}{2}}$. A related F-theory model was recently analyzed in \cite{Esole:2019rzq}.
The 4D anomaly cancellation conditions impose the constraints that the
chiral matter multiplicities must correspond to an integer multiple of
the family
\begin{equation}
(\chi_{\textbf{1}_{1}},
\chi_{\textbf{1}_{2}},
\chi_{\textbf{2}_{\tfrac{1}{2}}},
\chi_{\textbf{2}_{-\tfrac{3}{2}}})
= (2, -1, -3, -1) \,.
\label{eq:21-family}
\end{equation}
The characteristic data of
this class of F-theory models consists of the canonical class $K$ and the two divisor classes $ S_7, S_9$,
in
terms of which the $\SU(2)$ gauge divisor is given by $S_8 = -K +S_9 -
S_7$. Explicitly, the singular $F_6$ model $X_0$ is realized
as a hypersurface in an ambient $\mathbb{P}^2$ bundle over arbitrary smooth base
$B$, given by
\begin{equation}
s_1 u^3 + s_2 u^2 v + s_3 u v^2 + s_4 v^3 + s_5 u^2 w + s_6 uvw + s_7 v^2 w + s_8 u w^2 =0
\end{equation}
where $[u:v:w]$ are homogeneous coordinates of the ambient space fibers defined by the hyperplane classes
\begin{equation}
\label{eqn:F6fiberP2}
[u] = \boldsymbol{H} +\boldsymbol{K} + \boldsymbol{S}_9,~~~~[ v ] = \boldsymbol{H} - \boldsymbol{S}_7 + \boldsymbol{S}_9,~~~~ [w ] = \boldsymbol{H}
\end{equation}
(note $\boldsymbol{H}$ is the hyperplane class of the fibers and $\boldsymbol{K}, \boldsymbol{S}_7,\boldsymbol{S}_9$ are the pullbacks of the classes $K,S_7,S_9$ in the base to the Chow ring of the ambient space) and the divisor classes of the sections $s_m$ appearing the above hypersurface equation, namely $S_m=[s_m]$, are given by
\begin{equation}
S_1 = -3 K - S_7 - S_9,~~~~ S_4 = 2 S_7 - S_9
\end{equation}
along with
\begin{equation}
S_2 = \frac{1}{3} (2S_1 + S_4),~~~~ S_3 = \frac{1}{3} (S_1 + 2 S_4),~~~~ S_5 = \frac{1}{2} (S_1 + S_8) ,~~~~ S_6 = \frac{1}{2} (S_7 + S_8)\,.
\end{equation}
For a good model with gauge group $\mathsf{G}= (\SU(2) \times \U(1))/\mathbb{Z}_2$,
the characteristic data is constrained so that the divisor classes
$S_1,S_4,S_7, S_8$ are effective.\footnote{These divisor classes are associated with the vertices of the dual polytope of the toric fiber defining the $F_6$ model in \cite{KleversEtAlToric}.
}
\subsubsection{Resolution, Chern--Simons terms, chiral index}
\label{sec:21-rcc}
The resolution $X_2 \to X_0$ described in \cite{KleversEtAlToric} entails a sequence of two
blowups acting on the $\mathbb{P}^2$ fibers so that the resulting smooth
model $X_2$ may be viewed as a hypersurface in an ambient projective
bundle with fibers isomorphic to $\mathbb{P}_{F_6}$, i.e., the blowup of
$\mathbb{P}^2$ at two points. The hypersurface equation defining $X_2$ can be
computed systematically by exploiting the fact that to every two-dimensional toric variety $\mathbb{P}_{F_i}$ is associated a canonically
defined genus one curve in $\mathbb{P}_{F_i}$ that can be realized as a a
zero section of the anticanonical bundle.
In order to use the pushforward technology to compute intersection numbers and other relevant characteristic numbers associated to $X_2$, we regard the singular model $X_0$ as a hypersurface of the ambient projective bundle $Y_0 = \mathbb{P}(\mathscr{V}) \rightarrow B$, with $\mathbb{P}^2$ fibers described by \cref{eqn:F6fiberP2}. Combining this data with the classes of the generators of the centers of the blowups comprising the resolution $X_2 \to X_0$, it is straightforward to explicitly compute the pushforwards of the intersection numbers in a suitable basis and evaluate the geometric expressions for the fluxes using \labelcref{eq:omegabar}--\labelcref{eq:omegabar3}.
The computation of the 3D Chern--Simons terms following the strategy of matching intersection numbers of the form $W_{\bar{I} \bar{J} \bar{K} \alpha}$ to 5D Chern--Simons terms $k^{\text{5D}}_{\bar{I}\bar{J}\bar{K}}$ (where $\bar{I}, \bar{J}, \bar{K} = \bar{0}, \bar{1}, i$) in this case is more involved due to the fact that the resolved model $X_2$ has a rational, as opposed to holomorphic, zero section. This is because one needs to determine, in addition to the signs of the BPS central charges, the ratio of their magnitudes to the KK modulus $m_\text{KK}$. Fortunately there is a simple geometric computation one can do to determine which particles (descending from M2 branes wrapping irreducible holomorphic curves in the M-theory background) have nontrivial KK charge. Notice that the pushforward of the intersection of the zero section and generating section is given by
\begin{align}
\pi_*(\hat D_0 \cdot \hat D_1) = W_{01} = S_7\,.
\end{align}
From the above expression we can infer that the primitive BPS particles in the representation $\mathsf{R}'$ carrying nontrivial KK charge must be associated to matter loci of the schematic form
\begin{align}
C_{\mathsf{R}'} =S_7 \cdot (\cdots)\,.
\end{align}
Exploiting the fact that the spectrum of the $F_6$ model is known, we see that the classes of the relevant matter loci fitting this criterion are
\begin{align}
C_{\textbf{2}_{-\frac{3}{2}}} = S_7 \cdot ( -K - S_7 +S_9),~~~~ C_{\textbf{1}_2} = S_7 \cdot (2 S_7 -S_9)\,.
\end{align}
The above analysis implies that the BPS particles transforming in the representations $\mathsf{R}' = \textbf{2}_{-\frac{3}{2}},\textbf{1}_2$ have nontrivial KK charge. It follows that these are the only particles for which the KK mass is not larger than the Coulomb branch mass; we may therefore set $\floor{| r_\text{KK} \varphi \cdot w_i^{\mathsf{R}} |} =0$ for all other representations $\mathsf{R}$. Utilizing this simplifying assumption, we find a perfect match between the 5D Chern--Simons terms and the triple intersection numbers involving precisely three Cartan divisors, provided we use the signs in \cref{F6signs} and set
\begin{align}
\floor{|r_\text{KK} \varphi \cdot w^{\textbf{2}_{-\frac{3}{2}}}_+ |} = 1
\end{align}
where $w^{\textbf{2}_{-\frac{3}{2}}}_+ = (-\tfrac{3}{2},1)$ is the highest weight.
While we have not found a way to determine a general form for the the nullspace of $M_C$ for arbitrary
characteristic data without specifying $B$,
as discussed in \cref{sec:homologyrel}
we can
attempt to get a general picture of the
nullspace by restricting to the sublattice
$\Lambda_S \cap \{ \phi^{1\alpha}
=0\}$.
Combining the pushforwards of the intersection numbers with
formulae
for the constrained fluxes in \labelcref{eq:omegabar,eq:omegabar2,eq:omegabar3}
reveals that after imposing the additional restriction
$\phi^{1 \alpha} = 0$, the fluxes satisfy (the $\SU(2)$ Cartan index is $i=2$)
\begin{align}
\label{F6null}
2\Theta_{00} = -2 \Theta_{01} = 2\Theta_{11} =2 \Theta_{02} = \Theta_{22}\,.
\end{align}
Continuing to work in the restricted case $\phi^{1\alpha} =0$, comparing the fluxes with the corresponding 3D
Chern--Simons theory shows that the multiplicity of chiral matter
in $\textbf{2}_{-\tfrac{3}{2}}$ is
\begin{align}
2 \chi_{\textbf{2}_{-\tfrac{3}{2}}} = \Theta_{22}\,.
\end{align}
We thus expect in general that F-theory models can realize the
one-parameter family \cref{eq:21-family} of anomaly-free chiral matter fields for this
gauge group with generic matter.
Since in the unrestricted case $\phi^{1\alpha} \ne 0$ we cannot write down a completely general expression for the fluxes without specifying $B$, we next turn our attention to specific examples.
\subsubsection{Example: $B = \mathbb{P}^3$}
As a first simple example, we study the
case $B = \mathbb{P}^3, K = -4 H, S_7 = s_7 H, S_9 = s_9 H$. We define
$n =s_9 + 4- s_7$ for convenience, and parameterize the results in
terms of $n, s_7$ (the parameter $n$ corresponds to the parameter
$s_8$ in the $F_6$ Weierstrass model of \cite{KleversEtAlToric}, which
as discussed above
is the degree of the divisor class $S_8 = n H$ associated with the $\SU(2)$ factor; the $\U(1)$
factor is associated with the height pairing parameter $h :=-2 W_{\bar 1 \bar 1}\cdot H^2 = 16 +
4s_7-n$). Such a model is defined for integer values of $n, s_7$
satisfying the conditions $n, s_7, 16-n-2s_7 = 24 -3n/2-h/2, 4 + s_7-n
= (h-3n)/4 > 0$; from these conditions we see that the height pairing
also satisfies $h = 16 + 4s_7-n > 0$. In this set of cases, the matrix $M_{C_\text{na}(1H)(1H)} =
(n-4s_7-16)= -h/2$ would fail to have an inverse when $n =
4s_7 + 16$, but this does not happen in the parameter range of
interest as the height pairing divisor is always
positive/effective, so for these models the matrix $M_{C_\text{na}(1\alpha)
(1\beta)}$ is always invertible, making it possible in all cases to solve for $\phi^{1H}$ with this expression as a denominator. However, as we demonstrate below, in this case the formal rational expression for the flux (which must take integer values) is not an invariant property of the solution, but rather a feature of our choice of solution. As an alternative, we can solve the equation $\Theta_{1H} = 0$ by eliminating a different flux background parameter so as to produce a polynomial expression for $\Theta_{22}$ that is manifestly integer-valued.
As discussed in \cref{sec:fluxes-algebra}, we can analyze this model by following one of two approaches: either we first impose the symmetry constraints and then study the nullspace of $M_C$ in order to determine linear constraints on the fluxes, or we first quotient out the nullspace of $M$ and then impose the symmetry constraints.
Despite producing identical results, both approaches have their respective unique advantages. In the following discussion we briefly describe three
versions of the calculation (two of which follow the former approach, with the third version following the latter approach) in order to illustrate the
different aspects of the problem.
In particular, the full set of constraints determining the
quantization of the number of chiral matter fields is clearest in the
analysis beginning with $M_\text{red}$ in this class of examples,
though this may not be the case for other choices of $\mathsf{G}$ or $B$.
As stated more generally in \cref{sec:vertical-pairing}, throughout the analysis of this section we disregard the possible half-integer shift in $\phi$ that may be required when $c_2(X)$ is not even; this can easily be incorporated, as described for the $\SU(5)$ model in \cref{sec:su5}. \newline
\noindent \textbf{Rational solution.} Solving directly for $\phi^{1H}$ gives
\begin{align}
\begin{split}
\Theta_{22}& =
\chi_{\textbf{2}_{-\tfrac{3}{2}}} = \frac{2 n s_7 (16-n-2s_7) (4 +s_7-n)}{h} \ell(\phi^{\hat I \hat J} )\\
\ell(\phi^{\hat I \hat J} ) &= \phi^{00} - \phi^{01} + \phi^{02} + \phi^{11} +2 \phi^{22}\,,
\end{split} \label{eq:21-direct}
\end{align}
As in the discussion above, this is well-defined since $h >
0$.
As for the purely nonabelian cases described previously, the
symmetry constraints and integer conditions on the flux backgrounds $\phi^{IJ}$
are sufficient to guarantee that this expression is integer- (and
even-)valued, although we have not identified as simple a structure
underlying this integrality as the group theoretic structure
underlying e.g. the combination of flux background parameters appearing in \cref{SU5quantcond}.
In this case, the symmetry constraints $\Theta_{H 2} =0$ imply
additional conditions, including
\begin{align}
2 s_7(4 + s_7-n) \ell(\phi^{\hat I \hat J} ) \in
h\mathbb{Z} \,,
\end{align}
which in turn implies that $ \Theta_{22}$ is integer-valued, and is
in fact an integer multiple of $n (16-n-2s_7)$.
The full set of constraints is seen most easily after imposing the
homology equivalence conditions on $S_{IJ}$, as discussed further below. \newline
\noindent \textbf{Polynomial solution.} Alternatively, we can solve $\Theta_{1H} = 0$
in this case for the flux
$\phi^{22}$, which gives us the result
\begin{align}
\begin{split}
\Theta_{22}& =
\chi_{\textbf{2}_{-\tfrac{3}{2}}}
= \frac{2}{3} (16-n-2s_7) (4-n + s_7)
\ell'(\phi^{ I J} )\\
\ell'(\phi^{ I J} ) &= \phi^{1H} + (s_7-4)
\phi^{11} +n \phi^{12}\,.
\label{eq:21-indirect}
\end{split}
\end{align}
In this analysis, the condition $\Theta_{1H} = 0$ implies that
$(16-n + 4s_7)\ell'(\phi^{I J} ) = 3n s_7 k$ where $k$ is an
integer combination of fluxes, so generically we expect $\Theta_{22}$
to be an integer multiple of $2n s_7(16-n - 2s_7) (4-n+s_7)$.
The two presentations of the chiral multiplicity (\ref{eq:21-direct})
and (\ref{eq:21-indirect}) must give equivalent answers after all
quantization conditions are properly taken account of. The second
expression is simpler since only a factor of 3, and not the $\U(1)$
height pairing $h$, appears in the denominator. On the other hand,
the expression for $\ell$ is simpler than that of $\ell'$ as it does
not depend on the characteristic data.
For the full analysis of the quantization conditions we now turn to
the analysis using $M_\text{red}$. \newline
\noindent \textbf{Vertical homology and flux quantization.} An important difference in resolutions only admitting a rational zero
section from those with a holomorphic zero section such as those
associated with the purely nonabelian groups studied in previous
sections is that solving the symmetry conditions
\labelcref{eq:Poincare,eq:gauge} over the integers generically imposes
additional constraints on the parameters beyond those necessary
to ensure integrality of the solutions.
To complete the discussion of this simple example we describe the
complete quantization condition following from the integrality of the
fluxes $\phi^{IJ}$. We can first explicitly remove the nullspace of
$M$ by
dropping the fluxes $\phi^{01}, \phi^{02}, \phi^{11}, \phi^{12},
\phi^{22}$, which each appear with a coefficient of 1 in a nullspace
vector with all other entries integer. With this simplification,
the matrix $M_\text{red}$ in the basis $S_{0H}, S_{HH},
S_{H2}, S_{1H}, S_{00}$ becomes
\begin{align}
\label{21mr}
M_\text{red} = \begin{pmatrix} -4 & 1 &0 & s_7 & 16-ns_7
\\ 1 & 0 & 0 & 1 & -4 \\ 0 & 0 &-2 n &n & -ns_7\\
s_7 & 1 & n & -4 & s_7 (n-4)\\
16-ns_7 & -4 & -ns_7 & s_7 (n-4) & -64 + 12ns_7
-n^2 s_7-ns_7^2 \end{pmatrix}\,,
\end{align}
and
\begin{equation}
\det M_\text{red}= -n^2 s_7 (16-n - 2s_7) (4-n+s_7)\,.
\end{equation}
Note that the top left 4 $\times$ 4 block is
resolution-invariant
and
the top left 3 $\times$ 3 block corresponds to the generalization of \labelcref{SU2P3mat} to
arbitrary $n$. Given this form of the reduced matrix,
we can directly solve the constraint equations
$\Theta_{0H},\Theta_{HH}, \Theta_{H2} = 0$
for $\phi^{0H}, \phi^{HH}, \phi^{H2}$. The first two of
these each can be described as
an integer linear combination of remaining fluxes, and the third can
be solved as an integer whenever the flux combination
$\phi^{1H} -s_7 \phi^{00}$ is even. The remaining
equation $\Theta_{1H} = 0$ becomes
\begin{equation}
3ns_7 \phi^{00} = h \phi^{1H} \,.
\end{equation}
These are therefore the only nontrivial constraints on these fluxes.
With this simplification for removing the nullspace, the parameters
$\ell, \ell'$ become $\phi^{00}, \phi^{1H}$ respectively.
From these constraints for any fixed values of $n, s_7$ we can
explicitly determine the quantization of the chiral multiplicity
encoded by $\Theta_{22}$. For example, when $n$ is odd, $h$ is odd as
well and the even parity constraint on $\phi^{1H}-s_7 \phi^{00}$ is
automatically satisfied, so
when $h$
and $3ns_7$ furthermore
have no common divisors, it follows that $\Theta_{22}$ can
be an arbitrary integer multiple of $2ns_7 (16-n - 2s_7) (4-n+s_7)$,
up to bounds determined by the tadpole condition (and where, as noted above, to simplify the discussion we have ignored possible half-integer shifts for non-even $c_2(X)$).
As explicit examples, if $n = s_7 = 1\ (h = 19)$, the chiral index will be a
multiple of $2 \times 13 \times 4 = 104$, and if $n = 1, s_7 = 4\ (h =
43)$, the chiral index will be a multiple of 392. If, however, e.g., $n
= 3, s_7 = 2\ (h = 21)$, then $h$ has a common factor with $3ns_7$, in
particular, $4-n + s_7 = 3, 16-n-2s_7 = 9,$ and $\Theta_{22}$ can be
any multiple of $81$ (instead of 243), up to tadpole
constraints.
And when $n = 4, s_7 = 1\ (h = 16)$, the even parity constraint imposes the
additional condition that $\phi^{1H}$ must be even, so $\Theta_{22}$
is a multiple of $4ns_7 (16-n-2s_7) (4-n + s_7) = 160$.
\remove{As discussed in \cref{constrainedflux}, the lack of an invertible,
representation-theoretic expression for the matrix $MP_{(1\alpha)
(1\beta)}$ in \labelcref{eq:omegabar3} prevents us from writing down an
explicit expression for the fluxes associated to the
resolved $F_6$ model defined over an arbitrary base. Nonetheless, the
formulae for the constrained fluxes in
\labelcref{eq:omegabar}--\labelcref{eq:omegabar3} can be used to easily
obtain the fluxes for a specific choice of base. For
example, in the case $B = \mathbb{P}^3, K = -4 H, S_7 = s_7 H, S_8 =
s_8 H, S_9 = s_9 H$ we have
\begin{align}
\begin{split}
\Theta_{22}& = \frac{2 s_8 (s_8-2 s_9+8) (3 s_7+s_8-2 s_9-4) (2 s_7+s_8-s_9-4)}{12 s_7+7 s_8-8 s_9-16} \ell(\phi^{\hat I \hat J} )\\
\ell(\phi^{\hat I \hat J} ) &= \phi^{00} - \phi^{01} + \phi^{02} + \phi^{11} +2 \phi^{22}\,,
\end{split}
\end{align}
where the definition $S_8 = - K - S_7 + S_9$ implies in this case $s_8 = 4 -s_7 + s_9$. Comparing the above fluxes with the corresponding 3D Chern--Simons, we learn that the multiplicity of chiral matter in the $\textbf{2}_{-\tfrac{3}{2}}$ is
\begin{align}
\chi_{\textbf{2}_{-\tfrac{3}{2}}} = \Theta_{22}\,.
\end{align}
Note that invertibility of the matrix $MP_{(1\alpha)(1\beta)}$ in this example is equivalent to the condition $12 s_7 + 7 s_8 - 8 s_9 - 16 \ne 0$. The gauge symmetry constraints \labelcref{eq:gauge} are again sufficient to ensure integrality of the fluxes; in this example, the symmetry constraints $\Theta_{\alpha 2} =0$ imply
\begin{align}
\frac{2 (3 s_7+s_8-2 s_9-4) (2 s_7+s_8-s_9-4)}{12 s_7+7 s_8-8 s_9-16} \ell(\phi^{\hat I \hat J} ) \in \mathbb{Z}
\end{align}
which in turn implies $ \Theta_{22}/2$ is integer-valued. However, an
important difference in resolutions only admitting a rational zero
section is that solving the symmetry conditions
\labelcref{eq:Poincare,eq:gauge} over the integers generically imposes
additional constraints on the parameters beyond those necessary
to ensure integrality of the solutions.
}
\begin{table}
\begin{center}
$
\begin{array}{|cc|}
\hline
\textbf{1} & \textbf{2} \\\hline
\left(\begin{array}{c|cc} \frac{\varphi \cdot w}{|\varphi \cdot w |} & w_1 & w_2 \\\hline
+ & 1 & 0 \\\hline
+& 2 & 0 \end{array} \right) & \left( \begin{array}{c|cc} \frac{\varphi \cdot w}{|\varphi \cdot w |} & w_1 & w_2 \\\hline +& \frac{1}{2} & 1 \\ -& \frac{1}{2} & -1 \\\hline + & - \frac{3}{2} & 1 \\ + & -\frac{3}{2} & -1 \end{array}\right) \\\hline
\end{array}
$
\caption{Signs and Cartan charges associated to the BPS spectrum of the resolved $F_6$ model with gauge group $\mathsf{G} = (\SU(2) \times \U(1))/\mathbb{Z}_2$ analyzed in \cite{KleversEtAlToric}. The Cartan charges are the Dynkin coefficients of the weights in the representations $\textbf{1}_{1}, \textbf{1}_2, \textbf{2}_{\frac{1}{2}}, \textbf{2}_{-\frac{3}{2}}$ of $\SU(2)$ and the signs correspond to the signs of the BPS central charges $\varphi \cdot w$ for a given choice of Coulomb branch moduli $\varphi^i$.}
\label{F6signs}
\end{center}
\end{table}
\subsubsection{Example: $B = \tilde{\mathbb{F}}_n$}
\label{sec:21-exception}
We next consider a one-parameter family of examples where the F-theory base is taken to be a Hirzebruch threefold, $ B= \tilde{\mathbb{F}}_n$, in order to illustrate how the submatrix $M_{C_\text{na}(1\alpha)(1\beta)}$ can fail to be invertible for certain choices of characteristic data $K, S_7, S_9$. A Hirzebruch threefold is a generalization of a Hirzebruch surface $\mathbb{F}_n$ (i.e., a $\mathbb{P}^1$ fibration over a $\mathbb{P}^1$ base) in which the base of the $\mathbb{P}^1$ fibration is taken to be $\mathbb{P}^2$ instead of $\mathbb{P}^1$. For our purposes, we simply need to know the intersection theory of $\tilde{\mathbb{F}}_n$. To make an analogy, note that Hirzebruch $\mathbb{F}_n$ has two independent classes $F, E$, where $F$ is the class of the $\mathbb{P}^1$ fiber (meaning that $F$ is the divisor class of a point in the $\mathbb{P}^1$ base) and $E$ is the class of the $\mathbb{P}^1$ base (meaning that $E$ is the divisor class of a point in a $\mathbb{P}^1$ fiber). These two classes have the following intersection properties:
\begin{equation}
F^2 =0,~~~~ F \cdot E = 1,~~~~ E^2 = -n, ~~~~ n \in \mathbb{Z}_{\geq 0}\,.
\end{equation}
The threefold $\tilde{\mathbb{F}}_n$ similarly has two independent
divisor classes $D_2:= F, D_1:= E$ satisfying
\begin{equation}
\label{eqn:genFn}
D_2^3 =0 ,~~~~ D_2^2 \cdot D_1 = 1,~~~~ D_2\cdot D_1^2 = -n,~~~~ D_1^3 = n^2,~~~~ n \in \mathbb{Z}_{\geq 0}\,.
\end{equation}
Since $\tilde{\mathbb{F}}_n$ is a toric variety, the canonical class $K$ of $\tilde{\mathbb{F}}_n$ is as usual given by minus the sum of all divisors corresponding to one-dimensional cones of the toric fan:
\begin{equation}
K = - \sum D_{\alpha} = - ( D_2 + D_2 + D_2 + D_1 + ( D_1 + n D_2) ) = - (3 +n) D_2 - 2 D_1\,.
\end{equation}
(Note that the above results can easily be derived by adapting the pushforward technology described in \cref{pushapp} to the projectivization $\mathbb{P}(\mathscr{V}) \rightarrow B^{(2)}$ of a rank one vector bundle $\mathscr{V} = \mathscr{L} \oplus \mathscr{O}_{B^{(2)}}$.) We expand the divisors $S_m$ in the basis $D_\alpha$, $S_m = s_{m\alpha} D_\alpha$. In terms of this basis of divisors, the constraints on the characteristic data for a good
$(\SU(2) \times \U(1))/ \mathbb{Z}_2$ model are then
\begin{align}
\label{eqn:Fnconstraints}
(s_{71}, s_{72}) & > (0, 0) \\
(s_{81}, s_{82}) & > (0, 0) \nonumber\\
(8-2s_{71} -s_{81}, 12 + 4n-2s_{72} -s_{82}) & > (0, 0) \nonumber\\
(2+s_{71} -s_{81}, 3 + n+s_{72} -s_{82}) & > (0, 0) \,, \nonumber
\end{align}
where a (Weil) divisor $S_m$ is effective if $s_{m\alpha} \geq 0$ and either
$s_{m1} > 0$ or $s_{m2} > 0$.
Note that when $n > 3$ there is a non-Higgsable gauge factor on the
divisor $D_2$, which may lead to an enhancement of the gauge
symmetry in the class of universal $(\SU(2) \times \U(1))/ \mathbb{Z}_2$
models.
In this family of examples, (minus) the height pairing divisor is then given by
\begin{equation}
W_{\bar 1 \bar 1} = \frac{1}{2} S_8 + 2 (K - S_7) = \left( -4 -2 s_{71} + \frac{s_{81}}{2} \right) D_1 + \left( -2 (3 + n) - 2 s_{72}+ \frac{s_{82}}{2} \right) D_2.
\end{equation}
Combining the above expression for $W_{\bar 1 \bar 1}$ with the $\tilde{\mathbb{F}}_n$ intersection numbers in \cref{eqn:genFn} we find
\begin{equation}
[[M_{C_{\text{na}} (1\alpha)(1\beta)} ]] = \scalebox{.8}{$\left(
\begin{array}{cc}
n^2 (4 s_{71}-s_{81}+4)+n (-4 s_{72}+s_{82}-12) & n (-4 s_{71}+s_{81}-4)+4 s_{72}-s_{82}+12 \\
n (-4 s_{71}+s_{81}-4)+4 s_{72}-s_{82}+12 & 4 s_{71}-s_{81}+8 \\
\end{array}
\right)$}
\end{equation}
from which it follows
\begin{equation}
\det[[M_{C_\text{na} (1\alpha)(1\beta)} ]] =-\frac{1}{2} (4 (n+s_{72}+3)-s_{82}) (n (4 s_{71}-s_{81}+4)-4 (s_{72}+3)+s_{82})\,.
\end{equation}
Hence we see that if we choose the characteristic data such that
\begin{equation}
s_{82} = \begin{cases} 12 + 4n + 4 s_{72}\\
12 - 4n - 4n s_{71} + 4 s_{72} + n s_{81}
\end{cases}
\label{eq:special-cases}
\end{equation}
the matrix $M_{C_\text{na} (1\alpha)(1\beta)} $ will be singular.
We now turn our attention to some specific choices of $n \leq 3$ and
look in particular for flux compactifications on $(\SU(2) \times
\U(1))/ \mathbb{Z}_2$ models with characteristic data satisfying the
special conditions \cref{eq:special-cases} that lead to
non-invertibility of $M_{C_\text{na}(1\alpha)(1\beta) }$.
\paragraph{Example: $B = \tilde{\mathbb{F}}_0 \cong \mathbb{P}^2 \times \mathbb{P}^1$.} As a specific example, consider the case $n=0$. The matrix of intersection pairings with $W_{\bar 1 \bar 1}$ takes the form
\begin{equation}
[[M_{C_\text{na}(1\alpha)(1\beta) }]] = -\frac{1}{2} \left(
\begin{array}{cc}
0 &4 s_{72}-s_{82}+12 \\
4 s_{72}-s_{82}+12 & 4 s_{71}-s_{81}+8 \\
\end{array}
\right)
\end{equation}
with $\det[[M_{C_\text{na} (1\alpha)(1\beta)}]] =-(4
s_{72}-s_{82}+12)^2/4$.
This matrix is always invertible since $12 + 4s_{72}-s_{82} >
12-2s_{72}-s_{82} > 0$.
\paragraph{Example: $B = \tilde{\mathbb{F}}_3$.}
As another specific example consider the case $n=3$, for which
\begin{equation}
[[M_{C_\text{na}(1\alpha)(1\beta) }]] =-\frac{1}{2} \left(
\begin{array}{cc}
3 (12 s_{71}-4 s_{72}-3 s_{81}+s_{82}) & -12 s_{71}+4 s_{72}+3 s_{81}-s_{82} \\
-12 s_{71}+4 s_{72}+3 s_{81}-s_{82} & 4 s_{71}-s_{81}+8 \\
\end{array}
\right)\,.
\end{equation}
Generically the determinant of this matrix is non-vanishing, but there
is a family of allowed choices of characteristic data for which the
determinant vanishes. For example, making the choices
\begin{equation}
S_7 =S_8 = -K~~\Leftrightarrow ~~
(s_{71},s_{72}) = (s_{81},s_{82}) = (2,6)
\end{equation}
leads to a singular matrix. $M$ does not develop any additional null
vectors as a result of the above specialization,
so it is possible to fully solve the $\U(1)$ symmetry conditions by
eliminating distinctive parameters. In contrast to the previous
specific example $B = \tilde{\mathbb{F}}_0$, this choice for the
characteristic data is not forbidden by the constraints described at
the beginning of this section and hence it appears that such a choice
of parameters describes a consistent F-theory flux vacuum in which the
$\U(1)$ gauge symmetry can be preserved in 4D in spite of $M_{C_\text{na}(1\alpha)(1\beta) }$ being singular; therefore, an explicit solution must include at least one nontrivial flux background other than $\phi^{1\beta}$ as in e.g. the polynomial solution \labelcref{eq:21-indirect}.
This provides an explicit example of the kind of situation mentioned
at the end of \cref{constraintsolutions}.
\subsubsection{Resolution independence of $M_\text{red}$}
We collect some evidence supporting the conjecture that $M_\text{red}$ (and hence $H_{2,2}^{\text{vert}}(X,\mathbb{Z})$) is also resolution independent in the the more general setting of models with $\U(1)$ gauge factors. Here, we compare the resolution of the $(\SU(2) \times \U(1))/\mathbb{Z}_2$ model studied in the previous subsections, which we denote by $X_2$, and an alternative resolution $X'_3$ defined by the sequence of blowups
\begin{align}
X_3' \overset{(e_2,s_8|e_3)}{\longrightarrow} X_2' \overset{(u,v|e_2)}{\longrightarrow} X_1' \overset{(u,s_4v + s_7w|e_1)}{\longrightarrow} X_0
\end{align}
where we follow the notation of \cite{KleversEtAlToric}.
For simplicity, let us specialize again to the case $B= \mathbb{P}^3$, where we again denote the $\SU(2)$ gauge divisor by $S_8 = n H$. In a common basis $S_{0H},S_{HH},S_{H2},S_{1H},S_{11}$ we find
\begin{align}
\begin{split}
M_\text{red}(X_2)&=\scalebox{.85}{$\left(
\begin{array}{ccccc}
-4 & 1 & 0 & s_7 & s_7 (s_7-n) \\
1 & 0 & 0 & 1 & -4 \\
0 & 0 & -2 n & n & -4 n \\
s_7 & 1 & n & -4 & n s_7-s_7^2-4 s_7+16 \\
s_7 (s_7-n) & -4 & -4 n & n s_7-s_7^2-4 s_7+16 & -n^2 s_7+3 n s_7^2-4 n s_7-2 s_7^3+32 s_7-64 \\
\end{array}
\right)$}\\
M_\text{red}(X_3') &=\scalebox{.85}{$ \left(
\begin{array}{ccccc}
-4 & 1 & 0 & s_7 & s_7^2 \\
1 & 0 & 0 & 1 & -4 \\
0 & 0 & -2 n & n & -4 n \\
s_7 & 1 & n & -4 & -n s_7-s_7^2-4 s_7+16 \\
s_7^2 & -4 & -4 n & -n s_7-s_7^2-4 s_7+16 & -n^2 s_7-3 n s_7^2+12 n s_7-2 s_7^3+32 s_7-64 \\
\end{array}
\right)$}\,.
\end{split}
\end{align}
These two matrices are related by a change of basis
\begin{align}
M_\text{red}(X_2) = U^\ensuremath\mathrm{t} M_\text{red}(X_3') U,~~~~ U = \left(
\begin{array}{ccccc}
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
2 s_7 & n s_7 & -s_7 & 2 (4-s_7) & 1 \\
\end{array}
\right)\,.
\end{align}
An analogous change of basis holds for other choices of base we have checked.
\subsection{$\ensuremath(\SU(3) \times \SU(2) \times \U(1)) / \Z_6$ model}
\label{321model}
One of the initial motivations of this paper was to analyze the 4D
massless chiral spectrum of the universal $\ensuremath(\SU(3) \times \SU(2) \times \U(1)) / \Z_6$ model of
\cite{Raghuram:2019efb}. This model is believed to be the most general
F-theory model with tuned
$\ensuremath(\SU(3) \times \SU(2) \times \U(1)) / \Z_6$ gauge symmetry and generic matter spectrum,
consisting of the representations appearing in the MSSM as well as
three additional ``exotic'' matter representations. The gauge sector
of the 4D $\mathcal{N}=1$ supergravity describing this theory at low
energies admits three linearly independent families of anomaly free
combinations of chiral matter representations, so a flux
compactification of the $\ensuremath(\SU(3) \times \SU(2) \times \U(1)) / \Z_6$ F-theory model can be expected to yield
at most three independent combinations of chiral indices.
While the universal $\ensuremath(\SU(3) \times \SU(2) \times \U(1)) / \Z_6$ model can be defined by means of a
Weierstrass model, due to the presence of a $\U(1)$ gauge factor (much
like the $( \SU(2) \times \U(1))/\mathbb{Z}_2$ model), for the purpose
of computing a resolution it proves to be more convenient to start
with a construction of the singular F-theory background as a
hypersurface $X_0$ of an ambient $\mathbb{P}^2$ bundle where the
elliptic fiber of $X_0$ is realized as a general cubic in the $\mathbb{P}^2$ fibers of the ambient space. The hypersurface equation for $X_0$
can be obtained by unHiggsing the $\U(1)$ model with charge $q=4$
matter constructed in \cite{Raghuram34}. The characteristic data of
this model consists of the classes $K, \Sigma_2, \Sigma_3, Y$ where
$\Sigma_m$ is the gauge divisor class of the nonabelian factor
$\SU(m)$ and $Y =: W_{01}$ pulls back to the intersection of the
(rational) zero and generating sections of a resolution of $X_0$. One
special subclass of these models are those with $Y = 0$, which have
only MSSM-type matter and have been studied using the toric $F_{11}$
fiber \cite{KleversEtAlToric, CveticEtAlQuadrillion}.
In a forthcoming publication \cite{321-fluxes}, following the approach
of this paper we present a complete analysis of the lattice of 4D
symmetry-preserving vertical fluxes and associated 4D chiral
multiplicities of the universal $\ensuremath(\SU(3) \times \SU(2) \times \U(1)) / \Z_6$ model over an arbitrary
threefold base $B$. Consistent with the emerging picture of the
landscape of F-theory vertical flux vacua described in this paper, one
of the main results of \cite{321-fluxes} is that all three families of
chiral matter representations can be realized in F-theory---further
evidence suggesting that the linear constraints on 4D chiral matter
multiplicities imposed by F-theory geometry coincide with the linear
constraints implied by 4D anomaly cancellation.
These results are found for arbitrary bases using the simplified analysis
associated with the restricted class of flux backgrounds $\phi^{1
\alpha} = 0$, as well as for specific bases using the full analysis
of $M_{\text{red}}$ and keeping quantization conditions intact.
\section{Conclusions and future directions}
\label{sec:conclusions}
\subsection{Summary of results}
We have described a novel and coherent approach for analyzing 4D
vertical flux compactifications in F-theory (that is, flux backgrounds
belonging to $H_{2,2}^{\text{vert}}(X,\mathbb{Z})$) that preserve 4D local Lorentz and gauge symmetry. Our approach both offers
unique computational advantages, and sheds light on the
geometric nature of some of the resolution-invariant physics encoded
in the singularities of the F-theory background related to the 4D
massless chiral spectrum that has so far proven difficult to analyze
directly in the type IIB duality frame.
One of the key elements of our analysis is the integral lattice of
vertical 4-cycles, with symmetric bilinear form given by the symmetric matrix
of quadruple intersection numbers of the smooth CY fourfold $X$
interpreted as an intersection pairing on surfaces corresponding to the pairwise intersections of divisors.
By Poincar\'e duality the nondegenerate part of
this lattice is equivalent to the lattice
$\httv(X,\mathbb{Z})$ of vertical flux backgrounds.
We conjecture that this lattice, along with its nondegenerate inner
product given by the matrix $M_\text{red}$, is a resolution-invariant
structure for any singular elliptic CY fourfold encoding an F-theory
compactification. This conjecture seems natural from the point of
view of type IIB string geometry, and is satisfied by a wide range of
explicit examples that we have considered in this paper. The
resolution-independence of $M_\text{red}$ also implies that the symmetric bilinear form $M$ on the formal space of intersection surfaces $S_{IJ}$,
which contains a nullspace corresponding to homologically trivial
surfaces, is resolution-independent. This further implies the existence of nontrivial
relations among the set of quadruple intersection numbers of the resolved
CY fourfold, even though these quadruple intersection numbers are not in general
resolution-independent (i.e. equivalent under an integral linear change of basis of the
divisors.) Understanding the geometry of this conjecture better and
its ramifications for the intersection structure of singular CY
fourfolds is an interesting problem for further investigation.
The resolution-independence of $M$ and $M_\text{red}$ is a sufficient
condition for the chiral matter content of a given class of F-theory
flux compactifications to be resolution-invariant, but as far as we
can tell is not directly provable from this geometric condition.
The structure of $M_\text{red}$ we have studied here could be used to
further study F-theory flux compactifications both in situations
where the geometric gauge group remains unbroken in 4D by the fluxes, which is
the primary focus here, as well for cases where the geometric gauge
group is broken by vertical fluxes, which seems like an interesting
direction for further research.
In cases where the flux does not break the gauge group, additional
constraints are placed on the fluxes.
Conceptually, the approach we have taken here to studying such vacua
involves the interplay between two operations
applied to the formal intersection pairing matrix $M$.
The first of
these two operations entails restricting to a sublattice of flux
backgrounds satisfying the constraints necessary and sufficient to
preserve 4D local Lorentz and gauge symmetry. This operation
is central to the standard approach used in much of the previous literature to analyze vertical flux
backgrounds in F-theory; our methods for computing intersection
numbers combined with the rigid structure of the elliptic fibration
enable us to write a formal expression for the elements of this
sublattice.
In contrast, the second of these operations involves taking the
quotient of the
lattice of vertical 4-cycles by homologically trivial cycles, the
result of which is the lattice of vertical homology classes
$H_{2,2}^{\text{vert}}(X,\mathbb{Z})$ with inner product given by $M_\text{red}$.
While these two operations commute, studying the interplay between the
two different orders of these operations gives insight into the
structure of the connection between chiral matter and F-theory fluxes.
Computationally, the approach presented here is a synthesis of various techniques that have
appeared in the literature. Notably, we apply
recently developed algebrogeometric techniques for computing
intersection numbers of divisors in smooth elliptically fibered CY
varieties to classes of resolutions that can more easily be obtained
from geometric constructions of singular F-theory backgrounds in which
the elliptic fiber is realized as a general cubic in $\mathbb{P}^2$---this procedure therefore provides a means to analyze a
broader class of F-theory constructions than is encompassed by the
usual Weierstrass model construction. Moreover, since these techniques
(like those used in \cite{Cveti__2014, LinWeigandG4})
express the intersection numbers of divisors in terms of triple
intersections of certain divisors in the base of the
elliptic fibration (i.e., the characteristic data), this approach can be used to conveniently organize
the landscape of F-theory vertical flux compactifications into
families of vacua with fixed gauge symmetry and matter representations
over an arbitrary base.
We have demonstrated the utility of this approach by analyzing
vertical flux backgrounds in numerous examples with simple gauge
symmetry group and generic matter. We have also analyzed several examples of models with
a $\U(1)$ gauge factor, to illustrate the straightforward
generalization of these methods to models with
$\U(1)$ gauge factors; in principle a similar analysis is possible for
models with an arbitrary number of $\U(1)$ factors.
Of particular note among models with $\U(1)$
gauge factors is the universal $\ensuremath(\SU(3) \times \SU(2) \times \U(1)) / \Z_6$ model \cite{Raghuram:2019efb}
whose 4D massless chiral spectrum we analyze in a forthcoming
publication \cite{321-fluxes} using the methods described in this
paper. We find in all examples that the linear constraints on the
chiral matter multiplicities imposed by F-theory geometry exactly
match the 4D anomaly cancellation conditions, which suggests that it
may be possible to realize all anomaly-free combinations of 4D chiral
matter in F-theory, at least at the level of allowed linearly
independent families of the generic matter types for a given gauge
group.
\subsection{Future directions}
The existence of a resolution independent structure such as
$H_{2,2}^{\text{vert}}(X,\mathbb{Z})$ is consistent with the expectation that the kinematics
of F-theory vacua are captured entirely in the singular elliptic CY
geometry encoded by the axiodilaton over a general base in type IIB
string theory. To our
knowledge the conjecture that $H_{2,2}^{\text{vert}}(X,\mathbb{Z})$ is resolution independent
has not previously been explored in the literature and would be useful
to prove rigorously, as this points to several potential future
avenues of investigation related to the physics of F-theory flux
compactifications:
\begin{itemize}
\item{}
One of the outstanding challenges of F-theory is to give a
complete and mathematically precise definition of this formulation
of string theory. While this is often done by taking a limit of
M-theory (see, e.g., \cite{DenefLesHouches,Grimm:2010ks}), a more
intrinsic definition may be possible from the point of view of type
IIB string theory. The progress made here in understanding the
resolution-independent aspects of the singular elliptic
fourfold geometry $X_0$ may help in better understanding how matter
surfaces and chiral matter may be formulated and computed directly
from the type IIB point of view.
\item{} From a mathematical point of view, the
resolution-independence of $M_\text{red}$ indicates that there is
some intrinsic meaning to the lattice $\httv(X_0,\mathbb{Z})$ of integral
vertical surfaces and their intersection form on the singular
fourfold geometry $X_0$. This is particularly
intriguing as the surfaces $S_{ij}$ most relevant for chiral matter
in
F-theory flux
vacua project to trivial surfaces in the base and thus are hidden
in the singular elliptic CY given by the F-theory Weierstrass
model. Developing a clear mathematical picture of this aspect of
intersection theory of singular complex fourfolds poses an
interesting challenge on the mathematical side.
\item{} More concretely, the resolution-independence of $\httv(X,\mathbb{Z})$
suggests that there should be some way of directly computing the
intersection matrix $M_\text{red}$ without explicitly performing any
blowups at all. While many of the intersection numbers that form
this matrix are resolution-independent, others are not, so
identifying an organizing principle that would make possible a
resolution-independent statement of the form of this matrix would be
a significant step forward for the intrinsic understanding of
singular F-theory flux vacua.
\item{} We have focused in this paper on the intersection structure of CY fourfolds, which is relevant for 4D F-theory vacua. We may speculate, however, that the analogous homology group $H_{2,2}^{\text{vert}}(X^{(3)},\mathbb{Z})$ for a CY threefold $X^{(3)}$ is also resolution-invariant. It may be possible to prove this resolution-invariance in a more direct and explicit way, and this may further shed light on the structure of $H_{2,2}^{\text{vert}}(X,\mathbb{Z})$ for a CY fourfold.
\item{} While in this paper we have focused on fluxes that preserve
the geometric gauge group, so that the gauge invariance constraints
$\Theta_{ I\alpha}= 0$ are all satisfied, it would be interesting to
study flux vacua in which this condition is weakened. In
particular, as discussed in e.g. \cite{Raghuram:2019efb}, while
direct tuning of the Standard Model gauge group in F-theory is one
way to get semi-realistic physics models, the bulk of the moduli
space of CY fourfolds, and apparently the vast majority of the flux
vacua, are dominated by bases that force large numbers
of non-Higgsable gauge factors such as $E_6, E_7, E_8$ (see
e.g. \cite{HalversonLongSungAlg, TaylorWangLandscape, TaylorWangVacua}); for these
bases it is difficult or impossible to tune the Standard Model gauge
group, but the group $\ensuremath(\SU(3) \times \SU(2) \times \U(1)) / \Z_6$ may be realized by turning on fluxes that
break the gauge symmetry. Some preliminary work in this direction
for $E_8$ breaking was done in \cite{TianWangEString}, but the
methods developed here may provide a very useful tool in more
systematically pursuing this kind of analysis for flux breaking of
non-Higgsable groups like $E_6$ and $E_7$.
\item{} Intriguingly, in all models we study we find that the
symmetry-preserving fluxes appear to depend on resolution-invariant
linear combinations of triple intersections of characteristic
divisor classes in the base of the elliptic fibration, so that the
minimum magnitude of the fluxes appears to be controlled by certain
numbers of special points lying in the discriminant locus. Since the
chiral indices themselves can be expressed as linear combinations of
the fluxes, this suggests that the chiral indices in some sense ``count''
special points in the F-theory base. One very clear illustration of
this idea is given by $(4,6)$ points, as the symmetry-preserving
fluxes in all models we have studied receive contributions
proportional to the numbers of $(4,6)$ points in the
base \cite{46}.
More generally, in many cases the multiplicity of chiral matter in
fixed representations is proportional to the number of points in the
base in the intersection of the associated matter curve and another
characteristic divisor, suggesting some explicit direct connection
between chiral matter fields and base geometry.
While these observations are not
necessarily unique to our analysis, our computational methods have
enabled us to survey a large enough number of examples to reveal
patterns among different families of models, see, e.g., the
expressions for the fluxes in \cref{tab:fluxtable}. Along these
lines, it would be interesting and quite useful to understand how to
associate the chiral indices to certain types of singularities
visible directly from the F-theory limit, and it is possible that the
resolution independence of $H_{2,2}^{\text{vert}}(X,\mathbb{Z})$ will prove useful in this capacity.
\item{} Another direction in which this work could naturally be
extended involves the question of whether or not all families of
anomaly-free matter can be realized in F-theory. In all the
examples we have studied, of simple gauge groups and groups with a
single $\U(1)$ factor, we have found by considering both generic and
specific choices of base that F-theory imposes no linear
constraints on the chiral matter multiplicities beyond those
expected by anomaly cancellation. This has implications for the
analysis of the ``swampland,'' suggesting that at the level of
linear families of matter F-theory naturally realizes the full set of
possibilities that are consistent with low-energy constraints. It
would be good to check whether this continues to hold for more
complicated models with more abelian factors, or even to find some
general principle based on the resolution-invariance of
$M_\text{red}$ that can match the rank of this intersection form
with the number of expected families of chiral matter.
\item{} At finer level of detail, there are questions related to
the quantization and multiplicities of chiral matter that could be
explored further both mathematically and through more concrete
physics models.
As we have discussed here (see in particular \cref{sec:quantization-1}), the quantization conditions on matter from
purely vertical fluxes may be weakened when the other components of
middle homology are incorporated and/or fractional vertical flux
coefficients are included, since by Poincar\'{e} duality there
should be in principle cycles with a single unit of flux through any
primitive matter surface, even though in general the determinant $\det
M_\text{red}$ has magnitude greater than 1. Further analysis of the geometry and
associated physics of these kinds of questions could help elucidate
more detailed swampland type questions regarding which precise
multiplicities of matter can arise in given 4D supergravity models
realized from F-theory.
\item{} While in this paper we have focused on chiral matter in 4D
theories, a full understanding of the low-energy physics of a given
F-theory compactification also requires understanding the
vector-like matter. Though vector-like matter multiplicities are
subtler than chiral matter, some recent progress has been made
in this direction \cite{Bies:2014sra,Bies:2017fam,Bies:2020gvf,Bies:2021nje,Bies:2021xfh}. It would be interesting to investigate
whether there is resolution-independent structure, analogous to that
studied here, that can be used
to describe such vector-like multiplicities.
\item{} Finally, we note that for a pair of CY fourfolds related by
mirror symmetry, their respective vertical and horizontal
cohomologies are isomorphic \cite{Greene:1993vm,Braun:2014xka}. In this paper we
restricted our focus to vertical flux backgrounds and did not
attempt to explore the space of horizontal fluxes associated to a
given 4D F-theory model. However, a more complete analysis of
F-theory flux compactifications generically requires horizontal
fluxes to be included in the picture.
If it turns out that the
vertical homology of a given CY fourfold is indeed resolution
invariant, this would suggest that the corresponding horizontal
homology of the mirror CY fourfold is also an invariant structure
across certain regions of moduli space and may provide a strategy
for studying horizontal fluxes, which have received comparatively
less attention in the literature, and which may also give insight
into the quantization issues mentioned above.
The intersection form on the horizontal part of
$H^4 (X,\mathbb{Z})$ also plays an important role in recent work that uses asymptotic
Hodge theory to describe string vacua in large field limits
\cite{Grimm:2019bey,Grimm:2020ouv}, and it would be
interesting to understand if similar resolution-independent
structure is relevant there.
\end{itemize}
We thank Mirjam Cveti\v{c}, Mboyo Esole, Thomas Grimm, Jonathan
Heckman, David Morrison, Ling Lin, Shing Yan (Kobe) Li, Sakura Schafer-Nameki, and Timo Weigand for discussions and for
comments on earlier versions of this manuscript. This work was
supported by DOE grant DE-SC00012567. AT was also supported in part by
the Tushar Shah and Sara Zion fellowship and DOE (HEP) Award
DE-SC0013528. WT would like to thank the Aspen Center for Physics
(ACP) for hospitality during part of this work. The authors would all
like to thank the Witwatersrand (Wits) rural facility and the MIT
International Science and Technology Initiatives (MISTI)
MIT--Africa--Imperial College seed fund program for hospitality and
support during some stages of this project.
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Q: What do you call this type of plug? What do you call this plug as a whole? It has 2 pins and a grounding/earthing wire protuding out. As i do not know the name of this plug, I do not know how to properly use it or even know that its safe to use. Is it a fire hazard to use this kind of plug?
PS. It came inside the box when I bought a Philips monitor. Its supposed to be the mains cable that connects to the outlet
A: This seems to be a fork terminal for a grounding lead. One's supposed to mount that to a grounding post, as illustrated on this wikipedia page.
Do not forget to slide back the protective plastic cover, so the metal connection makes contact with the metal exposed by the grounding post. The screw of the grounding post goes between the two metal prongs.
Installing the grounding connection is generally recommended. Do consult the manual.
Alternatively, if the cord is removable from the monitor or its power supply, and you have grounded (3-hole) sockets available instead of the grounding posts, you can likely use the usual 3-wire grounded power cables in this cable's place.
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Branfman Mayfield Bustarde Reichenthal LLP
Business Contracts & Transactions
Buying & Selling of Businesses
Commercial Lease Negotiation & Review
Business Attorneys for the Professional
Business Dispute Resolution
Employment Law – For the Employer
Discrimination Claims
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Sexual Harassment Claims
Reasonable Accommodation Requests
Classifying Employees and Independent Contractors
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Small Business Subscription Plan
13Oct 2017 by bootnet
Employment Agreement to Arbitrate
Many employers prefer handling employment claims through binding arbitration because it's typically faster and more cost effective. However, there are several requirements that an Employment Agreement to Arbitrate must satisfy before it will be enforced by the court, such as:
A statutory prerequisite to compelling arbitration is demonstrating that the parties to an Employment Arbitration Agreement; i.e., both the employer and employee, agreed to submit their claims to arbitration. If the Employment Arbitration Agreement is not drafted and executed correctly, it may not be enforced by the court, and the employee will be allowed to proceed with litigating their claims.
An Employment Agreement to Arbitrate will also not be enforced if it's unconscionable. There are two types of unconscionability, procedural and substantive, both of which must be present before a court will declare an Employment Agreement to Arbitrate unconscionable. Procedural unconscionability focuses on oppression or surprise due to unequal bargaining power. Substantive unconscionability focuses on overly harsh or one-sided results. The more substantively oppressive a contract term, the less evidence of procedural unconscionability is required to determine if a term is unenforceable, and vice versa.
It's also important that an Employment Agreement to Arbitrate was not signed by the employee under duress. If an employee's assent to an Employment Agreement to Arbitrate is secured by the employer under duress, the Agreement is voidable by the employee.
While severance of unlawful provision(s) is an option that a court may exercise in its discretion, it's not wise for an employer to rely on this in the hope of correcting a poorly drafted Employment Arbitration Agreement. If it has multiple defects, the court may view it as a systematic effort by the employer to impose an inferior forum on the employee that works to the employer's advantage. In such a case, a trial court may conclude that the Employment Arbitration Agreement is permeated with an unlawful purpose and is, therefore, unenforceable.
The attorneys at Branfman Mayfield Bustarde Reichenthal, LLP have considerable experience in drafting enforceable Employment Arbitration Agreements and counseling employers on how they should be presented to their employees to avoid these pitfalls. It's far more cost effective to use Employment Arbitration Agreements that are drafted correctly then to find out after a claim has been lodged by an employee that it is unenforceable due to defects.
" I feel lucky to have been referred to such a wonderful attorney. Gayle Mayfield gives honest intelligent advise. She is trustworthy and has lots of integrity. She is caring and someone who will look after your best interests. I always receive prompt return phone calls and Gayle will always follow through on everything she says. Truly an attorney for life. "
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Call 858.793.8090 or fill out the form to schedule your consultation.
* Email*
858.793.8090 – Phone
2011 Palomar Airport Rd., Suite 306
Copyright © 2022 Branfman Mayfield Bustarde Reichenthal LLP | Business Web Hosting by Boot Networks Global Support
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{"url":"https:\/\/www.physicsforums.com\/threads\/x-0-x-0-1.81346\/","text":"# (x\/0) \/ (x\/0) = 1\n\neNathan\nI have came up with this conclusion after considering division by zero as being \"un-defined\".\n\n$$\\frac {1\/0} {1\/0} = 1$$\nOr maybe\n$$\\frac {1\/0} {1\/0} = undefined$$\nI choose the former\n\nHomework Helper\nIt's the latter. 1\/0 is undefined, so it's certainly wrong that:\n\n(meaningless symbol)\/(meaningless symbol) = 1\n\nDivision is not defined for meaningless symbols, so there's no way to compute the quotient, let alone compute that it is 1.\n\neNathan\nAKG said:\nIt's the latter. 1\/0 is undefined, so it's certainly wrong that:\n\n(meaningless symbol)\/(meaningless symbol) = 1\n\nDivision is not defined for meaningless symbols, so there's no way to compute the quotient, let alone compute that it is 1.\n\nIs there any reason, mathematicly, however, that undefined \/ undefined <> 1?\n\nI mean 1\/0 is undefined, but so is \"x\", but we all know x\/x = 1 even though we dont know the value of x. Maybe my definition of \"undefined\" is wrong though :tongue:\n\nHomework Helper\neNathan said:\nIs there any reason, mathematicly, however, that undefined \/ undefined <> 1?\n\nI mean 1\/0 is undefined, but so is \"x\", but we all know x\/x = 1 even though we dont know the value of x. Maybe my definition of \"undefined\" is wrong though :tongue:\nI'm guessing you don't know what undefined means. Tell me, is a;kljdfa;lkjasdf greater than ;kljasdf;kljsdf? Does that previous question even make sense? No, it doesn't. x is a variable, and it's value is unknown or unspecified, not undefined. x stands for a number, and normally, we want to figure out what the number is by solving for x. 1\/0 does not stand for any number. Division is an operation defined on numbers, so elephant\/rhinoceros doesn't make sense. Similarly, (1\/0)\/(1\/0) doesn't make sense, since 1\/0 is not a number.\n\nI'm assuming that you think (1\/0)\/(1\/0) is 1 because by inverse multiplication, it is equal to (1\/0)*(0\/1) and the 0's somehow cancel out. You're assuming, of course, that 0\/0 = 1.\n\nHomework Helper\neNathan said:\nIs there any reason, mathematicly, however, that undefined \/ undefined <> 1?\n\nI mean 1\/0 is undefined, but so is \"x\", but we all know x\/x = 1 even though we dont know the value of x. Maybe my definition of \"undefined\" is wrong though :tongue:\nIf one were being carefull one would say something like\nx\/x=1 when x is an elements of some division algebra and x is not zero.\n0\/0 is undefined as is 1\/0\nit must be because 0 has the property\n0x=0 for all x\nthus if we attempt to define\n1\/0\n0(1\/0)=0\nbut the whole point of defining 1\/0 was so that0(1\/0) was not 0\nso no definition could be consistient.\n\neNathan\nAKG, good point. I was thinking that perhaps 1\/0 had a value that is never known or something, which I did not belive, I was just woundering.\n\nWhile we are on this strange topic, what about\ninf \/ inf = 1\nTherefore inf - inf = 0\neh?\n\nHomework Helper\nDivision is also not defined for infinity, neither is subtraction (or addition, or multiplication).\n\nGold Member\neNathan said:\nWhile we are on this strange topic, what about\ninf \/ inf = 1\nTherefore inf - inf = 0\neh?\n\nAs AKG points out, $$\\infty$$ is a mathematical concept; it is not a number, thus cannot be used as one in arithmetical operations.\n\nGold Member\nMHB\nIn other threads it has been pointed out that basic algebra cannot be applied to concepts such as infinity and divide by 0.\n\nIs this true?\n\n$$100\\infty = 200\\infty$$\n\n$$100 = 200*\\frac{\\infty}{\\infty}$$\n\n$$100=200$$\n\nI think not.\n\nGlauberTeacher\nIn medieval times and before, both 0 and 1 used not to be considered numbers. Fractions with numerators other than one were not accepted as numbers for a long while. And infinity could very well be defined over all the operations if we choose to define it that way. But as we all agree, we see no reason to redefine in order to fit them in; the Indians came to the same conclusion in the 1300s.\n\nInfinitesmals have always interested me. The whole idea that the limit of (0)*infinity could approach any number based on the setup is astounding. (It also made me question the irrational number, e.)\n\nI still agree with the logic that Jameson, AKG, and Dave use to dispel eNathan's preponderances.\n\nquetzalcoatl9\nGlauberTeacher said:\nAnd infinity could very well be defined over all the operations if we choose to define it that way.\n\nhow could this be done?\n\nquetzalcoatl9\nJameson said:\nIn other threads it has been pointed out that basic algebra cannot be applied to concepts such as infinity and divide by 0.\n\nIs this true?\n\n$$100\\infty = 200\\infty$$\n\n$$100 = 200*\\frac{\\infty}{\\infty}$$\n\n$$100=200$$\n\nI think not.\n\nyes, but that is also not true if you used any number (except 0).\n\nHomework Helper\nquetzalcoatl9 said:\nhow could this be done?\nIt depends what is you want to do. For many purposes we wand to work with feilds. No feilds can be defined with zero division and infinite opererations. You could define a an extension so h^2=0 h not 0. Then we lose the zero product propert among other things (that is x*y can be zero without x or y being 0). Another approch is to define hyperreal numbers as is done in nonstandard analysis.\nHere is a link to a intro to calculus book that uses nonstandard analysis.\nhttp:\/\/www.math.wisc.edu\/~keisler\/calc.html\nThe problem adding yucky stuff to an algabraic structure is that you distroy the nice properties that that structure originally enjoyed. In particular nonstandard analysis introduces an infinite number of positive numbers less than all positive reals, a disturbing addition.\n\nStaff Emeritus\nGold Member\nNo feilds can be defined with zero division and infinite opererations.\n\nI presume things you mean things that behave like 1 ∞ = 2 ∞... because, for example, the hyperreals have lots of infinite numbers. (Meaning that they're larger in magnitude than any integer)\n\nHomework Helper\nHurkyl said:\nI presume things you mean things that behave like 1 \u221e = 2 \u221e... because, for example, the hyperreals have lots of infinite numbers. (Meaning that they're larger in magnitude than any integer)\nYes that sentence of mine was a bit unclear. I did mean infinity in the sense of a number where x=2x=3x=x^2. Hyper reals avoid those type of difficulties by introducing and infinite number of infinite and infinitessimal numbers instead of one infinity. And the fact that even with hyperreals one still cannot divide by zero, but can divide by an infinite number of numbers that are like zero.\n\nmathmike\nhave you heard of a guy named l'hopital?\n\nsay that limf(x) = 0 and lim g(x)= 0. if lim[f'(x)\/g'(x)] has a finite value or if this limit is + or _ inf, then lim f(x)\/g(x) = lim f'(x)\/g'(x)\n\nfor example: lim [(x^2 - 4) \/ (x-2) as x approches 2.\n\nlim [(x^2 - 4) \/ (x-2) lim [ d\/dx(x^2 -4) \/ d\/dx(x - 2) ] - lim 2x \/ 1 =4\n\nGold Member\nMHB\nYes, but L'Hopital's Rule applies with limits that are indeterminate. This is more about algebra with indeterminance.\n\nmathmike\nwell here is proff that (1 \/ 0 ) \/ (1 \/ 0) = 0\n\ntake lim (x \/ e^x) as e approachs + inf.\n\nlim (x \/ e^x) = lim [d\/dx (x) ] \/ [d\/dx (e^x)] = lim 1 \/ e^x = 0\n\nHomework Helper\nOne example is not a proof!\n\nGold Member\nMHB\nmathmike said:\nwell here is proff that (1 \/ 0 ) \/ (1 \/ 0) = 0\n\ntake lim (x \/ e^x) as e approachs + inf.\n\nlim (x \/ e^x) = lim [d\/dx (x) ] \/ [d\/dx (e^x)] = lim 1 \/ e^x = 0\n\nI think you meant as \"x\" approaches infinity, not e.\n\n$$\\lim_{x\\rightarrow\\infty}\\frac{x}{e^x}$$\n\nAnd I don't even see how this is an example... or a proof. I don't mean that negatively, I just don't see it.\n\nHessam\neNathan said:\nIs there any reason, mathematicly, however, that undefined \/ undefined <> 1?\n\nI mean 1\/0 is undefined, but so is \"x\", but we all know x\/x = 1 even though we dont know the value of x. Maybe my definition of \"undefined\" is wrong though :tongue:\n\nyou alsot have to consider that \"x\/x\" is a peice-wise defined function\n\nwhere\n\nf(x) = 1 if x =\/= 0, and undefined when x = 0\n\nmathmike\nyes your right i did mean as x approaches infinity.\n\nsay that lim f(x) = g(x) = inf.\n\nif lim [f'(x) \/ g'(x)] is + or - inf.\nthen lim f(x) \/ g(x) = lim f'(x) \/ g'(x)\n\nok there is the theorem;\n\nnow here is another example\n\nas x approaches 0 from the right lim ln(x) \/ csc(x);\n\nwe have lim ln(x) - - inf and lim csc(x) = + inf\n\nwhich is of type inf \/ inf\n\nso the lim ln(x_ \/ csc(x) = lim [(1\/x) \/ (-csc(x) cot(x)]\n\nwhich can be written as lim [- (sin(x) \/ x) * tan(x)] = -lim sin(x) \/ x * lim tan(x) = - (1) * (0) = 0\n\nthus lim ln(x) \/ csc(x) = 0\n\nGold Member\nMHB\nI'm well aware of L'Hopital's Rule, but your examples don't show that\n\n$$\\frac{\\frac{x}{0}}{\\frac{x}{0}}=0$$\n\nYour examples show that limits in indeterminate form can be modified to get a numerical result.\n\nLast edited by a moderator:\nmathmike\nyour right it dont show that (x\/0) \/ (x\/0) = 1\n\nit shows that it is 0. there is no way to show that it is 1 because it is not 1.\n\nand isnt inf \/inf indeterminite anyway. there is no way to show it isnt\n\nStaff Emeritus\nGold Member\nit shows that it is 0.\n\nNo, it doesn't show that either.\n\nmathmike\nyour right it doesnt show that x\/0 \/ x\/0 =0 it shows that the limit is zero, therfore it can be said that it is zero\n\nGold Member\nMHB\nWe all agree that L'Hopital's Rule works, but it just doesn't apply to the original topic.\n\nGold Member\nMHB\nmathmike said:\nyour right it doesnt show that x\/0 \/ x\/0 =0 it shows that the limit is zero, therfore it can be said that it is zero\n\nThat is not true. Just because a limit exists, it does not mean that the point that the limit approaches exists. We see this all the time.\n\n$$\\lim_{x\\rightarrow{0}}\\frac{\\sin{x}}{x}=1$$\n\nBut the point (0,1) does not exist on that graph.\n\nron damon\nAKG said:\nI'm guessing you don't know what undefined means. Tell me, is a;kljdfa;lkjasdf greater than ;kljasdf;kljsdf? Does that previous question even make sense?.\n\nbut can't you say that \"a;kljdfa;lkjasdf\" \/ \"a;kljdfa;lkjasdf\" equals one, since \"a;kljdfa;lkjasdf\" = \"a;kljdfa;lkjasdf\" ?\n\nLast edited:\nStaff Emeritus\nGold Member\nNo, you cannot. \"a;kljdfa;lkjasdf\" is undefined, and thus so is any expression involving it.\n\nYou also cannot say \"a;kljdfa;lkjasdf = a;kljdfa;lkjasdf\"\n\nron damon\nThen, how can an undefined entity differ from itself? If they are the same, comparing sames doesn't equal one?\n\nmathmike\nbut the implication of [x \/ 0] \/ [x \/ 0] = 1 is perposterous. but you are right in saying that it can be manipulated to get a numerical result\n\nron damon\nalso, can't you apply geomtric reasoning, since we DO know what happens to C\/X when X approaches 0?\n\nStaff Emeritus\nIt is just as invalid to say \"a;kljdfa;lkjasdf $\\neq$ a;kljdfa;lkjasdf\" as it is to say \"a;kljdfa;lkjasdf = a;kljdfa;lkjasdf\".","date":"2022-10-07 02:07:20","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8474909067153931, \"perplexity\": 1750.4883177002043}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-40\/segments\/1664030337906.7\/warc\/CC-MAIN-20221007014029-20221007044029-00644.warc.gz\"}"}
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{
"redpajama_set_name": "RedPajamaC4"
}
| 274
|
\section{Introduction}
The Casimir effect\cite{Casimir, Lifshitz, MT, MPnew,expt1} is a
manifestation of the quantum fluctuations of a quantum field at a
macroscopic level. Experiments on Casimir forces are precise tests
of one of the less intuitive predictions of field theory. For a
theoretician, predicting the outcome of these experiments is a
worthy challenge. Hence it seems somewhat astonishing that an
exact solution exists only for infinite, parallel plates
case\cite{Casimir}. Other formal solutions for geometries not made
of distinct rigid bodies free to move (like the wedge, the
interior of a sphere or of a rectangular box \cite{MT}) are
irrelevant for an experimental setup. Moreover in many such
solutions divergences have been discarded in a way that leaves the
result unrelated to practical materials and configurations
\cite{Graham:2002xq}. Interesting theoretical developments include
the method developed in \cite{Kardar} where the solution for
infinite periodic geometries is obtained as a series expansion in
the corrugation, and the numerical Montecarlo analysis in
\cite{Gies03}.
Amongst the various effects that an experimentalist must take in
account to interpret the data (e.g.\ finite conductivity,
temperature and roughness corrections) probably the most
challenging, interesting and full of connections with other
branches of physics and mathematics is the dependence of the force
on the geometry of the bodies. Calculating the Casimir force for
perfectly reflecting bodies in the end reduces to finding the
density of states (DOS) of the Scrh\"odinger Hamiltonian for the
equivalent billiard problem \emph{including the oscillatory ripple
on the averaged DOS}. This is an incredibly difficult problem in
spectral theory that still challenges mathematicians and
physicists today \cite{Uribe} and in essence is not solved beyond
the semiclassical approximation.
In this context we have introduced in Refs.\ \cite{pap1, opt1} a
method based on classical optics which has several virtues:
accuracy, uniform validity when a symmetry is born,
straightforward extension to higher spin fields, to non-zero
temperatures, to include finite reflectivity and, the main topic
of this paper, \emph{it can provide an approximation to local
observables.}
This paper is structured as follows: in Section
\ref{sec:enmomtens} we show how to cast the energy momentum tensor
into a sum over optical paths contributions and how to regulate
and analyze the divergences, ubiquitous in Casimir energy
calculations. Section III is dedicated to the analysis of the
three examples already studied in \cite{opt1} with pedagogical
intent. We study parallel plates, the Casimir torsion pendulum and
a sphere opposite a plate. In Section IV we show how to calculate
the same local observables and the free energy for a thermal state
and we prove (within the limits of our approximation) the
`classical limit' theorem \cite{MPnew,Feinberg}, which states that
at high $T$, Casimir forces become independent of $\hbar$ and
proportional to $T$. As far as we know this is the first time this
assertion can be generalized to geometries other than parallel
plates. We also study the example of parallel plates (finding the
known results) and of a sphere opposite a plate at non-zero
temperature. We find evidence, again within the framework of the
optical approximation, that the low $T$ behavior of the Casimir
force is a difficult problem, qualitatively different from the
$T=0$ and high temperature cases.
\section{Local Observables}
\setcounter{equation}{0}
\label{sec:enmomtens}
Local properties of the quantum vacuum induced by the presence of
boundaries are of broad interest in quantum field theory
\cite{local}. For example gravity couples locally to the
energy-momentum tensor. Vacuum polarization induces local charge
densities near boundaries, provided the symmetries of the theory
allow it. Also, local densities are free from some of the cutoff
dependencies that plague many other Casimir effects. Any local
observable that can be expressed in terms of the Greens function
can be estimated using the optical approach. In this section we
study the energy, momentum and stress densities for a scalar
field.
Some local observables are not unambiguously defined
\cite{Weinberg}. For example the charge density (in a theory with
a conserved charge) is unambiguously defined while the energy
density, in general, is not (while its integral over the volume,
the total energy, is). In this paper we use the Noether definition
of the energy-momentum tensor, similar results would be obtained
with other interesting definitions.
\subsection{Energy-momentum tensor}
We study the Noether energy-momentum tensor of a free, real scalar
field $\phi$ in a domain ${\cal D}$ with Dirichlet boundary conditions
(BC) on ${\cal S}=\partial{\cal D}$ made of (in general disconnected)
surfaces. Other BC\ (Neumann, Robin) can be discussed but for
simplicity we restrict ourselves to Dirichlet BC\ here.
The lagrangian is (we use $\hbar=c=1$)
\begin{equation}
{\cal L}=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi
-\frac{1}{2}m^{2}\phi^{2},
\end{equation}
where Greek letters are used for 4-dimensional indices while the
vector notation will be used for spatial vectors.
The Noether energy-momentum tensor for this real scalar field is
\begin{equation}
T_{\mu\nu}=\frac{\partial {\cal L}}{\partial(\partial^\mu
\phi)}\partial_\nu\phi-g_{\mu\nu}{\cal L}
\end{equation}
\begin{equation}
T_{\mu\nu}=
\partial_{\mu}\phi\ \partial_{\nu}\phi-g_{\mu\nu}\frac{1}{2}\left(\partial_{\alpha}\phi\ \partial^\alpha\phi
-m^{2}\phi^{2}\right)
\label{tmunu}
\end{equation}
from which we identify the energy density $T_{00}$, the momentum
density $T_{0i}$, and the stress tensor $T_{ij}$. The definition
of these quadratic operators involves divergences that we will
regulate by point splitting. We hence replace quadratic operators
like $\phi(x)^2$ by $\lim_{x'\to x}\phi(x')\phi(x)$. The energy
density operator, for example, is
\begin{eqnarray}
T_{00}(x,t)&=&\lim_{x'\to
x}\left[\frac{1}{2}\partial_0\phi(x',t)\partial_0\phi(x,t)+\frac{1}{2}\vec{\nabla}'\cdot\vec{\nabla}\phi(x',t)\phi(x,t)+\frac{1}{2}m^2\phi(x',t)
\phi(x,t)\right]\nonumber\\
&=&\lim_{x'\to
x}\bigg[\frac{1}{2}\partial_0\phi(x',t)\partial_0\phi(x,t)-\frac{1}{2}\phi(x',t)\vec{\nabla}^2\phi(x,t)+\nonumber\\
&&+\frac{1}{2}m^2\phi(x',t)\phi(x,t)+
\frac{1}{2}(\vec{\nabla}'+\vec{\nabla})\cdot\phi(x',t)\vec{\nabla}\phi(x,t)\bigg]
\label{eq:T00}
\end{eqnarray}
The field $\phi$ satisfies the free wave equation in ${\cal D}$
\begin{equation}
\partial^2\phi+m^2\phi=0
\end{equation}
and hence it can be decomposed into normal modes
\begin{equation}
\label{eq:decompo}
\phi(x,t)=\sum_j\frac{1}{\sqrt{2E_j}}\left(\psi_j(x)e^{-iE_jt}a_j+\psi_j^*(x)e^{iE_jt}a^{\dag}_j\right),
\end{equation}
where $\psi_j$ and $E_j$ are the eigenfunctions and eigenvalues of
the problem
\begin{eqnarray}
\label{eq:dirchletprobl}
(-\vec{\nabla}^2+m^2)\psi_j&=&E^2_j\psi_j\quad\mbox{for } x\in
{\cal D};\qquad \psi_j(x)=0 \quad \mbox{for }\; x\in{\cal S}.
\end{eqnarray}
We also use the definition $E(k)=\sqrt{k^2+m^2}$, and
$E_j=\sqrt{k_j^2+m^2}$ so that the eigenvalue equation reads
\begin{equation}
-\vec{\nabla}^2\psi_j=k^2_j\psi_j,
\label{eq:dirchletproblk}
\end{equation}
and because of the positivity of the operator $-\vec{\nabla}^2$, the
spectrum $\{ E_j\}$ is contained in the half-line $\{E\geq m\}$.
We now introduce the propagator $G(x',x,k)$, defined as in Ref.\
\cite{opt1} to be the Green's function of the problem
(\ref{eq:dirchletprobl}) or (\ref{eq:dirchletproblk}):
\begin{eqnarray}
(-\vec{\nabla}'^2-k^2)G(x',x,k)&=&\delta(x'-x)\nonumber\\
G(x',x)&=&0 \quad \mbox{for } x' \mbox{ or } x\in{\cal S},
\end{eqnarray}
which can be written using the spectral decomposition as
\begin{equation}
G(x',x,k)=\sum_n\frac{\psi_n(x')\psi_n(x)}{k^2_n-k^2-i\epsilon}
\end{equation}
In Ref.~\cite{pap1} we have developed an approximation for the
propagator $G(x',x,k)$ in terms of optical paths (closed, in the
limit $x'\to x$). The derivation can be found in Ref.~\cite{opt1},
the general result valid for $N$ spatial dimensions being
\begin{eqnarray}
G_{\rm opt}(x',x,k)&=&\sum_r\frac{(-1)^{n_r}}{2^{N/2+1}\pi^{N/2-1}}\left(\ell_r\Delta_r\right)^{1/2}k^{N/2-1}H^{(1)}_{\frac{N}{2}-1}
\left(k\ell_r\right),\nonumber \\
\label{eq:unifsemicl}
&\equiv&\sum_r G_r(x',x,k),
\end{eqnarray}
where $H$ is a Hankel function, $r$ labels the paths from $x$ to
$x'$, $n_r$ is the number of reflections of the path $r$,
$\ell_r(x',x)$ is its length and $\Delta_r(x',x)$ is the
\emph{enlargement factor} familiar from classical optics,
\begin{equation}
\label{eq:deltar}
\Delta_r(x',x)=\frac{d\Omega_{x}}{dA_{x'}}.
\end{equation}
$\Delta_r(x',x)$ is the ratio between the angular opening of a
pencil of rays at the point $x$ and the area spanned at the final
point $x'$ following the path $r$. For $N=3$ we have
\begin{equation}
\label{eq:Gopt3}
G_r(x',x,k)=(-1)^{n_r}\frac{\Delta^{1/2}_r(x',x)}{4\pi}e^{ik\ell_r(x',x)}.
\end{equation}
With this explicit form for the propagator $G$, we now have to
rewrite the elements of the quadratic operator $T_{\mu\nu}$ as
functions of $G$ and its derivatives. It is useful to pass from
the point-splitting to a frequency cutoff by inserting the latter
in the normal modes decomposition (\ref{eq:decompo}) as
\begin{equation}
e^{-k_j/\Lambda}=\int_0^\infty dke^{-k/\Lambda} 2k\;
\delta(k^2-k_j^2).
\end{equation}
The limit $x'\to x$ can then be exchanged with the $dk$ integral
and we get for the energy density,
\begin{equation}
\label{eq:T00opt}
\bra{0}T_{00}(x,t)\ket{0}=\int_0^\infty
dke^{-k/\Lambda}\frac{1}{2}E(k)\rho(x,k)+\int_0^\infty
dke^{-k/\Lambda} \frac{k}{2E(k)}\vec{\nabla}\cdot\vec{j}(x,k).
\end{equation}
The density $\rho$ and the vector $\vec{j}$ are defined as
\begin{eqnarray}
\label{eq:rhoxk}
\rho(x,k)&=&\frac{2k}{\pi}\Im\ G(x,x,k)\\
\vec{j}(x,k)&=&\lim_{x'\to x}\frac{1}{\pi}\Im \vec{\nabla}
G(x',x,k)=\frac{1}{2\pi}\Im\vec{\nabla}\ G(x,x,k).
\end{eqnarray}
${\cal E}$ is obtained by integrating $T_{00}$ over the whole volume
between the bodies:
\begin{equation}
\label{eq:volsurf}
{\cal E}=\int_{\cal D} d^3x\int_0^\infty dke^{-k/\Lambda}
\frac{1}{2}E(k)\rho(x,k)+\int_0^\infty
dke^{-k/\Lambda}\frac{k}{2E(k)}\int_{\cal S} d\vec{S} \cdot\vec{j}(x,k).
\end{equation}
We have turned the integral over the divergence of $\vec{j}$ into a
surface integral using Gauss's theorem. In the case of Dirichlet
or Neumann boundary conditions, since $d\vec{S}\propto \vec{n}$ we have
(here $\partial_{\vec{n}}\equiv \vec{n}\cdot\vec{\nabla}$ and
$j_{{\vec{n}}}=\vec{n}\cdot\vec{j}$)
\begin{equation}
j_{{\vec{n}}}(x,k)=\frac{1}{\pi}\Im\ \partial_{{\vec{n}}} G(x,x,k)=0, \qquad
x\in {\cal S}
\end{equation}
and the surface integral term disappears. It should be noted that
the vanishing of the $\vec{j}$ contribution to the total energy
relies on the continuity of the propagator for $x',x\in {\cal D}$. In
some approximations, including the optical one, this continuity is
lost. Hence spurious surface terms arise on the boundary of
certain domains ${\cal D}'\subset {\cal D}$. This region is what in wave
optics is called the `penumbra' region. Diffractive contributions
are also not negligible in this region and they cancel the
discontinuities in $G$, hence eliminating the surface
terms.\footnote{As an example see Kirchoff's treatment of the
diffraction from a hole in Ref.~\cite{BornWolf}.} The surface
terms in the energy are hence of the same order of the diffractive
contributions which define the error in our approximation.
The divergence $\vec{\nabla}\cdot\vec{j}$ could also be eliminated from
$T_{00}$ by changing the energy-momentum tensor according to
\begin{equation}
\tilde{T}_{\mu\nu}=T_{\mu\nu}+\partial^\alpha\psi_{\alpha\mu\nu},
\end{equation}
with
\begin{equation}
\psi_{\alpha\mu\nu}=\frac{1}{2}\phi\left(g_{\mu\nu}\partial_\alpha-g_{\alpha\nu}\partial_\mu\right)\phi.
\end{equation}
The total energy ${\cal E}$ and momentum are not affected by this
redefinition however the new tensor $T_{\mu\nu}$ is not symmetric.
It can be seen that the stress tensor $T_{ij}$ is normal on the
surface ${\cal S}$ (for both Dirichlet and Neumann BC) so locally the
force on the surface is given by the pressure alone
\begin{equation}
\frac{d\vec{F}}{dS}=\vec{n} P=\bra{0}T_{\vec{n},\vec{n}}\ket{0}.
\end{equation}
The operator $T_{{\vec{n}}\svn}$ regulated by point splitting is
\begin{eqnarray}
T_{{\vec{n}},{\vec{n}}}(x,t)&=&\lim_{x'\to
x}\left[\partial'_{{\vec{n}}}\phi'\partial_{{\vec{n}}}\phi-\frac{1}{2}
g_{{\vec{n}},{\vec{n}}}\left(\partial'_0\phi'\partial_0\phi-\vec{\nabla}'\phi'\cdot\vec{\nabla}\phi-m^2\phi^2\right)\right]\nonumber\\
&=&\lim_{x'\to
x}\Big[\partial'_{\vec{n}}\phi'\partial_{\vec{n}}\phi+\frac{1}{2}\left(\partial'_0\phi'\partial_0\phi+\phi'\vec{\nabla}^2\phi-m^2\phi^2\right)\nonumber\\
&-&\frac{1}{2} \left(\vec{\nabla}'+\vec{\nabla}\right)\phi'\vec{\nabla}\phi\Big]
\end{eqnarray}
where $\phi'$ is shorthand for $\phi(x',t)$. The second term in
brackets is zero when averaged over an eigenstate of the number
operator $\ket{\{n_j\}}$, by virtue of the equations of motion.
For Dirichlet BC\ the term $\phi\vec{\nabla}^2\phi=0$ on the
boundaries, so we have ($\vec{\nabla}=\vec{n}\partial_{{\vec{n}}}+\vec{\nabla}_t$)
\begin{equation}
\bra{0}T_{{\vec{n}},{\vec{n}}}\ket{0}=\lim_{x'\to
x}\sum_j\frac{1}{4E_j}\left(\partial'_{{\vec{n}}}\partial_{{\vec{n}}}-\vec{\nabla}'_{t}\cdot\vec{\nabla}_{t}+k_j^2\right)\psi_{j}(x')\psi_j(x).
\end{equation}
Since also $\vec{\nabla}_t\psi_j(x)=0$ on the boundaries this
expression simplifies to
\begin{equation}
P(x)=\lim_{x'\to
x}\sum_j\frac{1}{4E_j}\partial'_{{\vec{n}}}\partial_{{\vec{n}}}\psi_j(x')\psi_j(x).
\end{equation}
This expression can be rewritten, in terms of the propagator $G$,
regulated by a frequency cutoff as we did for $T_{00}$,
\begin{equation}
P(x)=\lim_{x'\to x}\partial'_{{\vec{n}}}\partial_{{\vec{n}}}\int_0^\infty dk
e^{-k/\Lambda}\frac{k}{2\pi E(k)}\Im G(x',x,k).
\end{equation}
In this regulated expression we can exchange the derivatives,
limit and integral safely. Below we discuss what the divergences
are when $\Lambda\to\infty$ and how to interpret and dispose them.
All the above expressions are exact. Once the propagator $G$ is
known, we can calculate the energy-momentum tensor components from
them. However as discussed above in the interesting cases it is
difficult to find an exact expression for $G$ and some
approximations must be used.
For smooth impenetrable bodies we use the optical approximation to
the propagator developed in Ref.\ \cite{pap1,opt1} and recalled in
eq.\ (\ref{eq:unifsemicl}). This gives $G$ as a series of optical
paths and hence the pressure $P$ as a sum of optical paths
contributions
\begin{eqnarray}
P&\simeq&\sum_r P_r\\
P_r&=&(-1)^{n_r}\lim_{x'\to
x}\partial'_{{\vec{n}}}\partial_{{\vec{n}}}\int_0^\infty dk
e^{-k/\Lambda}\frac{k}{2\pi
E(k)}\frac{\Delta_r^{1/2}(x',x)}{4\pi}\sin
\left(k\ell_r(x',x)\right),
\end{eqnarray}
An important feature of the optical approximation is that all
divergences are isolated in the low reflection terms whose
classical path length can vanish as $x',x\to{\cal S}$. In practice only
the zeroth and first reflection are potentially divergent. Before
performing the integral in $k$ and taking $\Lambda\to\infty$ then
we have to put aside the divergent zero and one reflection terms
$P_0$ and $P_1$ for a moment (in the next section we will show how
their contributions are to be interpreted).
For the remaining families of paths (that we will denote as
$r\in{\cal R}$) the integral over $k$ can be done and the limit
$\Lambda\to\infty$ taken safely. The result is finite and reads
\begin{equation}
P(x)= \sum_{r\in {\cal R}}\lim_{x'\to
x}\partial'_{{\vec{n}}}\partial_{{\vec{n}}}(-1)^{n_r}\frac{\Delta_r^{1/2}(x',x)}{8\pi^2\ell_r(x',x)}.
\label{eq:pressurepaths}
\end{equation}
We can further simplify this expression. For simplicity let us
call $z$ the normal direction. Notice that for any sufficiently
smooth function $f(z',z)$ vanishing for either $z'$ or $z$ on the
surface $z=0$
\begin{equation}
\label{eq:trick1}
\partial_{z'}\partial_z
f(z',z){\Big|}_{z'=z=0}=\frac{1}{2}\partial_z^2f(z,z){\Big|}_{z=0}.
\end{equation}
The proof is trivial: consider that the lowest order term in the
expansion of $f(z',z)$ near $z',z=0$ is $\propto z'z$. The
propagator $G(x',x,k)$ satisfies all these properties and hence we
can use this result to get rid of the limit $x'\to x$ and assume
$x'=x$ from the beginning. We can therefore rewrite
Eq.~(\ref{eq:pressurepaths}) as,
\begin{equation}
\label{eq:press2}
P(x)= \sum_{r\in
{\cal R}}(-1)^{n_r}\partial_{z}^2\frac{\Delta_r^{1/2}(x,x)}{16\pi^2\ell_r(x,x)}.
\end{equation}
Equation (\ref{eq:press2}) is one of the main results of this
paper. In Ref.\ \cite{opt1} we reduced the computation of Casimir
energy to a volume integral. The force is then found by taking the
derivative with respect to the distance between the bodies.
Calculating the pressure instead gives the force by means of just
a double integral of a local function. The problem is then
computationally lighter and sometimes (as we will see in the
examples) can even lead to analytic results.
Essentially the problem has been reduced to finding the lengths
and enlargement factors associated with the optical paths
\emph{for points close to the boundary}. In the case of the
pressure (eq.\ (\ref{eq:pressurepaths}) or (\ref{eq:press2})) it
is necessary to know their derivatives in the direction transverse
to the surfaces. We will see that this problem can be easily
tackled numerically when it cannot be solved analytically.
\subsection{Regulate and eliminate divergences}
As in the energy calculations \cite{opt1}, the only divergences
occurring in the pressure come from by paths whose lengths
$\ell\ll 1/\Lambda$, where $\Lambda$ is the plasma frequency of
the material. There are only two such families of paths: the zero
and one reflection paths. In this section we show that these
divergent contributions are independent of the distances between
the bodies. This fact is easily understood: in order for a path to
have arbitrarily small length all of its points must be on the
same body. So in order to study these terms we need only consider
a single, isolated body (and a massless field). We are also
careful in maintaining the double derivative $\partial^2_{z',z}$
since we are calculating the terms $P_0$ and $P_1$ separately.
For $r=0$, the zero reflection term, introducing an exponential
cutoff $\Lambda$ on the material reflection coefficient we obtain
\begin{equation}
\label{eq:pressure0}
P_{0}=\int_0^\infty e^{-k/\Lambda} dk \frac{k}{2\pi
E(k)}\lim_{x'\to x\in
{\cal S}}\partial_{z'}\partial_{z}\left(\frac{\sin
k|z'-z|}{4\pi|z'-z|}\right)=\frac{\Lambda^4}{4\pi^2}.
\end{equation}
The same calculation for the $r=1$ or one reflection term gives:
\begin{equation}
\label{eq:pressure1}
P_{1}=\int_0^\infty e^{-k/\Lambda} dk \frac{k}{2\pi
E(k)}\lim_{x'\to
x\in{\cal S}}\partial_{z'}\partial_{z}\left(-\frac{\sin
k|z'+z|}{4\pi|z'+z|}\right)=\frac{\Lambda^4}{4\pi^2}.
\end{equation}
Notice that these two terms are equal, so we could have
substituted $\partial_{z',z}\to\frac{1}{2}\partial^2_z$ for their
sum, after having properly regulated the divergence.
This positive, cut-off dependent pressure, $P_\Lambda\equiv
P_0+P_1$, must be dynamically balanced locally by a pressure
generated by the material, lest it collapse. Moreover the total
force obtained by integrating this quantity over the (closed)
surface ${\cal S}$ of the whole body gives zero. However, if the space
around the body were inhomogeneous, as in the presence of a
gravitational field, a finite term survives the surface
integration, giving rise to a ``vacuum Archimedes effect'' in
which the pressure on one side is, due to gravitational effects,
larger than on the other side, so the body feels a net force. We
have analyzed this effect in detail in Ref.\ \cite{buoy} and
called it ``Casimir buoyancy''.
Finally note that another important element of this class of
quadratic operators is the Feynman propagator. In studying a field
theory in a cavity or in between impenetrable bodies (for example
hadrons as bags, photons in cavities or Bose-Einstein condensates
in traps), we can consider expanding the Feynman propagator in a
series of classical optical paths reflecting off the boundaries.
The first term, related to the direct path is the familiar free
propagator, the others give the finite volume corrections.
\section{Examples}
\setcounter{equation}{0}
In this section we calculate the Casimir force from the pressure,
using the formalism developed in the previous section, for three
examples that were already addressed in Ref.\ \cite{opt1} using
the energy method.
\subsection{Parallel Plates}
\label{sec:parplates}
The parallel plates calculation is a classic example, whose result
is well known and constitutes the basis the widely used proximity
force approximation (PFA) \cite{Derjagin}. We use this standard
example to establish the rank among contributions to the total
pressure and show the similarity and differences with the energy
method \cite{opt1}.
We calculate the force acting on the lower plate, denoted by $d$
or \emph{down}, by calculating the pressure on its surface. We
discard the zero and $1d$ (one reflection on the lower plate
itself) reflection terms. The first term to be considered is the
path that bounces once on the upper plate ($u$ or \emph{up}) $1u$.
For parallel plates $\Delta=1/\ell^2$ and we have
\begin{equation}
\label{eq:pressplate}
P(x)= \sum_{r\geq
1u}(-1)^{n_r}\partial_{z}^2\frac{1}{16\pi^2\ell_r^2(x,x)}.
\end{equation}
The length $\ell_r(x,x)$ for the paths that bounce an even number
of times is a constant in $z$ and hence the derivatives vanish:
they do not contribute to the pressure. This seemingly innocuous
observation simplifies the calculations considerably and it is a
test for any other geometry which reduces to parallel plates in
some limit: in this limit the even reflections contributions must
vanish. Generically their contributions are small. This parallels
the role of the odd reflection paths in the energy method
\cite{opt1}.
Figure \ref{fig:paths} shows the odd reflection paths labelled
with our conventions.
\begin{figure}
\centerline{\asfigure{paths.eps}{450}{10cm}}
\caption{\label{fig:paths} Odd reflection paths that contribute to
the Casimir force between the two plates in the pressure
calculations with the optical approximation. The points $x'$ and
$x$ will eventually be taken coincident and lying on the lower
plate.}
\end{figure}
For the path $1u$ we have
\begin{equation}
\label{eq:plate1u}
P_{1u}(x)=\lim_{z\to
0}-\partial_z^2\frac{1}{16\pi^2(2a-2z)^2}=-\frac{3}{32\pi^2a^4}.
\end{equation}
The next path to be considered is the path that bounces 3 times,
first on $d$, then on $u$ and again on $d$, $dud=3u$ (3 stands for
3 reflections and $u$ for the plate where the middle reflection
occurs) which gives a contribution
\begin{equation}
\label{eq:plate3u}
P_{3u}(x)=\lim_{z\to
0}-\partial_z^2\frac{1}{16\pi^2(2a+2z)^2}=-\frac{3}{32\pi^2a^4}.
\end{equation}
The two contributions Eq.~(\ref{eq:plate1u}) and
Eq.~(\ref{eq:plate3u}) are equal. The reason is easily uncovered.
One can recover Eq.~(\ref{eq:plate3u}) from Eq.~(\ref{eq:plate1u})
sending $z\to-z$ but for the purpose of taking the second
derivative at $z=0$ this is irrelevant. In the same fashion
$P_{3d}=P_{5d}$, $P_{5u}=P_{7u}$ etc.\ and hence we find
\begin{equation}
P(x)=-2\frac{3}{32\pi^2a^4}-2\frac{3}{32\pi^2(2a)^4}-2\frac{3}{32\pi^2(3a)^4}+...=-\frac{3}{16\pi^2a^4}\frac{\pi^4}{90},
\end{equation}
which is the well-known result. Notice also that the rate of
convergence is the same as in the calculation making use of the
Casimir energy in Ref.\ \cite{opt1} ($n$-th term contributes
$1/n^4$ of the first term, in this case $1u+3u$). These
observations that allow us to determine the rank of the
contributions are fundamental, and they apply as well to the other
examples in this section.
\subsection{The Casimir Torsion Pendulum}
In this section we study a geometry already considered in Ref.\
\cite{opt1}: a plate inclined at an angle $\theta$ above another
infinite plate. We have called this configuration a `Casimir
torsion pendulum' because the Casimir force will generate a torque
which can be experimentally measured. The configuration is
analogous to the parallel plates case but the upper plate must be
considered tilted at an angle $\theta$ from the horizontal. The
length of the upper plate must be taken finite, we denote it by
$w$, while the length of the lower plate can be infinite which we
choose for simplicity. There is only one substantial difference
with the parallel plates case: the even reflection paths do
contribute in the pendulum, since their length varies as we move
the final points $x',x$.
We calculate the force exerted on the lower, infinite plate for
simplicity. We then obtain the energy ${\cal E}$, by integrating over
the distance along the normal to the lower plate and from this we
can calculate the torque as
\begin{equation}
{\cal T}=-\frac{\partial {\cal E}}{\partial \theta}.
\end{equation}
The lower plate is taken infinite, the upper plate width is $w$,
and the distance between the height at the midpoint of the upper
plate is $a$. We will choose as the origin of the coordinates one
point on the intersection line between the lower plate and the
line obtained by prolonging the upper plate. This defines a
fictitious wedge of opening angle $\theta$. We call $x$ the
horizontal and $z$ the vertical coordinate, the third direction,
along which one has translational symmetry, being $y$.
Since the surfaces are locally flat we have $\Delta=1/\ell^2$ as
in the case of the parallel plates, and again the odd reflections
are exactly as in the case of the parallel plates. However now the
even reflections contribute (the notation is the same as in the
parallel plates case, in the even reflections $2u$ means the first
reflection is on the upper plate etc.):
\begin{equation}
\label{eq:PserPend}
P=P_{1u+3u}+P_{2u+2d}+P_{3d+5d}+...\
\end{equation}
where we have grouped the terms with the symbolic notation
$P_{a+b}=P_a+P_b$ when $P_a=P_b$. It is useful to recapitulate
what we have learned about the rank of these contributions:
$P_{1u+3u}$ dominates, $P_{3d+5d}$ is smaller by $\sim 1/16$,
$P_{5u+7u}$ is smaller by $\sim 1/81$, {\it etc.\/} The even
reflections are generically much smaller than the odd reflections,
and vanish as $\theta\to 0$.
The first term in (\ref{eq:PserPend}) is
\begin{equation}
P_{1u+3u}=-2\frac{1}{16\pi^2}\partial_z^2\frac{1}{\ell_1^2(z,x)},
\end{equation}
with $\ell_1=2(x\sin\theta-z/\cos\theta)$, and an overall factor
of $2$ takes into account the identity $P_{1u}=P_{3u}$. Taking the
derivative and then setting $z=0$ we find
\begin{equation}
P_{1u+3u}=-\frac{3}{16\pi^2}\frac{1}{x^4\sin^4\theta\cos^2\theta},
\end{equation}
and integrating from $x_m=(a/\sin\theta-w/2)/\cos\theta$ to
$x_M=(a/\sin\theta+w/2)/\cos\theta$ we find the force per unit
length in the $y$ direction
\begin{equation}
F_{1u+3u}=-\frac{\cos\theta}{32\pi^2\sin^4\theta}\left(\frac{1}{(a/\sin\theta-w/2)^2}-\frac{1}{(a/\sin\theta+w/2)^2}\right).
\end{equation}
Since term by term $F=-\partial {\cal E}/\partial a$ we find the first
term in optical expansion of the Casimir energy ${\cal E}$ (the
arbitrary constant is chosen so that ${\cal E}\to 0$ when $a\to
\infty$) as
\begin{equation}
{\cal E}_{1u+3u}=-\frac{aw{\cos^4 \theta }}
{2{\pi }^2
{\left( 4a^2 -
w^2{\sin^2 \theta }
\right) }^2}
\end{equation}
and from this one obtains the torque
\begin{equation}
{\cal T}_{1u+3u}= \frac{2aw
\left(w^2 -4a^2 \right)
{\cos^3 \theta }
\sin \theta }{{\pi }^2
{\left( 4a^2 -
w^2{\sin^2 \theta }
\right) }^3}
\end{equation}
Analogously we can calculate the contribution to the pressure $P$
of the two reflections paths $2u$ and $2d$. Again the
contributions of the two paths are identical and the result
simplifies to
\begin{equation}
P_{2u+2d}=\frac{2}{8\pi^2}\frac{1}{2}\partial_z^2\frac{1}{\ell_2^2(z,x)},
\end{equation}
and using $\ell_2=2\sqrt{x^2+z^2}\sin\theta$ we find
\begin{equation}
P_{2u+2d}=-\frac{1}{16\pi^2\sin^2\theta\ x^4}
\end{equation}
which integrated from $x_m=(a/\sin\theta-w/2)/\cos\theta$ and
$x_M=(a/\sin\theta+w/2)/\cos\theta$ gives the force along the $z$
axis due to these paths:
\begin{equation}
F_{2u+2d}=-\frac{\cos^3\theta}{48\pi^2\sin^2\theta}\left(\frac{1}{(a/\sin\theta-w/2)^3}-\frac{1}{(a/\sin\theta+w/2)^3}\right).
\end{equation}
This expression can now be expanded for $\theta\ll 1$
(quasi-parallel plates)
\begin{equation}
F_{2u+2d}\simeq-\frac{1}{16\pi^2}\left(\frac{w}{a^4}\theta^2+\frac{5w^3-11wa^2}{6a^6}\theta^4+...\right).
\end{equation}
Notice that this expression vanishes when $\theta\to 0$, as it
should since for parallel plates all the contributions of even
reflections paths vanish.
The next term in the series is $F_{3d+5d}$, whose calculation is
performed in the same fashion. The result is:
\begin{eqnarray}
F_{3d+5d}&=&-\frac{3}{16\pi^2}\frac{\cos^5 2\theta}{\sin^4 2
\theta}\left(\frac{1}{(a/\sin\theta-w/2)^3}-\frac{1}{(a/\sin\theta+w/2)^3}\right),\\
&\simeq&-\frac{1}{16\pi^2}\left(\frac{3w}{16a^4}+\frac{5w^3-48a^2w}{32a^6}\theta^2+...\right).
\end{eqnarray}
We can also present the term given by the 4 reflections paths,
\begin{equation}
F_{4u+4d}=-\frac{\cos^3 2\theta}{48\pi^2\sin^2
2\theta}\left(\frac{1}{(a/\sin\theta-w/2)^3}-\frac{1}{(a/\sin\theta+w/2)^3}\right)
\simeq-\frac{1}{16\pi^2}\left(\frac{w}{4a^4}\theta^2+...\right).
\end{equation}
The terms independent of $\theta$ can be seen to reconstruct the
parallel limit case $F=-(1+1/16+1/81+...)3/16\pi^2a^4$.
Term by term, this series for the force reproduces the series in
Ref.~\cite{opt1}. The series for the energy and the torque agree
as well. The results of the pressure method then coincide with
those of the energy method (as for all the examples analyzed in
this paper). In Ref.~\cite{opt1} we discussed at some length the
predictions of the optical method for the Casimir torsion
pendulum. We will not repeat them here, referring the reader to
that paper for further details.
\subsection{Sphere and Plane}
\label{sec:sphereplane}
The sphere facing a plane is an important example for several
reasons: it has been analyzed theoretically with various exact or
approximate numerical techniques \cite{SandS,Gies03}; it is an
experimentally relevant configuration; the exact solution is
unknown and probably will escape analytical methods for a long
time to come. We have already calculated the optical approximation
to the Casimir energy in Ref.~\cite{opt1} up to $5$ reflections.
In this paper we study this problem for mainly pedagogical
purposes, leaving a more accurate and complete numerical analysis
for the future. We believe it is worth studying this example
because, contrary to the previous two examples, the enlargement
factor plays an important role and moreover we will reanalyze this
example with finite temperature in Section IV B 2.
We calculate the pressure (and by integrating, the force) exerted
on the plate by the sphere which, of course, equals the force
exerted by the plate on the sphere. We start from the qualitative
observation that the rank of the contributions is the same as in
the parallel plates case in the limit $a/R\to 0$. In all the
examples we have analyzed this rank is preserved for any value of
$a/R$. Moreover the ratios of the contributions to the force
$F_{3+5}(a,R)/F_{1+3}(a,R)$, $F_{4}(a,R)/F_{2}(a,R)$ etc.\
decrease quickly as $a/R$ increases, we believe due to the growing
importance of the enlargement factor.
In this paper we calculate analytically the $1s$ term (here $s$
stands for `sphere' and $p$ for `plate') and by using the relation
$P_{1s+3s}\equiv P_{1s}+P_{3s}=2P_{1s}$ proved in Section
\ref{sec:enmomtens} (the notation is the same as in that section)
we are able to include the $3s$ term as well.
Using the expressions for the length and enlargement factor for
the $1s$ path obtained in Ref.~\cite{opt1} we get
\begin{eqnarray}
P_{1s+3s}&=&-2\frac{R}{16\pi^2}\frac{\partial^2}{\partial
z^2}\frac{\Delta^{1/2}_{1s}}{\ell_{1s}}\nonumber\\
&=&-\frac{R}{32\pi^2}\frac{\partial^2}{\partial z^2}
\left( R - {\sqrt{{\left( a + R -
z \right) }^2 + {\rho }^2}}
\right)^{-2}\left({\left( a + R -
z \right) }^2 + {\rho }^2\right)^{-1/2}\bigg|_{z=0}.
\label{eq:P13sphere}
\end{eqnarray}
The final expression for the pressure $P_{1s+3s}$ obtained after
the derivatives are taken is rather long, however the contribution
to the force on the plate, $F_{1s+3s}$ (obtained by integration of
$P_{1s+3s}$ over the infinite plate) is quite simple:
\begin{equation}
\label{eq:F1s3s}
F_{1s+3s}=2\pi\int_0^\infty d\rho \rho P_{1s+3s}=-\frac{\hbar c
R}{8\pi a^3}.
\end{equation}
This is the largest of the contributions and increasing $a/R$
improves the convergence of the series due to the presence of the
enlargement factor, so the asymptotic behavior at large $a/R$
predicted by the optical approximation is that given by this
formula, \emph{i.e.}\ $F\propto R/a^3$ or $E\propto R/a^2$. This
asymptotic law is in accordance with the numerics of
Ref.~\cite{pap1} and the predictions of other semiclassical
methods \cite{SandS}. However, eq.~(\ref{eq:F1s3s}) is in
disagreement with the Casimir-Polder law \cite{CasimirPolder}
which predicts $E\propto R^3/a^4$ for $a\gg R$. This is no great
surprise, since our method is not valid for $a/R\gg 1$, the
semiclassical reflections being corrected and eventually
overshadowed by diffractive contributions
\cite{Keller,SandSdiffr}.
We have calculated the contribution of the two reflections paths
analytically as well. The calculation is more involved than the
one reflection term but a big simplification occurs if one notices
that, for the purpose of taking the second derivative with respect
to $z$ at $z=0$, one can leave the reflection point on the sphere
fixed. We could not prove a similar result for any other
reflection. It is certainly not true for \emph{odd} reflections
but one can conjecture it to be true for \emph{even} reflections.
In this paper we have not calculated the 4 reflection terms and
hence we could not check this conjecture for more than 2
reflections.
And finally, we have calculated the $3p$ (or $sps$) and hence
obtained the $5p$, or $pspsp$, paths contribution $P_{3p+5p}$;
$P_{3p+5p}$ in the parallel plates limit should account for $\sim
1/16$ of the total force. This contribution, unlike the previous
ones, must be calculated partly numerically, mainly because
finding the reflection point on the sphere requires the (unique)
solution of a transcendental equation. This task is achieved much
more quickly by a numerical algorithm than by patching together
the several branches of the analytic solution.
\begin{figure}
\centerline{\hspace{-3cm}\asfigure{pressure-sp1.eps}{550}{12cm}}
\caption{\label{fig:pressure-sp} The magnitude of the total
pressure up to reflection $5p$ in units of $\hbar c/R^4$ as a
function of the radial coordinate on the plate, $\rho/R$. Upward,
or red to blue $a/R=1$, $a/R=0.1$ and $a/R=0.01$. }
\end{figure}
\begin{figure}
\centerline{\hspace{-3cm}\asfigure{pressure-sp132235.eps}{550}{12cm}}
\caption{\label{fig:presspic} Contributions to the pressure in
units of $\hbar c/R^4$ as a function of $\rho/R$, for fixed
$a/R=0.1$. Downward or red to blue, we have $-P_{1s+3s}$,
$-P_{3p+5p}$ and $P_{2+2}$. Although unnoticeable in this figure,
the curve $P_{2+2}$ changes sign at around $\rho/R\simeq 0.4$ (see
Figure \ref{fig:P2negative} for a similar situation).}
\end{figure}
\begin{figure}
\centerline{\hspace{-3cm}
\asfigure{p2becomingnegative.eps}{550}{12cm}}
\caption{\label{fig:P2negative} Contribution of the two reflection
path(s) to the pressure in units of $\hbar c/R^4$ as a function of
$\rho/R$, for fixed $a/R=0.01$. The pressure becomes negative,
showing that the sign of the pressure is not determined by the
number of reflection only.}
\end{figure}
The total pressure is plotted in Fig.~in \ref{fig:pressure-sp}
while the various contributions (keeping in mind that $P_{1s+3s}$
and $P_{3p+5p}$ are negative and $P_{2+2}$ is mainly positive) are
shown in Fig.~\ref{fig:presspic}. Figure \ref{fig:pressure-sp}
reveals some interesting features of the pressure in this
geometry: the total pressure decays very quickly with the distance
as $P\sim \rho^{-\alpha}$: the exponent $\alpha$ seems to depend
upon the distance $a/R$, but for $a/R\leq 0.1$ a good fit is
obtained with $\alpha=6$, in accordance with the asymptotic
expansion of the $1+3$ reflection term Eq.\ (\ref{eq:P13sphere});
by decreasing the distance between the sphere and the plate, the
pressure becomes more and more concentrated near the tip, giving
us reasons to trust our approximation and supporting the use of
the PFA as a first approximation in the limit $a/R\to 0$.
Figure~\ref{fig:presspic} shows the relative importance of the
contributions due to the different paths. As expected the
contribution to the total pressure decreases quite fast by
increasing the number of reflections. In Fig.~\ref{fig:P2negative}
one can also see that the sign of the pressure is not determined
simply by the number of reflections of the underlying optical path
--- as for the contribution to the energy density.
\begin{figure}
\centerline{\hspace{-3cm}\asfigure{f.eps}{550}{12cm}}
\caption{\label{fig:f} The ratio between the optical force up to
the $5p$ reflection and the most divergent term in the PFA, as
defined by eq.~(\ref{def}).}
\end{figure}
By integrating the pressure over the whole plate we obtain the
force $F$. It is useful to factor out the most divergent term of
the force, as predicted by the PFA, so we define the quantity
$f(a/R)$ as
\begin{equation}
\label{def}
F(a)=-\frac{\pi^3 R}{720a^3}f(a/R).
\end{equation}
Since we include only a finite number of reflections it is
convenient to factor out the constant $\zeta(4)/(1+1/16)$ such
that $f$ is normalized with $f(0)=1$. The function $f(a/R)$,
calculated including paths $1s,\ 3s,\ 2,\ 3p$ and $5p$, is plotted
in Figure \ref{fig:f}. When $a/R\to 0$ $f$ is fitted by
\begin{equation}
\label{eq:fsphere}
f(a/R)=1-0.10\ a/R+\Ord{(a/R)^2}\ .
\end{equation}
By comparing to the results of \cite{pap1}
\begin{equation}
\label{eq:fsphereen}
f_{\rm energy}(a/R)=1+0.05\ a/R+\Ord{(a/R)^2}\,
\end{equation}
there is the difference in the sub-leading term.
By neglecting the $5s+7p$ reflection paths (which in the parallel
plates case contribute $\sim 2\%$ of the total force) we can only
assert that the functions $f$ in (\ref{eq:fsphere}) and
(\ref{eq:fsphereen}) represent the optical approximation with an
error of $2\%$. When plotted on the whole range of $a/R$ where the
optical approximation is to be trusted the pressure and energy
method curves never differ more than $2\%$. However there is no
such a bound on the sub-leading term which, on the contary,
depends on the higher reflections contributions which have not
been included in this calculations.\footnote{For example consider
that including only $1s,~3p$, and 2 and reflections would have
given a sub-leading term $-0.16a/R$ instead of $-0.10a/R$ in
Eq.~(\ref{eq:fsphere}). The sub-leading term then changes of
$50\%$ by adding the $3s+5p$ reflection terms which contributes
only up to $8\%$ of the total.} With the terms calculated at this
point, we cannot make a precise statement about the sub-leading
term. We can however safely say that the subleading term $a/R$
coefficient is quite small and our method disagrees with the PFA
prediction $-0.5a/R$. The sphere opposite plate is such an
experimentally relevant geometry that further, more accurate
studies need to be performed to compare with experimental data.
In conclusion, the lessons to be learned from this example are
two: 1) The calculations with the pressure method are even quicker
and simpler than the energy method and sometimes can give analytic
results for non-trivial geometries and 2) the sub-leading terms
must be compared only between calculations performed with the same
accuracy.\footnote{AS would like to thank M.\ Schaden and S.\
Fulling for conversations on this point during the workshop
`Semiclassical Approximations to Vacuum Energy' held at Texas A \&
M, College Station, TX, January 2005. The concerns about the
errors to be associated with the optical, semiclassical or
proximity force approximation is still open to debate and is
strictly connected to one of the most challenging open problems in
spectral theory \emph{i.e.\ }how to go beyond the semiclassical
approximation to the density of states of a positive Hermitian
operator.}
\section{Casimir Thermodynamics.}
\setcounter{equation}{0} \label{sec:thermo}
As measurements of Casimir forces increase in accuracy they become
sensitive to thermal effects. The natural scale for Casimir
thermodynamics is a distance, $\tilde\beta=\hbar c/\pi T$, which
at room temperature is about 2.5 microns. [To avoid confusion
with the wave number $k$, we set Boltzmann's constant equal to
unity and measure temperature in units of energy. We continue to
keep $\hbar$ and $c$ explicit.] So, assuming the corrections are
of $\Ord{(a/\tilde\beta)^\alpha}$, depending on the value of
$\alpha$ thermal effects might be expected between the $10\%$ (for
$\alpha=1$) and $0.3\%$ (for $\alpha=4$, the standard parallel
plates result) level for Casimir force measurements on the micron
scale. In open geometries, like the sphere and plane, even longer
distance scales are probed by Casimir effects, and this gives rise
to interesting changes in the temperature dependence of the
Casimir free energy in comparison with the case of parallel
plates\cite{MT}. The optical approximation is well suited for
discussion of thermodynamics since the thermodynamic observables,
like the Casimir energy, can be expressed in terms of the
propagator. Here we consider again a non-interacting, scalar
field outside rigid bodies on which it obeys Dirichlet boundary
conditions.
Before entering into a technical discussion of temperature
effects, it is useful to anticipate one of our central results
which follows from qualitative observations alone. As $T\to 0$
the temperature effects probe ever longer distances. Even at room
temperature the natural thermal scale is an order of magnitude
larger than the separation between the surfaces in present
experiments (see Ref.~\cite{expt1}). Since long paths contribute
little to the Casimir force, we can be confident that thermal
effects vanish quickly at low temperature. However, the leading
$T$-dependence at small $T$ comes from regions beyond the range of
validity of the optical (or any other) approximation, so we are
unable to say definitively how they vanish for geometries where no
exact solution is possible ({\it i.e.\/} other than infinite
parallel plates).
This section is organized as follows: First we discuss the free
energy and check our methods on the parallel plates geometry; then
we discuss the temperature dependence of the pressure, which we
apply to the sphere and plate case. Finally we discuss the
difficulties associated with the $T\to 0$ limit.
\subsection{Free Energy}
The free energy is all one needs to calculate both thermodynamic
corrections to the Casimir force and Casimir contributions to
thermodynamic properties like the specific heat and pressure.
However like the Casimir \emph{energy}, Casimir contributions to
the specific heat, pressure, {\it etc.\/}, are cutoff dependent
and cannot be defined (or measured) independent of the materials
which make up the full system. So we confine ourselves here to
the thermal corrections to the Casimir force. The problem of
parallel plates has been addressed before and our results agree
with those\cite{MT}.
\subsubsection{Derivation}
We start from the expression of the free energy for the scalar
field as a sum over modes
\begin{eqnarray}
{\cal F}_{\rm
tot}&=&-\beta^{-1}\sum_n\ln\left(\frac{e^{-\beta\frac{1}{2}\hbar\omega_{n}}}
{1-e^{-\beta(\omega_{n}-\mu)}}\right),\nonumber\\
&=&\beta^{-1}\sum_n\ln\left(1-e^{-\beta(\hbar\omega_{n}-
\mu)}\right)+\sum_{n}\frac{1}{2}\hbar\omega_n,\nonumber\\
&\equiv&{\cal F}+{\cal E},
\end{eqnarray}
where $\mu$ is the chemical potential, and the last term is the
Casimir energy, or the free energy at zero temperature, since
${\cal F}=0$ for $T=0$. The Casimir energy ${\cal E}$, being independent of
the temperature, does not contribute to the thermodynamic
properties of the system. It however does contribute to the
pressures and forces between two bodies. The force between two
bodies, say $a$ and $b$, is obtained by taking the gradient of the
free energy with respect to the relative distance $\vec{r}_{ab}$
\begin{equation}
\vec{f}_{ab}=-\vec{\nabla}_{ab} {\cal F}.
\end{equation}
At $T=0$ we recover the familiar result $\vec{f}=-\vec{\nabla} {\cal E}$.
Next we turn the sum over modes into a sum over optical paths.
Following the same steps that led from Eq.~(\ref{eq:T00}) to
Eq.~(\ref{eq:T00opt}) we obtain
\begin{equation}
{\cal F}=\beta^{-1}\int d^Nx\ \int_0^\infty dk\
\rho(x,k)\ln\left(1-e^{-\beta (\hbar\omega(k)-\mu)}\right).
\end{equation}
where $\rho(x,k)$ is given by Eq.\ (\ref{eq:rhoxk}). By
specializing to a massless field in $3$ dimensions with zero
chemical potential (to mimic the photon field), and substituting
the optical approximation for the propagator Eq.~(\ref{eq:Gopt3}),
we obtain the sum over paths
\begin{equation}
\label{eq:Fropt}
{\cal F}\equiv\sum_{r=0}{\cal F}_{r}
=\sum_{r}(-1)^{r}\frac{1}{2\pi^{2}\beta}\int_{{\cal D}_r}d^3x\ \Delta_r^{1/2}
\int_0^\infty
dk \ k \sin(k\ell_r) \ln\left(1-e^{-\beta\hbar c
k}\right).
\end{equation}
Here the term ${\cal F}_{0}$, the direct path, gives the usual free
energy for scalar black body radiation. Using the values for the
direct path, we have $\Delta_0=1/\ell_0^2$ and
$\ell_{0}=|x'-x|\to 0$ when taking $x'\to x$. We get the familiar
textbook expression
\begin{equation}
{\cal F}_{0}=V\int_0^\infty dk\ \frac{k^2}{2\pi^2}\
\beta^{-1}\ln\left(1-e^{-\beta\hbar
ck}\right)=-\frac{\pi^2}{90}\frac{VT^4}{(\hbar c)^3},
\label{eq:F0classic}
\end{equation}
where $V$ is the (possibly infinite) volume outside the bodies.
The general term ${\cal F}_{r}$ associated with the path $r$ is
calculated by performing the $k$ integral in Eq.~(\ref{eq:Fropt}):
\begin{equation}
\label{eq:Fr}
{\cal F}_r=(-1)^{r+1}\frac{\hbar c}{2\pi^{2}}\int_{{\cal D}_r}d^3x
\Delta_r^{1/2}\frac{1}{2\ell_r^3} \left[-2+ \tilde\ell_r
\left({\rm coth} \tilde\ell_r +\tilde\ell_r {\rm
csch}^2\tilde\ell_r \right)\right]
\end{equation}
where $\tilde\ell_{r}=\ell_{r}\pi T/\hbar c =
\ell_{r}/\tilde\beta$ measures the path length relative to the
thermal length scale.
Eq.~(\ref{eq:Fr}) is the fundamental result of this section and
gives a simple, approximate description of thermal Casimir effects
for geometries where diffraction is not too important. There are
no divergences in any of the ${\cal F}_{r}$, ultraviolet or otherwise,
even for the direct path (as we saw in eq.\ (\ref{eq:F0classic}))
and the first reflection path. All the ultraviolet divergences are
contained in the Casimir energy ${\cal E}$. Indeed, by expanding the
integrand of equation (\ref{eq:Fr}) at short distances,
\emph{i.e.} $\tilde\ell_{r}\ll 1$, we obtain
\begin{equation}
\label{eq:frsmallt}
\Delta_r^{1/2}
\frac{1}{2\ell_r^3}\left[-2+\tilde\ell_r \left({\rm
coth} \tilde\ell_r +\tilde\ell_r {\rm
csch}^{2}\tilde\ell_r \right)\right]
\simeq
\Delta_r^{1/2}\frac{1}{\tilde{\beta}^3}
\left[\frac{1}{45\tilde{\beta}}
\ell_r-\frac{4}{945\tilde{\beta}^3}\ell_r^3+...\right].
\end{equation}
Only the 1-reflection path length can go to zero to generate a
divergence. For this contribution $\Delta_{r}$ diverges like
$1/\ell_{r}^2$ as $\ell_{r}\to 0$, however this is compensated
by the $\ell_r$ term in (\ref{eq:frsmallt}) so the expression is
finite and then integrable.
To check for infrared divergences notice that at large distances,
$\tilde\ell_{r}\gg 1$, the integrand of (\ref{eq:Fr}) goes to
$\sim \Delta_{r}^{1/2}/\ell_{r}^{2}$. For an infinite flat plate
the $\Delta_{r}\sim 1/z^{2}$, where $z$ is the normal coordinate
to the plate, and the integral is hence $\sim dz/z^{3}$ at large
$z$. For finite plates the domain of integration is finite and for
curved plates the enlargement factor falls even faster than
$1/\ell^2$, and the integral remains convergent.
Since the integral converges in both the infrared and ultraviolet,
it is safe to estimate the important regions of integration by
naive dimensional analysis. This leads to the conclusion that
\emph{The paths that dominate the temperature dependence of the
Casimir force have lengths of order the thermal length
$\tilde\beta$}. High temperature implies short paths. Very low
temperatures are sensitive to very long paths. Long paths involve
both paths experiencing many reflections, which are sensitive to
the actual dynamics at and inside the metallic surface, or paths
making long excursions in an open geometry, which are sensitive to
diffraction. Either way, low temperatures will present a
challenge.
\subsubsection{Parallel Plates}
We know that in the limit of infinite, parallel plates the optical
approximation to the propagator becomes exact. Hence our method
gives another way to calculate the free energy of this
configuration of conductors. It is convenient to study this
example to check against known results and to prepare the way for
a study of the $T\to 0$ limit.
We recall that for this configuration the expression for the
enlargement factor is $\Delta=1/\ell^2$ and the lengths are given
by $\ell_{2n}=2na$ (where $a$ is the distance between the plates)
and $\ell_{2n+1,\ u}=2(a-z)+2na$, $\ell_{2n+1,\ d}=2z+2na$, the
notation being the same as in Section \ref{sec:parplates}, should
at this point be familiar to the reader.
As in the zero temperature case it is useful consider even and odd
reflection contributions separately and as for the zero
temperature case, the sum over odd reflections turns into an
integral over $z$ from $0$ to $\infty$
\begin{equation}
{\cal F}_{{\rm odd}}=\sum_{n=0}^\infty {\cal F}_{2n+1, d}+{\cal F}_{2n+1, u}=\frac{\hbar c}{2\pi^2\tilde{\beta}^3}S\int_0^\infty
dx \frac{1}{2x^4}\left[-2+x({\rm
coth} x +x\ {\rm csch}^2 x )\right],
\end{equation}
where $x=2z/\tilde{\beta}$ and $S$ is the area of the plate. The
definite integral can be easily performed numerically and its
value is $\nu=0.06089...$,
\begin{equation}
\label{eq:foddpp}
{\cal F}_{{\rm odd}}= 2\frac{\hbar
c}{4\pi^2\tilde{\beta}^3}S\nu=\frac{\pi T^3}{2(\hbar c)^2}S\nu
\end{equation}
which is independent of the separation, $a$, and therefore does
not contribute to the force.
Let us turn now to the even reflection paths. They have constant
length $2na$, so the volume integral simply yields the volume
between the surfaces $v=Sa$. We already calculated the
zero-reflection term ${\cal F}_0$ in Eq.~(\ref{eq:F0classic}). The
remaining even reflection contributions (2,4,6,...\ reflections)
${\cal F}_{{\rm even}, r\geq 2}$ can be written as an infinite sum
\begin{equation}
\label{eq:Freven}
{\cal F}_{{\rm even}, r\geq 2}=-2\frac{\hbar c}{2\pi^2}Sa\frac{1}{\tilde{\beta}^4}
\sum_{n=1}^\infty\frac{1}{2x_n^4}\left[-2+x_n\left({\rm coth}
x_n+x_n{\rm csch}^2 x_n\right)\right]
\end{equation}
where $x_n=2na/\tilde{\beta}\equiv n\tau$ (this defines the
dimensionless temperature $\tau$) and we have introduced an
overall factor of two to take into account the multiplicity of the
paths. Thus the total free energy for parallel plates is the sum
of ${\cal F}_{0}$ (eq.~(\ref{eq:F0classic}) and the results of
eqs.~(\ref{eq:foddpp}) and (\ref{eq:Freven})),
\begin{equation}
\label{eq:pptot}
{\cal F}_{\parallel} =-\frac{\pi^4}{90}\frac{VT^4}{(\hbar c)^3} +\frac{\pi T^3}{2(\hbar c)^2}S\nu - \frac{\hbar c}{ \pi^2\tilde {\beta}^4}Sa \sum_{n=1}^\infty
\frac{1}{2x_n^4}\left[-2+x_n\left({\rm coth}
x_n+x_n{\rm csch}^2 x_n\right)\right]
\end{equation}
It is not possible to rewrite ${\cal F}_{\parallel}$ in a closed form,
but the sum is easy to compute numerically and the high and low
temperature expansions are easy to obtain analytically. At high
temperatures (and fixed $a$) $\tau\to\infty$, and the summand
$g(n)$ in eq.~(\ref{eq:pptot}) falls rapidly enough with $n$
\begin{equation}
g(n)=\frac{1}{2(\tau n)^4}\left[-2+(\tau n)\left({\rm coth}
(\tau n)+(\tau n){\rm csch}^2 (\tau
n)\right)\right]=\frac{1}{2(\tau n)^4}\left[-2+\tau
n\right]+\Ord{e^{-\tau n}},
\end{equation}
that the limit may be taken under the summation, with the result,
\begin{equation}
{\cal F}_{{\rm even},r\geq 2}\simeq-\frac{\hbar
c}{\pi^2\tilde{\beta}^4}Sa\sum_{n=1}^\infty\left[
\frac{1}{2n^3\tau^3}-\frac{1}{\tau^4n^4}\right] =-\frac{\zeta(3)}{16\pi
a^2}ST+ \frac{\pi^2\hbar c}{1440a^3}S.
\end{equation}
Notice that the second term cancels the even paths contribution to
the Casimir energy. Hence the final expression for the high $T$
expansion of the free energy is particularly simple,
\begin{equation}
\label{eq:cFtothight}
{\cal F}_{\rm tot}= {\cal F} + {\cal E} = -\frac{\pi^{2}}{90\hbar c}VT^{4}+\nu\frac{\pi}{2(\hbar c)^{2}}ST^{3}
-\frac{\zeta(3)}{16\pi a^{2}}ST+\Ord{e^{-\pi T a/\hbar
c}}.
\end{equation}
The first term is usual black body contribution to the bulk free
energy. It does not contribute to the force. The second term is
also independent of $a$ and does not give rise to any force. The
third term instead gives the thermal Casimir force. Notice that
$\hbar c$ has disappeared from this expression. Called the
``classical limit'', this high temperature behavior has been noted
before and some early results are even due to Einstein (in
\cite{Milonni} pg.\ 2; see also \cite{Feinberg}). In the next
section, after the thermal corrections to the pressure are
calculated, we show how to extend this result to other geometries.
Note some interesting features of the $T\to\infty$ limit: First,
the sum over paths converges like the sum of $(1/n)^{3}$ as
indicated by the appearance of $\zeta(3)$. While slower than the
$T=0$ convergence, it is still rapid enough to obtain a good
approximation from low reflections. Second, note that the
$T\to\infty$ problem in 3-dimensions corresponds exactly to a
$T=0$ problem in 2-dimensions. This is an example of the familiar
dimensional reduction expected as $T\to\infty$. We can give a
short proof of this result. Let us first write:
\begin{equation}
F=-\frac{1}{\beta}\log Z
\end{equation}
where $Z$ is the partition function. We need to evaluate $Z$ to
the lowest order in $\beta$ when $\beta\to 0$. The thermal scalar
field theory can be written as a free theory on the cylinder
$\mathbb{R}^3\times [0,\beta)$. For $\beta\to 0$ the dynamics
along the thermal coordinate is frozen in the ground state, with
energy $E_0=0$, where $\phi$ does not depend on the thermal
coordinate. The partition function $Z$ is now
$Z=Z_{3}+\Ord{e^{-\beta E_1}}$ where $E_1$ is the first excited
state $E_1\propto 1/\beta^2$ and $Z_{3}$ is the partition function
of the remaining three-dimensional problem in $\mathbb{R}^3$. If
the conductors geometry is symmetric along one spatial coordinate,
say $x$ (in the parallel plates problem we have two of these
directions, $x$ and $y$) this can now be interpreted as an
Euclideanized time variable extending from $0$ to $L_x/c$. So we
will write $Z_3=Z_{2+1}=e^{-\frac{1}{\hbar}{\cal E}_2L_x/c}$ where
${\cal E}_2$ is the Casimir energy of the 2 dimensional problem of two
lines of length $L_y$, distant $a$. The free energy $F$ is then:
\begin{equation}
\label{eq:F21}
F=-\frac{1}{\beta}\log Z\simeq-\frac{1}{\beta}\log
Z_{2+1}=T\frac{1}{\hbar}\frac{L_x}{c}{\cal E}_{2}=-TL_xL_y\frac{\zeta(3)}{16\pi^2a^2}.
\end{equation}
Since $S=L_xL_y$ This is exactly the $a$-dependent term in
eq.~(\ref{eq:cFtothight}). If the geometry is not translational
invariant then we can only say from eq.~(\ref{eq:F21}) that the
free energy is linear in $T$ (since $Z_{2+1}$ is independent of
$\beta$). Later, by using the optical approximation we will find
an explicit analytic expression valid also for non-symmetric,
smooth geometries.
For low temperatures, $\tau\to 0$, the terms in the $n$-sum in
eq.~(\ref{eq:pptot}) differ very little from each other so we can
use the Euler-McLaurin formula\cite{AS},
\begin{equation}
\sum_{n=1}^\infty g(n)=\int_0^\infty dx\
g(x)-\frac{1}{2}g(0)-\frac{1}{12}g'(0)+... \ =
\frac{\nu}{\tau}-\frac{1}{90}+\Ord{\tau} .
\label{eq:em}
\end{equation}
Substituting into eq.~(\ref{eq:pptot}) we find that the first term
in eq.~(\ref{eq:em}) cancels the sum over odd reflections (the
second term in eq.~(\ref{eq:pptot})) and that the second term in
eq.~(\ref{eq:em}) combines with ${\cal F}_{0}$ to give a very simple
result,
\begin{equation}
\label{eq:excludeV}
{\cal F}_{\rm tot}={\cal E}-\frac{(V-Sa)\pi^2T^4}{90(\hbar c)^3}.
\end{equation}
at low temperatures. This has a simple physical interpretation:
the typical thermal excitations of the field at low temperature
have very long wavelengths, it is hence energetically inconvenient
for them to live between the two plates. As a result the only
modification of the $T=0$ result is to exclude from the standard
black body free energy the contribution from the volume between
the plates. One could imagine measuring this effect as a
diminished heat capacity for a stack of conducting plates inside a
cavity.
The low temperature result, eq.~(\ref{eq:excludeV}), is
deceptively simple. Its simplicity obscures an underlying problem
with the $T\to 0$ limit. We postpone further discussion until we
have explored the temperature dependence of the pressure. Suffice
it to say for the moment, that eq.~(\ref{eq:excludeV}) probably
does not apply to realistic conductor with finite absorption,
surface roughness, and other non-ideal characteristics.
\subsection{Temperature dependence of the pressure}
\label{sec:PT}
In this section we will obtain the temperature dependence of the
pressure within our approximation and apply it to a preliminary
study of the sphere and plate case. To begin, we calculate the
thermal average of an operator ${\cal O}$ quadratic in the real scalar
field $\phi$. The average of a generic operator ${\cal O}$ is given by
the trace over a complete set of eigenstates $\ket{\Psi_\alpha}$
of the Hamiltonian weighted by a Boltzmann factor:
\begin{equation}
\label{eq:tavg}
\tavg{{\cal O}}=\sum_\alpha e^{-\beta
{\cal E}_\alpha}\bra{\Psi_\alpha}{\cal O}\ket{\Psi_\alpha}.
\end{equation}
After some algebra we find
\begin{eqnarray}
\tavg{{\cal O}}&=&\sum_{j}{\cal O}_{j}\tavg{2n_{j}+1}\nonumber\\
&=&\sum_{j}{\cal O}_{j}\frac{1+e^{-\beta E_j}}{1-e^{-\beta E_j}}
\end{eqnarray}
where $\tavg{}$ denotes the thermal average, $j$ labels the normal
modes $\psi_j$ (cf.~Section II), $n_j$ is the occupation number of
the mode $j$ and $E_j$ its energy. The quantities ${\cal O}_j$ are read
from the decomposition of the diagonal part of the operator ${\cal O}$
written as ${\cal O}_{\rm diag}=\sum_j {\cal O}_j(a_j^\dag a_j+a_j
a^\dag_j)$ where $a_j$ is the annihilation operator of the mode
$j$.
The ${\cal O}_j$ for the pressure can be read easily from the analysis
in Section \ref{sec:enmomtens}:
\begin{equation}
P_j=\lim_{x'\to
x\in{\cal S}}\frac{1}{4E_{j}}\partial'_{\vec{n}}\partial_{\vec{n}}\psi_j(x')\psi_j(x)
\end{equation}
So we can write the pressure on the plate at non-zero temperature
as
\begin{eqnarray}
\label{eq:temppress}
P(x\in{\cal S})&=&\lim_{x'\to
x}\sum_j\frac{1}{4E_{j}}\partial'_n\partial_n\psi_j(x')\psi_j(x)\left(\frac{1+e^{-\beta
E_{j}}}{1-e^{-\beta E_{j}}}\right)\nonumber\\
&=& \lim_{x'\to x}\partial'_{\vec{n}}\partial_{\vec{n}}\int_0^\infty dk
e^{-k/\Lambda} \frac{k}{2\pi E(k)}\Im\
G(x',x,k)\left(\frac{1+e^{-\beta
E(k)}}{1-e^{-\beta E(k)}}\right)\nonumber \\
&=&\Im \int_0^\infty dk e^{-k/\Lambda}\frac{k}{2\pi
E(k)}\frac{1}{2}\partial^2_{\vec{n}}\ G(x,x,k)\left(\frac{1+e^{-\beta
E(k)}}{1-e^{-\beta E(k)}}\right)
\end{eqnarray}
where we have used Eq.~(\ref{eq:trick1}).
Next we introduce the optical approximation for the propagator and
limit ourselves to massless scalars $E(k)=\hbar c k$. The
discussion of the divergences parallels that of Section
\ref{sec:enmomtens} and needs not be repeated here. We remove
$P_0$ and $P_1$ and leave all the finite contributions $r\in{\cal R}$.
The optical approximation for the pressure exerted by a massless
scalar field reads
\begin{eqnarray}
\label{eq:Ptemp}
P(x)&=&\sum_{r\in{\cal R}}(-1)^{n_r}\partial^2_{\vec{n}}\frac{\Delta_r^{1/2}}{16\pi^2}\left[\frac{1}{\tilde{\beta}}
\coth\left(\ell_r/\tilde{\beta}\right)\right].
\end{eqnarray}
where it is understood that the zeroth and first reflection terms,
which contribute to the pressure on each surface individually, but
not to the force between surfaces, have been dropped.
Before applying this to the sphere and plate problem, let us again
look at the limiting behavior as $T\to\infty$ and $T\to 0$, and
draw some conclusions independent of the detailed geometry. First
consider $T\to\infty$. The shortest paths in the sum in
eq.~(\ref{eq:Ptemp}) are of order $a$, the intersurface
separation. [Remember that the optical approximation is accurate
as long as the important paths are short compared to $R$, a
typical radius of curvature of the surfaces.] At high $T$ we can
take the $\tilde\beta\to 0$ limit under the sum over reflections
since the resulting sum still converges. Therefore low
reflections dominate, and we can see, retrospectively, that the
high temperature approximation applies when $\tilde\beta/a\to 0$.
So as $T\to\infty$,
\begin{equation}
\label{eq:highT}
P=\sum_r(-1)^{n_r}\partial^2_{\vec{n}}\frac{\Delta_r^{1/2}}{16\pi^2}
\left[\frac{1}{\tilde{\beta}}+\Ord{\frac{1}{\tilde{\beta}}e^{-\ell_r/\tilde{\beta}}}\right].
\end{equation}
This limit has been called (it has been previously found for the
parallel plates case) the ``classical limit''
\cite{Milonni,Feinberg,MT}, since the final expression for high
temperatures, reinserting $\hbar$ and $c$,
\begin{equation}
\label{eq:classicallimit}
P\simeq\sum_r(-1)^{n_r}\partial^2_{\vec{n}}\frac{\Delta_r^{1/2}}{16\pi}T
\end{equation}
is independent of $\hbar$ and $c$ apart from exponentially small
terms. This expression amounts in neglecting the 1 in the
expression $\tavg{2n_j+1}$, corresponding to normal ordering or
neglecting the contribution of the vacuum state.
At low temperatures, $\tilde\beta\to\infty$, it is not possible to
interchange the limit with the sum. The relevant quantity is
$\frac{1}{\tilde\beta}\coth(\ell_{r}/\tilde\beta)$, which goes
like
\begin{equation}
\label{eq:lowT}
\frac{1}{\tilde\beta}\coth\left(\frac{\ell_{r}}{\tilde\beta}\right)=
\frac{1}{\ell_r} + \frac{\ell_r}{3\,{\tilde{\beta} }^2} -
\frac{\ell_r^3}{45\,{\tilde{\beta} }^4} +
\Ord{\frac{\ell_r^5}{\tilde{\beta}^4}}
\end{equation}
as $\tilde\beta\to\infty$. The first term yields the familiar
$T=0$ expression. The others would give divergent contributions
because of the factors of $\ell_{r}$ in the numerators (even after
the inclusion of the enlargement factor $\Delta_r$). Of course the
sum over reflections of the \emph{difference},
$\frac{1}{\tilde\beta}
\coth(\ell_{r}/\tilde\beta)-\frac{1}{\ell_{r}}$, converges to zero
as $\tilde\beta\to\infty$, so thermal corrections definitely
vanish for any geometry as $T\to 0$ as expected. Once again we
relegate more detailed consideration of the $T\to 0$ limit to a
later subsection.
\subsubsection{Sphere and plate}
\label{sec:sphplateT}
In this section we calculate the pressure and total force for the
configuration of a sphere facing a plane at non-zero temperature
within $5p$ reflections. The optical approximation should be
accurate if the important paths are short compared to $R$, the
radius of the sphere. On the other hand the thermal corrections
to the force are sensitive to paths with lengths of order
$\tilde\beta$. So we must have $R\gg \tilde\beta$ and $R \gg a$
in order to obtain reliable results from the optical
approximation. Fortunately this is a region of experimental
interest: present experiments use, for example, $a\approx 0.5\mu
m$, $R\approx 100\mu m$, and at room temperature,
$\tilde\beta\approx 2.5\mu m$. In this regime the optical
approximation should give a good description of the thermal
corrections to the force between perfectly reflective, perfectly
smooth conductors.
\begin{figure}
\centerline{\asfigure{p13temp.eps}{650}{12cm}}
\caption{\label{fig:p13temp} The $\rho$ dependence of the $1s+3s$
contribution to the pressure $P_{1s+3s}$ for the sphere and the
plate in units of $\hbar c/R^4$ for various temperatures. Two
effects must be noticed. The top 3 curves (in blue) show the
high-temperature region where the pressure is proportional to $T$
(notice the logarithmic scale). The two lower curves (in orange
and red) show the low-temperature region when increasing the
temperature changes the asymptotic behavior of $P$ for large
$\rho$ (\emph{i.e.\ }$\rho\gtrsim \tilde\beta$) while for small
$\rho$ the behavior reduces to the zero-temperature limit.}
\end{figure}
The expression for the pressure is given by Eq.~(\ref{eq:Ptemp}),
the enlargement factors and lengths are the same as in the $T=0$
case. By applying Eq.~(\ref{eq:Ptemp}) to the $1s+3s$ paths we
find the results in Figure \ref{fig:p13temp}. Notice that at high
temperatures increasing the temperature essentially scales the
whole plot proportionally to $T$. The force is then linearly
dependent on the temperature (this is the `classical limit'
already discussed in Section \ref{sec:PT}). More details are given
in the caption of Figure \ref{fig:p13temp}.
A dimensionless function $f(a/R,\tilde{\beta}/R)$ can again be
defined by rescaling the total force $F$ to extract the leading
term as $a\to 0$. The limiting behavior $a\to 0$ is not affected
by temperature effects so we stick to the old definition for $f$:
\begin{equation}
\label{def:ftherm}
F(a,\tilde{\beta},R)=-\frac{\hbar c \pi^3 R}{720
a^3}f(a/R,\tilde{\beta}/R).
\end{equation}
In Figure \ref{fig:fT} we present $f$ (up to 5 reflections) for 5
different values of $\tilde{\beta}/R$ (we choose 1, 1/2, 1/4, 1/8
and 1/16 recognizing that $\tilde\beta\sim 1$ strains the limits
of our approximations) and varying $a$. Notice that in a
neighborhood of $a/R=0$, shrinking as $\tilde\beta/R$ increases,
the function $f$ is very well approximated by the $T=0$ form,
already discussed in Section \ref{sec:sphereplane}, $f(a/R)\simeq
1-0.1 a/R$. It is not useful to study the derivative
$A(\tilde{\beta}/R)=\partial f(x,\tilde{\beta})/\partial x$ as
$x=a/R\to 0$ since this will take the constant value predicted by
the zero temperature analysis, or $-0.1$ in this approximation,
for any value of the temperature we choose.
It is also clear from the previous discussions leading to equation
(\ref{eq:highT}) that in the opposite regime, for
$a/\tilde{\beta}\gg 1$, we must have $F\propto R / a^2
\tilde{\beta}=R T/a^2$ (the `classical limit'). In fact, the first
term in the high temperature expansion (\ref{eq:highT}) integrated
over $\rho$ converges and gives a finite force linear in $T$. For
this problem, the first term in the reflection expansion for high
temperatures can even be calculated analytically:
\begin{equation}
F_{1s+3s}=-\hbar c\frac{R}{8\pi
a^2\tilde{\beta}}+\Ord{e^{-R/\tilde\beta}}\simeq-\hbar c\frac{R}{8
a^2} T.
\end{equation}
\begin{figure}
\centerline{\hspace{-3cm}\asfigure{fT132235.eps}{650}{12cm}}
\caption{\label{fig:fT} The function $f(a/R,\tilde{\beta}/R)$ as a
function of $a/R$ for $\tilde{\beta}/R$ (from red to violet or
down up) $=1,1/2,1/4,1/8,1/16$. $f(0)\simeq 0.98$ since we are
summing only up to reflection $5p$. The two lowermost curves, red
and orange ($\tilde{\beta}=1,1/2$) superpose almost exactly.}
\end{figure}
Unfortunately there is no such simple closed expression for higher
reflection terms (nor for this first term at arbitrary $T$).
However, if one believes that the rank of contributions is similar
to the parallel plates case one should feel safe to say that this
truncation captures the optical approximation within a
$\zeta(3)-1\simeq 20\%$. Hence our statements are at least
\emph{qualitatively} correct.
This expression for the force gives a prediction for the function
$f$, defined in Eq.\ (\ref{def:ftherm}). At this level of accuracy
($1s+3s$ reflection) and for $a/\tilde{\beta}\gtrsim 1$, apart for
exponentially small terms in the temperature expansion we have
\begin{equation}
\label{eq:highTf13}
f_{1s+3s}\simeq\frac{90}{\pi^4}\frac{a}{R}\frac{R}{\tilde{\beta}}
\end{equation}
which grows linearly in $a/R$ and is (interestingly enough)
independent of $R$. This is evident in Fig.\ \ref{fig:fT} for the
curves with $\tilde{\beta}=1/8,\ 1/16$. For higher $\tilde{\beta}$
the linear growth starts at higher values of $a$ not shown in
Fig.\ \ref{fig:fT}. Moreover the exponential accuracy manifests
itself in the sudden change of behavior from $f\simeq 1-0.1a/R$ to
$f\propto a/\tilde{\beta}$.
It is quite easy to extract a universal prediction from this data,
whatever the definitive numbers are, after the sum over optical
paths is carried to sufficiently high order: \emph{for any
non-zero temperature the function $f(a/R)$ will deviate from his
zero-temperature behavior at $a\gtrsim\tilde{\beta}\sim\hbar c/T$.
The deviation will be in the upward direction, increasing the
attractive force between the bodies.} Eventually, for sufficiently
large distances, the high temperature behavior given by eq.\
(\ref{eq:classicallimit}) (or (\ref{eq:highTf13}) for the
sphere-plane problem) will be recovered.
\subsection{Thermal corrections at low temperatures}
The preceding examples have made it clear that in the language of
the optical approximation, thermal corrections at low temperature
arise from very long paths, $\ell_{r}\sim \tilde\beta$. This can
be seen from the general form of the free energy,
eq.~(\ref{eq:Fr}), or in the attempt to take the
$\tilde\beta\to\infty$ limit under the summation in
eq.~(\ref{eq:Ptemp}), which fails because of the expansion,
eq.~(\ref{eq:lowT}). Here we examine this non-uniformity more
carefully in general and in particular for the parallel plate
case, where all the expressions are available. We then attempt to
draw some conclusions about the magnitude of corrections at low
temperature and the possibility of calculating them reliably in an
model that idealizes the behavior of materials.
We return to eq.~(\ref{eq:temppress}), which gives the exact
expression for the pressure, and separate out the thermal
contribution,
\begin{equation}
\label{eq:dPtot}
P(T)-P(0)\equiv\delta P=\Im\int_0^\infty
dk\frac{1}{2\pi}\partial^2_{n'n}{\cal G}(x',x,k)2\frac{e^{-\beta \hbar
c k}}{1-e^{-\beta \hbar c k}},
\end{equation}
still exact. Expanding the denominator in a geometric series, we
find
\begin{equation}
\delta P = \frac{1}{\pi}\Im\sum_{m=1}^{\infty}\int_0^\infty dk
\partial^2_{n'n}{\cal G}(x',x,k) e^{-m\beta \hbar c k}.
\end{equation}
Each term in the sum is a Laplace transform of the Greens
function. Clearly, as $\beta\to\infty$ the frequencies that
dominate this integral are $\propto 1/\beta\sim T$.
What are the low frequency contributions to ${\cal G}(x',x,k)$? In the
ideal case of infinite, perfectly conducting, parallel plates,
there is a gap in the spectrum at low $k$: $k\ge\frac{\pi}{a}$.
However \emph{in realistic situations} the plates are finite
and/or curved, the geometry is open, and there is no gap in the
spectrum. The low-$k$ part of the spectrum is sensitive to the
global geometry, including edges and curvature, and to the low
frequency properties of the material. If the conditions are close
to the ideal, the contributions to $\delta P$ from small $k$ may
be small. However as $T\to 0$, they dominate. We conclude that
the $T\to 0$ behavior of $\delta P$ cannot be calculated for
realistic situations.
The optical approximation does not take account of diffraction,
and cannot accurately describe the $T\to 0$ limit. Nevertheless
it is interesting to see how it fails, since this sheds light on
the problem in general. Substituting the optical expansion for the
Greens function (replacing
$\partial^2_{n'n}\to\frac{1}{2}\partial^2_z$ and setting
$\hbar=c=1$) we find
\begin{eqnarray}
\delta P&=&-\sum_{m=1}^{\infty}\sum_{r\geq
1}\frac{1}{8\pi^2}\partial^2_z\int_0^\infty dk
\Delta_r^{1/2}\sin(k\ell_r)e^{-m\beta k}\nonumber\\
&=&-\sum_{m=1}^{\infty}\sum_{r\geq
1}\frac{1}{8\pi^2}\partial^2_z\Delta_r^{1/2}\frac{\ell_r}{m^{2}\beta^2+\ell_r^2}.
\end{eqnarray}
The problems with $T\to 0$ are quite apparent: as
$\beta\to\infty$ all paths become important.
Next we specialize to parallel plates where $\ell_r=(2ar\pm 2z)$.
The derivative can be carried out explicitly. For simplicity we
focus on $m=1$ ($\delta P =\sum_{m=1}^{\infty}\delta P_{m}$),
\begin{equation}
\delta P_{1}=-
\frac{2}{\pi^2}\sum_{r=1}^{\infty}\frac{12(ar)^2-\beta^2}{\left(4(ar)^2+\beta^2\right)^3},
\end{equation}
which can be rewritten using the variable $\tau=2\pi a/\beta$
introduced earlier,
\begin{equation}
\label{sumform}
\delta P_{1}=- \frac{2\pi^{2}}{
\beta^{4}}\sum_{r=1}^{\infty}\frac{3\tau^2r^2-\pi^{2}}{(
\tau^2r^2+\pi^{2})^3}.
\end{equation}
The sum can be performed, giving
\begin{equation}
\delta
P_{1}=-\frac{1}{\pi^2}\Bigg(\frac{1}{\beta^4}-\frac{\pi^3}{8a^3\beta}\coth\left(\frac{\pi\beta}{2a}\right)
{\rm csch}^2\left(\frac{\pi\beta}{2a}\right)\Bigg).
\end{equation}
The second term in brackets is exponentially small as
$\beta\to\infty$. If we ignore it, restore the $m$-dependence,
and sum over $m$, we obtain
\begin{equation}
\label{eq:dPfin}
\delta P=-\frac{\pi^2}{90\beta^4}.
\end{equation}
which agrees with our earlier calculation, as it must.
However eq.~(\ref{sumform}) allows us to study the convergence of
the sum over reflections as $\beta\to\infty$. Instead of
performing the sum analytically, we sum up to some $r_{\rm
max}\equiv X$. Since $\tau\to 0$, we can once again use
Euler-Maclaurin,
to rewrite the sum over $r$ as
\begin{equation}
\label{partsum}
\delta
P_{1}=-\frac{2}{\pi^2}\frac{1}{\beta^4}\left[\frac{1}{2}-\frac{X}{(1+\tau^2X^2/\pi^{2})^2}+
\frac{1}{2}\frac{3\tau^2X^2/\pi^{2}-1}{(1+\tau^2X^2/\pi^{2})^3}+
...\right].
\end{equation}
where the omitted terms are higher Euler-Maclaurin contributions
that are unimportant as $\beta\to\infty$ (\emph{i.e.\ }$\tau\to
0$).
If the upper limit on the sum, $X$, is taken to $\infty$, only the
first term, $1/2$, survives and gives the expected result. The
question is: How large must $X$ be before the limiting behavior
set in? Dropping the third term in eq.~(\ref{partsum}), which is
subdominant, we can rewrite $\delta P_{1}$ as
\begin{equation}
\delta P_{1}
=-\frac{2}{\pi^2\beta^4}\left[\frac{1}{2}-\frac{X}{(1+
X^2\tau^2/\pi^{2})^2}\right]=-\frac{2}{\pi^2\beta^4}\left[\frac{1}{2}+\frac{1}{\tau}f(\tau
X)\right].
\end{equation}
The function $f(z)$ is negative definite and has a minimum at
$z=1/2\sqrt{3}\simeq 0.29$ where it takes the value
$-3^{3/2}/32\simeq -0.16$. So in order the result
Eq.~(\ref{eq:dPfin}) to be valid we must include $X\gg
X_c=\pi/\sqrt{3}\tau$ terms in the sum. For example in a typical
experimental situation we have $a=0.5\mu m$ and $T=300 K$ so
$\beta=8\mu m$, $\tau=8/\pi$ and $X_c=8/\sqrt{3}=4.6$. In this
case it is necessary to go to $X\sim 20$ before the contribution
of $|f(\tau X)/\tau|$ is smaller than 1/2. This means paths with
$\sim 40$ reflections and path lengths of order $20\mu m$. With 40
chances to sample the surface dynamics of the material and paths
of $20\mu m$ available to wander away from the parallel plate
regime, the idealizations behind the standard parallel plates
calculation must be called into question.
It must be said however that in the modern experiments the
temperature corrections are at most of the order of a few percent
at $a\sim 1\mu m$ and vanish when $a\to 0$. Nonetheless we want to
point out that there is a conceptual difference between
formulations based on the infinite parallel plates approximation,
extended to curved geometries by means of the PFA, and a
derivation (like ours) in which the curvature is inserted \emph{ab
initio}. The thermal and curvature scales interplay in a way that
the usual derivations \cite{MT,MPnew} could not possibly capture,
giving rise to different power law corrections in $a/\beta$. It is
worth reminding the reader that the usual numerical estimates of
thermal corrections are based on the infinite parallel plates
power law $(a/\beta)^4$. A smaller power like $(a/\beta)^2$ would
give a much bigger upper bound.
To summarize: temperature corrections are small at small $T$, but
the existing methods of calculating them, including both our
optical approximation and the traditional parallel plates
idealization, cannot be trusted to give a reliable estimate of the
$T$-dependence at small $T$.
\section{Conclusions}
In this paper we have shown how to adapt the optical approximation
to the study of local observables. We have illustrated the method
by studying the pressure, but the method applies as well to other
components of the stress tensor, to charge densities, or any
quantity that can be written in terms of the single particle
Greens function. The advantage of the optical approximation is to
extend the study of these local observables to novel geometries.
In particular we developed an expression for the Casimir pressure
on the bodies and applied our main result Eq.\ (\ref{eq:press2})
to the study of three important examples: parallel plates, the
Casimir pendulum and a sphere opposite a plate.
We have also shown how to calculate, within this approximation
scheme, thermodynamic quantities and thermal corrections to the
pressure in the general case and applied our results to the
example of parallel plates (retrieving the known results) and to
the case of a sphere opposite a plate. Along the way we have
given a proof of the ``classical limit'' of Casimir force for any
geometry (within our approximation), \emph{i.e.\ }the fact that
Casimir forces at high temperatures are proportional to the
temperature and independent of $\hbar$, a fact that previously was
known only for parallel plates.
Finally, we argued that all known methods of computing the
temperature dependence of the Casimir effect are suspect as $T\to
0$.
\section{Acknowledgments}
We would like to thank S.~Fulling and M.~Schaden for comments. AS
would like to thank M.~V.~Berry for useful discussions. RLJ would
like to thank the Rockefeller Foundation for a residency at the
Bellagio Study and Conference Center on Lake Como, Italy, where
much of this work was performed. This work is also supported in
part by funds provided by the U.S.~Department of Energy (D.O.E.)
under cooperative research agreement DE-FC02-94ER40818.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Zabo Raises $1.15M in Funding
Zabo, a Dallas-based cryptocurrency banking application that allows users to get paid in Bitcoin by allocating pay between cash and cryptocurrency, announced a $1.15m seed funding round raised last year.
Blockchange Ventures, a New York-based venture capital firm backing exclusively early stage blockchain companies, protocols and applications made the investment.
Co-founded by President Alex Treece and CEO Christopher Brown, Zabo lets users get paid in Bitcoin by allocating a pay between cash and cryptocurrency. Its licensed banking partner will provide FDIC-insured US Dollar bank accounts and be governed and compliant with applicable federal and state money transmission laws and regulations.
The product, currently under development, is intended for launch mid 2019.
Alex Treece serves as Director at the University of Texas at Dallas' Jindal School of Management. Previously, he was the lead Product Manager at Sentieo focused on building a modern cryptoasset data platform. Treece began his career as an investment banker at Stephens and later joined the global private equity firm Riverside Company.
Christopher Brown started his career in the US military, serving for 8 years in various capacities, including airspace design, air traffic control, systems security, and information security.
Tagged with: Blockchange Ventures, Zabo
Previous storyWeShareMD Closes Seed Funding Round
Next storyErasca Raises Series A Extension; Round Totals $64M
Braintrust Raises $18M In Strategic Growth Funding
By FinSMEs Published on October 5, 2020 October 5, 2020
CasperLabs Secures $14.5M in Series A Funding
Good Money Secures $30M in Series A Financing
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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\section{Introduction}
Finding exact solutions for compact objects (stars, black holes or more exotic structures)
in modified theories of gravity is in general a very difficult problem. Not only there exist many ``no-go'' theorems, extending the no-hair theorem of General Relativity (GR) which kill the hope of having new solutions, but also the equations of modified theories of gravity are, most of the time, much more complicated than the Einstein equations, even for symmetry reduced space-times. Nevertheless, it was possible to
circumvent these difficulties and new black hole (analytical and numerical) solutions have emerged these last years, in particular, in the context of Degenerate Higher Order Scalar Tensor (DHOST) theories \cite{Langlois:2015cwa,Langlois:2015skt,Achour:2016rkg,Motohashi:2016ftl,BenAchour:2016fzp,Crisostomi:2016tcp,Crisostomi:2016czh} which provide the most general class of viable scalar-tensor theories to date evading the Ostrogradsky ghosts~\cite{Woodard:2015zca,Motohashi:2014opa,Motohashi:2020psc}
(see also \cite{DeFelice:2018ewo} where DHOST theories have been extended to the so-called U-DHOST theories, and \cite{Motohashi:2020wxj} for further generalization).
However, most of these solutions are stealth\footnote{Stealth black hole solutions were first introduced in the context of three dimensional gravity in \cite{AyonBeato:2004ig}. } spherically symmetric black holes \cite{Babichev:2016rlq,Babichev:2017guv,Lehebel:2017fag,Chagoya:2018lmv, BenAchour:2018dap,Motohashi:2019sen,Takahashi:2019oxz,Minamitsuji:2019shy,BenAchour:2019fdf,Minamitsuji:2019tet}. (See also \cite{Mukohyama:2005rw} for an earlier stealth solution in the context of ghost condensate~\cite{ArkaniHamed:2003uy}.)
The first rotating stealth solution was discussed in \cite{Babichev:2017guv} in the context of beyond Horndeski theories, and only recently, a more general stealth Kerr black hole solution was obtained in \cite{Charmousis:2019vnf} in the class of DHOST theories where gravitational waves propagate at the speed of light.
On the other hand, the existence condition for any GR solutions with matter component minimally coupled to gravity was derived from a covariant analysis in \cite{Motohashi:2018wdq,Takahashi:2020hso} where the Kerr-Newman-de Sitter solution in quadratic DHOST theories was obtained.
Moreover, solution-generating method such as the Kerr-Schild procedure have newly been extended to DHOST theories opening new avenues to derive interesting solutions \cite{Babichev:2020qpr}. Finally, a non-stealth numerical rotating black hole solution has also been constructed recently in \cite{VanAelst:2019kku}.
Such rotating solutions represent an important step forward in order to investigate rotating compact objects in modified gravity.
However, up to now, analytic solutions describing a rotating compact object in DHOST theories \cite{Babichev:2017guv, Charmousis:2019vnf, Takahashi:2020hso} correspond to a stealth Kerr geometry and deviations from GR can only show up at the level of perturbations \cite{Charmousis:2019fre}.
Therefore it would be interesting to construct new exact solutions beyond the stealth sector in order to obtain analytic solutions which exhibit deviations from GR already at the background level. The goal of this work is to present such a construction using the disformal solution-generating method discussed first in the context of DHOST theories in \cite{BenAchour:2019fdf}\footnote{See also \cite{Filippini:2017kov} for previous use of disformal transformation to construct a rotating black hole solution in vector-tensor theories.}.
This method takes advantage of the properties of DHOST theories under disformal transformations of the metric, which were introduced in \cite{Bekenstein:1992pj}, to construct new exact solutions to the DHOST field equations from a known ``seed'' solution.
As the class of DHOST theories is stable under disformal transformations, any known solution $(\tilde{g}_{\mu\nu},\phi)$ of a theory associated to the action $\tilde{S}[\tilde{g}_{\mu\nu},\phi]$ enables us to
construct a ``new'' solution $(g_{\mu\nu},\phi)$ of a ``new'' theory whose action ${S}[{g}_{\mu\nu},\phi]$ is related to the previous one by,
\begin{eqnarray}
\label{Intro:disformal}
S[g_{\mu\nu},\phi] = \tilde{S}[\tilde{g}_{\mu\nu},\phi] \, , \qquad
\tilde{g}_{\mu\nu} = A(\phi,X) g_{\mu\nu} + B(\phi,X) \phi_\mu \phi_\nu \, ,
\end{eqnarray}
where we are using the standard notations $\phi_\mu \equiv \partial_\mu \phi$ for the partial derivative of $\phi$ and $X=g^{\mu\nu} \phi_\mu \phi_\nu$
for its kinetic density. The functions $A$ and $B$ are, a priori, free but we will restrict our study to the cases where the
disformal transformation is invertible and where the two metrics are non-degenerate. Although the construction of new exact solutions appears to be straightforward using this procedure, it is worth emphasizing that the obtained solution is not physically equivalent to the seed one in general.
Indeed, in vacuum, the two actions $S$ and $\tilde S$
are equivalent in the sense that they have equivalent spaces of solutions provided that the disformal transformation is invertible.
However, in the presence of matter minimally coupled to the metric (either $g_{\mu\nu}$ for the action $S$ or $\tilde{g}_{\mu\nu}$ for the disformed action $\tilde{S}$),
they become inequivalent.
Hence, a particle falling into a black hole solution of the theory $S$ will have, in general,
a different trajectory from that of a particle falling into the ``disformed black hole'' of the theory $\tilde S$ (see \cite{Deffayet:2020ypa} for a recent investigation on the fate of matter coupling under disformal transformations)\footnote{Disformal transformations also allow one to consider black hole singularity from a different perspective, as recently discussed in \cite{Domenech:2019syf}.}.
Therefore, this mehtod allows one to construct new exact solutions which encode interesting deviations from GR. Consequently, the resulting solutions provide an interesting arena to investigate modification of the shadow of rotating compact objects induced by modified theories of gravity. Recent investigations in this direction have been reported in \cite{Khodadi:2020jij}.
In this work, we shall extend the investigations initiated in \cite{BenAchour:2019fdf} to the axisymmetric framework. Concretely, we will apply the solution-generating method here to produce a new rotating solution in DHOST theories. Transforming the stealth Kerr ``seed'' solution by a disformal transformation \eqref{Intro:disformal} where the functions $A$ and $B$ are
constants, we present a new non-stealth rotating analytic solution, given by Eq.~(\ref{newsol}) below, which depend on three parameters: the mass $M$, the spin parameter $a$ and the disformal parameter $\alpha$ which encodes deviations form the Kerr metric.
This new geometry is different from Kerr (it is no more Ricci flat for instance) but has similarities: it is asymptotically flat, it admits the two same Killing vectors as the ones in Kerr, it has ergoregions very similar to those of Kerr, and it comes with the same time-dependent scalar field (with constant kinetic term) as in the stealth Kerr solution.
We explore geometrical properties of this family of novel rotating solutions: we argue that there is no new singularity compared to the usual Kerr black hole, we discuss the existence of event horizons computing the disformed null vectors, and finally we study some
properties of its geodesics.
The paper is organized as follows. In the following \S\ref{sec2}, we review basic results on stealth solutions in DHOST theories: we
give the conditions for such solutions to exist and we quickly present the stealth Kerr solution. In \S\ref{sec3},
we construct and study solutions obtained from a disformal transformation \eqref{Intro:disformal}, where $A=A_0$ (without loss of generality we can fix $A_0=1$) and $B=B_0$
are constant, of this stealth Kerr solution.
We start with reviewing useful results on disformal transformations which are the main
tool of our method.
Then, we compute explicitly the novel solution and discuss some of its geometrical properties.
In \S\ref{sec4} we conclude with a summary of the results and a discussion on open issues.
In addition, to illustrate again the potentiality of the disformal solution-generating method, we present and discuss in Appendix~\ref{AppB} another axisymmetric solution for DHOST theories obtained from a disformal transformation of the generalized Kerr solution of Einstein-Scalar gravity.
\medskip
\underline{Note added.}
As we were writing this article, we learned that a similar (but complementary) article \cite{Anson}
was about to be posted on the arXiv.
In \cite{Anson}, the authors study in more details the geometry of the disformed Kerr solutions and propose candidates for the event horizons.
In the present paper, on the other hand, we clarify a class of DHOST theories in which the disformed Kerr solutions are exact solutions.
Our results and their results agree where we overlap.
\section{Stealth solutions in DHOST theories}
\label{sec2}
This section is devoted to review important results on stealth solutions in DHOST theories. We start, in \S\ref{sec2A}, with giving
the conditions that a DHOST theory must satisfy to have a stealth solution whose metric is also a solution of GR. In \S\ref{sec2B},
we describe more specifically the stealth Kerr solution which will be the ``seed'' to construct new rotating solutions in DHOST
theories in the subsequent section.
\subsection{Conditions of existence}
\label{sec2A}
The most general theory of quadratic DHOST theory \cite{Langlois:2015cwa} is described by the action
\begin{equation}
\label{DHOST}
S=\int d^4x\sqrt{-g}\left(P(X,\phi)+Q(X,\phi)\, \Box \phi+F(X,\phi)\,R+\sum_{i=1}^{5}A_{i}(X,\phi)\, L_{i}\right)
\end{equation}
where the functions $A_{i},\,F,\,Q$ and $P$ depend on the scalar
field $\phi$ and its kinetic term $X\equiv \phi_\mu \phi^\mu$ with $\phi_\mu \equiv \nabla_{\mu}\phi$,
and $R$ is the Ricci scalar.
The five elementary Lagrangians $L_{i}$ quadratic in second derivatives of $\phi$ are defined by
\begin{eqnarray}
&&L_1 \equiv \phi_{\mu\nu} \phi^{\mu\nu} \, , \quad
L_2 \equiv (\Box \phi)^2 \, , \quad
L_3 \equiv \phi^\mu \phi_{\mu\nu} \phi^\nu\Box \phi \, , \quad \nonumber \\
&&L_4 \equiv \phi^\mu \phi_{\mu\nu} \phi^{\nu\rho} \phi_\rho \, , \quad
L_5 \equiv (\phi^\mu \phi_{\mu\nu} \phi^\nu)^2 \, ,
\end{eqnarray}
where we are using the standard notations $\phi_\mu \equiv \nabla_\mu \phi$ and $\phi_{\mu\nu} \equiv \nabla_\nu \nabla_\mu \phi$ for the first and second (covariant) derivatives of $\phi$.
For the theory to propagate only one extra scalar degree of freedom in addition to the usual tensor modes of gravity, the functions $F$ and $A_i$ have to satisfy the so-called degeneracy conditions \cite{Langlois:2015cwa,Achour:2016rkg} while $P$ and $Q$ are totally free.
The degeneracy conditions can be derived for general higher-derivative theories in a systematic way~\cite{Motohashi:2016ftl,Motohashi:2017eya,Motohashi:2018pxg}.
It has been shown
in \cite{Achour:2016rkg} that these DHOST theories can be classified into three classes which are
stable under general disformal transformations, i.e.\ transformations of the metric of the form
\begin{equation}
\label{disf}
g_{\mu\nu} \longrightarrow \tilde g_{\mu\nu}= A(X, \phi) g_{\mu\nu}+ B(X,\phi) \phi_\mu \, \phi_\nu\,,
\end{equation}
where $A$ and $B$ are arbitrary functions with the conditions that the two metrics are not degenerate. Notice that, when the disformal
transformation is not invertible, the DHOST theory after the transformation falls in the class of mimetic theories of gravity \cite{Chamseddine:2013kea,Takahashi:2017pje,Langlois:2018jdg}. See also \cite{Sebastiani:2016ras, Casalino:2018wnc, Gorji:2019rlm} for further details on mimetic gravity.
The theories belonging to the first class, named class Ia in \cite{Achour:2016rkg}, can be mapped into a Horndeski form by applying a disformal transformation.
The other two classes are not physically viable (either gradient instabilities of cosmological perturbations develop or tensor modes have pathological behavior)
\cite{Langlois:2017mxy}
and will not be considered in the present work.
Theories in class Ia are labelled by the three free functions $F,A_1$ and $A_3$ (in addition to $P$ and $Q$) and
the three remaining functions are given by the relations
\cite{Langlois:2015cwa}
\begin{eqnarray}
A_{2} & = & -A_{1} \, ,\label{deg1}
\\
A_{4} & = & \frac{1}{8\left(F+XA_{2}\right)^2}\Bigl(A_{2}A_{3}\left(16X^{2}F_{X}-12XF\right)+4A_{2}^{2}\left(16XF_{X}+3F\right) \nonumber\\
& & +16A_{2}\left(4XF_{X}+3F\right)F_{X} +16XA_{2}^{3}+8A_{3}F\left(XF_{X}-F\right)-X^{2}A_{3}^{2}F+48FF_{X}^{2}\Bigr) \, \label{funA4},\\
A_{5} & = & \frac{1}{8\left(F+XA_{2}\right)^2}\Bigl(2A_{2}+XA_{3}-4F_{X}\Bigr)\Bigl(3XA_{2}A_{3}-4A_{2}F_{X}-2A_{2}^{2}+4A_{2}^{3}F\Bigr) \, ,
\label{deg3}
\end{eqnarray}
where $F_X$ denotes the derivative of $F(X,\phi)$ with respect to $X$. Similarly $F_\phi$ denotes the partial derivative of $F$
with respect to $\phi$ and the same notations will be used for all other functions as well.
The above relations (\ref{deg1}-\ref{deg3}) are a direct consequence of the three degenerate conditions that
guarantee
only one scalar degree of freedom is present
\cite{Langlois:2015cwa,Langlois:2015skt}.
In conclusion, this means that all the DHOST theories we study here are characterized by five free functions of $X$ and $\phi$, which are $P$, $Q$, $F$, $A_1$ and $A_3$. Notice that we have implicitly supposed the condition $F+XA_{2}\neq 0$. Theories where
$F+XA_{2}= 0$ belong to the sub-class Ib which is not physically relevant \cite{Achour:2016rkg}. Finally, coupling to external fields (perfect fluids, scalar fields, vector fields, etc.) can be done by adding to the DHOST action an action $S_m$ where the external degrees of freedom are minimally coupled to the metric $g_{\mu\nu}$ (which is assumed to be the physical one).
\medskip
The Euler-Lagrange equations of DHOST theories are very complicated. Even though only one scalar degree of freedom comes with the usual two tensorial degrees of freedom of GR, these equations are higher order and can be up to fourth order in $\phi$ and third order in
$g_{\mu\nu}$. It is only in the Horndeski frame (where the DHOST theory falls in the Horndeski class) where the equations of motion are second order, but the external fields are no more minimally coupled to the metric and, even in that case, the equations still have a very complex structure compared to GR.
While the reduction of the higher-order Euler-Lagrange equations to a system of second-order differential equation for the case of static spherically symmetric space-time was performed explicitly in \cite{Takahashi:2019oxz}, the process is more involved for more general space-time.
Hence, finding exact solutions in DHOST theory is far from being an easy task and one thus usually makes assumptions to simplify the problem.
Here we consider the following assumptions. First, we impose shift-symmetry which means that the DHOST action \eqref{DHOST} is unchanged by the transformation $\phi \rightarrow \phi+c$ where $c$ is a constant, and thus all the functions entering in the definition of \eqref{DHOST} depend on $X$ only. Second, one
assumes the solution is such that $X=X_0$ is a constant which drastically simplifies the modified Einstein equations.
And finally, one looks for
stealth solutions where the metric $g_{\mu\nu}$ is also a solution of the vacuum Einstein equations with a cosmological constant $\Lambda$,
\begin{eqnarray}
G_{\mu\nu}+\Lambda g_{\mu\nu} = 0\, ,
\end{eqnarray}
where $G_{\mu\nu} \equiv R_{\mu\nu} - R g_{\mu\nu}/2$ is the Einstein tensor. One can go further
and requires that a given DHOST theory admits all GR solutions, and not only some of them, as the metric part of stealth solutions. This is the case if the following conditions hold \cite{Takahashi:2020hso},
\begin{eqnarray}
\label{condStealth}
P + 2 \Lambda F = 0 \, , \quad
P_X + \Lambda (4 F_X - X_0 A_{1X}) = 0 \, , \quad
Q_X=0 \, , \quad
A_1 = 0 \, \quad
A_3 + 2 A_{1X} = 0 \, ,
\end{eqnarray}
where all these functions are evaluated on the solution $X=X_0$.
These conditions have been recently generalized to non-shift symmetric theories and to the case where matter is coupled to gravity minimally
\cite{Takahashi:2020hso}.
Notice that these conditions are very strong and drastically restrict the set of DHOST theories.
It was also argued that some stealth solutions lead to a problem of strong coupling \cite{Minamitsuji:2018vuw,deRham:2019gha} at the level of linear perturbations.
Further, using the effective field theory framework it was shown in \cite{Motohashi:2019ymr} that perturbations about stealth solutions are strongly coupled, for de Sitter background in the decoupling limit, and for the Minkowski background even away from the decoupling limit, so long as we require evolution equation of perturbations to be second order.
Thus, in general the strong coupling is inevitable for asymptotically de Sitter or flat stealth solutions.
Moreover, even if spacetime is different from de Sitter or Minkowski on superhorizon scales, the strong coupling is inevitable on subhorizon scales where the spacetime is nearly flat and hence the analysis of \cite{Motohashi:2019ymr} applies.
However, we can introduce a controlled detuning of the degeneracy condition, dubbed the scordatura mechanism in \cite{Motohashi:2019ymr}, to render the perturbations weakly coupled all the way up to a sufficiently high scale, as in the ghost condensate~\cite{ArkaniHamed:2003uy}.
The Ostrogradsky ghosts associated with the scordatura is adjusted to show up only above the cutoff scale of the effective field theory.
It is also important to note that the scordatura does not change the properties of the stealth solutions of degenerate theories at astrophysical scales (similarly to the stealth solution \cite{Mukohyama:2005rw} in the ghost condensate).
Thus, below we focus on stealth solutions in degenerate theories.
\subsection{The stealth Kerr solution in DHOST theories}
\label{sec2B}
The stealth Kerr solution in DHOST theories we shall use in this work as a seed was obtained in \cite{Charmousis:2019vnf}.
It was derived for theories within the class Ia with no cubic galileon term where gravitational waves propagate at the speed of light ($c_{\rm GW}=c$), i.e.\
\begin{eqnarray}
\label{cequal1}
A_1=A_2=0 \, , \qquad Q=0 \, .
\end{eqnarray}
Here, the condition $c_{\rm GW}=c$ implies $A_1=0$~\cite{Langlois:2017dyl}, and we have used the degeneracy condition $A_2=-A_1$ in \eqref{deg1}.
Therefore, the stealth conditions \eqref{condStealth} simplify and become,
\begin{eqnarray}
P(X_0) + 2 \Lambda F(X_0) = 0 \, , \quad
P_X(X_0) + 4 \Lambda F_X(X_0) = 0 \, , \quad
A_3(X_0) = 0 \, .
\end{eqnarray}
The metric is the usual Kerr solution of GR, or the de Sitter (dS)/Anti-de Sitter (AdS) Kerr solution
in the presence of a non-zero cosmological constant $\Lambda=3/\ell^2$ ($\ell^2 >0 $ for dS or $\ell^2<0$ for AdS). In Boyer-Lindquist
coordinates $(t,r,\theta,\psi)$, it reads
\begin{align}
ds^2 = - \frac{\Delta_r}{\Xi^2 \rho} \left( dt - a \sin^2{\theta} \, d\psi \right)^2 + \rho \left( \frac{dr^2}{\Delta_r} + \frac{d\theta^2}{\Delta_{\theta}}\right) + \frac{\Delta_{\theta} \sin^2{\theta}}{\Xi^2 \rho} \left( a dt - \left(r^2 + a^2 \right) d\psi \right)^2
\label{Kerrmetric}
\end{align}
where $\Xi \equiv 1 + {a^2}/{\ell^2}$ is a constant, and the different functions entering in the metric are defined by
\begin{align}
\Delta_r & = \left( 1 - \frac{r^2}{\ell^2}\right) \left( r^2 + a^2\right) - 2 Mr \;, \qquad \Delta_{\theta} = 1 + \frac{a^2}{\ell^2} \cos^2{\theta} \;,\quad \rho = r^2 + a^2 \cos^2{\theta} \, ,
\end{align}
while $M$ is the mass of the black hole and $a$ the angular momentum parameter satisfying the condition $a \leq M$.
The stealth Kerr black hole comes with a scalar hair whose highly non-trivial profile \cite{Charmousis:2019vnf} coincides with the
Hamilton-Jacobi potential associated to the Kerr geodesic equation and is given by,
\begin{eqnarray}
\label{profilekerr}
\phi(t,r,\theta) = -Et + S_r(r) + S_{\theta} (\theta) \, , \qquad
S_r \equiv \pm \int dr \, \frac{\sqrt{\cR}}{\Delta_r} \;, \quad S_{\theta} \equiv \pm \int d\theta \, \frac{\sqrt{\Theta}}{\Delta_{\theta}}
\end{eqnarray}
where $E$ is a constant while the two functions $S_r$ and $S_\theta$ are defined, up to a sign ambiguity, as integrals involving the
radial and angular functions
\begin{equation}
\label{func}
\cR(r) \equiv m^2 \left( r^2 + a^2\right) \left[ \eta^2 \left( r^2 + a^2\right) - \Delta_r\right]\;, \qquad \Theta(\theta) \equiv a^2 m^2 \sin^2{\theta} \left( \Delta_{\theta} - \eta^2 \right) \, ,
\end{equation}
with $X_0=-m^2$ and $\eta \equiv \Xi E /m$. In fact, there are four different branches for the scalar field $\phi$ because of the freedom to choose the signs
of $S_r$ and $S_\theta$. It has been shown in \cite{Charmousis:2019vnf} that one can make use of these branches to construct a scalar field solution which is regular and finite in an untrapped region as well as a trapped region (either a black hole region or a white hole region but not both), and in particular on the the black hole horizon (as well as on the cosmological horizons when $\ell^2 > 0$).
In the particular case where there is no cosmological constant $\ell \rightarrow \pm \infty$, we have $\eta = 1$, then $\Theta$ vanishes, and
finally the scalar field does not depend on the variable $\theta$ anymore. As we are going to see in the following section, this is the case we
will focus on when we consider disformal Kerr solutions to avoid several issues. Furthermore, the radial function $\cR(r)$ simplifies as well and becomes
\begin{equation}
\label{funcR0}
\cR(r) \equiv 2 M m^2 r \left( r^2 + a^2\right)\; ,
\end{equation}
with the condition $E=m$ which identifies the kinetic energy $X_0=-m^2$ to $E^2$ .
\section{Rotating solutions beyond the stealth sector}
\label{sec3}
In this section, we will construct the new rotating solution in DHOST theories obtained from a disformal transformation
of the stealth Kerr black hole we have just described above.
We will start, in \S\ref{sec3A}, by reviewing useful and general properties of disformal
transformations \eqref{Intro:disformal} on DHOST theories. Then, we will concentrate on such transformations that $A$ and $B$
are constant, and we will show how the stealth conditions \eqref{condStealth} transform under these ``constant'' disformal transformations giving the conditions for any DHOST theory to have the disformed Kerr black hole as a solution.
Finally, in \S\ref{sec3B} we will transform the stealth Kerr black hole and in \S\ref{sec3C} we study some geometrical properties of the disformed geometry, which is no more stealth.
\subsection{Constant disformal transformations and stealth conditions}
\label{sec3A}
When the metric $g_{\mu\nu}$ comes with a scalar field
$\phi$, one can define the ``disformed'' metric $\tilde{g}_{\mu\nu}$ by~\cite{Bekenstein:1992pj}
\begin{eqnarray}
\tilde{g}_{\mu\nu} = A(\phi,X) g_{\mu\nu} + B(\phi,X) \phi_\mu \phi_\nu \, ,
\end{eqnarray}
where $A$ and $B$ are arbitrary functions. One can show that the transformation is invertible when the condition $\partial_X(A/X+ B ) \neq 0$
(and $A \neq 0$) is satisfied.
We will always consider invertible disformal transformations here. Furthermore, as we restrict our study to shift-invariant DHOST theories, the functions $A$ and $B$ are supposed to depend on $X$ only.
Anticipating on the next section, we point that these two assumptions of i) invertibility and ii) shift symmetry lead to drastic simplifications when one considers the disformal mapping of seed solutions with a constant kinetic term $\tilde{X}=\tilde{X_0}$. Indeed, this automatically imposes that the disformal potentials $A$ and $B$ are constant
\cite{BenAchour:2019fdf}.
Disformal transformations on the metric induce transformations on DHOST actions. Given an action $\tilde S[\tilde{g}_{\mu\nu},\phi]$,
one defines a new action $S[g_{\mu\nu},\phi]$ by the identification,
\begin{eqnarray}
S[g_{\mu\nu},\phi] = \tilde S[A(X) g_{\mu\nu} + B(X) \phi_\mu \phi_\nu,\phi] \, .
\end{eqnarray}
Interestingly, DHOST theories are stable under disformal transformations \cite{Achour:2016rkg} and the transformation
rules between the functions (of $\tilde{X} \equiv \tilde{g}^{\mu\nu} \phi_\mu \phi_\nu$) $\tilde{P}$, $\tilde{Q}$,
$\tilde{F}$ and $\tilde{A}_I$ entering in the definition of the action $\tilde{S}[\tilde{g}_{\mu\nu},\phi]$ on one side,
and the functions (of $X = g^{\mu\nu} \phi_\mu \phi_\nu$) $P$, $Q$, $F$ and $A_I$ defining $S[g_{\mu\nu},\phi]$ on the other side
are given in \cite{Achour:2016rkg}.
As it turns out, these rules, which are rather complicated, simplify drastically when one considers
constant disformal transformations where $A=A_0$ and $B=B_0$ do not depend on $X$ anymore. As pointed above, this is the case when considering invertible and shift symmetric disformal mapping of seed solution with constant kinetic term. After a straightforward
calculation, one shows that the k-essence, the cubic galileon and the Ricci terms transform as follows,
\begin{eqnarray}
P = \tilde{P} \, , \qquad
Q = A_0 \int dX \, N {\tilde Q}_X \, , \qquad F =\frac{A_0}{N}\tilde{F} \, ,
\end{eqnarray}
while the functions $A_I$ entering in the quadratic part of the Lagrangian transform as,
\begin{eqnarray}
&&A_1 = N (B_{0} \tilde{F} + N^2 \tilde{A}_1) \, , \quad
A_2 = N(- B_{0} \tilde{F} + N^2 \tilde{A}_2) \, ,\nonumber \\
&& A_3 = \frac{N}{A_0} \left[ - 4 A_0 B_0 \tilde{F}_X - 2 B_0 N^4\tilde{A}_2 +N^6 \tilde{A}_3\right] \, , \nonumber \\
&&A_4 = \frac{N}{A_0} \left[- N^2 B_0^2 \tilde{F} + 4 A_0 B_0 \tilde{F}_X - 2 N^4 B_0 \tilde{A}_1+ {N^6} \tilde{A}_4 \right] \, , \nonumber \\
&&A_5 = \frac{N^7}{A_0^2}\left[ B^2_{0} ( \tilde{A}_1 + \tilde{A}_2) + N^2 B_{0}( \tilde{A}_3 - \tilde{A}_4) +N^4 \tilde{A_5} \right] \, ,
\end{eqnarray}
where we introduced the factor
\begin{eqnarray}
N \equiv {A_0}^{1/2}{(A_0 + X B_0)}^{-1/2} \, .
\end{eqnarray}
We recall that ``tilde'' functions $\tilde{P}$, $\tilde{Q}$, $\tilde{F}$, $\tilde{A}_I$, in the right-hand side of the previous equations are viewed as functions of $X$ via the relation,
\begin{eqnarray}
\label{X}
\tilde{X} = \frac{X}{ A_0 + X B_0} \, .
\end{eqnarray}
\medskip
Now, we assume that the theory $\tilde{S}[\tilde{g}_{\mu\nu},\phi]$ satisfies the conditions \eqref{condStealth} to have a stealth
solution where $\tilde{X}_0$ is constant and $\tilde{g}_{\mu\nu}$ is the Kerr metric recalled above \eqref{Kerrmetric}. Then, we
want to translate these conditions in terms of the functions entering into the action $S[g_{\mu\nu},\phi]$.
We first remark that under constant disformal transformation $X_0$ is also constant when $\tilde{X}_0$ is constant.
After a direct calculation, from \eqref{condStealth} we obtain
\begin{eqnarray}
&&P + \frac{2 \Lambda N}{A_0} F = 0 \, , \qquad
{\partial_X} \left[ P + \frac{\Lambda}{A_0} \left( 4 + \frac{B_0 X_0}{N}\right) F - \frac{\Lambda X_0}{N^3} A_1\right] = 0 \, , \label{cond1}\\
&& Q_X = 0 \, , \qquad A_1 - \frac{N^2 B_0}{A_0} F = 0 \, , \label{cond2}
\end{eqnarray}
together with the remaining more complicated condition
\begin{eqnarray}
\frac{A_0}{2N} A_3 + \left( 2 B_0 N_X + \frac{B_0 N^2}{A_0}\right)F + 2 B_0 N F_X + B_0 N A_2 + \frac{N^8}{A_0}
\partial_X \left( \frac{A_1}{N^3} - \frac{B_0}{A_0} \frac{F}{N}\right)=0 \, , \label{cond3}
\end{eqnarray}
which comes from the last equation of \eqref{condStealth}. Let us recall that these equations holds only when they are
evaluated on $X_0$. As a consequence, any DHOST theories which satisfy all these conditions admit disformal stealth solutions, which
are in general non-stealth. Obviously, these conditions reduce to \eqref{condStealth} for a trivial disformal transformation where
$A_0=1$ and $B_0=0$.
In the special case where the theory $\tilde{S}$ satisfies, in addition, the conditions \eqref{cequal1} which insure that gravitational waves
propagate at the speed of light, the disformed theory $S$ satisfies in turn,
\begin{eqnarray}
A_1= - A_2 = N^2 \frac{B_0}{A_0} F \, , \qquad Q=0 \, .
\end{eqnarray}
Hence, \eqref{cond2} are automatically satisfied and the first equation in \eqref{cond1} is unchanged.
The last conditions in \eqref{cond1} and \eqref{cond3} are also simplified according to
\begin{eqnarray}
\partial_X \left( P + \frac{4 \Lambda}{A_0} F\right)=0 \, , \qquad
\frac{A_0}{2 N B_0} A_3 + \left(2N_X + \frac{N^2}{A_0}(1-NB_0) \right) F + 2N F_X =0 \, .
\end{eqnarray}
In that case, the theory $S$ admits the disformed Kerr black hole we are going to describe now as a solution.
\subsection{Disformal Kerr solution: construction and preliminary properties}
\label{sec3B}
Considering the previous stealth Kerr-(A)dS seed solution with a constant kinetic term, we turn now to generate a new non-stealth solution whose metric takes the form
\begin{eqnarray}
\label{disKerrformal}
g_{\mu\nu} \; = \; \tilde{g}_{\mu\nu} - B_0 \, \phi_\mu \phi_\nu \, ,
\end{eqnarray}
where $\tilde{g}_{\mu\nu} $ is the Kerr metric \eqref{Kerrmetric}. Without loss of generality, we have fixed $A_0=1$, otherwise the metric would simply get a global physically irrelevant constant conformal factor. If the scalar field $\phi$ depends on the angular variable $\theta$, then
the disformed Kerr metric \eqref{disKerrformal} acquires new components, among which
\begin{eqnarray}
g_{t \theta} = \pm B_0 E \frac{\sqrt{\Theta}}{\Delta_\theta} \, ,
\end{eqnarray}
where the expression of $\Theta(\theta)$ and $\Delta_\theta(\theta)$ has been recalled in \eqref{func}.
Such a term depends on the radial variable $r$ and then they do not vanish at infinity. As a consequence, one cannot expect
that the disformed metric is asymptotically flat, dS or AdS. To avoid this pathological behavior, we require that the scalar field does not
depend on $\theta$ which implies necessarily the vanishing of the cosmological constant $\ell \rightarrow \pm \infty$, then $\eta=1$ and $E=m$ (as
a consequence of $\Theta=0$). Hence, from now on, we consider only this case which, as recalled before \eqref{funcR0}, corresponds to scalar field of the form
\begin{eqnarray}
\label{profilekerr}
\phi(t,r) = -m t + S_r(r) \, , \qquad
S_r = \pm \int dr \, \frac{\sqrt{\cR}}{\Delta} \, , \qquad
\Delta = r^2+a^2-2Mr \; ,
\end{eqnarray}
where, for simplicity, we have omitted the subscript $r$ in $\Delta$ (as there is no more possible ambiguity).
Thus, the disformal transformation (with $A_{0} =1$) leads to the new solution
\begin{eqnarray}
\label{newsol}
ds^2 &=&- \frac{\Delta}{\rho} \left( dt - a \sin^2{\theta} \, d\psi \right)^2 + \frac{\rho}{\Delta} {dr^2} + \rho \, {d\theta^2} + \frac{\sin^2{\theta}}{\rho} \left(a \, dt - \left(r^2 + a^2 \right) d\psi \right)^2 \nonumber \\
&& + \alpha \left( dt \pm {\sqrt{2Mr(r^2+a^2)}}/{\Delta} \, dr\right)^2 \, ,
\end{eqnarray}
with $\alpha \equiv -B_0 m^2$ while the scalar field profile remains unchanged (\ref{profilekerr}).
The inverse disformed Kerr metric is given by
\begin{eqnarray}
\label{inversedisf}
g^{\mu\nu} \, = \, \tilde{g}^{\mu\nu} + \frac{\alpha}{m^2(1-\alpha)} \phi^\mu \phi^\nu \, ,
\end{eqnarray}
where $\tilde{g}^{\mu\nu}$ is the inverse Kerr metric while the only non-vanishing components of $\phi^\mu = \tilde{g}^{\mu\nu} \phi_\nu$
are,
\begin{eqnarray}
\phi^t = \frac{m}{\Delta}\left( r^2+a^2 + \frac{2M r a^2 \sin^2\theta}{\rho}\right) \, , \quad
\phi^r = m \frac{\sqrt{2M r (r^2+a^2)}}{\rho}\,, \quad
\phi^\varphi =m\frac{2aMr }{\Delta \rho} \, .
\end{eqnarray}
Therefore, the disformal transformation of the stealth Kerr solution provides a new \textit{non-stealth} exact solution which is parametrized, in addition to the mass and angular momentum parameters $(M, a)$ of the Kerr family, by one new deformation parameter $ \alpha$ which encodes precisely the deviations from GR. The apparent $\pm$ ambiguity in \eqref{newsol} can be absorbed thanks to simple redefinitions of
$t$ and $a$ which are replaced by $\pm t$ and $\pm a$. Hence, we can safely fix the sign to $+$ from now on without loss of generality.
\medskip
At infinity where $r\rightarrow + \infty$, the disformed Kerr metric becomes equivalent to,
\begin{eqnarray}
\label{asymptomet}
ds^2 &\simeq& -\left( 1-\frac{2M_1}{r}\right)dt^2 + \left( 1-\frac{2M_2}{r}\right)^{-1} dr^2 + r^2 (d\theta^2 + \sin^2\theta \, d\varphi^2) \nonumber \\
&+& 2\alpha \sqrt{\frac{2M_1}{r}} dr\,dt + {\cal O}\left( \frac{1}{r^2}\right) \, ,
\end{eqnarray}
where we introduced the notations,
\begin{eqnarray}
M_1 \equiv \frac{M}{1-\alpha} \, , \quad M_2\equiv(1+\alpha)M \, ,
\end{eqnarray}
and we rescaled the time coordinate $t$ by $\sqrt{1-\alpha}$. Note that the coefficient $\alpha$ modifies the black hole mass in the matrix elements $g_{tt}$ and $g_{rr}$ in a different way in Schwarzschild coordinates as $M_1 \neq M_2$. These masses agree at the first order in the parameter $\alpha$. Moreover, while the cross term $g_{tr}dt dr$ induced by the disformal transformations decays in the asymptotic regime, one can show that it cannot be removed by a coordinate change without introducing new off-diagonal terms, such that the new solution is not circular. This property appears a the key novelty of this new exact solution. See \cite{Anson} for more details on this point. Hence, the metric is asymptotically flat but, contrary to the Kerr metric, the disformed one is not equivalent to the Schwarzschild metric at infinity
essentially because the difference between the masses $M_1$ and $M_2$. Nevertheless, the deviations introduced by the presence of the cross term proportional to $dr dt$ in the metric become manifest only at next-to-leading order in the asymptotic expansion \cite{Anson}.
\subsection{Some properties of the disformed Kerr space-time}
\label{sec3C}
In this section, we quickly discuss geometrical properties of the disformed Kerr space-time. First of all, we say a few words on its singularities. Even though the metric is
singular in Boyer-Lindquist coordinates when $\Delta=0$ (at the values $r_\pm = M \pm \sqrt{M^2-a^2}$ of the radial coordinate), this is
not a physical singularity but only a coordinate singularity exactly as in the usual Kerr black hole. Indeed, this can be seen immediately from the expressions of the curvature invariants,
\begin{align}
R & = \frac{ \alpha }{1-\alpha} \frac{6 a^2 M r}{\rho^3} \left(\cos^2{\theta} - \frac{1}{3} \right)\, , \\
\label{CI2}
R_{\mu\nu} R^{\mu\nu} & = - \frac{\alpha^2}{\left( 1-\alpha \right)^2} \frac{18 a^4 M^2 }{\left( r^2 + a^2\right)\rho^6} \; P_1(r, \theta, a) \, ,\\
\label{CI3}
R_{\mu\nu\rho\sigma} R^{\mu\nu\rho\sigma} & = \frac{48 M^2}{ \left( 1-\alpha \right)^2 \left( r^2 + a^2\right) \rho^6} \; P_2 \left( r, \theta,a\right) \, ,
\end{align}
where the functions $P_1(r, \theta, a)$ and $P_1(r, \theta, a)$ are given in the Appendix~\ref{AppA}.
As for the Kerr metric, the disformed Kerr geometry is singular at $\rho=0$ only. Nonetheless, one important difference between the disformed and the usual Kerr metric is that the disformed geometry ($\alpha \neq 0$) is, interestingly, no longer Ricci flat.
Then, we see immediately that the disformed geometry admits the two same Killing vectors $\xi_t \equiv \partial_t$ and $\xi_\varphi \equiv \partial_\varphi$ as in the Kerr black hole because none of the
coefficients of the metric depend on $t$ and $\varphi$. As a consequence, we can look at the positions of the ergospheres, i.e.\ the hypersurfaces where the Killing vector field $\xi_t$ is null, i.e.\
\begin{eqnarray}
\xi_t \cdot \xi_t = g_{tt} = 0 \qquad \Longleftrightarrow \qquad r^2 -2 {M_1} r + a^2 \cos^2\theta = 0 \, ,
\end{eqnarray}
where $M_1=M/(1-\alpha)$ as above \eqref{asymptomet}.
As a consequence, the disformed Kerr metric admits, as the usual Kerr metric, an outer and an inner ergospheres denoted
respectively by ${\cal E}^+$ and ${\cal E}^-$ whose positions are given by the same formulae as the Kerr ones,
\begin{eqnarray}
r=r_{{\cal E}^\pm}(\theta)= M_1 \pm \sqrt{M_1^2 - a^2 \cos^2\theta} \, ,
\end{eqnarray}
with the difference that the mass of the black hole has now been rescaled. The ergoregions are defined similarly and one expects the possibility for a Penrose process (with an energy extraction mechanism) to exist in this geometry as well.
Now, let us consider the null directions. Indeed, computing the null directions is particularly interesting to understand the causal structure of a metric and to see whether a metric $g_{\mu\nu}$ describes a black hole (or more generally possesses horizons). These vectors enable us, in particular, to compute light rays (the principal null geodesics) in the space-time and also to characterize the properties of horizons. The normalized (future directed) principal null vectors are denoted by
$\ell_\pm^\mu$ and satisfy the normalization conditions,
\begin{eqnarray}
g_{\mu\nu} \, \ell_\pm ^\mu \, \ell_\pm^\nu \; = \; 0 \; , \qquad g_{\mu\nu} \, \ell_+^\mu \, \ell_-^\nu \; = \; - 1 \, .
\end{eqnarray}
These conditions do not define completely (and then uniquely) the null vectors which can be rescaled according to $\ell_\pm \rightarrow \mathcal{N}^{\pm 1} \ell_\pm$ where $\mathcal{N}$ is an arbitrary (non-vanishing) function.
In the case of the Kerr metric $\tilde{g}_{\mu\nu}$, the null vectors are well-known and are given by,
\begin{eqnarray}
\tilde{\ell}_+^\mu \partial_\mu \equiv \frac{r^2+a^2}{\Delta} \partial_t + \partial_r + \frac{a}{\Delta} \partial_\varphi \,, \qquad
\tilde{\ell}_-^\mu \partial_\mu \equiv \frac{r^2+a^2}{2 \rho} \partial_t - \frac{\Delta}{2\rho}\partial_r + \frac{a}{2 \rho} \partial_\varphi \, .
\end{eqnarray}
Interestingly, we see that, at the horizons where $\Delta=0$, these two null vector fields are proportional to the Killing vector
\begin{eqnarray}
\frac{\Delta}{r^2+a^2} \tilde{\ell}_+ = \frac{2\rho}{r^2+a^2} \tilde{\ell}_- =\partial_t + \Omega_H \, \partial_\varphi \, ,
\end{eqnarray}
where $\Omega_H=a/(2M r_\pm)$ is a constant whose value depends whether we are considering the outer ($r=r_+$) or the inner ($r=r_-$)
horizon. In fact, requiring that a Killing vector is null characterizes completely the horizons in the Kerr geometry where the event horizons are thus also Killing horizons.
One can easily see that the principal directions $\ell_\pm$ of the disformed metric $g_{\mu\nu}$ are also ``disformed'' in the sense
that they are now given by,
\begin{eqnarray}
\ell_\pm^\mu = \tilde{\ell}_\pm^\mu + \beta (\phi_\alpha \tilde{\ell}_\pm^\alpha) \, \phi^\mu \, , \qquad \beta \equiv \frac{(1- B_0 X)^{-1/2} -1}{X} = \frac{1-(1- \alpha)^{-1/2} }{m^2} \, ,
\end{eqnarray}
where $\phi^\mu = \tilde{g}^{\mu\nu} \phi_\nu$ and $X=-m^2$ here. Notice that these formulae generalize immediately to any disformal transformation (when $X$ and $B_0$ are not necessarily constant) of an arbitrary metric $g_{\mu\nu}$.
Interestingly the effect of the disformal transformation on the null directions is a shift of the usual Kerr null vectors in the direction of the gradient of the scalar field $\phi^\mu$. Everything happens as if the scalar field is somehow drifting the light rays.
The explicit expressions of the null directions of the
disformed Kerr metric can be easily written from the relations
\begin{eqnarray}
\phi_\alpha \tilde \ell_+^\alpha & = & -\frac{m \sqrt{r^2+a^2}}{\sqrt{r^2+a^2} + \sqrt{2Mr}} \, , \qquad
\phi_\alpha \tilde \ell_-^\alpha =- m\frac{r^2+a^2 + \sqrt{2Mr(r^2+a^2)}}{2 \rho} \, ,
\end{eqnarray}
which enable us to obtain, after a direct calculation,
\begin{eqnarray}
\ell_+ & = & \left[ \frac{r^2+a^2}{\Delta} + \beta m (\phi_\alpha \tilde \ell_+^\alpha) \left(1+ \frac{\cR}{m^2 \rho \Delta} \right)\right] \partial_t \nonumber \\
&+& \left[ 1 + \beta (\phi_\alpha \tilde \ell_+^\alpha) \frac{\sqrt{\cR}}{\rho}\right] \partial_r - \frac{a}{\Delta}\left[1 + \beta m (\phi_\alpha \tilde \ell_+^\alpha) \frac{2 M r}{\rho} \right] \partial_\varphi \, , \\
\ell_- & = & \left[ \frac{r^2+a^2}{2\rho} + \beta m (\phi_\alpha \tilde \ell_-^\alpha) \left(1+ \frac{\cR}{m^2 \rho \Delta} \right)\right] \partial_t
\nonumber \\
&+ & \left[ -\frac{\Delta}{2 \rho} + \beta (\phi_\alpha \tilde \ell_-^\alpha) \frac{\sqrt{\cR}}{\rho}\right] \partial_r - \frac{a}{2 \rho}\left[1 + \beta m (\phi_\alpha \tilde \ell_-^\alpha) \frac{2 M r}{\Delta} \right] \partial_\varphi .
\end{eqnarray}
If we proceed as in the Kerr case, we would look at the regions where $\ell_+$ or $\ell_-$ becomes proportional to Killing vectors. For
$\ell_+$, we obtain the condition,
\begin{eqnarray}
\Delta \left[ 1 + \beta (\phi_\alpha \tilde \ell_+^\alpha) \frac{\sqrt{\cR}}{\rho}\right] \; = \; 0 \, ,
\end{eqnarray}
which fixes $r$. Interestingly, $r_\pm$ are solutions but there are extra non-trivial solutions which are given by $r=F(\theta)$, i.e. $r$
is a function of $\theta$ and whose limit $\alpha \rightarrow 0$ is not defined.
For $\ell_-$, we obtain the condition,
\begin{eqnarray}
-{\Delta} + 2 \beta (\phi_\alpha \tilde \ell_-^\alpha) \sqrt{\cR} \; = \; 0 \, ,
\end{eqnarray}
which also fixes $r$ at some non trivial function of $\theta$. In both cases, both null vectors reduce to a vector field proportional to,
\begin{eqnarray}
\partial_t + \Omega_\pm \, \partial_\varphi \, ,
\end{eqnarray}
where $\Omega_\pm$ is no more a constant and depends on $\theta$.
Therefore, none of the principal null directions reduce to Killing vectors in some hypersurfaces, and hence the horizons
(if they exist) cannot be Killing vectors\footnote{This has also been observed in another but equivalent way in \cite{Anson}}. This is an important difference with the Kerr geometry.
Furthermore, one can easily check that the hypersurfaces of constant $r$ are not null because the norm of their normal vector $\partial_r$ is given by \eqref{inversedisf}
\begin{eqnarray}
g^{rr}= \frac{\Delta}{\rho} + \frac{\alpha}{1-\alpha}\frac{2Mr(r^2+a^2)}{\rho^2} \, ,
\end{eqnarray}
and therefore depends on $\theta$ through $\rho=r^2+a^2 \cos^2\theta$. As a consequence, the horizons of the disformed Kerr metric cannot be obtained in the way we get the
Kerr horizons.
The problem of finding event horizons seems complicated but it has been initiated very recently in \cite{Anson} where first candidates has been proposed and analyzed. The basic idea is rather simple and consists in looking at null hypersurfaces defined by an equation of the form $F(r,\theta)=0$ with a $\theta$-dependency contrary to the Kerr case. We assume that we can locally solve $r$ as a function of $\theta$ and restrict ourselves to separable functions of the form $F(r,\theta)=r+F(\theta)$.
The condition that such an hypersurface is null implies that its normal vector
$(0,1, \partial_\theta F,0)$ is also null (by definition), which leads to a non-linear differential equation for $F(\theta)$,
\begin{eqnarray}
g^{rr} + g^{\theta \theta} \left(\frac{dF}{d\theta}\right)^2 \, = \, 0 \, \Longleftrightarrow \, \left[{\Delta} + \frac{\alpha}{1-\alpha}\frac{2Mr(r^2+a^2)}{\rho} \right] + \left(\frac{dF}{d\theta}\right)^2 \, = \, 0 \, ,
\end{eqnarray}
where we used \eqref{inversedisf} for the coefficients of the inverse disformed metric and $r=-F(\theta)$ everywhere in this equation. It is the
same equation as Eq.(23) in \cite{Anson}. The geometry of this null hypersurface is subtle as the detailed and very interesting analysis in \cite{Anson} shows, but it is still an open issue to show whether it is an event horizon or not. Computing its expansions may help and we hope to study this issue in details in a future work. Nevertheless, it is worth emphasizing that the characterization of quasi-local horizon through the expansions of the null directions is slicing dependent, and the choice of the null directions and thus of the $2$-surface foliating our geometry is therefore ambiguous as different choices might allow to identify different quasi-local (not necessarily null) horizons. See \cite{Faraoni:2016xgy, Schnetter:2005ea} for detailed discussions on this point.
\medskip
We finish with a quick discussion on the geodesic equations in the disformed Kerr background. Following the same method as in the case
of the Kerr metric, the geodesic equations can be obtained from the Hamilton-Jacobi equation for the ``action'' $S$,
\begin{eqnarray}
H(x^\mu, {\partial_\mu S}) + \frac{\partial S}{\partial \lambda} = 0 \, , \qquad H(x^\mu,p_\mu) \equiv \frac{1}{2} g^{\mu\nu} p_\mu p_\nu \, ,
\end{eqnarray}
where $\lambda$ is the affine parameter along the geodesic. Due to the invariance of the disformed metric (whose components do not depend neither on $t$ nor on $\varphi$), the action $S$ takes the form
\begin{eqnarray}
S(t,r,\theta,\varphi) \; = \; \frac{1}{2} \mu^2 \lambda + p_t t + p_\varphi \varphi + \Phi(r,\theta) \, ,
\end{eqnarray}
where $\mu$, $p_t$ and $p_\varphi$ are the standard constants of motion. A straightforward calculation shows that $\Phi(r,\theta)$
satisfies the differential equation,
\begin{eqnarray}
0&=&\left[ \mu^2 r^2 + \Phi_r^2 \Delta - \frac{(r^2+a^2)^2}{\Delta} - \frac{4Mra}{\Delta} p_t p_\varphi - \frac{a^2}{\Delta} p_\varphi^2 \right] \nonumber \\
&+&\left[ \mu^2 a^2 \cos^2\theta + \Phi_\theta^2 + a^2 p_t^2 \sin^2\theta + \frac{p_\varphi^2}{\sin^2\theta}\right] \label{HJeq} \\
&-&\frac{\alpha}{(1-\alpha m^2)\rho} \left[ \frac{m}{\Delta} (r^2+a^2)^2 p_t + \sqrt{\cR} \Phi_r - ma^2 p_t \sin^2\theta \right] ^2 \, ,
\nonumber
\end{eqnarray}
where $\Phi_r\equiv \partial \Phi/\partial r$ and $\Phi_\theta\equiv \partial \Phi/\partial \theta$.
In the case where $\alpha=0$, the equation is clearly separable as the first line depends only on $r$ while the second one depends on $\theta$ only. This makes the geodesic equation integrable. Furthermore, the separability of the Hamilton-Jacobi equation is intimately linked to the existence of the famous Carter constant and of a hidden symmetry of the Kerr metric (associated to a Killing tensor)~\cite{Carter:1968rr}.
When $\alpha \neq 0$, the Hamilton-Jacobi equation is no more separable in the Boyer-Lindquist coordinates and it is very likely that the geodesic equation is no more integrable and one cannot find a ``disformed'' Carter constant associated to the
disformed Kerr metric. Interestingly, there is an obvious solution of the equation \eqref{HJeq} given by
\begin{eqnarray}
\Phi= z \int dr \, \frac{\sqrt{\cR}}{\Delta} \, ,
\end{eqnarray}
where $z$ is a
constant when the integration constants $\mu,p_t$ and $p_\varphi$ coincide with those of the scalar field according to,
\begin{eqnarray}
p_t=-zm \, , \quad
p_\theta=0 \, , \quad
p_\varphi=0 \, , \quad
\mu^2 = \frac{z^2m^2}{1-\alpha m^2} \, .
\end{eqnarray}
In that case, the geodesic follows exactly the gradient of the scalar field.
\section{Discussion and perspectives}
\label{sec4}
The disformal-generating method that was recently introduced in \cite{BenAchour:2019fdf} appears to be very useful to construct new exact
solutions in DHOST theories. It enables us, in this paper, to construct the first non-stealth rotating solution in DHOST theories where the geometry is given by a disformed Kerr metric while the scalar field $\phi(t,r)$ has a non-trivial profile with a constant kinetic density $X$. Even
though this solution is equivalent to the stealth Kerr solution of \cite{Charmousis:2019vnf} in vacuum, it becomes physically inequivalent when one considers coupling to matter (for instance, the geodesic motion of test particles is different from the geodesic motion in the Kerr black hole as we briefly showed in the last part of the paper).
We started with a quick review on quadratic DHOST theories and the conditions of existence of stealth solutions where the metric is
also a solution of GR. Then, we found the general conditions for a DHOST theory to have a ``disformed'' stealth solution
where the metric is a ``disformed'' solution of GR while the scalar field has a non-trivial profile. This is an important result because
it allows one to identify DHOST theories which admit a disformed stealth metric as a solution. Then, we performed a disformal transformation of the theories which admits a stealth Kerr solution and we obtained a family of scalar-tensor theories which admits disformed Kerr solutions. We have restricted ourselves to invertible and shift symmetry disformal transformation. As a consequence, working with a seed stealth Kerr solution with constant kinetic term imposes that $A=A_0$
and $B=B_0$ are constant, providing a drastic simplification. Under these assumptions, we have obtained the first non-stealth rotating solution in DHOST theories where the metric depends on
three parameters which are the usual mass $m$ and spin parameter $a$ together with a new deformation parameter $\alpha$.
We analyzed some geometrical properties of the new metric. It is easy to see that, contrary to the Kerr metric, the disformed Kerr metric is not Ricci flat but (in the case where the scalar field does not depend on $\theta$ in the Boyer-Lindquist coordinates system) it remains asymptotically flat, the metric still has two Killing vectors associated to the fact that the metric components are independent of
$t$ and $\varphi$ (still in Boyer-Lindquist coordinates), there is the same ring singularity at $\rho=0$ as in Kerr, and the space-time
admits ergospheres and ergoregions very similar to those in Kerr space-time. However, there are important differences with Kerr. First,
we showed that the null directions are disformed, which lead to modifications of the structure of the horizons. In particular, if horizons exist,
there can no longer be Killing horizons and cannot be given by $r=\rm const$ in Boyer-Lindquist coordinates. So far, we have no proof that the disformed solution is a black hole. These issues have been analyzed
in the recent paper \cite{Anson} where a candidate for the event horizons has been proposed. However, understanding the geometry and more particularly the causal structure of the disformed Kerr metric deserve to be studied in details, what we hope to do next.
Many other questions, that we hope to address in future works, remain open: analyzing the geodesic motion, studying the thermodynamics, etc.
Finally, to illustrate again the potentiality of the disformal solution-generating method, we present in addition another axisymmetric solution for DHOST theories obtained from a disformal transformation of the generalized Kerr solution of Einstein-Scalar gravity in the Appendix~\ref{AppB}.
\medskip
\acknowledgments
The work of JBA was supported by Japan Society for the Promotion of Science (JSPS) Grants-in-Aid for Scientific Research (KAKENHI) No.\ 17H02890.
HM was supported by JSPS KAKENHI No.\ 18K13565.
The work of SM was supported by JSPS KAKENHI No.\ 17H02890, No.\ 17H06359, and by World Premier International Research Center Initiative, MEXT, Japan.
KN acknowledges support from the CNRS grant 80PRIME and thanks the Laboratory of Physics at the ENS in Paris for hospitality during this exceptional period.
We would like to thank the authors of \cite{Anson} for letting us know about their work prior to submission to arXiv and KN is especially grateful to Christos Charmousis and Eugeny Babichev for our discussions on the disformed Kerr geometry.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,410
|
Detto di foglie, pesci o rettili, elementi contigui in parte sovrapposti come le tegole di un tetto.
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,334
|
Q: Can Java Web Start install a Windows driver, and also copy dll and jar into the Java dirs? My Java FX2 application configures a serial device and as part of deployment, it has to install a Windows device driver and also copy native libs (RxTx serial port libs) over into the Java dirs.
Can Java Web Start handle it? Or should user do it in stages: install driver manually, then copy over libs, then install application via Web Start.
Update:
MyApp.jar below is self-signed. I put the dll under the project name. I added this line for the dll in my jnlp. I am using Netbeans 7.3 and so put the VM args line.
My jnlp looks like this:
<resources>
<j2se version="1.6+" java-vm-args="-Djava.library.path=. " href="http://java.sun.com/products/autodl/j2se"/>
<jar href="MyApp.jar" size="216992" download="eager" />
<jar href="lib/RXTXcomm.jar" size="66493" download="eager" />
<nativelib href="rxtxSerial.dll" size="122880" download="eager" />
</resources>
<security>
<all-permissions/>
</security>
I can run the app in the Netbeans IDE. However, clicking the jnlp, I get the popup, 'Runtime Error'. It asks me to click for details, but there's nothing shown.
I deleted earlier jnlps in the Temporary Internet Files in Java Control Panel. In the Java console I see nothing.
@AndrewThompson I tried Janela to check the jnlp. Useful tool! I see that the dll did not make it into the dist folder. The jnlp is generated by NB 7.3. Is there any way to get it to put in the line for nativelib? I have to do it manually each time.
`
JaNeLA Report - version 11.05.17
Report for
file:/D:/Profiles/Anil/My%20Documents/NetBeansProjects/MyApp/dist/MyApp.jnlp
Content type application/xml does not equal expected type of
application/x-java-jnlp-file cvc-complex-type.2.4.a: Invalid content
was found starting with element 'jfx:javafx-runtime'. One of '{java,
j2se, jar, nativelib, extension, property, package}' is expected.
cvc-complex-type.2.4.a: Invalid content was found starting with
element 'jfx:javafx-runtime'. One of '{java, j2se, jar, nativelib,
extension, property, package}' is expected. cvc-complex-type.2.4.a:
Invalid content was found starting with element 'security'. One of
'{resources, application-desc, applet-desc, component-desc,
installer-desc}' is expected. cvc-complex-type.2.4.a: Invalid content
was found starting with element 'security'. One of '{resources,
application-desc, applet-desc, component-desc, installer-desc}' is
expected. XML encoding not known, but declared as utf-8 Codebase not
specified. Defaulting to
file:/D:/Profiles/Anil/My%20Documents/NetBeansProjects/MyApp/dist/ The
resource download at MyApp.jar can be optimized by removing the
(default) value of download='eager'. The resource download at
MyApp.jar can be optimized by removing the (default) value of
main='false'. It might be possible to optimize the start-up of the
app. by specifying download='lazy' for the MyApp.jar resource. Lazy
downloads might not work as expected for MyApp.jar unless the download
'part' is specified. Resource 'lib/RXTXcomm.jar' declared as size
'66490' but is actually '66512'. The resource download at
lib/RXTXcomm.jar can be optimized by removing the (default) value of
download='eager'. The resource download at lib/RXTXcomm.jar can be
optimized by removing the (default) value of main='false'. It might be
possible to optimize the start-up of the app. by specifying
download='lazy' for the lib/RXTXcomm.jar resource. Lazy downloads
might not work as expected for lib/RXTXcomm.jar unless the download
'part' is specified. Problem fetching resource rxtxSerial.dll.
D:\Profiles\Anil\My
Documents\NetBeansProjects\MyApp\dist\rxtxSerial.dll (The system
cannot find the file specified)
`
I put the dll into a jar, and added
System.loadLibrary("./rxtxSerial");
That too failed.
A: As far as you remain in the sandbox (unsigned app) you won't access to the local file system. Regarding the device driver installation i suppose a signed app could do it too, but it will probably trigger Windows confirmation.
See http://docs.oracle.com/javase/tutorial/deployment/doingMoreWithRIA/security.html
A: What worked was this
Using a simpler approach (confirmed to work when using NetBeans or
Borland JBuilder):
Add RXTXcomm.jar as a library
compile (build) your application
add the rxtxSerial.dll to the root of the distribution folder
(projectname/dist when using NetBeans)
from the wiki http://rxtx.qbang.org/wiki/index.php/Deploying_JAVA_with_RXTX
The advice (from Andrew) to put a 'nativelib' directive in the jnlp, did not work.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,760
|
Bathyaulax appelatrix — вид паразитических наездников рода Bathyaulax из семейства Braconidae.
Распространение
Встречается в Африке (ЮАР).
Описание
Бракониды среднего размера, длина тела около 1 см (тело 13 мм, переднее крыло 12 мм, яйцеклад 11 мм). Усики тонкие, нитевидные От близких родов отличается следующими признаками: 4-й тергит гладкий; срединная область 3-го тергита полосатая, но не приподнятая и не выемчатая; лицо гранулированное . Основная окраска оранжево-коричневая, кроме чёрных усиков, лица, вершин мандибул и яйцеклада. Предположительно, как и близкие виды, паразитоиды личинок древесных жуков. Вид был впервые описан в 1909 году, а его валидный статус подтверждён входе ревизии, проведённой 2007 году энтомологами Austin Kaartinen (University of Helsinki, Финляндия) и Donald Quicke (Chulalongkorn University, Бангкок, Таиланд).
См. также
Mama mariae
Примечания
Литература
Quicke D. L. J., Laurenne N. M., Barclay M. 2005. A new host record for the Afrotropical parasitic wasp genus Bathyaulax Szepligeti (Hymenoptera: Braconidae: Braconinae) confirmed using DNA sequence data. Journal of Hymenoptera Research 14:96-101.
Ссылки
waspweb.org: Bathyaulax
eol.org: Bathyaulax
Bathyaulax
Животные, описанные в 1909 году
Насекомые Африки
Эндемики Африки
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,876
|
The Brit Hotel Aux Hortensias offers you 19 spacious and comfortable rooms. We have 6 rooms with a kitchenette and a separate lounge and 9 have a balcony. Our hotel also has a room for disabled guests.
At the Brit Hotel Aux Hortensias , we strive to make you feel at home! Private bathroom, flat screen TV with Canal + and Being SPORTS and an internet connection Wi-Fi and free throughout the hotel.
This offer is not cancellable or refundable.
. Gouffre de Plougrescant, 20 km.
Take Paris-Rennes motorway (A11), then Rennes-Brest expressway (RN12).
Take Lannion exit towards Perros Guirec.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,691
|
/**
* @file sf1_msgpack_serialization_types.h
* @author Zhongxia Li
* @date Jul 28, 2011
* @brief Make SF1 data types can be serialized by MSGPACK.
*/
#ifndef SF1_MSGPACK_SERIALIZATION_TYPES_H_
#define SF1_MSGPACK_SERIALIZATION_TYPES_H_
#include <3rdparty/msgpack/msgpack/object.hpp>
#include <3rdparty/msgpack/msgpack/type/int.hpp>
#include <3rdparty/msgpack/msgpack/type/float.hpp>
#include <3rdparty/msgpack/msgpack/type/string.hpp>
#include <3rdparty/msgpack/msgpack/type/UString.hpp>
#include <ranking-manager/RankingEnumerator.h>
#include <query-manager/SearchingEnumerator.h>
#include <query-manager/QueryTypeDef.h>
#include <common/PropertyValue.h>
#include <util/ustring/UString.h>
namespace msgpack {
/// RankingType
inline sf1r::RankingType::TextRankingType& operator>>(object o, sf1r::RankingType::TextRankingType& v)
{
signed int iv = type::detail::convert_integer<signed int>(o);
v = static_cast<sf1r::RankingType::TextRankingType>(iv);
return v;
}
template <typename Stream>
inline packer<Stream>& operator<< (packer<Stream>& o, const sf1r::RankingType::TextRankingType& v)
{ o.pack_int(static_cast<int>(v)); return o; }
inline void operator<< (object& o, sf1r::RankingType::TextRankingType v)
{ o.type = type::POSITIVE_INTEGER, o.via.u64 = static_cast<int>(v); }
inline void operator<< (object::with_zone& o, sf1r::RankingType::TextRankingType v)
{ static_cast<object&>(o) << static_cast<int>(v); }
/// SearchingModeType
inline sf1r::SearchingMode::SearchingModeType& operator>>(object o, sf1r::SearchingMode::SearchingModeType& v)
{
signed int iv = type::detail::convert_integer<signed int>(o);
v = static_cast<sf1r::SearchingMode::SearchingModeType>(iv);
return v;
}
template <typename Stream>
inline packer<Stream>& operator<< (packer<Stream>& o, const sf1r::SearchingMode::SearchingModeType& v)
{
o.pack_int(static_cast<int>(v));
return o;
}
inline void operator<< (object& o, sf1r::SearchingMode::SearchingModeType v)
{
o.type = type::POSITIVE_INTEGER,
o.via.u64 = static_cast<int>(v);
}
inline void operator<< (object::with_zone& o, sf1r::SearchingMode::SearchingModeType v)
{
static_cast<object&>(o) << static_cast<int>(v);
}
/// SuffixMatchFilterMode
inline sf1r::SearchingMode::SuffixMatchFilterMode& operator>>(object o, sf1r::SearchingMode::SuffixMatchFilterMode& v)
{
signed int iv = type::detail::convert_integer<signed int>(o);
v = static_cast<sf1r::SearchingMode::SuffixMatchFilterMode>(iv);
return v;
}
template <typename Stream>
inline packer<Stream>& operator<< (packer<Stream>& o, const sf1r::SearchingMode::SuffixMatchFilterMode& v)
{
o.pack_int(static_cast<int>(v));
return o;
}
inline void operator<< (object& o, sf1r::SearchingMode::SuffixMatchFilterMode v)
{
o.type = type::POSITIVE_INTEGER,
o.via.u64 = static_cast<int>(v);
}
inline void operator<< (object::with_zone& o, sf1r::SearchingMode::SuffixMatchFilterMode v)
{
static_cast<object&>(o) << static_cast<int>(v);
}
///// QueryFiltering::FilteringOperation
//inline sf1r::QueryFiltering::FilteringOperation& operator>>(object o, sf1r::QueryFiltering::FilteringOperation& v)
//{
// signed int iv = type::detail::convert_integer<signed int>(o);
// v = static_cast<sf1r::QueryFiltering::FilteringOperation>(iv);
// return v;
//}
//
//template <typename Stream>
//inline packer<Stream>& operator<< (packer<Stream>& o, const sf1r::QueryFiltering::FilteringOperation& v)
// { o.pack_int(static_cast<int>(v)); return o; }
//
//inline void operator<< (object& o, sf1r::QueryFiltering::FilteringOperation v)
// { o.type = type::POSITIVE_INTEGER, o.via.u64 = static_cast<int>(v); }
//
//inline void operator<< (object::with_zone& o, sf1r::QueryFiltering::FilteringOperation v)
// { static_cast<object&>(o) << static_cast<int>(v); }
//
///// QueryFiltering::InterFilteringLogic
//inline sf1r::QueryFiltering::InterFilteringLogic& operator>>(object o, sf1r::QueryFiltering::InterFilteringLogic& v)
//{
// signed int iv = type::detail::convert_integer<signed int>(o);
// v = static_cast<sf1r::QueryFiltering::InterFilteringLogic>(iv);
// return v;
//}
//
//template <typename Stream>
//inline packer<Stream>& operator<< (packer<Stream>& o, const sf1r::QueryFiltering::InterFilteringLogic& v)
// { o.pack_int(static_cast<int>(v)); return o; }
//
//inline void operator<< (object& o, sf1r::QueryFiltering::InterFilteringLogic v)
// { o.type = type::POSITIVE_INTEGER, o.via.u64 = static_cast<int>(v); }
//
//inline void operator<< (object::with_zone& o, sf1r::QueryFiltering::InterFilteringLogic v)
// { static_cast<object&>(o) << static_cast<int>(v); }
///// PropertyValue
//static const char STR_TYPE_STRING = 's';
//static const char STR_TYPE_USTRING = 'u';
//
//inline sf1r::PropertyValue& operator>>(object o, sf1r::PropertyValue& v)
//{
// if(o.type == type::POSITIVE_INTEGER)
// {
// if (o.raw_type == type::RAW_UINT64)
// {
// v = type::detail::convert_integer<uint64_t>(o); return v;
// }
// else
// {
// v = type::detail::convert_integer<int64_t>(o); return v;
// }
// }
// else if(o.type == type::NEGATIVE_INTEGER)
// {
// v = type::detail::convert_integer<int64_t>(o); return v;
// }
// else if (o.type == type::DOUBLE)
// {
// v = (double)o.via.dec;
// }
// else if (o.type == type::RAW)
// {
// std::string s;
// o >> s;
//
// if (s.length() <= 1)
// {
// v = s;
// }
// else if (s[0] == STR_TYPE_USTRING)
// {
// std::string str = s.substr(1);
// izenelib::util::UString ustr(str, izenelib::util::UString::UTF_8);
// v = ustr;
// }
// else
// {
// std::string str = s.substr(1);
// v = str;
// }
// }
//
// return v;
//}
//
//template <typename Stream>
//inline packer<Stream>& operator<< (packer<Stream>& o, const sf1r::PropertyValue& cv)
//{
// sf1r::PropertyValue& v = const_cast<sf1r::PropertyValue&>(cv);
// if (int64_t* p = boost::get<int64_t>(&v.getVariant()))
// {
// o.pack_fix_int64(*p);
// }
// else if (uint64_t* p = boost::get<uint64_t>(&v.getVariant()))
// {
// o.pack_fix_uint64(*p);
// }
// else if (const float* p = boost::get<float>(&v.getVariant()))
// {
// o.pack_double(*p);
// }
// else if (double* p = boost::get<double>(&v.getVariant()))
// {
// o.pack_double(*p);
// }
// else if (std::string* p = boost::get<std::string>(&v.getVariant()))
// {
// std::string packstr;
// packstr += STR_TYPE_STRING;
// packstr += *p;
// o.pack_raw(packstr.size());
// o.pack_raw_body(packstr.c_str(), packstr.size());
// }
// else if (izenelib::util::UString* p = boost::get<izenelib::util::UString>(&v.getVariant()))
// {
// std::string packstr;
// packstr += STR_TYPE_USTRING;
// std::string s;
// (*p).convertString(s, izenelib::util::UString::UTF_8);
// packstr += s;
// o.pack_raw(packstr.size());
// o.pack_raw_body(packstr.data(), packstr.size());
// }
// else
// {
// // reserved
// //std::vector<izenelib::util::UString>,
// //std::vector<uint32_t>
//
// throw type_error();
// }
//
// return o;
//}
//
//inline void operator<< (object& o, sf1r::PropertyValue v)
//{
// if (int64_t* p = boost::get<int64_t>(&v.getVariant()))
// {
// *p < 0 ? (o.type = type::NEGATIVE_INTEGER, o.via.i64 = *p, o.raw_type = type::RAW_INT64)
// : (o.type = type::POSITIVE_INTEGER, o.via.u64 = *p);
// }
// else if (uint64_t* p = boost::get<uint64_t>(&v.getVariant()))
// {
// o.type = type::POSITIVE_INTEGER, o.via.u64 = *p, o.raw_type = type::RAW_UINT64;
// }
// else if (float* p = boost::get<float>(&v.getVariant()))
// {
// o << *p;
// }
// else if (double* p = boost::get<double>(&v.getVariant()))
// {
// o << *p;
// }
// else if (std::string* p = boost::get<std::string>(&v.getVariant()))
// {
// std::string packstr;
// packstr += STR_TYPE_STRING;
// packstr += *p;
// o << packstr;
// }
// else if (izenelib::util::UString* p = boost::get<izenelib::util::UString>(&v.getVariant()))
// {
// std::string packstr;
// packstr += STR_TYPE_USTRING;
// std::string s;
// (*p).convertString(s, izenelib::util::UString::UTF_8);
// packstr += s;
// o << packstr;
// }
// else
// throw type_error();
//}
//
//inline void operator<< (object::with_zone& o, sf1r::PropertyValue v)
//{
// if (int64_t* p = boost::get<int64_t>(&v.getVariant()))
// {
// static_cast<object&>(o) << *p;
// }
// else if (uint64_t* p = boost::get<uint64_t>(&v.getVariant()))
// {
// static_cast<object&>(o) << *p;
// }
// else if (float* p = boost::get<float>(&v.getVariant()))
// {
// static_cast<object&>(o) << *p;
// }
// else if (double* p = boost::get<double>(&v.getVariant()))
// {
// static_cast<object&>(o) << *p;
// }
// else if (std::string* p = boost::get<std::string>(&v.getVariant()))
// {
// std::string packstr;
// packstr += STR_TYPE_STRING;
// packstr += *p;
// static_cast<object&>(o) << packstr;
// }
// else if (izenelib::util::UString* p = boost::get<izenelib::util::UString>(&v.getVariant()))
// {
// std::string packstr;
// packstr += STR_TYPE_USTRING;
// std::string s;
// (*p).convertString(s, izenelib::util::UString::UTF_8);
// packstr += s;
// static_cast<object&>(o) << packstr;
// }
// else
// throw type_error();
//}
} // namespace
#endif /* SF1_MSGPACK_SERIALIZATION_TYPES_H_ */
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,937
|
The effects of repeated intra-articular PRP injections on clinical outcomes of early osteoarthritis of the knee
Knee Surg Sports Traumatol Arthrosc. 2015 Aug;23(8):2170-2177. doi: 10.1007/s00167-014-2987-4. Epub 2014 Apr 20.
Alberto Gobbi 1 , Dnyanesh Lad 2 , Georgios Karnatzikos 2
1 O.A.S.I. Bioresearch Foundation, Via Amadeo 24, 20133, Milan, Italy. gobbi@cartilagedoctor.it.
2 O.A.S.I. Bioresearch Foundation, Via Amadeo 24, 20133, Milan, Italy.
Purpose: To assess the outcome of intra-articular platelet-rich plasma (PRP) injections into the knee in patients with early stages of osteoarthritis (OA) and to determine whether cyclical dosing would affect the end result.
Methods: This is a prospective, randomized study in which 93 patients (119 knees) were followed up for a minimum of 2 years. Fifty knees were randomly selected prior to the first injection, to receive a second cycle at the completion of 1 year. A cycle consisted of three injections, each given at a monthly interval. The outcome was assessed using Knee Injury and Osteoarthritis Outcome Score (KOOS), Visual Analogue Scale (VAS), Tegner and Marx scoring systems, recorded prior to the first injection and then at 12, 18 and 24 months.
Results: There was a significant improvement in all scores over time compared to the pre-treatment value (p < 0.001). At 12 months, both groups showed similar and significant improvement. At 18 months, except for KOOS (Symptoms) and Tegner score, all other parameters showed a significant difference between the two groups in favour of the patients who had received the second cycle (p < 0.001). At 2 years, the scores declined in both groups but remained above the pre-treatment value with no significant difference between the groups despite the patients with two cycles showing higher mean values for all the scores.
Conclusion: Intra-articular PRP injections into the knee for symptomatic early stages of OA are a valid treatment option. There is a significant reduction in pain and improvement in function after 12 months, which can be further improved at 18 months by annual repetition of the treatment. Although the beneficial effects are ill sustained at 2 years, the results are encouraging when compared to the pre-treatment function.
Level of evidence: II.
Injections, Intra-Articular
Osteoarthritis, Knee / therapy*
Platelet-Rich Plasma*
Visual Analog Scale
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,279
|
package patcher
import (
"context"
"encoding/json"
"fmt"
"log"
"os"
"path/filepath"
"runtime"
"strings"
"testing"
"time"
"github.com/gamejolt/joltron/concurrency"
"github.com/gamejolt/joltron/fs"
"github.com/gamejolt/joltron/game/data"
"github.com/gamejolt/joltron/launcher"
"github.com/gamejolt/joltron/test"
)
var (
bigDownloadXZ = &test.Fixture{
Name: ".gj-bigTempFile.tar.xz",
Checksum: "ca292a1cfa2d93f6e07feffa6d53e836",
}
bigDownloadGZ = &test.Fixture{
Name: ".gj-bigTempFile.tar.gz",
Checksum: "9c48bcb8e17b16e835b1b7c051ce4ef0",
}
patch1File = &test.Fixture{
Name: ".gj-testPatcher1.tar.xz",
Checksum: "f530baefafcc5e2158b3d680b60df62d",
}
patch2File = &test.Fixture{
Name: ".gj-testPatcher2.tar.xz",
Checksum: "a49fbb734965fb92ffbf990e7cfa9c14",
}
build1File = &test.Fixture{
Name: ".gj-testDiff-Build1-clean.tar.xz",
Checksum: "039652f51d6cf49328bffa19f2d18743",
}
currentFile = &test.Fixture{
Name: ".gj-testDiff-Build1.tar.xz",
Checksum: "bb008c0036e7aaa2781417c995c45ad2",
}
diffFile = &test.Fixture{
Name: ".gj-testDiff-Diff.tar.xz",
Checksum: "c81516f083d574f166618c6cbc123305",
}
)
var (
oldMetadata data.BuildMetadata
newMetadata data.BuildMetadata
diffMetadata data.DiffMetadata
)
func getUpdateMetadata(version, url, checksum string, remoteSize int64, sideBySide bool) *data.UpdateMetadata {
updateMetadata := &data.UpdateMetadata{
GameUID: "game-" + version,
URL: url,
Checksum: checksum,
RemoteSize: remoteSize,
OS: runtime.GOOS,
Arch: runtime.GOARCH,
Executable: "./",
SideBySide: &sideBySide,
}
if *updateMetadata.SideBySide == true {
updateMetadata.DataDir = fmt.Sprintf("data-%s", updateMetadata.GameUID)
} else {
updateMetadata.DataDir = "data"
}
return updateMetadata
}
func TestMain(m *testing.M) {
test.EnsureFixtures(bigDownloadXZ, bigDownloadGZ, patch1File, patch2File, currentFile, diffFile)
oldMetadata = data.BuildMetadata{
Files: map[string]data.FileMetadata{
"assets/icon/game-icon (identical).txt": data.FileMetadata{
Size: 259,
Checksum: "35afb487d7a128c9f3006ab1b2ae8e6a",
},
"assets/maps/map1 (deleted).txt": data.FileMetadata{
Size: 259,
Checksum: "49136019440a92076f106396cf87d010",
},
"assets/sounds/jump (2 similar).txt": data.FileMetadata{
Size: 259,
Checksum: "f8929a0ae50c3561a9be1de4da9d6e74",
},
"assets/sprites/enemy (renamed).txt": data.FileMetadata{
Size: 259,
Checksum: "b08c52291b88b33911d82569588ecb0f",
},
"assets/sprites/player (identical).txt": data.FileMetadata{
Size: 260,
Checksum: "cc369cef611012ff4b40664d9d14f7a5",
},
"framework/current/license_mit (2 identical).txt": data.FileMetadata{
Size: 259,
Checksum: "32ae4d79f8ef9233f90e8f8a76ac217a",
},
"framework/current/runner (similar).txt": data.FileMetadata{
Size: 259,
Checksum: "718999ce58d76eeeb998730c3b6a8cc8",
},
"framework/v1/license_mit (2 identical).txt": data.FileMetadata{
Size: 259,
Checksum: "32ae4d79f8ef9233f90e8f8a76ac217a",
},
"framework/v1/runner (similar).txt": data.FileMetadata{
Size: 259,
Checksum: "718999ce58d76eeeb998730c3b6a8cc8",
},
"settings (invalid).txt": data.FileMetadata{
Size: 259,
Checksum: "5b12b816453874500c8fbc515609e9f7",
},
},
Dirs: []string{"assets", "assets/icon", "assets/maps", "assets/sounds", "assets/sprites", "framework", "framework/v1"},
Symlinks: map[string]string{
"framework/current": "v1/",
"game (indirect)": "framework/current/runner (similar).txt",
"icon (unchanged)": "assets/icon/game-icon (identical).txt",
"license_mit (2 identical) (removed).txt": "framework/v1/license_mit (2 identical).txt",
},
}
newMetadata = data.BuildMetadata{
Files: map[string]data.FileMetadata{
"assets/backgrounds/forest (created).txt": data.FileMetadata{
Size: 259,
Checksum: "341750ed999d0409493b3e39c054a76e",
},
"assets/icon/game-icon (identical).txt": data.FileMetadata{
Size: 259,
Checksum: "35afb487d7a128c9f3006ab1b2ae8e6a",
},
"assets/sounds/jump1 (2 similar).txt": data.FileMetadata{
Size: 268,
Checksum: "1391fc6089d2c7b49d4e023319dbde09",
},
"assets/sounds/jump2 (2 similar).txt": data.FileMetadata{
Size: 268,
Checksum: "4d80eb06f7528743d6c73c311ae8b371",
},
"assets/sprites/ogre (created).txt": data.FileMetadata{
Size: 259,
Checksum: "c12294dd3ed128ca486ee4ceec62d54d",
},
"assets/sprites/player (identical).txt": data.FileMetadata{
Size: 260,
Checksum: "cc369cef611012ff4b40664d9d14f7a5",
},
"assets/sprites/slime (renamed).txt": data.FileMetadata{
Size: 259,
Checksum: "b08c52291b88b33911d82569588ecb0f",
},
"framework/current/license_mit (2 identical).txt": data.FileMetadata{
Size: 259,
Checksum: "32ae4d79f8ef9233f90e8f8a76ac217a",
},
"framework/current/runner (similar).txt": data.FileMetadata{
Size: 267,
Checksum: "4addc5bec539ad76738585193783879a",
},
"framework/v2/license_mit (2 identical).txt": data.FileMetadata{
Size: 259,
Checksum: "32ae4d79f8ef9233f90e8f8a76ac217a",
},
"framework/v2/runner (similar).txt": data.FileMetadata{
Size: 267,
Checksum: "4addc5bec539ad76738585193783879a",
},
"license_mit (2 identical).txt": data.FileMetadata{
Size: 259,
Checksum: "32ae4d79f8ef9233f90e8f8a76ac217a",
},
"metadata (invalid).txt": data.FileMetadata{
Size: 259,
Checksum: "2ef852c4936c7d2feac1bbbcd38338d8",
},
"settings (invalid).txt/settings (invalid).txt": data.FileMetadata{
Size: 268,
Checksum: "958341cbfcfdf1b672cc995a5cb3abd3",
},
},
Dirs: []string{"assets", "assets/backgrounds", "assets/icon", "assets/sounds", "assets/sprites", "framework", "framework/v2", "settings (invalid).txt"},
Symlinks: map[string]string{
"framework/current": "v2/",
"game (indirect)": "framework/current/runner (similar).txt",
"icon (unchanged)": "assets/icon/game-icon (identical).txt",
},
}
diffMetadata = data.DiffMetadata{
Identical: map[string][]string{
"assets/icon/game-icon (identical).txt": []string{"assets/icon/game-icon (identical).txt"},
"assets/sprites/enemy (renamed).txt": []string{"assets/sprites/slime (renamed).txt"},
"assets/sprites/player (identical).txt": []string{"assets/sprites/player (identical).txt"},
"framework/v1/license_mit (2 identical).txt": []string{"framework/v2/license_mit (2 identical).txt", "license_mit (2 identical).txt"},
},
Similar: map[string][]data.SimilarFile{
"assets/sounds/jump (2 similar).txt": []data.SimilarFile{
data.SimilarFile{
ChunkSize: 51200000,
New: "assets/sounds/jump1 (2 similar).txt",
Size: 268,
Patches: []data.SimilarFilePart{
data.SimilarFilePart{
File: "assets/sounds/jump1 (2 similar).txt",
Size: 169,
Checksum: "f622d518c1a4fa27e9f1a32d667d359e",
},
},
Tails: []data.SimilarFilePart{},
DiffSize: 169,
},
data.SimilarFile{
ChunkSize: 51200000,
New: "assets/sounds/jump2 (2 similar).txt",
Size: 268,
Patches: []data.SimilarFilePart{
data.SimilarFilePart{
File: "assets/sounds/jump2 (2 similar).txt",
Size: 169,
Checksum: "77c98073e183101706dbb60af74aa335",
},
},
Tails: []data.SimilarFilePart{},
DiffSize: 169,
},
},
"framework/v1/runner (similar).txt": []data.SimilarFile{
data.SimilarFile{
ChunkSize: 51200000,
New: "framework/v2/runner (similar).txt",
Size: 267,
Patches: []data.SimilarFilePart{
data.SimilarFilePart{
File: "framework/v2/runner (similar).txt",
Size: 164,
Checksum: "6a7116149401e7d9c99219c4de2d4b13",
},
},
Tails: []data.SimilarFilePart{},
DiffSize: 164,
},
},
"settings (invalid).txt": []data.SimilarFile{
data.SimilarFile{
ChunkSize: 51200000,
New: "settings (invalid).txt/settings (invalid).txt",
Size: 268,
Patches: []data.SimilarFilePart{
data.SimilarFilePart{
File: "settings (invalid).txt/settings (invalid).txt",
Size: 168,
Checksum: "f0945928560bebbf5785046a42b81220",
},
},
Tails: []data.SimilarFilePart{},
DiffSize: 168,
},
},
},
Created: []string{"assets/backgrounds/forest (created).txt", "assets/sprites/ogre (created).txt", "metadata (invalid).txt"},
Removed: []string{"assets/maps/map1 (deleted).txt"},
Dirs: []string{"assets", "assets/backgrounds", "assets/icon", "assets/sounds", "assets/sprites", "framework", "framework/v2", "settings (invalid).txt"},
Symlinks: map[string]string{
"framework/current": "v2/",
"game (indirect)": "framework/current/runner (similar).txt",
"icon (unchanged)": "assets/icon/game-icon (identical).txt",
},
}
os.Exit(m.Run())
}
func TestPatchFirstSameDir(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, false)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
dataDir := filepath.Join(dir, "data")
if patch.newDataDir != dataDir {
t.Fatalf("Expecting patch.newDataDir to be:\n%s\nreceived:\n%s\n", dataDir, patch.newDataDir)
}
assertFirstPatchState(t, updateMetadata, patch, false)
}
// TODO this test is broken for Windows. Binary diff functionality works, the test is likely wrong.
// TODO - updated: This test is actually completely broken. It's not handling the result of extractions
// when doing binary diffs
// func TestPatchDiffExistingSameDir(t *testing.T) {
// if runtime.GOOS == "windows" {
// t.Skipf("Test isn't ready for windows because we need a symlink-less version")
// }
// _, dir := test.PrepareNextTest(t)
// updateMetadata := getUpdateMetadata("v1", build1File.DownloadURL(), build1File.Checksum, 0, false)
// patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
// if err != nil {
// t.Fatal(err.Error())
// }
// <-patch.Done()
// if err := patch.Result(); err != nil {
// t.Fatal(err)
// }
// test.GetNextPort()
// updateMetadata = getUpdateMetadata("v2", diffFile.DownloadURL(), diffFile.Checksum, 0, false)
// updateMetadata.OldBuildMetadata, updateMetadata.NewBuildMetadata, updateMetadata.DiffMetadata = &oldMetadata, &newMetadata, &diffMetadata
// patch, err = NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
// if err != nil {
// t.Fatal(err.Error())
// }
// <-patch.Done()
// if err := patch.Result(); err != nil {
// t.Fatal(err)
// }
// }
func TestPatchGzip(t *testing.T) {
_, dir := test.PrepareNextTest(t)
// Requiring this fixture will allow us to test on a bigger file without having to download it every time.
bigDownloadGZ.Require(t, filepath.Join(dir, ".tempDownload"), nil)
updateMetadata := getUpdateMetadata("v1", bigDownloadGZ.DownloadURL(), bigDownloadGZ.Checksum, 0, false)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
dataDir := filepath.Join(dir, "data")
if patch.newDataDir != dataDir {
t.Fatalf("Expecting patch.newDataDir to be:\n%s\nreceived:\n%s\n", dataDir, patch.newDataDir)
}
if err := patch.Result(); err != nil {
t.Fatal(err)
}
dir = patch.Dir
dataDir = patch.newDataDir
expectedExtractedFiles := []string{
"./bigfile",
"./fToPreserve",
"./fToPreserveCase",
"./fToRemove",
"./fToRemoveCase",
"./fToUpdate",
"./fToUpdateCase",
"./toAdd/file1",
"./toClear/file1",
"./toRemove/file1",
"./toRemove/file2",
}
extractResult := patch.extractor.Result()
if extractResult.Err != nil {
t.Fatal(extractResult.Err)
}
test.AssertSlicesEqual(t, extractResult.Result, expectedExtractedFiles)
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToPreserve"), "")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToPreserveCase"), "")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToRemove"), "test\n")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToRemoveCase"), "test\n")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToUpdate"), "test\n")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToUpdateCase"), "test\n")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "toAdd", "file1"), "")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "toClear", "file1"), "")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "toRemove", "file1"), "")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "toRemove", "file2"), "")
test.AssertFolderExists(t, filepath.Join(dataDir, "empty"))
// Attempt to read manifest file
manifest := test.ReadManifest(t, dir)
if manifest.Info.Dir != updateMetadata.DataDir ||
manifest.Info.GameUID != updateMetadata.GameUID ||
manifest.IsFirstInstall != false ||
manifest.PatchInfo != nil {
bytes, _ := json.Marshal(manifest)
t.Fatalf("Game manifest is invalid:\n%s", string(bytes))
}
test.AssertSlicesEqual(t, manifest.Info.ArchiveFiles, expectedExtractedFiles)
test.AssertFileNotExist(t, filepath.Join(dir, ".tempDownload"))
}
func TestPatchExistingSameDir(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, false)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertFirstPatchState(t, updateMetadata, patch, false)
dataDir := patch.newDataDir
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamic"), "test\n")
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamicCase"), "test\n")
test.GetNextPort()
updateMetadata = getUpdateMetadata("v2", patch2File.DownloadURL(), patch2File.Checksum, 0, false)
patch, err = NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertSecondPatchState(t, updateMetadata, patch, false)
}
func TestPatchExistingSameDirSameBuild(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, false)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertFirstPatchState(t, updateMetadata, patch, false)
dataDir := patch.newDataDir
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamic"), "test\n")
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamicCase"), "test\n")
test.GetNextPort()
updateMetadata = getUpdateMetadata("v1", patch2File.DownloadURL(), patch2File.Checksum, 0, false)
patch, err = NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertFirstPatchState(t, updateMetadata, patch, true)
}
func TestPatchUninstallSameDir(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, false)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
if err := patch.Result(); err != nil {
t.Fatal(err)
}
test.GetNextPort()
patch, err = NewUninstall(nil, dir, nil, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
if err := patch.Result(); err != nil {
t.Fatal(err)
}
test.AssertFileNotExist(t, filepath.Join(dir, "data"))
test.AssertFileNotExist(t, filepath.Join(dir, ".manifest"))
test.AssertFileNotExist(t, filepath.Join(dir, ".tempDownload"))
}
func TestPatchFirstBuildDir(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, true)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
dataDir := filepath.Join(dir, fmt.Sprintf("data-%s", updateMetadata.GameUID))
if patch.newDataDir != dataDir {
t.Fatalf("Expecting patch.newDataDir to be:\n%s\nreceived:\n%s\n", dataDir, patch.newDataDir)
}
assertFirstPatchState(t, updateMetadata, patch, false)
}
func TestPatchExistingBuildDir(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, true)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertFirstPatchState(t, updateMetadata, patch, false)
dataDir := patch.newDataDir
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamic"), "test\n")
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamicCase"), "test\n")
test.GetNextPort()
updateMetadata = getUpdateMetadata("v2", patch2File.DownloadURL(), patch2File.Checksum, 0, true)
patch, err = NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertSecondPatchState(t, updateMetadata, patch, false)
test.AssertFileNotExist(t, patch.dataDir)
}
func TestPatchExistingSameBuildDirSameBuild(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, true)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertFirstPatchState(t, updateMetadata, patch, false)
dataDir := patch.newDataDir
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamic"), "test\n")
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamicCase"), "test\n")
test.GetNextPort()
updateMetadata = getUpdateMetadata("v1", patch2File.DownloadURL(), patch2File.Checksum, 0, true)
patch, err = NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertFirstPatchState(t, updateMetadata, patch, true)
}
func TestPatchUninstallBuildDir(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, true)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
if err := patch.Result(); err != nil {
t.Fatal(err)
}
test.GetNextPort()
patch, err = NewUninstall(nil, dir, nil, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
if err := patch.Result(); err != nil {
t.Fatal(err)
}
log.Println(patch.dataDir)
test.AssertFileNotExist(t, filepath.Join(dir, "data-game-v1"))
test.AssertFileNotExist(t, filepath.Join(dir, ".manifest"))
test.AssertFileNotExist(t, filepath.Join(dir, ".tempDownload"))
}
func TestPatchRelocateSameBuild(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, false)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertFirstPatchState(t, updateMetadata, patch, false)
dataDir := patch.newDataDir
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamic"), "test\n")
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamicCase"), "test\n")
test.GetNextPort()
//TODO this unit test might be wrong, do we need to use patch 2 or patch 1 here?
// the original unit test attempted to upgrade to patch 2 but assert the first patch state.
// is it because the game uid is the exact same so it doesnt attempt doing an update?
updateMetadata = getUpdateMetadata("v1", patch2File.DownloadURL(), patch2File.Checksum, 0, true)
patch, err = NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertFirstPatchState(t, updateMetadata, patch, true)
}
func TestPatchFirstRecoverCrashDuringDownload(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, false)
// Mock the update dir to look like it crashed during first installation at download
patch1File.Require(t, filepath.Join(dir, ".tempDownload"), nil)
launchOptions := &data.LaunchOptions{
Executable: updateMetadata.Executable,
}
manifest := &data.Manifest{
Info: &data.Info{
Dir: "data",
GameUID: updateMetadata.GameUID,
},
LaunchOptions: launchOptions,
OS: updateMetadata.OS,
Arch: updateMetadata.Arch,
IsFirstInstall: true,
PatchInfo: &data.PatchInfo{
Dir: "data",
GameUID: updateMetadata.GameUID,
IsDirty: false,
LaunchOptions: launchOptions,
DownloadSize: updateMetadata.RemoteSize,
DownloadChecksum: updateMetadata.Checksum,
},
}
test.WriteManifest(t, manifest, dir)
// Attempt to patch normally
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertFirstPatchState(t, updateMetadata, patch, false)
}
func TestPatchFirstRecoverCrashDuringExtract(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, false)
// Mock the update dir to look like it crashed during first installation at extract
patch1File.Require(t, filepath.Join(dir, ".tempDownload"), nil)
launchOptions := &data.LaunchOptions{
Executable: updateMetadata.Executable,
}
manifest := &data.Manifest{
Info: &data.Info{
Dir: "data",
GameUID: updateMetadata.GameUID,
},
LaunchOptions: launchOptions,
OS: updateMetadata.OS,
Arch: updateMetadata.Arch,
IsFirstInstall: true,
PatchInfo: &data.PatchInfo{
Dir: "data",
GameUID: updateMetadata.GameUID,
DynamicFiles: []string{},
IsDirty: true,
LaunchOptions: launchOptions,
DownloadSize: updateMetadata.RemoteSize,
DownloadChecksum: updateMetadata.Checksum,
},
}
test.WriteManifest(t, manifest, dir)
// Mock extracting some files
test.AssertMkdirAll(t, filepath.Join(dir, "data"))
test.AssertWriteFile(t, filepath.Join(dir, "data", "fToPreserve"), "")
test.AssertWriteFile(t, filepath.Join(dir, "data", "fToUpdate"), "test\n")
// Attempt to patch normally
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertFirstPatchState(t, updateMetadata, patch, false)
}
func TestPatchCancelFirstAfterRecoverCrashDuringDownload(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, false)
// Mock the update dir to look like it crashed during first installation at download
patch1File.Require(t, filepath.Join(dir, ".tempDownload"), nil)
launchOptions := &data.LaunchOptions{
Executable: updateMetadata.Executable,
}
manifest := &data.Manifest{
Info: &data.Info{
Dir: "data",
GameUID: updateMetadata.GameUID,
},
LaunchOptions: launchOptions,
OS: updateMetadata.OS,
Arch: updateMetadata.Arch,
IsFirstInstall: true,
PatchInfo: &data.PatchInfo{
Dir: "data",
GameUID: updateMetadata.GameUID,
IsDirty: false,
LaunchOptions: launchOptions,
DownloadSize: updateMetadata.RemoteSize,
DownloadChecksum: updateMetadata.Checksum,
},
}
test.WriteManifest(t, manifest, dir)
// Attempt to patch normally
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
assertPatchStateTransition(t, patch, StateDownload, 3*time.Second)
patch.Cancel()
<-patch.Done()
if patch.Result() != context.Canceled {
t.Fatalf("Expected patch result to be the context cancelation, received %s\n", patch.Result().Error())
}
test.AssertFileNotExist(t, patch.newDataDir)
test.AssertFileNotExist(t, filepath.Join(dir, ".manifest"))
test.AssertFileNotExist(t, filepath.Join(dir, ".tempDownload"))
}
func TestPatchCancelFirstAfterRecoverCrashDuringExtract(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, false)
// Mock the update dir to look like it crashed during first installation at extract
patch1File.Require(t, filepath.Join(dir, ".tempDownload"), nil)
launchOptions := &data.LaunchOptions{
Executable: updateMetadata.Executable,
}
manifest := &data.Manifest{
Info: &data.Info{
Dir: "data",
GameUID: updateMetadata.GameUID,
},
LaunchOptions: launchOptions,
OS: updateMetadata.OS,
Arch: updateMetadata.Arch,
IsFirstInstall: true,
PatchInfo: &data.PatchInfo{
Dir: "data",
GameUID: updateMetadata.GameUID,
DynamicFiles: []string{},
IsDirty: true,
LaunchOptions: launchOptions,
DownloadSize: updateMetadata.RemoteSize,
DownloadChecksum: updateMetadata.Checksum,
},
}
test.WriteManifest(t, manifest, dir)
// Mock extracting some files
test.AssertMkdirAll(t, filepath.Join(dir, "data"))
test.AssertWriteFile(t, filepath.Join(dir, "data", "fToPreserve"), "")
test.AssertWriteFile(t, filepath.Join(dir, "data", "fToUpdate"), "test\n")
// Attempt to patch normally
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
assertPatchStateTransition(t, patch, StateExtract, 3*time.Second)
patch.Cancel()
<-patch.Done()
if patch.Result() != context.Canceled {
t.Fatalf("Expected patch result to be the context cancelation, received %s\n", patch.Result().Error())
}
test.AssertFileNotExist(t, patch.newDataDir)
test.AssertFileNotExist(t, filepath.Join(dir, ".manifest"))
test.AssertFileNotExist(t, filepath.Join(dir, ".tempDownload"))
}
func TestPatchExistingRecoverCrashDuringDownload(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, false)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertFirstPatchState(t, updateMetadata, patch, false)
dataDir := patch.newDataDir
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamic"), "test\n")
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamicCase"), "test\n")
// Mock the update dir to look like it crashed during an update to the second build at download
patch2File.Require(t, filepath.Join(dir, ".tempDownload"), nil)
manifest := test.ReadManifest(t, dir)
updateMetadata = getUpdateMetadata("v2", patch2File.DownloadURL(), patch2File.Checksum, 0, false)
manifest.PatchInfo = &data.PatchInfo{
Dir: manifest.Info.Dir,
GameUID: updateMetadata.GameUID,
IsDirty: false,
LaunchOptions: &data.LaunchOptions{
Executable: updateMetadata.Executable,
},
DownloadSize: updateMetadata.RemoteSize,
DownloadChecksum: updateMetadata.Checksum,
}
test.WriteManifest(t, manifest, dir)
test.GetNextPort()
patch, err = NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertSecondPatchState(t, updateMetadata, patch, false)
}
func TestPatchExistingRecoverCrashDuringExtract(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, false)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertFirstPatchState(t, updateMetadata, patch, false)
dataDir := patch.newDataDir
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamic"), "test\n")
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamicCase"), "test\n")
// Mock the update dir to look like it crashed during an update to the second build at extract
patch2File.Require(t, filepath.Join(dir, ".tempDownload"), nil)
manifest := test.ReadManifest(t, dir)
updateMetadata = getUpdateMetadata("v2", patch2File.DownloadURL(), patch2File.Checksum, 0, false)
manifest.PatchInfo = &data.PatchInfo{
Dir: manifest.Info.Dir,
GameUID: updateMetadata.GameUID,
DynamicFiles: []string{"./fDynamic", "./fDynamicCase"},
IsDirty: true,
LaunchOptions: &data.LaunchOptions{
Executable: updateMetadata.Executable,
},
DownloadSize: updateMetadata.RemoteSize,
DownloadChecksum: updateMetadata.Checksum,
}
test.WriteManifest(t, manifest, dir)
// Mock extracting some files
test.AssertMkdirAll(t, filepath.Join(dir, "data"))
test.AssertWriteFile(t, filepath.Join(dir, "data", "fToUpdate"), "update\n")
test.GetNextPort()
patch, err = NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertSecondPatchState(t, updateMetadata, patch, false)
}
func TestPatchCancelFirstBuildDirAfterRecoverCrashDuringDownload(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, true)
// Mock the update dir to look like it crashed during first installation at download
patch1File.Require(t, filepath.Join(dir, ".tempDownload"), nil)
launchOptions := &data.LaunchOptions{
Executable: updateMetadata.Executable,
}
manifest := &data.Manifest{
Info: &data.Info{
Dir: "data-" + updateMetadata.GameUID,
GameUID: updateMetadata.GameUID,
},
LaunchOptions: launchOptions,
OS: updateMetadata.OS,
Arch: updateMetadata.Arch,
IsFirstInstall: true,
PatchInfo: &data.PatchInfo{
Dir: "data-" + updateMetadata.GameUID,
GameUID: updateMetadata.GameUID,
IsDirty: false,
LaunchOptions: launchOptions,
DownloadSize: updateMetadata.RemoteSize,
DownloadChecksum: updateMetadata.Checksum,
},
}
test.WriteManifest(t, manifest, dir)
// Attempt to patch normally
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
assertPatchStateTransition(t, patch, StateDownload, 3*time.Second)
patch.Cancel()
<-patch.Done()
if patch.Result() != context.Canceled {
t.Fatalf("Expected patch result to be the context cancelation, received %s\n", patch.Result().Error())
}
test.AssertFileNotExist(t, patch.newDataDir)
test.AssertFileNotExist(t, filepath.Join(dir, ".manifest"))
test.AssertFileNotExist(t, filepath.Join(dir, ".tempDownload"))
}
func TestPatchCancelFirstBuildDirAfterRecoverCrashDuringExtract(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, true)
// Mock the update dir to look like it crashed during first installation at extract
patch1File.Require(t, filepath.Join(dir, ".tempDownload"), nil)
launchOptions := &data.LaunchOptions{
Executable: updateMetadata.Executable,
}
manifest := &data.Manifest{
Info: &data.Info{
Dir: "data-" + updateMetadata.GameUID,
GameUID: updateMetadata.GameUID,
},
LaunchOptions: launchOptions,
OS: updateMetadata.OS,
Arch: updateMetadata.Arch,
IsFirstInstall: true,
PatchInfo: &data.PatchInfo{
Dir: "data-" + updateMetadata.GameUID,
GameUID: updateMetadata.GameUID,
DynamicFiles: []string{},
IsDirty: true,
LaunchOptions: launchOptions,
DownloadSize: updateMetadata.RemoteSize,
DownloadChecksum: updateMetadata.Checksum,
},
}
test.WriteManifest(t, manifest, dir)
// Mock extracting some files
test.AssertMkdirAll(t, filepath.Join(dir, "data-"+updateMetadata.GameUID))
test.AssertWriteFile(t, filepath.Join(dir, "data-"+updateMetadata.GameUID, "fToPreserve"), "")
test.AssertWriteFile(t, filepath.Join(dir, "data-"+updateMetadata.GameUID, "fToUpdate"), "test\n")
// Attempt to patch normally
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
assertPatchStateTransition(t, patch, StateExtract, 3*time.Second)
patch.Cancel()
<-patch.Done()
if patch.Result() != context.Canceled {
t.Fatalf("Expected patch result to be the context cancelation, received %s\n", patch.Result().Error())
}
log.Println(patch.dataDir)
log.Println(patch.newDataDir)
test.AssertFileNotExist(t, patch.newDataDir)
test.AssertFileNotExist(t, filepath.Join(dir, ".manifest"))
test.AssertFileNotExist(t, filepath.Join(dir, ".tempDownload"))
}
func TestPatchExistingBuildDirRecoverCrashDuringDownload(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, true)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertFirstPatchState(t, updateMetadata, patch, false)
dataDir := patch.newDataDir
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamic"), "test\n")
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamicCase"), "test\n")
// Mock the update dir to look like it crashed during an update to the second build at download
patch2File.Require(t, filepath.Join(dir, ".tempDownload"), nil)
manifest := test.ReadManifest(t, dir)
updateMetadata = getUpdateMetadata("v2", patch2File.DownloadURL(), patch2File.Checksum, 0, true)
manifest.PatchInfo = &data.PatchInfo{
Dir: "data-" + updateMetadata.GameUID,
GameUID: updateMetadata.GameUID,
IsDirty: false,
LaunchOptions: &data.LaunchOptions{
Executable: updateMetadata.Executable,
},
DownloadSize: updateMetadata.RemoteSize,
DownloadChecksum: updateMetadata.Checksum,
}
test.WriteManifest(t, manifest, dir)
test.GetNextPort()
patch, err = NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertSecondPatchState(t, updateMetadata, patch, false)
}
func TestPatchExistingBuildDirRecoverCrashDuringExtract(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, true)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertFirstPatchState(t, updateMetadata, patch, false)
dataDir := patch.newDataDir
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamic"), "test\n")
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamicCase"), "test\n")
// Mock the update dir to look like it crashed during an update to the second build at extract
patch2File.Require(t, filepath.Join(dir, ".tempDownload"), nil)
manifest := test.ReadManifest(t, dir)
updateMetadata = getUpdateMetadata("v2", patch2File.DownloadURL(), patch2File.Checksum, 0, true)
manifest.PatchInfo = &data.PatchInfo{
Dir: "data-" + updateMetadata.GameUID,
GameUID: updateMetadata.GameUID,
DynamicFiles: []string{"./fDynamic", "./fDynamicCase"},
IsDirty: true,
LaunchOptions: &data.LaunchOptions{
Executable: updateMetadata.Executable,
},
DownloadSize: updateMetadata.RemoteSize,
DownloadChecksum: updateMetadata.Checksum,
}
test.WriteManifest(t, manifest, dir)
// Mock extracting some files
test.AssertMkdirAll(t, filepath.Join(dir, "data-"+updateMetadata.GameUID))
test.AssertWriteFile(t, filepath.Join(dir, "data-"+updateMetadata.GameUID, "fToUpdate"), "update\n")
test.GetNextPort()
patch, err = NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertSecondPatchState(t, updateMetadata, patch, false)
}
func TestPatchFirstManualSameDirWithoutChildren(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, false)
patch, err := NewInstall(nil, dir, true, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
// It should reach the extract phase eventually
assertPatchStateTransition(t, patch, StateExtract, 10*time.Second)
<-patch.Done()
dataDir := filepath.Join(dir, "data")
if patch.newDataDir != dataDir {
t.Fatalf("Expecting patch.newDataDir to be:\n%s\nreceived:\n%s\n", dataDir, patch.newDataDir)
}
assertFirstPatchState(t, updateMetadata, patch, false)
}
func TestPatchFirstManualBuildDirWithoutChildren(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, true)
patch, err := NewInstall(nil, dir, true, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
// It should eventually finish the patch in a reasonable time
select {
case <-patch.Done():
case <-time.After(10 * time.Second):
t.Fatalf("Expecting patch to finish. It may be stuck waiting for a signal from a child that will never arive")
}
dataDir := filepath.Join(dir, "data-"+updateMetadata.GameUID)
if patch.newDataDir != dataDir {
t.Fatalf("Expecting patch.newDataDir to be:\n%s\nreceived:\n%s\n", dataDir, patch.newDataDir)
}
assertFirstPatchState(t, updateMetadata, patch, false)
}
func TestPatchExistingManualSameDirWithoutChildren(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, false)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertFirstPatchState(t, updateMetadata, patch, false)
dataDir := patch.newDataDir
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamic"), "test\n")
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamicCase"), "test\n")
test.GetNextPort()
updateMetadata = getUpdateMetadata("v2", patch2File.DownloadURL(), patch2File.Checksum, 0, false)
patch, err = NewInstall(nil, dir, true, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
// It should reach the extract phase eventually
assertPatchStateTransition(t, patch, StateExtract, 10*time.Second)
<-patch.Done()
assertSecondPatchState(t, updateMetadata, patch, false)
}
func TestPatchExistingManualBuildDirWithoutChildren(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, true)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertFirstPatchState(t, updateMetadata, patch, false)
dataDir := patch.newDataDir
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamic"), "test\n")
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamicCase"), "test\n")
test.GetNextPort()
updateMetadata = getUpdateMetadata("v2", patch2File.DownloadURL(), patch2File.Checksum, 0, true)
patch, err = NewInstall(nil, dir, true, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
// It should eventually finish the patch in a reasonable time
select {
case <-patch.Done():
case <-time.After(10 * time.Second):
t.Fatalf("Expecting patch to finish. It may be stuck waiting for a signal from a child that will never arive")
}
assertSecondPatchState(t, updateMetadata, patch, false)
test.AssertFileNotExist(t, patch.dataDir)
}
func TestPatchExistingManualSameDirWithChildren(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, false)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertFirstPatchState(t, updateMetadata, patch, false)
dataDir := patch.newDataDir
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamic"), "test\n")
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamicCase"), "test\n")
test.GetNextPort()
updateMetadata2 := getUpdateMetadata("v2", patch2File.DownloadURL(), patch2File.Checksum, 0, false)
// Creating and tracking a mock launcher would make the installation halt right before extraction
mockLauncher := &launcher.Launcher{}
launcher.TrackInstance(mockLauncher)
patch, err = NewInstall(nil, dir, true, nil, updateMetadata2, test.OS)
if err != nil {
t.Fatal(err.Error())
}
// It should stop the installation waiting for launched children
assertPatchStateTransition(t, patch, StateUpdateReady, 10*time.Second)
// Make sure it stays in that state and doesn't transition away
assertPatchStateNoTransition(t, patch, StateUpdateReady, 5*time.Second)
// The manifest shouldn't have updated yet, so we need to make sure it is correct for the old build data
manifest := test.ReadManifest(t, dir)
if manifest.Info.Dir != updateMetadata.DataDir ||
manifest.Info.GameUID != updateMetadata.GameUID ||
manifest.IsFirstInstall != false ||
manifest.PatchInfo.Dir != updateMetadata2.DataDir ||
manifest.PatchInfo.GameUID != updateMetadata2.GameUID ||
// This should still be false because the game hasn't actually starting extracting until after the mocked launched instance is closed.
manifest.PatchInfo.IsDirty != false {
bytes, _ := json.Marshal(manifest)
t.Fatalf("Game manifest is invalid:\n%s", string(bytes))
}
// Killing the mock launcher should remove it from track list, allowing patch to detect it and continue patching.
mockLauncher.Kill()
// It should eventually finish the patch in a reasonable time
select {
case <-patch.Done():
case <-time.After(10 * time.Second):
t.Fatalf("Expecting patch to finish. It may be stuck waiting for a signal from a child that will never arive")
}
assertSecondPatchState(t, updateMetadata2, patch, false)
}
func TestPatchExistingManualBuildDirWithChildren(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, true)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertFirstPatchState(t, updateMetadata, patch, false)
dataDir := patch.newDataDir
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamic"), "test\n")
test.AssertWriteFile(t, filepath.Join(dataDir, "fDynamicCase"), "test\n")
test.GetNextPort()
updateMetadata2 := getUpdateMetadata("v2", patch2File.DownloadURL(), patch2File.Checksum, 0, true)
// Creating and tracking a mock launcher would not halt the installation in this case because in a build dir update it prepares and extracts everything in a new directory before pausing.
mockLauncher := &launcher.Launcher{}
launcher.TrackInstance(mockLauncher)
patch, err = NewInstall(nil, dir, true, nil, updateMetadata2, test.OS)
if err != nil {
t.Fatal(err.Error())
}
// It should stop the installation waiting for launched children.
assertPatchStateTransition(t, patch, StateUpdateReady, 10*time.Second)
// Make sure it stays in that state and doesn't transition away.
assertPatchStateNoTransition(t, patch, StateUpdateReady, 5*time.Second)
// TODO: verify that the patcher already went through the extraction state because this is a build dir update
// When patching in a build dir the patcher pauses after already preparing the next directory completely, so we can test the validity of the second patch state.
if err := patch.Result(); err != nil {
t.Fatal(err)
}
dataDir = patch.newDataDir
extractResult := patch.extractor.Result()
if extractResult.Err != nil {
t.Fatal(extractResult.Err)
}
expectedExtractedFiles := []string{
"./fDynamiccase",
"./fToPreserve",
"./fToPreservecase",
"./fToUpdate",
"./fToUpdatecase",
"./toAdd/file1",
"./toAdd/file2",
"./toRemove/file1",
}
test.AssertSlicesEqual(t, extractResult.Result, expectedExtractedFiles)
// Common
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fDynamic"), "test\n")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToPreserve"), "")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToUpdate"), "update\n")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "toAdd", "file1"), "")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "toRemove", "file1"), "")
// Files haven't been cleared yet
test.AssertFileContentEquals(t, filepath.Join(dataDir, "toClear", "file1"), "")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "toRemove", "file2"), "")
test.AssertFolderExists(t, filepath.Join(dataDir, "toClear"))
test.AssertFolderExists(t, filepath.Join(dataDir, "empty"))
test.AssertFolderExists(t, filepath.Join(dataDir, "newEmpty"))
caseSensitive, err := fs.IsCaseSensitive(dataDir, test.OS)
if err != nil {
t.Fatal(err)
}
if caseSensitive {
// For case sensitive systems, if the exact same filename doesn't exist in the new archive we expect it to be removed.
// Dynamic files are not part of the old build so we expect to see them even if theres a file in the new archive with the the same name but different case.
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fDynamiccase"), "update\n")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fDynamicCase"), "test\n") // This is the old dynamic file which shouldn't be updated
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToPreservecase"), "")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToUpdatecase"), "update\n")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToPreserveCase"), "")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToUpdateCase"), "test\n")
} else {
// For case insensitive systems, we expect to see the exact old filename even if it has a different case in the new archive
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fDynamicCase"), "update\n")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToPreserveCase"), "")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToUpdateCase"), "update\n")
}
// The manifest shouldn't have updated yet, so we need to make sure it is correct for the old build data
manifest := test.ReadManifest(t, dir)
if manifest.Info.Dir != updateMetadata.DataDir ||
manifest.Info.GameUID != updateMetadata.GameUID ||
manifest.IsFirstInstall != false ||
manifest.PatchInfo.Dir != updateMetadata2.DataDir ||
manifest.PatchInfo.GameUID != updateMetadata2.GameUID ||
// This should still be false because the directory isn't updated in a build dir operation
manifest.PatchInfo.IsDirty != false {
bytes, _ := json.Marshal(manifest)
t.Fatalf("Game manifest is invalid:\n%s", string(bytes))
}
// Archive files should not have changed from the first patch until after the patch is finished.
test.AssertSlicesEqual(t, manifest.Info.ArchiveFiles, []string{
"./fToPreserve",
"./fToPreserveCase",
"./fToRemove",
"./fToRemoveCase",
"./fToUpdate",
"./fToUpdateCase",
"./toAdd/file1",
"./toClear/file1",
"./toRemove/file1",
"./toRemove/file2",
})
// Dynamic files should have been written to contain the files created between patch1 and patch2.
test.AssertSlicesEqual(t, manifest.PatchInfo.DynamicFiles, []string{
"./fDynamic",
"./fDynamicCase",
})
// Download file should still exist
test.AssertFileExists(t, filepath.Join(dir, ".tempDownload"))
// The folder of the current patch data dir should still exist tho.
test.AssertFolderExists(t, patch.dataDir)
// Killing the mock launcher should remove it from track list, allowing patch to detect it and continue patching.
mockLauncher.Kill()
// It should eventually finish the patch in a reasonable time.
select {
case <-patch.Done():
case <-time.After(10 * time.Second):
t.Fatalf("Expecting patch to finish. It may be stuck waiting for a signal from a child that will never arive")
}
// Now that the patch is finished, some files should have cleaned up.
test.AssertFileNotExist(t, filepath.Join(dataDir, "toClear", "file1"))
test.AssertFileNotExist(t, filepath.Join(dataDir, "toRemove", "file2"))
if caseSensitive {
test.AssertFileNotExist(t, filepath.Join(dataDir, "fToPreserveCase"))
test.AssertFileNotExist(t, filepath.Join(dataDir, "fToUpdateCase"))
}
test.AssertFileNotExist(t, filepath.Join(dir, ".tempDownload"))
// Check again for redundancy.
assertSecondPatchState(t, updateMetadata2, patch, false)
// This time the folder of the current patch data dir should not exist because cleanup should have run.
test.AssertFileNotExist(t, patch.dataDir)
}
func TestPatchInvalidRelativeDir(t *testing.T) {
test.PrepareNextTest(t)
var invalidDir string
if runtime.GOOS == "windows" {
invalidDir = ".\\test-dir\\patch"
} else {
invalidDir = "./test-dir/patch"
}
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, false)
patch, err := NewInstall(nil, invalidDir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
err = patch.Result()
if err == nil {
t.Fatal("Expecting patch to error out on invalid dir, received no error")
} else if err.Error() != "Patch dir is invalid: not absolute" {
t.Fatalf("Expecting patch to error out on invalid dir, received: %s\n", err.Error())
}
}
// TODO: this fails on Windows because it can't get an absolute path at all from the given test string.
// I'm not sure if this should be an issue or intended behaviour.
func TestPatchInvalidPathDir(t *testing.T) {
test.PrepareNextTest(t)
invalidDir := filepath.Join(test.Dir, "\x00invalid?")
dir, err := filepath.Abs(invalidDir)
if err != nil {
t.Skipf("Could not get absolute path to %s", invalidDir)
}
invalidDir = dir
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, false)
patch, err := NewInstall(nil, invalidDir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
err = patch.Result()
if err == nil {
t.Fatal("Expecting patch to error out on invalid dir, received no error")
} else if !strings.HasPrefix(err.Error(), "Patch dir is invalid:") {
t.Fatalf("Expecting patch to error out on invalid dir, received: %s\n", err.Error())
}
}
func TestPatchInvalidDirOutOfScope(t *testing.T) {
test.PrepareNextTest(t)
invalidDir := filepath.Join(test.Dir, "..")
dir, err := filepath.Abs(invalidDir)
if err != nil {
t.Fatalf("Could not get absolute path to %s", invalidDir)
}
invalidDir = dir
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, false)
patch, err := NewInstall(nil, invalidDir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
err = patch.Result()
if err == nil {
t.Fatal("Expecting patch to error out on invalid dir, received no error")
} else if err.Error() != "Failed to prepare patch: File out of scope" {
t.Fatalf("Expecting patch to error out on out of scope, received: %s\n", err.Error())
}
}
func TestPatchInvalidChecksum(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), "wrong", 0, false)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
err = patch.Result()
if err == nil {
t.Fatal("Expecting patch to error out on invalid checksum, received no error")
} else if err.Error() != fmt.Sprintf("Failed to download update: Invalid checksum. Expected wrong, got %s", patch1File.Checksum) {
t.Fatalf("Expecting patch to error out on invalid checksum, received: %s\n", err.Error())
}
}
func TestPatchNoChecksum(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), "", 0, false)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
assertFirstPatchState(t, updateMetadata, patch, false)
}
func TestPatchCancelImmediately(t *testing.T) {
_, dir := test.PrepareNextTest(t)
// Create a resumable that gets canceled even before patch starts
// This'll mean the patcher should never even start preparation and finish right away
r := concurrency.NewResumable(nil)
r.Cancel()
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, false)
patch, err := NewInstall(r, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
<-patch.Done()
if patch.Result() != context.Canceled {
t.Fatalf("Expected patch result to be the context cancelation, received %s\n", patch.Result().Error())
}
log.Println(patch.Dir)
test.AssertFileNotExist(t, patch.Dir)
test.AssertFileNotExist(t, filepath.Join(dir, ".manifest"))
test.AssertFileNotExist(t, filepath.Join(dir, ".tempDownload"))
}
func TestPatchCancelFirstInstallationDownload(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, false)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
assertPatchStateTransition(t, patch, StateDownload, 10*time.Second)
patch.Cancel()
<-patch.Done()
if patch.Result() != context.Canceled {
t.Fatalf("Expected patch result to be the context cancelation, received %s\n", patch.Result().Error())
}
test.AssertFileNotExist(t, patch.newDataDir)
test.AssertFileNotExist(t, filepath.Join(dir, ".manifest"))
test.AssertFileNotExist(t, filepath.Join(dir, ".tempDownload"))
}
func TestPatchCancelFirstInstallationExtract(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, false)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
assertPatchStateTransition(t, patch, StateExtract, 10*time.Second)
patch.Cancel()
<-patch.Done()
if patch.Result() != context.Canceled {
t.Fatalf("Expected patch result to be the context cancelation, received %s\n", patch.Result())
}
test.AssertFileNotExist(t, patch.newDataDir)
test.AssertFileNotExist(t, filepath.Join(dir, ".manifest"))
test.AssertFileNotExist(t, filepath.Join(dir, ".tempDownload"))
}
func TestPatchCancelFirstInstallationCleanup(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, false)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
assertPatchStateTransition(t, patch, StateCleanup, 10*time.Second)
patch.Cancel()
<-patch.Done()
if patch.Result() != nil {
t.Fatalf("Expected patch result to be successful even tho it was canceled. Received %s\n", patch.Result().Error())
}
// Even tho we canceled, we're expecting the installation to go through because it was canceled at the last state, which is not meant to be cancellable.
assertFirstPatchState(t, updateMetadata, patch, false)
}
func TestPatchPauselImmediately(t *testing.T) {
_, dir := test.PrepareNextTest(t)
// Create a resumable that gets paused even before patch starts
// This'll mean the patcher should not even start preparation until it is resumed
r := concurrency.NewResumable(nil)
r.Pause()
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, false)
patch, err := NewInstall(r, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
// Wait for one second to make sure preparation didn't start and respected being paused.
assertPatchStateNoTransition(t, patch, StateStart, 1*time.Second)
r.Resume()
<-patch.Done()
assertFirstPatchState(t, updateMetadata, patch, false)
}
func TestPatchPauseDownload(t *testing.T) {
_, dir := test.PrepareNextTest(t)
updateMetadata := getUpdateMetadata("v1", patch1File.DownloadURL(), patch1File.Checksum, 0, false)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
// Wait for it to start downloading
assertPatchStateTransition(t, patch, StateDownload, 10*time.Second)
patch.Pause()
// Wait for 5 seconds to make sure download didn't go through.
assertPatchStateNoTransition(t, patch, StateDownload, 5*time.Second)
patch.Resume()
<-patch.Done()
assertFirstPatchState(t, updateMetadata, patch, false)
}
// TODO this is a risky test since the extractor is incapable of really stopping in the middle of extraction of a single file yet.
func TestPatchPauseExtract(t *testing.T) {
_, dir := test.PrepareNextTest(t)
// Requiring this fixture will allow us to test on a bigger file without having to download it every time.
bigDownloadGZ.Require(t, filepath.Join(dir, ".tempDownload"), nil)
updateMetadata := getUpdateMetadata("v1", bigDownloadGZ.DownloadURL(), bigDownloadGZ.Checksum, 0, false)
patch, err := NewInstall(nil, dir, false, nil, updateMetadata, test.OS)
if err != nil {
t.Fatal(err.Error())
}
// Wait for it to start extracting
assertPatchStateTransition(t, patch, StateExtract, 10*time.Second)
patch.Pause()
log.Println("Paused extraction...")
// Wait for 5 seconds to make sure extract didn't go through.
assertPatchStateNoTransition(t, patch, StateExtract, 5*time.Second)
log.Println("Resuming extraction...")
patch.Resume()
<-patch.Done()
if patch.Result() != nil {
t.Fatal(patch.Result())
}
}
func waitForPatchState(p *Patch, state State, timeout time.Duration) <-chan bool {
ch := make(chan bool)
go func() {
sub, err := p.Subscribe()
if err != nil {
ch <- false
close(ch)
return
}
defer sub.Close()
expire := time.After(timeout)
defer close(ch)
// We loop until we get the state changed message we are waiting for.
// The loop also breaks on the subscriber closing or reaching the time limit.
for {
select {
// Attempt to get a value from the subscriber
case msg, open := <-sub.Next():
// Break if the subscriber closed
if !open {
ch <- false
return
}
// We need to check if the message type we received is a state change message
switch msg.(type) {
case StateChangeMsg:
newState := msg.(StateChangeMsg)
log.Printf("Received new state %d, waiting state %d", newState.State, state)
if state == newState.State {
ch <- true
return
}
// We don't care about any other message type
default:
}
// Break if we reached the time limit
case <-expire:
ch <- false
return
}
}
}()
return ch
}
func assertPatchStateTransition(t *testing.T, p *Patch, state State, timeout time.Duration) {
if !<-waitForPatchState(p, state, timeout) {
t.Fatalf("Expecting patch state to transition to %d after %s, but patch state is %d\n", state, timeout.String(), p.state)
}
}
func assertPatchStateNoTransition(t *testing.T, p *Patch, state State, timeout time.Duration) {
<-time.After(timeout)
if p.state != state {
t.Fatalf("Expecting patch state to remain %d after %s, but patch state is %d\n", state, timeout.String(), p.state)
}
}
// assertFirstPatchState assets wether the given patch and info was correctly executed as the first patch.
// if noop is true, we're checking if the update operation ended up being a no-op (if upgrading to the same build for example).
func assertFirstPatchState(t *testing.T, updateMetadata *data.UpdateMetadata, patch *Patch, noop bool) {
if err := patch.Result(); err != nil {
t.Fatal(err)
}
dir := patch.Dir
dataDir := patch.newDataDir
expectedExtractedFiles := []string{
"./fToPreserve",
"./fToPreserveCase",
"./fToRemove",
"./fToRemoveCase",
"./fToUpdate",
"./fToUpdateCase",
"./toAdd/file1",
"./toClear/file1",
"./toRemove/file1",
"./toRemove/file2",
}
// When noop is true we shouldn't reach the extraction step
if !noop {
extractResult := patch.extractor.Result()
if extractResult.Err != nil {
t.Fatal(extractResult.Err)
}
test.AssertSlicesEqual(t, extractResult.Result, expectedExtractedFiles)
} else {
if patch.extractor != nil {
t.Fatal("Expecting the patch extractor to be nil")
}
}
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToPreserve"), "")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToPreserveCase"), "")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToRemove"), "test\n")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToRemoveCase"), "test\n")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToUpdate"), "test\n")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToUpdateCase"), "test\n")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "toAdd", "file1"), "")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "toClear", "file1"), "")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "toRemove", "file1"), "")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "toRemove", "file2"), "")
test.AssertFolderExists(t, filepath.Join(dataDir, "empty"))
// Attempt to read manifest file
manifest := test.ReadManifest(t, dir)
if manifest.Info.Dir != updateMetadata.DataDir ||
manifest.Info.GameUID != updateMetadata.GameUID ||
manifest.IsFirstInstall != false ||
manifest.PatchInfo != nil {
bytes, _ := json.Marshal(manifest)
t.Fatalf("Game manifest is invalid:\n%s", string(bytes))
}
test.AssertSlicesEqual(t, manifest.Info.ArchiveFiles, expectedExtractedFiles)
test.AssertFileNotExist(t, filepath.Join(dir, ".tempDownload"))
}
// assertSecondPatchState assets wether the given patch and info was correctly executed as the second patch.
// if noop is true, we're checking if the update operation ended up being a no-op (if upgrading to the same build for example).
func assertSecondPatchState(t *testing.T, updateMetadata *data.UpdateMetadata, patch *Patch, noop bool) {
if err := patch.Result(); err != nil {
t.Fatal(err)
}
dir := patch.Dir
dataDir := patch.newDataDir
expectedExtractedFiles := []string{
"./fDynamiccase",
"./fToPreserve",
"./fToPreservecase",
"./fToUpdate",
"./fToUpdatecase",
"./toAdd/file1",
"./toAdd/file2",
"./toRemove/file1",
}
// When noop is true we shouldn't reach the extraction step
if !noop {
extractResult := patch.extractor.Result()
if extractResult.Err != nil {
t.Fatal(extractResult.Err)
}
test.AssertSlicesEqual(t, extractResult.Result, expectedExtractedFiles)
} else {
if patch.extractor != nil {
t.Fatal("Expecting the patch extractor to be nil")
}
}
// Common
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fDynamic"), "test\n")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToPreserve"), "")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToUpdate"), "update\n")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "toAdd", "file1"), "")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "toRemove", "file1"), "")
test.AssertFileNotExist(t, filepath.Join(dataDir, "toClear", "file1"))
test.AssertFileNotExist(t, filepath.Join(dataDir, "toRemove", "file2"))
test.AssertFolderExists(t, filepath.Join(dataDir, "toClear"))
test.AssertFolderExists(t, filepath.Join(dataDir, "empty"))
test.AssertFolderExists(t, filepath.Join(dataDir, "newEmpty"))
caseSensitive, err := fs.IsCaseSensitive(dataDir, test.OS)
if err != nil {
t.Fatal(err)
}
log.Printf("CASE SENSITIVE: %t\n", caseSensitive)
if caseSensitive {
// For case sensitive systems, if the exact same filename doesn't exist in the new archive we expect it to be removed.
// Dynamic files are not part of the old build so we expect to see them even if theres a file in the new archive with the the same name but different case.
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fDynamiccase"), "update\n")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fDynamicCase"), "test\n") // This is the old dynamic file which shouldn't be updated
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToPreservecase"), "")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToUpdatecase"), "update\n")
test.AssertFileNotExist(t, filepath.Join(dataDir, "fToPreserveCase"))
test.AssertFileNotExist(t, filepath.Join(dataDir, "fToUpdateCase"))
} else {
// For case insensitive systems, we expect to see the exact old filename even if it has a different case in the new archive
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fDynamicCase"), "update\n")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToPreserveCase"), "")
test.AssertFileContentEquals(t, filepath.Join(dataDir, "fToUpdateCase"), "update\n")
}
manifest := test.ReadManifest(t, dir)
if manifest.Info.Dir != updateMetadata.DataDir ||
manifest.Info.GameUID != updateMetadata.GameUID ||
manifest.IsFirstInstall != false ||
manifest.PatchInfo != nil {
bytes, _ := json.Marshal(manifest)
t.Fatalf("Game manifest is invalid:\n%s", string(bytes))
}
test.AssertSlicesEqual(t, manifest.Info.ArchiveFiles, expectedExtractedFiles)
test.AssertFileNotExist(t, filepath.Join(dir, ".tempDownload"))
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,325
|
require 'spec_helper'
describe Bankscrap::Transaction do
describe '#initialize' do
context 'the received amount is a Money object' do
subject { Bankscrap::Transaction.new(amount: Money.new(1000, 'EUR')) }
it 'raise an exception' do
expect { subject }.not_to raise_error
end
end
context 'the received amount is not a Money object' do
subject { Bankscrap::Transaction.new(amount: 100) }
it 'raise an exception' do
expect { subject }.to raise_error(Bankscrap::NotMoneyObjectError)
end
end
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 958
|
package org.jnode.fs.ext2.test.command;
import java.io.FileNotFoundException;
import java.io.FileOutputStream;
import java.io.FileWriter;
import java.io.IOException;
/**
* Sorry, this is not a proper JNode command...
* @author Andras Nagy
*/
public class WriteTest {
public WriteTest(String fname) throws FileNotFoundException, IOException {
byte[] bbuf = new byte[20];
for (byte i = 0; i < 20; i++)
bbuf[i] = (byte) (i + 65);
FileOutputStream fos = new FileOutputStream(fname, false);
fos.write(bbuf);
fos.close();
}
public WriteTest(String fname, String text) throws FileNotFoundException, IOException {
FileWriter writer = new FileWriter(fname);
writer.write(text.toCharArray());
writer.close();
}
public static void main(String args[]) {
String fname;
if (args.length > 0)
fname = args[0];
else {
System.out.println("writeTest filename [some_text]");
return;
}
try {
if (args.length > 1)
new WriteTest(fname, args[1]);
else
new WriteTest(fname);
} catch (IOException e) {
e.printStackTrace();
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,688
|
Disney Channel es un canal de televisión por suscripción infanitl, propiedad de The Walt Disney Company y con cobertura para Chequia, Eslovaquia y Hungría. Fue lanzado originalmente como Fox Kids y después como Jetix.
Historia
En Hungría, el canal fue lanzado en septiembre de 2000 originalmente como Fox Kids, expandiéndose a Chequia y Eslovaquia en febrero de 2001. En julio del mismo año, The Walt Disney Company adquiere las operaciones de Fox Family Worldwide, empresa propietaria de los canales Fox Kids en Europa (incluyendo a la señal en Europa Central) bajo la sociedad Fox Kids Europe N.V. El 18 de abril de 2004, todas las señales europeas de Fox Kids lanzaron la marca Jetix como un bloque de programación de series animadas de acción y, después, el 1 de enero del 2005, la mayor parte de estas (incluida la señal centroeuropea) son relanzadas como Jetix.
El 11 de agosto de 2008, Jetix Europa Central empezó a emitir un bloque de programación llamado Disney Stars, en las que aparecían series de Disney como Kim Possible, Phineas y Ferb, American Dragon Jake Long, Hannah Montana y Wizards of Waverly Place.
Después del lanzamiento de Disney XD en Estados Unidos, la Disney-ABC Television Group dio luz verde al lanzamiento de Disney XD en Francia el 1 de abril de 2009 en reemplazo de Jetix y se esperaba de que el canal se lanzase en otros países europeos a lo largo de 2009. Sin embargo, Disney anunció que Jetix en ciertos países (específicamente Chequia, Eslovaquia, Hungría, Rumania, Moldavia, Bulgaria y Rusia) sería relanzado como Disney Channel, marcando la primera vez que el canal entra en emisión en la región. El 19 de septiembre de 2009, Disney Channel Europa Central comenzó sus emisiones en reemplazo de la señal local de Jetix.
El 3 de mayo de 2011, Disney Channel Europa Central estrenó un nuevo paquete gráfico (que incluía nuevos bumpers y un nuevo logotipo) que fue lanzado primero en Estados Unidos, en conjunto con el resto de señales europeas del canal (a excepción de Rusia). El nuevo logo debutó en esta señal primero antes de aparecer en las señales del Reino Unido, Alemania, Francia, Países Bajos y España. El bloque de programación preescolar Playhouse Disney fue relanzado bajo la marca Disney Junior en esta señal el 1 de junio del 2011.
Desde 2012, la mayoría de los comerciales del canal empezaron a ser producidas en la relación de aspecto 16:9, siendo emitidas con un letterbox en 14:9. En diciembre de 2012, Disney Channel Europa Central empezó a usar el mismo logo de transmisión y gráficas—con pequeñas diferencias—usadas por la señal británica del canal. Esta actualización fue completamente implementada meses después, en 2013.
En junio del 2014, Disney Channel Europa Central cambió de logotipo y renovó su paquete gráfico, el cual se lanzó primero en Alemania. El canal cambió su relación de aspecto de 4:3 a 16:9 el 29 de enero del 2015.
Programación
Programación actual
Acción real
Diario de amigas (Prázdninové deníky)
Alex & Co. (Alex & Spol.)
Bia
Best Friends Whenever (Kámošky s časem)
Bizaardvark (Bizaardvark)
Bunk'd (Táborníci z Kikiwaka)
Elena de Ávalor (Elena z Avaloru)
Gamer's Guide to Pretty Much Everything (Průvodce všehoschopného hráče)
Girl Meets World (Riley ve velkém světě)
Hank Zipzer
Kickin' It (Nakopni to!)
Kirby Buckets (Kirby Buckets)
Liv and Maddie (Liv a Maddie)
Mako Mermaids (Mako mermaids: Mořské víly z ostrova Mako)
O11CE (Jedenáctka)
Soy Luna (Soy Luna)
The Lodge
Series animadas
Harvey Street Kids (Dêti á Harvey Street)
Kitty Is Not a Cat (Kitty neni kočka)
LoliRock
Miraculous: Las aventuras de Ladybug (Kouzelná Beruška a Černý kocour)
Phineas y Ferb (Phineas a Ferb)
Star vs. the Forces of Evil (Star proti silám zla)
Star Wars Rebels (Star Wars: Povstalci)
The ZhuZhus (Zhu Zhu)
Pokémon (Pokémon)
Skyler (Sky)
Disney Junior
Doc McStuffins (Doctor Hračky)
Handy Manny (Mistr Manny)
Jake and the Never Land Pirates (Jake a piráti ze země Nezemě)
Jungle Junction (Středo džungle)
Mickey Mouse Clubhouse (Mickeyho klubík)
The New Adventures of Winnie the Pooh (Nová Dobrodružství Medvídka Pú)
The Owl (Sova)
Little Bear (Malý Medvěd)
Special Agent Oso (Zvláštní Agent Oso)
Referencias
Enlaces externos
Sitio en checo
Sitio en húngaro
Disney Channel
Canales y estaciones de televisión fundados en 2000
Canales de televisión de la República Checa
Canales de televisión de Eslovaquia
Canales de televisión de Hungría
|
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"redpajama_set_name": "RedPajamaWikipedia"
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| 7,262
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{"url":"http:\/\/annals.math.princeton.edu\/2006\/164-3\/p02","text":"# Orbit equivalence rigidity and bounded cohomology\n\n### Abstract\n\nWe establish new results and introduce new methods in the theory of measurable orbit equivalence, using bounded cohomology of group representations. Our rigidity statements hold for a wide (uncountable) class of groups arising from negative curvature geometry. Amongst our applications are (a) measurable Mostow-type rigidity theorems for products of negatively curved groups; (b) prime factorization results for measure equivalence; (c) superrigidity for orbit equivalence; (d) the first examples of continua of type $II_1$ equivalence relations with trivial outer automorphism group that are mutually not stably isomorphic.\n\n## Authors\n\nNicolas Monod\n\nDepartment of Mathematics\nUniversity of Chicago\nChicago, IL 60637\nUnited States\n\nYehuda Shalom\n\nSchool of Mathematical Sciences\nTel Aviv University\nTel Aviv 69978\nIsrael","date":"2017-10-17 18:44:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.4114554822444916, \"perplexity\": 4451.621516479649}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-43\/segments\/1508187822480.15\/warc\/CC-MAIN-20171017181947-20171017201947-00802.warc.gz\"}"}
| null | null |
\section{Introduction}
Checkpoint-restart is now a mature technology with a variety of
robust packages~\cite{ansel2009dmtcp,BLCR06,criu}. This work concentrates
on the DMTCP (Distributed MultiThreaded CheckPointing) package
and its sophisticated plugin model that enables
process virtualization~\cite{arya2016design}.
This plugin model has been used recently to demonstrate
checkpointing of 32,752 MPI processes on a supercomputer
at TACC (Texas Advanced Computing Center)~\cite{cao2016system}.
DMTCP itself is free and open source. The DMTCP publications
page~\cite{dmtcpPublications} lists approximately 50~refereed publications
by external groups that have used DMTCP in their work.
This work concentrates on the recent advances in the DMTCP programming model
that were motivated by work with Intel Corporation. While Intel
works with multiple vendors of hardware emulators, this work
reflects the three-way collaboration between the DMTCP team,
Intel, and Mentor Graphics, a vendor of hardware emulators for EDA.
Further information specific to EDA (Electronic Design Automation)
is contained in~\cite{dac2017}.
In particular, the ability to save the state of a simulation
{\em including the state of a back-end hardware emulator} is a key to using
checkpoint-restart in EDA.
For background
on how DMTCP is used generally at Intel, see~\cite{hpec2014}.
The focuses of the ongoing work at Intel is best described
by their statement of future work:
\begin{quotation}
\noindent
``Within Intel IT, we will focus on the development and
enhancement of the DMTCP technology for use with
graphical EDA tools, with strong network dependencies.
$\ldots$
There is also additional engagement with third-party vendors
to include native DMTCP support in their tools, as well as
engagement with super-computing development teams on
enabling DMTCP for the Xeon Phi family of products.''
\end{quotation}
A hardware emulator may entail a thousand-fold
slowdown, as compared to direct execution in silicon.
There are two natural use cases of checkpointing in the context of EDA.
In both cases, the natural strategy is to run until reaching
the region of logic of interest. Then checkpoint. Later,
one can repeatedly restart and test the logic, without worrying
about the long initialization times under a hardware emulator.
Restarting under DMTCP is extremely fast, especially when the
{\tt --fast-restart} flag is used that takes advantage of {\tt mmap()}
to load pages into memory on-demand at runtime (after the initial restart).
The two use cases follow.
\begin{LaTeXdescription}
\item[Testing of silicon logic:] run until reaching the logic to
be tested; then repeatedly restart and follow different
logic branches; and
\item[Fault injection in silicon logic:] run until reaching the logic
to be tested; then repeatedly restart, inject faults in the
emulated (or simulated) silicon model and run along a pre-determined
logic branch to determine the level of fault tolerance for that
silicon design.
\end{LaTeXdescription}
For this work, the second case is of greater interest. This requires
running arbitrary code either immediately at the point of restart by
injecting faults in the logic design,
or by interposing on later logic functions of the simulator/emulator
so as to inject transient faults.
The first use case above has been extensively studied
using DMTCP
in domains as varied as architecture simulation~\cite{ShinaEtAl12},
formal verification of embedded control
systems~\cite{resmerita2012verification},
network simulation~\cite{HarriganRiley14},
and software model checking~\cite{LeungwattanakitEtAl14}. While the
two use cases are closely related, this work
highlights the second use case, by including the possibility of
interposing at runtime. Section~\ref{sec:processVirtualization}
presents the tools for such interposition, including the creation
of global barriers at an arbitrary point in the program.
Section~\ref{sec:caseStudies} presents three particular extensions
of checkpointing that were added to the DMTCP plugin model specifically
motivated by the concerns observed in our general collaboration on EDA.
The DMTCP plugin model is critical in this latter application.
One must stop a computation at a pre-defined location in the
simulation, save additional state information (such as the
state of a hardware emulator being used~\cite{dac2017}), and
then inject additional code (such as fault injection) at restart time.
A contribution of the DMTCP plugin model is the ability to virtualize
multiple aspects of the computation. These include: pathnames (for example,
the subdirectory corresponding to the current ``run slot'' of the
emulator);
environment variables (for example, modification of
the DISPLAY environment variable, or other environment variables
intrinsic to the running of the simulation);
interposition of the simulation by a third-party plugin (for example,
for purposes of measuring timings since restart at multiple levels
of granularity, or programmatically creating additional checkpoints
for analysis of interesting states leading to logic errors);
and third-party programmable barriers across all processes (enabling
the acceleration of simulations through the use of parallel
processes and even distributed processes within a single computation).
The DMTCP plugin model is an example of {\em process virtualization}:
virtualization of external abstractions from within a process.
It is argued here that the DMTCP plugin model sets it apart from
other checkpointing approaches. To this end, a brief survey
of existing checkpointing approaches and process virtualization
is provided at the end.
In the rest of this paper, Section~\ref{sec:processVirtualization}
motivates the need for a model of process virtualization with a simple
example concerning process ids. It also reviews the DMTCP plugin model.
Section~\ref{sec:caseStudies} presents a series of micro-case studies
in which DMTCP was extended to support the applications at Intel,
along with third-party DMTCP plugins developed by Mentor Graphics
for use by Intel and other customers..
Section~\ref{sec:relatedWork} the provides a survey of DMTCP
and some other related approaches to checkpointing and process
virtualization.
Section~\ref{sec:conclusion} then presents the conclusions.
\section{User-Space Process Virtualization}
\label{sec:processVirtualization}
Application-specific checkpointing and system-level transparent checkpointing
are two well-known options for checkpointing. Unfortunately, neither
one fits the requirements for the proposed use case for simulating
fault injection in silicon logic. Application-specific checkpointing
is error-prone and difficult to maintain. System-level transparent
checkpointing generally does not provide enough control at runtime
to dynamically adjust the type of fault injection.
In particular, it is often necessary to
capture control of the target application dynamically at runtime in
order to inject faults. Here we show how that can be incorporated
in a modular DMTCP plugin, rather than incorporated directly into
the simulator/emulator.
For a more thorough introduction to the DMTCP plugin model, see
either~\cite{arya2016design} or the DMTCP documentation~\cite{dmtcpPlugins}.
This section highlights those aspects most likely to assist in adding
fault injection through a DMTCP plugin.
The primary features of the model of interest for fault injection are:
\begin{enumerate}
\item interposition on function/library calls, and their use
in virtualization;
\item programmatically defined barriers across all processes on
a computer; and
\item programmatically defined choices of when to checkpoint and
when to avoid checkpointing.
\end{enumerate}
\subsection{Process Virtualization through Interposition and Layers:
A Simple Example with Pids}
\label{sec:pidVirtExample}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\columnwidth]{pid-virt.eps}
\end{center}
\caption{Process virtualization for pids.}
\label{fig:pid-virt}
\end{figure}
Figure~\ref{fig:pid-virt} succintly describes the philosophy of
process virtualization. Some invariant (in this case the pid (process~id)
of a process may have a different name prior to checkpoint and after
restart. A virtualized process will interact only with virtual
process ids in the base code. A DMTCP plugin retains a translation
table between the virtualized pid known to the base code and
the real pid known to the kernel.
Since the base code and the kernel interact primarily through system calls,
the DMTCP plugin defines a wrapper function around that system call.
The wrapper function translates between virtual and real pids both
for arguments to the system call and for the return value. This is
illustrated both in Figure~\ref{fig:pid-virt} and in the example
code of Listing~\ref{lst:pidWrapperExample}.
\begin{lstlisting}[belowskip=-20pt, float=ht,
escapechar=@, label={lst:pidWrapperExample},
morekeywords={WRAPPER,enable_ckpt,disable_ckpt,REAL_},
caption={A simplified function wrapper for pid virtualization}]
WRAPPER int kill(pid_t pid, int sig) {
disable_ckpt();
real_pid = virt_to_real(pid);
int ret = REAL_@@kill(real_pid, sig);
enable_ckpt();
return ret;
}
\end{lstlisting}
\vskip10pt
Additionally, pid's may be passed as part of the proc filesystem,
and through other limited means. To solve this, DMTCP implements
virtualization of filenames as well as pid names, and so the ``open''
system call will also be interposed upon to detect names such
as \texttt{/proc/PID/maps}.
In this way, a collection of wrapper functions can be collected
together within a DMTCP plugin library. Such a library implements
a virtualization layer. The ELF library standard implements
a library search order such that symbols are searched in order as follows:
\newline
\centerline{
EXECUTABLE $\rightarrow$ LIB1 $\rightarrow$ LIB2 ... LIBC $\rightarrow$ KERNEL
}
\newline
where the symbol is finally replaced by a direct kernel call.
This sequence can also be viewed as a sequence of layers, consistent
with the common operating system implementation through layers. A DMTCP
plugin for pids then presents a virtualization layer in which all higher
layers see only virtual pids, and all lower layers see only real pids.
This is analogous to an operating system design in which a higher layer
sees the disk as a filesystem, and a lower layer sees the disk as a
collection of disk blocks.
In a similar way, DMTCP provides layers to virtualize filenames,
environment variables and myriad other names.
In this way, an end
user can implement a fault injection plugin layer such that
all code below that layer sees injected faults, while higher layers
do not see the injected faults. Additionally, such a layer can
be instrumented to gather information such as the cumulative number
of faults.
DMTCP also provides an API for the application or a plugin to
either request a checkpoint or to avoid a checkpoint. Upon checkpoint,
each plugin is notified of a checkpoint barrier, and similarly upon
restart. Thus, it is feasible to create successive checkpoints
available for restart or available as a snaphot for later forensics
on the cause of a later error. Optimizations such as forked checkpointing
(fork a child and continue in the parent) are available in order to
take advantage of the kernel's copy-on-write in order to
make checkpointing/snapshotting extremely fast.
\subsection{Checkpointing Distributed Resources with the Help of Barriers}
\label{sec:checkpointingDistributedResources}
Checkpointing in a distributed application context requires coordination
between multiple processes at different virtualization layers. The use
of programmable barriers enables this coordination. In addition to
the checkpoint and restart events, each plugin (or virtualization layer)
can define its own set barriers and a callback to execute at a barrier. A
centralized DMTCP coordinator forces the application processes to execute
the barriers in sequence.
Further, a hardware resource, for example, the interface to a hardware
emulator, might be shared among multiple processes that share parent-child
relationships. To get a semantically equivalent state on restart, the barriers
can be used to elect a leader to save and restore the connection to the
hardware emulator on restart.
\section{Case Studies along the Way to Extending DMTCP}
\label{sec:caseStudies}
This section describes three specific real-world use cases where DMTCP
was extended to support hardware emulation and simulation software. The
examples are motivated by our work with various hardware and EDA tool
vendors.
\subsection{External connections}
GUI-based simulation software presents a unique challenge in
checkpointing. The front-end software communicates with an X server
via a socket. The X server runs in a privileged mode and outside of
checkpoint control. While the connection could be blacklisted for the
checkpointing, application's GUI context and state is part of the X server
and cannot be checkpointed. The context does not exist at restart time
and needs to be restored. DMTCP was extended to transparently support
checkpointing of VNC~\cite{richardson1998virtual}
and XPRA~\cite{xpra}. The two tools allow
X command forwarding to a local X server that can be run under checkpoint
control. \cite{KazemiGC13} presents an alternate record-prune-replay
based approach using DMTCP to checkpoint GUI-based applications.
Authentication and license services is an important issue for protecting
the intellectual property of all the parties. Often, the authentication
protocols and software are proprietary and specific to a vendor. Further,
the licensing services are not run under checkpoint control, which makes
it difficult to get a ``complete'' checkpoint of the software. Extensions
were added to DMTCP to allow a vendor to hook into the checkpoint
and restart events and mark certain connections as ``external'' to
the computation. At checkpoint time, the connections marked external
are ignored by DMTCP and instead the responsibility of restoring
these connections is delegated to the vendor-specific extension. The
vendor-specific plugin also allows the application to check back with
the licensing service at restart time in order to not violate a licensing
agreement that restricts the number of simultaneous ``seats''.
\subsection{Virtualizing an application's environment}
The ability to migrate a process among the available resources is critical
for efficient utilization of hardware emulator resources. However, the
environment variables, the file paths, and the files that are saved as
part of a checkpoint image make such migrations challenging. We added
DMTCP extensions (plugins) to virtualize the environment and the file
paths. This allows a process to be restarted on a different system
by changing the values and the paths. Another extension that we added
to DMTCP allows a user to explicitly control the checkpointing of
files used by their application at the granularity of a single file.
\subsection{Interfacing with hardware and closed-source, third-party libraries}
Hardware emulators communicate with the host software via high-speed
interfaces. Any in-flight transactions at checkpoint time can result
in the data being lost and inconsistent state on restart. Thus,
it is important to bring the system to a quiescent state and drain
the in-flight data on the buses before saving the state. Further,
checkpointing while the software is in a critical state (like holding
a lock on a bus) can lead to complications on restart. To help mitigate
such issues, DMTCP was extended to allow fine-grained programmatic control
over checkpointing. This enables the hardware/EDA tool vendor to tailor
the checkpointing for their specific requirements. In particular, it
allows a user to invoke checkpointing from within their code, disable
checkpointing for critical sections, or delay the resuming of user
threads until the system reaches a well-behaved state.
The software toolchain used for simulation and emulation is often put
together by integrating various third-party components. The components
may be closed-source and may use proprietary protocols for interfacing
with each other and the system. For example, many software toolchains
rely on legacy 32-bit code that's difficult to port to 64-bits, and so,
support for mixed 32-/64-~bit processes was an important consideration.
Checkpointing while holding locks was another interesting issue. While
the locks and their states are a part of the user-space memory (and hence,
a part of the checkpoint image), an application can also choose to use
an error-checking lock that disallows unlocking by a different thread
than the one that acquired it. On restart, when new thread ids would be
assigned by the system, the locks would become invalid and the unlock
call would fail. We extended DMTCP by adding wrapper functions for lock
acquisition and release functions to keep track of the state of locks. At
restart time, a lock's state is patched with the newer thread ids.
More generally, the problem described above is about the state that's
preserved when a resource is allocated at checkpoint time and needs to
be deallocated at restart time. While the restarted process inherits
its state from the checkpoint image, its environment (thread ids, in
the above case) might have changed on restart. An application author
with domain expertise can extend the DMTCP checkpointing framework to
recognize and virtualize these resources. The state could be a part of
the locks that are acquired by a custom thread-safe malloc library, or
the guard regions created by a library to guard against buffer overflows,
or the libraries that are loaded temporarily.
\section{Survey of Existing Approaches to Checkpointing
and Process Virtualization}
\label{sec:relatedWork}
High performance computing (HPC) is the traditional domain in which
checkpoint-restart is heavily used. It is used for the sake of fault
tolerance during a long computation, for example of days. For a
survey of checkpoint-restart implementations in the context of high
performance computing, see Egwutuoha\hbox{et al.}~\cite{Egwutuoha2013}.
In the context of HPC, DMTCP and BLCR~\cite{BLCR06,BLCR03} are the most
widely used examples of transparent, system-level checkpoint-restart
parallel computing. (A transparent checkpointing package is one
that does not modify the target application.)
\subsection{DMTCP}
\label{sec:relatedWorkDMTCP}
DMTCP (Distributed MultiThreaded CheckPointing) is a purely user-space
implementation. In addition to being transparent, it also does not
require any kernel modules and its installation and execution does
not require root privilege or the use of special Linux capabilities.
It achieves its robustness by trying to stay as close to the POSIX
standard as possible in its API with the Linux kernel.
The first version of DMTCP was later described in~\cite{ansel2009dmtcp}.
That version did not provide the plugin model for process virtualization.
For example, virtualization of network addresses did not exist,
as well as a series of other constructs, such as
timers, session ids, System~V shared memory, and other features.
These features were added later due to the requirements of
high performance computing. Eventually, the current procedure
for virtualizing process ids (see Section~\ref{sec:pidVirtExample}
was developed. To the best of our knowledge, DMTCP is unique
in its approach toward process id virtualization.
Eventually, the plugin model was developed, initially for transparent
support of the InfiniBand network fabric~\cite{cao2014transparent}.
the current extension of that plugin model is described
in~\cite{arya2016design}.
Still later, the requirements for robust support of EDA in collaboration
with Intel led to the development of reduction of runtime overhead
graphic support using XPRA, path virtualization (for virtualization
of the runtime slot and associated directory of a run using
a hardware emulator, including different mount points on
the restart computer), virtualization of environment variables
including the X-Windows DISPLAY variable (for similar reasons), robustness
across a variety of older and newer Linux kernels and GNU libc versions,
mixed multi-architecture (32- and 64-bit) processes within a single
computation, low-overhead support for malloc-intensive programs,
re-connection of a socket to a license server on restart, and whitelist
and blacklist of special temporary files that many or may not be present
on the restart computer.
\subsection{BLCR}
\label{sec:relatedWorkBLCR}
BLCR supports only single-node standalone checkpointing. In particular,
it does not support checkpointing of TCP sockets, InfiniBand
connections, open files, or SysV shared memory objects.
BLCR is often used in HPC clusters, where one has full control over
the choice of Linux kernel and other systems software. Typically,
a Linux kernel is chosen that is compatible with BLCR, a BLCR
kernel module is installed, and when it is time to checkpoint,
it is the responsibility of an MPI checkpoint-restart service
to temporarily disconnected the MPI network layer, then checkpoint
locally on each node, and finally re-connect the MPI network
layer.
Note that BLCR is limited in what features it supports, notably
including a lack of support for sockets and System~V shared memory.
Quoting from the BLCR User's Guide:
\begin{quote}
``However, certain applications are not supported because they use
resources not restored by BLCR: $\ldots$
Applications which use sockets (regardless of address family). $\ldots$;
Applications which use character or block devices (e.g. serial ports
or raw partitions). $\ldots$;
Applications which use System~V IPC mechanisms including shared memory,
semaphores and message queues.''~\cite{BLCRshmem}
\end{quote}
The lack of BLCR support for shared memory also prevents its use
in OpenSHMEM~\cite{openshmem}.
\subsection{ZapC and CRUZ}
ZapC and CRUZ represent two other checkpointing approaches
that are not currently widely used.
ZapC~\cite{Laadan2005} and CRUZ~\cite{CRUZ2005} were earlier
efforts to support distributed checkpointing, by modifying the kernel to
inserting hooks into the network stack using netfilter to translate source
and destination addresses. ZapC and CRUZ are no longer in active use.
They were designed to virtualize primarily two resources:
process ids and IP network addresses. They did not support
SSH, InfiniBand, System~V IPC, or POSIX timers, all of which are commonly
used in modern software implementation.
\subsection{CRIU}
CRIU~\cite{criu} leverages Linux namespaces for transparently
checkpointing on a single host (often within a Linux container),
but lacks support for distributed computations. Instead of directly
virtualizing the process id, CRIU relies on extending the kernel API
through a much larger proc filesystem and a greatly extended ``prctl''
system call. For example, the ``PR\_SET\_MM'' has 13~additional parameters
that can be set (e.g., beginning end end of text, data, and stack).
In another example, CRIU relies on the ``CONFIG\_CHECKPOINT\_RESTORE''
kernel configuration to allow a process to directly modify the kernel's
choice of pid for the next process to be created~\cite{criuSetPid}.
In a general context, there is a danger that the desired pid to be
restored may already be occupied by another process, but CRIU is also
often used within a container where this restriction can be avoided.
Finally, CRIU has a more specialized plugin facility~\cite{criuPlugins}.
Some examples are: ability to save and restore the contents of particular
files; and the means to save and restore pointers to external sockets,
external links, and mount points that are outside the filesystem namespace
of an LXC (Linux Container). Recall that CRIU does not try to support
distributed computations. Perhaps it is for this reason that CRIU did
not have the same pressure to develop a broader plugin system capable
of supporting generic external devices such as hardware emulators.
\subsection{Process Virtualization}
The term {\em process virtualization} was used in~\cite{ValleeEtAl05}.
That work discusses kernel-level support for such process virtualization,
while the current work emphasizes an entirely user-space approach
within unprivileged processes.
Related to process virtualization is the concept of a Library~OS,
exemplified by the Drawbridge Library~OS~\cite{PorterEtAl11} and
Exokernel~\cite{EnglerEtAl95}. However, such systems are concerned
with providing {\em extended or modified} system services that are not
natively present in the underlying operating system kernel.
Both process-level virtualization and the Library~OS approach employ a
user-space approach (ideally with no modification to the application
executable, and no additional privileges required). However, a Library~OS
is concerned with providing {\em extended or modified} system services
that are not natively present in the underlying operating system kernel.
Process virtualization is concerned with providing a semantically
equivalent system object using the {\em same} system service. This need
arises when restarting from a checkpoint image, or when carrying out a
live process migration from one computer to another. The target computer
host is assumed to provide the same system services as were available
on the original host.
Although process-level virtualization and a Library~OS both operate
in user space without special privileges, the goal of a Library~OS
is quite different. A Library~OS modifies or extends the system
services provided by the operating system kernel. For example,
Drawbridge~\cite{PorterEtAl11} presents a Windows~7 personality,
so as to run Windows~7 applications under newer versions of Windows.
Similarly, the original exokernel operating system~\cite{EnglerEtAl95}
provided additional operating system services beyond those of a small
underlying operating system kernel, and this was argued to often be more
efficient that a larger kernel directly providing those services.
\begin{comment}
The CIFTS project~\cite{GuptaEtAl09} provides a fault-tolerance
backplane (FTB), based on an interface specification that
allows libraries and runtime systems to jointly manage faults.
\end{comment}
\vskip -12pt
\section{Conclusion}
\label{sec:conclusion}
In order to develop a successful plugin model for checkpointing
in the context of EDA, one required modularity that enabled
the DMTCP team, Intel, and Mentor Graphics to each write their
own modular code. Further, the Intel and Mentor Graphics DMTCP-based plugins
and other code were of necessity proprietary. This work has shown
how the DMTCP plugin model can be used to provide a flexible
model enabling full cooperation, while avoiding the more extreme
roadmaps of either fully application-specific code or transparent,
system-level checkpointing with no knowledge of the proprietary
aspects of the Mentor Graphics hardware emulator.
|
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| 4,049
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Gap's season comes to an end with loss to East Rock
Despite 329 yards from Carter Rivenburg, Buffalo Gap was no match for top-seed East Rockingham Friday.
Gap's season comes to an end with loss to East Rock Despite 329 yards from Carter Rivenburg, Buffalo Gap was no match for top-seed East Rockingham Friday. Check out this story on newsleader.com: https://www.newsleader.com/story/sports/high-school/2018/11/16/football-buffalo-gap-east-rockingham-vhsl/2015208002/
Patrick Hite, Staunton News Leader Published 9:32 p.m. ET Nov. 16, 2018
Buffalo Gap's Seth Fitzgerald has the ball as he is wrapped up and brought down during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. (Photo: Mike Tripp/The News Leader)Buy Photo
HARRISONBURG — Carter Rivenburg went out with a big performance on a big stage, but it wasn't enough. Not nearly enough.
The Buffalo Gap senior had 329 yards in his final game in the Bison backfield.
But East Rockingham came out on top, beating the Bison 51-22 in the Region 2B semifinals Friday at James Madison University's Bridgeforth Stadium. The Eagles remain unbeaten on the season, heading into the regional championship with a 12-0 record.
The game was moved from East Rock to JMU because of icy weather.
J'wan Evans, East Rock's all-district running back, had 220 yards in the game on just eight carries, all in the first half. He sat the final two quarters.
And East Rock's quarterback was 7-of-7 for 202 yards passing and four touchdowns in the first half.
Buffalo Gap vs East Rock - Region 2B semifinals
Buffalo Gap's Seth Fitzgerald has the ball as he is wrapped up and brought down during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
East Rockingham's Isaac Kisling catches a pass intended for him as Buffalo Gap's Tucker Kiracofe tries to make the stop during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap's Trevor Staton runs the football during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap's Trevor Staton runs the football as East Rockingham's Trenton Morris goes for the tackle during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap's Tucker Kiracofe has the football as he is surrounded by East Rockingham's Taylor Spencer, Isaac Kisling and Colton Dean during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap's Carter Rivenburg runs the ball after breaking free of a tackle during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap's Carter Rivenburg runs the ball along the sidelines for a first down during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap's Carter Rivenburg is brought down with the football during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap's Carter Rivenburg goes down with the ball during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap's Seth Fitzgerald looks to pass during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap's Carter Rivenburg runs the ball through a hole opened by teammates in the East Rockingham defensive line during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap's Carter Rivenburg runs the ball during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap's Carter Rivenburg spins and breaks free of a tackle attempt with the ball during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap's Carter Rivenburg makes it into the end zone for a touchdown during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap's Trevor Staton has the ball slip away as he tried to grab onto it in a kick return during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap's Carter Rivenburg protects the ball as he is brought down during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap's Carter Rivenburg runs the football during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap's Carter Rivenburg moves the football in the backfield as he looks to pass during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap's Carter Rivenburg still looks to pass as he heads to the sideline while an East Rockingham defender tries to make the stop during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap's Carter Rivenburg tries to squeeze through the pack with the ball during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
A Buffalo Gap defender moves to tackle East Rockingham's Darrias Brown during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap's Tucker Kiracofe paces the East Rockingham ball carrier as he closes in for a tackle during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap quarterback Seth Fitzgerald hands the ball off to Carter Rivenburg who will run it during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap's Cody McCray is brought down with the ball during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap'sSeth Fitzgerald is brought down with the ball during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap's Cody McCray breaks free of a tackle attempt as he runs the ball during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap's Cody McCray runs the ball during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap's Cody McCray is shoved out of bounds as he has the ball during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap quarterback Seth Fitzgerald passes the ball during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
Buffalo Gap's Christian Elkins goes down after failing to catch a pass intended for him during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. Mike Tripp/The News Leader
"They're a really good football team," Buffalo Gap coach Andy Cline said. "We had no answer defensively tonight. They're really fast and have some weapons. We knew that."
Williams had scoring passes of 18 to Isaac Kisling, 72 to Tyce McNair, 83 to Darrias Brown and 3 to McNair.
East Rock's other scores came on a 62-yard punt return by Brown, and 89-yard run by Evans, and a 26-yard run by Chandler Breeden.
Buffalo Gap's Christian Elkins goes down after failing to catch a pass intended for him during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. (Photo: Mike Tripp/The News Leader)
Gap's one score in the first half came on a Rivenburg 16-yard run. That was set up by a pass interference call against East Rock. Gap had gone for it on fourth-and-8, attempting a pass. The penalty moved Gap from the 31 to the 16 and Rivenburg scored on the next play to get Gap within 21-6.
But East Rock finished the half by scoring 30 unanswered points.
Rivenburg had 260 yards at halftime, including four runs of more than 30 yards. But the Bison couldn't find the end zone to cap many of those long runs.
Buffalo Gap added two second-half touchdowns. Rivenburg had a touchdown on a 2-yard run. Colin Bowers added the two-point conversion with 5:18 left in the third quarter.
Buffalo Gap's Carter Rivenburg runs the football during a Region 2B semifinal game played at James Madison University on Friday, Nov. 16, 2018. (Photo: Mike Tripp/The News Leader)
Gap junior Tucker Kiracofe then scored on a 3-yard run. Seth Fitzgerald hit Ryan Benitez for the two-point conversion. Those were the final points of the game.
"It is what it is," Cline said. "Our kids battled and played hard all year. I'm proud of them. It did not turn out like we wanted it to, certainly, but they're a good football team and so are we. It just comes to an end."
The game marked the end of high school football for 14 seniors listed on the Gap roster.
"They're a special group," Cline said. "They have been to the playoffs ever year they've been in high school and have done a lot for this program. I couldn't be prouder of them. There are some special kids out there."
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,097
|
<component name="libraryTable">
<library name="support-vector-drawable-25.3.1">
<CLASSES>
<root url="file://$USER_HOME$/.android/build-cache/7a1745d32a82ccf18d36e2a189b595a405eff6a3/output/res" />
<root url="jar://$USER_HOME$/.android/build-cache/7a1745d32a82ccf18d36e2a189b595a405eff6a3/output/jars/classes.jar!/" />
</CLASSES>
<JAVADOC />
<SOURCES>
<root url="jar://E:/Android_SDK/extras/android/m2repository/com/android/support/support-vector-drawable/25.3.1/support-vector-drawable-25.3.1-sources.jar!/" />
</SOURCES>
</library>
</component>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,013
|
/**
*
*/
package ie.ibuttimer.widget;
import ie.ibuttimer.pmat.util.Logger;
import java.util.List;
import android.content.Context;
import android.view.LayoutInflater;
import android.view.View;
import android.view.ViewGroup;
import android.widget.ArrayAdapter;
import android.widget.CheckBox;
import android.widget.LinearLayout;
/**
* @author Ian Buttimer
*
*/
public class CheckBoxListAdapter<T> extends ArrayAdapter<T> {
private int layoutId; // id of layout containing CheckBox
private int checkBoxViewResourceId; // id of CheckBox in layout
private boolean[] state; // selected state of items in list
/**
* Constructor
* @param layoutResourceId The resource ID for a layout file containing a layout to use when instantiating views
* @param viewResourceId The id of the CheckBox within the layout resource to be populated
* @param objects The objects to represent in the ListView
* @param state Initial state of the CheckBox for the objects
*/
public CheckBoxListAdapter(Context context, int layoutId, int checkBoxViewResourceId, List<T> objects, boolean[] state) {
super(context, checkBoxViewResourceId, objects);
this.layoutId = layoutId;
this.checkBoxViewResourceId = checkBoxViewResourceId;
this.state = new boolean[state.length];
System.arraycopy(state, 0, this.state, 0, state.length);
}
/**
* Return the selected state of the items in the list
* @return
*/
public boolean[] getSelections() {
return state;
}
/**
* Clear all the selected check boxes
*/
public void clearSelections() {
final int N = state.length;
for ( int i = 0; i < N; ++i ) {
state[i] = false;
}
notifyDataSetChanged();
}
/* (non-Javadoc)
* @see android.widget.ArrayAdapter#getView(int, android.view.View, android.view.ViewGroup)
*/
private View getItemView(int position, View convertView, ViewGroup parent) {
LinearLayout itemView;
T item = getItem(position);
if ( convertView == null ) {
itemView = new LinearLayout(getContext());
String inflater = Context.LAYOUT_INFLATER_SERVICE;
LayoutInflater vi = (LayoutInflater)getContext().getSystemService(inflater);
vi.inflate(layoutId, itemView, true);
}
else {
itemView = (LinearLayout)convertView;
}
// set the text
CheckBox checkBox = (CheckBox)itemView.findViewById(checkBoxViewResourceId);
try {
checkBox.setText( ((TextViewAdapterInterface) item).toDisplayString() );
}
catch ( ClassCastException e ) {
String msg = item.getClass().getName() + " supplied object does not implement TextViewAdapterInterface";
Logger.wtf(msg, e);
throw ( e );
}
// set the state
checkBox.setChecked(state[position]);
// add listener to remember the state of the check box
checkBox.setOnClickListener( new View.OnClickListener() {
public void onClick(View v) {
CheckBox checkBox = (CheckBox) v;
int position = (Integer) v.getTag();
if(checkBox.isChecked())
state[position] = true;
else
state[position] = false;
}
});
// store the position, so the listener can retrieve it and update the correct position in the state array
checkBox.setTag(position);
return itemView;
}
/* (non-Javadoc)
* @see android.widget.ArrayAdapter#getView(int, android.view.View, android.view.ViewGroup)
*/
@Override
public View getView(int position, View convertView, ViewGroup parent) {
return getItemView(position, convertView, parent);
}
/* (non-Javadoc)
* @see android.widget.ArrayAdapter#getDropDownView(int, android.view.View, android.view.ViewGroup)
*/
@Override
public View getDropDownView(int position, View convertView, ViewGroup parent) {
return getItemView(position, convertView, parent);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,000
|
import gulp from 'gulp'
import uglify from 'gulp-uglify'
import cssnano from 'gulp-cssnano'
import htmlmin from 'gulp-htmlmin'
import gulpif from 'gulp-if'
import scripts from './scripts'
import styles from './styles'
import templates from './templates'
import images from './images'
import fonts from './fonts'
import critical from './critical'
const build = () => {
gulp.src(['.tmp/**/*', 'src/*.*'])
.pipe(gulpif('*.js', uglify()))
.pipe(gulpif('*.css', cssnano()))
.pipe(gulpif('*.html', htmlmin({
collapseWhitespace: true,
minifyCSS: true,
minifyJS: true,
minifyURLs: true,
removeAttributeQuites: true,
removeComments: true,
removeEmptyAttributes: true,
removeOptionalTags: true,
removeRedundantQuotes: true
})))
.pipe(gulp.dest('dist'))
}
gulp.task('build', [
'scripts',
'styles',
'templates',
'critical',
'images',
'fonts'
], build)
export default build
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,384
|
If you play the Michigan Lottery online and win, expect a phone call.
If you don't, keep an eye on your ticket and the phone number on the back.
Anthony Frabotta, 66, of Rochester Hills played online, he kept getting phone calls until someone from Lansing got him on the phone earlier this month and informed him he had won $1 million in the Jan. 24 Mega Millions drawing.
Online jackpot winners are the only ones contacted by phone, said Jeff Holyfield, a spokesman for the Michigan Lottery Commission.
"What happens is with the online games where players have registered ... we know immediately who it is," Holyfield said. "If we see a person has won a prize of that sort then we'll give them a few days. If we don't see them logging into their account, then we'll make that outreach to let them know that they have won a million dollars or more.
Winning is one thing; how much you take home is another.
If a player wins up to $5,000 there are no withholdings.
If the jackpot is more than $5,000 the Lottery Commission is required by law to withhold 25% for Federal income tax and 4.25% for state income taxes.
Another thing the Lottery Commission does is check anyone winning $1,000 or more to see if they owe any debt to the state. That would include back taxes and child support. If they owe the state money is withheld to satisfy the debt.
"There are instances where people would come in with the winning ticket and we'll process it and instead of giving them a check they get a letter that says you have this debt and we've withheld this amount which was registered with the Dept. of Treasury,'' said Holyfield.
Winners who have to go to Lansing to collect their winnings must have the winning ticket, a valid photo ID, usually a driver's license, and a social security card. The names on the social security card and the photo ID must match.
That information would be important if someone won a record jackpot.
The Powerball record was $1.6 billion on Jan. 13, 2016.
It was the first jackpot to hit $1 billion and "everybody in the industry threw up their hands and said that's the world record,'' said Holyfield.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,210
|
by Project Title
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The National Academies
Room KECK 219
Email: paro@nas.edu
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Print Feedback PARO
Decadal Assessment and Outlook Report on Atomic, Molecular, and Optical Science
The committee will be charged with producing a comprehensive report on the status and future directions of atomic, molecular, and optical (AMO) science. The committee's report shall:
1. Review the field of AMO science as a whole, emphasize recent accomplishments, and identify new opportunities and compelling scientific questions.
2. Use case studies in selected, non-prioritized fields in AMO science to describe the impact that AMO science has on other scientific fields, identify opportunities and challenges associated with pursuing research in these fields because of their interdisciplinary nature, and inform recommendations for addressing these challenges.
3. Identify the impacts of AMO science, now and in the near future, on emerging technologies and in meeting national needs.
4. Evaluate recent trends in investments in AMO research in the United States relative to similar research that is taking place internationally, and provide recommendations for either securing leadership in the United States for certain subfields of AMO science, where appropriate, or for enhancing collaboration and coordination of such research support, where appropriate.
5. Identify future workforce, societal, and educational needs for AMO science.
6. Make recommendations on how the U.S. research enterprise might realize the full potential of AMO science.
In carrying out its charge, the committee might consider issues such as the state of the AMO research community, international models for support and collaboration, and institutional and programmatic barriers.
PIN: DEPS-BPA-16-01
Project Duration (months): 18 month(s)
RSO: Jones, Christopher
Division(s):
Division on Engineering and Physical Sciences
Board(s)/Committee(s):
Board on Physics and Astronomy
Math, Chemistry, and Physics
Geographic Focus:
Committee Post Date: 08/02/2018
Ray Beausoleil
RAY BEAUSOLEIL is the HPE Senior Fellow for Information and Quantum Systems at Hewlett Packard Laboratories. He leads the Large-Scale Integrated Photonics Research Group at Hewlett Packard Labs, where he is responsible for research on the applications of optics at the micro/nanoscale to high performance classical and quantum information processing. Beausoleil's research interests includes solid-state laser physics, nonlinear optics, quantum optics, quantum information science and technology, nanophotonics, embedded computer algorithms, and image processing. He received his Ph.D. from Stanford University in physics.
Louis F. DiMauro
LOUIS DIMAURO is the Edward E. and Sylvia Hagenlocker Chair of Physics at the Ohio State University (OSU). Before joining OSU in 2007, he was a senior scientist at Brookhaven National Laboratory. His research interest is in experimental ultra -fast and strong-field physics. His current work is focused on the generation, measurement and application of attosecond x-ray pulses and the study of fundamental scaling of strong field physics. Dr. DiMauro received his BA (1975) from Hunter College, CUNY and his Ph.D. from University of Connecticut in 1980. He was a postdoctoral fellow at SUNY at Stony Brook before arriving at AT&T Bell Laboratories in 1981.
Mette Gaarde
METTE GAARDE is a Professor of Physics at Louisiana State University (LSU). Previously, Dr. Gaarde was a research assistant professor at Lund University in Sweden. Dr. Gaarde is an expert on the theory of ultrafast and strong-field laser-matter interactions in atomic, molecular, and solid systems. In particular, she is interested in the interplay between the microscopic (quantum) effects and the macroscopic (classical) effects that govern this interaction. She recently served on the Committee on Atomic, Molecular, and Optics Sciences, on the Editorial Board of Physics Revision-A, and in the chair-line of the APS National Organizing Committee for the Conferences for Undergraduate Women in Physics. She is currently serving on the executive committee for the APS Division of Atomic, Molecular, and Optical Physics (DAMOP) as well as on the DAMOP Fellowship Committee. Dr. Gaarde earned her Ph.D. in physics from Copenhagen University, Denmark.
Steven M. Girvin
STEVE GIRVIN [NAS] is a Eugene Higgins Professor of Physics and Applied Physics at Yale University. He is a theoretical physicist who studies the quantum mechanics of large collections of atoms, molecules, and electrons such as are found in superconductors, magnets, and transistors. Dr. Girvin is interested in quantum many-body physics, and quantum and classical phase transitions, particularly in disordered systems. Much of his work has been on the quantum Hall effect, but he has also worked on the superconductor-insulator transition, the vortex glass transition in high Tc superconductors, superfluid helium in fractal aerogel, the Anderson localization problem, the Coulomb blockade problem in mesoscopic device physics, and on quantum spin chains. He received his Ph.D. in 1977 from Princeton University.
Chris H. Greene
CHRIS H. GREENE is a Professor of Physics at Purdue University. Previously he was at Louisiana State University and the University of Colorado at Boulder before joining Purdue University. His research concentrates on theoretical atomic, molecular, and optical physics. His expertise has been on novel treatments of few-body quantum systems, such as universal Efimov physics, ultra-long-range "trilobite" Rydberg molecules, collisions in Bose-Einstein condensates, atomic/molecular collision, and photo-absorption processes. He has served in the past as chair of JILA at the University of Colorado. He received his Ph.D. from the University of Chicago in theoretical atomic physics in 1980. In 1981, he was a postdoctoral research associate at Stanford University.
Taekjip Ha
TAEKJIP HA [NAS] is Bloomberg Distinguished Professor of Biophysics and Biophysical Chemistry at Johns Hopkins University. He is also an Investigator for Howard Hughes Medical Institute. Ha's research is focused on pushing the limits of single-molecule detection methods to study complex biological systems. His group develops state-of-the-art biophysical techniques and applies them to study diverse protein-nucleic acid and protein-protein complexes, and mechanical perturbation and response of these systems both in vitro and in vivo. Dr. Ha received his B.S. (1990) from Seoul National University and Ph.D. from University of California-Berkeley (1996) in physics.
Mark A. Kasevich
MARK KASEVICH a Professor of Applied Physics at Stanford University. Prior to Stanford University, he was at Yale University. His research interests are centered on the development of quantum sensors of rotation and acceleration based on cold atoms (quantum metrology), the application of these sensors to the tests of General Relativity, the investigation of many-body quantum effects in Bose-condensed vapors (including quantum simulation), and the investigation of ultra-fast laser-induced phenomena. He graduated from Dartmouth College in 1985 with a B.A. in physics and received his Ph.D. from Stanford University in applied physics in 1992.
Mikhail D. Lukin
MIKHAIL LUKIN [NAS] is a Professor of Physics at Harvard University, where he is also a co-director of the Harvard-MIT Center for Ultracold Atoms. His research interests include quantum optics, quantum control of atomic and nanoscale solid-state systems, quantum metrology, nanophotonics, and quantum information science. He has co-authored over 300 technical papers and has received a number of awards, including the Alfred P. Sloan Fellowship, the David and Lucile Packard Fellowship for Science and Engineering, the NSF Career Award, the Adolph Lomb Medal of the Optical Society of America, the AAAS Newcomb Cleveland Prize, the APS I.I.Rabi Prize, the Vannevar Bush Faculty Fellowship, the Julius Springer Prize for Applied Physics, and the Willis E. Lamb Award for Laser Science and Quantum Optics. He received his Ph.D. from Texas A&M University in 1998.
A. Marjatta Lyyra
A. MARJATTA LYYRA is a Professor of Physics at Temple University. Prior to joining Temple University, she was a research scientist at the University of Iowa. Her field of interest is in experimental atomic, molecular and optical physics. Dr. Lyyra received her B.S. and M.S. from the University of Helsinki, Finland (1972, 1974), and Ph.D. from the University of Stockholm, Sweden, in 1979.
Marianna Safronoa
MARIANNA SAFRONOVA is a Professor of Physics at the University of Delaware and an adjunct fellow of the Joint Quantum Institute, NIST, and the University of Maryland. Marianna Safronova is currently the chair-elect of the American Physical Society Division of the Atomic, Molecular, and Optical Physics (DAMOP) and a member of Physical Review-A Editorial Board (2012-2018). Her diverse research interests include the study of fundamental symmetries and search for physics beyond the standard model of elementary particles and fundamental interactions; development of high-precision methodologies for calculating atomic properties and exploring their applications; atomic clocks, ultra-cold atoms, and quantum information; long-range interactions; superheavy atoms; highly-charged ions; atomic anions; and other topics. In 2001, she received her Ph.D. from University of the University Norte Dame.
Peter Zoller
PETER ZOLLER [NAS] is a Professor of Physics at the University of Innsbruck in Austria, and Scientific Director for the Institute for Quantum Optics and Quantum Information (IQOQI) of the Austrian Academy of Sciences. His interest and expertise are in the field of theoretical quantum optics, in particular the description of interaction of light with matter, and various aspects of quantum noise. During the last ten years the focus of his work has been on the interface between quantum optics and quantum information, and condensed matter physics with cold atoms. Peter Zoller received his Ph.D. in physics at the University of Innsbruck.
Nergis Mavalvala - (Co-Chair)
NERGIS MAVALVALA [NAS] is the Curtis and Kathleen Marble Professor of Astrophysics at MIT. She is working on the detection of gravitational waves and quantum measurement science. She is a longtime member of the scientific team that announced in 2016 the first direct detection of gravitational waves from colliding black holes by the Laser Interferometer Gravitational-wave Observatory (LIGO). In the quest for ever greater sensitivity in the LIGO detectors, Mavalvala has also conducted pioneering experiments on generation and application of exotic quantum states of light, and on laser cooling and trapping of macroscopic objects to enable observation of quantum phenomena, that usually manifest at the atomic scale, in human-scale systems. Mavalvala received a B.A. from Wellesley College and a Ph.D. from MIT. She was a postdoctoral fellow and research scientist at the California Institute of Technology before joining the Physics faculty at MIT in 2002. She was appointed Associate Department Head of Physics in February 2015. Mavalvala is recipient of numerous honors, including a MacArthur "genius" award in 2010 and election to the National Academy of Sciences in 2017.
Jun Ye - (Co-Chair)
JUN YE [NAS] is currently a fellow of JILA and NIST at the University of Colorado Boulder. At JILA, his research focuses on the frontiers of light-matter interactions and includes precision measurement, quantum physics and ultracold matter, optical frequency metrology, and ultrafast science. Dr. Ye is a recipient of many awards and honors, including the Rabi Award (IEEE), U.S. Presidential Rank (Distinguished) Award, three Gold Medals from the U.S. Commerce Department, Foreign Member of the Chinese Academy of Sciences, Frew Fellow of the Australian Academy of Science, I. I. Rabi Prize of the American Physical Society, European Frequency and Time Forum Award, Carl Zeiss Research Award, William F. Meggers Award and Adolph Lomb Medal from the Optical Society of America, Arthur S. Flemming Award, Presidential Early Career Award for Scientists and Engineers, Friedrich Wilhem Bessel Award of the Alexander von Humboldt Foundation, Samuel Wesley Stratton Award, and Jacob Rabinow Award from NIST. He earned his Ph.D. in physics from the University of Colorado in 1997. He served as a member of the Academies Committee on Atomic, Molecular and Optical Sciences (CAMOS).
Patricia M. Dehmer
PATRICIA M. DEHMER is the former Deputy Director for Science Programs in the Office of Science (SC) in the U.S. Department of Energy (DOE) and the former Director of the Office of Basic Energy Sciences (BES) within SC. As the deputy director, Dr. Dehmer was the senior career science official in SC and was the acting director between Senate-confirmed Presidential appointees, most recently for three years from 2013 to 2015. As director of the BES Program, Dr. Dehmer was known for her broad support of physical science research and for the planning, design, and construction phases of a dozen major scientific construction projects totaling more than $3 billion. Previously, Dr. Dehmer was a distinguished fellow at Argonne National Laboratory with research activities in atomic, molecular, optical, and chemical physics. Since her retirement from federal service in 2016, Dr. Dehmer works as a management consultant with additional service on boards, science advisory committees, and professional society committees. During her federal service, Dr. Dehmer was awarded three Presidential Rank Awards and, in 2016, the James R. Schlesinger Award – the highest recognition in DOE – for management of SC's portfolio in the physical sciences and for outstanding management of the Department of Energy's largest-scale scientific construction projects. She is a fellow of the American Physical Society and the American Association for the Advancement of Science. She earned her Ph.D. in chemical physics from the University of Chicago. Dr. Dehmer served on and was vice chair of the Academies Committee on Atomic, Molecular, and Optical Sciences.
Michal Lipson
MICHAL LIPSON is the Eugene Higgins Professor in Electrical Engineering and Professor of Applied Physics at Columbia University. Prior to joining Columbia University, she was the Given Foundation Professor of Engineering at Cornell University. Her research interests are in silicon photonics, inventor of GHz silicon modulator, novel on-chip nanophotonics devices, and novel micron-size photonic structures for light manipulation. In 2014, she was named by Thomson Reuters as a top 1% highly cited researcher in the field of physics. She completed her B.S., M.S., and Ph.D. degrees in physics at the Technion (Israel Institute of Technology) followed by a postdoctoral position at M.I.T. in the Materials Science Department until 2001.
Peter J. Reynolds
PETER J. REYNOLDS is a senior research scientist at the Army Research Office. He is responsible for setting the direction of the U.S. Army's Army Research office, particularly the scientific program of the Physical Sciences Directorate as well as in-house programs of the Army Research Laboratory (ARL). The programs in his direct purview include those at ARO in the Physics, Chemistry, and Life Sciences Division, and support in particular emerging areas of research. Prior to joining ARO, Dr. Reynolds was a research professor at Georgetown University from 1996-2005 and a program manager at the Office of Naval Research from 1988-2003. He has received the U.S. Presidential Rank Award as a Distinguished Senior Scientist in 2015, and is a member of APS, MRS, and OSA. Dr. Reynolds obtained his Ph.D. in physics from MIT in 1979 and his AB in physics from the University of California at Berkeley in 1971.
Committee Membership Roster Comments
Update (5/21/18): Please note that there has been a change in the committee membership with the appointment of all committee members aside from Dr. Ye and Dr. Mavalvala.
Update (8/2/18): Please note that there has been a change in the committee membership with the appointment of Dr. Dehmer and Dr. Reynolds.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,517
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This call deletes an admin user on a Relying Party.
Note. The API call will not delete an admin if they are the only (non-pending) admin for that RP. The call will respond with an error in this case. In the dashboard, choosing to delete the last admin of an RP will cause the RP to be deleted. The delete API call should be used for that purpose when using the API.
email string Yes Email address of the administrator.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 39
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{"url":"http:\/\/www.ein-unfall-aendert-alles.de\/rpqv6bm\/academo-wavelength-to-color-9d9d11","text":"The aim is for a suite of pages for a series of Tuition, Teaching or Group sessions. (\u00a0Log\u00a0Out\u00a0\/\u00a0 Academo. Using the term \"cool\" to describe something glowing red hot could be thought of as a slight misnomer, but it helps you to understand that compared to blue-hot objects, red-hot objects certainly are cool! Source code available at GitHub.com, Algorithm for generating RGB colours created by. (If you're interested in the exact way in which this occurs, please head over to our demo of Planck's Law of Blackbody Radiation.). The color is also written in RGB and hexadecimal format. In very simple terms, a hotter object emits more high frequency radiation than a less hot one. This web-blog is also for working locally with other people or a community and to balance this with other resource creation and actual work, the timing of input is essentially important. The color is also written in RGB and hexadecimal format. Jul 20, 2015 - Wavelength to Colour Relationship | Academo.org - Free, interactive, education. As with our Wavelength-Colour demo, the perception of colour by the the human eye depends not only on the wavelength of the incoming radiation but also on a number of additional factors (including psychological ones), so this scale should best be thought of as an approximation. Sunlight looks white because it contains all visible colors. Fill in your details below or click an icon to log in: Email (required) (Address never made public). The wavelength of light must be even higher than this. \u00a9 Academo.org 2020. So, hexatriene absorbs light at approximately 258 nanometers. .... and we have to sleep sometime. Maths Geometry Polar plot parametric. It goes up to approximately 258 nanometers. Name (required) A tool to convert a temperature in Kelvin into a RGB colour, As an object heats up, it begins to emit light. For the sake of this explanation, a \"hot\" object will have a temperature of around 15,000 Kelvin, a \"warm\" object will be at approximately 6,500 Kelvin and a \"cool\" object will be around 1,500 Kelvin. Redder light, with longer wavelengths, bends less. (\u00a0Log\u00a0Out\u00a0\/\u00a0 As with our Wavelength-Colour demo, the perception of colour by the the human eye depends not only on the wavelength of the incoming radiation but also on a number of additional factors (including psychological ones), so this scale should best be thought of as an approximation. The violet\/purple wavelength starts at about 400nm, which goes into 800, therefore the red conecells are a \u2026 In order for something to have a color, it has to absorb light in the visible region. (\u00a0Log\u00a0Out\u00a0\/\u00a0 And this is the key point. About; Search; Contact; Submit an idea; Sponsor; Browse all demos; All; Engineering; Geography; Maths; Music; Physics; Ad. Bluer light has a shorter wavelength and bends more. Wavelengths that are a bit outside this range will only excite the red conecells a bit. .... For example there has to be enough time to actually Tutor or Teach and yet internet speed is quickest in the very early mornings. (\u00a0Log\u00a0Out\u00a0\/\u00a0 As we become faster and have all the essential in mind, the initial project will become perfect as it is created. https:\/\/academo.org\/demos\/wavelength-to-colour-relationship\/. The wavelengths that excites the red conecells the best range from 700nm to 646nm. Change\u00a0), You are commenting using your Facebook account. Rhodonea Curves (Roses) An interactive demonstration of Rhodonea curves, also known as Roses. Change\u00a0), You are commenting using your Twitter account. The slider above allows you to control the temperature, which in turn changes the display to the color of the light that would be emitted from an object at that temperature. Change\u00a0), You are commenting using your Google account. A rhodonea curve is a graph of the following polar equation: $r = A cos (k \\theta)$ where $$k = \\frac{m}{n}$$. Fill in your details below or click an icon to log in: You are commenting using your WordPress.com account. This creates a dilema in awake time and effectiveness time - when other folk nearby are just going to work - the internet input has only just been completed. Change\u00a0). That's still in the UV region of the electromagnetic spectrum. Create a free website or blog at WordPress.com. Blue light has a higher frequency than red light, so it follows that hot objects will glow bluish, warm objects will glow white (made up from a combination of blue and red light), and cool objects will glow red. So, hexatriene doesn't have a color. After Newton, other scientists discovered that the light's wavelength is what determines how much it will refract, and what color it will be. Initially lets create some ideas - and if we like them - then it's the time to perfect.","date":"2021-02-26 09:52:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.19189812242984772, \"perplexity\": 1165.7359984167194}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178356456.56\/warc\/CC-MAIN-20210226085543-20210226115543-00257.warc.gz\"}"}
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\section*{Introduction}
F. Enriques and D. W. Babbage \cite{En,B} proved that a nonhyperelliptic smooth canonical curve is the set theoretic intersection of hyperquadrics, unless it is trigonal or isomorphic to a plane quintic. It can be read off from their works the following statement: ``a regular, integral and projective curve is trigonal if and only if it is isomorphic to a canonical curve which lies on a nonsingular two-dimensional rational normal scroll"
From this perspective, two generalizations of such a characterization are quite natural, and, in fact, were actually done. One may consider higher gonality, or, rather, allow trigonal curves to have singularities.
In the first vein above, one finds, for instance, in Schreyer's \cite[Sec. 6]{Sc}, a detailed study of the relation between a $d$-gonal canonical curve $C$ and the $(d-1)$-fold scroll $S$ it lies on, specially when $d=4,5$ (and $d=3$ as well). This envolves, e.g., verifying the uniqueness of the $g_d^1$; finding the resolution of $C$ inside $S$, and also determining upper and lower bounds for the invariants of $S$ in terms of the genus of $C$.
On the other way, in \cite{RS}, St\"ohr and Rosa devoted their study to the case where both the trigonal curve and the surface scroll are singular. Their results perfectly match the statement above replacing ``regular" by ``Gorenstein". A key point in their approach was that the linear series could possibly admit non-removable base points. This relaxation on the standard notion of a pencil turned out to be necessary once they proved that a canonical (Gorenstein) curve lying on a cone always meets the vertex, which cannot be removed, otherwise the curve would be hyperelliptic. Actually, systems with non-removable base points appear earlier in the literature, introduced by Coppens in \cite{Cp}. Essentially, one is allowing torsion free sheaves of rank $1$ (rather than bundles) in the definition of linear series,
which, by its turn, allows pencils (rather than morphisms) to compute gonality (see Section \ref{secs21}).
When this study is led to non-Gorenstein curves, soon we come accross the following question: what exactly would be a generalization of the statement above in such a case? The terms ``isomorphic" and ``canonical" leads now to different ways, depending on which choice we make. Indeed, given a $d$-gonal non-Gorenstein curve $C$, one may look for a $(d-1)$-fold scroll containing: (i) either an isomorphic copy of $C$ by means of a suitable embedding; (ii) or, rather, a curve $C'$ which could be naturally called a ``canonical model" for $C$.
In Example \ref{exedif} we make few remarks about the difficulties of getting (i) above since the standard methods of inducing inclusions on scrolls by pencils may fail to reach the expected dimension when $C$ is not Gorenstein. On the other hand, if (ii) stands for an option, one may deal, for instance, with the notion of a \emph{canonical model} $C'$ introduced by Rosenlicht in \cite{R} and also studied by S. L. Kleiman along with the third named author in \cite{KM} (see Section \ref{seccan}). Moreover, within this framework, \cite[Thm. 3.4]{KM} which states that $C'$ is the (non-degenerate) rational normal curve (of minimal degree) if and only if $C$ is either hyperelliptic or rational nearly normal,
can be rephrased as: $C$ is $2$-gonal if and only if $C'$ lies on a $1$-fold scroll. So the mere formalism of considering rational normal curves as scrolls turns here into the first step of a general result, proved in Theorem \ref{thmrat} for monomial curves.
We start discussing the general case in Section 2. We show in Theorem \ref{thmscr} how to get an inclusion $C'\subset S$ from any pencil on $C$. In particular we get that $S$ is $(d-1)$-dimensional if $C$ is $d$-gonal. The result extends St\"ohr-Rosa's \cite[Thm. 2.1, Lem 2.3]{RS} from trigonal curves to any gonality, using similar methods based on Andreotti-Mayer's \cite{AM}. We also give an upper bound for the dimension of the singular set of $S$ in terms of some invariants of the pencil, and look for sufficient conditions for $S$ to be in fact singular.
In Section 3 we do the reverse engeneering, that is, we assume $C'$ lies on a given scroll $S$ with prescribed dimension $d$ and intersection number $\ell$ with a generic fiber of $S$. Varrying $\ell$, we are able to relate some important properties of $C$ with $d$ and other invariants of $S$. Our main concern is gonality, but we also study the number of non-Gorenstein points of $C$ and then check when the curve happens to be Kunz, nearly Gorenstein or nearly normal. These concepts, based on local principles introduced by Barucci and Fr\"oberg in \cite{BF}, got a geometric characterization in \cite[Thms. 5.10, 6.5]{KM} where they were connected to projective and arithmetic normality of the canonical model. They seem to be an essential tool when making first distinctions among non-Gorenstein curves. We summarize the results we got in Theorem \ref{thmth2}, which is a generalization to arbitrary $d$ of \cite[Thms. 2.1, 4.1]{LMM}, proved to $d=2,3$ by Marchesi along with the first and last named author. We close the section with Theorem \ref{thmth3}, which deals with a particular case.
However, we do not obtain a converse for the assertion of prior section. In fact, it will be clear the difficulty of adjusting the arguments of, for instance, \cite{Sc, BS, CE}, even for small $d$, when the dualizing sheaf fails to be a bundle. On the other hand, in Section 4, we prove that $C$ is $d$-gonal if and only if $C'$ lies on a $(d-1)$-fold scroll if $C$ is a rational monomial curve in Theorem \ref{thmrat}. It generalizes \cite[Thms. 3.3, 5.1]{LMM} which was proved assuming $d=2,3$ and $C$ with just one singular point. The key point is a combinatorial description of $C'$ in Theorem \ref{prprat}, which is an extension of \cite[Prp 3.1]{LMM} and by means of slightly different arguments.
\
\noindent{\bf Acknowledgments.} The third named author is partially supported by CNPq grant number 306914/2015-8.
\section{Preliminaries}
For the remainder, a \emph{curve} is an integral and complete one-dimensional scheme over an algebraically closed ground field. Let $C$ be a curve of (arithmetic) genus $g$ with structure sheaf $\oo_C$, or simply $\oo$, and $k(C)$ the field of rational functions. Let $\pi :\overline{C}\rightarrow C$ be the normalization map, set $\overline{\oo}:=\pi _{*}(\oo _{\overline{C}})$ and call $\ccc:=\mathcal{H}\text{om}(\overline{\oo},\oo)$, the conductor of $\overline{\oo}$ into $\oo$. Let also $\ww_C$, or simply $\ww$, denote the dualizing sheaf of $C$. A point $P\in C$ is said to be \emph{Gorenstein} if $\ww_P$ is a free $\oo_P$-module. The curve is said to be \emph{Gorenstein} if all of its points are so, or equivalently, $\ww$ is invertible. It is said to be \emph{hyperelliptic} if there is a morphism $C\rightarrow\pp^1$ of degree $2$.
\subsection{Linear Systems and Gonality}
\label{secs21}
A \emph{linear system of dimension $r$ in $C$} is a set of the form
$$
{\rm L}:=\dd(\aaa ,V):=\{x^{-1}\aaa\ |\ x\in V\setminus 0\}
$$
where $\mathcal{F}$ is a coherent fractional ideal sheaf on $C$ and $V$ is a vector subspace of $H^{0}(\aaa )$ of dimension $r+1$. The sheaf above is given by
$$
(x^{-1}\aaa)(U):=x^{-1}(\aaa(U))
$$
for any open set $U$ on $C$, which makes sense since $x^{-1}\in k(C)$ and $\aaa(U)\subset k(C)$. In other words, we are using the language of sheaves to the approach of ``divisors by product" of \cite{S}. That is, linears systems are regularly defined by
$$
\{D+\text{div}(x)\,|\, x\in V\subset L(D)\}
$$
for a (Weil) divisor $D$. On the other hand, here, ``divisors" are ``fractional ideal sheaves" so ``product" plays the role of ``sum" and ``inclusion" the role of ``inequality" (see \cite{S} for more details). So as $x\in L(D)$ if and only if $0\leq D+\text{div}(x)$, similarly, $x\in H^0(\aaa)$ if and only if $x\op\in\aaa(U)$ if and only if $\op\subset x^{-1}\aaa(U)$ for any open set $U$. So the structure sheaf $\oo$ is $0$ and $x^{-1}\oo$ is $\text{div}(x)$ compared to the divisors theory. This approach happens to be useful specially for describing linear systems with non-removable base points, which we define below. The \emph{degree} of the linear system is the integer
$$
d:=\deg \aaa :=\chi (\aaa )-\chi (\oo)
$$
Note, in particular, that if $\oo\subset\aaa$ then
$$
\deg\aaa=\sum_{P\in C}\dim(\aaa_P/\op).
$$
The notation $g_{d}^{r}$ stands for ``linear system of degree $d$ and dimension $r$". The linear system is said to be \emph{complete} if $V=H^0(\aaa)$, in this case one simply writes ${\rm L}=|\aaa|$. The \emph{gonality} of $C$ is the
smallest $d$ for which there exists a $g_{d}^{1}$ in $C$, or equivalently, the smallest $d$ for which there exists a torsion free sheaf $\aaa$ of rank $1$ on $C$ with degree $d$ and $h^0(\aaa)\geq 2$. A point $P\in C$ is called a \emph{base point of ${\rm L}$} if $x\op\subsetneq\aaa_P$ for every $x\in V$. A base point is called \emph{removable} if it is not a base point of $\dd(\oo\langle V\rangle,V)$, where $\oo\langle V\rangle$ is the subsheaf of the constant sheaf of rational functions generated by all sections in $V\subset k(C)$. So $P$ is a non-removable base point of ${\rm L}$ if and only if $\aaa_P$ is not a free $\op$-module; in particular, $P$ is singular if so.
\subsection{The Canonical Model}
\label{seccan}
Given any integral scheme $A$, any map $\varphi :A\to C$ and a sheaf $\mathcal{G}$ on $C$, set
$$\mathcal{O}_A\mathcal{G}:= \varphi^*\mathcal G/\text{Torsion}(\alpha^*\mathcal G).$$
Given any coherent sheaf $\mathcal{F}$ on $C$ set $\mathcal{F}^n:=\text{Sym}^n\mathcal{F}/\text{Torsion}(\text{Sym}^n\mathcal{F})$. If $\mathcal{F}$ is invertible then clearly $\mathcal{F}^{n}=\mathcal{F}^{\otimes n}$.
Call $\widehat{C}:=\text{Proj}(\oplus\,\ww ^n)$ the blowup of $C$ along $\ww$. If $\widehat{\pi} :\widehat{C}\rightarrow C$ is the natural morphism, set $\widehat{\oo}=\widehat{\pi}_*(\oo_{\widehat{C}})$ and $\widehat{\oo}\ww:=\widehat{\pi}_*(\oo _{\widehat{C}}\ww)$. In \cite[p\,188\,top]{R} Rosenlicht showed that the linear system $\sys(\oo_{\overline{C}}\ww,H^0(\ww))$ is base point free. He considered then the morphism $\kappa :\overline{C}\rightarrow\pp^{g-1}$ induced by it and called $C':=\kappa(C)$ the \emph{canonical model} of $C$. He also proved in \cite[Thm\,17]{R} that if $C$ is nonhyperelliptic, the map
$\pi :\overline{C}\rightarrow C$ factors through a map $\pi' : C'\rightarrow C$. So set $\oo':=\pi'_*(\oo_{C'})$ in this case. In \cite[Dfn\,4.9]{KM} one finds another characterization of $C'$. It is the image of the morphism $\widehat{\kappa}:\widehat{C}\rightarrow\pp^{g-1}$ defined by the linear system $\sys(\oo_{\widehat{C}}\ww,H^0(\ww))$. By Rosenlicht's Theorem, since $\ww$ is generated by global sections, we have that $\widehat{\kappa}:\widehat{C}\rightarrow C'$ is an isomorphism if $C$ is nonhyperelliptic.
Now set $\overline{\oo}\ww:=\pi_*(\oo_{\overline{C}}\ww)$ and take $\lambda\in H^0(\ww)$ such that $(\overline{\oo}\ww)_P=\overline{\oo}_P\lambda$ for every singular point $P\in C$. Such a differential exists because $H^0(\ww)$ generates $\overline{\oo}\ww$ as proved in \cite[p\,188 top]{R}, and because the singular points of $C$ are of finite number and $k$ is infinite since it is algebraically closed. Set
\begin{equation}
\label{equvvv}
\vv=\vv_{\lambda}:=\ww/\lambda
\end{equation}
If so, we have
$$
\ccc_P\subset
\mathcal{O}_P \subset \vvp\subset\oph=\op'\subset\obp
$$
for every singular point $P\in C$, where the equality makes sense if and only if $C$ is nonhyperelliptic.
\begin{defi} \label{defnng}
\emph{Let $P\in C$ be any point. Set
$$
\eta_P:=\dim(\vvp/\op)\ \ \ \ \ \ \ \ \ \ \ \mu_P:=\dim({\widehat{\oo}_{P}}/\vvp)
$$
and also
$$
\eta:=\sum_{P\in C}\eta_P\ \ \ \ \ \ \ \ \ \ \mu:=\sum_{P\in C}\mu_P
$$
In particular, letting $g'$ be the genus of $C'$, we have
\begin{equation}
\label{equget}
g=g'+\eta+\mu
\end{equation}
Following \cite[Prps. 20, 21, 28]{BF}, we call $P$ \emph{Kunz} if $\eta_P=1$ and,
accordingly, we say that $C$ is \emph{Kunz} if all of its non-Gorenstein points are Kunz; we call $P$ \emph{almost Gorenstein} if $\mu_P=1$ and,
accordingly, we say that $C$ is \emph{almost Gorenstein} if all of its points are so. Following \cite[Dfn. 5.7]{KM}, we call $C$ \emph{nearly Gorenstein} if $\mu=1$, i.e., $C$ is almost Gorenstein with just one non-Gorenstein point. Finally, following \cite[Dfn. 2.15]{KM}, we call $C$ \emph{nearly normal} if $h^0(\oo/\mathcal{C})=1$.}
\end{defi}
\begin{rem}
\label{remrel}
\emph{The relevance of the concepts above are summarized in three properties: (i) $C$ is nearly Gorenstein if and only if it is non-Gorenstein and $C'$ is projectively normal, owing to \cite[Thm. 6.5]{KM}; (ii) $C$ is nearly normal if and only if $C'$ is arithmetically normal, due to \cite[Thm. 5.10]{KM}; (iii) $P$ is Gorenstein if and only if $\eta_P=\mu_P=0$, and $P$ is non-Gorenstein if and only if $\eta_P,\mu_P>0$ by \cite[p. 438 top]{BF}; besides, if $\eta_P=1$ then $\mu_P=1$, by \cite[Prp. 21]{BF}. In particular, a Kunz curve with only one non-Gorenstein point is as close to being Gorenstein as it gets.}
\end{rem}
\subsection{Semigroup of Values}
Now we establish few notations on evaluations. Given a unibranch point $P\in C$ and any function $x\in k(C)^*$, set
$$
v_{P}(x):=v_{\pb}(x)\in\zz
$$
where $\pb$ is the point of $\cb$ over $P$. The \emph{semigroup of values} of $P$ is
$$
\ssp:=v_{P}(\op ).
$$
We also feature two elements of $\sss$, namely:
\begin{equation}
\label{equaab}
\alpha_P :={\rm min}(\sss\setminus\{ 0\})\ \ \ \ \text{and}\ \ \ \beta_P :={\rm min}(v_P(\cp)).
\end{equation}
The set of \emph{gaps} of $\sss_P$ is
$$
{\rm G}_P:=\nn\setminus\sss_P
$$
and one defines the local invariant
$$
\delta_P:=\#({\rm G}_P)
$$
which agrees with the singularity degree of $P$, that is, $\delta_P=\dim(\obp/\op)$. The \emph{Frobenius vector} of $\sss_P$ is
$$
\gamma_P :=\beta_P -1
$$
and one sets
\begin{equation}
\label{equkkp}
\kk_{P}:=\{ a\in\zz\ |\ \gamma_P -a\not\in\sss_P\}
\end{equation}
whose importance will appear later on.
\subsection{Scrolls}
\label{secscr}
A \emph{rational normal scroll} $S:=S_{m_1,\ldots,m_d}\subset\pp^{N}$ with $m_1\leq\ldots\leq m_d$, is a projective variety of dimension $d$ which, after a suitable choice of coordinates, is the set of points $(x_0:\ldots: x_N)\subset\mathbb{P}^N$ such that the rank of
\begin{equation}
\label{equscr}
\bigg(
\begin{array}{cccc}
x_0 & x_1 & \ldots & x_{m_1-1} \\
x_1 & x_2 & \ldots & x_{m_1}
\end{array}
\begin{array}{c}
\big{|} \\
\big{|}
\end{array}
\begin{array}{ccc}
x_{m_1+1} & \ldots & x_{m_1+m_2} \\
x_{m_1+2} & \ldots & x_{m_1+m_2+1}
\end{array}
\begin{array}{c}
\big{|} \\
\big{|}
\end{array}
\begin{array}{c}
\ldots \\
\ldots
\end{array}
\begin{array}{c}
\big{|} \\
\big{|}
\end{array}
\begin{array}{cc}
\ldots & x_{N-1} \\
\ldots & x_N
\end{array}
\bigg)
\end{equation}
is smaller than 2. So, in particular,
\begin{equation}
\label{equnnn}
N=e+d-1
\end{equation}
where $e:=m_1+\ldots+m_d$
Note that $S$ is the disjoint union of $(d-1)$-planes determined by a (parametrized) choice of a point in each of the $d$ rational normal curves of degree $m_d$ lying on complementary spaces on $\mathbb{P}^N$. We will refer to any of these $(d-1)$-planes as a \emph{fiber}. So $S$ is smooth if $m_i>0$ for all $i\in\{1,\ldots,d\}$.
From this geometric description one may see that
\begin{equation}
\label{equdgs}
\deg(S)=e
\end{equation}
The scroll $S$ can also naturally be seen as the image of a projective bundle. In fact, taking $\mathcal{E}:=\oo_{\pum}(m_1)\oplus\ldots\oplus\oo_{\pum}(m_d)$, one has a birational morphism
\begin{equation*}
\mathbb{P}(\mathcal{E})\longrightarrow S\subset\mathbb{P}^{N}
\end{equation*}
defined by $\oo_{\mathbb{P}(\mathcal{E})}(1)$. The morphism is such that any fiber of $\mathbb{P}(\mathcal{E})\to\pum$ is sent to a fiber of $S$. It is an isomorphism if $S$ is smooth. One can check, for instance, \cite{EH,Rd} for more details.
In this case, one may describe the Picard group of the scroll as
$$
\text{Pic}(S)=\mathbb{Z}H\oplus\mathbb{Z}F
$$
where $F$ is the class of a fiber, and $H$ is the hyperplane class. One may also compute its Chow ring as
\begin{equation}
\label{equchw}
A(S)=\frac{\mathbb{Z}[H,F]}{(F^2\, ,\, H^{d+1}\, ,\, H^{d}F\, ,\, H^{d}-eH^{d-1}F)}
\end{equation}
From (\ref{equdgs}) we get the relations
\begin{equation}
\label{equrel}
H^d=e\ \ \ \ \ \text{and}\ \ \ \ \ H^{d-1}F=1
\end{equation}
The canonical class in $S$ is given by
\begin{equation}
\label{equccs}
K_S=-dH+(e-2)F
\end{equation}
By \cite[Lem. 3.1, Cor. 3.2]{Mr}, we also have the formulae
\begin{equation}
\label{equhhz}
h^0(\oo_S(aH+bF))=
\begin{cases}
\displaystyle (b+1)\binom{a+d-1}{d-1}+e\binom{a+d-1}{d}
&
{\rm if}\ a\geq 0\ \text{and}\ b\geq-am_1
\\
0 & \text{otherwise}
\end{cases}
\end{equation}
and
\begin{equation}
\label{equhh1}
h^i(\oo_S(aH+bF))=0\ \ \ \ \ \text{if}\ i\geq 1,\ a\geq 0\ \text{and}\ b\geq -(am_1+1)
\end{equation}
which are important in the analisys of effective divisors on $S$.
\section{Canonical Models on Scrolls via Gonality}
In this section we analyze the relation between pencils on a curve and scrolls it might lie on. We are particularly interested when this curve happens to be a canonical model. So let $\dd(\aaa,V)$ be a pencil on a curve $C\subset\pp^n$; and assume $\aaa=\oo_C(D)$ where $D$ is an effective Weil divisor on $C$ supported outside ${\rm Sing}(C)$. Let also $H$ be a hyperplane divisor on $C$ and suppose the curve is linearly normal, that is, the hyperplane sections cut out a complete linear series. In this framework, one may adjust, for instance, Schreyer's survey in \cite[pp. 113-115]{Sc} to the singular case, in order to induce, by means of $\dd$, an inclusion $C\subset S\subset\pp^{n}$ where $S$ is a rational normal scroll.
In fact, consider then the multiplication map
\begin{equation}
\label{eqummp}
V\otimes H^0(\oo_C(H-D))\longrightarrow H^0(\oo_C(H))
\end{equation}
and assume $f:=h^0(\oo_C(H-D))\geq 2$. Then (\ref{eqummp}) yields a matrix in $M_{2\times f}(H^0(\oo_C(H)))$ whose $2\times 2$ minors vanish on $C$. Thus $C$ is contained in the rational normal scroll $S$ defined by these minors, which is such that
\begin{equation}
\label{equdsc}
\dim(S)=h^0(\oo_C(H))-h^0(\oo_C(H-D))
\end{equation}
We may apply this construction to the following example.
\begin{exem}
\label{exedif}
\emph{We already know from \cite{RS} that any trigonal canonical (Gorenstein) curve lies on a $2$-fold scroll. So consider, for example, the curve
$$
C=(1:t^3:t^6:t^7:t^9:t^{10})\subset\pp^5
$$
It is a rational monomial curve with just one singular point $P=(1:0:0:0:0:0)$. So its genus agrees with the singularity degree of $P$. Now $\sss_P=\langle 3,7\rangle$, hence $C$ has genus $g=\delta_P=6$. Moreover, $\deg(C)=10$ so $C$ is canonical. One can check that the curve is also trigonal, with the gonality computed by the linear series $|\oo_C(3Q)|$, where $Q=(0:0:0:0:0:1)$. So we may use the theory sketched above to place $C$ within a $2$-fold scroll. We have that $V=\langle 1,t^3\rangle$, $H=10\,Q$ and
$$
H^0(\oo_C(H-3Q))=H^0(\oo_C(7Q))=\langle 1,t^3,t^6,t^7\rangle\subset k(t)=k(C)
$$
So, according to (\ref{equdsc}), $C$ lies on a $2$-fold scroll $S$ since
$$
h^0(\oo_C(H))-h^0(\oo_C(7Q))=6-4=2
$$
In order to see this scroll in a way that it is defined by a matrix just like (\ref{equscr}), one may reorder the coordinates as
$$
C=(t^7:t^{10}:1:t^3:t^6:t^9)\subset\pp^5=\{(x_0:x_1:x_2:x_3:x_4:x_5)\}
$$
If so, $S$ is the surface of $\mathbb{P}^5$ cut out by the $2\times 2$ minors of
$$
\bigg(
\begin{array}{cccc}
x_0 & x_2 & x_3 & x_4 \\
x_1 & x_3 & x_4 & x_5
\end{array}
\bigg)
$$
Note that the scroll is of the form $S_{13}$. Note also that the sections of the first row (when restricted to $C$) generate $H^0(\oo_C(7Q))= \langle 1\rangle \otimes H^0(\oo_C(7Q))$, while the sections of the second row generate $\langle t^3\rangle \otimes H^0(\oo_C(7Q))$ exactly as in the map (\ref{eqummp}).}
\emph{A small disturb on the above example is enough to realize how things get worse when one goes through the non-Gorenstein case. For instance, let now $C$ be a rational monomial curve with just one singularity whose semigroup is the same above up to the removal of $7$. Namely, a linearly normal model for such a curve with smaller possible dimension of the ambient space is
$$
C=(1:t^3:t^6:t^9:t^{10}:t^{12}:t^{13}:t^{14})\subset\pp^7
$$
The singular point is $P=(1:0:\ldots :0)$. Since $\sss_P=\langle 3,10,14\rangle$, the curve has genus $g=\delta_P=7$. And since $\beta_P=12\neq 2\delta_P$, it follows that $P$ is non-Gorenstein, and so is $C$. Moreover, the curve is also trigonal, with gonality computed by $|\oo_C(3Q)|$, where $Q=(0:\ldots:0:1)$, similar to the prior example. However, this pencil does not induce an inclusion of $C$ on a $2$-fold scroll, as would be expected. In fact, we have that $H=14Q$ and
$$
H^0(\oo_C(H-3Q))=H^0(\oo_C(11Q))=\langle 1,t^3,t^6,t^9,t^{10}\rangle
$$
So, according to (\ref{equdsc}), $C$ lies on a scroll of dimension
$$
h^0(\oo_C(H))-h^0(\oo_C(11Q))=8-5=3
$$
In order to achieve the expected dimension, one may deal with the canonical model $C'$ instead. Although $C'$ is not isomorphic to $C$ (since the latter is non-Gorenstein), it does preserve this desired property related to the gonality of $C$. Indeed, by Theorem \ref{prprat} ahead, we have that
$$
C'=(1:t^3:t^4:t^6:t^7:t^9:t^{10})\subset\pp^6
$$
which is clearly contained on a $2$-fold scroll of the form $S_{23}$. This inclusion of the canonical model $C'\subset S_{23}$ can be obtained, for instance, dealing with the pullback $|(\pi')^*(\oo_{C}(3Q))|$ and following the same steps above.}
\end{exem}
A general result relating pencils and scrolls for an arbitrary rational singular curve (not necessarily monomial) with prescribed degree and ambient space dimension can be found, for instance, in \cite{CFM2} (motivated by \cite{CFM1}). However, if the concern is gonality (and hence canonical models), then the way of inducing an inclusion of $C'$ on a scroll by means of a $g_d^1$ on $C$ should be slightly modified.
A more intrinsic approach is required. This was done by St\"ohr and Rosa in \cite{RS} based on Andreotti and Mayer's \cite{AM} in the case $d=3$. The following result uses similar arguments to extend \cite[Thm. 2.1, Lem 2.3]{RS} to higher degree.
\begin{teo}
\label{thmscr}
Let $C$ be an integral and projective curve of arithmetic genus $g$ over an algebraically closed field $k$, and $C'$ be its canonical model. Let $\dd(\aaa,V)$ be a $g_d^1$ on $C$. Then $\dd$ induces an inclusion $C'\subset S\subset\pp^{g-1}$ where $S$ is an $m$-dimensional rational normal scroll, such that:
\begin{itemize}
\item[(I)] $m\leq d-1$, with equality if and only if $\dd$ is complete;
\item[(II)] If $S$ is singular then
$$
\dim({\rm Sing}(S))< d-2h^0(\aaa)+1+\frac{\deg(\aaa\cap x^{-1}\aaa)}{2}
$$
with $x\in V\setminus k$; in particular,
\begin{itemize}
\item[(i)] if the expression above is not extrictly positive, then $S$ is smooth;
\item[(ii)] if $\dd$ is complete and base point free, then $\dim({\rm Sing}(S))\leq m-3$;
\end{itemize}
\item[(III)] If $C$ is Gorenstein and $\dd$ is complete with a base point then $S$ is singular.
\end{itemize}
\end{teo}
\begin{proof}
Without loss in generality one may write $V=\langle 1,x\rangle\subset H^0(\aaa)\subset k(C)$. Consider then the map
\begin{gather*}
\begin{matrix}
\varphi: & H^1(\aaa) & \longrightarrow & H^1(\oo_C) \\
& f & \longmapsto & xf
\end{matrix}
\end{gather*}
defined for any $f\in{\rm Hom}(\aaa,\ww)$. Note that $\varphi$ is non-stable, i.e., for any subspace $W\in H^1(\aaa)$, if $\varphi(W)\subset W$, then $W=0$, because $\{x^if\}_{i\in\nn}$ form a linear independent set (viewd in $k(C)$). As $\oo_C$ is a subsheaf of $\aaa$, we have that $H^1(\aaa)\subset H^1(\oo_C)$. So, by \cite[Lem. 5, Cor. 1]{AM}, one may write
$$
H^1(\aaa)=\bigoplus_{i=1}^{r}\bigg(\bigoplus_{j=0}^{m_i-1}kx^jf_i \bigg)
$$
and
\begin{equation}
\label{equwws}
H^0(\ww)\cong H^1(\oo_C)=\bigg(\bigoplus_{i=1}^{r}\bigg(\bigoplus_{j=0}^{m_i}kx^jf_i \bigg)\bigg)\oplus\bigg(\bigoplus_{i=1}^{s}kh_i \bigg)
\end{equation}
where $f_i\in H^1(\aaa)\setminus\varphi( H^1(\aaa))$ for $1\leq i\leq r$ and $h_i\in H^1(\oo_C)\setminus(H^1(\aaa)+\varphi(H^1(\aaa)))$ for $1\leq i\leq s$. So by (\ref{equwws}) the canonical model $C'$ is the image of the morphism
$$
(f_1:\ldots:x^{m_1}f_1:\ldots\ldots:f_r:\ldots:x^{m_r}f_r:h_1:\ldots:h_s):\cb\longrightarrow\pp^{g-1}
$$
and, in particular,
$$
C'\subset S:=S_{m_1,\ldots,m_r,0,\ldots,0}\subset \pp^{g-1}
$$
Besides,
\begin{align*}
\dim(S) &=r+s =\dim(H^1(\oo_C)/\varphi(H^1(\aaa)))\\
&=h^1(\oo_C)-h^1(\aaa)\\
&=g-(h^0(\aaa)-\deg(\aaa)-1+g)\\
&=\deg(\aaa)-(h^0(\aaa)-1)
\end{align*}
where the third equality holds since $\varphi$ is injective. Now $\deg(\aaa)=d$ and
$$
\dim(\dd)=2\leq h^0(\aaa)
$$
with equality holding if and only if $\dd$ is complete. So (I) is proved.
To prove (II), set $\G:=x^{-1}\aaa$, and note that $\varphi(H^1(\aaa))=xH^1(\aaa)=H^1(\G)$. Now, for any subsheaves $\aaa$ and $\G$ of the constant sheaf $\mathcal{K}$ of rational functions of the curve $C$, one may form the short exact sequence
\begin{equation}
\label{equext}
0\longrightarrow \aaa\cap\G\longrightarrow \aaa\oplus\G\longrightarrow \aaa+\G\longrightarrow 0
\end{equation}
and apply the left exact functor $H^0(\mathcal{H}{\rm om}(\bullet,\ww))$ to get
$$
H^0(\mathcal{H}{\rm om}(\aaa,\ww))\cap H^0(\mathcal{H}{\rm om}(\G,\ww))=H^0(\mathcal{H}{\rm om}(\aaa+\G,\ww))
$$
from which we conclude that
\begin{equation}
\label{equhh1}
H^1(\aaa)\cap H^1(\G)=H^1(\aaa+\G)
\end{equation}
From (\ref{equext}) we also have that
$
\chi(\aaa)+\chi(\G)=\chi(\aaa+\G)+\chi(\aaa\cap\G)
$
which yields
\begin{equation}
\label{equdeg}
\deg(\aaa+\G)=\deg(\aaa)+\deg(\G)-\deg(\aaa\cap\G)
\end{equation}
For $\aaa$ as in the statement of the theorem, and $\G$ as priorly set, it is easily seen by the definition of the latter that
\begin{equation}
\label{equhgf}
h^0(\G)=h^0(\aaa)\ \ \ \ \ \ \text{and}\ \ \ \ \ \deg(\G)=\deg(\aaa)=d
\end{equation}
With this in mind, we have
\begin{align*}
\dim({\rm Sing}(S))&=s-1=\dim(H^1(\oo_C)/(H^1(\aaa)+\varphi(H^1(\aaa)))-1\\
&=\dim(H^1(\oo_C)/(H^1(\aaa)+H^1(\G)))-1\\
&=h^1(\oo_C)-(h^1(\aaa)+h^1(\G)-h^1(\aaa+\G))-1\\
&=g-(2(h^0(\aaa)-d-1+g)-h^1(\aaa+\G))-1\\
&=2(d-h^0(\aaa))-g+1+h^1(\aaa+\G)\\
&< 2(d-h^0(\aaa))-g+1+g-\frac{\deg(\aaa+\G)}{2}\\
&= 2(d-h^0(\aaa))+1-\bigg(\frac{2d-\deg(\aaa\cap\G)}{2}\bigg)\\
&=d-2h^0(\aaa)+1+\frac{\deg(\aaa\cap \G)}{2}
\end{align*}
where the fourth equality holds from (\ref{equhh1}), the fifth is due to (\ref{equhgf}) and Riemann-Roch, the unequality follows from \cite[App.]{EHKS}, and the seventh equality owes to (\ref{equdeg}). So item (II).(i) follows. Now, if $\dd$ is base point free, then $\aaa\cap\G=\oo$, and if it is complete, then $h^0(\aaa)=2$, so (II).(ii) follows as well.
To prove (III), the natural isomorphism
$
\mathcal{H}{\rm om}(\aaa\cap\G,\ww)=\mathcal{H}{\rm om}(\aaa,\ww)+\mathcal{H}{\rm om}(\G,\ww)
$
yields the inclusion $H^1(\aaa)+H^1(\G)\subset H^1(\aaa\cap\G)$, thus
\begin{align*}
\dim(H^1(\oo_C)/(H^1(\aaa)+H^1(\G))) &\geq \dim(H^1(\oo_C)/H^1(\aaa\cap\G))\\
&=g-(h^0(\aaa\cap\G)-\deg(\aaa\cap\G)-1+g)\\
&=\deg(\aaa\cap\G)+1-h^0(\aaa\cap\G)
\end{align*}
If $\dd$ is complete, then $h^0(\aaa)=2$; but since $H^0(\aaa\cap\G)\subset H^0(\aaa)$ and $x\not\in H^0(\aaa\cap\G)$ it follows that $h^0(\aaa\cap G)=1$. On the other hand, if $\dd$ is has a base point $P$, which is Gorenstein since $C$ is so, then $\aaa_P\cap\G_P\supsetneq \oo_P$, thus $\deg(\aaa\cap\G)>0$ and hence $s>0$, i.e., $S$ is singular.
\end{proof}
As a consequence of the above result we have the following.
\begin{coro}
\label{corida}
Let $C$ be an integral and projective curve of gonality $d$ and $C'$ be its canonical model. Then $C'$ lies on a $(d-1)$-fold scroll.
\end{coro}
\begin{proof}
By item (I) of the prior theorem, it suffices to prove that if a linear series $\dd$ computes the gonality of $C$, then it is complete. So write $\dd=(\aaa,V)$ where $\aaa$ is torsion free of rank $1$. Choose any regular point $P\in C$ and consider the sequence
$$
0\longrightarrow \aaa(-P)\longrightarrow \aaa \longrightarrow \aaa/\aaa(-P)\longrightarrow 0
$$
Taking Euler characteristic yields
$$
\big(h^0(\aaa)-h^0(\aaa(-P))\big)+\big(h^1(\aaa(-P))-h^1(\aaa)\big)=1
$$
Now both summands are nonnegative; besides, $h^0(\aaa)\geq \dim(V)=2$ and we also have that $h^0(\aaa(-P))\leq 1$ because this sheaf is of degree $d-1$ and the gonality of $C$ is $d$. Thus $h^0(\aaa)=2$ as desired, i.e., $\dd$ is complete.
\end{proof}
\section{Gonality via Canonical Models on Scrolls}
The point of departure of this section is the assumption that the canonical model $C'$ is a complete intersection inside a smooth scroll $S$. From this, we derive some properties about $C$ in terms of invariants of $S$ and $C'$.
To begin with, let $X$ be a curve lying on a $d$-dimensional smooth variety $S$ as
$$
X=D_1\cdot\ldots\cdot D_{d-1}
$$
where the $D_i$'s are divisors on $S$, and set $\mathcal{E}:=\oplus_{i=1}^{d-1}\oo_S(D_i)$.
In order to compute the arithmetic genus of $X$, consider its resolution inside $S$ via Koszul Complex given by
$$
0\rightarrow\bigwedge^{d-1}\mathcal{E}^{\vee}\to\ldots\to
\bigwedge^2\mathcal{E}^{\vee}\to\mathcal{E}^{\vee}\to\oo_S\to\oo_X\to 0
$$
It yields
\begin{equation}
\label{equerr}
p_a(X)=1-\chi(\oo_S)+\chi(\mathcal{E}^{\vee})-\chi(\wedge^2\mathcal{E}^{\vee})+\ldots+ (-1)^{d}\chi(\wedge^{d-1}\mathcal{E}^{\vee})
\end{equation}
Now
\begin{equation}
\label{equext}
\bigwedge^j\mathcal{E}^{\vee}=
\bigoplus_{1\leq i_1<\ldots <i_j \leq d-1}\oo_S(-D_{i_1}-\ldots -D_{i_j})
\end{equation}
and for an arbitrary divisor $D\in{\operatorname{Pic}}(S)$, Hirzebruch-Riemann-Roch Theorem yields
\begin{equation}
\label{equeul}
\chi(\oo_S(D))=t_d+ D\cdot t_{d-1} + \ldots +\frac{1}{(d-2)!}D^{d-2}\cdot t_2+\frac{1}{(d-1)!}D^{d-1}\cdot t_1+\frac{1}{d!}D^{d}
\end{equation}
where $t_i$ is the $i$th degree component of the Todd class of the tangent bundle $\mathcal{T}_S$.
On the other hand, note that whatever are $\alpha_1,\ldots ,\alpha_s\in A$, a commutative ring, we have that
\begin{equation}
\label{equcrg}
\sum_{j=1}^{s}\bigg((-1)^{j-1}\sum_{1\leq i_1<\ldots <i_j \leq s}(\alpha_{i_1}+\ldots+\alpha_{i_j})^r\bigg) =
\begin{cases}
0 & r<s\\
(-1)^{s+1}s!\,\alpha_1...\,\alpha_s& r=s \\
\displaystyle\frac{(-1)^r r!}{2}\sum_{i=1}^{s}\alpha_1...\alpha_i^2...\alpha_s &r=s+1
\end{cases}
\end{equation}
So apply (\ref{equeul}) to (\ref{equext}) using the linearity of the Euler characteristic. Then write the sum envolving the exterior powers of $\mathcal{E}^{\vee}$ in (\ref{equerr}) as a linear polynomial on the variables $t_i$ and apply (\ref{equcrg}) to each of its coefficients. If so, we are reduced to
$$
p_a(X) = 1 - \chi(\oo_S)+\bigg (t_d-D_1 \cdot\ldots\cdot D_{d-1}\,t_{1}+\frac{\sum_{i=1}^{d-1}D_1\cdot\ldots\cdot D_{i}^2\cdot\ldots D_{d-1}}{2}\bigg)
$$
Now recall that $t_d=\chi(\oo_S)$ and $t_1=c_1(\mathcal{T}_S)/2$, which yields the formula
\begin{equation}
\label{equger}
2p_a(X)-2= D_1 \cdot\ldots\cdot D_{d-1}\cdot\big(D_1+\ldots+D_{d-1}-c_1(\mathcal{T}_S)\big)
\end{equation}
Now assume $S$ is a rational normal scroll and recall the settings of Section \ref{secscr}. To any curve $X\subset S$ consider the parameter
\begin{equation}
\label{equlll}
\ell:=X\cdot F
\end{equation}
which will be widely studied here. If $X$ is a complete intersection inside $S$ write
$$
D_i=a_iH+b_iF
$$
as above. Then, from (\ref{equchw}) and (\ref{equrel}), we easily get
\begin{equation}
\label{equcll}
\ell=a_1\cdot\ldots\cdot a_{d-1}
\end{equation}
and one may also use these relations to compute $X\cdot H$ and obtain
\begin{equation}
\label{equcdd}
\deg(X)=a_1\cdot\ldots\cdot a_{d-1}\cdot e+\sum_{i=1}^{d-1}a_1\cdot\ldots\cdot b_{i}\cdot\ldots\cdot a_{d-1}
\end{equation}
For the remainder and for the sake of simplicity, write
$$
a:= a_1+\ldots+a_{d-1}\ \ \ \ \ \ \ \ \ \ \ \ b:= b_1+\ldots+b_{d-1}
$$
If so, we will refer to $X$ as being of \emph{$(a,b)$-type}, and note that
$$
D_1+\ldots+D_{d-1}=aH+bF
$$
while from (\ref{equccs}) we have that
$$
c_1(\mathcal{T}_S)=dH+(2-e)F
$$
So to compute the arithmetic genus of $X$, we get
\begin{align*}
D_1 \cdot\ldots\cdot D_{d-1}\cdot c_1(\mathcal{T}_S) & =(de+2-e)\,a_1\cdot\ldots\cdot a_{d-1}+d\,\sum_{i=1}^{d-1}{a_1\cdot\ldots\cdot b_i \cdot\ldots\cdot a_{d-1}}\\
& = d\deg(X)+(2-e)\ell
\end{align*}
and similarly
$$
D_1 \cdot\ldots\cdot D_{d-1}\cdot (D_1+\ldots+D_{d-1})=a\deg(X)+ b\,\ell
$$
which, combined with (\ref{equger}), yield
\begin{equation}
\label{equgsc}
2p_a(X)-2= \deg(X)(a-d)+\ell(b+e-2)
\end{equation}
that is a helpful tool to be used here applied to canonical models of curves.
In the following result we generalize, to arbitrary dimension, \cite[Thms. 2.1, 4.1]{LMM} which study curves $C$ for which the canonical model $C'$ lies on a $d$-fold scroll for $d=2,3$. The idea is to put the statement within a way that both cases can be deduced from general formulae involving $d$; and the reason why focusing just on the cases $\ell=1,2,d,d+1$ in item (II) will be clear after Theorem \ref{thmth3}.
\begin{teo}
\label{thmth2}
Let $C$ be a nonhyperelliptic curve of genus $g\geq d+3$, whose canonical model $C'$, of genus $g'$, lies on a smooth $d$-dimensional rational normal scroll $S$ as a complete intersection of $(a,b)$-type. Let $\ell$ be the number of points of $C'$ in a generic fiber of the scroll. Then the following hold:
\begin{itemize}
\item[(I)] $\operatorname{gon}(C)\leq \ell+{g-g'}$
\item[(II)] If $b=-(g-(d+2))$ then either $C$ is Gorenstein at most tetragonal or, else, the equality holds if and only if $\ell=2$ and $C'$ is elliptic. Otherwise
$$
\ell= \frac{(a-d-1)(2g-2-\eta)+\eta +2\mu}{b-d-2+g}
$$
in particular, $\ell\leq 3$ if $d=2$.
\item[(III)] The following hold:
\begin{itemize}
\item[(i)] if $\ell=1$ then $C'\cong\mathbb{P}^{1}$ and $C$ is rational with all singular points non-Gorenstein;
\item[(ii)] if $\ell=2$ and $C'$ is not elliptic, then $C'\cong\mathbb{P}^1$ iff $b=-(g-(d+1))$ or else $b\geq -(g-(d+3))$.
\item[(iii)] if $\ell=d$, then $b=-(g-(d+2))-(\eta+2\mu +\tau(2g-2-\eta))/d$; where $\tau\in\{-1,0,1,\ldots,d-3\}$. In particular, if $d=2$ then $C$ is nearly Gorenstein; and if $d=3$, then $C$ is Kunz with just one non-Gorenstein point iff $b=-(g-4)$.
\item[(iv)] if $\ell=d+1$, then $b=-(g-(d+2))-(\eta+2\mu +\tau(2g-2-\eta))/(d+1)$; where $\tau\in\{0,1,\ldots,d-2\}$. In particular, if $d=2$ then $C$ is almost Gorenstein if and only if it is Kunz; and if $d=3$, then $C$ is Kunz with just one non-Gorenstein point iff $b=-(3g-2)/2$ with $g$ even.
\end{itemize}
\item[(IV)] If $C$ is non-Gorenstein and $\ell\geq 3$, writing $S=S_{m_1,\ldots,m_d}$ we have
$$
\frac{g-d-1}{\ell+d-2}+\frac{\nu (g-1)+3-\ell}{\ell(d+\ell-2)}\leq m_1\leq\ldots\leq m_{d}\leq\frac{2g-2-\eta}{\ell}
$$
where $\nu:=(d-1)\sqrt[d-1]{\ell}-d-1$; unless $a=d+1$ where $m_1\geq (g-d-1)/(d+1)$; in particular, both formulae extend the case $d=2$ where $m_1\geq (g-3)/3$.
\end{itemize}
\end{teo}
\begin{proof}
We start by pointing out that item (I) and and the upper bound in (IV) do not actually depend on $C'$ being a complete intersection in $S$. So we begin our proof by these parts. To see (I), first note that, since $S$ is nonsingular, $\ell$ agrees with the intersection number $C'\cdot F$ priorly defined. Moreover, the fibers of $S$ cut out a $g_{\ell}^1$ which we may write as $\dd(\oo_{C'}(D),V)$. We may assume $D$ is effective, and considering the following sequence
$$
0\longrightarrow \oo_C\longrightarrow \pi'_*(\oo_{C'}(D)) \longrightarrow \pi'_*(\oo_{C'}(D))/\oo_C\longrightarrow 0
$$
from which we get, after taking Euler characteristic, that
$$
\deg(\pi'_*(\oo_{C'}(D)))=h^0(\pi'_*(\oo_{C'}(D))/\oo_C)
$$
We may further assume that $D$ is supported outside $(\pi')^{-1}({\rm Sing}(C))$ to see that
$$
h^0(\pi'_*(\oo_{C'}(D))/\oo_C)=\deg(D)+h^0(\pi'_*(\oo_{C'})/\oo_C)=\ell+g-g'
$$
which is the degree of the pencil $\dd(\pi'_*(\oo_{C'}(D)),V)$ on $C$ and, in particular, beats its gonality. To check the upper bound in (IV), follow \cite[pp. 113-115]{Sc} to see that $\oo_{C'}(m_dD)\subset \oo_{C'}(H)$. Now $\deg(\oo_{C'}(m_dD))=m_d\ell$; while, on the other hand, $\oo_{C'}(H)=\oo_{C'}\ww=\oo_{\widehat{C}}\ww$ since $C'\cong\widehat{C}$. But $\deg(\oo_{\widehat{C}}\ww)=2g-2-\eta$ as proved in \cite[Cor. 4.9]{KM} and the bound follows comparing degrees.
To prove the remaining items, we will take $X = C'$ in (\ref{equgsc}). If so, note that $p_a(C')=g-\eta-\mu$ due to (\ref{equget}), and $\deg(C')=\deg(\oo_{C'}(H))=2g-2-\eta$ as we just have seen. Besides, the ambient dimension is $N=g-1$ which implies that $e=g-d$ by (\ref{equnnn}). Thus (\ref{equgsc}) reduces to
\begin{equation}
\label{equpacan}
(a - d -1)(2g - 2 - \eta) + \eta + 2\mu - \ell(d+2-b-g) = 0
\end{equation}
If $b \neq d+2-g$, the above equality provides the formula of (II). Otherwise
\begin{equation}
\label{equgn1}
\eta + 2\mu = (d+1 -a)(2g - 2 - \eta)
\end{equation}
Since the left hand side of the equation is positive, we have that $a\leq d+1$. Now note that $C'$ is given by the intersection of effective divisors, so all $a_i\geq 0$ due to (\ref{equhhz}). Moreover, by its very construction, $C'$ is nondegenerate; therefore $\ell\geq 1$, which implies that all $a_i\geq 1$ and hence $a\geq d-1$. Being nondegenerate in $\mathbb{P}^{g-1}$, it follows that $C'$ has degree at least $g-1$. So if all $a_i=1$, then (\ref{equcdd}) yields
\begin{align*}
\deg(C')&=e+b=(g-d)-(g-d-2)=2
\end{align*}
which is precluded since $g\geq d+3\geq 5$. Therefore at least one $a_i\neq 1$ and hence $a\geq d$. Now $a=d$ if and only if there is exactly one $a_i\neq 1$, and it must be $2$, which holds if and only if $\ell=2$; besides (\ref{equgn1}) establishes the relation
$$
g-\eta-\mu=1
$$
which, by (\ref{equget}), is equivalent to $g'=1$, i.e., $C'$ is elliptic. If $a=d+1$, then either there are just two $a_i\neq 1$ both with value $2$, so $\ell=4$, or there is exactly one $a_i\neq 1$ and with value $3$, so $\ell=3$; besides the vanishing of the left hand side of (\ref{equgn1}) implies that $C$ is Gorenstein. In this case, $C\cong C'$ and, recalling that the fibers cut out a $g_{\ell}^1$ in $C'$, the gonality of $C$ is at most $\ell$, which, as just seen, should be $3$ or $4$ in such a case. So the first statement of (II) is proved up to sufficiency, which will be analyzed right away when we deal with the case $\ell=2$.
To carry out (II), it remains proving that $\ell\leq 3$ if $d=2$. In fact, $d=2$ is a special case where $a=\ell$ and $b=d'-\ell(g-2)$ with, say, $d':=\deg(C')$, owing to (\ref{equcdd}). So (\ref{equgsc}) turns into
\begin{equation}
\label{equll2}
2g'=p(\ell)=-(g-2)\ell^2+(2d'+g-4)\ell-2(d'-1)
\end{equation}
Since $y=p(x)$ is concave downwards with roots $1$ and $2(d'-1)/(g-2)$, and $g'$ is positive, it follows that
$$
\ell\leq \frac{2(d'-1)}{g-2}=\frac{2(2g-\eta-3)}{g-2}=4-\frac{2(\eta-1)}{g-2}
$$
So $\ell\leq 4$ since $g\geq 5$. If $C$ is Gorenstein then $g'=g$, so taking $\ell=4$ in (\ref{equll2}) yields $g=3$ which is precluded. And if $C$ is not Gorenstein and $\ell=4$ then $\eta=1$ and thus $g'=0$; but by Remark \ref{remrel}.(iii), $\mu=1$ as well so $g=2$ which cannot happen either. It follows that $\ell\leq 3$ in any case.
To prove (III), assume $\ell=1$. If so, the fibers of $S$ cut out a $g_1^1$ on $C'$, so $C' \cong \mathbb{P}^1$. Since $C$ is nonhyperelliptic, $C$ and $C'$ are birationally equivalent, so $C$ is rational. Moreover, as $C'$ is nonsingular, then all singular points of $C$ must be non-Gorenstein, because $\oo_{C',P}\cong\oo_{C,P}$ if $P$ is Gorenstein and $C$ nonhyperelliptic.
If $\ell=2$ then, as said above, $a=d$ and (\ref{equpacan}) yields
$$
\eta + \mu + b - d -1 = 0
$$
So one may write $b = d-g+g'+1$. If $g'=0$ then $C' \cong \mathbb{P}^1$ with $b =d+1-g$; if $g'= 1$, that is, $C'$ is elliptic, then $b =d+2-g$ and the equivalence in (II) is now accomplished. And, finally, if $g'\geq 2$, it follows that $b>d+2-g$ as desired.
If $\ell=d$ then (the formula in) (iii) follows from (\ref{equpacan}) setting $\tau:=a-d-1$ and observing that $a\geq d$, as seen above, and $a\leq \ell+d-2=2d-2$. In particular, if $d=2$ then $\tau=-1$, and, as said above, $b=d'-\ell(g-2)=2-\eta$, so replacing this in (iii) yields $\mu=1$, which is equivalent to saying that $C$ is nearly Gorenstein. And if $d=3$ then either $\tau=-1$, which implies that $a=d$ from which we get $\ell=2$ as priorly discussed; but this is precluded since $\ell=d=3$; hence $\tau=0$ which yields $b=-(g-4)+(\eta+2\mu)/3$; the result follows since $C$ is Kunz with just one non-Gorenstein point iff $\eta=\mu=1$.
If $\ell=d+1$ then, similarly, (iv) follows from (\ref{equpacan}) setting agian $\tau:=a-d-1$ and observing that $a\geq d+1$ and $a\leq \ell+d-2=2d-1$. In particular, if $d=2$ then $\tau=0$, and now $b=d'-\ell(g-2)=4-g-\eta$, so replacing this in (iv) yields $\eta=\mu$, which implies that $C$ is almost Gorenstein iff it is Kunz. And if $d=3$ then either $\tau=0$, which implies that $b=-g+5-(\eta+2\mu)/4$ which precludes the case $\eta=\mu=1$ since $b$ is an integer, or, else $\tau=1$ and $b=-(3g-2)/2$ with $g$ even for $C$ to be Kunz with just one non-Gorenstein point.
To finish the proof, recall that $m_1\geq -b/a$ by (\ref{equhhz}). Since we are assuming $\ell\geq 3$, we may take $a\geq d+1$ as discussed above. If equality holds then (\ref{equpacan}) yields
$$
m_1\geq \frac{g - d - 1}{d+1} + \frac{\eta + 2\mu}{\ell}-\frac{1}{d+1}
$$
and the lower bound follows disregarding the sum of the last two terms, which is positive since $C$ is non-Gorenstein. On the other hand, if $a> d+1$ then (\ref{equpacan}) yields
$$
m_1 \geq \frac{g - d - 2}{a}+\frac{(a - d - 1)d' + \eta + 2\mu}{a\ell}
$$
and the bound follows since $d'\geq g-1$ as seen above, $\eta+2\mu\geq 3$ because $C$ is non-Gorenstein, and $(d-1)\sqrt[d-1]{\ell} \leq a \leq \ell+d-2$ by definition.
\end{proof}
As pointed out in \cite[Thm 4.1]{LMM}, if $d=3$ a similar result can be obtained relaxing the hypothesys to $C'$ being just a local complete intersection. This was done via Hartshorne-Serre correspondence for varieties of codimension $2$ lying on a smooth ambient. So we may complete the tableaux in \cite[Sec. 5]{LMM} of all tetragonal rational monomial curves by exhibiting chart equations adjusting the methods of \cite{Br}. The parameter $\ell$ defined above can be computed independently for such a curve. In fact, one may write an affine chart of the scroll $S_{m_1\ldots m_{d}}$ where $C'$ lies as
$$
(1:x:\ldots :x^{m_{d}}:y_1:y_1x:\ldots:y_1x^{m_{d-1}}:\ldots :y_{d-1}:y_{d-1}x:\ldots: y_{d-1}x^{m_1})\cong\mathbb{A}^{d}
$$
This yields morphisms
$$
\varphi_i: C'\longrightarrow \mathbb{P}^1\cong (1:x:\ldots:x^{m_1})
$$
and hence $\ell$ agrees with the generic number of points in a fiber of $\varphi_i$ no matter is $i\in\{1,\ldots,d\}$. If $C$ is rational monomial, then one may write $x=t^r$ and $y_i=t^{s_i}$. So the number of points in the pre-image of $(1:a:\ldots:a^{m_i})$ is precisely the number of solutions of the equation $t^r=a$ because any $t$ determines a unique point of $C'$ since it is parametrized. Thus $\ell=r$ and, by construction, this $r$ agrees with the same common difference of the arithmetic progression of Lemma \ref{Danielle}. The following tableau improves then \cite[pp. 18-19]{LMM}, where, for any rational monomial tetragonal curve $C$ with just one singular point of genus at most $8$, is given the canonical model $C'$ and a scroll $S$ it lies on. Here we also give the equations of $C'$ in $S$, the parameter $\ell$, and the kind of singularities.
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
\multicolumn{5}{|c|}{\textbf{genus 6}}\\
\hline
$C$ and $C'$ & & eqs for $C'$ & $\ell$ & scr \\
\hline
$(1:t^5:t^6:t^8:t^{13}:t^{14})$ & -- & $f=x^3-z$ & $2$ & $S_{111}$ \\
$(1:t^2:t^5:t^6:t^7:t^8)$ & & $h=y^6-3x^2y^4z$ & & \\
& & \ \ \ \ \ \ \ \ \ \ \ \ \ $+3x^4y^2z^2-z^5$ & & \\ \hline
\multicolumn{5}{|c|}{\textbf{genus 7}}\\
\hline
$(1:t^4:t^7:t^{12}:t^{13})$ & -- & $f=x^4-y$ & $1$ & $S_{112}$ \\
$(1:t:t^4:t^5:t^7:t^8:t^9)$ & & $h=x^7-z$ & & \\
\hline
$(1:t^4:t^{10}:t^{11}:t^{12}:t^{13})$ & -- & $f=y^2-x$ & $4$ & $S_{112}$ \\
$(1:t^2:t^3:t^4:t^6:t^7:t^8)$ & & $h=z^4-2xyz^2+x^3$ & & \\ \hline
$(1:t^5:t^8:t^{11}:t^{12}:t^{13}:t^{14})$ & -- & $f=x^3-z$ & $2$ & $S_{112}$ \\
$(1:t^2:t^3:t^5:t^6:t^7:t^8)$ & & $h=y^2-z$ & & \\
\hline
$(1:t^5:t^7:t^8)$ & -- & $f=x^4-z$ & $2$ & $S_{112}$ \\
$(1:t^2:t^5:t^7:t^8:t^9:t^{10})$ & & $h=y^8-4xy^6z+6x^2y^4z^2$ & & \\
& &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $-4x^3y^2z^3+z^5$ & & \\
\hline
$(1:t^6:t^7:t^8:t^{10})$ & -- & $f=x^3-y$ & $2$ & $S_{112}$ \\
$(1:t^2:t^6:t^7:t^8:t^9:t^{10})$ & & $h=y^7-3x^2y^5z^2$ & & \\
& & \ \ \ \ \ \ \ \ \ \ $+3x^4y^2z^4+z^6$ & & \\
\hline
\multicolumn{5}{|c|}{\textbf{genus 8}}\\
\hline
$(1:t^4:t^{10}:t^{13}:t^{14}:t^{15})$ & -- & $f=x^3-z$ & $2$ & $S_{222}$ \\
$(1:t^2:t^4:t^5:t^6:t^7:t^8:t^9:t^{10})$ & & $h=y^6-3x^2y^4z$ & & \\
& &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $+3x^4y^2z^2+z^5$ & & \\
\hline
$(1:t^6:t^9:t^{11}:t^{13}:t^{14}:t^{15}:t^{16})$ & -- & $f=x^3-z$ & $2$ & $S_{113}$ \\
$(1:t^2:t^3:t^5:t^6:t^7:t^8:t^9)$ & & $h=y^2-z$ & & \\
\hline
$(1:t^4:t^9:t^{14}:t^{15})$ & -- & $f=x^4-y$ & $1$ & $S_{122}$ \\
$(1:t:t^4:t^5:t^6:t^8:t^9:t^{10})$ & & $h=y^2-z$ & & \\
\hline
$(1:t^4:t^9:t^{14}:t^{15})$ & -- & $f=x^4-y$ & $4$ & $S_{122}$ \\
$(1:t:t^4:t^5:t^6:t^8:t^9:t^{10})$ & & $h=x^6-z$ & & \\
\hline
$(1:t^5:t^7:t^{13}:t^{15}:t^{16})$ & -- & $f=x^4-z$ & $2$ & $S_{113}$ \\
$(1:t^2:t^3:t^5:t^7:t^8:t^9:t^{10})$ & & $h=y^2-x^3$ & & \\
\hline
$(1:t^5:t^7:t^9:t^{10})$ & -- & $f=x^5-z$ & $2$ & $S_{113}$ \\
$(1:t^2:t^5:t^7:t^9:t^{10}:t^{11}:t^{12})$ & & $h=y^2-z$ & & \\
\hline
$(1:t^4:t^{10}:t^{11}:t^{16}:t^{17})$ & NG & $f=x^3-y^2$ & $4$ & $S_{113}$ \\
$(1:t^4:t^6:t^7:t^8:t^{10}:t^{11}:t^{12})$ & & $h=y^7-3xy^4z$\ \ \ \ \ \ \ \ \ & & \\
& & \ \ \ \ \ \ \ \ \ $+3x^2yz^4+z^6$ & & \\
\hline
$(1:t^4:t^9:t^{11}:t^{15}:t^{16})$ & K & $f=x^4-yz$ & $4$ & $S_{113}$ \\
$(1:t^4:t^7:t^8:t^9:t^{11}:t^{12}:t^{13})$ & & $h=y^9-4x^3y^6z$\ \ \ \ \ \ \ \ \ \ \ & & \\
& &\ \ \ \ \ \ $+6x^6y^3z^2-4x^9z^3+z^7$& & \\
\hline
$(1:t^4:t^{11}:t^{13}:t^{14})$ & -- & $f=x^3-y$ & $1$ & $S_{122}$ \\
$(1:t:t^3:t^4:t^5:t^7:t^8:t^9)$ & & $h=x^7-z$ & & \\
\hline
$(1:t^4:t^{11}:t^{13}:t^{14})$ & -- & $f=x^3-y$ & $4$ & $S_{122}$ \\
$(1:t:t^3:t^4:t^5:t^7:t^8:t^9)$ & & $h=x^4-z$ & & \\
\hline
$(1:t^5:t^8:t^{12}:t^{13}:t^{14})$ & -- & $f=x^4-z$ & $2$ & $S_{122}$ \\
$(1:t^2:t^4:t^5:t^7:t^8:t^9:t^{10})$ & & $h=y^8-4xy^6z+6x^2y^4z^2$& & \\
& &\ \ \ \ \ \ \ \ \ \ \ \ \ $-4x^3y^2z^3+z^5$ & & \\
\hline
\end{tabular}
\end{center}
\
Note that in the tableau of \cite[p. 11]{LMM} one finds no trigonal nearly Gorenstein (or Kunz) curve $C$ for which the canonical model $C'$ lies on a smooth surface scroll with $\ell=1$. And in the above tableau, one also finds any tetragonal nearly Gorenstein $C$ with $C'$ lying on a $3$-fold scroll with $\ell=1,2$. This motivates the following result.
\begin{teo}
\label{thmth3}
Let $C$ be a nearly Gorenstein curve, whose canonical model $C'$ lies on a smooth $d$-dimensional rational normal scroll $S$. If the fibers of $S$ cut out a complete $g_{\ell}^1$ on $C'$, then $d\leq \ell\leq d+1$.
\end{teo}
\begin{proof}
Since $S$ is smooth, the $g_{\ell}^1$ is base point free and can be written as $\dd(\oo_{C'}(D),V)$. Now, according to \cite[Thm. 6.5]{KM}, $C$ is nearly Gorenstein if and only if $C'$ is linearly normal. And if $C'$ is linearly normal and lies on $S$, then, by \cite[Sec. 2]{Sc}, we have that $h^0(\oo_{C'}(H-D))=\deg(S)$. Using \cite[Cor. 4.9]{KM} along with Riemann-Roch in $C'$ for the left hand side of the latter equality, while using (\ref{equnnn}) and (\ref{equdgs}) for the right hand side we get
\begin{equation}
\label{equff1}
2g-2-\eta-\ell+1-g'+h^1(\oo_{C'}(H-D))=g-d
\end{equation}
Recalling that $g'=g-\eta-\mu$ and that $C$ is nearly Gorenstein if and only if $\mu=1$, we have that (\ref{equff1}) turns into
\begin{equation}
\label{equff2}
\ell=d+h^1(\oo_{C'}(H-D))
\end{equation}
Set $\aaa:=\pi'_{*}(\oo_{C'}(D))$. Pushing forward and using Serre duality on $C$ we get
\begin{align*}
H^1(\oo_{C'}(H-D))&=H^1(\pi'_{*}(\oo_{C'}(H-D)))\\
&=H^1(\ooh\ww\otimes\aaa^{\vee})\\
&=H^0(\mathcal{H}{\rm om}(\ooh\ww,\ww)\otimes\aaa)
\end{align*}
Set $\G:=\mathcal{H}{\rm om}(\ooh\ww,\ww)\otimes\aaa$. Clearly, $H^0(\G)\subset H^0(\aaa)=H^0(\oo_{C'}(D))$. Now, since $C$ is nearly Gorenstein, it has a unique non-Gorenstein point, say $P$. We may assume that $D$ is supported outside $(\pi')^{-1}(P)$. So, by construction, $\G_Q=\aaa_Q$ if $Q\neq P$, while $\G_P={\rm Hom}_{\op}(\ooh_P,\op)$. We may further assume that $D$ is effective, so $k\subset H^0(\aaa)$; but note that $k\cap\G_P=0$, so $H^0(\G)\subsetneq H^0(\aaa)$. Therefore
$$
h^1(\oo_{C'}(H-D))=h^0(\G)<h^0(\aaa)=h^0(\oo_{C'}(D))=2
$$
where the last equality holds since the $g_{\ell}^1$ is complete. The result follows by (\ref{equff2}).
\end{proof}
As observed in \cite{LMM}, the efforts we make here to charcterize gonality via scrolls, though general, are still incipient. A detailed study of syzygies in the same way of, e.g., \cite{Sc} or \cite{CS} (and references therein), will likely be required; or even adjusting the techniques of the recent \cite{HU} (and references therein) to canonical models, and also scrolls as the ambient space.
\section{The Gonality of Rational Monomial Curves}
Now we set the objects we'll be dealing with in this section. Consider the morphism
\begin{gather*}
\begin{matrix}
\mathbb{P}^1 & \longrightarrow & \mathbb{P}^n \\
(s:t) & \longmapsto & (s^{a_n}:t^{a_1}s^{a_n-a_1}:\ldots:t^{a_{n-1}}s^{a_n-a_{n-1}}:t^{a_n})
\end{matrix}
\end{gather*}
The image $C$ of such a map is what we call a \emph{rational monomial curve}, which, for simplicity, we denote by
$$
C=(1:t^{a_1}:\ldots:t^{a_{n-1}}:t^{a_n})
$$
with $a_1<\ldots < a_n$. Note that $C$ admits at most two singular points which are
$$
P=(1:0:\ldots:0)
$$
the origin of the affine space $\mathbb{A}^n\subset \mathbb{P}^n$, and
$$
Q=(0:0:\ldots : 1)
$$
the point at the infinity under the parametrization given by $t$. The following result generalizes \cite[Prp. 3.1]{LMM} and by means of rather different arguments.
\begin{teo}
\label{prprat}
Let $C$ be a rational monomial curve. Then its canonical model is
$$
C'=(1:t^{b_2}:\ldots :t^{b_{\delta_P}}:t^{c_1}:\ldots:t^{c_{\delta_Q}})
$$
where $\{0,b_2,\ldots,b_{\delta_P}\}=\gamma_P-G_P$ and $\{c_1,\ldots,c_{\delta_Q}\}=\gamma_P+G_Q$.
\end{teo}
\begin{proof}
Let $g$ be the genus of $C$ and let $\vv$ be a torsion free subsheaf of rank $1$ of the constant sheaf of rational functions $\mathcal{K}$ on $C$. If $h^0(\vv)\geq g$ and $\deg(\vv)=2g-2$ then there exists an embedding of the dualizing sheaf $\ww\hookrightarrow\mathcal{K}$ whose image is $\vv$. In fact, this can be seen, for instance, rephrasing \cite[p. 110 top]{S} in terms of sheaves. So consider the subsheaf of $\mathcal{K}$ defined by
$$
\vv:=\oo_C\langle 1,t^{b_1},\ldots,t^{b_{\delta_P-1}},t^{c_1},\ldots,t^{c_{\delta_Q}}\rangle
$$
that is, it is generated by the (global) sections $1,t^{b_i},t^{c_j}\in k(t)=k(C)$ defined by the statement of the theorem. Since the $b_i$ and $c_j$ all differ to one another, these sections are linear independent, so $h^0(\vv)\geq \delta_P+\delta_Q=g$. We claim that $\deg(\vv)=2g-2$ as well.
In order to prove this, first note that $1\in H^0(\vv)$ thus $\vv\supset\oo$ and hence
$$
\deg(\vv)=\sum_{R\in C}\dim(\vv_R/\oo_R).
$$
Now if $R\neq P,Q$, then, clearly, $\vv_R=\oo_R$ and any such an $R$ gives no contribution to the degree. Let us compute the dimension for the point $Q$. Recall, for instance, from \cite[Prp. 1]{M} that
\begin{equation}
\label{equdwq}
\dim(\vv_Q/\oo_Q)=\#(v_Q(\vv_Q)\setminus\sss_Q)
\end{equation}
We assert that
\begin{equation}
\label{equvwq}
v_Q(\vv_Q)\setminus\sss_Q=-(\gamma_P+G_Q)\,\bigcup\,\{-\gamma_P+1,-\gamma_P+2,\ldots,-1\}\,\bigcup\, G_Q
\end{equation}
Indeed, to prove ``$\supset$" note that $\gamma_P+G_Q=\{c_i\}_{i=1}^{\gamma_P}$, and that $t^{c_i}\in\vv_Q$ by construction; on the other hand, $v_Q(t^{c_i})=-c_i$ so the inclusion of the first set in the above union holds. Now let $r\in\zz$ be such that $r\leq\gamma_P-1$; then, in particular, $c_i-r\geq 2$ for any $i$; since the number of $c_i$'s is the number of gaps of $\sss_Q$ and $1$ is a gap, we have that $c_j-r\in\sss_Q$ for some $j$, so take $f\in\oo_Q$ for which $v_Q(f)=c_j-r$; thus $t^{c_j}f\in\vv_Q$ and $v_Q(t^{c_j}f)=-c_j+c_j-r=-r$ so the inclusions of the second and third sets above hold as well.
To prove ``$\subset$" we use the fact that $C$ is monomial and, in particular, so is the local ring of $Q$. That is, if one writes
\begin{equation}
\label{equun3}
\sss_Q=\{0,s_1,\ldots,s_n,\beta_Q,\rightarrow\}
\end{equation}
then we have that
\begin{equation}
\label{equooq}
\oo_Q=k\oplus kt^{-s_1}\oplus\ldots\oplus kt^{-s_n}\oplus\mathcal{C}_Q
\end{equation}
since $t^{-1}$ is a local parameter for $\overline{Q}$. From this we conclude that any $a\in v_Q(\vv_Q)$ is of the form $a=v_Q(t^dt^{-s})=s-d$ where $d\in\{ 0,b_2,\ldots,b_{\delta_P},c_1,\ldots,c_{\delta_Q}\}$ and $s\in\sss_Q$. So it suffices to show that if $s-d\leq -\gamma_P$ then $d-s-\gamma_P\not\in S_Q$. Now the inequality implies that $d\in\{c_1,\ldots,c_{\delta_Q}\}=\gamma_P+G_Q$. Hence there is an $\ell\in G_P$ for which $d=\gamma_P+\ell$ yielding that $d-s-\gamma_P=\ell-s$ which cannot be in $\sss_Q$ otherwise so would be $\ell$. Thus ``$\subset$" is proved too.
From (\ref{equdwq}) and (\ref{equvwq}) we conclude that
\begin{equation}
\label{equdmq}
\dim(\vv_Q/\oo_Q)=2\delta_Q+\beta_P-2
\end{equation}
Now let us compute the dimension for the point $P$. We claim that
\begin{equation}
\label{equwpl}
\vv_P=k\oplus kt^{b_2}\oplus\ldots\oplus kt^{b_{\delta_P}}\oplus\cp
\end{equation}
Indeed, ``$\supset$" trivially comes from the fact that
\begin{equation}
\label{equwwp}
\vv_P=\oo_P+t^{b_2}\oo_P+\ldots+t^{b_{\delta_P}}\oo_P
\end{equation}
To prove ``$\subset$" note first that
$$
v_P(k\oplus kt^{b_2}\oplus\ldots\oplus kt^{b_{\delta_P}}\oplus\cp)=\kk_P
$$
where the latter was defined in (\ref{equkkp}). Since ``$\supset$" holds, it is enough showing that
\begin{equation}
\label{equvwp}
v_P(\vv_P)=\kk_P
\end{equation}
as well. To see so, recall again that $C$ is monomial thus $\oo_P$ also satisfies (\ref{equooq}) replacing $Q$ by $P$ and $t^{-1}$ by $t$. Combining this with (\ref{equwwp}) it suffices to prove that $\sss_P+\kk_P\subset \kk_P$ to get (\ref{equvwp}); or, in the language of semigroups, that $\kk_P$ is an $\sss_P$ relative ideal; but this is known, for instance, by \cite{J}, so ``$\subset$" holds and the claim follows.
Thus, from (\ref{equwpl}), we have that $\dim(\vv_P/\cp)=\delta_P$. Therefore one may write
\begin{align*}
\dim(\vv_P/\op)&=\dim(\obp/\oo_P)+\dim(\vv_P/\cp)-\dim(\obp/\cp) \\
&=2\delta_P-\beta_P
\end{align*}
Combining this along with (\ref{equdmq}) yields
\begin{align*}
\deg(\vv)&=\dim(\vv_P/\oo_P)+\dim(\vv_Q/\oo_Q)\\
&=2\delta_P-\beta_P+2\delta_Q+\beta_P-2\\
&=2g-2
\end{align*}
So $\vv$ is an isomorphic image of $\ww$ in $\mathcal{K}$ as desired. Call
$$
V:=k\oplus kt^{b_2}\oplus\ldots\oplus kt^{b_{\delta_P}}\oplus kt^{c_1}\oplus\ldots\oplus kt^{c_{\delta_Q}}
$$
Since $\vv\cong\ww$, in particular, $h^0(\vv)=g$ and hence $H^0(\vv)=V$. Now $\cb=\pum$ so the canonical model $C'$ is the image of the morphism $\kappa :\pum\to\mathbb{P}^{g-1}$ defined by the linear system $\mathcal{L}(\oo_{\cb}\vv,H^0(\vv))=\mathcal{L}(\oo_{\pum}\langle V\rangle,V)$, that is, $C'=\kappa(\pum)$ where
\begin{gather*}
\begin{matrix}
\kappa : & \mathbb{P}^1 & \longrightarrow & \mathbb{P}^{g-1} \\
& (1:t) & \longmapsto & (1:t^{b_2}:\ldots: t^{b_{\delta_P}}:t^{c_1}:\ldots :t^{c_{\delta_Q}})
\end{matrix}
\end{gather*}
and we are done.
\end{proof}
Now we recall a result for general scrolls which will be helpful for our purposes. We just warn the reader that the statement -- as any involving monomiality -- depends on a choice of coordinates of the ambient space, though one is allowed at least to reodering them.
\begin{lema}
\label{Danielle}
A rational monomial curve $(1:t^{a_1}:\ldots :t^{a_{N}})\subset \pp^{N}$ lies on a $d$-fold scroll $S_{m_1m_2\ldots m_d}$ if and only if there is a partition of the set $\{0=a_0,a_1,\ldots,a_{N}\}$ into $d$ subsets, with, respectively, $m_1+1, m_2+1,\ldots,m_d+1$ elements, such that the elements of all of these subsets can be reordered within an arithmetic progression with the same common difference.
\end{lema}
\begin{proof}
See \cite[Lem 3.2]{LMM}
\end{proof}
With this in mind, we are ready to prove the following result.
\begin{teo}
\label{thmrat}
Let $C$ be a rational monomial curve of genus at least $1$. Then $\operatorname{gon}(C)=d$ if and only if its canonical model $C^{\prime}$ lies on a $(d-1)$-fold scroll, and does not lie on any scroll of smaller dimension.
\end{teo}
\begin{proof}
We will proceed by induction on $d$. Since $g\geq 1$, then $\operatorname{gon}(C)\geq 2$. So first note that the statement of the theorem for $d=2$ corresponds to the following sentence: $C$ has gonality $2$ if and only if $C'$ is the rational normal curve of degree $g-1$ in $\mathbb{P}^{g-1}$. But this holds from \cite[Thm. 3.4]{KM} and \cite[Thm. 2.1]{Mt}.
For the remainder, write
$$
C^{\prime} = (1:t^{b_2}:\cdots:t^{b_{\delta_P}}:t^{c_1}:\cdots:t^{c_{\delta_Q}})\subset\mathbb{P}^{g-1}
$$
as in Theorem \ref{prprat} and set
$$
A:=\{0,b_2,\ldots, b_{\delta_P},c_1,\ldots, c_{\delta_Q}\}
$$
Now we prove the result for a general $d$ assuming that it holds for any smaller integer. To check sufficiency, suppose $C^{\prime}$ lies on a $(d-1)$-fold scroll. By Lemma \ref{Danielle}, there is a partition of $A$ into $d-1$ subsets, say $A_1$, $\ldots$, $A_{d-1}$, all forming an arithmetic progression with the same common difference, say $r$.
Write
\begin{align*}
A_1&=\{0, r, \ldots, e_1\}\\
A_2&=\{a_2, a_2 +r, \ldots, e_2\}\\
&\ \, \vdots\\
A_{d-1}&=\{a_{d-1}, a_{d-1}+r, \ldots, e_{d-1}\}
\end{align*}
Supposing also that $d-1$ is the smallest dimension of a scroll on which $C'$ can lie, we see from Lemma \ref{Danielle} that
any ending element $e=e_i$ of $A_i$ satisfies $e+r\notin A$.
Now consider the subsheaf of $\mathcal{K}$ on $C$ defined by
$$
{\mathcal F}:={\mathcal O}_C\left\langle 1, t^r\right\rangle
$$
generated by the (global) sections $1,t^r\in k(C)=k(t)$. We claim that
$$
\deg(\aaa)=d
$$
In order to prove so, note that since $C$ is monomial and $t$ (resp. $t^{-1}$) is a local parameter for $P$ (resp. $Q$) we have that
$$
v_R(\aaa_R)=
\begin{cases}
\sss_P\,\bigcup\, (\sss_P+r) & \text{if}\ R=P\\
\nn & \text{if}\ R\neq P,Q \\
\sss_Q\,\bigcup\,(\sss_Q-r) & \text{if}\ R=Q
\end{cases}
$$
Thus, from what was said in the proof of the prior theorem, we conclude that
$$
\deg(\aaa)=\#(\sss_P+r\setminus\sss_P)+\#(\sss_Q-r\setminus\sss_Q)
$$
Now let $e=e_i\in A_i$ for some $i\in\{1,\ldots,d-1\}$. Consider the following cases:
\noindent (i) if $e=\gamma_P-\ell$ with $\ell\in G_P$ and $\ell\geq r$ then $\ell-r\in S_P$ since $e+r\not\in A$, which implies that $\ell\in\sss_P+r\setminus\sss_P$;
\noindent (ii) if $e=\gamma_P-\ell$ with $\ell\in G_P$ and $\ell<r$ then $r-\ell\in S_Q$ since $e+r\not\in A$, which implies that $-\ell\in\sss_Q-r\setminus\sss_Q$;
\noindent (iii) if $e=\gamma_P+\ell$ with $\ell\in G_Q$ then $\ell+r\in S_Q$ since $e+r\not\in A$, thus $\ell\in\sss_Q-r\setminus\sss_Q$;
Moreover, $-r\in \sss_Q-r\setminus\sss_Q$. Therefore, combining this along with the three statements above we get that $\deg(\aaa)\geq d$. Let us now prove that equality holds.
If $s\in \sss_P$ with $s+r\notin S_P$, then $\gamma_P-(s+r)\in A$ and $\gamma_P-s \notin A$; therefore
$\gamma_P-(s+r)=e_i$ for some $i$.
If $s\in S_Q$ with $s-r\notin S_Q$ we have two cases to analyze. First, if $s>r$, then $\gamma_P+(s-r)\in A$ and $\gamma_P+s\not\in A$; therefore $\gamma_P+(s-r)=e_i$ for some $i$.
Otherwise, if $s<r$, we break down this case within two new ones. If $r-s\not\in\sss_P$, then $\gamma_P-(r-s)\in A$ and $\gamma_P+s\notin A$ therefore $\gamma_P-(r-s)=e_i$ for some $i$.
If not, i.e., if $r-s\in\sss_P$, take a new partition of $A$ with subsets all forming an arithmetic progression with same common difference $r-s$. Note that one can always do that no matter if no elements of $A$ will be linked to one another. But we claim that, in our case, the new partition splits $A$ into $d-1$ disjoint sets as well. In fact, if $\gamma_P-\ell\in A$ for $\ell\in G_P$ with $\ell\geq r-s$ then $\gamma_P-l+r-s \in A$ because otherwise $l-r+s\in\sss_P$ which implies that $\ell\in\sss_P$ which cannot occur. So if $\gamma_P-\ell$, with $\ell\in G_P$, ends a subset of $A$ in the new partition then $\ell < r-s$ and $\gamma_P+r-s-\ell\not\in A$; thus $r-s-\ell\in\sss_Q$ and hence $r-\ell\in\sss_Q$. But this implies that $\gamma_P-\ell$ ends a subset of $A$ in the first partition as well. And if $\gamma_P+\ell$, with $\ell\in G_Q$, ends a subset of $A$ in the new partition then $\gamma_P+\ell+r-s\not\in A$, which implies that $\ell+r-s\in\sss_Q$ and so $\ell+r\in\sss_Q$; thus $\gamma_P-\ell$ also ends a subset of $A$ in the first partition. It follows that if the new partion subdivide $A$ in $d'$ subsets, then $d'\leq d-1$. But equality should hold otherwise, by Lemma \ref{Danielle}, $C'$ lies on a $d'$-fold scroll with $d'<d-1$, which contradicts the hypothesys. So one may replace $r$ by $r-s$ and restart the proof over and over again until we make sure that $r-s\not\in\sss_P$ for any $s\in\sss_Q$ with $s-r\not\in\sss_Q$. This will surely happen as $r$ decreases each step.
So $\deg({\mathcal F})=d$ and $\operatorname{gon}(C)\leq d$. But it has to be $d$, because otherwise $C^{\prime}$ would lie on a scroll of dimension smaller than $d-1$ due to our induction hypothesis.
Conversely, necessity follows from Theorem \ref{thmscr}.(I) and induction.
\end{proof}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Twitter users are pouncing on Barr's claim that a sleeping pill led her to write the offensive tweet that prompted her show's cancellation.
Roseanne Barr unleashed a fresh batch of controversial tweets overnight and into Wednesday, blaming a prescription sleeping pill for the racist comment that got her show canceled — and lashing out at her co-stars for throwing her "under the bus."
In a tweet posted just after midnight ET, which has since been deleted, Barr wrote: "guys I did something unforgiveable [sic] so do not defend me. It was 2 in the morning and I was ambien tweeting-it was memorial day too-I went 2 far & do not want it defended-it was egregious indefensible. I made a mistake I wish I hadn't but...don't defend it please. ty [thank you]."
It came after a profuse series of apologies from Barr for her tweet early Tuesday which referred to former adviser to President Barack Obama Valerie Jarrett as a "child" of the "Muslim Brotherhood" and "Planet of the Apes." Jarrett, who is black, was born in Iran to American parents.
That caused ABC to cancel Barr's show in record time, leaving her out of a job before the close of business Tuesday.
Barr also criticized her co-stars overnight, including Michael Fishman, who played her son D.J. on her hit show "Roseanne" and its revival, for speaking out against her.
Fishman tweeted that he felt "devastated" by Barr's words, especially given that the staff of "Roseanne" had been so dedicated to storylines that strived for inclusiveness.
In response, Barr shot back: "i created the platform for that inclusivity and you know it. ME. You throw me under the bus. nice!"
She also furiously called out fellow co-star Sara Gilbert, who played her daughter Darlene, for expressing her disappointment.
Her tweets kept coming Wednesday morning, touching on everything from religion ("I'm a Jew making fun of Hitler, bigot," she defensively responded to a user who shared a photo of her from 2009, when she was slammed for posing as Hitler for a satirical magazine) to backing off from fully blaming the Ambien that she said she had taken. "i blamed myself. not ambien," she wrote.
Ambien, the brand name for zolpidem, and other sleeping pills are some of the most widely prescribed drugs in the United States. Zolpidem is classified by the Food and Drug Administration as a hypnotic and a host of serious side effects are associated with it, including memory problems; sleepwalking; and sleep eating, sleep cooking, and sleep driving.
And on Wednesday afternoon, Barr insisted she was "not a racist," tweeting: "I never was & I never will be."
"One stupid joke in a lifetime of fighting 4 civil rights 4 all minorities, against networks, studios. at the expense of my nerve system/family/wealth will NEVER be taken from me," she tweeted.
Twitter users immediately pounced on Barr's claim that her sleeping pill led her to write the offensive tweet.
"Ask your doctor if blaming Ambien for your racism is right for you," one wrote.
And Sanofi, the maker of Ambien, tweeted: "While all pharmaceutical treatments have side effects, racism is not a known side effect of any Sanofi medication."
"Bob Iger [president] of ABC called Valerie Jarrett to let her know that "ABC does not tolerate comments like those" made by Roseanne Barr. Gee, he never called President Donald J. Trump to apologize for the HORRIBLE statements made and said about me on ABC. Maybe I just didn't get the call?"
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,115
|
20 Jan 2022, Edition - 2382, Thursday
செய்திகள் தமிழில்
Kangana Ranaut had earlier approached the HC seeking that the FIR registered against her to be quashed.
Ghana to fine airlines $3,500 for unvaccinated passengers
Governor Arif Mohammed Khan junks Kerala CM Pinarayi Vijayan's claims on VC postings
PM calls for free & open Indo-Pacific
Home > Businesswire
Merck Foundation Provides 'Health Media Training' in Partnership With the First Lady of Chad
by businesswireindia.com
Merck Foundation, the philanthropic arm of Merck KGaA Germany organized their first "Merck Foundation Health Media Training" recently in N'Djamena, Chad in partnership with H.E. HINDA DEBY ITNO, The First Lady of Chad and Ambassador of Merck More Than a Mother together with Ministry of Health and Ministry of Communication and Ministry of Public Health to break the stigma around infertility, raise awareness about prevention and male infertility and empower infertile women in Chad and rest of Africa.
Dr. Rasha Kelej, CEO of Merck Foundation and President of Merck More Than a Mother emphasized "Today we underscore our long-term commitment to build healthcare capacity in Chad. Merck Foundation has already provided and will continue to provide specialty training to Chad doctors in the fields of Oncology, Fertility and Diabetes Care. Moreover, we conducted the first health media training in Chad to educate media on how to break the stigma of infertility and raise awareness about male infertility and infertility prevention through their day to day work".
The training program is a part of 'Merck More than a Mother' community awareness Program and was organized for the first time in Chad for local media representatives and media students.
H.E. HINDA DEBY ITNO, The First Lady of Chad and Ambassador of Merck More Than a Mother emphasized, "We are happy to conduct this important training in partnership with Merck Foundation in our country. Media can take the messages to the community and bring a change in the current scenario where women solely blamed for infertility. I urge our media partners to work together for this cause to make a difference."
"I am delighted to initiate this important training as I strongly believe that media is a critical partner that plays a significant role to influence our society to create a cultural shift. It has the capacity and ability to empower infertile women and couples in our communities" Rasha Kelej added.
The training was addressed by the stalwarts of media industry, including international faculty and infertility experts.
Moreover, it provided a great opportunity for the journalists to meet the experts and also to network with each other and work as a unit to eradicate the stigma around infertility in Chad and the rest of Africa. It was attended by journalists working for Print, TV, Radio and Online media and journalism students.
"The Merck Health Media Training program focused on the international standards and media ethics for reporting sensitive issues like infertility in Africa. It was designed to benefit the journalists in understanding the infertility issues in African communities and to learn the best media practices to cover such issues" added Dr. Rasha Kelej.
Merck Foundation also announced Call for Application for "Merck More than a Mother" 'Media Recognition Awards' for Chad and the rest of Africa. The "Merck More than a Mother" 'Media Recognition Awards' were launched in 2017 with the aim to emphasize the role of media in enhancing the public engagement and understanding of infertility stigma and the need to change its social perception in African communities.
The applications are invited by media professionals to showcase their work to raise awareness about infertility prevention and breaking infertility stigma in Chad and the rest of Africa.
Who can apply? Journalists from print, online, radio and multimedia platforms from Chad and the rest of Africa.
Last date of submission: Entries can be submitted till 15th June 2020.
How to apply? Entries can be submitted via email to [email protected]
Categories and prize money for winners:
Category TV Radio Print Media Online Media
Prize Money USD 1000 USD 500 USD 500 USD 500
About 'Merck More Than a Mother' campaign
"Merck More Than a Mother" is a strong movement that aims to empower infertile women through access to information, education and change of mind-sets. This powerful campaign supports governments in defining policies to enhance access to regulated, safe and effective fertility care. It defines interventions to break the stigma around infertile women and raises awareness about infertility prevention, management and male infertility. In partnership with African First Ladies, Ministries of Health, Information, Education & Gender, academia, policymakers, International fertility societies, media and art, the initiative also provides training for fertility specialists and embryologists to build and advance fertility care capacity in Africa and developing countries.
With "Merck More than a mother", we have initiated a cultural shift to de-stigmatize infertility on all levels: By improving awareness, training local experts in the fields of fertility care and media, building advocacy in cooperation with African First Ladies and women leaders and by supporting childless women in starting their own small businesses. It's all about giving every woman the respect and the help she deserves to live a fulfilling life, with or without a child.
The Ambassadors of "Merck More Than a Mother" are:
H.E. NEO JANE MASISI, The First Lady of Botswana H.E. FATOUMATTA BAH-BARROW, The First Lady of The Gambia H.E. MONICA GEINGOS, The First Lady of Namibia
H.E DENISE NKURUNZIZA, The First Lady of Burundi H.E. CONDÉ DJENE, The First Lady of Guinea Conakry H.E AÏSSATA ISSOUFOU MAHAMADO, The First Lady of Niger
H.E. BRIGITTE TOUADERA, The First Lady of Central African Republic H.E. CLAR WEAH, The First Lady of Liberia H.E. AISHA BUHARI, The First Lady of Nigeria
H.E. HINDA DEBY ITNO, The First Lady of Chad H.E. PROFESSOR GERTRUDE MUTHARIKA, The First Lady of Malawi H.E FATIMA MAADA, The First Lady of Sierra Leone
H.E. ANTOINETTE SASSOU-NGUESSO, The First Lady of Congo Brazzaville H.E. DR. MAESAIAH THABANE, The First Lady of Lesotho H.E. AUXILLIA MNANGAGW, The First Lady of Zimbabwe
H.E. DENISE NYAKERU TSHISEKEDI, The First Lady of DRC H.E. KEÏTA AMINATA MAIGA, The First Lady of Mali H.E. ESTHER LUNGU, The First Lady of Zambia
H.E. REBECCA AKUFO-ADDO, The First Lady of Ghana H.E. ISAURA FERRÃO NYUSI, The First Lady of Mozambique
Merck Foundation is making history in many African countries where they never had fertility specialists or specialized fertility clinics before 'Merck More Than a Mother' intervention, to train the first fertility specialists such as; in Sierra Leone, Liberia, The Gambia, Niger, Chad, Guinea, Ethiopia and Uganda.
Merck Foundation launched new innovative initiatives to sensitize local communities about infertility prevention, male infertility with the aim to break the stigma of infertility and empowering infertile women as part of Merck more than a Mother COMMUNITY AWARENESS CAMPAIGN, such as;
Merck More than a Mother media recognition award and health media training
Merck More than a Mother fashion award
Merck More than a Mother film award
Local songs with local artists to address the cultural perception of infertility and how to change it
Children storybook, localized for each country
Join the conversation on our social media platforms below and let your voice be heard
Facebook: Merck Foundation
Twitter: @Merckfoundation
YouTube: MerckFoundation
Instagram: Merck Foundation
Flickr: Merck Foundation
Website: www.merck-foundation.com
Source: Businesswire
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,418
|
Thousands of school websites went down after ransomware strikes vendor
Several school districts around the U.S. have been hit with ransomware as a difficult school year begins. (Getty Images / Scoop News Group)
Jan 7, 2022 | STATESCOOP
Thousands of schools in the United States and around the world have had their websites and some other functions knocked offline as a result of a ransomware attack targeting a major web host for K-12 institutions just as students prepared to return from the winter holidays.
The company, Finalsite, said Thursday it "identified the presence of ransomware on certain systems in our environment," which led to its clients, spanning 110 countries, losing access to their websites and other online services.
The outages began on Tuesday, when Finalsite's customers, like the school district in Oakwood, Ohio, began notifying students and parents that their websites were down.
The Oakwood Schools district website is currently down. The FinalSite team is working on the issue. In the meantime, f you need to report a student or staff member who has tested positive for COVID-19 please go to https://t.co/GUfEyOaDyq pic.twitter.com/5GDV9T27pj
— Oakwood Schools (@Oakwood_Schools) January 5, 2022
Many other public and private schools that depend on Finalsite for their web hosting posted similar messages, though the company did not disclose that the cause was ransomware until Thursday evening.
"We immediately took steps to secure our systems and to contain the activity. We quickly launched an investigation into the event with the assistance of third-party forensic specialists, and began proactively taking certain systems offline," reads an update on the Glastonbury, Connecticut, company's website.
Finalsite also said it believes that none of the company's nor its clients' data was stolen by malicious actors.
"We have full access to our files and data," another update read. "The forensic investigation is ongoing and at this time, we have no evidence that our data or client data has been taken. If we determine otherwise through the course of the investigation, we'll act swiftly to notify you and will take all appropriate actions."
The company also said last night that it has been able to restore a majority of the thousands of public-facing websites that were knocked out earlier this week. But some Finalsite clients still have not regained access to all their applications. Holy Ghost Preparatory School, a private high school in Bensalem, Pennsylvania, said Friday that while its website is back, its email system is still down, TechCrunch reported.
Ransomware remains a nagging problem for the K-12 sector, with at least 102 publicly disclosed incidents targeting schools last year, according to Recorded Future. But vendors are also susceptible: According to a March 2021 report from the K-12 Cybersecurity Resource Center and the K12 Security Information Exchange, at least three-fourths of all data breaches at schools were related to vendor compromises.
President Joe Biden last October signed legislation ordering the Department of Homeland Security to study the cyber risks and vulnerabilities against K-12 schools.
Finalsite, ransomware
Georgia says new voter registration system will cut down long lines
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Interstate cybersecurity operations center is on the way
by Colin Wood • 2 days ago
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,173
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Q: Trying to hide/show a div however .slideToggle() is applying to all divs on page - Advanced Custom Fields Repeater I have a hidden div that I want to show on a button press however because the ACF repeater is repeating the id it's opening all the hidden divs at once.
//This is inside a repeater field causing the #buy to repeat
<button id="button">
<img src="images/right-arrow.png" width="35" class="button__icon">
</button>
</div>
<div id="buy" style="display:none" class="tour-event__wrapper--hidden tour-event__wrapper--hidden-bottom">
<div class="widget-containter">
<?php the_sub_field('eventbrite_widget'); ?>
</div>
</div>
// JQuery
$('button').click(function () {
$( "#buy" ).slideToggle("slow");
});
I think I need to find #buy using .find() or .next() however I haven't had much luck using those.
A: This should work
$('#button').click(function (event) {
var button = $(event.currentTarget)
button.parent().find("#buy").slideToggle("slow")
});
But IDs must be unique. I recommend, selecting based on class.
Also the code above is really fragile and rather just a hack. If your HTML structure changes, this will not work anymore.
Ideally you have to be able to uniquely identify each iteration of the repeater and append the index to the id (so it will look like id="buy001"
A: It's possible to get unique IDs on an ACF repeater, which should solve your problem:
<?php if(get_field('repeater_field_name')): $i = 0; ?>
<ul>
<?php while(has_sub_field('repeater_field_name')): $i++; ?>
<li class="section-<?php echo $i; ?>">
<button id="button<?php echo $i; ?>">
<img src="images/right-arrow.png" width="35" class="button__icon">
</button>
<div id="buy<?php echo $i; ?>" style="display:none" class="tour-event__wrapper--hidden tour-event__wrapper--hidden-bottom">
<div class="widget-containter">
<?php the_sub_field('eventbrite_widget'); ?>
</div>
</div>
</li>
<?php endwhile; ?>
</ul>
<?php endif; ?>
Source
Your jQuery snippet would then be
$('[id^=button]').click(function(evt) {
var buyId = evt.target.id.replace("button","buy");
$("#"+buyId).slideToggle("slow");
});
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 483
|
\section{Introduction}
WSNs are currently being considered for many applications; including industrial, security surveillance, medical, environmental and weather monitoring. Due to limited battery lifetime at each sensor node; minimizing transmitter $P_{level}$ to increase energy efficiency and network lifetime is useful. Sensor nodes consist of three parts; sensing unit, processing unit and transceiver~\cite{1}. Limited battery requires low power sensing, processing and communication system. Energy efficiency is of paramount interest and optimal WSN should consume minimum amount of power.
In WSNs, sensor nodes are widely deployed in different environments to collect data. As sensor nodes usually operate on limited battery, so each sensor node communicate using a low power wireless link and link quality varies significantly due to environmental dynamics like temperature, humidity etc. Therefore, while maintaining good link quality between sensor nodes we need to reduce energy consumption for data transmission to extend network lifetime~\cite{2}, ~\cite{3}, ~\cite{4}. IEEE802.15.4 is a standard used for low energy, low data rate applications like WSN. This standard operate at frequency 2.45 GHz with channels up to 16 and data rate 250 kbps.
To efficiently compensate link quality changes due to temperature variations, we propose a new scheme for $P_{level}$ control EAST, that improves network lifetime while achieving required reliability between sensor nodes. This scheme is based on combination of open-loop and closed-loop feedback processes in which we divide network into three regions on basis of threshold on $RSSI_{loss}$ for each region. In open-loop process, each node estimates link quality using its temperature sensor. Estimated link quality degradation is then effectively compensated using closed-loop feedback process by applying propose scheme. In closed-loop feedback process, appropriate transmission $P_{level}$ control is obtained which assign substantially less power than those required in existing transmission power control schemes.
Rest of the paper is organized as follows: section II briefs the related existing work and motivation for this work. In section III, we provide the readers with our proposed scheme. In section IV, we model our proposed scheme. Experimental results have been given in section V.
\section{Related Work and Motivation}
To transmit data efficiently over wireless channels in WSNs, existing schemes set some minimum transmission $P_{level}$ for maintaining reliability. These schemes either decrease interference among sensor nodes or increase unnecessary energy consumption. In order to adjust transmission $P_{level}$, reference node periodically broadcasts a beacon message. When nodes hear a beacon message from a reference node, nodes transmit an ACK message. Through this interaction, reference node estimate connectivity between nodes.
In Local Mean Algorithm (LMA), a reference node broadcasts LifeMsg message. Nodes transmit LifeAckMsg after they receive LifeMsg. Reference nodes count number of LifeAckMsgs and transmission $P_{level}$ to maintain appropriate connectivity. For example, if number of LifeAckMsgs is less than NodeMinThresh; transmission $P_{level}$ is increased. In contrast, if number of LifeAckMsgs is more than NodeMaxThresh transmission; $P_{level}$ is decreased. As a result, they provide improvement of network lifetime in a sufficiently connected network. However, LMA only guarantees connectivity between nodes and cannot estimate link quality~\cite{5}.
Local Information No Topology/Local Information Link-state Topology (LINT/LILT) and Dynamic Transmission Power Control (DTPC) use $RSSI_{loss}$ to estimate transmitter $P_{level}$. Nodes exceeding threshold $RSSI_{loss}$ are regarded as neighbor nodes with reliable links. Transmission $P_{level}$ also controlled by Packet Reception Ratio (PRR) metric. As for the neighbor selection method, three different methods have been used in the literature: connectivity based, PRR based and $RSSI_{loss}$ based. In LINT/LILT, a node maintains a list of neighbors whose $RSSI_{loss}$ values are higher than the threshold $RSSI_{loss}$, and it adjusts the radio transmission $P_{level}$ if number of neighbors is outside the predetermined bound. In LMA/LMN, a node determines its range by counting how many other nodes acknowledged to the beacon message it has sent ~\cite{6}.
Adaptive Transmission Power Control (ATPC) adjusts transmission $P_{level}$ dynamically according to spatial and temporal effects. This scheme tries to adapt link quality that changes over time by using closed-loop feedback. However, in large-scale WSNs, it is difficult to support scalability due to serious overhead required to adjust transmission $P_{level}$ of each link. The result of applying ATPC is that every node knows the proper transmission $P_{level}$ to use for each of its neighbors, and every node maintains good link qualities with its neighbors by dynamically adjusting the transmission $P_{level}$ through on-demand feedback packets. Uniquely, ATPC adopts a feedback-based and pairwise transmission $P_{level}$ control. By collecting the link quality history, ATPC builds a model for each neighbor of the node. This model represents an in-situ correlation between transmission $P_{level}$ and link qualities. With such a model, ATPC tunes the transmission $P_{level}$ according to monitored link quality changes. The changes of transmission $P_{level}$ reflect changes in the surrounding environment ~\cite{7}.
Existing approaches estimate variety of link quality indicators by periodically broadcasting a beacon message. In addition, feedback process is repeated for adaptively controlling transmission $P_{level}$. In adapting link quality for environmental changes, where temperature variation occur, packet overhead for transmission $P_{level}$ control should be minimized. Reducing number of control packets while maintaining reliability is an important technical issue ~\cite{8}.
Radio communication quality between low power sensor devices is affected by spatial and temporal factors. The spatial factors include the surrounding environment, such as terrain and the distance between the transmitter and the receiver. Temporal factors include surrounding environmental changes in general, such as weather conditions (Temperature). To establish an effective transmission $P_{level}$ control mechanism, we need to understand the dynamics between link quality and $RSSI_{loss}$ values. Wireless link quality refers to the radio channel communication performance between a pair of nodes. PRR is the most direct metric for link quality. However, the PRR value can only be obtained statistically over a long period of time. $RSSI_{loss}$ can be used effectively as binary link quality metrics for transmission $P_{level}$ control~\cite{9}.
Radio irregularity results in radio signal strength variation in different directions, but the signal strength at any point within the radio transmission range has a detectable correlation with transmission power in a short time period. There are three main reasons for the fluctuation in the $RSSI_{loss}$. First, fading causes signal strength variation at any specific distance. Second, the background noise impairs the channel quality seriously when the radio signal is not significantly stronger than the noise signal. Third, the radio hardware doesn't provide strictly stable functionality~\cite{10}.
Since the variation is small, this relation can be approximated by a linear curve. The correlation between $RSSI_{loss}$ and transmission $P_{level}$ is approximately linear. Correlation between transmission $P_{level}$ and $RSSI_{loss}$ is largely influenced by environments, and this correlation changes over time. Both the shape and the degree of variation depend on the environment. This correlation also dynamically fluctuates when the surrounding environmental conditions change. The fluctuation is continuous, and the changing speed depends on many factors, among which the degree of environmental variation is one of the main factors~\cite{11}.
Propose energy efficient transmission scheme EAST helps efficiently compensate link quality changes due to temperature variation. To reduce packet overhead for adaptive power control temperature measured by sensors is utilized to adjust transmission $P_{level}$ for all three regions based on $RSSI_{loss}$. Compared to single region in which large control packets overhead occur even due to small change in link quality. Closed-loop feedback process is executed to minimize control packets overhead and required transmitter $P_{level}$.
\section{Proposed Energy Efficient Transmission Scheme}
In this section, we present energy efficient transmission scheme that maintains link quality during temperature variation in wireless environment. It utilizes open-loop process based on sensed temperature information according to temperature variation. Closed-loop feedback process based on control packets is further used to accurately adjust transmission $P_{level}$. By adopting both open-loop and closed-loop feedback processes we divide network into three regions A, B, C for high, medium and low $RSSI_{loss}$ respectively.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.9]{BD.eps}
\caption{Block Diagram}
\end{center}
\end{figure}
In order to assign minimum and reachable transmission $P_{level}$ to each link EAST is designed. EAST has two phases that is initial and run-time. In initial phase reference node build a model for nodes in network. In run-time phase based on previous model EAST adapt the link quality to dynamically maintain each link with respect to time. In a relatively stable network, control overhead occurs only in measuring link quality in initial phase. But in a relatively unstable network because link quality is continuously changing initial phase is repeated and serious overhead occur. Before we present block diagram for proposed scheme some variables are defined as follows (1)Current nodes in a region $n_{c}(t)$ (2) Desired nodes in a region $n_{d}(t)$ (3) Error: e(t) = $n_{d}(t) - n_{c}(t)$,(4) $P_{level}$.
\begin{figure*}[t]
\begin{center}
\includegraphics[height=18cm,width=15cm]{AEETS.eps}
\caption{Flow chart of Reference Node}
\end{center}
\end{figure*}
Fig1 shows system block diagram of proposed scheme. PRR, ACK, and $RSSI_{loss}$ used to determine connectivity. ACK estimates connectivity but it cannot determine link quality. PRR estimates connectivity accurately but it causes significant overhead ~\cite{8}. In our scheme, we use $RSSI_{loss}$ for connectivity estimation, which measures connectivity with relatively low overhead.
Power controller adjusts transmission $P_{level}$ by utilizing both number of current nodes and temperature sensed at each node. Since power controller is operated not merely by comparing number of current nodes with desired nodes but by using temperature-compensated $P_{level}$, so that it can reach to desired $P_{level}$ rapidly. If temperature is changing then temperature compensation is executed on basis of relationship between temperature and $RSSI_{loss}$. Network connectivity maintained with low overhead by reducing feedback process between nodes which is achieved due to logical division of network.
Transmission power loss due to temperature variation formulated using relationship between $RSSI_{loss}$ and temperature experimented in Bannister et al.. Mathematical expression for $RSSI_{loss}$ due to temperature variation is as follows ~\cite{12}:
\begin{equation}
RSSI_{loss}[dBm]=0.1996*(T[C^{o}]-25[C^{o}])
\end{equation}
To compensate $RSSI_{loss}$ estimated from Eq.(1) we have to control output $P_{level}$ of radio transmitter accordingly. Relationship between required transmitter $P_{level}$ and $RSSI_{loss}$ is formulated by Eq.(2) using least square approximation ~\cite{12}:
\begin{equation}
P_{level}=[(RSSI_{loss}+40)/12]^{2.91}
\end{equation}
Based on Eqs (1, 2), we obtain appropriate $P_{level}$ to compensate $RSSI_{loss}$ due to temperature variation. To compensate path loss due to distance between each sensor node in WSN, free space model helps to estimate actual required transmitter power. After addition of $RSSI_{loss}$ due to temperature variation in Eq.(3), we estimate actual required transmitter power between each sensor node. For free space path loss model we need number of nodes in a network (N), distance between each node (d), ($E_{b}/N_{o}$) depends upon ($SNR$), spectral efficiency ($\eta$), frequency ($f$) and receiver noise figure ($(RNF)$):
\begin{equation}
P_{t}[dBm]=[\eta*(E_{b}/N_{0})*mkTB*(4\pi d /\lambda )^2+RNF]+RSSI_{loss}
\end{equation}
Parameters for propose scheme are,(1) Threshold $RSSI_{loss}$ for each region. (2) Desired nodes in each region $n_{d}(t)=n_{c}(t)-5$, (3) Transmission power level $P_{level}$ for each region.
Threshold $RSSI_{loss}$ is minimum value required to maintain link reliability. Reference node broadcasts beacon message periodically to nodes and wait for ACKs. If ACKs are received from nodes then $RSSI_{loss}$ is estimated for logical division of network, number of nodes with high $RSSI_{loss}$ considered in region A, medium $RSSI_{loss}$ considered in region B, and with low $RSSI_{loss}$ in region C. If ($RSSI_{loss}$ $\geq$ $RSSI_{loss}$ Threshold) and ($n_{c}(t)\geq n_{d}(t)$) then Threshold transmitter $P_{level}$ assigned if for similar case ($n_{c}(t)< n_{d}(t)$) then similar transmitter $P_{level}$ assigned and if ($RSSI_{loss}$ $<$ $RSSI_{loss}$ Threshold) then by default keep same transmitter $P_{level}$. Given below is an algorithm for EAST.
\begin{algorithm}
\caption{EAST Algorithm}
\begin{algorithmic}[1]
\STATE $r \gets$ $Number$ $of$ $rounds$
\STATE $N \gets$ $Number$ $of$ $nodes$ $in$ $Network$
\STATE $d \gets$ $Distance$ $between$ $each$ $node$ $and$ $reference$ $node$
\STATE $T \gets$ $Temperature$ $for$ $each$ $node$
\STATE $RSSI_{loss} \gets$ $Transmission$ $power$ $loss$ $for$ $each$ $node$
\STATE $P_{level} \gets$ $Power$ $level$ $for$ $each$ $node$
\STATE $P_{t} \gets$ $Transmitter$ $power$ $for$ $each$ $node$
\STATE $Region$ $A \gets$ $High RSSI_{loss}$
\STATE $Region$ $B\gets$ $Medium RSSI_{loss}$
\STATE $Region$ $C\gets$ $Low RSSI_{loss}$
\STATE $n_{c}(t) \gets$ $Current$ $number$ $of$ $nodes$
\STATE $n_{d}(t) \gets$ $Desired$ $number$ $of$ $nodes$
\IF{$RSSI_{loss}(A,B,C)\geq RSSI_{loss}(Threshold)$}
\IF{$n_{c}(t)(A,B,C)\geq n_{d}(t)(A,B,C)$}
\STATE $RSSI_{loss}(new)(A,B,C)=RSSI_{loss}(Threshold)$
\ELSE
\STATE $RSSI_{loss}(new)(A,B,C)=RSSI_{loss}(A,B,C)$
\ENDIF
\ENDIF
\IF{$RSSI_{loss}(A,B,C)< RSSI_{loss}(Threshold)$}
\STATE $RSSI_{loss}(new)(A,B,C)=RSSI_{loss}(A,B,C)$
\ENDIF
\STATE $P_{levsl}(Save)(A,B,C) = P_{level}- P_{level}(new)(A,B,C)$
\end{algorithmic}
\end{algorithm}
Fig2 shows complete flow chart for reference node. Node senses temperature by using locally installed sensor and checks if temperature change detected. If there is any temperature change, compensation process is executed on the basis of Eqs (1, 2). Nodes send an ACK message including temperature change information with a newly calculated $P_{level}$. Apply ing this temperature-aware compensation scheme we can reduce overhead caused by conventional scheme in changing temperature environments.
\section{Mathematical Representation of the Proposed Scheme}
Let suppose we have 100 nodes in a network that are randomly deployed represented as ($N_{i}$). Nodes are placed at different locations in a square area of 100*100m and distance ($d_{i}$) between them is from 1 to 100m. For given environment temperature ($T_{i}$) can have values in range
-10$C^{0}$ $\leq$ $T_{i}$ $\leq$ 53 $C^{0}$ $\forall$ i $\epsilon$ N.\\
$RSSI_{loss}$ due to the temperature variation can be formulated using the relation between $RSSI_{loss}$ and the temperature experimented in Bannister et al ~\cite{12}. Equation for the $RSSI_{loss}$ for the temperature variation is as follows:\\
\begin{equation}
RSSI_{loss}(i)[dBm]=0.1996*(T_{i}[C^{o}]-25[C^{o}])
\end{equation}
Relation between$P_{level}$ and $RSSI_{loss}$ is formulated by using a least square approximation~\cite{12}:\\
\begin{equation}
P_{level}(i)=[(RSSI_{loss}(i)+40)/12]^{2.91}
\end{equation}
Maximum, minimum and average value of $RSSI_{loss}$ for all nodes in network can be formulated as:\\
\begin{equation}
RSSI_{loss}(min)= min(RSSI_{loss}(i))
\end{equation}
\begin{equation}
RSSI_{loss}(max)= max(RSSI_{loss}(i))
\end{equation}
\begin{equation}
RSSI_{loss}(avg)= (min(RSSI_{loss}(i))+max(RSSI_{loss}(i)))/2
\end{equation}
After finding maximum and minimum values of $RSSI_{loss}$ we will define upper and lower limit of $RSSI_{loss}$ to divide network into three regions and also set counter to count number of nodes in each region. Let suppose we have set counter zero initially and then define upper and lower bound and check condition, nodes that follow this condition are considered to be in region A $\forall$ i $\epsilon$ N.\\
\begin{equation}
RSSI_{loss}(Amax)= max(RSSI_{loss}(i))
\end{equation}
\begin{equation}
RSSI_{loss}(Amin)=RSSI_{loss}(avg)+2
\end{equation}
count=0;\\
$count_{A}$=count+1\\
Given that $\forall$ i $\epsilon$ N;\\
$RSSI_{loss}(i)$ $\leq$ $RSSI_{loss}(Amax)$ and $RSSI_{loss}(i)$ $>$ $RSSI_{loss}(Amin)$\\
Similarly we define upper and lower limits for region B and C and also check nodes that follow given conditions are said to be in region B and C respectively.\\
\begin{equation}
RSSI_{loss}(Bmax)=RSSI_{loss}(avg)+2
\end{equation}
\begin{equation}
RSSI_{loss}(Bmin)=RSSI_{loss}(avg)-2
\end{equation}
count=0;\\
$count_{B}$=count+1\\
Given that $\forall$ i $\epsilon$ N;\\
$RSSI_{loss}(i)$ $\leq$ $RSSI_{loss}(Bmax)$ and $RSSI_{loss}(i)$ $>$ $RSSI_{loss}(Bmin)$\\
\begin{equation}
RSSI_{loss}(Cmin)= min(RSSI_{loss}(i))
\end{equation}
\begin{equation}
RSSI_{loss}(Cmax)=RSSI_{loss}(avg)-2
\end{equation}
count=0;\\
$count_{C}$=count+1\\
Given that $\forall$ i $\epsilon$ N;\\
$RSSI_{loss}(i)$ $\leq$ $RSSI_{loss}(Cmax)$ and $RSSI_{loss}(i)$ $\geq$ $RSSI_{loss}(Cmin)$\\
To apply our proposed scheme $EAST$ we need to define threshold on $RSSI_{loss}$ for each region for energy efficient communication between sensor nodes. Threshold on $RSSI_{loss}$ for each region depends upon $RSSI_{loss}$ of all nodes in a particular region and number of nodes in that region. Threshold on $RSSI_{loss}$ for each region is defined as:\\
\begin{equation}
RSSI_{loss}(Threshold_{A})=\sum_{i=1}^{count_{A}} (RSSI_{loss}(i))/count_{A}
\end{equation}
\begin{equation}
RSSI_{loss}(Threshold_{B})=\sum_{i=1}^{count_{B}} (RSSI_{loss}(i))/count_{B}
\end{equation}
\begin{equation}
RSSI_{loss}(Threshold_{C})=\sum_{i=1}^{count_{C}} (RSSI_{loss}(i))/count_{C}
\end{equation}
$PRR$ is also an important metric to measure link reliability. Here $count_{A}$ are $n_{d}(t)$ and $count_{\bar {A}}$ number of nodes not present in region due to mobility and ($count_{A}$-$count_{\bar {A}}$) are $n_{c}(t)$. It is defined as number of nodes present in a region at particular time $n_{c}(t)$ to number of desired nodes $n_{d}(t)$ in a region. Similarly we can define $PRR$ for regions B and C. $PRR$ for all three regions is defined as given below:\\
\begin{equation}
PRR_{A}=(count_{A}-count_{\bar {A}})/count_{A}
\end{equation}
\begin{equation}
PRR_{B}=(count_{B}-count_{\bar {B}})/count_{B}
\end{equation}
\begin{equation}
PRR_{C}=(count_{C}-count_{\bar {C}})/count_{C}
\end{equation}
Here $PRR_{A}$, $PRR_{B}$ and $PRR_{C}$ are packet reception ratio for regions A, B, C respectively. $RSSI_{loss}$ for each region on basis of propose scheme for given conditions like threshold $RSSI_{loss}$ and $n_{c}(t)$ is formulated as: \\
\begin{equation}
RSSI_{loss}(\tilde {A}, \tilde {B}, \tilde {C})(i)=RSSI_{loss} (Threshold A, B, C)
\end{equation}
Given that $\forall$ i $\epsilon$ N:\\
$RSSI_{loss}(Threshold A, B, C)$ $\leq$ $RSSI_{loss}(A, B, C)(i)$ and $n_{c}(t)(A, B, C)$ $\geq$ $n_{d}(t)(A, B, C)$\\
\begin{equation}
RSSI_{loss}(\tilde {A}, \tilde {B}, \tilde {C})(i)=RSSI_{loss}(A, B, C)(i)
\end{equation}
Given that $\forall$ i $\epsilon$ N:\\
$RSSI_{loss}(Threshold A, B, C)$ $\leq$ $RSSI_{loss}(A, B, C)(i)$ and $n_{c}(t)(A, B, C)$ $\leq$ $n_{d}(t)(A, B, C)$ or $RSSI_{loss}(Threshold A, B, C)$ $>$ $RSSI_{loss}(A, B, C)(i)$\\
Estimation of $P_{level}$ for new $RSSI_{loss}$ is formulated as $\forall$ i $\epsilon$ N:\\
\begin{equation}
P_{level}(\tilde {A}, \tilde {B}, \tilde {C})(i)=[(RSSI_{loss}(\tilde {A}, \tilde {B}, \tilde {C})(i)+40)/12]^{2.91}
\end{equation}
$P_{save}$ is defined as the difference between $P_{level}$ assigned before applying propose scheme and after applying propose scheme:\\
\begin{equation}
P_{save}(A, B, C)=\sum_{i=1}^{N} (P_{level}(A, B, C)(i))- \sum_{i=1}^{N} (P_{level}(\tilde {A}, \tilde {B}, \tilde {C})(i))
\end{equation}
Network life time can be enhanced by maximizing $P_{save}$. Aim of proposed scheme is to save maximum power with link reliability. Objective function formulation for $P_{save}$ is defined $\forall$ i $\epsilon$ N:\\
\begin{equation}
Maximize \sum_{i=1}^{N} (P_{save}(i))
\end{equation}
Constraints to save maximum power are given below $\forall$ i $\epsilon$ N:\\
\begin{equation}
\sum_{i=1}^{N} RSSI_{loss}(A, B, C)(i)\geq RSSI_{loss}(Threshold A, B, C)
\end{equation}
\begin{equation}
\sum_{i=1}^{N} n_{c}(t)(A, B, C)(i)\geq \sum_{i=1}^{N} n_{d}(t)(A, B, C)(i)
\end{equation}
\begin{equation}
\sum_{i=1}^{N} count_{AT}(A, B, C)(i)\geq \sum_{i=1}^{N} count_{BT}(A, B, C)(i)
\end{equation}
Here $count_{AT}$ and $count_{BT}$ are number of nodes above and below threshold in each region respectively.
\section{Results and Discussions}
In this section, we describe simulation results of proposed technique for energy efficient transmission in WSNs. Simulation parameters are; rounds 1200, temperature -10-53 $C^{0}$, distance (1-100)m, nodes 100, regions A,B,C, $\eta$ 0.0029, SNR 0.20dB, bandwidth 83.5MHz, frequency 2.45GHz, RNF 5dB, T 300k, $E_{b}/N_{0}$ 8.3dB. In Fig3 we have shown values of meteorological temperature for one round that each sensor node have sensed. Let suppose we have 100 nodes in 100*100 $m^{2}$ square region and temperature can have values in range (-10 - 53)$C^{o}$ ~\cite{13} for given meteorological condition of Pakistan. Reference node is placed at edge of this region.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.32]{Temperature.eps}
\caption{Temperature sensed at each sensor node}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.32]{RssiLoss.eps}
\caption{Estimated Transmission power loss}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.32]{Plevel.eps}
\caption{Required Power level}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.32]{Pt.eps}
\caption{Transmitter Power}
\end{center}
\end{figure}
Different values of temperature for each sensor node based on meteorological condition helps to estimate $RSSI_{loss}(dBm)$. Fig4 shows $RSSI_{loss}(dBm)$ due to temperature variation in any environment using the relationship between $RSSI_{loss}(dBm)$ and temperature $(C^{o})$ given by Bannister et al. High $RSSI_{loss}(dBm)$ means that sensor node placed in region where temperature is high so link not have good quality. For temperature (-10 - 53)$C^{o}$ $RSSI_{loss}(dBm)$ have value in range (-6dBm) - (5dBm).
\begin{table}[h!]
\centering
\caption{Estimated Parameters}
\tiny
\begin{tabular}{|c|c|}\hline
N (A,B,C) & 46,30,24 \\ \hline
$n_{d}$ (A,B,C) & 41,25,19 \\ \hline
$n_{c}$ (A,B,C) & 41,22,17 \\ \hline
Threshold power level (A,B,C) & 43.24,31.77,22.21 \\ \hline
Nodes above threshold $RSSI_{loss}$ (A,B,C) & 23,11,8 \\ \hline
Nodes below threshold $RSSI_{loss}$ (A,B,C) & 18,11,9 \\ \hline
PRR (A,B,C) & (80-98),(70-96),(63-97) $\%$ \\ \hline
Threshold $RSSI_{loss}$ ( A,B,C) & 3.78,-0.61,-5.17 dBm \\\hline
\end{tabular
\label{tab:addlabel
\end{table
From Fig4 it is also clear that link quality and $RSSI_{loss}$ have inverse relation, when temperature is high $RSSI_{loss}$ has high value means low quality link and vise versa. After estimating $RSSI_{loss}$ for each node in WSN we compute corresponding transmitter $P_{level}$ to compensate $RSSI_{loss}$. Fig5 shows range of $P_{level}$ on y-axis for given $RSSI_{loss}$ that is between (20- 47) and also variation of required $P_{level}$ for sensor node with changing temperature that is at low temperature required $P_{level}$ is low and for high temperature required $P_{level}$ is high.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.32]{plevelabc.eps}
\caption{Power level using Classical Approach for regions A, B, and C}
\end{center}
\end{figure}
As we have earlier estimated $RSSI_{loss}$ for each sensor node on the basis of given meteorological temperature that helps to estimate required $P_{level}$ to compensate $RSSI_{loss}$. That power level only helps to compensate $RSSI_{loss}$ due to temperature variations. To compensate path loss due to distance between each sensor node in WSNs, free space model helps to estimate actual required transmitter power. After addition of required $P_{level}$ due to temperature variation and distance, we estimate actual required $P_{t}$ between each sensor node. Fig6 shows required $P_{t}$ including both $RSSI_{loss}$ due to temperature variation and free space path loss for different nodes. We clearly see from figure that $P_{t}$ lies between (-175 - 90)$dBm$ and most of times it is above -120$dBm$ .
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.32]{plevelabcnew.eps}
\caption{ Power level using $EAST$ for regions A, B, and C}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.32]{psavea.eps}
\caption{Difference of Power level save between Classical Technique and EAST for region A}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.32]{psaveb.eps}
\caption{Difference of Power level save between Classical Technique and EAST for region B}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.32]{psavec.eps}
\caption{Difference of Power level save between Classical Technique and EAST for region C}
\end{center}
\end{figure}
In Fig7, we have shown $P_{level}$ using classical approach for three regions and in Fig8, $P_{level}$ for the proposed technique; EAST. We can clearly see the difference between $P_{level}$ assigned. To show $P_{level}$ for each region, we take the difference between the assigned $P_{level}$s using EAST and classical technique, as can be seen in the figures 9, 10, 11. As we know that in classical approach, there is no concept of sub regions, so, for the sake of comparison with the proposed technique; EAST, we have shown $P_{level }$ for different regions using classical approach.
After estimating $RSSI_{loss}$ for nodes of each region, we have estimated required $P_{level}$ for nodes of each region that we clearly see in Fig7, in region A, $P_{level}$ lies between (40-45), for region B (30-35) and for region C (20-25). It means that for region A required $P_{level}$ high than both other region that also shows that for that region temperature and $RSSI_{loss}$ is large. For region B required $P_{level}$ is between both region A and C and for C region required $P_{level}$ is less than both other two regions. We have earlier seen in Fig7 $P_{level}$ for each region assigned using classical approach. After applying proposed technique we see what $P_{level}$ required for each region. We can clearly see difference between $P_{level}$ as shown in Fig8, that required $P_{level}$ decreases for each region and for region A it decreases maximum. Fig9,10,11 respectively shows required $P_{save}$ for region A,B and C after implanting proposed technique. $P_{save}$ up to 2.3 for region A, 1.7 for B and 1.5 for C.
Fig12 describes the effect of reference node mobility on $P_{save}$ for region A. Reference node move around boundaries of square region and nodes in a region considered to be static. When reference node is at center location (50, 50) of network maximum nodes around reference node have large $RSSI_{loss}$ than threshold so we need to reduce $P_{level}$ to meet threshold $P_{level}$ requirement that cause maximum $P_{save}$. We can clearly see maximum $P_{save}$ 12dBm to 20dBm for center location. When reference node move from center to one of the corner (0, 0) of square region $P_{save}$ remains constant approximately around 1dB, fact is that number of nodes near reference node region having same $RSSI_{loss}$ mean constant temperature and they need approximately same $P_{level}$ near threshold. $P_{save}$ for reference node movement from (0, 0) to (0, 100) fluctuate between -5dBm - 6dBm and at two moments we observe maximum $P_{save}$ because number of nodes near reference node have to increase their $P_{level}$ to meet threshold is minimum.
Movement of reference node from (0, 100) to (100, 100) causes $P_{save}$ between -4dBm - 12dBm and only one time peak $P_{save}$. Similarly when reference node move from (100, 100) to (100, 0) $P_{save}$ remains in limits between -4dBm- 7dBm and only one time maximum $P_{save}$ . From this figure it is also clear that for region A reference node location at center gives maximum $P_{save}$ that enhances network lifetime. We can also see variation of $P_{save}$ with respect to time that basically depends upon nodes near reference node have what $RSSI_{loss}$ if nodes have less $RSSI_{loss}$ then threshold then we have to increase $P_{level}$ that decrease $P_{save}$ and if nodes have large $RSSI_{loss}$ then threshold then we need to decrease $P_{level}$ that enhances $P_{save}$. It is also clear from result that peak maximum and minimum $P_{save}$ comes at same time.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.32]{pa.eps}
\caption{ Transmitter power save in region A for different Reference Node Locations}
\end{center}
\end{figure}
Similarly we can see $P_{save}$ for similar pattern of reference node mobility considering regions B and C. For region B in Fig13 when reference node at center location (50, 50) $P_{save}$ remains between 14dBm-20dBm, from center to (0, 0) $P_{save}$ remains between 0 - 1dBm. When reference node moves from (0, 0) to one of the corner of square region (0, 100) $P_{save}$ fluctuate between 0 - 4dBm. Reference node movement from (0, 100) to (100, 100) cause $P_{save}$ -1dBm-5dBm. Reference node movement from (100, 100) to (100, 0) $P_{save}$ -4dBm-5dBm.
This figure also indicates that $P_{save}$ for region B is maximum when reference node at center location. For reference node mobility from center to (0, 0) $P_{save}$ remains constant due to constant $RSSI_{loss}$ near reference node region. For other reference node movements $P_{save}$ remains approximately constant due to less variations in $RSSI_{loss}$. Compared to region A where $P_{save}$ goes to peak maximum and minimum value in region B $P_{save}$ remains on average approximately constant and less variation occurs, fact is that nodes in region B have approximately same $RSSI_{loss}$ near threshold.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.32]{pb.eps}
\caption{ Transmitter Power save in region B for different Reference Node Locations}
\end{center}
\end{figure}
$P_{save}$ for reference node mobility in region C around square as shown in Fig14. When reference node is at center (50, 50) $P_{save}$ fluctuates between 8dBm-50dBm. From center to edge (0, 0) reference node mobility cause $P_{save}$ around 0dBm. When reference node move from corner of square (0, 0) to corner (0, 100) $P_{save}$ -5dBm-12dBm. Similarly from (0, 100) to (100, 100) $P_{save}$ remains between -10dBm-18dBm. Finally when reference node location changes from (100, 100) to (100, 0) $P_{save}$ goes to maximum value 60dBm that shows that nodes near reference node have large $RSSI_{loss}$ than threshold $RSSI_{loss}$ at that moment. This figure also elaborates that on average $P_{save}$ maximum for reference node location at center. Compared to region B in this region peak maximum and minimum $P_{save}$ exists reason is that nodes in this region have large $RSSI_{loss}$ than threshold at that moment.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.32]{pc.eps}
\caption{ Transmitter Power save in region C for different Reference Node Locations}
\end{center}
\end{figure}
\section{Conclusion and Future Work}
In this paper, we presented a new proposed technique EAST. It shows that temperature is one of most important factors impacting link quality. Relationship between $RSSI_{loss}$ and temperature has been analyzed for our transmission power control scheme. Proposed scheme uses open-loop control to compensate for changes of link quality according to temperature variation. By combining both open-loop temperature-aware compensation and close-loop feedback control, we can significantly reduce overhead of transmission power control in WSN, we further extended our scheme by dividing network into three regions on basis of threshold $RSSI_{loss}$ and assign $P_{level}$ to each node in three regions on the basis of current number of nodes and desired number of nodes, which helps to adapt $P_{t}$ according to link quality variation and increase network lifetime. We have also evaluate the performance of propose scheme for reference node mobility around square region that shows $P_{save}$ up to 60dBm. But in case of static reference node $P_{save}$ goes maximum to 2dBm.
In future, firstly, we are interested to work on Internet Protocol (IP) based solutions in WSNs~\cite{14}. Secondly, as sensors are usually deployed in potentially adverse environments~\cite{15}, so, we will address the security challenges using the intrusion detection systems because they provide a necessary layer for the protection.
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{"url":"http:\/\/www.darwinproject.ac.uk\/letter\/DCP-LETT-3635F.xml","text":"# From John Murray\u00a0\u00a0 [1\u00a0July \u2013 23\u00a0August\u00a01862]1\n\nDarwin on Orchids2\n\nDr.\u00a0Cr.\u00a01862 1862\u00a0June 30\u00a0To Printing\u00a01500\u00a0No Clowes3 89 1 9\u00a0May By\u00a01500\u00a0copies\n\n\u3003\u00a048 $\\frac{1}{4}$ Rms. Sht $\\frac{1}{2}$ Post4 51 17 4 5\u00a0Stationers\u2019 Hall5\n\n\u3003\u00a0Drawing & Engraving6 74 2 12\u00a0allowed Author\n\n\u3003\u00a0Binding\u00a01000\u00a0copies 30 4 2 30\u00a0Presented Reviews7\n\n\u3003\u00a0Commission allowed. Agents\u00a02 3 732 685\u00a0on hand June\u00a030\/62\n\n\u3003\u00a0Advertising\u00a027 4 768\u00a0Sold Viz\n\n\u3003\u00a0Entering Stationers\u2019 Hall8 5 358\u00a0Trade\u00a025\u00a0as 24 6\/ 103 4\n\n410\u00a0Do \u3003\u00a0 \u3003\u00a06\/59 126 8 2\n\n768 229 12 2\n\nJune\u00a030\n\nBy Balance Deficiency\u00a045 5 1\n\n274 17 3 274 17 3 1862 1862\u00a0June\u00a030\u00a0To Balance Deficiency\u00a045 5 1\u00a0June\u00a030\u00a0By\u00a0685\u00a0on hand\n\n## CD annotations\n\nBottom of page: \u2018Feb\u201327\u2014 1866. Still about 600\u00a0copies on Hand & 30\u00a3 deficient.\u2014\u201910 pencil\nVerso: \u20181875\u2019 mauve crayon; \u2018Orchis Book & 4th Edit of Origin.\u2014 \u201911 pencil; \u2018Nov\u00a01869\u2019 blue crayon; \u2018Orchis Book & 4th Edit of Origin\u2019 pencil\n\n## Footnotes\n\nThe date range is established by the dates mentioned in the account and by the relationship between this document and the letter to John Murray, 24\u00a0August [1862] (Correspondence vol.\u00a010).\nFor a discussion of the publication of Orchids, see Correspondence vol.\u00a010, Appendix IV.\nWilliam Clowes & Sons, printers.\nMurray ordered 48$\\frac{1}{4}$ reams of sheet-and-a-half post paper. Sheet-and-a-half post was one and a half times the size of ordinary post; each sheet was about 500\u00a0mm by 600\u00a0mm, accommodating 12\u00a0leaves of an octavo size book (about 200\u00a0mm by 125\u00a0mm uncut) such as Orchids (Gaskell\u00a01972, p.\u00a0224, and Nicholas Smith, CUL, personal information).\nUnder the 1842\u00a0Copyright Act, four copies of any new work published in the United Kingdom and the British Dominions were to be delivered to an officer of the Stationers\u2019 Company for distribution on demand to the Bodleian Library in Oxford, the \u2018Public Library\u2019 (now the University Library) in Cambridge, the Faculty of Advocates at Edinburgh, and the library of Trinity College, Dublin. A copy was also to be delivered to the British Museum in London. Before the 1842\u00a0Act, the Stationers\u2019 Company had been responsible for handling the British Museum\u2019s copy as well as those of a number of other libraries legally entitled to copies. See Seville\u00a01999, pp.\u00a0233, 262.\nThe illustrations for Orchids were made by George Brettingham Sowerby Jr.\nFor CD\u2019s presentation list for Orchids, see Correspondence vol.\u00a010, Appendix IV.\u00a0Sixty-five individuals and three institutions are listed. CD presumably paid for the extra copies himself: see Correspondence vol.\u00a010, letter to John Murray, 24\u00a0August [1862] and n.\u00a02. See also ibid., letters to John Murray, 9\u00a0April [1862] and 2\u00a0May [1862].\nUnder the 1842\u00a0Copyright Act, five shillings was the cost of making an entry in the Book of Registry of the Stationers\u2019 Company, whose headquarters were at Stationers\u2019 Hall, London. Registration did not affect copyright, but was \u2018a necessary preliminary to an action at law for copyright infringement\u2019 (Seville 1999, p.\u00a0235).\nTrade purchasers received one free copy in every twenty-five, as was the general practice (Plant\u00a01965, p.\u00a0405), and were charged 6s. or 6s. 5d. Orchids went on sale to the public priced at 9s. See Freeman\u00a01977, p.\u00a0113.\nThis information was conveyed to CD in the letter from John Murray, 24\u00a0February [1866] (Correspondence vol.\u00a014).\nThe fourth edition of Origin was published in 1866.\n\n## Summary\n\nAccount of Orchids.\n\n## Letter details\n\nLetter no.\nDCP-LETT-3635F\nFrom\nJohn Murray\nTo\nCharles Robert Darwin\nSource of text\nDAR 171: 525","date":"2018-04-21 21:21:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 3, \"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\": 0.43599599599838257, \"perplexity\": 7030.615388254492}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-17\/segments\/1524125945448.38\/warc\/CC-MAIN-20180421203546-20180421223546-00618.warc.gz\"}"}
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Kids Home > What Is the Varicella Vaccine Used For?
The varicella vaccine is approved for preventing chickenpox in both adults and children. Healthcare providers may also occasionally recommend off-label uses for the varicella vaccine, such as for the prevention of chickenpox (or to reduce the severity) in people who have recently been exposed to chickenpox within the past three to five days.
What Is the Varicella Vaccine Used For?
The varicella vaccine (Varivax®) is the chickenpox vaccine. It can be given to children older than 12 months, adolescents, and adults. Individuals who have already had chickenpox do not need to get the varicella vaccine (vaccination of individuals who have had chickenpox is not dangerous but is unnecessary).
Some people may question the benefit of the varicella vaccine, since chickenpox was once a common (and seemingly mild) childhood illness. While it is true that most cases of chickenpox are not dangerous, chickenpox can cause severe skin infection, scars, pneumonia, brain damage, or even death. Before the vaccine, 11,000 people were hospitalized every year and 100 people died every year from chickenpox in the United States alone.
Like all vaccines, the varicella vaccine is not 100 percent effective for preventing chickenpox. While most people who get the vaccine will never get chickenpox, the few cases that do occur are generally mild. As is common with relatively new vaccines, it is unknown exactly how long this protection will last and if a booster will be necessary.
Varicella Vaccine and Shingles
At this time, it is not exactly clear how the varicella vaccine affects the risk of having shingles. Shingles (also known as herpes zoster) is caused by the same virus that causes chickenpox. After a chickenpox infection, the body never completely gets rid of the virus, and the virus remains inactive in certain nerve cells in the body. Later in life, often triggered by stress or illness, the virus can become active again, causing shingles.
It is not yet clear how the varicella vaccine may impact the risk of getting shingles. Although early research indicated that the vaccine decreases the risk of shingles, population surveys have shown inconclusive results.
There is also a concern that the varicella vaccine may indirectly increase the risk of shingles, particularly in people who have never been vaccinated (who actually had chickenpox). Exposure to people with chickenpox (typically through contact with young children) serves as a "booster," providing some protection against shingles.
As actual chickenpox cases are becoming rarer (due to increased vaccination), this natural immune boosting is less likely to occur, and shingles cases may increase.
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Varivax [package insert]. Philadelphia, PA: Wyeth Pharmaceuticals, Inc.; 2008 October.
Centers for Disease Control and Prevention (CDC). Vaccine information statement: chickenpox vaccine (3/18/08). CDC Web site. Available at: http://www.cdc.gov/vaccines/pubs/vis/downloads/vis-varicella.pdf. Accessed July 29, 2009.
Centers for Disease Control and Prevention (CDC). Varicella vaccine Q&A (6/22/07). CDC Web site. Available at http://www.cdc.gov/vaccines/vpd-vac/varicella/vac-faqs-clinic.htm. Accessed July 29, 2009.
Civen R, Chaves SS, Jumaan A, et al. The incidence and clinical characteristics of herpes zoster among children and adolescents after implementation of varicella vaccination. Pediatr Infect Dis J 2009. [Epub ahead of print].
Goldman GS. Universal varicella vaccination: efficacy trends and effect on herpes zoster. Int J Toxicol 2005 ;24(4):205-13.
Side Effects of the Varicella Vaccine
Varicella Vaccine Dosage
Drug Interactions With the Varicella Vaccine
Precautions and Warnings With the Varicella Vaccine
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{"url":"https:\/\/jacbel.github.io\/virtem_code\/behavioral-analysis.html","text":"# 5 Behavioral Analysis\n\nWe are now ready to analyze and plot the behavioral data. Let\u2019s start with a quick recap of the data and tasks we have:\n\nThe day sorting task (Figure 1D) was performed in front of a computer screen. The 20 event images from the day learning task were presented on the screen in a miniature version. They were arranged in a circle around a central area displaying 4 rectangles. Participants were instructed to drag and drop all events of the same sequence into the same rectangle with a computer mouse. Participants freely chose which rectangle corresponded to which sequence as the sequence were not identifiable by any label and were presented in differing orders across mini-blocks during learning.\n\nThus, in analysis, we took the grouping as provided by the rectangles and assigned the four groups of events to the four days in a way that there was maximal overlap between actual days and sorted days. We found the best solution for this by trying all combinations in a preparation step.\n\nIn this task, participants saw a timeline ranging from 6 a.m. to midnight together with miniature versions of the five event images belonging to one sequence (Figure 1E). Participants were instructed to drag and drop the event images next to the timeline so that scene positions reflected the event times they had inferred in the day learning task. To facilitate precise alignment to the timeline, event images were shown with an outward pointing triangle on their left side, which participants were instructed to base their responses on.\n\nParticipants responses were read out from the logfiles of this task and converted to virtual hours. The data are saved in the text file including all behavioral data (virtem_behavioral_data.txt).\n\nLet\u2019s begin by loading the data into a tibble (dataframe) with the following columns:\n\n\u2022 sub_id = subject identifier for all rows of this subject\n\u2022 day = virtual day. There are 4 virtual days\n\u2022 event = number of event in a day, i.e.\u00a0order. Each day has 5 events.\n\u2022 pic = picture identifier. Pictures were randomly assigned to events and days for each subject\n\u2022 virtual_time = true virtual time of an event\n\u2022 real_time = real time of an event\n\u2022 memory_time = data from the timeline task, where participants indicated remembered (virtual) time of each event\n\u2022 memory_order = remembered ordinal position of an event\n\u2022 sorted_day = data from the day sorting task, where participants sorted all event pictures into the 4 days.\nfname <- file.path(dirs$data4analysis, \"behavioral_data.txt\") col_types_list <- cols_only( sub_id = col_factor(), day = col_factor(), event = col_factor(), pic = col_integer(), virtual_time = col_double(), real_time = col_double(), memory_time = col_double(), memory_order = col_double(), sorted_day = col_integer() ) beh_data <- as_tibble(read_csv(fname, col_types = col_types_list)) head(beh_data) Table 5.1: sub_iddayeventpicvirtual_timereal_timememory_timememory_ordersorted_day 3111129.257.819.4511 31122011.8 23.4 11.5 21 3113214.8 42.2 14.5 31 3114816.2 51.6 16.5 41 31151918.2 64.1 19 51 312119.5 9.389.5312 ## 5.1 Sorting Task For analysis of the sorting task, we took the grouping of event images as provided by the participants and assigned them to the four sequences to ensure maximal overlap between actual and sorted sequence memberships. While the assignment of groupings to sequences is unambiguous when performance is, as in our sample, high, this procedure is potentially liberal at lower performance levels. We then calculated the percentage of correctly sorted event images (Figure 2A). ### Calculate percentage of correct responses So to calculate participants\u2019 accuracy in this task we need to figure out how often the true day label matches the label of the quadrant into which an event was sorted. # calculate the number and percentages of correct sorts day_sorting <- beh_data %>% group_by(sub_id) %>% summarise( n_correct = sum(day == sorted_day), prcnt_correct = sum(day == sorted_day)\/(n_days*n_events_day)*100, group = as.factor(1), .groups = \"drop\" ) # print a simple summary of descriptive statistics summary(day_sorting$prcnt_correct)\n## Min. 1st Qu. Median Mean 3rd Qu. Max.\n## 40.00 75.00 95.00 86.43 100.00 100.00\n\nResults of the sorting task: 86.43 \u00b1 16.82 mean\u00b1standard deviation of correct sorts\n\n### Plot sorting performance\n\n# raincloud plot\nf2a <- ggplot(day_sorting,aes(x=1,y=prcnt_correct, fill = group, colour = group)) +\n# plot the violin\ngghalves::geom_half_violin(position=position_nudge(0.1),\nside = \"r\", fill=ultimate_gray, color= NA) +\n# single subject data points with horizontal jitter\ngeom_point(aes(x = 1-.2), alpha = 0.7,\nshape=16, colour=ultimate_gray, size = 1,\nposition = position_jitter(width = .1, height = 0)) +\n# box plot of distribution\ngeom_boxplot(width = .1, colour = \"black\") +\n# add plot of mean and SEM\nstat_summary(fun = mean, geom = \"point\", size=1, shape = 16,\nposition = position_nudge(.1), colour = \"black\") +\nstat_summary(fun.data = mean_se, geom = \"errorbar\",\nposition = position_nudge(.1), colour = \"black\", width = 0, size = 0.5) +\n# edit axis labels and limit\nylab('% correct') + ylim(0, 100) +\n# make it pretty\ntheme_cowplot() + theme(legend.position = \"none\",\naxis.ticks.x = element_blank(), axis.text.x = element_blank()) +\n# change color\nscale_color_manual(values = ultimate_gray) +\nscale_fill_manual(values = ultimate_gray) +\nguides(color=FALSE, fill=FALSE)\nf2a\n\n### Visualize the raw data\n\nFirst, let\u2019s get an overview of participants\u2019 behavior in this task. For this, we plot the responses from the timeline task for all participants. In these plots, each row is one virtual day. The circles along the gray lines represent the true virtual times that participants were supposed to learn. For each event on each day, we then add a full plot of the behavior. This includes the single-subject data points (colored circles), their mean and standard error across subjects (black circle and line) as well as the boxplot and kernel density plot of the distribution.\n\n# reduce data frame to the times specified by the design\ndesign_temp_struct <- beh_data %>% group_by(day, event) %>%\nsummarise(virtual_time = unique(virtual_time), .groups = \"drop\")\n\n# raincloud plot\nf2b <- ggplot(beh_data, aes(x=day,y=memory_time,\nfill = virtual_time, colour = virtual_time,\ngroup=paste(day, event, sep = \"_\"))) +\ngghalves::geom_half_violin(position = \"identity\", side = \"r\", color = NA) +\n# plot the violin\n#geom_flat_violin(position = position_nudge(x = 0.1, y = 0),adjust = 1, trim = TRUE) +\n# single subject data points with horizontal jitter\ngeom_point(aes(x = as.numeric(day)-0.25, y = memory_time, fill = virtual_time), alpha = 0.7,\nposition = position_jitter(width = .05, height = 0),\nshape=16, size = 1) +\n# box plot of distribution\ngeom_boxplot(position = position_nudge(x = -0.1, y = 0), aes(x = day, y = memory_time),\nwidth = .05, colour = \"black\", outlier.shape = NA) +\n# add plot of mean and SEM\nstat_summary(fun = mean, geom = \"point\", size = 1, colour = \"black\", shape = 16) +\nstat_summary(fun.data = mean_se, geom = \"errorbar\",\ncolour = \"black\", width = 0, size = 0.5)+\n# plot point and line for true virtual times\ngeom_line(data = design_temp_struct, aes(x = day, y = virtual_time, group = day),\nposition = position_nudge(-0.45), size = 1, colour = ultimate_gray)+\ngeom_point(data = design_temp_struct, aes(x = day, y = virtual_time, fill = virtual_time),\nposition = position_nudge(-0.45), size = 2, color = ultimate_gray, stroke = 0.5, shape = 21) +\nylab('virtual time') + xlab('sequence') +\nscale_y_continuous(limits = c(6, 24), breaks=seq(6,24,4)) +\nscico::scale_fill_scico(begin = 0.1, end = 0.7, palette = \"devon\") +\nscico::scale_color_scico(begin = 0.1, end = 0.7, palette = \"devon\") +\ncoord_flip() +\nguides(color=guide_colorbar(title.position = \"top\", direction = \"horizontal\",\ntitle = element_blank(),\nbarwidth = unit(20, \"mm\"), barheight = unit(3, \"mm\")),\nfill=FALSE)+\ntheme_cowplot() + theme(text = element_text(size=10), axis.text = element_text(size=8),\nlegend.position = c(1,1), legend.justification = c(1,1))\nf2b\n\n### Accuracy of remembered times\n\nWe analyzed how well participants constructed the event times based on the day learning task. We quantified absolute errors across all events (Figure 2C) as well as separately for the five sequence positions (Figure 2D).\n\n#### Average absolute error per participant\n\nNow, let\u2019s look at the average error per participant collapsed across all trials.\n\n# calculate signed timeline error as difference between virtual time and response\nbeh_data <- beh_data %>%\nmutate(timeline_error = virtual_time - memory_time)\n\n# average absolute timeline error across subjects\ntimeline_group <- beh_data %>%\ngroup_by(sub_id) %>%\nsummarise(\navg_error = mean(abs(timeline_error)),\n.groups = \"drop\")\n\n# print summary of the average error\nsummary(timeline_group$avg_error) ## Min. 1st Qu. Median Mean 3rd Qu. Max. ## 0.2377 0.5625 0.9093 0.9103 1.1221 2.0963 f2c <- ggplot(timeline_group, aes(x=1,y=avg_error), colour= time_colors[1], fill=time_colors[1]) + # plot the violin gghalves::geom_half_violin(position=position_nudge(0.1), side = \"r\", fill=time_colors[1], color = NA) + # single subject data points with horizontal jitter geom_point(aes(x = 1-.2), alpha = 0.7, position = position_jitter(width = .1, height = 0), shape=16, colour=time_colors[1], size = 1)+ # box plot of distribution geom_boxplot(width = .1, fill=time_colors[1], colour = \"black\", outlier.shape = NA) + # add plot of mean and SEM stat_summary(fun = mean, geom = \"point\", size = 1, shape = 16, position = position_nudge(.1), colour = \"black\") + stat_summary(fun.data = mean_se, geom = \"errorbar\", position = position_nudge(.1), colour = \"black\", width = 0, size = 0.5)+ # edit axis labels ylab('absolute error') + xlab('timeline task') + guides(color=FALSE, fill=FALSE)+ theme_cowplot() + theme(legend.position = \"none\", axis.text.x = element_blank(), axis.ticks.x = element_blank()) f2c Timeline task: 0.91 \u00b1 0.47 mean\u00b1standard deviation of average absolute errors. #### Event-wise average absolute errors We can also aggregate the data across days by group the events as a function of event position (i.e. the order). Let\u2019s look at the absolute errors in the timeline task for the events at positions 1-5, averaged across days. # calculate mean absolute error per event for each subject timeline_per_event <- beh_data %>% group_by(sub_id, event) %>% summarise( mean_abs_error =mean(abs(timeline_error)), .groups = \"drop\") # raincloud plot f2d <- ggplot(timeline_per_event,aes(x=event,y=mean_abs_error, fill = event)) + gghalves::geom_half_violin(position=position_nudge(0.1), side = \"r\", colour=NA) + geom_point(aes(x = as.numeric(event)-.2, y = mean_abs_error, colour=event), alpha = 0.7, position = position_jitter(width = .1, height = 0), shape=16, size = 1) + geom_boxplot(position = position_nudge(x = 0, y = 0), aes(x = event, y = mean_abs_error), width = .1, colour = \"black\", outlier.shape = NA, outlier.size = 1) + stat_summary(fun = mean, geom = \"point\", size=1, shape = 16, position = position_nudge(.1), colour = \"BLACK\") + stat_summary(fun.data = mean_se, geom = \"errorbar\", position = position_nudge(.1), colour = \"BLACK\", width = 0, size = 0.5)+ ylab('absolute error')+ xlab('event') + scale_colour_manual(values = event_colors, name=\"event position\") + scale_fill_manual(values = event_colors) + guides(fill=FALSE, color=guide_legend(override.aes=list(fill=NA, alpha = 1, size=2))) + theme_cowplot() + theme(legend.position = \"none\") f2d ### Time metrics underlying remembered times Further, using two approaches we tested whether virtual time drove participants\u2019 responses rather than the sequence order or objectively elapsing time. While participants were asked to reproduce the virtual time, any of the other two factors could have an impact on their behavioral responses. If participants had, for example, only memorized the order of scenes in a given day, they would probably do reasonably well on the timeline task by distributing the scenes evenly along the timeline in the correct order. #### Summary Statistics For the summary statistics approach, we ran a multiple regression analysis for each participant with virtual time, sequence position (order), and real time since the first event of a day as predictors of responses in the timeline task. To test whether virtual time indeed explained participants\u2019 responses even when competing for variance with order and real time, included in the model as control predictors of no interest, we compared the participant-specific t-values of the resulting regression coefficients against null distributions obtained from shuffling the remembered times against the predictors 10,000 times. We converted the resulting p-values to Z-values and tested these against zero using a permutation-based t-test (two-sided; \u03b1=0.05; 10,000 random sign-flips, Figure 2E). As a measure of effect size, we report Cohen\u2019s d with Hedges\u2019 correction and its 95% confidence interval as computed using the effsize-package106. set.seed(115) # set seed for reproducibility # do RSA using linear model and calculate z-score for model fit from permutations fit <- beh_data %>% group_by(sub_id) %>% # run the linear model do(z = lm_perm_jb(in_dat = ., lm_formula = \"memory_time ~ virtual_time + event + real_time\", nsim = n_perm)) %>% mutate(z = list(setNames(z, c(\"z_intercept\", \"z_virtual_time\", \"z_order\", \"z_real_time\")))) %>% unnest_longer(z) %>% filter(z_id != \"z_intercept\") %>% mutate(z_id = factor(z_id, levels = c(\"z_virtual_time\", \"z_order\", \"z_real_time\"))) # run group-level t-test for virtual time stats <- fit %>% filter(z_id ==\"z_virtual_time\") %>% select(z) %>% paired_t_perm_jb (., n_perm = n_perm) # Cohen's d with Hedges' correction for one sample using non-central t-distribution for CI d<-cohen.d(d=(fit %>% filter(z_id ==\"z_virtual_time\") %>% select(z))$z, f=NA, paired=TRUE, hedges.correction=TRUE, noncentral=TRUE)\nstats$d <- d$estimate\nstats$dCI_low <- d$conf.int[[1]]\nstats$dCI_high <- d$conf.int[[2]]\n\n# print results\nhuxtable(stats) %>% theme_article()\n(#tab:multiple regression timeline)\nestimatestatisticp.valuep_permparameterconf.lowconf.highmethodalternativeddCI_lowdCI_high\n2.4410.63.82e-110.0001271.972.91One Sample t-testtwo.sided1.951.382.7\n# raincloud plot of the results\nf2e <- ggplot(fit, aes(x=z_id, y=z, fill = z_id, colour = z_id)) +\n# plot the violin\ngghalves::geom_half_violin(position=position_nudge(0.1), side = \"r\", colour=NA) +\n# single subject data points with horizontal jitter\ngeom_point(aes(x = as.numeric(z_id)-.2, y = z, colour = z_id), alpha = 0.7,\nposition = position_jitter(width = .1, height = 0), shape = 16, size = 1) +\ngeom_boxplot(position = position_nudge(x = 0, y = 0), aes(x = z_id, y = z),\nwidth = .1, colour = \"black\", outlier.shape = NA, outlier.size = 2) +\nstat_summary(fun = mean, geom = \"point\", size=1, shape = 16,\nposition = position_nudge(.1), colour = \"BLACK\") +\nstat_summary(fun.data = mean_se, geom = \"errorbar\",\nposition = position_nudge(.1), colour = \"BLACK\", width = 0, size = 0.5) +\nylab('Z multiple regression')+xlab('time metric') +\nscale_x_discrete(labels = c(\"virt. time\", \"order\", \"real time\")) +\ncoord_cartesian(ylim = c(-2,4.5))+\nscale_color_manual(labels = c(\"virtual time\", \"order\", \"real time\"),values=time_colors) +\nscale_fill_manual(labels = c(\"virtual time\", \"order\", \"real time\"),values=time_colors) +\nguides(fill=FALSE, color=guide_legend(override.aes=list(fill=NA, alpha = 1, size=2),\ntitle = element_blank(), direction =\"vertical\",\ntitle.position = \"bottom\")) +\nannotate(geom = \"text\", x = 1, y = 4.2, label = \"***\", hjust = 0.5, size = 8\/.pt, family=font2use) +\ntheme_cowplot() + theme(text = element_text(size=8), axis.text = element_text(size=8),\nlegend.position = c(1,1), legend.justification = c(1,1),\nlegend.spacing.x = unit(0, 'mm'), #legend.spacing.y = unit(2, 'mm'),\nlegend.key.size = unit(3,\"mm\"),\nlegend.box.margin = margin(0,0,0,0,\"mm\"), legend.background = element_rect(size=0))\nf2e\n\nSummary Statistics: t-test against 0 for virtual time when order and real time are in the model\nt27=10.62, p=0.000, d=1.95, 95% CI [1.38, 2.70]\n\n#### Linear mixed effects\n\nA potentially more elegant way of testing the above is to use linear mixed effects models. However, drawing statistical inferences from these data is less straight forward and care needs to be taken with respect to the precise hypotheses that are tested.\n\nSecond, we addressed this question using linear mixed effects modeling. Here, we included the three z-scored time metrics as fixed effects. Starting from a maximal random effect structure (Barr et al., 2013), we simplified the random effects structure to avoid convergence failures and singular fits. The final model included random intercepts and random slopes for virtual time for participants. The model results are visualized by dot plots showing the fixed effect parameters with their 95% confidence intervals (Figure 2F) and marginal effects (Figure 2G) estimated using the ggeffects package (L\u00fcdecke, 2018). To assess the statistical significance of virtual time above and beyond the effects of order and real time, we compared this full model to a nested model without the fixed effect of virtual time, but including order and real time, using a likelihood ratio test. Supplemental Table 1 provides an overview of the final model and the model comparison.\n\nThe main point we want to make is that virtual time explains variance above and beyond order. To test this, we run the full LMM and a reduced version of the model without virtual time. These two models are then compared using a likelihood ratio test to obtain a p-value.\n\nFollowing Barr et al.\u00a0(2013), we want to implement a maximal random effect structure. However, for the models to converge and to avoid singular fits, we have to simplify the random effects structure to include only random intercepts for each subject and a random slope for the effect of virtual time for each subject. Barr et al.\u00a0(2013, p.\u00a0275) \u201cpropose the working assumption that it is not essential for one to specify random effects for control predictors to avoid anti-conservative inference, as long as interactions between the control predictors and the factors of interest are not present in the model\u201d. However, \u201cin the common case where one is interested in a minimally anti-conservative evaluation of the strength of evidence for the presence of an effect, our results indicate that keeping the random slope for the predictor of theoretical interest is important\u201d (Barr et al., 2013, p.\u00a0276). Thus, we keep the random slopes for virtual time in the model.\n\nMore specifically, we follow these steps:\n\n\u2022 we start with a model that includes random intercepts and slopes for all 3 fixed effects for each subject. This results in a singular fit\n\u2022 Thus, we drop the random effects of real time and order.\n\nWe do not incorporate random intercepts for the individual events as this sampling unit cannot be dissociated from the predictors. See the description of the exception by Brauer & Curtin (Psych Methods, 2018, p.\u00a013).\n\n# center the predictor variables (set scale to true for z-score)\nbeh_data <- beh_data %>%\ngroup_by(sub_id) %>%\nmutate(\norder_z = scale(as.numeric(event), scale = TRUE),\nvirtual_time_z = scale(virtual_time, scale = TRUE),\nreal_time_z = scale(real_time, scale = TRUE)\n) %>%\nungroup()\n\n# set RNG\nset.seed(27)\n\n# define the full model with all 3 time metrics as fixed effects and\n# by-subject random intercepts and random slopes for each time metric --> singular fit\nformula <- \"memory_time ~ virtual_time_z + order_z + real_time_z + (1 + virtual_time_z + order_z + real_time_z | sub_id)\"\nlmm_full <- lme4::lmer(formula, data = beh_data,\nREML = FALSE, control=lmerControl(optCtrl=list(maxfun=20000)))\n## boundary (singular) fit: see ?isSingular\n# remove random slopes for the fixed effects of non-interest (order and real time)\n# by-subject random intercepts and random slopes for virtual time\nset.seed(212) # set seed for reproducibility\nformula <- \"memory_time ~ virtual_time_z + order_z + real_time_z + (1 + virtual_time_z | sub_id)\"\nlmm_full <- lme4::lmer(formula, data = beh_data,\nREML = FALSE, control=lmerControl(optCtrl=list(maxfun=20000)))\nsummary(lmm_full)\n## Linear mixed model fit by maximum likelihood ['lmerMod']\n## Formula: memory_time ~ virtual_time_z + order_z + real_time_z + (1 + virtual_time_z | sub_id)\n## Data: beh_data\n## Control: lmerControl(optCtrl = list(maxfun = 20000))\n##\n## AIC BIC logLik deviance df.resid\n## 1940.0 1974.6 -962.0 1924.0 552\n##\n## Scaled residuals:\n## Min 1Q Median 3Q Max\n## -7.3874 -0.4034 0.0429 0.4259 7.8191\n##\n## Random effects:\n## Groups Name Variance Std.Dev. Corr\n## sub_id (Intercept) 0.04928 0.2220\n## virtual_time_z 0.02742 0.1656 0.23\n## Residual 1.75541 1.3249\n## Number of obs: 560, groups: sub_id, 28\n##\n## Fixed effects:\n## Estimate Std. Error t value\n## (Intercept) 14.01002 0.06996 200.252\n## virtual_time_z 3.06932 0.25997 11.807\n## order_z 1.66763 0.43023 3.876\n## real_time_z -0.33226 0.47331 -0.702\n##\n## Correlation of Fixed Effects:\n## (Intr) vrtl__ ordr_z\n## virtul_tm_z 0.017\n## order_z 0.000 -0.112\n## real_time_z 0.000 -0.426 -0.841\n# tidy summary of the fixed effects that calculates 95% CIs\nlmm_full_bm <- broom.mixed::tidy(lmm_full, effects = \"fixed\", conf.int=TRUE, conf.method=\"profile\")\n## Computing profile confidence intervals ...\n# tidy summary of the random effects\nlmm_full_bm_re <- broom.mixed::tidy(lmm_full, effects = \"ran_pars\")\n\nTo test whether virtual time is relevant to explaining the data even when order and real time are in the model, we compare the full model defined above against a reduced model. In this reduced model, we do not include the fixed effect of virtual time. We compare the full against the nested (reduced) model using a likelihood ratio test.\n\n# to test for significance let's by comparing the likelihood against a simpler model.\n# Here, we drop the effect of virtual time and run an ANOVA. See e.g. Bodo Winter tutorial\n# this is the current best way of testing the effect of virtual time on behavior\nformula <- \"memory_time ~ order_z + real_time_z + (1 + virtual_time_z | sub_id)\" # random intercepts for each subject and random slopes for virtual time\nset.seed(213) # set seed for reproducibility\nlmm_no_vir_time <- lme4::lmer(formula, data = beh_data, REML = FALSE, control=lmerControl(optCtrl=list(maxfun=20000)))\n\n# because the model fails to converge with a warning that the scaled gradient at the fitted (RE)ML estimates\n# is large, we restart the model as described here (https:\/\/rdrr.io\/cran\/lme4\/man\/troubleshooting.html)\n# obtaining consistent results (with no warning) suggests a false positive\nlmm_no_vir_time <- update(lmm_no_vir_time, start=getME(lmm_no_vir_time, \"theta\"))\n\n# run the ANOVA\nlmm_aov <- anova(lmm_no_vir_time, lmm_full)\nlmm_aov\nTable 5.2:\nnparAICBIClogLikdevianceChisqDfPr(>Chisq)\n72.05e+032.08e+03-1.02e+032.04e+03\n81.94e+031.97e+03-962\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a01.92e+0311614.88e-27\n\nMixed Model: Fixed effect of virtual time time with order and real time in the model of remembered times $$\\chi^2$$(1)=115.95, p=0.000\n\nMake mixed model summary table that includes overview of fixed and random effects as well as the model comparison to the nested (reduced) model.\n\nfe_names <- c(\"intercept\", \"virtual time\", \"order\", \"real time\")\nre_groups <- c(rep(\"participant\",3), \"residual\")\nre_names <- c(\"intercept\", \"virtual time (SD)\", \"correlation random intercepts and random slopes\", \"SD\")\n\nlmm_hux <- make_lme_huxtable(fix_df=lmm_full_bm,\nran_df = lmm_full_bm_re,\naov_mdl = lmm_aov,\nfe_terms =fe_names,\nre_terms = re_names,\nre_groups = re_groups,\nlme_form = gsub(\" \", \"\", paste0(deparse(formula(lmm_full)),\ncollapse = \"\", sep=\"\")),\ncaption = \"Mixed Model: Virtual time explains constructed times with order and real time in the model\")\n## Warning: select_vars() is deprecated as of dplyr 0.8.4.\n## Please use tidyselect::vars_select() instead.\n## This warning is displayed once every 8 hours.\n## Call lifecycle::last_warnings() to see where this warning was generated.\n# convert the huxtable to a flextable for word export\nstable_lme_memtime_time_metrics <- convert_huxtable_to_flextable(ht = lmm_hux)\n\n# print to screen\ntheme_article(lmm_hux)\nfixed effects term estimate SE t-value 14.010019 0.069962 200.25 13.868056 14.151981 3.069324 0.259967 11.81 2.558874 3.579774 1.667630 0.430230 3.88 0.822785 2.512476 -0.332261 0.473306 -0.70 -1.261696 0.597173 0.221991 0.232089 0.165592 1.324919 2053.90 -1019.95 1939.95 -961.98 115.95 1 4.88e-27\n\nTo visualize the model, we use the confidence intervals to create dot plots for the model coefficients. Further, we estimate marginal means for each fixed effect while holding the other parameters constant. We do this for the quartiles (including minimum and maximum values).\n\n# make the time metrics a factor\nlmm_full_bm <- lmm_full_bm %>%\nmutate(term=as.factor(term) %>%\nfactor(levels = c(\"virtual_time_z\", \"order_z\", \"real_time_z\")))\n\n# dot plot of Fixed Effect Coefficients with CIs\nf2f <- ggplot(data = lmm_full_bm[2:4,], aes(x = term, color = term)) +\ngeom_hline(yintercept = 0, colour=\"black\", linetype=\"dotted\") +\ngeom_errorbar(aes(ymin = conf.low, ymax = conf.high, width = NA), size = 0.5) +\ngeom_point(aes(y = estimate), size = 1, shape = 16) +\nscale_fill_manual(values = time_colors) +\nscale_color_manual(values = time_colors, labels = c(\"Virtual Time\", \"Order\", \"Real Time\")) +\nlabs(#title = \"Fixed Effect Coefficients\",\nx = element_blank(), y=\"fixed\\neffect estimate\",color = \"Time Metric\") +\ntheme_cowplot() +\n#coord_fixed(ratio = 3) +\ntheme(plot.title = element_text(hjust = 0.5), axis.text.x=element_blank()) +\nguides(color=FALSE, fill=FALSE)+\nannotate(geom = \"text\", x = 1, y = 3.8, label = \"***\", hjust = 0.5, size = 8\/.pt, family=font2use)\nf2f\n\n# estimate marginal means for each model term by omitting the terms argument\nlmm_full_emm <- ggeffects::ggpredict(lmm_full, ci.lvl = 0.95) %>% get_complete_df\n# convert the group variable to a factor to control the order of facets below\nlmm_full_emm$group <- factor(lmm_full_emm$group, levels = c(\"virtual_time_z\", \"order_z\", \"real_time_z\"))\n\n# plot marginal means\nf2g <- ggplot(data = lmm_full_emm, aes(color = group)) +\ngeom_line(aes(x, predicted)) +\ngeom_ribbon(aes(x, ymin = conf.low, ymax = conf.high, fill = group), alpha = .3, linetype=0) +\nscale_color_manual(values = time_colors, name=element_blank(),\nlabels = c(\"virtual time\", \"order\", \"real time\")) +\nscale_fill_manual(values = time_colors, labels = c(\"Virtual Time\", \"Order\", \"Real Time\")) +\nylab('estimated\\nmarginal means') +\nxlab('z time metric') +\nscale_y_continuous(breaks = c(9, 14, 19), labels = c(9, 14, 19)) +\nguides(fill = FALSE, color=FALSE) +\ntheme_cowplot() +\ntheme(plot.title = element_text(hjust = 0.5), strip.background = element_blank(),\nstrip.text.x = element_blank())\nf2g\n\n##### LME model assumptions\nlmm_diagplots_jb(lmm_full)\n\n\u2022 Absence of collinearity: The different time metrics are correlated. This correlation is slightly reduced by z-scoring the predictors. In any case, these correlations are inherent to the design and not really a problem as long as estimated coefficients are interpreted correctly. For more info, see e.g.\u00a0this opinion piece.\n\n## Compose Figure for Behavioral Data\n\nThis figure consists of the plots showing the results of the sorting task and the timeline task. It will probably be figure 2 of the manuscript.\n\nlayout = \"\nAABBBBBBDDDDDD\nAABBBBBBDDDDDD\nAABBBBBBDDDDDD\nAABBBBBBDDDDDD\nAABBBBBBDDDDDD\nAABBBBBBDDDDDD\nCCBBBBBBEEEEFF\nCCBBBBBBEEEEFF\nCCBBBBBBEEEEFF\nCCBBBBBBEEEEGG\nCCBBBBBBEEEEGG\nCCBBBBBBEEEEGG\"\n\nf2 <- f2a + f2b + f2c + f2d + f2e + f2f + f2g +\nplot_layout(design = layout, guides = \"keep\") &\ntheme(text = element_text(size=10, family=font2use),\naxis.text = element_text(size=8),\nlegend.text=element_text(size=8),\nlegend.title=element_text(size=8)\n) &\nplot_annotation(theme = theme(plot.margin = margin(t=0, r=0, b=0, l=-5, unit=\"pt\")),\ntag_levels = 'A')\n\n# save as png and pdf and print to screen\nfn <- here(\"figures\", \"f2\")\nggsave(paste0(fn, \".pdf\"), plot=f2, units = \"cm\",\nwidth = 17.4, height = 16, dpi = \"retina\", device = cairo_pdf)\nggsave(paste0(fn, \".png\"), plot=f2, units = \"cm\",\nwidth = 17.4, height = 16, dpi = \"retina\", device = \"png\")\nf2\n\nFigure 2. Participants learn the temporal structure of the sequences relative to the virtual clock. A. Plot shows the percentage of correctly sorted event images in the sorting task. B. Constructed event times were assessed in the timeline task. Responses are shown separately for each of the five events (color coded according to true virtual time) of each of the four sequences (rows). Colored circles with gray outline at the bottom of each row show true event times. C, D. Mean absolute errors in constructed times (in virtual hours) are shown (C) averaged across events and sequences and (D) averaged separately for the five event positions. E. Z-values for the effects of different time metrics from participant-specific multiple regression analyses and permutation tests show that virtual time explained constructed event times with event order and real time in the model as control predictors. F. Likewise, parameter estimates and 95% confidence intervals for the fixed effects of the three time metrics from a linear mixed model indicate that virtual time relates to constructed event times beyond the effects of order and real time. G. Estimated marginal means (model predictions) illustrate the effects of the three time metrics. A-E. Circles are individual participant data; boxplots show median and upper\/lower quartile along with whiskers extending to most extreme data point within 1.5 interquartile ranges above\/below the upper\/lower quartile; black circle with error bars corresponds to mean\u00b1S.E.M.; distributions show probability density function of data points. *** p<0.001\n\n## 5.3 Generalization bias\n\nIf participants use structural knowledge about the sequences when constructing times of events, then we might expect biases in their behavior: Errors in constructed event times could be non-random. Specifically, when constructing the time of one specific event, participants could be biased in their response by the times of the events from other sequences at that sequence position. This would indicate that knowledge about the other sequences in generalized to influence specific mnemonic constructions, resulting in a bias.\n\n### Quantify relative time of other events\n\nTo explore whether general time patterns bias the construction of event times, we assessed errors in remembered event times. Specifically, when constructing the time of one specific event, participants could be biased in their response by the times of the events from other sequences at that sequence position. For each event, we quantified the average time of events in the other sequences at the same sequence position (Figure 8A). For example, for the fourth event of the first sequence, we calculated the average time of the fourth events of sequences two, three and four.\n\nTo test for such a bias, we quantify, for each event, the relative time of the other events at that sequence position. We then calculate by how much virtual time each individual event time deviates from the average virtual time of the other events at that sequence position (e.g.\u00a0the difference in virtual time for event 1 from sequence 1 compared to the average virtual time of events 1 from sequences 2-4). We do this so that positive values of this relative time measure indicate that the other events happened later than the event of interest.\n\n# quantify the deviation in virtual time for each event relative to other events\n# at the same sequence position\nbeh_data$rel_time_other_events<-NA for (i_day in 1:n_days){ for (i_event in 1:n_events_day){ # find the events at this sequence position from all four sequences curr_events <- beh_data[beh_data$event == i_event,]\n\n# find the average virtual time of the other events at this sequence position\navg_vir_time_other_events <- curr_events %>%\nfilter(day != i_day) %>%\nsummarise(avg_vir_time = mean(virtual_time)) %>%\nselect(avg_vir_time) %>%\nas.numeric(.)\n\n# for the given event, store the relative time of the other events,\n# as the difference in virtual time (positive values mean other events happen later)\nevent_idx <- beh_data$day==i_day & beh_data$event==i_event\nbeh_data$rel_time_other_events[event_idx] <- avg_vir_time_other_events - beh_data$virtual_time[event_idx]\n}\n}\n\n### Test Generalization Bias\n\nWe then asked whether the deviation between the average time of other events and an event\u2019s true virtual time was systematically related to signed errors in constructed event times. A positive relationship between the relative time of other events and time construction errors indicates that, when other events at the same sequence position are relatively late, participants are biased to construct a later time for a given event than when the other events took place relatively early.\n\nThe crucial test is then whether over- and underestimates of remembered time can be explained by this deviation measure.\n\n# calculate timeline error so that positive numbers mean overestimates (later than true virtual time)\nbeh_data <- beh_data %>%\nmutate(timeline_error=memory_time-virtual_time)\n\n#### Summary statistics\n\nIn the summary statistics approach, we ran a linear regression model for each participant (Figure 8B, Supplemental Figure 8) and tested the resulting coefficients for statistical significance using the permutation-based procedures described above (Figure 8C).\n\nIn the summary statistics approach, we run a linear regression model for each participant and test the resulting coefficients against a permutation-based null distribution. The resulting z-scores are then tested against 0 on the group level.\n\n# SUMMARY STATISTICS\nset.seed(117) # set seed for reproducibility\n\n# do RSA using linear model and calculate z-score for model fit from permutations\nfit_beh_bias <- beh_data %>% group_by(sub_id) %>%\n# run the linear model (also without permutation tests to get betas)\ndo(model = lm(timeline_error ~ rel_time_other_events, data=.),\nz = lm_perm_jb(in_dat = .,\nlm_formula = \"timeline_error ~ rel_time_other_events\",\nnsim = n_perm)) %>%\n# store beta estimates and their z-values\nmutate(beta_rel_time_other_events = coef(summary(model))[2,\"Estimate\"],\nt_rel_time_other_events = coef(summary(model))[2,\"t value\"],\nz = list(setNames(z, c(\"z_intercept\", \"rel_time_other_events\")))) %>%\n# get rid of intercept z-values and model column\nunnest_longer(z) %>%\nfilter(z_id != \"z_intercept\") %>%\nselect (.,-c(model))\n\n# run group-level t-tests on the RSA fits from the first level in aHPC for within-day\nstats <- fit_beh_bias %>%\nfilter(z_id ==\"rel_time_other_events\") %>%\nselect(z) %>%\npaired_t_perm_jb (., n_perm = n_perm)\n\n# Cohen's d with Hedges' correction for one sample using non-central t-distribution for CI\nd<-cohen.d(d=fit_beh_bias$z, f=NA, paired=TRUE, hedges.correction=TRUE, noncentral=TRUE) stats$d <- d$estimate stats$dCI_low <- d$conf.int[[1]] stats$dCI_high <- d$conf.int[[2]] # print results huxtable(stats) %>% theme_article() Table 5.4: estimatestatisticp.valuep_permparameterconf.lowconf.highmethodalternativeddCI_lowdCI_high 1.385.321.3e-050.0001270.8491.92One Sample t-testtwo.sided0.9760.5521.48 Summary Statistics: t-test against 0 for generalization bias t27=5.32, p=0.000, d=0.98, 95% CI [0.55, 1.48] We observe a significant positive effect of the virtual time of other events at the same sequence position on remembered virtual time. That means that when other events at the same sequence position are later than a given event, participants are likely to overestimate the event time. Conversely, when the other events at this sequence position relatively early, participants underestimate. This demonstrates an across-sequence effect of virtual time. Virtual time is generalized across sequences to bias remembered times at similar sequence positions. To illustrate this effect, lets plot the data for one example subject (data from all subjects will be plotted below). For this, we pick a subject with an average fit. # pick average subject based on t-value of regression example_sub <- fit_beh_bias$sub_id[sort(fit_beh_bias$t_rel_time_other_events, decreasing = TRUE)[n_subs\/2]==fit_beh_bias$t_rel_time_other_events]\n\nf8b <- ggplot(beh_data%>%filter(sub_id==example_sub), aes(x=rel_time_other_events, y=timeline_error)) +\ngeom_smooth(method='lm', formula= y~x,\ncolor=aHPC_colors[\"across_main\"],\nfill=aHPC_colors[\"across_main\"])+\ngeom_point(size = 1, shape = 16) +\nscale_x_continuous(breaks = c(-2.5, 0, 2.5), labels= c(\"-2.5\", \"0\", \"2.5\")) +\nxlab(\"relative time\\nof other events\") +\nylab(\"timeline error\") +\ntheme_cowplot() +\ntheme(strip.background = element_blank(),\nstrip.text = element_blank(),\ntext = element_text(size=10, family=font2use),\naxis.text = element_text(size=8))\nf8b\n\nTo visualize the effect from the summary statistics approach on the group-level, we create a raincloud plot of the z-values from the linear model permutations for each subject.\n\n# plot regression z-values\nf8c<-ggplot(fit_beh_bias, aes(x=as.factor(1),y=z), colour= aHPC_colors[\"across_main\"], fill=aHPC_colors[\"across_main\"]) +\n# plot the violin\ngghalves::geom_half_violin(position=position_nudge(0.1),\nside = \"r\", fill=aHPC_colors[\"across_main\"], color = NA) +\ngeom_point(aes(x = 1-.2), alpha = 0.7,\nposition = position_jitter(width = .1, height = 0),\nshape=16, colour=aHPC_colors[\"across_main\"], size = 1)+\ngeom_boxplot(width = .1, fill=aHPC_colors[\"across_main\"], colour = \"black\", outlier.shape = NA) +\nstat_summary(fun = mean, geom = \"point\", size = 1, shape = 16,\nposition = position_nudge(.1), colour = \"black\") +\nstat_summary(fun.data = mean_se, geom = \"errorbar\",\nposition = position_nudge(.1), colour = \"black\", width = 0, size = 0.5)+\nylab('Z regression') + xlab('generalization\\n bias') +\nannotate(geom = \"text\", x = 1, y = 4.2, label = \"***\", hjust = 0.5, size = 8\/.pt, family=font2use) +\nguides(color=FALSE, fill=FALSE)+\ntheme_cowplot() +\ntheme(axis.text.x = element_blank())\nf8c\n\n#### Mixed Model\n\nFurther, we analyzed these data using the linear mixed model approach (Figure 8DE, Supplemental Table 13).\n\nWe also want to this effect using a linear mixed model. We use the maximal random effects structure with random intercepts and random slopes for participants.\n\nLet\u2019s fit the model and get tidy summaries.\n\n# z-score the relative time of other events for each participant\nbeh_data <- beh_data %>%\ngroup_by(sub_id) %>%\nmutate(rel_time_other_events_z = scale(rel_time_other_events)) %>%\nungroup()\n\n# Fit the model\nset.seed(245) # set seed for reproducibility\nlmm_full <- lme4::lmer(\"timeline_error ~ rel_time_other_events_z + (1+rel_time_other_events_z|sub_id)\",\ndata=beh_data, REML=FALSE)\n\n# tidy summary of the fixed effects that calculates 95% CIs\nlmm_full_bm <- broom.mixed::tidy(lmm_full, effects = \"fixed\", conf.int=TRUE, conf.method=\"profile\")\n## Computing profile confidence intervals ...\n# tidy summary of the random effects\nlmm_full_bm_re <- broom.mixed::tidy(lmm_full, effects = \"ran_pars\")\n\nCompare against a reduced model without the fixed effect of interest.\n\n# fit reduced model\nset.seed(248) # set seed for reproducibility\nlmm_reduced <- lme4::lmer(\"timeline_error ~ 1 + (1+rel_time_other_events_z|sub_id)\",\ndata=beh_data, REML=FALSE)\nlmm_aov<-anova(lmm_full, lmm_reduced)\nlmm_aov\nTable 5.5:\nnparAICBIClogLikdevianceChisqDfPr(>Chisq)\n51.96e+031.98e+03-9741.95e+03\n61.94e+031.97e+03-9651.93e+0317.912.32e-05\n\nMixed Model: Fixed effect of relative time of other events on timeline errors\n$$\\chi^2$$(1)=17.90, p=0.000\n\nMake summary table.\n\nfe_names <- c(\"intercept\", \"relative time other events\")\nre_groups <- c(rep(\"participant\",3), \"residual\")\nre_names <- c(\"intercept\", \"relative time other events (SD)\", \"correlation random intercepts and random slopes\", \"SD\")\n\nlmm_hux <- make_lme_huxtable(fix_df=lmm_full_bm,\nran_df = lmm_full_bm_re,\naov_mdl = lmm_aov,\nfe_terms =fe_names,\nre_terms = re_names,\nre_groups = re_groups,\nlme_form = gsub(\" \", \"\", paste0(deparse(formula(lmm_full)),\ncollapse = \"\", sep=\"\")),\ncaption = \"Mixed Model: Behavioral generalization bias\")\n\n# convert the huxtable to a flextable for word export\nstable_lme_beh_gen_bias <- convert_huxtable_to_flextable(ht = lmm_hux)\n\n# print to screen\ntheme_article(lmm_hux)\nfixed effects term estimate SE t-value -0.352481 0.069962 -5.04 -0.494444 -0.210518 0.337262 0.067360 5.01 0.200579 0.473945 0.220016 -0.114173 0.183681 1.331485 1958.57 -974.29 1942.67 -965.34 17.90 1 2.32e-05\n\nTo visualize the mixed model we create a dot plot of the fixed effect coefficient and the estimated marginal means.\n\n# dot plot of Fixed Effect Coefficients with CIs\nf8d <- ggplot(data = lmm_full_bm[2,], aes(x = term, color = term)) +\ngeom_hline(yintercept = 0, colour=\"black\", linetype=\"dotted\") +\ngeom_errorbar(aes(ymin = conf.low, ymax = conf.high, width = NA), size = 0.5) +\ngeom_point(aes(y = estimate), size = 1, shape = 16) +\nscale_fill_manual(values = unname(aHPC_colors[\"across_main\"])) +\nscale_color_manual(values = unname(aHPC_colors[\"across_main\"]), labels = c(\"across sequence bias\")) +\nlabs(x = element_blank(), y=\"fixed\\neffect estimate\") +\ntheme_cowplot() +\ntheme(plot.title = element_text(hjust = 0.5), axis.text.x=element_blank()) +\nguides(color=FALSE, fill=FALSE)+\nannotate(geom = \"text\", x = 1, y = 0.5, label = \"***\", hjust = 0.5, size = 8\/.pt, family=font2use)\n\n# estimate marginal means for each model term by omitting the terms argument\nlmm_full_emm <- ggeffects::ggpredict(lmm_full, ci.lvl = 0.95) %>% get_complete_df\n\n# plot marginal means\nf8e <- ggplot(data = lmm_full_emm, aes(color = group)) +\ngeom_line(aes(x, predicted)) +\ngeom_ribbon(aes(x, ymin = conf.low, ymax = conf.high, fill = group), alpha = .3, linetype=0) +\nscale_color_manual(values = unname(aHPC_colors[\"across_main\"]), name=element_blank()) +\nscale_fill_manual(values = unname(aHPC_colors[\"across_main\"])) +\nscale_x_continuous(breaks = c(-2.5, 0, 2.5), labels= c(\"-2.5\", \"\", \"2.5\")) +\nylab('estimated\\nmarginal means') +\nxlab('relative time\\nof other events') +\nguides(fill = FALSE, color=FALSE) +\ntheme_cowplot()\nf8d+f8e\n\nLastly, we show that the effect is quite visible from the single-subject plots: The slopes of the least-squares lines are, on average, positive.\n\n# plot the relationship for each subject\nsfig_bias_single_sub <- ggplot(beh_data, aes(x=rel_time_other_events, y=timeline_error)) +\ngeom_smooth(method='lm', formula= y~x,\ncolor=aHPC_colors[\"across_main\"],\nfill=aHPC_colors[\"across_main\"])+\ngeom_point(size = 1, shape = 16) +\nfacet_wrap(~sub_id, scales=\"free_y\", nrow=4) +\nscale_x_continuous(breaks = c(-2.5, 0, 2.5), labels= c(\"-2.5\", \"0\", \"2.5\")) +\nxlab(\"relative time of other events (virtual hours)\") +\nylab(\"timeline error\") +\ntheme_cowplot() +\ntheme(strip.background = element_blank(),\nstrip.text = element_blank(),\ntext = element_text(size=10, family=font2use),\naxis.text = element_text(size=8))\nsfig_bias_single_sub\n\n## 5.4 Replication of Generalization Bias\n\nTo replicate the results from this exploratory analysis, we conducted the same analysis in an independent group of participants. These participants (n=46) constituted the control groups of a behavioral experiment testing the effect of stress induction on temporal memory (Montijn et al., 2021). They underwent the same learning task as described above with the only difference being the duration of this learning phase (4 rather than 7 mini-blocks of training). The timeline task was administered on the day after learning. The procedures are described in detail in Montijn et al.\u00a0(2021). The data from this independent sample are shown in Figure 8F-H and Supplemental Figure 8B.\n\nfname <- file.path(dirs$data4analysis, \"beh_dataNDM.txt\") col_types_list <- cols_only( sub_id = col_factor(), day = col_factor(), event = col_factor(), virtual_time = col_double(), memory_time = col_double() ) beh_data_replication <- as_tibble(read_csv(fname, col_types = col_types_list)) head(beh_data_replication) Table 5.7: sub_iddayeventvirtual_timememory_time P001119.259.21 P0011211.8 11.1 P0011314.8 16.1 P0011416.2 15 P0011518.2 18.4 P001219.5 8.5 ### Replication data: Quantify relative time of other events To test for such a bias, we quantify, for each event, the relative time of the other events at that sequence position. We then calculate by how much virtual time each individual event time deviates from the average virtual time of the other events at that sequence position (e.g. the difference in virtual time for event 1 from sequence 1 compared to the average virtual time of events 1 from sequences 2-4). We do this so that positive values of this relative time measure indicate that the other events happened later than the event of interest. # quantify the deviation in virtual time for each event relative to other events # at the same sequence position beh_data_replication <- beh_data_replication %>% add_column(rel_time_other_events=NA) for (i_day in 1:n_days){ for (i_event in 1:n_events_day){ # find the events at this sequence position from all four sequences curr_events <- beh_data_replication[beh_data_replication$event == i_event,]\n\n# find the average virtual time of the other events at this sequence position\navg_vir_time_other_events <- curr_events %>%\nfilter(day != i_day) %>%\nsummarise(avg_vir_time = mean(virtual_time)) %>%\nselect(avg_vir_time) %>%\nas.numeric(.)\n\n# for the given event, store the relative time of the other events,\n# as the difference in virtual time (positive values mean other events happen later)\nevent_idx <- beh_data_replication$day==i_day & beh_data_replication$event==i_event\nbeh_data_replication$rel_time_other_events[event_idx] <- avg_vir_time_other_events - beh_data_replication$virtual_time[event_idx]\n}\n}\n\n### Replication data: Test Generalization Bias\n\nThe crucial test is then whether over- and underestimates of remembered time can be explained by this deviation measure.\n\n# calculate timeline error so that positive numbers mean overestimates (later than true virtual time)\nbeh_data_replication <- beh_data_replication %>%\nmutate(timeline_error=memory_time-virtual_time)\n\n#### Replication data: Summary statistics\n\nIn the summary statistics approach, we run a linear regression model for each participant and test the resulting coefficients against a permutation-based null distribution. The resulting z-scores are then tested against 0 on the group level. Both of the tests rely on the custom stats function defined above.\n\n# SUMMARY STATISTICS\nset.seed(120) # set seed for reproducibility\n\n# do RSA using linear model and calculate z-score for model fit from permutations\nfit_beh_bias_replication <- beh_data_replication %>% group_by(sub_id) %>%\n# run the linear model (also without permutation tests to get betas)\ndo(model = lm(timeline_error ~ rel_time_other_events, data=.),\nz = lm_perm_jb(in_dat = .,\nlm_formula = \"timeline_error ~ rel_time_other_events\",\nnsim = n_perm)) %>%\n# store beta estimates and their z-values\nmutate(beta_rel_time_other_events = coef(summary(model))[2,\"Estimate\"],\nt_rel_time_other_events = coef(summary(model))[2,\"t value\"],\nz = list(setNames(z, c(\"z_intercept\", \"rel_time_other_events\")))) %>%\n# get rid of intercept z-values and model column\nunnest_longer(z) %>%\nfilter(z_id != \"z_intercept\") %>%\nselect (.,-c(model))\n\n# run group-level t-tests on the RSA fits from the first level in aHPC for within-day\nstats <- fit_beh_bias_replication %>%\nfilter(z_id ==\"rel_time_other_events\") %>%\nselect(z) %>%\npaired_t_perm_jb (., n_perm = n_perm)\n\n# Cohen's d with Hedges' correction for one sample using non-central t-distribution for CI\nd<-cohen.d(d=fit_beh_bias_replication$z, f=NA, paired=TRUE, hedges.correction=TRUE, noncentral=TRUE) ## Warning in pt(q = t, df = df, ncp = x): full precision may not have been achieved in 'pnt{final}' ## Warning in pt(q = t, df = df, ncp = x): full precision may not have been achieved in 'pnt{final}' stats$d <- d$estimate stats$dCI_low <- d$conf.int[[1]] stats$dCI_high <- d\\$conf.int[[2]]\n\n# print results\nhuxtable(stats) %>% theme_article()\n\nTable 5.8:\nestimatestatisticp.valuep_permparameterconf.lowconf.highmethodalternativeddCI_lowdCI_high\n1.6611.39.76e-150.0001451.371.96One Sample t-testtwo.sided1.641.232.13\nSummary Statistics: t-test against 0 for generalization bias in replication sample t45=11.3, p=0.000, d=1.64, 95% CI [1.23, 2.13]\n\nWe observe a significant positive effect of the virtual time of other events at the same sequence position on remembered virtual time. That means that when other events at the same sequence position are later than a given event, participants are likely to overestimate the event time. Conversely, when the other events at this sequence position relatively early, participants underestimate. This demonstrates an across-sequence effect of virtual time. Virtual time is generalized across sequences to bias remembered times at similar sequence positions.\n\nTo visualize this effect from the summary statistics approach, we create a raincloud plot of the z-values from the linear model permutations for each subject.\n\n# plot regression z-values\nf8f<-ggplot(fit_beh_bias_replication, aes(x=as.factor(1),y=z), colour= aHPC_colors[\"across_main\"], fill=aHPC_colors[\"across_main\"]) +\n# plot the violin\ngghalves::geom_half_violin(position=position_nudge(0.1),\nside = \"r\", fill=aHPC_colors[\"across_main\"], color = NA) +\ngeom_point(aes(x = 1-.2), alpha = 0.7,\nposition = position_jitter(width = .1, height = 0),\nshape=16, colour=aHPC_colors[\"across_main\"], size = 1)+\ngeom_boxplot(width = .1, fill=aHPC_colors[\"across_main\"], colour = \"black\", outlier.shape = NA) +\nstat_summary(fun = mean, geom = \"point\", size = 1, shape = 16,\nposition = position_nudge(.1), colour = \"black\") +\nstat_summary(fun.data = mean_se, geom = \"errorbar\",\nposition = position_nudge(.1), colour = \"black\", width = 0, size = 0.5)+\nylab('Z regression') + xlab('generalization\\nbias') +\nannotate(geom = \"text\", x = 1, y = 3.7, label = \"***\", hjust = 0.5, size = 8\/.pt, family=font2use) +\nguides(color=FALSE, fill=FALSE)+\ntheme_cowplot() +\ntheme(axis.text.x = element_blank())\nf8f\n\n#### Replication data: Mixed Model\n\nWe also want to this effect using a linear mixed model. We use the maximal random effects structure with random intercepts and random slopes for participants.\n\nLet\u2019s fit the model and get tidy summaries.\n\n# z-score the relative time of other events for each participant\nbeh_data_replication <- beh_data_replication %>%\ngroup_by(sub_id) %>%\nmutate(rel_time_other_events_z = scale(rel_time_other_events)) %>%\nungroup()\n\n# Fit the model with full random effect structure -> singular\nlmm_full <- lme4::lmer(\"timeline_error ~ rel_time_other_events_z + (1+rel_time_other_events_z|sub_id)\",\ndata=beh_data_replication, REML=FALSE)\n## boundary (singular) fit: see ?isSingular\n# fit again after removing correlation of slope and intercept\nlmm_full <- lme4::lmer(\"timeline_error ~ rel_time_other_events_z + (1+rel_time_other_events_z||sub_id)\",\ndata=beh_data_replication, REML=FALSE)\n## boundary (singular) fit: see ?isSingular\n# fit again after removing correlation of also random intercept (only random slopes left)\nset.seed(278) # set seed for reproducibility\nlmm_full <- lme4::lmer(\"timeline_error ~ rel_time_other_events_z + (0+rel_time_other_events_z|sub_id)\",\ndata=beh_data_replication, REML=FALSE)\n## boundary (singular) fit: see ?isSingular\n# tidy summary of the fixed effects that calculates 95% CIs\nlmm_full_bm <- broom.mixed::tidy(lmm_full, effects = \"fixed\", conf.int=TRUE, conf.method=\"profile\")\n## Computing profile confidence intervals ...\n# tidy summary of the random effects\nlmm_full_bm_re <- broom.mixed::tidy(lmm_full, effects = \"ran_pars\")\n\nCompare against a reduced model without the fixed effect of interest.\n\n# fit reduced model\nset.seed(221) # set seed for reproducibility\nlmm_reduced <- lme4::lmer(\"timeline_error ~ 1 + (0+rel_time_other_events_z|sub_id)\",\ndata=beh_data_replication, REML=FALSE)\nlmm_aov<-anova(lmm_full, lmm_reduced)\nlmm_aov\nTable 5.9:\nnparAICBIClogLikdevianceChisqDfPr(>Chisq)\n34.5e+03\u00a04.52e+03-2.25e+034.5e+03\n44.45e+034.47e+03-2.22e+034.44e+0353.712.29e-13\n\nMixed Model: Fixed effect of relative time of other events on timeline errors\n$$\\chi^2$$(1)=53.74, p=0.000\n\nMake summary table.\n\nfe_names <- c(\"intercept\", \"relative time other events\")\nre_groups <- c(rep(\"participant\",1), \"residual\")\nre_names <- c(\"relative time other events (SD)\", \"SD\")\n\nlmm_hux <- make_lme_huxtable(fix_df=lmm_full_bm,\nran_df = lmm_full_bm_re,\naov_mdl = lmm_aov,\nfe_terms =fe_names,\nre_terms = re_names,\nre_groups = re_groups,\nlme_form = gsub(\" \", \"\", paste0(deparse(formula(lmm_full)),\ncollapse = \"\", sep=\"\")),\ncaption = \"Mixed Model: Behavioral generalization bias (replication)\")\n\n# convert the huxtable to a flextable for word export\nstable_lme_beh_gen_bias_replication <- convert_huxtable_to_flextable(ht = lmm_hux)\n\n# print to screen\ntheme_article(lmm_hux)\nfixed effects term estimate SE t-value -0.320564 0.089155 -3.60 -0.495488 -0.145640 0.863631 0.091472 9.44 0.684152 1.043110 0.000000 2.704218 4501.04 -2247.52 4449.30 -2220.65 53.74 1 2.29e-13\n\nTo visualize the mixed model we create a dot plot of the fixed effect coefficient and the estimated marginal means.\n\n# dot plot of Fixed Effect Coefficients with CIs\nf8g <- ggplot(data = lmm_full_bm[2,], aes(x = term, color = term)) +\ngeom_hline(yintercept = 0, colour=\"black\", linetype=\"dotted\") +\ngeom_errorbar(aes(ymin = conf.low, ymax = conf.high, width = NA), size = 0.5) +\ngeom_point(aes(y = estimate), size = 1, shape = 16) +\nscale_fill_manual(values = unname(aHPC_colors[\"across_main\"])) +\nscale_color_manual(values = unname(aHPC_colors[\"across_main\"]), labels = c(\"across sequence bias\")) +\nlabs(x = element_blank(), y=\"fixed\\neffect estimate\") +\ntheme_cowplot() +\ntheme(plot.title = element_text(hjust = 0.5), axis.text.x=element_blank()) +\nguides(color=FALSE, fill=FALSE)+\nannotate(geom = \"text\", x = 1, y = 1.1, label = \"***\", hjust = 0.5, size = 8\/.pt, family=font2use)\n\n# estimate marginal means for each model term by omitting the terms argument\nlmm_full_emm <- ggeffects::ggpredict(lmm_full, ci.lvl = 0.95) %>% get_complete_df\n\n# plot marginal means\nf8h <- ggplot(data = lmm_full_emm, aes(color = group)) +\ngeom_line(aes(x, predicted)) +\ngeom_ribbon(aes(x, ymin = conf.low, ymax = conf.high, fill = group), alpha = .3, linetype=0) +\nscale_color_manual(values = unname(aHPC_colors[\"across_main\"]), name=element_blank()) +\nscale_fill_manual(values = unname(aHPC_colors[\"across_main\"])) +\nscale_x_continuous(breaks = c(-2.5, 0, 2.5), labels= c(\"-2.5\", \"\", \"2.5\")) +\nylab('estimated\\nmarginal means') +\nxlab('relative time\\nof other events') +\nguides(fill = FALSE, color=FALSE) +\ntheme_cowplot()\nf8g+f8h\n\nLastly, we show that the effect is quite visible from the single-subject plots: The slopes of the least-squares lines are, on average, positive.\n\n# plot the relationship for each subject\nsfig_bias_replication_single_sub <- ggplot(beh_data_replication, aes(x=rel_time_other_events, y=timeline_error)) +\ngeom_smooth(method='lm', formula= y~x,\ncolor=aHPC_colors[\"across_main\"],\nfill=aHPC_colors[\"across_main\"])+\ngeom_point(size = 1, shape = 16) +\nfacet_wrap(~sub_id, scales=\"free_y\", ncol=7) +\nscale_x_continuous(breaks = c(-2.5, 0, 2.5), labels= c(\"-2.5\", \"0\", \"2.5\")) +\nxlab(\"relative time of other events (virtual hours)\") +\nylab(\"timeline error\") +\ntheme_cowplot() +\ntheme(strip.background = element_blank(),\nstrip.text = element_blank(),\ntext = element_text(size=10, family=font2use),\naxis.text = element_text(size=8))\n\nlayout = \"\nA\nA\nA\nA\nB\nB\nB\nB\nB\nB\"\n\nsfig_bias_single_sub_both_samples <- sfig_bias_single_sub + sfig_bias_replication_single_sub +\nplot_layout(design = layout, guides = \"keep\") &\ntheme(text = element_text(size=10, family=font2use),\naxis.text = element_text(size=8),\nlegend.text=element_text(size=8),\nlegend.title=element_text(size=8),\nlegend.position = \"none\"\n) &\nplot_annotation(tag_levels=\"A\",\ntheme = theme(plot.margin = margin(t=0, r=0, b=0, l=-5, unit=\"pt\")))\n\n# save as png and pdf and print to screen\nfn <- here(\"figures\", \"sf08\")\nggsave(paste0(fn, \".pdf\"), plot=sfig_bias_single_sub_both_samples, units = \"cm\",\nwidth = 17.4, height = 22.5, dpi = \"retina\", device = cairo_pdf)\nggsave(paste0(fn, \".png\"), plot=sfig_bias_single_sub_both_samples, units = \"cm\",\nwidth = 17.4, height = 22.5, dpi = \"retina\", device = \"png\")\nsfig_bias_single_sub_both_samples\n\nSupplemental Figure 8. Generalization bias in individual participants. A, B. Each panel shows the data from one participant. Each circle corresponds to one event. The x-axis indicates the average relative time of the events occupying the same sequence position in other sequences. The y-axis shows the signed error of constructed event times as measured in the timeline task. The regression line and its confidence interval are overlaid in red. Positive slopes of the regression line indicate that constructed event times are biased by the average time of events in the other sequences. A shows data from the main sample; B from the replication sample.","date":"2021-09-21 12:11:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"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\": 0.4111475348472595, \"perplexity\": 11323.335433367314}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780057202.68\/warc\/CC-MAIN-20210921101319-20210921131319-00188.warc.gz\"}"}
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Hsu Shao-chang 许绍昌 (* 9. April 1913 in der Provinz Zhejiang; † 1999 in Kalifornien) war ein taiwanischer Diplomat.
Persönliches
Er heiratete Tsai, Wan-chun und sie hatten eine Tochter.
Bildung
Er studierte Politikwissenschaft am Central Institute of Political Sciences der University of Nanking und an der University of Southern California.
Karriere
In den auswärtigen Dienst der Republik China trat er 1935 ein und wurde bis 1937 in der Abteilung Europa und Amerika des Außenministeriums in Nanking beschäftigt. Vizekonsul in Los Angeles war er von 1937 bis 1945. Die Personalabteilung im Außenministerium in Peking leitete er von 1945 bis 1947. Von 1947 bis 1948 hatte er Exequatur als Generalkonsul in Rangoon in Britisch-Burma. In Rangoon in Britisch-Burma war er 1948 Gesandtschaftssekretär. In Französisch-Indochina hatte er von 1948 bis 1949 Exequatur. Von 1951 bis 1954 was er Konsul und Geschäftsträger in Teheran. Bei zehn Sitzungsperiode des Economic and Social Council der Vereinten Nationen leitete er 1954 die taiwanesische Delegation. Die Abteilung Amerika leitete er von 1954 bis 1958. Als Staatssekretär wurde er von 1960 bis 1962 eingesetzt. 1966 war er Delegierter bei der 21 Generalversammlung der Vereinten Nationen in New York City. Von Juni 1963 bis 1968 war er Botschafter in Rio de Janeiro. In Rom und gleichzeitig in La Valletta war er von 1968 bis November 1970 akkreditiert. Von 1971 bis 1973 war er Botschafter zuerst in Buenos Aires und dann in Saigon.
Einzelnachweise
Botschafter der Republik China
Botschafter im Iran
Botschafter in Brasilien
Botschafter in Italien
Botschafter in Argentinien
Botschafter in Südvietnam
Chinese
Taiwaner
Geboren 1913
Gestorben 1999
Mann
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Home/Health/Bob Saget: 'Full House' father dies at 65
Bob Saget: 'Full House' father dies at 65
mccadminJanuary 11, 2022
The comedian was found dead in a hotel room in Orlando, Florida on Sunday, January 9th.
He was the narrator of "How I Met Your Mother".
The popular actor, who became known as Danny Tanner in "Full House" in the late 1980s, was found dead by US authorities in a hotel in Orlando, Florida. The cause of death is not yet known, but the Orange County Sheriff's office said authorities had found no evidence of drug use or crime.
A few hours before the death announcement, Bob Saget himself, who was touring his latest stand-up comedy show, shared a message on Twitter saying he was excited about the live show he had just staged . Jacksonville, also in Florida.
Loved today's show @PV_ConcertHall in Jacksonville. Appreciative audience. Thanks again to @RealTimWilkins for opening it. Little did I know I was doing a 2 hour set tonight. I'm happily addicted to this shit again. Check https://t.co/nqJyTiiezU for my data in 2022. pic.twitter.com/pEgFuXxLd3
– Bob saget (@bobsaget) January 9, 2022
The comedian's family has already commented on the incident in a statement. "We are devastated over the death of our beloved Bob," it says. "He was everything to us and we want you to know how much he loved his fans, performed live and brought people from all backgrounds together through humor and laughter."
In the course of his acting career, Saget also became famous as the narrator of the series "How I Met Your Mother". However, the role of a newly widowed father on Full House who tried to look after his three daughters with the help of his brother-in-law (John Stamos) and best friend (Dave Coulier) would end up being the one with the greatest recognition in the world Publicity.
The series that starred twins Mary-Kate and Ashley Olsen had a Netflix-produced sequel with much of the original cast. There were five seasons from 2016 to 2020.
At the same time, the moderator was also characterized by his provocative humor.
Worries about high corona figures in Great Britain | free press
of visitors swallowed by a full-size Gozdilla in an amusement park
Aiwanger: Greens practice "bullying against men" | free press
Federal Constitutional Court: Six percent tax rate unconstitutional | free press
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Anthophora plumipes est une espèce d'abeilles sauvages de la famille des apidés appelée l'Anthophore aux pattes poilues. C'est une abeille solitaire, à longue langue qui butine généralement les fleurs à corolles profondes (lamiacées, fabacées, boraginacées).
Description
Avec une pilosité importante, cette espèce ressemble à un bourdon des champs, mais elle vole plus rapidement. La longueur du corps atteint 14 à . L'aile possède 3 cellules cubitales de même taille.
Le dimorphisme sexuel est important : les poils tirent vers le gris chez le mâle alors que les femelles sont brunes ou noires. Les pattes médianes des mâles sont très allongées avec de longues franges de poils sur les tarses. La langue est longue et fine, ce qui leur permet de visiter des fleurs à corolle tubulaire.
Le mâle brun possède une marque blanc crème sur la face : le clypeus et le labre sont très clairs. Il est facilement reconnaissable par ses poils aux pattes médianes. Il patrouille autour des fleurs et poursuit les femelles. Ce comportement territorial est essentiellement limité à des zones fleuries.
La femelle est noirâtre ou brune sauf les brosses à pollen orange sur les pattes postérieures.
Le nid peut être approprié par l'abeille-coucou Melecta albifrons.
Période de vol
Mars à juin mais surtout en avril-mai. Anthophora plumipes fait partie des premiers hyménoptères à butiner.
Habitat et distribution
C'est une espèce commune des milieux ouverts qui nidifie en creusant des galeries dans des talus argileux ou dans le mortier friable des constructions. L'extrémité de ces galeries est ramifiée en plusieurs cellules ovoïdes tapissées d'une sécrétion blanche. À l'intérieur des cellules de nidification du miel et du pollen sont stockés.
Cette espèce est largement répartie. On la trouve presque partout en France mais aussi en Angleterre, en Allemagne...
Notes et références
Liens externes
Anthophora plumipes sur site Discover Life
Anthophora plumipes comparée à espèces proches sur site aramel.free.fr
Bibliographie
Espèce d'Abeilles (nom scientifique)
Apidae
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{"url":"http:\/\/dsp.stackexchange.com\/questions","text":"# All Questions\n\n13 views\n\n### Does the enlarged photo benefit from shrinking it?\n\nTake a look at the following diagrams: The original picture: The upscaled (2x, bilinear) picture: In both cases the first item in first row is the input image. There is it's 2d fft next to it. ...\n12 views\n\n### What's the use case and explanation of frequency-domain subsampling?\n\nIn my solution to a certain problem I use subsampling in the frequency domain. Amazingly, it produces good results. Is it really that awkward to see subsampling in frequency domain? What the ...\n5 views\n\n### Kalman filter for velocity estimation from acceleration and displacement: constant-acceleration assumption\n\nI have implemented a Kalman filter to estimate the velocity knowing the acceleration and the positon measurements as explained in this Q\/A: Estimating velocity from known position and acceleration ...\n10 views\n\n### homography matrix problem\n\ni have mini project to do about homography , but i am really a beginner in matlab . i have calculated the homography matrix easily by taking a set pixels from both images but the problem is how to ...\n7 views\n\n### Inferring space domain signal from 2D DFT\n\nBy just looking at the 2D Fourier Transform of a signal, can it every be known precisely which values in the space domain are zero?\n18 views\n\n### Upscaling detection approach\n\nI'm trying to implement a solution that would, to some extent, evaluate a picture telling what are the chances that it was upscaled from the smaller image. My approach is as follows: Take a picture ...\n12 views\n\n27 views\n\n### Circle detection opencv\n\nI Would like an explanation of the parameters for opencv's HoughCircles function. I'm new to image processing in general and I have recently started using opencv. ...\n28 views\n\n### How does sampling and subsequent resampling affect aliased harmonics?\n\nIf I sample a signal with a harmonic higher than half my sample frequency I can predict the aliased frequency easily as $$|F_h-kF_s|$$ But what if after sampling the signal, I re-sample at a lower ...\n29 views\n\n### What preprocessing technique can help me to filter the background text noises using matlab\u2026the image is shown below?\n\nCan anyone help me to improve the degradation of the following image....i am working on OCR system in Amharic language and what is the best image filtering technique that can help me to separate ...\n17 views\n\n### How to determine stopband of discrete Gaussian, stdev sigma, support N\n\nI would like to analyse a gaussian smoothing kernel with a set standard deviation and support (let's say, in MATLAB, fspecial('gaussian', [5 1], 1.3) so sigma is 1.3 and support is 5) in the DTFT ...\n51 views\n\n### Is image reconstruction possible using its magnitude only?\n\nLet us take any image like Lena or cameraman. I know that when Fourier transform is applied on image,spatially global features like amplitude and phase are obtained locally. My question is whether it ...\n9 views\n\n### How to programatically perform image segmentation in real-time\n\nI am currently trying to segment ultrasound images of blood vessels (video frames), such as the one directly below. Using a simple binary filter on the saturation channel (code below), I can get a ...\n18 views\n\n### Correction of signal after High-pass RC-filter\n\nI have one question about digital correction of signal after RC filter which is high-pass. Let me explain detail. I have one simple signal conditioner. It has RC filter on the signal input with ...\n24 views\n\n### DFT and Inverse DFT in Image Processing\n\nI have implemented DFT from this link. 1) Tried taking dft for the three Channels(R, G and B) and reconstructed the original image by taking inverse dft for all three channels and merged them ...\n45 views\n\n### I saw this question on one of the sites related to DFT\n\nThe analog signal x(t) is band-limited to 40 Hz. Suppose the signal is sampled at the rate of 100 samples per second and that at this rate 200 samples are collected. Then 200 zeros are appended to the ...\n44 views\n\n### Multi-image denoising\n\nI have two signals Sa and Sb, both affected by the same noise. I don't know how much noise is mixed into each one, and the signal+noise is transformed using some (unknown) nonlinear function before I ...\n18 views\n\n### Determining the point in a temporal signal when a periodic behavior begins to occur\n\nI'm a newcomer to digital signal processing, so I'm looking for some general advice and direction on this problem I have. I have a signal like that shown below. Starting around x=40000, you can see a ...\n37 views\n\n15 views\n\n### Significance of Lambda in Basis pursuit\n\nIn basis Pursuit,l1 minimization is done to perform compressed sensing.In the literature there is a lamba parameter used as a regulizer. What is its significance?","date":"2015-03-01 20:54:18","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\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8711113929748535, \"perplexity\": 1718.8722076056633}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-11\/segments\/1424936462555.21\/warc\/CC-MAIN-20150226074102-00082-ip-10-28-5-156.ec2.internal.warc.gz\"}"}
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\section{Introduction}\label{sec:Intro}
In nuclear physics, the symmetry energy is first encountered within the elementary
Bethe-Weizs\"{a}cker formula for nuclear energy:
\begin{equation}
\label{eq:BW}
{ E = - a_V \, A + a_S \, A^{2/3}
+ a_C \, \frac{Z^2}{A^{1/3}} +} { a_A \, \frac{(N
- Z)^2}{A} } { + \Delta} \, ;
\end{equation}
it is the change in nuclear energy associated with changing neutron-proton asymmetry
$(N-Z)/A$. In nuclear matter, the energy per nucleon, dependent on neutron $\rho_n$ and proton $\rho_p$
densities, may be represented as a sum of the
energy $E_0$ for symmetric nuclear matter and the correction $E_1$ associated with the asymmetry,
\begin{equation}
E(\rho_n, \rho_p) = E_0 (\rho) + E_1 (\rho_n, \rho_p) \, ,
\end{equation}
where $\rho=\rho_n +\rho_p$.
The charge symmetry of nuclear interactions, which is the symmetry under the interchange of neutrons and protons,
requires that the correction be quadratic in the asymmetry, for small asymmetries:
\begin{equation}
E_1 = E - E_0 \simeq { S(\rho)} \left( \frac{\rho_n-\rho_p}{\rho} \right)^2 \, .
\end{equation}
Microscopic calculations, such as \cite{bom91}, indicate that the quadratic approximation yields
a very good representation for the energy in nuclear matter, up to the limit of neutron matter with
asymmetry of~1, over a wide range of net densities. As a consequence, the energy in nuclear matter
over a broad range of parameters can be described exclusively in terms of the energy of
symmetric matter $E_0(\rho)$ and the symmetry-energy coefficient $S(\rho)$.
Much effort has been dedicated in the past to the understanding of $E_0(\rho)$ and less so to
$S(\rho)$ whose features remain more obscure \cite{bro00}. The energy $E_0(\rho)$ minimizes at the
normal density $\rho_0$, reaching there the value of $E_0 \approx - 16.0$~MeV. The uncertainties in $S$ hamper
predictions for neutron stars whose structure depends on pressure
in neutron matter \cite{lat01}.
In pure neutron matter, the energy is $E=E_0+S$ and the pressure is
${ P = \rho^2 \, dE/d\rho \simeq \rho^2 \, dS/d\rho}$ close to $\rho_0$, as $E_0$ minimizes at $\rho_0$.
In the calculations of neutron-star structure \cite{lat01}, a correlation is found,
${R \, P^{-1/4} \approx \mbox{const}}$, between the radius $R$ of a neutron star of a given mass and
the pressure $P$ in neutron matter at a given density $\rho \sim \rho_0$.
\section{Binding Formula}
When examining the standard energy formula (\ref{eq:BW}), we notice that the symmetry energy
has a volume character: it changes as $A$ when the neutron and proton numbers are scaled
by one factor. The formula lacks a surface symmetry term that would change as $A^{2/3}$ with the
change in nucleon number. A question to ask is whether there should be such a term.
Let us look at the surface energy. We can write it as $E_S = a_S \, A^{2/3} =
\frac{a_S}{4\pi \, r_0^2} \, 4\pi \,
r_0^2 \, A^{2/3}
=
\frac{a_S}{4\pi \, r_0^2} \, {\mathcal S}$. The ratio
$\frac{E_S}{\mathcal S} = \sigma = \frac{a_S}{4\pi \, r_0^2}$ is surface tension, the work that needs
to be done per unit area when changing the surface area of the nucleus,
such as in deforming the nucleus. The work needs to be done to compensate lost binding, as nucleons
close to the surface are less bound, due to fewer neighbors, than in the interior. The formula (\ref{eq:BW})
states that in the interior the nucleons are less bound in a more asymmetric nucleus. In that case,
the energetic price for increasing the surface should drop, i.e.\
$\sigma = \frac{\partial E_S}{\partial {\mathcal S}}$
should decrease with asymmetry, in the more general definition of tension.
As intensive, $\sigma$ should be expressed in terms an intensive quantity associated with the asymmetry,
which is the asymmetry chemical potential:
\begin{equation}
\mu_A = \frac{\partial \, E}{\partial \, (N-Z)} = \frac{1}{2} \left( \mu_n - \mu_p \right)
\, .
\end{equation}
To lowest order, under charge symmetry, the tension needs to be quadratic in $\mu_A$,
\begin{equation}
\label{eq:sigma}
\sigma = \sigma_0 - \gamma \, \mu_A^2 \, .
\end{equation}
If the tension depends on asymmetry, so must surface energy. On examining the
function $\Phi=\mu_A(N-Z)-E$, with the derivative $\partial \Phi/\partial \mu_A=N-Z$, the last
dependence is seen to produce the following
apparent paradox: some of the nuclear asymmetry $N-Z$ must be associated with the surface and not
the interior. To answer the question on how particles can be attributed to the surface, one needs
to adapt a systematic approach the separation of quantities into volume and surface contributions.
Gibbs \cite{gib48} proposed to consider two copies of the system, actual
and an idealized reference copy where the interior densities of different quantities extend up to the
surface position, cf.\ Fig.~\ref{fig:gibbs}.
\begin{figure}
\centerline{\includegraphics[width=.52\linewidth]{gibs}}
\caption{\label{fig:gibbs}
Gibbs' \cite{gib48} construction for defining volume and surface contributions.}
\end{figure}
The idealized system represents one with only volume contributions to different quantities, while
the difference between the systems can be associated with the surface, $F_S = F - F_V$. The
separation, however, depends on the position of the surface which must be set utilizing some auxiliary
condition. For nuclei, it is natural to demand a vanishing surface nucleon surface number $A_S=0$, i.e.\
set the surface position at the sharp-edged sphere radius~$R$. Since nuclei, though,
are binary systems, the surface positions that might be separately attributed
to neutrons and protons will be,
generally, displaced relative to each other, cf. \ Fig.~\ref{fig:gibbs2}.
\begin{figure}
\centerline{\includegraphics[width=.52\linewidth]{gibs2}}
\caption{\label{fig:gibbs2}
In a two-component system, the surfaces for the two components will be, generally, displaced
relative to each other.}
\end{figure}
In consequence, in spite of a vanishing surface nucleon number, $A_S = N_S + Z_S = 0$, we may have a
finite surface asymmetry $N_S-Z_S \ne 0$.
Having resolved the apparent paradox, from (\ref{eq:sigma}) we find
\begin{equation}\label{eq:ES}
E_S = \sigma_0 \, {\cal S} + \gamma \, \mu_A^2 \, {\cal S} =
E_S^0 + \frac{1}{4 \gamma} \, \frac{(N_S - Z_S)^2}{\cal S}
= E_S^0 + a_A^S \, \frac{(N_S - Z_S)^2}{A^{2/3}} \, .
\end{equation}
Here, the index 0 refers to symmetric matter and we have introduced a surface symmetry
coefficient $a_A^S$, with dimension of energy. For the volume energy, within the standard formula,
we have
\begin{equation}\label{eq:EV}
E_V = E_V^0 + a_A^V \, \frac{(N_V - Z_V)^2}{A} \, ,
\end{equation}
where $a_A^V$ is the volume symmetry energy coefficient. The Coulomb energy is temporarily
ignored. The net energy and asymmetry are, respectively, $E=E_S+E_V$ and $N-Z=N_S-Z_S+N_V-Z_V$. In the ground
state, the asymmetry should partition itself into the surface and volume contributions, in such a way as to
minimize the energy. The result of the energy minimization can be, actually, written right away once one
notices that the surface and volume energies, quadratic in asymmetry, are analogous to the energies of
capacitors quadratic in the charge. The energy of the coupled capacitors is quadratic in the net charge, with the square
divided by the net capacitance, yielding:
\begin{equation}\label{eq:Enet}
E=E^0 + \frac{q^2}{2C}=E^0 + \frac{(N-Z)^2} {\frac{A}{a_A^V} + \frac{A^{2/3}}{a_A^S}} \, .
\end{equation}
Adding now the Coulomb term, we arrive at the modified energy formula
\begin{equation}\label{eq:BWmod}
E = - a_V \, A + a_S \, A^{2/3}
+ a_C \, \frac{Z^2}{A^{1/3}} + \frac{a_A^V}{1+ A^{-1/3} \, {a_A^V}/{a_A^S}
} \, \frac{(N
- Z)^2}{A} \, .
\end{equation}
Compared to the standard formula (\ref{eq:BW}), the symmetry coefficient becomes now mass dependent,
$a_A(A)= a_A^V/(1+ a_A^V/(a_A^S \, A^{1/3}))$. The standard formula is recovered when ${a_A^V}/{a_A^S}=0$,
i.e.\ the surface does not accept asymmetry excess, or in the limit of $A \rightarrow \infty$. Within
the modified formula, the symmetry coefficient weakens at low $A$ \cite{mye74,dan03}. Whether or not the
coefficient may be replaced by the constant $a_A^V$ depends on the ratio $a_A^V/a_A^S$.
That ratio can be determined from the ratio of surface to volume asymmetry partitioning for
the energy minimum in proportion to the capacitances:
\begin{equation}
\frac{N_S-Z_S}{N_V-Z_V}= \frac{C_S}{C_V}
= \frac{A^{2/3}/a_A^S}{A/a_A^V} = A^{-1/3} \, \frac{a_A^V}{a_A^S} \, .
\label{eq:NZS}
\end{equation}
\section{Asymmetry Skins}
Establishing the relatively small differences in the distribution of neutrons and protons in nuclei has been,
generally, difficult experimentally.
Probes with different sensitivities to protons and neutrons had been utilized, such as
electrons and protons, negative and positive pions, or protons and neutrons, with different associated
systematic errors. The results have not been expressed
in terms of the surface excess, but rather in terms of the
difference in the r.m.s.\ radii between neutrons and protons. The conversion from the excess (\ref{eq:NZS})
to the difference of radii is relatively straightforward \cite{dan03}, if the surface diffuseness is similar
for neutrons and protons. Another issue to consider theoretically is that,
for heavy nuclei, Coulomb forces compete with
symmetry-energy effects, pushing the proton radius out against neutrons and polarizing the nuclear interior.
That competition is easily taken into account by minimizing the sum of three energies with respect to the
asymmetry:
\begin{equation}\label{eq:E3sum}
E=E_V+E_S+E_C \, ,
\end{equation}
where
\begin{equation}\label{eq:Ec=}
E_C= \frac{e^2}{4 \pi \epsilon_0} \, \frac{1}{R} \left( \frac{3}{5}
\, Z_V^2 + Z_V \, Z_S + \frac{1}{2}\, Z_S^2\right) \, .
\end{equation}
From the energy minimization, an analytic formula for the difference of the radii follows,
\begin{eqnarray}
\frac{\langle r^2 \rangle_n^{1/2} -
\langle r^2 \rangle_p^{1/2}}
{\langle r^2 \rangle^{1/2}}
&
=
&
\frac{A}{6NZ} \, \frac{N-Z}{1 + A^{1/3} \, { {a_A^S}/{a_A^V}}
}
\nonumber \\ &&
- \frac{a_C}{168 { a_A^V}} \, \frac{A^{5/3}}{N} \,
\frac{\frac{10}{
3} + A^{1/3}\, { {a_A^S}/{a_A^V}}}{1 + A^{1/3}\, {a_A^S}/{a_A^V}}
\, .
\label{eq:rdif}
\end{eqnarray}
The first term on the r.h.s.\ represents the effects of symmetry energy only, from Eq.\ (\ref{eq:NZS}),
while the second term represents the Coulomb correction. It should be mentioned that the impact
of the Coulomb-symmetry energy competition is much weaker onto the net energy than onto the skin size.
Before trying to draw conclusions from data with Eq.~(\ref{eq:rdif}), it may be worthwhile to test
the macroscopic theory against the microscopic. In their nonrelativistic Hartree-Fock and
relativistic Hartree calculations, Typel and Brown \cite{typ01} observed correlations
between the sizes of asymmetry skins in different nuclei, when utilizing different effective
interactions. Those correlations are shown in Fig.~\ref{fig:typel} together with the predictions
of Eq.~(\ref{eq:rdif}) when changing the ratio ${a_A^S}/{a_A^V}$. The accuracy of the macroscopic
theory in reproducing correlations from the microscopic theory appears to be at the level of $0.01$~fm!
\begin{figure}
\centerline{\includegraphics[width=.77\linewidth]{rralx1}}
\caption{\label{fig:typel}
Correlation between the asymmetry skins for $^{208}$Pb and $^{132}$Sn and $^{138}$Ba,
in the nonrelativistic and relativistic mean-field calculations \cite{typ01} (symbols) and
predicted by macroscopic Eq.~(\protect\ref{eq:rdif}) (lines).}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=.64\linewidth]{narnp1}}
\caption{\label{fig:narnp}
Asymmetry skin for Na
isotopes as a function of the mass number, from the data analysis
of Ref.\
\protect\cite{suz95} (symbols) and
from Eq.\ (\protect\ref{eq:rdif}), for the indicated values of
${a_A^S}/{a_A^V}$.}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=.77\linewidth]{rabd23}}
\caption{\label{fig:rab}
Constraints on the symmetry energy parameters in the plane of ${a_A^S}/{a_A^V}$
vs ${a_A^V}$. The sloped horizontal lines represent constraints on ${a_A^S}/{a_A^V}$ from fitting
skin data at an assumed value of ${a_A^V}$. The elliptical contours represent constraints obtained
from the fit linear in $A^{-1/3}$ to the values of $a_A^{-1}(A)$ from IAS.}
\end{figure}
We next turn to the implications of skin data. Figure \ref{fig:narnp} shows a comparison of
the data by Suzuki {\em et al.} \cite{suz95} from Na isotopes to the predictions of
Eq.~(\ref{eq:rdif}) for different values of ${a_A^S}/{a_A^V}$. The comparison suggests a value
of ${a_A^S}/{a_A^V} \sim 3$. Figure \ref{fig:rab} shows, with sloped parallel lines, the constraints
on ${a_A^S}/{a_A^V}$ from fitting a variety of skin data (for references to the experiments see \cite{dan03}).
Without Coulomb effects the constraint lines would have been horizontal; the weak sensitivity of the fits
to $a_A^V$ results from the second term on the r.h.s.\ of (\ref{eq:rdif}). The favored values of the
ratio, ${a_A^S}/{a_A^V} \sim 2.8$, imply that $A^{-1/3} \, {a_A^S}/{a_A^V}$ is never small. Neither can
the $A$-dependent symmetry coefficient be replaced by $a_A^V$, nor even expanded linearly in $A^{-1/3}$.
\section{Isobaric Analogue States}
To find absolute values of the symmetry coefficients, one might try to fit the binding formula to measured
energies. However, this is treacherous as conclusions on details in different isospin-dependent terms,
including Coulomb, Wigner and pairing get interrelated when drawn from a global fit to the energies. In
addition, the correlation between mass number and asymmetry along the line of stability correlates the
conclusions on details in the isospin dependent and isospin independent terms. The conclusions on the
symmetry coefficients change depending on what is done to the other terms in the formula \cite{dan03}.
Optimal for determining the symmetry parameters would be a study of the symmetry term in the binding formula
in isolation from the formula remainder, which might seem impossible. However, one can take advantage
of the extension of charge symmetry of nuclear interactions to charge invariance. Under charge invariance
the symmetry term should be a scalar in isospin space \cite{jan03} and can be, thus, generalized with
\begin{eqnarray}
E_A & = & a_A(A) \, \frac{(N-Z)^2}{A} = 4 \, a_A(A) \, \frac{T_z^2}{A} \nonumber \\
& \rightarrow &
4 \, a_A(A) \, \frac{T^2}{A}= 4 \, a_A(A) \, \frac{T(T+1)}{A} \, ,
\label{eq:EAgeneral}
\end{eqnarray}
where we also happen to absorb most of the Wigner term into the symmetry term.
Under the generalization, the binding formula may be applied to the lowest state of a given isospin $T$ in
a nucleus. When excited, such a state is an isobaric analogue state (IAS) of the ground state of
a neighboring nucleus. In the formula generalization, the pairing contribution depends on the evenness of
$T$.
For an isospin of the same evenness as the ground, the change in the formula in the excitation occurs only in the
symmetry term:
\begin{equation}\label{eq:Eexc}
E_2(T_2)-E_1(T_1) = \frac{4 \, a_A}{A} \big\lbrace T_2(T_2+1) - T_1 (T_1+1) \big\rbrace \, ,
\end{equation}
and the excitation energy can be used to the determine the symmetry energy nucleus by nucleus from
\begin{equation}\label{eq:aAA}
a_A(A) = \frac{A \, \Delta E}{4 \, \Delta T^2} \, .
\end{equation}
In the context of the previous considerations, the question to ask is whether the deduced $A$-dependent
symmetry coefficient weakens for light nuclei and whether the inverse of the coefficient is linear
in $A^{-1/3}$:
\begin{equation}\label{eq:ainv}
a_A^{-1} (A) \stackrel{?}{=} (a_A^V)^{-1} + (a_A^S)^{-1} \, A^{-1/3} \, .
\end{equation}
\begin{figure}
\centerline{\includegraphics[width=.77\linewidth]{delta4}}
\caption{\label{fig:delta}
Inverse of the $A$-dependent symmetry coefficient as a function of $A^{-1/3}$. Circles
represent values extracted with (\protect\ref{eq:Eexc})
from extremal IAS excitation energies in \protect\cite{ant97}.
Circle size is proportional to the factor $\Delta T^2/A$ in the coefficient
determination. The line and squares show results of the fits to the experimental results following either
Eq.~(\protect\ref{eq:ainv}) or Thomas-Fermi theory \protect\cite{dan04}.
}
\end{figure}
Inverse values of symmetry coefficients, extracted according to Eq.~(\ref{eq:Eexc}) from IAS data \cite{ant97},
are shown in Fig.~\ref{fig:delta}. It is seen that the inverse coefficient changes with $A^{-1/3}$
in a roughly linear fashion, although significant shell effects are present. The line across the figure
represents best fit with Eq.~(\ref{eq:ainv}). The 1- and 2-$\sigma$ constraints on the symmetry
coefficients from the fit are further indicated with elliptical contours in Fig.~\ref{fig:rab}.
Combining the constraints from the fits, we conclude that
$30.0 \, \mbox{MeV} \lesssim a_A^V \lesssim 32.5 \, \mbox{MeV}$
and
$2.6 \lesssim a_A^V/a_A^S \lesssim 3.0$.
\section{Consequences and Conclusion}
The emergence of the surface capacitance for asymmetry may be tied to the weakening of the symmetry energy
with density, see e.g.\ \cite{bod58,dan03}.
Due to the weakening, it becomes advantageous for the nucleus to push its asymmetry to the
surface to lower energy. The ratio of the symmetry coefficients, specifically, can be tied to the
shape of the symmetry energy dependence on density as, in the local-density approximation to the symmetry
energy, the ratio is found to be
\begin{equation}\label{eq:aVS}
\frac{a_A^V}{a_A^S} = \frac{3}{r_0} \int dr \, \frac{\rho(r)}{\rho_0}
\, \left[\frac{S(\rho_0)}{S(\rho(r))} - 1 \right] \, .
\end{equation}
Here, the integration is across the nuclear surface and $\rho(r)$ is
the density as a function of position.
For density-independent symmetry energy, $S(\rho)= S(\rho_0) \equiv a_A^V$, the surface does not accept
the asymmetry, ${a_A^V}/{a_A^S}=0$!
Using the correlations between the coefficient ratio, skins and drop of the symmetry energy with $\rho$,
within the relativistic and nonrelativistic calculations by Fuhrnstahl \cite{fur02}, one can arrive at
limits at on the drop, either expressed in terms of the value of symmetry energy at half of the
normal density or in terms of the power of density in parameterization of the symmetry energy \cite{dan04}.
Specifically, one finds
$0.58 \lesssim S(\rho_0/2)/a_A^V \lesssim 0.69$ and $0.54 \lesssim \gamma \lesssim 0.77$ in
$S(\rho) \simeq a_A^V (\rho/\rho_0)^\gamma$. These further imply limits on pressure in neutron
matter at normal density and, with results of Ref.~\cite{lat01}, produce limits on neutron-star
radius of $11.5 \, \mbox{km} \lesssim R \lesssim 13.5 \, \mbox{km}$ for
$1.4 \, M_{\astrosun}$ mass.
To conclude, the requirement of macroscopic consistency brings in the surface symmetry energy into the nuclear
binding formula. The volume and surface symmetry energies combine as energies of coupled
capacitors. The extension of the binding formula implies emergence of the asymmetry skins for nuclei
and weakening of the symmetry term in light nuclei. The systematic of the asymmetry skins restricts
ratio of the symmetry coefficients. The charge invariance allows to study variation of the symmetry coefficient
nucleus by nucleus. Combination of the fits to skins and IAS yields
$30.0 \, \mbox{MeV} \lesssim a_A^V \lesssim 32.5 \, \mbox{MeV}$
and
$2.6 \lesssim a_A^V/a_A^S \lesssim 3.0$. The surface symmetry energy is associated with weakening of the
symmetry energy with density. The $a_A^V/a_A^S$ ratio implies limits on drop characteristics, such as
$0.58 \lesssim S(\rho_0/2)/a_A^V \lesssim 0.69$. Implications for neutron stars follow.
Current direction is to incorporate shell corrections into the IAS analysis.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,388
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\section*{\label{sec:intro}Introduction}
"Logic-In-Memory" (LIM) architectures have recently emerged as an alternative to Von Neumann architectures, where the constant shuttle of data between logic and memory units leads to a critical rise in energy consumption and delay when downscaling. This has led to the search for novel technologies that combine memory and logic functionalities. Recently, magnetic skyrmions have been proposed as the building block of such logic-in-memory technologies. Magnetic skyrmions are local chiral whirling of the magnetization. Their small lateral dimensions, down to the nanometer size and topological stability grant them particle-like properties, which, combined with the possibility to be manipulated via electrical currents, can be exploited to code data and achieve computation at the nanoscale. These textures appear promising for logic-in-memory applications since they intrinsically merge high density non-volatile storage and logic capabilities. However, while a number of memory and logic concepts have been proposed based on skyrmions~\cite{song2021logic,gnoli2021skyrmion,zhang2020stochastic,chauwin2019skyrmion,zhang2019skyrmion,he2017current,xing2016skyrmion,zhang2015magnetic,luo2018reconfigurable,yan2021logic16,mankalale2019skylogic,Fattouhi2021PhysRevApplied}, several major issues have hindered their technological advancement. In the first concepts of the skyrmion racetrack~\cite{fert_skyrmions_2013}, data was encoded through the distance between neighboring skyrmions and the memory operation relies on the synchronous shift of skyrmion trains, where the skyrmion interdistance remains constant. However, the latter can be easily perturbed by thermal activation or small variation in the skyrmion velocities, induced for instance by local changes in the magnetic properties in the material or process variations.
Different pathways have been proposed to solve this issue such as double lanes~\cite{muller_magnetic_2017} or local gate control~\cite{kang2016complementary}. However, these solutions appear difficult to implement. For instance, local gate control requires numerous gate contacts leading to increased periphery area, limited density and increasing complexity. Regarding skyrmion based logic, the fundamental building blocks need to fulfill a set of criteria, including cascading, fan-out, logic level restoration, immunity to noise, mitigation of potential loss of information, and input-to-output isolation. However, concepts proposed so far required complex operations and faced serious limitations. These include skyrmion-charge inter-conversion which limits the advantage of a full skyrmionic signal~\cite{mankalale2019skylogic,xing2016skyrmion,he2017current,yan2021logic16}, large skyrmion circuits requiring complex synchronization of data~\cite{chauwin2019skyrmion}, multiple gate voltages with complex implementation~\cite{zhang2020stochastic,zhang2019skyrmion}, etc. In addition, several proposals do not fulfill the aforementioned basic requirements of logic, namely cascading, input and clock synchronization, etc \dots Thus, there is currently no concept for a full skyrmionic computer, incorporating logic, memory and interconnects using exclusively magnetic signals without the need for intermediate conversion to charge signals.
\begin{figure*}[!htb]
\includegraphics[width=0.9\linewidth]{Fig1_nucleatev8.pdf
\caption{\label{fig:nucleate}(a) Schematic of a skyrmion generator with regions of different anisotropy values. Current can be sent in-between $C_1-C_2$ or $C_2-C_3$. (b) Energy as the skyrmion moves between two bit cells $R_2$ and $R_3$. $E_{\rm 0}$ is the energy of the skyrmions inside the bit cell. The anisotropy fields are indicated in the different regions. (c) Protocol for data encoding: a skyrmion is first nucleated in the low anisotropy region (row 1) by sending current between $C_1-C_2$ (J=15 MA/m$^2$ for 0.3 ns) and then pushed ("fired") to the 80nm cells (row 2) which work as memory bits ($J=35\rm ~MA/cm^{2}$ for $0.55\rm~ns$). A shift operation is carried out by sending current between $C_2-C_3$ (row 3) ($J=15\rm~MA/cm^{2}$ for $0.9\rm~ns$). We show only the top layer magnetization of the SAF as the bottom layer is magnetized symmetrically opposite to the top layer due to strong RKKY coupling. (d) Full 8-bit skyrmion data encoding of an 8-bit binary number ``10001011" using operations given in (c). Note that the data is encoded starting first from the least significant digit.}
\end{figure*}
Besides the aforementioned limitations, moving skyrmions are subject to the skyrmion Hall effect (skHE), namely a motion transverse to the driving force. To address this problem, topological spin textures with vanishing topological charges and thus skHE have been proposed as alternatives of skyrmions~\cite{zhang2016antiferromagnetic,jin2016dynamics,2018_SciRep_KolesnikovSamardakOgnev,sisodia_droplet_PRB_2021,dohi2019formation,legrand2020room,juge_skyrmions_2021,saf_zhang2016magnetic}. Of particular interest are skyrmions in synthetic antiferromagnets (SAF), which have been recently demonstrated to be stable at room temperature~\cite{dohi2019formation,legrand2020room,juge_skyrmions_2021}. These skyrmions are resilient against external stray magnetic fields and possess velocities which are much higher than their ferromagnetic counterparts, making them good candidates for elementary building blocks in storage and logic. In addition to using magnetic configurations with zero topological charge, skHE can also be suppressed with different confinement techniques~\cite{indent3d_pathak,albisetti2016NatNano_nanopatterning,albisetti2017AIPnanopatterning,guang2020creating,juge2021helium,domainwallconfinement}. In particular, we have shown that skyrmion channels can be defined by a local modification of the magnetic properties using light ion irradiation. This can be exploited to guide the skyrmion dynamics and suppress the skyrmion Hall effect~\cite{juge2021helium}. This technique can also be leveraged for a reliable control of the skyrmion position in racetracks as well as guide their motion in complex geometries to achieve logic operations.
In this work, using micromagnetic simulations, we propose a Logic-In-Memory (LIM) device based on SAF skyrmions which leverages skyrmion confinement using local variation of the anisotropy. It combines two main innovations: Firstly, a novel concept of racetrack shift register where the skyrmion position is defined by low anisotropy dots and moved reliably using current induced spin-orbit torques. This provides an elegant solution to the longstanding issue of reliability of data retention and shift operation in racetracks. We develop elementary protocols for basic ``nucleate" and ``shift" operations on a train of skyrmions. Secondly, a compact Full Adder logic gate extendable to n-bit FA by cascading is designed. The designed FA can be re-programmed to perform different logic operations (e.g. NOT, BUFFER, AND, OR, XOR, NXOR, NAND). Our proposal allows a simple and intuitive synchronization scheme which in turn enables us to seamlessly cascade logic design to implement large-scale networks without any additional electronic circuitry. The designed logic architecture can also tolerate deviations in the amplitude or width of the current pulses which may arise from the electrical part of the design.
\section*{\label{sec:nucleation}Skyrmion Nucleation and Confinement}
As shown in our previous work~\cite{juge2021helium}, it is possible to artificially create an energy barrier for a skyrmion using focused He$^+$ ion-irradiation which locally modifies the material properties (anisotropy and Dzyaloshinskii-Moriya interaction (DMI)) of the ferromagnet. Using this feature, we can engineer regions of different anisotropy values specifically tuned for two different tasks, namely nucleation and confinement of skyrmions.
For the micromagnetic simulations, a SAF structure is assumed composed of two Co layers with a thickness of 0.9nm antiferromagnetically coupled by RKKY interaction and separated by a thin spacer [see Appendix.~\ref{sec:methods} for details regarding the micromagnetic model and parameters]. The magnetic parameters for the irradiated and non-irradiated areas are close to the ones of the Pt/Co/MgO stacks measured experimentally in Ref.~\cite{juge2021helium}. The He$^+$ ion irradiation leads to a decrease of the perpendicular magnetic anisotropy as well as the DMI.
In Fig.~\ref{fig:nucleate}(a), we show the schematic of a skyrmion generator device coupled with a memory (bit) cell which can physically confine and hold a skyrmion inside of it. There are three different regions of anisotropy. The larger $\rm150 nm$ square-shaped cell ($R_1$) with a very small effective anisotropy of $B_{\rm K,eff}=5\rm mT$ is used to nucleate the skyrmion. The anisotropy of this region is intentionally kept smaller to reduce the current density required to nucleate the skyrmion. The second region is composed of the $\rm80 nm$ bit cells ($R_2$ and $R_3$) with $B_{\rm K,eff}=100\rm mT$, which are surrounded by high anisotropy regions of $B_{\rm K,eff}=165\rm mT$. These act as a barrier for the skyrmion, effectively trapping the skyrmion inside the bit cells. In Fig.~\ref{fig:nucleate}(b), we show the energy of the skyrmion as a function of its position $x$ as it passes from region $R_2$ to $R_3$. The energy is highest in the middle of regions $R_2$ and $R_3$. The energy barrier is $\sim33k_BT$ providing reasonable stability to the device against thermal noise. More information on the calculation and optimization of the energy barrier is given in Appendix.~\ref{sec:barrier}.
To nucleate and shift the skyrmions, three separate contacts $C_1$, $C_2$ and $C_3$ are used (Fig.~\ref{fig:nucleate}(a)).
To nucleate a skyrmion in $R_1$ (first panel of Fig.~\ref{fig:nucleate}(c)), a short ($0.3\rm~ns$) current pulse ($J=15\rm ~MA/cm^{2}$) is first injected in between contacts $C_1-C_2$. A second current pulse with a higher current density ($J=35\rm ~MA/cm^{2}$ for $0.55\rm~ns$) allows the skyrmion to overcome the barrier and move from $R_1$ towards $R_2$. Simultaneously, due to the high current density, another skyrmion nucleates in the region $R_1$. This whole operation can thus be dubbed as ``fire+nucleate" as one skyrmion is nucleated and another skyrmion is simultaneously fired (panel 2 of Fig.~\ref{fig:nucleate}(c)). Once nucleated, the shift operation is performed by injecting a current between contacts $C_2$ and $C_3$ such that the skyrmions move between $R_2-R_3$ without influencing the nucleation region $R_1$ (third row panel in Fig.~\ref{fig:nucleate}(c)). Overall, to encode a full data stream, the following protocol is adopted: 1.) Nucleate in region $R_1$, 2.) If Bit 1 is required, use ``fire+nucleate" followed by ``shift" 3.) If Bit 0 is required, use only ``shift". We show in Fig.~\ref{fig:nucleate}(d) and in video SV1 the encoding of an 8-bit binary number ``10001011" in a racetrack composed of 9 cells using this protocol. After nucleation, the skyrmion train shifts synchronously along the cell track when injecting the current pulses. Thus, our proposed racetrack design based on confinement cells and current induced tunneling provides large bit stability and natural synchronization of the bit train, solving an important issue of initial skyrmion racetrack design.
\begin{figure*}[!htb]
\includegraphics[width=0.7\linewidth]{Fig2_onebit_figure_v4.pdf
\caption{\label{fig:1bit}Design and operation of a 1-bit Full-Adder (FA) based on irradiated SAF. (a) shows the schematic of the FA and the current pulses which are designed to move the skyrmions from their respective bits on the left to the middle region where the operation takes place and then to the output bits on the right. The current pulses $J_1$ and $J_2$ have an amplitude and duration of $J_1=15\rm~MA/cm^{2}$ and $t_1=0.8\rm~ns$ and $J_2=5\rm~MA/cm^{2}$ and $t_2=0.9\rm~ns$. (b)-(f) show the micromagnetic simulation of the 1-bit FA gate for various test cases. Snapshot of the magnetization at different times are overlayed to show the complete trajectories of the skyrmions (black solid curves). (g) Serial FA operation on a stream of $C_{in}$, $A$ and $B$ inputs. }
\end{figure*}
\section*{\label{sec:1BitFA}Full Adder Design}
A Compact full Adder (FA) logic gate can also be designed using anisotropy energy barriers in irradiated SAF stacks. A Full Adder is considered as the basic unit in Arithmetic Logic Unit (ALUs) performing bitwise addition of two binary numbers. It has three inputs, $A$, $B$ and $carry-in~(C_{\rm in})$ and produces two outputs, $Sum$ and $carry-out~(C_{\rm out})$. The logic gate is divided into three parts (see Figure.~\ref{fig:1bit}(a)). The left part of the device is made of three bit cells, which are the inputs $A$, $B$ and $C_{\rm in}$. Similarly, at the right-hand side of the logic gate, three other bit cells hold the two outputs $Sum$ and $C_{\rm out}$ of the FA. The middle region of the design represents the part of the device which performs the logic operation. All three regions and the individual bit cells are separated from each other by anisotropy energy barriers.
We show the operation of the designed Full Adder gate in Figs.~\ref{fig:1bit}(b)-(f) and in video SV2: The skyrmions which are present inside the bit cells inputs $A$, $B$ and $C_{\rm in}$ on the left region are pushed inside the middle region after passing over the energy barrier by sending a current pulse uniformly through the device ($J_1=15\rm~MA/cm^{2}$). When skyrmions are inside the middle region, different possibilities arise depending on the total number of skyrmions inside the middle region. For only one skyrmion [Figs.~\ref{fig:1bit}(b)-(c)], the most stable state for the skyrmion is to stay at the center of the middle region. However, when there are two skyrmions [Figs.~\ref{fig:1bit}(d)-(e)], they repel each other and would stay near the top and bottom edge of the middle region to minimize their mutual interaction. When the total number of skyrmions in the middle region is three [Figs.~\ref{fig:1bit}(f)], the skyrmions will maximize the individual distances between them and will acquire a position at the top, middle and bottom parts.
To achieve the relaxation of the interacting skyrmions in the middle part, two pulses with smaller amplitude $J_2=5\rm~MA.cm^{-2}$ are then injected. The skyrmions are then pushed into the rightmost region by sending a current pulse with a larger amplitude $J_1$. At the output end, the middle cell will only have a skyrmion if the total number of skyrmions on the input side is either one or three. This represents the $Sum$ output of the Full Adder. Similarly, the top and bottom outputs will only have a skyrmion if the total number of input skyrmions is either two or three. This is the same as the $C_{\rm out}$ (carry-out) output of the Full Adder. We thus achieve an entire full adder operation with a single logic gate.
We have specifically designed our FA logic gate in Fig.~\ref{fig:1bit}(a) keeping in mind an easy integration with racetrack storage such as the one in Fig.~\ref{fig:nucleate}(a). Using this architecture, the FA logic gate can be easily extended to perform logic operations with a continuous stream of data to serially compute $Sum$ and $C_{\rm out}$ outputs for each input set of data bits. Such an operation is shown in Fig.~\ref{fig:1bit}(g), where we show serial 1-bit FA calculation on streams of input $A$, $B$ and $C_{\rm in}$ [see also video SV3]. The current pulses, in general, follow the same protocol and can be easily extended to perform $n$ operations, where $n$ is the total number of bits in the bit-stream.\footnote{The only modification needed in the current pulses is the merging of $J_1$ pulse of outgoing operation with the $J_1$ pulse of incoming operation as both of these operations can be carried out simultaneously without affecting the logic operation.}
\begin{figure*}
\includegraphics[width=\linewidth]{Fig3_full_3bit_v10.pdf
\caption{\label{fig:3bit}(a) Design of a 3-bit Full-Adder (FA) by cascading three 1-bit FAs. Additional components have been added to synchronize inputs application and the entire operation (b) shows the micromagnetic simulation for the case of (101+111=1100). The current pulses used to perform the logic operation are exactly the same as the 1-bit FA operation repeated 3-times. (c) and (d) show the design and operation of an FA where the carry-bit can be sent back using current of negative polarity with density $J=-10\rm~MA/cm^{2}$ for $t=3.2\rm~ns$ followed by an off-pulse ($J=0\rm~MA/cm^{2}$) of $1\rm ~ns$. (e) Operation of a 3-bits FA using the modified FA shown in (c). The operation corresponds to the case (011+111=1010). (f)-(h) shows the comparison of both approaches for 3-bits addition in terms of device area, energy consumption and operation time, respectively.
}
\end{figure*}
\section*{\label{sec:3bitFA}Full Adder Multi-bit operation}
One limitation of the FA gate is that it only operates on 1-bit inputs. For instance, the actual operation on the bitstreams shown in Fig.~\ref{fig:1bit}(g) is a 1-bit FA operation done serially on each arriving set of input bits, since the carry-bit ($C_{\rm out}$) is not taken forward in each successive computation. We show here that $N$-bit FA gates can be easily designed by properly cascading 1-bit FA gates. As an example, we present in Fig.~\ref{fig:3bit}(a) the design of a 3-bit FA gate composed of three cascaded FA gates where the carry bit from each computation is used in the next computation. The bottom $C_{\rm out}$ from the FA1 is connected to the top bit cell of FA2, such that the output $C_{\rm out}$ from FA1 operation is used as one of the inputs of FA2 operation. To ensure that the two other inputs of FA2 reach at the same time as the output $C_{\rm out}$ of FA1, we add some delay gates Sync-In (green boxes) before FA2. These synchronization gates can be cascaded to vary the delay time and provide on-demand synchronization between various segments of logic operations. Similar gates are also added to synchronize the outputs (Sync-Out, black).
In Fig.~\ref{fig:3bit}(b) and video SV4, we show the operation of a 3-bit FA using cascaded 1-bit FAs. The calculation performed is ($a_3~a_2~a_1$) + ($b_3~b_2~b_1$) = ($c_4~c_3~c_2~c_1$), where ($a_3~a_2~a_1$)=(101) and ($b_3~b_2~b_1$)=(111). The expected output is ($c_4~c_3~c_2~c_1$)=(1100) which is correctly obtained by bit cells $c_4$, $c_3$, $c_2$ and $c_1$ shown by dotted yellow boxes.
The design of Fig.~\ref{fig:3bit}(a) is very promising for fast n-bit FA operation and for further modifications using cascading and synchronization. However, a more compact and low power design can be implemented by directly computing the n-bits FA operation using streams of skyrmions in two racetracks and a single FA gate. This would be similar to the operation of FA shown in Fig.~\ref{fig:1bit}(g), however, with bit-streams replaced with $N$-bit numbers. For this purpose, we modify our FA design to include another region of different anisotropy value as shown in Fig.~\ref{fig:3bit}(c). This new region (light blue) which covers the area joining the input bits to middle region and the area to the left of top $C_{\rm out}$ bit has an anisotropy in-between the anisotropy of the regular bit cells and the barrier region. This enables us to send a negative polarity current pulse which is enough to move only the $C_{\rm out}$ bit while keeping the other skyrmions at their respective places. This pulse is shown alongside in Fig.~\ref{fig:3bit}(c) [highlighted part]. In Fig.~\ref{fig:3bit}(d), we show how the $C_{\rm out}$ bit is moved back to the input end by this current pulse of negative polarity. During the entire operation, none of the other skyrmions are affected. Using this, we can now compute the addition operation of two $N$-bit numbers by successively sending them through the top and bottom inputs of the FA (middle input is left empty). The design and operation for a 3-bit FA using this strategy is shown in Fig.~\ref{fig:3bit}(e) and video SV5. It may be noted that we need to add an empty bit between each successive bit as no operation can take place during the movement of $C_{\rm out}$ bit using negative polarity current. The snapshots in Fig.~\ref{fig:3bit}(e) correspond to the initial and final stage of the addition operation ($a_3~a_2~a_1$) + ($b_3~b_2~b_1$) = ($c_4~c_3~c_2~c_1$), where, ($a_3~a_2~a_1$)=(011), ($b_3~b_2~b_1$)=(111) and the expected output is ($c_4~c_3~c_2~c_1$)=(1010) which is correctly computed with our implementation. The inputs and outputs are shown in dotted yellow boxes.
\begin{figure*}[!htb]
\includegraphics[width=0.7\linewidth]{Fig4_nand_v2.pdf
\caption{\label{fig:nand}Different logic gate designs derived from the FA. (a) shows a NOT/COPY gate design. Some input bits have been fixed to either "0" or "1". Similarly, (b) and (c) show XOR/AND and NXOR/OR gates derived from FA. (d) shows a universal NAND gate designed by cascading AND and NOT gates, both of which are modified versions of FA. (e) shows the operation of NAND for both inputs as 1.
}
\end{figure*}
The performance of the two methods for $N$-bit FA computation described above can be compared on the basis of three important characteristic metrics: (i) Device Area, (ii) Energy consumption and (iii) Operation time. In Fig.~\ref{fig:3bit}(f), we show the variation of the device area with the number of bits ($N$). For the cascaded FA shown in Fig.~\ref{fig:3bit}(a), both dimensions in the $x-y$ plane increase linearly with $N$ leading to an `$\mathcal{O}\rm~(N^2)$ variation of device area. However, since only one FA gate is needed for the alternative approach (without cascading) shown in Figs.~\ref{fig:3bit}(c)-(e), the device area remains the same as `$N$' increases. Note that in Fig.~\ref{fig:3bit}(e), the increase in the length along x-axis is not included in the calculation as the extended regions are considered as a part of the racetrack storage rather than logic. In Fig.~\ref{fig:3bit}(g), we show the total energy consumption of the FA logic gate as a function of number of bits. For a 1-bit operation, the energy of the operation is $9\rm~fJ$. In the case of cascaded FA operation, the energy dramatically increases with the number of bits as $\mathcal{O}\rm~(N^3)$ due to linear increase in operation time and quadratic increase in the device area. However, when using the FA design shown in Fig.~\ref{fig:3bit}(c)-(e), the total energy only increases linearly with the number of bits. For a 32-bit operation, the total energy consumption is $\sim0.8~\rm pJ$ which is more than two orders lower than the cascaded FA operation. However, this design suffers from a larger operation time compared to cascaded FA gate since each operation is delayed by $4.2\rm~ns$ to move the carry-out bit back to the input position [Fig.~\ref{fig:3bit}(h)]. Due to the Joule dissipation in the metallic conductors, the proposed device consumes 1-2 orders of magnitude more energy for individual logic operations compared to state-of-art CMOS full adder~\cite{energycomp}. However, owing to its intrinsic logic-in-memory approach, a large gain in energy and delay is expected at the device level, since it minimizes the energy dissipated in interconnects and the stand-by power. Note that the speed of both designs can be further increased by using skyrmions with higher velocity ($>1000\rm m/s$ is expected for SAF~\cite{tomasello2017performance} which is 6 times the maximum velocity of skyrmions in this work).
\section*{\label{sec:derivative}Re-programming FA logic}
Full adder logic gates can be easily modified in order to build a number of basic logic functions, which in turn can be combined to perform complex logic operations.
The output of the full adder gate is given as: \(Sum=C_{\rm in}\oplus (A \oplus B)\) ; \(C_{\rm out}= AB+BC_{\rm in}+AC_{\rm in}\). By fixing one or more of the inputs, different logic gates can be achieved. Note that the inputs of a FA logic gate are all interchangeable, so it does not matter which of the inputs are fixed. Figures~\ref{fig:nand}(a)-(c) show the modified FA gates which can perform COPY, NOT, AND, XOR, OR and XNOR operations.
The universal NAND gate can also be designed by cascading two modified FA gates [Fig.~\ref{fig:nand}(d)]. The first gate is modified by setting one of the inputs to ``0" such that the $C_{\rm out}$ represents the output of an AND gate. The second FA gate is modified to perform an inversion operation by setting two of the inputs to ``0'' and ``1''. The operation of this gate for a test case of $A=1$ and $B=1$ is shown in Fig.~\ref{fig:nand}(e). The output $\overline{A.B}=0$ corresponding to the NAND gate is highlighted. It may be noted that we also obtain AND and XOR outputs along with the NAND output.
\section*{\label{sec:FAtolerance}Electrical Tolerances and failure mechanisms}
\begin{figure}[!htb]
\includegraphics[width=\linewidth]{Fig5_tolerance_figv4.pdf
\caption{\label{fig:tolerance}(a) Current pulses used for FA operation in Fig.~\ref{fig:1bit}. (b) Acceptable change (\%) in different parameter values in the current pulse. (c) Test case with inputs (0,1,1) for FA operation with random anisotropy grains.}
\end{figure}
For practical application of our design, it is important to probe its tolerance against variations in the current amplitude or pulse width. In Fig.~\ref{fig:tolerance}(a), we show the set of pulses used for the 1-bit Full Adder operation. There are two different current density values used in the circuit : $J_1=15\rm~MA/ cm^{2}$ and $J_2=5\rm~MA /cm^{2}$ with pulse widths of $t_1=0.8\rm~ns$ and $t_2=0.9\rm~ns$, respectively. There is also an off-pulse of width $t_0=0.5\rm~ns$. We individually vary each of these parameters by fixing other parameters at their reference values. We also try to vary $J_1$, $J_2$ or $t_0$, $t_1$ and $t_2$, simultaneously. The results are shown in Fig.~\ref{fig:tolerance}(b) depicting the acceptable level of variation for each case for which the logic operations are executed without any errors. Note that these results are also valid for the NAND as well as other derived logic gates. These tolerance values provide valuable input for further modeling with large-scale electrical circuit designing tools.
We also performed simulations to check the robustness of device against variations in anisotropy. We varied the effective anisotropy field of the barrier region and found that the Full Adder works correctly for the range of values $\rm 155mT-167mT$. Further, we also perform simulations by adding grains of size $\rm 15nm$ with random $0.5\%$ variation of anisotropy constant (equivalent to $\sim5\%-8\%$ change in effective anisotropy field), similar to previous works~\cite{juge2018magnetic,gross_skyrmion_2018} and show that the Full Adder works correctly. The motion of skyrmions for a test case with inputs (0,1,1) is shown in Fig.~\ref{fig:tolerance}(c). However, we note that a stronger anisotropy variation may lead to incorrect operations. In our device, this happens as the skyrmions are stabilized for the parameter values which are close to in-plane to out-of-plane transition regime. Therefore, a small change in the anisotropy constant rapidly changes the skyrmion energy and hence its trajectory. A possible strategy to mitigate this is to use higher anisotropy thin films as the pristine sample and then irradiating them to obtain the tracks/barriers with the same anisotropy difference as used in this work.
An important part of the logic gate design involves device testing against a possible failure during operation. To investigate this point, we include thermal noise corresponding to $\rm T=350~K$ in our simulations \footnote{As we are already doing simulations with experimental parameters extracted at T=300K, we only include thermal noise corresponding to T=50K to avoid overestimating the effect of temperature}. We simulate 20 instances for each of the possible 8 test cases for the Full Adder (total 160 attempts). Despite high thermal noise, the Full Adder worked correctly for $\sim85\%$ of the total attempts. For the cases in which it failed, three different failure mechanisms were found. The first one is caused by the annihilation of input skyrmions during the operation. We predict that a stronger RKKY coupling could counter the thermal noise and improve the overall stability of the SAF skyrmions. The second failure mechanism is due to the presence of skyrmions at the wrong output after the operation has ended. This happens when the skyrmion diffusion due to thermal noise significantly perturbs the intended trajectory of the skyrmions. Higher Gilbert damping is expected to minimize this effect. However, the trade-off will be an increased energy consumption as higher current densities will be required to move the skyrmions.
The third possibility is a skyrmion getting stuck in between two bit cells. This occurs when the skyrmion finds a local minima in between two memory bits. Our barrier has already been optimized to avoid this situation [see Appendix.~\ref{sec:barrier}], however, further optimization is needed to comply with high-temperature conditions. This will involve increasing the difference of anisotropy between the low and high anisotropy regions which will increase the barrier height.
\section*{\label{sec:future}Conclusion}
To conclude, we proposed a logic-in-memory device based on skyrmion confinement and channeling by anisotropy energy barriers in synthetic antiferromagnets. We first designed a racetrack shift register memory based on skyrmions confined by anisotropy energy barriers as memory bits, allowing reliable data storage with large stability and robust synchronous shift operations on skyrmion train. This design naturally solves the stability and synchronization issues of the initial racetrack concept without introducing additional gates or complex geometries. We then combine our racetrack storage with a newly designed 1-bit Full Adder (FA) gate extendable to $N$-bits FA by cascading. The designed FA is reprogrammable and can also be used to perform AND/OR/NOT/NAND/XOR/NXOR operations. The device is expected to perform well even with some fluctuations in the amplitude/width of the injected current pulses and in presence of thermal noise. The simplicity and compactness of the design combined with the minimal use of electrical circuitry brings the proposed device to the same level as the current technological capabilities of fabrication/manufacturing processes, thus providing an important advance for the development of skyrmion-based logic-in-memory technology.
\begin{acknowledgments}
The authors acknowledge financial support from the French national research agency (ANR) (Grant Nos. ANR-15-CE24-0015-01 and ANR-17-CE24-0045) and the American defense advanced research project agency (DARPA) TEE program (Grant No. MIPR HR0011831554). This work has been partially supported by MIAI@Grenoble Alpes, (ANR-19-P3IA-0003).
\end{acknowledgments}
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{"url":"https:\/\/physics.stackexchange.com\/questions\/470305\/why-do-people-not-have-gravitational-attraction","text":"Why do people not have gravitational attraction? [closed]\n\nGravitational theory says every thing that has mass attract each other. So why don't people attract each other and overlap\n\nclosed as unclear what you're asking by Cosmas Zachos, SRS, Kyle Kanos, Jon Custer, GiorgioPApr 5 at 22:19\n\nPlease clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it\u2019s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.\n\n\u2022 You should notice that in order to have overlapping, you don't even need to have an attractive force. People can just run into each other and overlap. This doesn't happen due to the fact that matter is made out of Fermions which have an anti-symmetric many-particles state. Or, in other words, they follow the Pauli exclusion principle. See: en.wikipedia.org\/wiki\/Pauli_exclusion_principle \u2013\u00a0Dvij Mankad Apr 3 at 14:25\n\nThey do attract. Just with very little force.\n\nThe entire planet Earth pulls in you with only some $$\\sim 800 \\;\\mathrm N$$ of gravitational force. So, you can imagine how small the pull is from the tiny mass of another person. You can calculate it with the formula:\n\n$$F_g=G\\frac{m_1m_2}{r^2}$$\n\nwhere $$G=6.67 \u00d7 10^{-11} \\mathrm{\\frac{m^3}{s^2kg}}$$. $$r$$ is the distance and the $$m$$s are the masses of you and the person. Just stand one meter apart and plug in the masses in this formula. You might get a force in the order of one-billionth of a Newton. You can blow at people with a larger force than that, so it is simply not detectable in practical circumstances. It will have to be measured with very delicate methods and tools; for instance have a look at the Cavendish Experiment which might even be possible to perform in your own bedroom.\n\n\u2022 +1: The OP asked as to why they don't attract and overlap. I think you might want to say \"They do (attract)\" to avoid implausible but possible confusion. \u2013\u00a0Dvij Mankad Apr 3 at 14:28\n\u2022 Thanks @DvijMankad, good point, added. \u2013\u00a0Steeven Apr 3 at 14:32\n\u2022 Re, \"very little force,\" Yeah, very little force between two people, but gravitational interactions are mutual. Your body attracts the whole Earth with exactly the same force as the Earth's attraction to you. \u2013\u00a0Solomon Slow Apr 3 at 16:36\n\u2022 @Steeven Since the force is insignificant, relative acceleration is also. And there is also friction which is stronger between our feel and the ground. \u2013\u00a0SRS Apr 4 at 9:00\n\u2022 @SolomonSlow Indeed \u2013\u00a0Steeven Apr 4 at 9:02\n\nFirst, gravity is too weak, we don't have significant gravitational fields enough to pull other people towards us... although there is a very small gravitational tug between two people. The reason why people do not overlap, or even other objects merging into each other, is due to electrostatic repulsion.","date":"2019-08-21 03:04:09","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 5, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5638259649276733, \"perplexity\": 674.9290824572327}, \"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-2019-35\/segments\/1566027315750.62\/warc\/CC-MAIN-20190821022901-20190821044901-00399.warc.gz\"}"}
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Sphecodes fukuiensis is een vliesvleugelig insect uit de familie Halictidae. De wetenschappelijke naam van de soort is voor het eerst geldig gepubliceerd in 1983 door Tsuneki.
fukuiensis
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Q: How to show an objects traits when it is called from another piece of code as an instance I am just starting out with TraitsUI, and I am beginner Python programmer. I hope this question is not too low-level.
I want to display a button set up in ControlPanel, calling it from Main. When i do the following, i just a window with a button that says "Panel". If I click that button, i get another window with the "Start" button I want. How do i just get the window with the Start button?
Thanks,
Cosmo
Main:
from enthought.traits.api import *
from enthought.traits.ui.api import *
class ControlPanel(HasTraits):
""" This object is the core of the traitsUI interface. It hosts the method for
interaction between the objects and the GUI.
"""
start = Button("Start Measurements")
view = View(Item('start', show_label=False, style='custom' ))
class MainWindow(HasTraits):
""" The main window, here go the instructions to create and destroy the application. """
panel = Instance(ControlPanel)
def _panel_default(self):
return ControlPanel()
view = View(Item('panel'))
if __name__ == '__main__':
MainWindow().configure_traits()
A: In MainWindow, change this:
view = View(Item('panel'))
to this:
view = View(Item('panel', style='custom'))
See the InstanceEditor() documentation for more information. The relevant part of that document is the paragraph below the screenshot. The simple style for an InstanceEditor (which is the default style) creates a button that when clicked opens up a new window containing a view of the instance. The custom style embeds the view of the instance in the same window that contains the item. The screenshot in Fig. 36 shows an example. The top of the screenshot shows the simple style, and below that is the custom style. (And below that are the text and readonly styles, but they are not very useful except perhaps for debugging.)
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MOHOMBI to release new single 'In Your Head' on March 5th - Choose which mix to listen to here!
Having already worked with some of the biggest names in music and the first act to be signed to Lady Gaga producer RedOne, Mohombi is set to launch his assault on the UK with new single 'In Your Head' released March 5th through Island Records.
The catchy feel-good anthem showcases Mohombi's distinctive and powerful vocals, rhythmic sound and a smashing chorus that you'll be singing 'In your Head' for days.
With an irresistible party sound he calls 'Afro-Viking', the 25-year-old half-Swedish/half-African singer songwriter/producer – who has collaborated with the likes of Nelly, Akon and Nicole Scherzinger - certainly sounds like no-one else.
While many a breathless, newly anointed pop idol will gasp about their 'journey' - when in reality it entailed little more than a trip from home to the audition - Mohombi's tale of escaping civil war in Africa to where he is today, working with the hottest producer on the planet, really sets him apart.
Mohombi is the first signing to the new label of uber-producer RedOne, a name which has become synonymous with quality pop worldwide since he penned hits like Pokerface, Bad Romance and Just Dance with Lady Gaga which led him to work with everyone from Michael Jackson and Usher to Akon and Jennifer Lopez.
Mohombi has already built a devoted following thanks to his dancehall sound and infectious dance routines - His most recent videos have amounted a jaw-dropping 100 million You Tube views. Mohombi has also had multi-platinum selling singles in several European countries including France, Spain, Holland, Belgium, Switzerland and Sweden, and Top Ten hits in Canada and Japan....not bad for a 25 year old.
Choose which video to watch below:
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Many North Americans are surprised to learn that Muslims have a long history on their continent. Historians estimate that between 10 and 20 percent of the slaves who came from West Africa were Muslim. Thomas Jefferson, to take a noted figure in American history, purchased a translation of the Qurʾan in 1765, more than a decade before he drafted the Declaration of Independence. The first Muslim immigrants to North America other than slaves were from the Ottoman Empire in the late 19th century and the first half of the 20th century. Many were itinerants who came to make money and then return to their countries of origin. Some, however, were farmers and settled permanently. Mosques sprang up in 1915 (Maine), 1919 (Connecticut), 1928 (New York), and 1937 (North Dakota). In the late 19th century, the first Muslims came to Canada as Arab merchants, who often landed in the east but wandered west to the frontier selling goods to remote farms, and to the north selling to fur traders. This early population was small, with the first Canadian census of 1871 listing thirteen Muslims. The first established Muslim settlement was in Lac Labiche in northern Alberta. The descendants of those settlers helped build the first Canadian mosque, the Al-Rashid Mosque in Edmonton in 1938. The immigration policies of Canada in the 1970s meant that many of the Muslim immigrants were professionals or well-qualified business people. They often did well in their new country. Most of these Muslims emigrated either from South Asia or from the Arab world. In addition, however, there are Canadian Muslims whose ethnic backgrounds reflect immigration from almost every part of the world, from Bosnia to Indonesia. The 2011 National Household Survey counted over 1 million Muslims in Canada, meaning that Islam had become the second-largest religious tradition in Canada—well behind Christianity but ahead of Judaism. In the last half-century, the Muslim population of the United States increased dramatically through immigration (especially following the Immigration and Nationality Act of 1965), strong birth rates, and conversion. The US census does not ask the question of religious affiliation, so there is less certainty about the size of the US Muslim population. Some estimates are as low as 2 million people and as high as 10 million. Research into America's immigration patterns, birth rates, and conversion rates—similar to those of Canada—indicates that both of these estimates are extreme. Instead, many researchers estimate that there are between 6 and 7 million American Muslims as of the second decade of the 21st century.
Introductory work on Islam in North America includes Haddad 1991, Haddad and Smith 1994, and Waugh, et al. 1983. Curtis 2008 is a collection of primary source documents, and Alan Godlas has an excellent website called Islam, the Modern World, and the West: Contemporary Topics.
Cesari, Jocelyne, ed. Encyclopedia of Islam in the United States. 2 vols. Westport, CT: Greenwood, 2007.
This is a two-volume encyclopedia that is a good introductory reference work.
Curtis, Edward E., IV, ed. The Columbia Sourcebook of Muslims in the United States. New York: Columbia University Press, 2008.
Edward Curtis has compiled a collection of primary source documents written by American Muslims.
Godlas, Alan, comp. Islam, the Modern World, and the West: Contemporary Topics.
Alan Godlas, who teaches about Islam at the University of Georgia, has the best website for the academic study of Islam. This section of his website provides information about North American Muslims as well as links to North American Muslim groups.
Haddad, Yvonne Yazbeck, ed. The Muslims of America. New York: Oxford University Press, 1991.
Yvonne Haddad has done the most scholarly work on Muslims in the United States. All who work in this area are indebted to her. This is a good collection of essays about Muslims in the United States.
Haddad, Yvonne Yazbeck, and Jane Idleman Smith, eds. Muslim Communities in North America. Albany: State University of New York Press, 1994.
Yvonne Haddad and Jane Smith have edited this volume that updates the earlier work of Waugh, et al. 1983.
Hussain, Amir. Oil and Water: Two Faiths, One God. Kelowna, BC: Wood Lake/Copper House, 2006.
This is a basic introduction to Islam with a focus on North America. It helps introduce Islam to a North American Christian audience. It has also been adopted as a textbook for several courses on Islam and Islam in North America.
Kepel, Gilles. Allah in the West: Islamic Movements in America and Europe. Translated by Susan Milner. Stanford, CA: Stanford University Press, 1997.
A noted French scholar of Islam writes an introduction to Islam in the United States with comparisons to Europe.
Smith, Jane Idleman. Islam in America. New York: Columbia University Press, 1999.
An introduction to Islam in the United States. This could be used as a textbook for a course on Islam in America.
Waugh, Earle H., Baha Abu-Laban, and Regula Qureshi, eds. The Muslim Community in North America. Edmonton: University of Alberta Press, 1983.
One of the best early edited collections on Muslims in North America. The book is invaluable for information on early Muslim communities.
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| 4,568
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{"url":"http:\/\/mathhelpforum.com\/advanced-math-topics\/4141-question-print.html","text":"# It is a Question!\n\n\u2022 July 14th 2006, 02:53 PM\nbabygirl\nIt is a Question!\nUnder what conditions would kinetic energy not be conserved? Explain each response and back up your response using principles of physics.\n\u2022 July 14th 2006, 09:30 PM\ntopsquark\nQuote:\n\nOriginally Posted by babygirl\nUnder what conditions would kinetic energy not be conserved? Explain each response and back up your response using principles of physics.\n\nConsider the Work-Energy Theorem. $W = \\Delta KE$ For KE to be conserved, W must be zero. Under what conditions would W not be equal to zero?\n\n-Dan","date":"2016-08-27 19:47:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 1, \"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\": 0.5523661971092224, \"perplexity\": 2432.784698045317}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-36\/segments\/1471982924728.51\/warc\/CC-MAIN-20160823200844-00268-ip-10-153-172-175.ec2.internal.warc.gz\"}"}
| null | null |
#include <aws/location/model/SearchPlaceIndexForSuggestionsResult.h>
#include <aws/core/utils/json/JsonSerializer.h>
#include <aws/core/AmazonWebServiceResult.h>
#include <aws/core/utils/StringUtils.h>
#include <aws/core/utils/UnreferencedParam.h>
#include <utility>
using namespace Aws::LocationService::Model;
using namespace Aws::Utils::Json;
using namespace Aws::Utils;
using namespace Aws;
SearchPlaceIndexForSuggestionsResult::SearchPlaceIndexForSuggestionsResult()
{
}
SearchPlaceIndexForSuggestionsResult::SearchPlaceIndexForSuggestionsResult(const Aws::AmazonWebServiceResult<JsonValue>& result)
{
*this = result;
}
SearchPlaceIndexForSuggestionsResult& SearchPlaceIndexForSuggestionsResult::operator =(const Aws::AmazonWebServiceResult<JsonValue>& result)
{
JsonView jsonValue = result.GetPayload().View();
if(jsonValue.ValueExists("Results"))
{
Array<JsonView> resultsJsonList = jsonValue.GetArray("Results");
for(unsigned resultsIndex = 0; resultsIndex < resultsJsonList.GetLength(); ++resultsIndex)
{
m_results.push_back(resultsJsonList[resultsIndex].AsObject());
}
}
if(jsonValue.ValueExists("Summary"))
{
m_summary = jsonValue.GetObject("Summary");
}
return *this;
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,688
|
Q: SQS batching for Lambda trigger doesn't work as expected I have 2 Lambda Functions and an SQS queue inbetween.
The first Lambda sends the messages to the Queue.
Then second Lambda has a trigger for this Queue with a batch size of 250 and a batch window of 65 seconds.
I expect the second Lambda to be triggered in batches of 250 messages after about every 65 seconds. In the second Lambda I'm calling a 3rd party API that is limited to 250 API calls per minute (I get 250 tokens per minute).
I tested this setup with for 32.000 messages being added to the queue and the second Lambda didn't pick up the messages in batches as expected. At first it got executed for 15k messages and then there were not enough tokens so it did not process those messages.
The 3rd party API is based on a token bucket with a fill rate of 250 per minute and a maximum capacity of 15.000. It managed to process the first 15.000 messages due to the bucket capacity and then didn't have enough capacity to handle the rest.
I don't understand what went wrong.
A: The misunderstanding is probably related to how Lambda handles scaling.
Whenever there are more events than a single Lambda execution context/instance can handle, Lambda just creates more execution contexts/instances to process these events.
What probably happened is that Lambda saw there are a bunch of messages in the queue and it tries to work on these as fast as possible. It created a Lambda instance to handle the first event and then talked to SQS and asked for more work. When it got the next batch of messages, the first instance was still busy, so it scaled out and created a second one that worked on the second batch in parallel, etc. etc.
That's how you ended up going through your token budget in a few minutes.
You can limit how many functions Lambda is allowed to execute in parallel by using reserved concurrency - here are the docs for reference. If you set the reserved concurrency to 1, there will be no parallelization and only one Lambda is allowed to work on the messages.
This however opens you up to another issue. If that single Lambda takes less than 60 seconds to process the messages, Lambda will call it again with another batch ASAP and you might go over your budget again.
At this point a relatively simple approach would be to make sure that your lambda function always takes about 60 seconds by adding a sleep for the remaining time at the end.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,677
|
Whenever Kessig Forgemaster blocks or becomes blocked by a creature, Kessig Forgemaster deals 1 damage to that creature.
At the beginning of each upkeep, if no spells were cast last turn, transform Kessig Forgemaster.
Whenever Flameheart Werewolf blocks or becomes blocked by a creature, Flameheart Werewolf deals 2 damage to that creature.
At the beginning of each upkeep, if a player cast two or more spells last turn, transform Flameheart Werewolf.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,347
|
\section{Conclusion and Future Work}
\label{sec:conclusion}
In this paper, we conduct empirical studies to
understand the prediction variation estimated by
ensembles under various randomness control settings.
Our experiments on two public datasets
({MovieLens}\ and {Criteo}) demonstrate that
with more variation sources, ensemble tends to
produce more accurate point estimates
with higher prediction variations.
More importantly, we demonstrate strong predictive power of neuron activation strength to infer ensemble prediction variations, which provides an efficient way to estimate prediction variation without the need to run inference multiple times as in ensemble methods.
In the future, we are interested in exploring the proposed activation
strength based methods in various applications, such as
model-based reinforcement learning, and curriculum learning.
In addition, we plan to make two further improvements to our
activation strength based approach.
First, we would like to study additional neuron activation patterns,
such as adjacent neuron paths, to further
improve the variation estimation model.
Second, currently we re-train the variation
estimation model for each new target task.
Activation pattern is a universal and general feature,
and we hope to find
a universal model for prediction variation estimation
without re-training each individual target task.
\subsection{Model Ensemble Sizes}
\label{sec:ensemble_size}
Our prediction variation is estimated using the ensemble method, and
we are interested in finding out how many models to ensemble to estimate prediction variation accurately. In this section, we
conduct empirical experiments on {MovieLens-R},
and {Criteo}\ to answer this question.
For each target task, using the same training data and model configuration,
we first train 1000 models as the ensemble universe $M_{gt}$
to obtain ground-truth prediction variations.
We used the R3 randomness setting here, as it is the most commonly
used setting in practice.
Given an example $x$, we obtain its
prediction variation from the 1000 ensemble models as $PV_{gt}(x)$ .
We calculate the mean prediction variation for all the examples
as $\bar{PV}_{gt}$.
Then we evaluate the prediction variation difference between an ensemble $M_N$ of a smaller size $N$ and
$M_{gt}$.
We use {\em delta ratio} to quantify the difference
between the prediction variation estimated from the
two ensembles $M_N$ and $M_{gt}$ as follows.
\begin{definition} (Delta Ratio)
Let prediction variation delta
$\delta_{M_N}(x)$ be the absolute difference of the estimated prediction variation
between a model ensemble $M_N$ of size $N$ and
the ground-truth model ensemble $M_{gt}$, as
$\delta_{M_N}(x) = |PV_{M_N}(x) - PV_{gt}(x)|$.
We obtain the average prediction variation delta for all the examples
in a dataset $D$ as
$\delta_{M_N} =\frac{1}{|D|}\sum_{x \in D}\delta_{M_N}(x)$.
We define delta ratio $dr_{M_N}$
to be the ratio of prediction variation delta $\delta_{M_N}$
to the average prediction variation in the ground-truth ensemble models $\bar{PV}_{gt}$, as
$dr_{M_N} = \delta_{M_N} / \bar{PV}_{gt}$
\end{definition}
In Figure~\ref{fig:ensemble_size},
we show the delta ratio of different ensemble sizes for
the {MovieLens}\ regression task and the {Criteo}\ task.
For each ensemble size $N$, we sample N models without replacement
from the 1000 ground-truth model universe and obtain its delta ratio.
We repeat this sampling process for 20 times, and
obtain the mean and standard deviation of the delta ratio for the given $N$.
We plot the delta ratio as shown in Figure~\ref{fig:ensemble_size}.
We can see the delta ratio decreases when the ensemble size increases.
The delta ratio statistics is similar on both datasets, {MovieLens}\ and {Criteo}.
When the ensemble size is 100, the delta ratio is about 7\% which
indicates 93\% of prediction variation from the ground-truth ensemble
of 1000 models is captured.
As a result, in this paper, we use 100 as the default
ensemble size for all experiments.
\section{Introduction}
Deep neural networks (DNNs) have gained widespread adoption in recent years across many domains.
Despite their impressive performance in various applications,
most DNNs today only generate point predictions.
And it is well known that a set of DNN models trained with the same model
specification and the same data can produce very different
predictions~\cite{lakshminarayanan2017simple, ovadia2019can, wen2020batchens}.
Researchers realize that point predictions do not
tell the whole story and raise questions about whether DNNs
predictions can be trusted~\cite{jiang2018trust, schulam2019can}.
In response, a growing number of researches are looking into quantifying
prediction uncertainties for DNNs.
Ensemble method is a state-of-the-art benchmark for prediction
uncertainty estimation to consolidate agreements
among the ensemble members and produce better point predictions~\cite{breiman1996bagging, dietterich2000ensemble, lakshminarayanan2017simple, ovadia2019can}.
Many researches focus on
whether these point predictions are well-calibrated on either
in-distribution or out-of-distribution (OOD)
data~\cite{ovadia2019can, guo2017calibration, lee2017training,liu2020simple}.
However, there exist disagreements among the predictions
in the ensemble, which we call {\bf \em prediction variation}.
For example, different models in the ensemble often
yield different prediction results even on the same input example.
Although prediction variation contributes to model
uncertainty~\cite{lakshminarayanan2017simple},
there is no comprehensive study to advance the understanding of it.
Ensembles provide us with a good approximation of
prediction variation, but they are computationally expensive as they
require training multiple copies of the same model.
At inference time, they require computing predictions on every
example for every ensemble member, which can be infeasible for real-time large-scale machine learning systems.
Researches propose various resource-saving
techniques \cite{gal2016dropout, huang2017snapshot, wen2020batchens, lu2020uncertainty}.
However, as far as we know, none of these works studies whether we can
infer prediction variation from neuron activation
strength collected from the DNN directly,
without running predictions on the same data multiple times.
Here, we use {\bf \em neuron activation strength} to indicate DNN's neuron output
strength, {e.g.}, the value of the neuron after activation and whether the neuron is activated.
We hypothesize that neuron activation strength could
be directly used to infer prediction variation.
Our intuition is based on neuroscience's
Long-Term Potentiation (LTP) process~\cite{nicoll2017brief}
which states that connections between neurons become stronger
with more frequent activation.
LTP is considered as one of the underlying mechanisms for learning and
memorization.
If we imagine deep networks learn like the brain,
some groups of neurons will be more frequently and/or strongly activated, i.e.
strengthened neurons. During learning,
these strengthened neurons represent where the network has learned or memorized better.
Therefore, we hypothesize that deep neural networks predict more
confidently if an input activates through strengthened neurons,
and less so if the input goes through weaker neurons.
\vspace{0.1cm}
\noindent
{\bf Our Goal} ---
In this paper, we aim to advance the understanding of
prediction variations
estimated by ensemble models, and look into
the predictive power of neuron activation strength on
prediction variations.
To the best of our knowledge, we are the first to
conduct comprehensive studies on prediction variations from different ensemble models,
and we are also the first to demonstrate that we are able to infer
prediction variation from neuron activation strength.
\vspace{0.1cm}
\noindent
{\bf Challenges} --- We face the following challenges:
{\em Variation Quantification} ---
There are a variety of prediction problems. For example,
predicting the target rating for a user on a given movie could be a regression task
or a multi-class
classification task by dividing the movie ratings into multiple buckets;
predicting click-through rate is often modeled as a binary classification task.
There is no standard way
to quantify prediction variation for such a variety of
tasks.
{\em Variation Sources} ---
Training a set of models with the same model specification and
the same data can produce very different results, and
the prediction disagreements are inherently caused by the nonconvex
nature of DNN models in which multiple local minima exist.
Multiple randomness sources could lead to the disagreements, such as
random initialization of DNN parameters, random shuffling of training data,
sub-sampling of training data, optimization algorithms, and even the hardware itself.
It is often hard to identify the contribution of each randomness source
to prediction variation.
{\em Neuron Activation Strength} ---
We hypothesize that a deep network's
prediction variations are strongly correlated with neuron activation
strength throughout training and at inference time.
However, it is not straightforward to demonstrate this relationship for
different prediction problems and randomness sources.
\vspace{0.1cm}
\noindent
{\bf Our Approach} ---
In this paper, we investigate sources for prediction variation, and by
controlling the randomness sources explicitly, we
demonstrate that neuron activation strength has strong
prediction power to infer ensemble prediction variations
in almost all the randomness-controlled settings.
We demonstrate our findings on two popular datasets used to
evaluate recommender systems, MovieLens and Criteo.
First, we quantify prediction variation
across different types of target tasks, including
regression, binary classification, and multi-class classification tasks.
We use standard deviation of the ensemble predictions
to quantify prediction variation for regression
and binary classification tasks, and use
KL divergence based measurement to quantify prediction distribution
disagreement for multi-class classification tasks.
Second, we identify and examine three variation sources of randomness
(data shuffling, weight initialization, and data re-sampling). By
explicitly controlling randomness sources,
we study their contributions to the performance of
point prediction and prediction variation.
Our results show that every variation source
exhibits a non-negligible contribution towards the
total prediction variation.
The prediction variation
may have different sensitivities to different types of
variation sources on different target tasks.
When we include more variation sources,
the ensemble's prediction mean tends to be more accurate, while the
prediction variations are higher.
Finally, we demonstrate that neuron activation strength
has strong prediction power in estimating prediction variation of
ensemble models, while the neuron activation strength information
can be obtained from a single DNN. We obtain the neuron activation strength information
from the neural network. With this strength information, we can add a cheap auxiliary
task to estimate prediction variation directly.
Our experiment results show that
our activation strength based method estimates prediction variation fairly well
as a regression task.
The average $R^2$ on {MovieLens}\ is 0.43 and 0.51 with
different task definitions,
and on {Criteo}\ is 0.78.
Our method is especially good at detecting the lowest and highest variation
bucket examples, on average with 0.92 AUC score for the lowest bucket and 0.89 AUC score
for the highest bucket on both datasets.
Our approach is complementary and orthogonal to
many other resource-saving or single model prediction
variation estimation techniques, as it doesn't alter the target task's
optimization objectives and process.
\vspace{0.1cm}
\noindent
{\bf Applications} ---
Prediction variation quantification is
a fundamental problem and
our activation strength based approach opens up new opportunities
for a lot of interesting applications.
For example,
in model-based reinforcement learning, prediction variation
has to be quantified for exploration~\cite{zhou2019neural,chua2018deep}.
In curriculum learning~\cite{bengio2009curriculum},
prediction variation can be used
as a way for estimating example difficulty.
In medical domain, prediction variation can be used to capture
significant variability in patient-specific predictions~\cite{dusenberry2020analyzing}.
Our proposed activation strength based method provides
a simple and principled way to serve prediction variation
estimate by deploying a cheap auxiliary task,
instead of using an expensive ensemble model, during inference time.
We explore applying our method in
the above scenarios.
\vspace{0.1cm}
\noindent {\bf Contributions} --- Our contributions are four fold:
\begin{itemize}
\item Framework for prediction variation estimation using activation
strength in Section~\ref{sec:preliminary}.
\item Formal quantification on prediction variation for various
target tasks in Section~\ref{sec:variation_definition}.
\item Prediction variation understanding by explicitly controlling
various randomness sources in Section~\ref{sec:prediction_sources}.
\item Empirical experiments to demonstrate strong predictive
power from neuron activation strength to estimate ensemble prediction variation
in Section~\ref{sec:variation_estimation}.
\end{itemize}
We cover the related work in Section~\ref{sec:related_work},
and conclude with a discussion of future work in Section~\ref{sec:conclusion}.
\section{Appendix}
\input{ensemble_size.tex}
\bibliographystyle{ACM-Reference-Format}
\section{Variation Estimation Framework}
\label{sec:preliminary}
Similar to the work on model uncertainty
estimation~\cite{nix1994estimating,nix1995learning,su2018tight},
we build two components for the prediction variation estimation framework:
target task, and variation estimation task,
as shown in Figure~\ref{fig:framework}. Before discussing
the two tasks in detail,
we first introduce the two experiment datasets that
we use throughout this paper.
\subsection{Datasets}
\label{sec:datasets}
Our studies are based on two datasets: {MovieLens}\ and {Criteo}.
\vspace{0.1cm}
\noindent {\bf {MovieLens}} ---
The {MovieLens}~\footnote{\url{http://files.grouplens.org/datasets/movielens/ml-1m-README.txt}} contains 1M movie ratings from 6000 users on 4000
movies. This data also contains
user related features and movie related features.
\vspace{0.1cm}
\noindent {\bf {Criteo}} ---
The Criteo Display Advertising challenge
~\footnote{https://www.kaggle.com/c/criteo-display-ad-challenge}
features a binary classification task to predict click-through rate
(clicked event's label is 1, otherwise 0).
The {Criteo}\ data consists of around 40M examples
with 13 numerical and 26 categorical features.
\begin{figure}[t!]
\includegraphics[width=0.9\linewidth]{pic/framework.pdf}
\caption{Our framework for prediction variation
estimation using activation strength.}
\label{fig:framework}
\end{figure}
\subsection{Target Task}
\label{sec:target_tasks}
The target task is defined by the original
prediction problem,
such as the rating prediction task on {MovieLens},
and the click-through prediction task on {Criteo}.
The target task
takes in the input features from the dataset,
and predicts the target.
In this paper, we focus on the multi-layer perceptron
architecture (MLP),
with ReLU as the activation function for all layers.
Furthermore, we define three target tasks
on {MovieLens}\ and {Criteo}.
\vspace{0.1cm}
\noindent {\bf {MovieLens}\ Regression ({MovieLens}-R)} ---
The target task takes in
user-related features ({i.e.}, id, gender, age, and occupation)
and movie-related features ({i.e.}, id, title and genres), and
predicts movie rating as a regression task.
The movie ratings are integers from 1 to 5.
We use mean squared error (MSE) as the loss function.
MSE is a standard metric for evaluating the
performance of rating prediction in
recommenders~\cite{herlocker2004evaluating,saadati2019movie,bennett2007netflix}.
For example, \cite{bennett2007netflix} used 100 million
anonymous movie ratings
and reported their Root Mean Squared Error (RMSE)
performance on a test dataset as 0.95.
Each model trains for 20 epochs with early-stopping.
We only use the observed ratings in {MovieLens}\ as training data.
Similar to the neural collaborative filtering
framework~\cite{he2017neural},
we use fully connected neural network for the rating
prediction task and ReLU as the activation function.
We set the fully connected neuron layer sizes to be
[50, 20, 10].
We set the user id and item id embedding size to 8~\cite{he2017neural},
the user age embedding size to 3, and user
occupation embedding size to 5.
\vspace{0.1cm}
\noindent {\bf {MovieLens}\ Classification ({MovieLens}-C)} ---
Similar to the {MovieLens}\ regression task, we predict
movie ratings as 5 integer values from 1 to 5 and
model this problem as a multi-class classification
task with softmax cross entropy as the loss function.
We experiment with temperature scaling
values
$T=$[0.1, 0.2, 0.5, 1, 2, 5, 10] with batch size of 1024 to make sure predictions are well
calibrated~\cite{guo2017calibration}. We pick $T=0.2$ which gives the best Brier score\footnote{
\url{https://en.wikipedia.org/wiki/Brier_score}} while
achieving similar accuracy compared to other settings.
\vspace{0.1cm}
\noindent {\bf {Criteo}} --- This target task uses
a set of numerical and categorical features to predict the click-through rate.
The label for the task is either 0 or 1 representing whether an ad is clicked or not.
We model this problem as a binary classification
task with sigmoid cross entropy loss function.
The trained model outputs a float between 0 and 1
representing the predicted click-through probability.
We use the same model setting as described in~\cite{ovadia2019can}, except
for ReLU layer sizes. In the beginning, we set ReLU layer sizes to
[2572, 1454, 1596] as in~\cite{ovadia2019can},
but found that only around 80 neurons are activated at least once on a 10k sample data.
As a result, for the experiments in this paper, we use ReLU layer sizes of [50, 20, 10] and
find the prediction performance is similar to the model with much larger ReLU layer sizes. Each model is trained for 1 epoch.
\subsection{Variation Estimation Task}
In this paper, we focus on using neuron activation strength to estimate
prediction variation for each input example.
We use ensemble to estimate prediction variation as the
ground-truth label.
We define prediction variation
formally in Section~\ref{sec:variation_definition}.
As shown in
Figure~\ref{fig:framework},
we build a neural network model taking the
neuron activation strength features to estimate
prediction variation.
During the inference time, we
directly output the estimated
prediction variation using activation strength
as an auxiliary task.
In our current setup, we collect the activation strength features
from all the neurons in the target task. Also, we find
that it is possible to identify important neurons
in order to reduce the number of activation strength features.
Due to the space limit, we will not discuss feature reduction
in this paper.
The detailed setup of the variation estimation task is discussed in
Section~\ref{sec:variation_estimation}.
\section{Related Work}
\label{sec:related_work}
In machine learning literature, researchers mostly
focus on two distinct types of uncertainties:
aleatoric uncertainty and epistemic uncertainty~\cite{der2009aleatory}.
Aleatoric uncertainty is due to the
stochastic variability inherent in the data generating
process~\cite{liu2019accurate}.
Aleatoric uncertainty corresponds to
{\em data uncertainty}, which describes uncertainty for a given
outcome due to incomplete information~\cite{knight1921risk}.
Epistemic uncertainty is due to our lack of knowledge about the data generating mechanism~\cite{liu2019accurate}, and
corresponds to {\em model uncertainty}, which
can be viewed as uncertainty regarding the
true function underlying the observed
process~\cite{bishop2006pattern}.
In this paper, we focus on studying model uncertainty,
especially prediction variations or disagreements.
There has been extensive research on methodologies for estimating model uncertainty and discussions on their comparisons~\cite{ovadia2019can}. Principled approaches include Bayesian
approaches~\cite{neal2012bayesian,mackay1992practical,hinton1993keeping,louizos2017multiplicative,zhu2018bayesian} and ensemble-based approaches~\cite{lakshminarayanan2017simple}.
Bayesian methods provide a mathematically grounded framework to model uncertainty, through learning the deep neural network as Gaussian processes~\cite{neal2012bayesian}, or learning approximate posterior distributions for all or some weights of the network ~\cite{blundell2015weight,kwon2018uncertainty}. Ensemble methods~\cite{lakshminarayanan2017simple}, on the other hand, is a conceptually simpler way to estimate model uncertainty. There are multiple ways to create ensembles of neural networks: bagging~\cite{breiman1996bagging}, Jackknife~\cite{mcintosh2016jackknife}, random parameter initialization, or random shuffling of training examples. The resulting ensemble of neural networks contains some diversity, and the variation of their predictions can be used as an estimate of model uncertainty.
A lot of research work results are based on the ensemble
method~\cite{berthelot2019mixmatch, de2018clinically, chua2018deep, leibig2017leveraging,ovadia2019can}.
For example,
\cite{de2018clinically} demonstrates promising results
uses deep ensembles for diagnosis and referral in retinal disease.
\cite{chua2018deep} proposes a new algorithm for
model-based reinforcement learning
by incorporating uncertainty via ensemble.
In this paper, we use ensemble as the ground truth to produce prediction variations
in different scenarios ({i.e.}, different randomness settings).
Estimating model uncertainty through Bayesian modeling or ensemble usually incurs significant computation cost. For example, Bayesian neural networks that perform variational learning on the full network~\cite{blundell2015weight} significantly increase the training and serving cost. The cost for ensemble methods scales by the number of models in the ensemble, which can be prohibitive for practical use. To this end, researchers have proposed various techniques to reduce the cost for Bayesian modelling and ensemble methods. For example, single-model approaches are proposed to quantify model uncertainty by modifying the output layer~\cite{tagasovska2019single, liu2020simple}, deriving tractable posteriors from last layer output only~\cite{riquelme2018deep, snoek2015scalable}, or constructing pseudo-ensembles that can be solved and estimated analytically~\cite{lu2020uncertainty}. Our proposed method in this paper can also be viewed as a single-model approach for model uncertainty estimation. However, we do not impose any Bayesian assumptions on the network or any distributional assumptions on the ensembles. Instead, we build an empirical model to learn the association between activation strength and model uncertainty, and use it to estimate model uncertainty for new examples, which offers a relatively simple, robust and computationally efficient way to estimate prediction variation from a single model.
\section{Prediction Variation Sources}
\label{sec:prediction_sources}
In this section, we diagnose the prediction variation sources, and
we are interested in seeing their effects on the total prediction variation by
controlling type of each randomness source.
There are many sources contributing to prediction variation.
Random initialization of DNN parameters contributes randomness to model predictions.
Randomness can also come from training data shuffling
or sub-sampling.
In addition, asynchronous or distributed training could lead to training order randomness.
More surprisingly, we observe the hardware itself contributes to the model prediction
variation: we find that by fixing all the other settings, training a model
on different CPUs might produce different models.
In this paper, we consider three types of randomness sources:
\vspace{0.1cm}
\noindent {\bf 1. Shuffle (S)} --- Whether randomly shuffles
input data, {i.e.}, randomizes input data order.
\vspace{0.1cm}
\noindent {\bf 2. RandInit (R)} ---
Whether randomly initializes model parameters,
including DNN weights and embeddings.
We can fix the initialization by setting
a global random seed in Tensorflow.
\vspace{0.1cm}
\noindent {\bf 3. Jackknife (J)} ---
Whether randomly sampling input data by
applying delete-1
Jackknife~\cite{mcintosh2016jackknife}.
We split data into N Jackknife sub-samples and
each ensemble member randomly leaves one Jackknife
sample out.
We use 100 models for each ensemble as discussed in Appendix~\ref{sec:ensemble_size},
and we split the data into 100 unique Jackknife sub-samples.
Another popular data sampling method is bootstrap~\cite{wichmann2001psychometric}.
For this work, we pick Jackknife due to its simplicity
to implement.
\vspace{0.1cm}
We set up the prediction variation randomness control experiments as follows:
First, if incorporating Jackknife randomness, we obtain the delete-1 Jackknife
sub-samples; otherwise, we use all the training data.
Second, if incorporating Shuffle randomness,
we shuffle the training data; otherwise we do not.
Finally, if incorporating RandInit randomness,
we randomly initialize all the parameters without a fixed global seed
for all the ensemble members; otherwise we use a fixed global seed.\footnote{When RandInit is enabled, we use a fixed set of 100 random seeds.}
We use an ensemble of 100 models for each of the randomness settings:
as discussed in Appendix~\ref{sec:ensemble_size},
93\% of prediction variation in the
ensemble of 1000 models can be captured with size 100 ensemble.
On each dataset of {MovieLens}\ and {Criteo}, we randomly split the data into
training and testing.
On {MovieLens}\ we split the 1M data into 60\% for training,
and 40\% for testing. On {Criteo},
same to~\cite{ovadia2019can}, we use 37M data for training
and 4.4M for testing.
We obtain the prediction variation estimation
on all the testing examples for further analysis.
\vspace{0.1cm}
\noindent {\bf Randomness Source Comparison} ---
Table~\ref{tab:randomness_control} shows the
accuracy and prediction variation statistics
for different combinations of the three randomness sources.
For each type of randomness combination ({e.g.}, (R3) R+S
means using RandInit and Shuffle only),
we train 100 models of the same setting, and
obtain the ensemble mean prediction for accuracy evaluation
and report the prediction variations.
For accuracy evaluation, we report
Mean Squared Error (MSE) and accuracy (ACC) for {MovieLens-R},
ACC for {MovieLens-C}, and AUC score for {Criteo}.
We obtain ACC for {MovieLens-R}\ by rounding the ratings to the closest integers.
For prediction variation metrics,
we report prediction variation (PV) mean and standard deviation,
and PV coefficient (coeff).
We obtain the PV coefficient for each example
$x$ as PV(x) divided by the ensemble mean prediction.
From the tables, we can see that
each type of the randomness sources exhibits a
non-negligible and different contribution towards
the total prediction variations.
As we add more randomness sources, the mean prediction of the ensembles
tends to become more accurate,
and the prediction variations get higher.
On all three target tasks, the R7 setting appears to exhibit
the best or close to best accuracy score, and its prediction variations are also
the highest or close to the highest among all the randomness settings.
It also seems that different target tasks or datasets
are sensitive to different types
of randomness sources.
We observe that the {Criteo}\ data is more sensitive to Shuffle
and Jackknife randomness
while the {MovieLens}\ data is more sensitive to RandInit.
We verify that
by fixing all the randomness sources, we do not observe any prediction variation
in the R0 settings. The PV Mean and PV std is always 0 for R0.
According to PV coefficient,
we notice that {MovieLens}\ shows around 5-6\% of
prediction sway from the mean prediction
while {Criteo}\ has around 20\% of prediction sway.
\vspace{0.1cm}
\noindent {\bf Randomness Source Correlations} ---
Under each randomness setting, we are able to obtain the
prediction variation for each example.
Figure~\ref{fig:randomness_correlation} shows
the Pearson correlation of the prediction variations of all
the examples between each pair of the randomness settings.
The randomness setting can be found at Table~\ref{tab:randomness_control}.
We do not consider R0 because it eliminates all the randomness
in the model and PV is always 0.
As we can see from Figures~\ref{fig:randomness_correlation},
the prediction variation correlation patterns on {MovieLens}\ is quite
different from {Criteo}.
For example, while the lowest Pearson correlation score on {MovieLens}\ is around 0.7,
all the randomness settings on {Criteo}\ are quite correlated with the
lowest Pearson correlation score to be around 0.92.
\vspace{0.1cm}
\noindent {\bf Regression vs Classification} ---
On the {MovieLens}\ dataset, we are able to predict ratings
as regression or classification.
Table~\ref{tab:randomness_control} shows that
the prediction accuracy is higher when we
predict ratings as a classification task than as a regression task
almost for all the randomness settings,
as classification optimizes for the accuracy metric directly.
We also compare the prediction variations obtained
through regression and classification, and
Figure~\ref{fig:correlation_movielens_tasks} shows the Pearson correlation
of prediction variations
for the two tasks on various randomness settings for all the testing
examples. We find that whether the variations are strongly
correlated depends on the randomness settings.
As shown in the figure, the prediction variations
are highly correlated (with Pearson correlation more than 0.8) when
we add the RandInit randomness source, otherwise the two tasks are
less correlated.
we think the reason is that RandInit controls model parameters and the loss function
plays an important role,
while underlying data properties affect Shuffle and Jackknife more.
\section{Prediction Variation Estimation}
\label{sec:variation_estimation}
In this section, we study the problem of using neuron activation strength to
infer prediction variation. We first discuss the prediction variation estimation
task setup in Section~\ref{sec:variation_estimation_setup}, and then show
the experiment results on using neuron activation strength to estimate
prediction variation in Section~\ref{sec:variation_estimation_results}.
\begin{figure}
\centering
\includegraphics[width=.85\linewidth]{pic/movielens_tasks_correlation.pdf}
\vspace{-0.4cm}
\caption{Pearson correlation of prediction variations between {MovieLens-R}\ and {MovieLens-C}.}
\label{fig:correlation_movielens_tasks}
\end{figure}
\subsection{Variation Estimation Task Setup}
\label{sec:variation_estimation_setup}
As shown in Figure~\ref{fig:framework},
the variation estimation task takes in the neuron activation information
collected during the target task inference time.
We use the prediction variation estimated by the ensemble model for training,
and then during inference time, the variation estimation model is able
to infer the prediction variation as a cheap auxiliary task.
We set up the variation estimation task as follow.
\vspace{0.1cm}
\noindent {\bf Ground-truth Labels} ---
We first obtain the prediction variation ground-truth from the ensemble.
On both {MovieLens}\ or {Criteo},
we first split the data into training $D_{t}$
and testing $D_{e}$ for the target task.
On {MovieLens}, we split the 1M data
into 60\% for training and 40\% for testing;
on {Criteo}, we use the same setting as in~\cite{ovadia2019can},
37M data for training and 4.4M for testing.
We train an ensemble of 100 models to
obtain $PV(x)$ for each $x \in D_{e}$.
\vspace{0.1cm}
\noindent {\bf Evaluation Procedure} ---
We further split $D_e$ into 2 sets: 50\% as
$D_{e1} = (x_{e1}, PV(x_{e1}))$ for training the
variation estimation model; and another 50\% as
$D_{e2} = (x_{e2}, PV(x_{e2}))$ for testing.
Given a trained target task model $m_t$,
we build a neural network model $m_v$ to estimate prediction variations. $m_v$ collects the activation
strength information from $m_t$'s neurons
during $m_t$'s inference time.
$m_v$ trains on $D_{e1}$ and tests on $D_{e2}$ with fully connected layers of size $[100, 50]$,
batch size 256, Adam optimizer
with learning rate 0.001, and 150 training epochs with early stopping.
We find that $m_v$ takes less than one epoch to converge on {Criteo},
but takes longer to converge on {MovieLens}\ due to its
much smaller data size.
Now we explain $m_v$'s input features and objective in detail.
\vspace{0.1cm}
\noindent {\bf Input Features} ---
We consider two types of input features
collected from neuron activation strength.
We use ReLU~\cite{nair2010rectified} as the activation function.
We believe our activation strength feature is general and
can be applied to other activation functions, such as
Softplus~\cite{glorot2011deep}, ELU~\cite{clevert2015fast},
GELUs~\cite{hendrycks2016gaussian},
and Swish~\cite{ramachandran2017searching}.
Due to space limitations, we only experiment with ReLU.
{\em Binary} --- On ReLU neurons, we consider whether a neuron is activated as
the input feature. This binary feature represents whether
the neuron output is greater than 0.
{\em Value} --- The raw value of a neuron's output directly
represents the strength of an activated neuron. Therefore, we experiment with normalized activation value as the input feature.
We normalize the neuron outputs according to the neuron output
mean and standard deviation collected from the training data.
\vspace{0.1cm}
\noindent {\bf Objective} ---
We estimate the prediction variation in two ways.
{\em Regression} ---
In this setting, the model directly estimates the prediction variation
as a regression task. We use Mean Squared Error (MSE)
as the loss function. However, by directly optimizing for MSE,
this regression task's output range could be huge. As a result, we limit the
minimum output of the model to be 0 as
prediction variation is always positive,
and limit the maximum output to be $mean + 3*std$ where the mean
and std are estimated on the training data's prediction variations.
$mean + 3*std$ should be able to cover 99.7\% of the data in
a Guassian distribution.\footnote{
\url{https://en.wikipedia.org/wiki/68-95-99.7_rule}}
{\em Classification} ---
In this setting, we divide prediction variation
into multiple buckets according to the percentile,
and then predict which variation bucket it belongs to.
We set the bucket number to be 5, and use cross entropy as
the loss function for the prediction variation classification model.
\begin{table}[t!]
\begin{tabular}{l|cc|cc|cc}
\toprule
& \multicolumn{2}{c|}{MovieLens-R} & \multicolumn{2}{c|}{MovieLens-C} & \multicolumn{2}{c}{Criteo} \\ \cline{2-7}
& MSE & $R^2$ & MSE & $R^2$ & MSE & $R^2$ \\ \hline
(R1) R
& 0.0022 & 0.5416 & 3.6159 & 0.4586
& 0.0063 & 0.7617
\\
(R2) S
& 0.0011 & 0.5636 & 1.5015 & 0.4683
& 0.0068 & 0.7863
\\
(R3) R+S
& 0.0019 & 0.5514 & 3.4288 & 0.4569
& 0.0062 & 0.8100
\\ \hline
(R4) J
& 0.0025 & 0.3885 & 2.5986 & 0.3386
& 0.0086 & 0.7817
\\
(R5) R+J
& 0.0024 & 0.5219 & 3.7727 & 0.4646
& 0.0085 & 0.7868
\\
(R6) S+J
& 0.0017 & 0.4938 & 2.3175 & 0.4125
& 0.0091 & 0.7739
\\
(R7) R+S+J
& 0.0022 & 0.5123 & 4.0496 & 0.4368
& 0.0092 & 0.7761
\\ \hline
Average
&0.0020 &0.5104 &3.0407 &0.4338
&0.0078 &0.7824 \\
\bottomrule
\end{tabular}
\caption{Performance of variation estimation as regression
on 7 randomness settings.
}
\label{tab:uncertainty_prediction_regression}
\end{table}
\subsection{Variation Estimation Performance}
\label{sec:variation_estimation_results}
In this section, we show the variation estimation performance using
neuron activation strength on {MovieLens}\ and {Criteo}.
\vspace{0.1cm}
\noindent {\bf Regression Performance} ---
When we run the prediction variation estimation as a regression task,
we directly output a score as the estimated prediction variation.
We use both binary and value input features.
In Table~\ref{tab:uncertainty_prediction_regression},
we show the Mean Squared Error (MSE) and $R^2$ for
the three target tasks on the 7 randomness control settings.
From the table,
on all the three target tasks and all the 7 randomness control
settings, we observe strong prediction power of neuron activation strength for ensemble prediction variations.
The average $R^2$ on {MovieLens-R}\ is 0.51, on {MovieLens-C}\ is 0.43,
and on {Criteo}\ is 0.78.
The variation estimation performance is the best
on the {Criteo}\ data,
while {MovieLens-R}\ is better than {MovieLens-C}. The reasons could be:
First, {Criteo}\ has more training data than {MovieLens}.
To train the variation estimation model,
we have 2.2M (50\% of 4.4M) training data on {Criteo},
while only 0.2M (50\% of 0.4M) training data on {MovieLens};
Second, {Criteo}\ has a larger relative range of prediction variations, compared to {MovieLens}.
As shown in Table~\ref{tab:randomness_control},
{Criteo}\ shows around 20\% of prediction variation sway from the mean prediction, which is much higher than ~4-6\% on {MovieLens};
Finally, the {Criteo}\ task is probably the easiest task among
the three: it is a binary classification task,
while {MovieLens-R}\ is a regression task and {MovieLens-C}\ is a multi-class
classification task.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{pic/variation_estimation_auc.pdf}
\caption{AUC for variation bucket prediction
with the randomness setting R3.}
\label{fig:classification_auc}
\end{figure}
\vspace{0.1cm}
\noindent {\bf Classification Performance} ---
We also run the prediction variation estimation as a classification task,
by predicting which variation bucket it should be in.
We use both binary and value input features. Due to the space limit, we
only show the results on the R3 randomness setting, as
R3 uses training data shuffling and parameter random initialization which is the most common setting in practice.
Figure~\ref{fig:classification_auc} shows the AUC scores and
Figure~\ref{fig:confusion_matrix} shows the confusion matrix
for the 5-bucket prediction variation classification
on both {MovieLens}\ and {Criteo}.
The numbers in Figure~\ref{fig:confusion_matrix}
are normalized by the actual example number in each bucket.
Bucket 1 represents the lowest variation slice, while
bucket 5 represents the highest.
As we can see from Figure~\ref{fig:classification_auc},
our variation estimation model is fairly good at
distinguishing examples at different variation buckets, especially
for the lowest and highest buckets.
The average AUC score for the three tasks on bucket 1 is about 0.92
and on bucket 5 is about 0.89.
Figure~\ref{fig:confusion_matrix} shows
that the classification errors mostly
happen on adjacent buckets.
For example, on {Criteo}, most of the mis-classifications assign bucket 1 examples to bucket 2.
When we divide the prediction variation buckets on training data,
we notice that the bucket thresholds are close.
For example, under the randomness control setting R3,
the thresholds of the 5 buckets
for {MovieLens-R}\ are [0.1420, 0.1672, 0.1950, 0.2366],
and the thresholds for {Criteo}\ is [0.0194, 0.0287, 0.0398, 0.0515].
As show in the figures,
{Criteo}\ seems to have the best performance among the three tasks.
Again the reasons could be that the {Criteo}\ task
has more training data, is probably the easiest
among the three tasks,
and it has much larger relative prediction variation range.
\begin{figure*}
\includegraphics[width=.33\linewidth]{pic/movielens_activation_prediction_cm_regression.pdf}
\includegraphics[width=.33\linewidth]{pic/movielens_activation_prediction_cm_classification.pdf}
\includegraphics[width=.33\linewidth]{pic/criteo_activation_prediction_cm.pdf}
\vspace{-0.8cm}
\caption{Confusion matrix for variation estimation as classification for the randomness setting R3.}
\label{fig:confusion_matrix}
\end{figure*}
\begin{table}[t!]
\begin{tabular}{c|cc|cc|cc}
\toprule
& \multicolumn{2}{c|}{MovieLens-R} & \multicolumn{2}{c|}{MovieLens-C} & \multicolumn{2}{c}{Criteo} \\ \cline{2-7}
& MSE & $R^2$ & MSE & $R^2$ & MSE & $R^2$ \\ \hline
B & 0.0025 & 0.4196 & 4.1616 & 0.3408
& 0.0124 & 0.6210
\\
BV & 0.0019 & 0.5514 & 3.4288 & 0.4569
& 0.0062 & 0.8100
\\
\bottomrule
\end{tabular}
\caption{Activation feature study for variation estimation as
regression with the randomness setting R3.}
\label{tab:feature_ablation}
\vspace{-0.4cm}
\end{table}
\begin{table}[t!]
\begin{tabular}{c|cc|cc|cc}
\toprule
& \multicolumn{2}{c|}{MovieLens-R} & \multicolumn{2}{c|}{MovieLens-C} & \multicolumn{2}{c}{Criteo} \\ \cline{2-7}
Run & MSE & $R^2$ & MSE & $R^2$ & MSE & $R^2$ \\ \hline
1 & 0.0019 & 0.5514 & 3.4288 & 0.4569
& 0.0062 & 0.8100 \\
2 & 0.0020 & 0.5160 & 3.3243 & 0.4734
& 0.0061 & 0.8140 \\
3 & 0.0019 & 0.5418 & 3.1825 & 0.4959
& 0.0064 & 0.8036 \\
4 & 0.0019 & 0.5384 & 3.2375 & 0.4872
& 0.0070 & 0.7863 \\
5 & 0.0021 & 0.4994 & 3.2631 & 0.4831
& 0.0066 & 0.7969 \\ \hline
Std & 0.00008 & 0.0190 & 0.0842 & 0.0133
&0.0003 &0.0098 \\
\bottomrule
\end{tabular}
\caption{Reproducibility test for variation estimation
as regression with the randomness setting R3.}
\vspace{-0.4cm}
\label{tab:reproducibility}
\end{table}
\vspace{0.1cm}
\noindent {\bf Activation Feature Study} ---
In Table~\ref{tab:feature_ablation}, we show the contribution of
the two input activation strength features.
We try two feature settings: {\em B} refers to using the binary feature only; and
{\em BV} refers to using both the binary and value features.
As shown in the table, each type
of features makes a non-negligible contribution towards
prediction variation estimation.
Thus, it is beneficial to have both the binary and value features
for prediction variation estimation.
\vspace{0.1cm}
\noindent {\bf Reproducibility} ---
When we evaluate the variation estimation model, the performance is calculated
based on one target task model.
To check whether the performance is reproducible,
we train 5 new target task models.
To simplify the problem, we also use the randomness setting R3, which
is the most commonly used setting in practice.
Table~\ref{tab:reproducibility} shows the MSE and $R^2$ for each run on
both {MovieLens}\ and {Criteo}. As shown in the table,
the standard deviation of MSE and $R^2$ for the 5 runs is small for
each of the three tasks.
Thus, we conclude that activation strength is useful for estimating
prediction variation and it is reproducible.
\vspace{0.1cm}
\noindent {\bf Comparison with Dropout~\cite{gal2016dropout}} ---
Dropout is also a standard way to estimate model uncertainty.
We estimate prediction variation using dropout as follows:
We train one model by randomly dropping 20\% of the neurons on the ReLU
layers using the randomness settings R3. During inference time, we
keep the dropout turned on to obtain the predicted results
for all the testing data.
We run the inference 100 times, and
obtained the prediction variation for each testing example.
We find that the prediction variation estimated by dropout is not very correlated
with the variation estimated by the ensemble method.
On {MovieLens-R}, Pearson correlation of prediction variations for dropout
and ensemble is 0.25, RMSE is 0.0798, and $R^2$ is -0.5010.
On {Criteo}, Pearson correlation of prediction variations for dropout
and ensemble is 0.37, RMSE is 0.0279, and $R^2$ is -1.3709.
As a result, we did not conduct further comparison with
our activation strength based method.
\section{Prediction Variation Quantification}
\label{sec:variation_definition}
In this section, we formally quantify prediction
variation in different problem settings.
Ensemble is one state-of-the-art benchmark for prediction
uncertainty estimation~\cite{breiman1996bagging, dietterich2000ensemble, lakshminarayanan2017simple, ovadia2019can}.
We use ensemble to estimate model prediction variation, that is how much
disagreement there is among ensemble model predictions.
Given the same training data and model configuration,
we train an ensemble of $N$ models $M = \{m\}$, where
$M$ is the set of models, and $N$ is the ensemble size.
Let $\{x\}$ be the testing data, and
$x$ represents the feature vector.
Each of the trained model $m \in M$ makes a prediction on an example
$x \in \{x\}$ as $y^\prime_{m}(x)$.
For {\em regression and binary classification} tasks, the model output is a float
value, thus we define prediction variation as
{\em value prediction variation} based on the standard deviation
of the predicted float values across different models
in the ensemble.
For {\em multi-class classification} tasks, the model output is
a probability distribution over different categories.
Thus, we define prediction variation
as {\em distribution prediction variation} based on the
KL-disagreement or generalized Jensen-Shannon
divergence~\cite{lakshminarayanan2017simple}
on the predicted probability distributions.
Now we define prediction variation formally.
\begin{definition}{(Value Prediction Variation)}
\label{def:regression_uncertainty}
Given an example $x$ that represents the feature vector,
we define its prediction variation $PV(x)$ to be
the standard deviation of predictions from the set of models $M$ as
$PV(x) = \sqrt{\frac{\sum_{m \in M}{(y^\prime_{m}(x) - \bar{y}(x))^2}}{|M|-1}}$,
where $\bar{y}(x) = \frac{1}{|M|} \sum_{m \in M} y^\prime_{m}(x)$
\end{definition}
\begin{definition}{(Distribution Prediction Variation~\cite{lakshminarayanan2017simple})}
\label{def:classification_uncertainty}
Given the example $x$ that represents the feature vector,
let the prediction distribution for the example $x$ be $p(y|x)$.
We define prediction variation $PV(x)$
to be the sum of the Kullback-Leibler (KL) divergence
from the prediction distribution of each model $m \in M$
to the mean prediction distribution of the ensemble.
$ PV(x) = \sum_{m=1}^{M} KL(p_m(y|x) || p_E(y|x))$ where $p_E(y|x) = M^{-1} \sum_{m} p(y|x)$ is the mean prediction of the ensemble.
\end{definition}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,516
|
\section{Introduction}
Graphene, a two-dimensional hexagonal lattice of carbon atoms, has
attracted considerable attention due to its unusual electronic properties,
characterized by massless Dirac Fermions.\cite{GN07,Kts06rev,NGPpw}
It was first produced via micromechanical cleavage on top of a SiO$_{2}$
substrate\cite{NGM+04,pnas} and its hallmark is the half integer
quantum Hall effect.\cite{NGM+05,ZTS+05}
In addition to graphene, few-layer graphene can also be produced.
Of particular interest to us is the double layer graphene system,
where one encounters two carbon layers placed on top of each other
according to usual Bernal stacking of graphite (see Fig.~\ref{Fig_bilayer}).
The low-energy properties of this so-called bilayer graphene are then
described by massive Dirac Fermions.\cite{MF06} These new quasi-particles
have a quadratic dispersion close to the neutrality point and have
recently been identified in Quantum Hall measurements\cite{NMcCM+06}
and in Raman spectroscopy.\cite{FMS+06,GME+06}
In a graphene bilayer it is possible to have the two planes at different
electrostatic potentials.\cite{OBS+06,CNM+06} As a consequence,
a gap opens up at the Dirac point and the low energy band acquires
a Mexican hat relation dispersion.\cite{GNP06} This system is called
a biased graphene bilayer. The potential difference created between
the two layers can be obtained by applying a back gate voltage to
the bilayer system and covering the exposed surface with some chemical
dopant, as for example Potassium\cite{OBS+06} or NH$_{3}$.\cite{CNM+06}
In addition, it is also possible to control the potential difference
between the layers by using back and top gate setups.\cite{OHL+07}
The opening of the gap at the Dirac point in the biased bilayer system
was demonstrated both by angle resolved photoemission experiments
(ARPES)\cite{OBS+06} and Hall effect measurements.\cite{CNM+06}
The electronic gap in the biased system has been also observed in
epitaxially grown graphene films on SiC crystal surfaces.\cite{BSL+04}
Due to the Mexican hat dispersion relation the density of states close
to the gap diverges as the square root of the energy. The possibility
of having an arbitrary large density of states at the Fermi energy
poses the question whether this system can be unstable toward a
ferromagnetic ground state. The question of magnetism in carbon based
systems already has a long history. Even before the discovery of
graphene, highly oriented pyrolytic graphite (HOPG) has attracted a
broad interest due to the observation of anomalous properties, such as
magnetism and insulating behavior in the direction perpendicular to
the planes. \cite{ESH+02,KEK02,KSE+03,KTS+03,OTH+07} The research of
$s-p$ based magnetism\cite{RGC+04,TNS+91,SDL+98} was especially
motivated by the technological use of nanosized particles of graphite,
which show interesting features depending on their shape, edges, and
applied field of pressure.\cite{EK05}
Microscopic theoretical models of bulk carbon magnetism include
nitrogen-carbon compositions where ferromagnetic ordering of spins
could exist in $\pi$ delocalized systems due to a lone electron pair
on a trivalent element\cite{Ovch78} or intermediate graphite-diamond
structures where the alternating $sp^{2}$ and $sp^{3}$ carbon atoms
play the role of different valence elements. \cite{OS91} More general
models focus on the interplay between disorder and
interaction. \cite{SGV05,VSS+05} Further, midgap states due to zig-zag
edges play a predominant role in the formation of magnetic moments
\cite{japonese,PCM+07} which support flat-band
ferromagnetism.\cite{Mielke91,Tasaki98,KM03} A generic model based on
midgap states was recently proposed in
Refs. \cite{WSG08,WSSG08}. Magnetism is also found in fullerene based
metal-free systems.\cite{CMG+04} For a recent overview on metal-free
Carbon-based magnetism see Ref.~\cite{MPbook06}.
To understand carbon-based magnetism in graphite, one may start with
the simplest case of one-layer, i.e., graphene. Because the density
of states of intrinsic graphene vanishes at the Dirac point, the simple
Stoner-like argument predicts an arbitrary large value of the Coulomb
on-site energy needed to produce a ferromagnetic ground state.\cite{PAB04,AP06}
In fact, because of the vanishing density of states, the Coulomb interaction
is not screened and the Hubbard model is not a good starting point
to study ferromagnetism in clean graphene. One, therefore, has to
consider the exchange instability of the Dirac gas due to the bare,
long-range Coulomb interaction in two dimensions. This study shows
that for a clean, doped or undoped graphene layer, a spin-polarized
ground-state due to the gain of exchange energy is only favorable
for unphysical values of the dimensionless coupling constant of graphene.\cite{PGN05}
The paramagnetic ground-state of clean graphene is thus stable against
the exchange interaction. If the system is disordered, e.g., due to
vacancies or edge states, a finite density of states builds in at
the Dirac point. As a consequence, a finite Hubbard interaction for
driving the system to a ferromagnetic ground state is obtained.\cite{PGN06}
In this case, the exchange interaction favors a ferromagnetic ground
state for reasonable values of the dimensionless coupling parameter.\cite{PGN05}
The presence of itinerant magnetism due to quasi-localized states
induced by single-atom defects in graphene, such as vacancies,\cite{PGS+06}
has also been obtained recently using first-principles.\cite{YH07}
The situation is quite different in a bilayer system. There, a finite
density of states exists at the neutrality point producing some amount
of screening in the system. Moreover, in the case of a biased bilayer
and for densities close to the energy gap, the density of states is
very large producing very effective screening. As a consequence, for
this system the Hubbard model is a good starting point to study the
tendency toward ferromagnetism. From the point of view of the exchange
instability of the bilayer system, it is found that the system is
always unstable toward a ferromagnetic ground state for low enough
particle densities. \cite{NNP+05,Sta07,CPSS08}
In the present paper, we want to explore the fact that the Hubbard
model is a good starting point to describe the Coulomb interactions
in the regime where the Fermi energy is close to the band edge of
the biased bilayer system. In particular we want to study the phase
diagram of the system as function of the doping. We further want to
determine the mean field critical temperature.
The paper is organized as followed. In section \ref{sec_model}, we
introduce the model and define the mean-field decoupling which allows
for different electronic density and magnetization in the two layers.
In section \ref{sec_ground_state}, we set up the mean-field equations
and present the numerical results in section \ref{sec_results}. We
close with conclusions and future research directions.
\section{Model Hamiltonian and mean field approximation}
\label{sec_model}
The Hamiltonian of a biased bilayer Hubbard model is the sum of two
pieces $H=H_{TB}+H_{U}$, where $H_{TB}$ is the tight-binding part
and $H_{U}$ is the Coulomb on-site interaction part of the Hamiltonian.
The tight-binding Hamiltonian is itself a sum of four terms describing
the tight-binding Hamiltonian of each plane, the hopping term between
the planes, and the electrostatic bias applied to the two planes.
We therefore have, \begin{equation}
H_{TB}=\sum_{\iota=1}^{2}H_{TB,\iota}+H_{\perp}+H_{V}\,,\end{equation}
with \begin{eqnarray}
H_{TB,\iota}= & - & t\sum_{\bm R,\sigma}[a_{\iota\sigma}^{\dag}(\bm R)b_{\iota\sigma}(\bm R)+a_{\iota\sigma}^{\dag}(\bm R)b_{\iota\sigma}(\bm R-\bm a_{1})\nonumber \\
& + & a_{\iota\sigma}^{\dag}(\bm R)b_{\iota\sigma}(\bm R-\bm a_{2})+H.c.]\,,\end{eqnarray}
\begin{equation}
H_{\perp}=-t_{\perp}\sum_{\bm R,\sigma}[a_{1\sigma}^{\dag}(\bm R)b_{2\sigma}(\bm R)+b_{2\sigma}^{\dag}(\bm R)a_{1\sigma}(\bm R)]\,,\end{equation}
and \begin{equation}
H_{V}=\frac{V}{2}\sum_{\bm R,\sigma}[n_{a1\sigma}(\bm R)+n_{b1\sigma}(\bm R)-n_{a2\sigma}(\bm R)-n_{b2\sigma}(\bm R)]\,.\label{HV}\end{equation}
As regards the bias term in Eq.~(\ref{HV}), we assume here that~$V$
can be externally controlled and is independent of the charge density
in the system. This situation can be realized with a back and top
gate setup.\cite{OHL+07} The on-site Coulomb part is given by, \begin{eqnarray}
H_{U}= & U & \sum_{\bm R}[n_{a1\uparrow}(\bm R)n_{a1\downarrow}(\bm R)+n_{b1\uparrow}(\bm R)n_{b1\downarrow}(\bm R)\nonumber \\
& + & n_{a2\uparrow}(\bm R)n_{a2\downarrow}(\bm R)+n_{b2\uparrow}(\bm R)n_{b2\downarrow}(\bm R)]\,,\end{eqnarray}
where $n_{x\iota\sigma}(\bm R)=x_{\iota\sigma}^{\dag}(\bm R)x_{\iota\sigma}(\bm R)$,
with $x=a,b$, $\iota=1,2$ and $\sigma=\uparrow,\downarrow$.
\begin{figure}[htf]
\begin{centering}
\includegraphics[clip,width=7.5cm]{fig1}
\par\end{centering}
\caption{(Color online) The unit cell a of graphene bilayer in the Bernal stacking.
The dashed hexagons are on top of the solid ones. The unit cell vectors
have coordinates $\bm a_{1}=a(3,\sqrt{3})/2$ and $\bm a_{2}=a(3,-\sqrt{3})/2.$}
\label{Fig_bilayer}
\end{figure}
The problem defined by the Hamiltonian $H_{TB}+H_{U}$ can not be
solved exactly and therefore we have to rest upon some approximation.
Here we adopt a mean field approach, neglecting quantum fluctuations.
Since we are interested in studying the existence of a ferromagnetic
phase we have to introduce a broken symmetry ground state. There is
however an important point to remark: since the two planes of the
bilayer are at different electrostatic potentials one should expect
that the electronic density and the magnetization will not be evenly
distributed among the two layers. Therefore our broken symmetry ground
state must take this aspect into account. As a consequence we propose
the following broken symmetry ground state: \begin{equation}
\langle n_{x1\sigma}(\bm R)\rangle=\frac{n+\Delta n}{8}+\sigma\frac{m+\Delta m}{8}\,,\end{equation}
and \begin{equation}
\langle n_{x2\sigma}(\bm R)\rangle=\frac{n-\Delta n}{8}+\sigma\frac{m-\Delta m}{8}\,,\end{equation}
where $n$ is the density per unit cell and $m=n_{\uparrow}-n_{\downarrow}$
is the spin polarization per unit cell. The quantities $\Delta n$
and $\Delta m$ represent the difference in the electronic density
and in the spin polarization between the two layers, respectively.\cite{foot1}
We note that $m$ and $\Delta m$ are independent parameters, being
in principle possible to have a ground state where $m=0$ but $\Delta m\ne0$.
When transformed to momentum space the mean field Hamiltonian obtained
from the above reads \begin{eqnarray}
H_{MF} & = & \sum_{\bm k,\sigma}\Psi_{\bm k,\sigma}^{\dag}H_{\bm k,\sigma}\Psi_{\bm k,\sigma}\nonumber \\
& - & \frac{N_{c}U}{32}[(n+\Delta n)^{2}-(m+\Delta m)^{2}]\nonumber \\
& - & \frac{N_{c}U}{32}[(n-\Delta n)^{2}-(m-\Delta m)^{2}]\,,\label{Eq_Hamilt}\end{eqnarray}
with $\Psi_{\bm k,\sigma}^{\dag}=[a_{1\bm k\sigma}^{\dag},b_{1\bm k\sigma}^{\dag},a_{2\bm k\sigma}^{\dag},b_{2\bm k\sigma}^{\dag}]$
and $H_{\bm k,\sigma}$ given by \begin{equation}
H_{\bm k,\sigma}=\left(\begin{array}{cccc}
s_{\sigma} & -t\phi_{\bm k} & 0 & -t_{\perp}\\
-t\phi_{\bm k}^{\ast} & s_{\sigma} & 0 & 0\\
0 & 0 & p_{\sigma} & -t\phi_{\bm k}\\
-t_{\perp} & 0 & -t\phi_{\bm k}^{\ast} & p_{\sigma}\end{array}\right)\,,\end{equation}
with $s_{\sigma}=\frac{V}{2}+\left(\frac{n+\Delta n}{8}-\sigma\frac{m+\Delta m}{8}\right)U$,
$p_{\sigma}=-\frac{V}{2}+\left(\frac{n-\Delta n}{8}-\sigma\frac{m-\Delta m}{8}\right)U$,
and $\phi_{\bm k}=1+e^{i\bm k\cdot\bm a_{1}}+e^{i\bm k\cdot\bm a_{2}}$.
The energy eigenvalues are given by, \begin{eqnarray}
E_{\sigma}^{j,l}(\bm k,m,\Delta m)&=&\left(\frac{n}{8}-\sigma\frac{m}{8}\right)U\\&+&
\frac{l}{2}\sqrt{2t_{\perp}^{2}+V_{\sigma}^{2}+4t^{2}\vert\phi_{\bm k}\vert^{2}+j2\sqrt{t_{\perp}^{4}+4t^{2}(t_{\perp}^{2}+V_{\sigma}^{2})\vert\phi_{\bm k}\vert^{2}}}\,,\nonumber\end{eqnarray}
where $l,j=\pm$ and $V_{\sigma}$ is given by \begin{equation}
V_{\sigma}=V+U\Delta\tilde{n}-\sigma U\Delta\tilde{m}\,,\label{Vs}\end{equation}
where we have introduced the definitions $\Delta n=4\Delta\tilde{n}$
and $\Delta m=4\Delta\tilde{m}$. It is clear that as long as $\Delta n$
and $\Delta m$ are finite the system has an effective $V_{\sigma}$
that differs from the bare value $V$. The momentum values are given
by, \begin{equation}
\bm k=\frac{m_{1}}{N}\bm b_{1}+\frac{m_{2}}{N}\bm b_{2}\,,\label{Eq_mom}\end{equation}
with $m_{1},m_{2}=0,1,\ldots\, N-1$, the number of unit cells given
by $N_{c}=N^{2}$, and $\bm b_{1}$ and $\bm b_{2}$ given by, \begin{equation}
\bm b_{1}=\frac{2\pi}{3a}(1,\sqrt{3})\,,\hspace{0.5cm}\bm b_{2}=\frac{2\pi}{3a}(1,-\sqrt{3})\,.\end{equation}
The Brillouin zone of the system is represented in Fig.~\ref{Fig_BZ}.
\begin{figure}[htf]
\begin{centering}
\includegraphics[clip,width=7.5cm]{fig2}
\par\end{centering}
\caption{(Color online) Brillouin zone of the bilayer problem. The Dirac point
$\bm K$ has coordinates $2\pi(1,\sqrt{3}/3)/(3a)$ and the $\bm M$
point has coordinates $2\pi(1,0)/(3a)$.}
\label{Fig_BZ}
\end{figure}
\section{Free energy and mean-field equations}
\label{sec_ground_state}
The free energy per unit cell, $f$, of Hamiltonian~(\ref{Eq_Hamilt})
is given by, \begin{eqnarray}
f=&-&\frac{k_{B}T}{N_{c}}\sum_{\bm k,\sigma}\sum_{l,j=\pm}\ln\left(1+e^{-(E_{\sigma}^{l,j}(\bm k)-\mu)/(k_{B}T)}\right)\\\nonumber
&-&\frac{U}{16}\big[n^{2}-m^{2}+(\Delta n)^{2}-(\Delta m)^{2}\big]+\mu n\,,\label{Eq_free_energy}\end{eqnarray}
where $\mu$ is the chemical potential.
Let us introduce the density of states per spin per unit cell $\rho(E)$
defined as \begin{equation}
\rho(E)=\frac{1}{N_{c}}\sum_{\bm k}\delta(E-t\vert\phi_{\bm k}\vert)\,.\label{Eq_rho}\end{equation}
The momentum integral in Eq. (\ref{Eq_rho}) is over the Brillouin
zone defined in Fig. \ref{Fig_BZ}, using the momentum definition
(\ref{Eq_mom}). The integral can be performed leading to, \begin{equation}
\rho(E)=\frac{2E}{t^{2}\pi^{2}}\left\{ \begin{array}{ccc}
\frac{1}{\sqrt{F(E/t)}}\mathbf{K}\left(\frac{4E/t}{F(E/t)}\right)\,, & & 0<E<t\,,\\
\\\frac{1}{\sqrt{4E/t}}\mathbf{K}\left(\frac{F(E/t)}{4E/t}\right)\,, & & t<E<3t\,,\end{array}\right.\label{eq:DOS1L}\end{equation}
where $F(x)$ is given by \begin{equation}
F(x)=(1+x)^{2}-\frac{(x^{2}-1)^{2}}{4}\,,\label{eq:Fe}\end{equation}
and $\mathbf{K}(m)$ is defined as, \begin{equation}
\mathbf{K}(m)=\int_{0}^{1}dx[(1-x^{2})(1-mx^{2})]^{-1/2}\,.\label{eq:K}\end{equation}
Using Eq.~(\ref{Eq_rho}), the free energy in Eq.~(\ref{Eq_free_energy})
can be written as a one-dimensional integral, \begin{eqnarray}
f & =\nonumber \\
- & k_{B}T & \sum_{\sigma}\sum_{l,j=\pm}\int dE\rho(E)\ln\left(1+e^{-(E_{\sigma}^{l,j}(E)-\mu)/(k_{B}T)}\right)\nonumber \\
& - & \frac{U}{16}\big[n^{2}-m^{2}+(\Delta n)^{2}-(\Delta m)^{2}\big]+\mu n\,.\label{Eq_FE}\end{eqnarray}
The mean field equations are now obtained from the minimization of
the free energy (\ref{Eq_FE}). The doping, $\delta n$, relative
to the situation where the system is at half filling is defined as,
\begin{eqnarray}
\delta n=\sum_{\sigma}\sum_{l,j=\pm}\int dE\rho(E)f[E_{\sigma}^{l,j}(E)-\mu]-4\,,\label{Eq_MF1}\end{eqnarray}
where $f(x)=(1+e^{x/(k_{B}T)})^{-1}$. The spin polarization per
unit cell obeys the mean field equation, \begin{eqnarray}
m=\sum_{\sigma}\sum_{l,j=\pm}\sigma\int dE\rho(E)f[E_{\sigma}^{l,j}(E)-\mu]\,.\label{Eq_MF2}\end{eqnarray}
The difference in the electronic density between the two layers is
obtained from, \begin{equation}
\Delta\tilde{n}=\frac{1}{2}\sum_{\sigma}\sum_{l,j=\pm1}\int dE\rho(E)f[E_{\sigma}^{l,j}(E)-\mu]v_{\sigma}^{l,j}(E)\,,\label{eq:Dn}\end{equation}
where $v_{\sigma}^{l,j}(E)$ is given by \begin{equation}
v_{\sigma}^{l,j}(E)=\frac{l}{2}\frac{V_{\sigma}}{\sqrt{\ldots}}\left(1+\frac{j4E^{2}}{\sqrt{t_{\perp}^{4}+4E^{2}(t_{\perp}^{2}+V_{\sigma}^{2})}}\right)\,,\end{equation}
and \begin{equation}
\sqrt{\ldots}=\sqrt{2t_{\perp}^{2}+V_{\sigma}^{2}+4E^{2}+j2\sqrt{t_{\perp}^{4}+4E^{2}(t_{\perp}^{2}+V_{\sigma}^{2})}}\,.\end{equation}
The difference in the magnetization between the two layers is obtained
from \begin{equation}
\Delta\tilde{m}=\frac{1}{2}\sum_{\sigma}\sum_{l,j=\pm1}\sigma\int dE\rho(E)f[E_{\sigma}^{l,j}(E)-\mu]v_{\sigma}^{l,j}(E)\,,\label{eq:Dm}\end{equation}
Let us now assume that the system supports a ferromagnetic ground
state whose magnetization vanishes at some critical value $U_{c}$
at zero temperature. Additionally we assume that $\Delta m=0$ when
$m=0$, which will be shown to be the case in this system. The value
of $U_{c}$ is determined from expanding (\ref{Eq_MF2}) to first
order in $m$, leading to, \begin{eqnarray}
1 & = & \frac{U_{c}}{4}\sum_{l,j=\pm1}\int dE\rho(E)\delta[E_{\sigma}^{l,j}(E,0,0)-\mu]\nonumber \\
& = & \frac{U_{c}}{4}\sum_{l,j,k=\pm1}^{\ast}\frac{\rho(E_{k}^{\ast})}{\vert f'_{l,j}(E_{k}^{\ast})\vert}\theta(3t-E_{k}^{\ast})\theta(E_{k}^{\ast})\nonumber \\
& = & \frac{U_{c}}{4}\rho_{b}(\tilde{\mu},U_{c})=U_{c}\tilde{\rho}_{b}(\tilde{\mu},U_{c})\,,\label{Eq_stoner}\end{eqnarray}
where $\rho_{b}(\tilde{\mu},U_{c})$ is the density of states per
unit cell per spin
for a biased bilayer at the energy $\tilde{\mu}=\mu-nU_{c}/8$ and
$\tilde{\rho}_{b}(\tilde{\mu},U_{c})$ is the density of states per
spin per lattice point. Although Eq.~(\ref{Eq_stoner}) looks like
the usual Stoner criterion the fact that
the bias $V_{\sigma}$ given in Eq.~(\ref{Vs}) depend on $U$ due
to the difference in the electronic density $\Delta n$ makes Eq.~(\ref{Eq_stoner})
a non-linear equation for $U_{c}$ which must be solved self-consistently.
For low doping $\delta n$ the product $U_{c}\Delta\tilde{n}$ is
a small number when compared to $V$ and therefore it can be neglected
in Eq.~(\ref{Vs}). In this case Eq.~(\ref{Eq_stoner}) reduces
to the usual Stoner criterion: \begin{equation}
U_{c}\simeq1/[\tilde{\rho}_{b}(\tilde{\mu})]\,.\label{scapp}\end{equation}
The quantities $E_{k}^{\ast}$ in Eq.~(\ref{Eq_stoner}) are the
roots of the delta function argument, \begin{equation}
E_{\sigma}^{l,j}(E_{k}^{\ast})-\mu=0\,.\label{eq:roots_eq}\end{equation}
The quantity $f'_{l,j}(E_{k}^{\ast})$ is the derivative in order
to the energy $E$ of Eq.~(\ref{eq:roots_eq}) evaluated at the roots
$E_{k}^{\ast}$. The roots $E_{k}^{\ast}$ are given by \begin{equation}
E_{k}^{\ast}=\frac{1}{2}\sqrt{4\tilde{\mu}^{2}+V_{\sigma}^{2}+k2\sqrt{4\tilde{\mu}^{2}(t_{\perp}^{2}+V_{\sigma}^{2})-t_{\perp}^{2}V_{\sigma}^{2}}}\,,\label{roots}\end{equation}
with $k=\pm$. Equation~(\ref{eq:roots_eq}) cannot be solved for
all bands: the existence of a solution is determined by $\mu$. As
a consequence we added the~$\ast$ symbol in the summation of Eq.~(\ref{Eq_stoner}),
which means that only bands for which Eq.~(\ref{eq:roots_eq}) can
be solved (two at the most) contribute to the summation. It also means
that for the contributing bands only real roots in Eq.~(\ref{roots})
are taken into account to the summation. The number of real roots
in Eq.~(\ref{roots}) depends on the particular band an $\mu$ through
Eq.~(\ref{eq:roots_eq}). As the function $f'_{l,j}(E)$ is given
by \begin{equation}
f'_{l,j}(E)=\frac{2lE}{\sqrt{\ldots}}\left(1+j\frac{t_{\perp}^{2}+V_{\sigma}^{2}}{\sqrt{t_{\perp}^{4}+4E^{2}(t_{\perp}^{2}+V_{\sigma}^{2})}}\right)\,.\end{equation}
it is clear that both roots are imaginary for $\tilde{\mu}$ in the
range \begin{equation}
-\frac{t_{\perp}V_{\sigma}}{2\sqrt{t_{\perp}^{2}+V_{\sigma}^{2}}}<\tilde{\mu}<\frac{t_{\perp}V_{\sigma}}{2\sqrt{t_{\perp}^{2}+V_{\sigma}^{2}}}\,,\end{equation}
which means that the system has an energy gap of value \begin{equation}
\Delta_{g}=\frac{t_{\perp}V_{\sigma}}{\sqrt{t_{\perp}^{2}+V_{\sigma}^{2}}}\,.\label{gap}\end{equation}
We finally note that since we have assumed $\Delta m=0$, $V_{\sigma}$
does not effectively depend on $\sigma$.
\section{Results and Discussion}
\label{sec_results}
We start with the zero temperature phase diagram in the plane $U$~vs.~$\delta n$.
An approximate analytic treatment is possible in this limit, which
is used to check our numerical results. The effect of temperature
is considered afterwards.
\subsection{Zero temperature}
\label{subsec_zeroT}
\subsubsection{Approximate solution}
\label{subsubsec_appT0} In Fig.~\ref{Fig_DOS} we represent the
density of states of a biased bilayer with $U=0$ together with the
low doping critical value $U_{c}$, as given by Eq.~(\ref{scapp}).
In panel~(b) of Fig.~\ref{Fig_DOS} a zoom in of the density of
states close to the gap is shown. It is clear that the density of
states diverges at the edge of the gap. As consequence the closer
to edge of the gap the chemical potential is the lower will be the
critical $U_{c}$ value. This quantity is shown in panel~(c) of Fig.~\ref{Fig_DOS}
as function of the chemical potential $\tilde{\mu}$ and in panel~(d)
as a function of doping $\delta n$. The lowest represented value
of $U_{c}$ is about $U_{c}\simeq2.7$ \texttt{eV} to which corresponds
an electronic doping density $\delta n\simeq2.5\times10^{-5}$ electrons
per unit cell. The step like discontinuity shown in panels~(c) and~(d)
for $U_{c}$ occurs when the Fermi energy equals $V/2$, signaling
the top of the Mexican hat dispersion relation.
\begin{figure}[t]
\begin{centering}
\includegraphics[clip,width=0.9\columnwidth]{fig3}
\par\end{centering}
\caption{(Color online) (a)~--~Density of states $\rho(\tilde{\mu})$ per
unit cell per spin of the bilayer problem with $U=0$. (b)~--~Zoom
of~(a) near the gap region. (c)~--~Critical value $U_{c}$ for
ferromagnetism in the low doping, $\delta n$, regime. (d)~--~The
same as in~(c) as a function of doping. The parameters are $t=2.7$~\texttt{eV},
$t_{\perp}=0.2t$, $V=0.05$~\texttt{eV}. The edge of the gap is
located at $\Delta_{g}/(2t)\simeq0.00922$. }
\label{Fig_DOS}
\end{figure}
It is clear from panel~(d) of Fig.~\ref{Fig_DOS} that in the low
doping limit $U_{c}$ is a linear function of doping $\delta n$.
This limit enables for an approximate analytic treatment which not
only explains the linear behavior but also provides a validation test
of our numerical results. Firstly we note that for very low doping
the density of states in Eq.~(\ref{scapp}) is close to the gap edge,
$|\tilde{\mu}|\sim\Delta_{g}/2$, where $\Delta_{g}$ is the size
of the gap Eq.~(\ref{gap}). In this energy region the density of
states has a 1D like divergence,\cite{GNP06} behaving as, \begin{equation}
\rho_{b}(\tilde{\mu})\propto\frac{1}{\sqrt{|\tilde{\mu}|-\Delta_{g}/2}}\,.\label{dos1D}\end{equation}
Using this approximate expression to compute the doping, $\delta n\propto\textrm{sign}(\tilde{\mu})\times\int_{\Delta_{g}/2}^{|\tilde{\mu}|}\textrm{d}x~\rho_{b}(x)$,
we immediately get $\delta n\propto\textrm{sign}(\tilde{\mu})/\rho_{b}(\tilde{\mu})$
and thus $U_{c}\propto|\delta n|$. In order to have an analytic expression
for $U_{c}$ in the low doping limit we have to take into account
the proportionality coefficient in Eq.~(\ref{dos1D}). After some
algebra it can be shown that the density of states per spin per lattice
point near the gap edge can be written as, \begin{equation}
\rho_{b}(\tilde{\mu})\approx\frac{1}{t^{2}4\pi^{2}}\sqrt{\frac{\Delta_{g}(t_{\perp}^{2}+V^{2})}{F(\chi)}}\mathbf{K}\left(\frac{4\chi}{F(\chi)}\right)\frac{1}{\sqrt{|\tilde{\mu}|-\Delta_{g}/2}}\,,\label{eq:dosapp}\end{equation}
where $\chi=[(\Delta_{g}^{2}+V^{2})/(4t^{2})]^{1/2}$, with $F(x)$
and $\mathbf{K}(m)$ as in Eqs.~(\ref{eq:Fe}) and~(\ref{eq:K}).
The doping $\delta n$, measured with respect to half filling in units
of electrons per unit cell, can be written as, \begin{eqnarray}
\delta n & =&\textrm{sign}(\tilde{\mu})\times8\int_{\Delta_{g}/2}^{|\tilde{\mu}|}\mbox{d}x\,\rho_{b}(x)\nonumber \\
& \approx&\frac{4}{t^{2}\pi^{2}}\sqrt{\frac{\Delta_{g}(t_{\perp}^{2}+V^{2})}{F(\chi)}}\mathbf{K}\left(\frac{4\chi}{F(\chi)}\right)\sqrt{|\tilde{\mu}|-\Delta_{g}/2}.\label{eq:dnapp}\end{eqnarray}
Inserting Eq.~(\ref{eq:dosapp}) into Eq.~(\ref{scapp}), and taking
into account Eq.~(\ref{eq:dnapp}), we are able to write, \begin{equation}
U_{c}\approx t^{4}\pi^{4}\frac{F(\chi)}{\Delta_{g}(t_{\perp}^{2}+V^{2})}\left[\mathbf{K}\Big(\frac{4\chi}{F(\chi)}\Big)\right]^{-2}\delta n\,.\label{eq:Ucapp}\end{equation}
In panel~(d) of Fig.~\ref{Fig_DOS} both the numerical result of
Eq.~(\ref{scapp}) and the analytical result of Eq.~(\ref{eq:Ucapp})
are shown. The agreement is excellent.
\begin{figure}[t]
\begin{centering}
\includegraphics[clip,width=0.98\columnwidth]{fig4}
\par\end{centering}
\caption{(Color online)
Panels~(a), (b), and~(c) show the zero temperature self-consistent
solution for~$m$, $\Delta m$, and $\Delta n$, respectively. The
zero temperature phase diagram of the biased bilayer in the $U$ vs.
$\delta n$ plane is shown in panel~(d). Symbols in panel~(d) are
inferred from panel~(a) and signal a \textit{first-order} phase transition;
the solid {[}Eq.~(\ref{Eq_stoner})] and dashed {[}Eq.~(\ref{eq:Ucapp})]
lines stand for a \textit{second-order} phase transition. The constant
parameters are $V=0.05$ \texttt{eV}, $t_{\perp}=0.2t$, and $t=2.7$
\texttt{eV}.}
\label{Fig_PD_n}
\end{figure}
\subsubsection{Self-consistent solution}
\label{subsubsec_scT0} We now need to solve the mean field equations
in order to obtain the zero temperature phase diagram of the biased
bilayer.
In order to achieve this goal we study how $m$, $\Delta m$, and
$\Delta n$ depend on the interaction $U$, for given values of the
electronic doping $\delta n$.
In panel~(a) of Fig.~\ref{Fig_PD_n} it is shown how $m$ depends
on $U$ for different values of $\delta n$. The chosen values of
$\delta n$ correspond to the chemical potential being located at
the divergence of the low energy density of states. The lower the
$\delta n$ is the more close to the gap edge is the chemical potential
and therefore the larger the density of states is. As a consequence,
$m$ presents a smaller critical $U_{c}$ value for smaller $\delta n$
values. It is interesting to note that the magnetization saturation
values correspond to full polarization of the doping charge density
with $m=\delta n$, also found within a one-band model.\cite{Sta07}
In panel~(b) of Fig.~\ref{Fig_PD_n} we plot the $\Delta m$ mean
field parameter. Interestingly the value of $\Delta m$ vanishes at
the same $U_{c}$ as $m$. For finite values of $m$ we have $\Delta m>m$,
which means that the magnetization of the two layers is opposite.
We therefore have two ferromagnetic planes that possess opposite and
unequal magnetization. In panel~(c) of Fig.~\ref{Fig_PD_n} we show
the value of $\Delta n$ as function of $U$. It is clear that $|\delta n|<|\Delta n|$,
which implies that the density of charge carriers is above the Dirac
point in one plane and below it in the other plane. This means that
the charge carriers are electron like in one plane and hole like in
the other.
In panel~(d) of Fig.~\ref{Fig_PD_n} we show the phase diagram of
the system in the $U$ vs. $\delta n$ plane. Symbols are inferred
from the magnetization behavior in panel~(a). They signal a \textit{first-order}
phase transition when $m$ increases from zero to a finite value {[}see
panel~(a)]. The full (red) line is the numerical self-consistent
result of Eq.~(\ref{Eq_stoner}), and the dashed (blue) line is the
approximate analytic result given by Eq.~(\ref{eq:Ucapp}). The discrepancy
between lines and symbols has a clear meaning. In order to obtain
both Eqs.~(\ref{Eq_stoner}) and~(\ref{scapp}) we assumed that
a \textit{second-order} phase transition would take place, i.e., the
magnetization $m$ would vanish continuously when some critical $U_{c}$
is approached from above. This is not the case, and the system undergoes
a first-order phase transition for smaller $U$ values than those
for the second-order phase transition case. There are clearly two
different regimes in panel~(d) of Fig.~\ref{Fig_PD_n}: one at densities
lower than $\delta n\lesssim1\times10^{-4}$, where the dependence
of $\delta n$ on $U_{c}$ is linear, and another regime for $\delta n>1\times10^{-4}$
where a plateau like behavior develops. This plateau has the same
physical origin as the step like discontinuity we have seen in panels~(c)
and~(d) of Fig.~\ref{Fig_DOS}. Clearly, as the density $\delta n$
grows the needed value of $U_{c}$ for having a ferromagnetic ground
state increases. This is a consequence of the diverging density of
states close to the gap edge. As regards the limit $\delta n\rightarrow0$
it is obvious from panel~(d) of Fig.~\ref{Fig_PD_n} that we have
$U_{c}\rightarrow0$. It should be noted, however, that lowering the
density $\delta n$ leads to a decrease of $m$ and $\Delta m$, as
can be seen in panels~(a) and~(b) of Fig.~\ref{Fig_PD_n}. Therefore,
even though we have $U_{c}\rightarrow0$ in the limit $\delta n\rightarrow0$,
we have also $m\rightarrow0$ and $\Delta m\rightarrow0$, which implies
a \emph{paramagnetic} ground state for the undoped ($\delta n=0$)
biased bilayer. Only $\Delta n$ remains finite at zero doping, in
agreement with the observations that screening is still possible at
the neutrality point ($\delta n=0$ ).\cite{McC06,CNM+06,MSB+06}
\begin{figure}
\begin{centering}
\includegraphics[clip,width=0.98\columnwidth]{fig5}
\par\end{centering}
\caption{\label{fig:PDgen} (Color online) Effect of $t_{\perp}$ and $V$
on the zero temperature $U_{c}$ vs. $\delta n$ phase diagram: (a)~fixed
$t_{\perp}=0.2t$ and varying $V$; (b)~fixed $V=0.05\,\mathtt{eV}$
and varying $t_{\perp}$. For a given $\delta n$ the \emph{ferromagnetic}
phase establishes for $U>U_{c}$ and the \emph{paramagnetic} phase
for $U<U_{c}$.}
\end{figure}
So far we have analyzed the system for fixed values of the bias voltage,
$V$, and interlayer coupling, $t_{\perp}$. In Fig.~\ref{fig:PDgen}
we show the effect of the variation of these two parameters on the
zero temperature phase diagram. In panel~(a) we have fixed the interlayer
coupling, $t_{\perp}=0.2t$, and varied the bias voltage, $V(\mathtt{eV})=\{0.01,0.05,0.1\}$;
in panel~(b) we did the opposite, with $V=0.05\,\mathtt{eV}$ and
$t_{\perp}/t=\{0.05,0.1,0.2\}$. Essentially, raising either $V$
or $t_{\perp}$ leads to a decrease of the critical interaction, $U_{c}$,
needed to establish the ferromagnetic phase for a given $\delta n$.
The order of the transition, however, remains \emph{first-order}:
for a given $\delta n$, the critical interaction $U_{c}$ predicted
by Eq.~(\ref{Eq_stoner}), which is valid for a \emph{second-order}
phase transition, is always higher than what is obtained by solving
self-consistently the mean-field equations, meaning that a \emph{first-order}
transitions is occurring at a lower $U_{c}$. It is interesting to
note that the effect of $V$ and $t_{\perp}$ on the \emph{first-order}
critical $U_{c}$ line is similar to what is expected for the usual
Stoner criterion, where increasing either $V$ or $t_{\perp}$ gives
rise to an increase in the density of states at the Fermi level and
a lower $U_{c}$ thereof.
The bias voltage and the interlayer coupling have also interesting
effects on the magnetization, $m$, and spin polarization difference
between layers, $\Delta m$. Decreasing $t_{\perp}$ leads to a decrease
in $\Delta m$, and below some $t_{\perp}$ we have $\Delta m<m$,
as opposed to the case discussed above ($V=0.05\,\mathtt{eV}$ and
$t_{\perp}=0.2t$). In particular, for $V=0.05\,\mathtt{eV}$, we
have already found $\Delta m<m$ for $t_{\perp}\leq0.1t$. A similar
effect has been observed when $V$ is increased. For $t_{\perp}=0.2t$
we have found $\Delta m<m$ for $V\geq0.1\,\mathtt{eV}$. It should
be noted, however, that $m$ and $\Delta m$ are $U$-dependent. Increasing
$U$ leads $m$ to saturate while $\Delta m$ keeps growing, as can
be seen in panels~(a) and~(b) of Fig.~\ref{Fig_PD_n} for the particular
case of $V=0.05\,\mathtt{eV}$ and $t_{\perp}=0.2t$. This means that,
depending on the value of the parameters $V$ and $t_{\perp}$, we
can go from $\Delta m<m$ to $\Delta m>m$ just by increasing the
interaction strength $U$. It can also be seen in panel~(a) of Fig.~\ref{Fig_PD_n}
that $m$ is completely saturated at the transition for $\delta n<\delta n_{c}\approx6\times10^{-5}$~electrons
per unit cell, while for $\delta n>\delta n_{c}$ it saturates only
at some $U>U_{c}$. Even though this behavior seems to be general
for any $V$ and $t_{\perp}$, the value of $\delta n_{c}$ is not.
In particular, we have found $\delta n_{c}$ to depend strongly on
$V$~--~it seems to vary monotonically with $V$, increasing when
$V$ increases. Let us finally comment on the effect of $V$ and $t_{\perp}$
on the charge imbalance between planes, $\Delta n$. Irrespective
of $V$ and $t_{\perp}$ we have always observed $|\delta n|<|\Delta n|$,
which means that charge carriers are always electron like in one plane
and hole like in the other. As expected, increasing/decreasing either
$V$ or $t_{\perp}$ leads to an increase/decrease of $\Delta n$.
\subsubsection{Understanding the asymmetry between planes}
\label{subsubsec_assymT0}
\begin{figure}[t]
\begin{centering}
\includegraphics[width=0.8\columnwidth]{fig6}
\par\end{centering}
\caption{\label{fig:bs} (Color online) Hartree-Fock bands for \emph{up} (full
lines) and \emph{down} (dashed lines) spin polarizations. Three different
cases are considered (from left to right): $U<U_{c}$, $U\gtrsim U_{c}$,
and $U\gg U_{c}$.}
\end{figure}
The asymmetry between planes regarding both charge and spin polarization
densities can be understood based on the Hartree-Fock bands shown
in Fig.~\ref{fig:bs}. The figure stands for $V=0.05\,\mathtt{eV}$
and $t_{\perp}=0.2t$, but can easily be generalized for other parameter
values.
It should be noted firstly that in the biased bilayer the weight of
the wave functions in each layer for near-gap states is strongly dependent
on their valence band or conduction band character.\cite{CNM+06,McC06,MSB+06}
Valence band states near the gap have their amplitude mostly localized
on layer~2, due to the lower electrostatic potential $-V/2$ {[}see
Eq.~(\ref{HV})]. On the other hand, near-gap conduction band states
have their highest amplitude on layer~1, due to the higher electrostatic
potential $+V/2$ for this layer {[}see Eq.~(\ref{HV})].
The case $U<U_{c}$ shown in Fig.~\ref{fig:bs} (left) stands for
the paramagnetic phase. The values $m=0$ and $\Delta m=0$ seen in
this phase are an immediate consequence of the degeneracy of \emph{up}
and \emph{down} spin polarized bands. The presence of a finite gap,
however, leads to the abovementioned asymmetry between near-gap valence
and conduction states. As a consequence, a half-filled bilayer would
have $n_{2}=(4+\Delta n)/2$ electrons per unit cell on layer~2 (electron
like charge carriers) and $n_{1}=(4-\Delta n)/2$ electrons per unit
cell on layer~1 (hole like charge carriers), with $\Delta n\neq0$.
Even though the system studied here is not at half-filling, as long
as $|\delta n|<|\Delta n|$ the carriers on layers~1 and~2 will
still be hole and electron like, respectively. Note that as $U$ is
increased the charge imbalance $\Delta n$ is suppressed in order
to reduce the system Coulomb energy, as can be seen in panel~(c)
of Fig.~\ref{Fig_PD_n}. From the band structure point of view a
smaller $\Delta n$ is the result of a smaller gap $\Delta_{g}$,
which means that increasing $U$ has the effect of lowering the gap.
Let us now consider the case $U\gtrsim U_{c}$ shown in Fig.~\ref{fig:bs}
(center). The degeneracy lifting of spin polarized bands gives rise
to a finite magnetization, $m\neq0$. Interestingly enough, the degeneracy
lifting is only appreciable for conduction bands, as long as $U$
is not much higher than $U_{c}$. This explains why the total polarization~$m$
and the difference in polarization between layers~$\Delta m$ have
similar values, $m\approx\Delta m$, as shown in panels~(a) and~(b)
of Fig.~\ref{Fig_PD_n}~--~as only conduction bands are contributing
to $\Delta m$, the spin polarization density is almost completely
localized in layer~1, where $m_{1}=(m+\Delta m)/2\approx m$, while
the spin polarization in layer~2 is negligible, $m_{2}=(m-\Delta m)/2\approx0$.
It is only when $U\gg U_{c}$ that valence bands become non-degenerate,
as seen in Fig.~\ref{fig:bs} (right). This implies that near-gap
valence states with \emph{up} and \emph{down} spin polarization have
different amplitudes in layer~2. As the valence band for \emph{down}
spin polarization has a lower energy the near-gap valence states with
spin \emph{down} have higher amplitude in layer~2 than their spin
\emph{up} counterparts. Consequently, the magnetization in layer~2
is effectively opposite to that in layer~1, i.e., $\Delta m>m$.
This can be observed in panels~(a) and~(b) of Fig.~\ref{Fig_PD_n},
where as $U$ is increased the magnetization of the two layers becomes
opposite.
We note, however, that the cases $U\gtrsim U_{c}$ and $U\gg U_{c}$
mentioned above are parameter dependent. For instance, the valence
bands can show an appreciable degeneracy lifting already for $U\gtrsim U_{c}$,
especially for small values of the $t_{\perp}$ parameter ($t_{\perp}\lesssim0.05t$).
In this case the magnetization of the two layers is no longer opposite,
with $\Delta m<m$. This can be understood as due to the fact that
as $t_{\perp}$ is decreased the weight of near-gap wave functions
becomes more evenly distributed between layers, leading not only to
a decrease in $\Delta n$ but also in $\Delta m$. As $U$ is further
increased the energy splitting between \emph{up} and \emph{down} spin
polarized bands gets larger, enhancing $\Delta m$. For $U\gg U_{c}$,
and depending on the parameters $V$ and $t_{\perp}$, the magnetization
of the two layers may become opposite even for small $t_{\perp}$
values.
\begin{figure}[t]
\begin{centering}
\includegraphics[width=0.98\columnwidth]{fig7}
\par\end{centering}
\caption{(Color online) Panels~(a), (b), and~(c) show the finite temperature
self-consistent solution for~$m$, $\Delta m$, and $\Delta n$,
respectively, with temperature measured in~\texttt{K}. The finite
temperature phase diagram of the biased bilayer in the $U$ vs. $T$
plane is shown in panel~(d). The constant parameters are $V=0.05$
\texttt{eV}, $t_{\perp}=0.2t$, $t=2.7$ \texttt{eV}, and $\delta n=0.00005$
\texttt{e$^{-}$/unit cell}.}
\label{Fig_PD_T}
\end{figure}
\subsection{Finite temperature}
\label{subsec_T}
Next we want to describe the phase diagram of the bilayer in the temperature
vs. on-site Coulomb interaction $U$ plane. This is done in Fig.~\ref{Fig_PD_T}
for a charge density $\delta n=5\times10^{-5}$ electrons per unit
cell. For temperatures ranging from zero to $T=$1.1~\texttt{K} we
studied the dependence of $m$, $\Delta m$ and $\Delta n$ on the
Coulomb on-site interaction $U$. First we note that the minimum critical
value $U_{c}$ is not realized at zero temperature. There is a reentrant
behavior which is signaled by the smallest $U_{c}$ for $T=0.06\pm0.02$\texttt{~K}.
For temperatures above $T\approx0.1$\texttt{~K} we have larger $U_{c}$
values for the larger temperatures, as can be seen in panel~(a).
The same is true for $\Delta m$, panel~(b). As in the case of Fig.~\ref{Fig_PD_n},
the value of $\Delta m$, at a given temperature and $U$ value, is
larger than $m$. Also the value of $\Delta n$, shown in panel~(c),
is larger than $\delta n$. Therefore we have the two planes presenting
opposite magnetization and the charge carriers being hole like in
one graphene plane and electron like in the other plane. In panel~(d)
of Fig.~\ref{Fig_PD_T} we present the phase diagram in the $T$
vs. $U$. Except at very low temperatures, there is a linear dependence
of $T$ on $U_{c}$. It is clear that at low temperatures, $T\simeq$
0.2\texttt{~K}, the value of $U_{c}$ is smaller than the estimated
values of $U$ for carbon compounds.\cite{Parr50,Baeriswyl86}
\subsection{Disorder}
Crucial prerequisite in order to find ferromagnetism is a high DOS at
the Fermi energy. The presence of disorder will certainly cause a
smoothing of the singularity in the DOS and the band gap
renormalization, and can even lead to the closing of the gap. We note,
however, that for small values of the disorder strength the DOS still
shows an enhanced behavior at the band gap edges.\cite{NN06,NNG+07}
The strong suppression of electrical noise in bilayer
graphene\cite{LA08} further suggests that in addition to a high
crystal quality -- leading to remarkably high mobilities\cite{MNK+07}
-- an effective screening of random potentials is at work. Disorder
should thus not be a limiting factor in the predicted low density
ferromagnetic state, as long as standard high quality BLG samples are
concerned.
Let us also comment on the next-nearest interlayer-coupling $\gamma_{3}$,
which in the unbiased case breaks the spectrum into four pockets for
low densities.\cite{MF06} In the biased case, $\gamma_{3}$ still
breaks the cylindrical symmetry, leading to the trigonal distortion
of the bands, but the divergence in the density of states at the edges
of the band gap is preserved.\cite{NNG+07} Therefore, the addition
of $\gamma_{3}$ to the model does not qualitatively change our result.
\section{Summary}
We have investigated the tendency of a biased bilayer graphene towards
a ferromagnetic ground state. For this, we used a mean-field theory
which allowed for a different carrier density and magnetization in
the two layers. We have found that in the ferromagnetic phase the
two layers have unequal magnetization and that the electronic density
is hole like in one plane and electron like in the other. We have
also found that at zero temperature, where the transition can be driven
by doping, the phase transition between paramagnetic and ferromagnetic
phases is \emph{first-order}.
\ack
TS, EVC, and NMRP acknowledge the financial support
from POCI 2010 via project PTDC/FIS/64404/2006, and the financial
support of Funda\c{c}\~ao para a Ci\^{e}ncia e a Tecnologia through Grant No.~SFRH/BD/13182/2003. Support from ESF via INSTANS is also acknowledged.
\section{References}
\bibliographystyle{unsrt}
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Q: Cathode bypass capacitor value in a Fender 5F1 Champ-style amplifier I am building a 5F1 Champ-style amplifier and I am wondering about what value of capacitor to strap to the cathode resistor? I have seen suggestions of 22 uF and 25 uF and I am not sure what value I should use.
A: I assume you are talking about the Cathode Resistor Bypass Capacitor. The purpose of these resistors in any transistor or tube based amplifier is to allow AC signals to "bypass" the cathode resistor and go directly to ground.
This results in a higher gain due to no "local cathode feedback". Typically this capacitor just requires. \$Z_c << R_c \$.
If this capacitor is to small, it will create a low cut filter, and remove frequency below a certain point. This is often unwanted behavior, but occasionally this effect is used for tone shaping, especially in some lead channels of guitar amplifiers.
For Tube amplifiers like this you will commonly see values of 22u to 100u.
Source:https://robrobinette.com/How_Amps_Work.htm
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\section{Introduction}
A new framework for the dynamical description of the late phase of gravitational collapse
has been recently proposed \cite{Kozameh10,Kozameh11}. In this framework one introduces physical null
coordinates based on the assumption that a suitable family of null surfaces are caustic free
in a neighbourhood of timelike infinity containing a portion of the black hole horizon $H$ and
future null infinity ${\mfs I}}\newcommand{\sO}{{\mfs O}^{+}$.
We consider an asymptotically flat spacetime at future null infinity $({\mfs M},g_{ab})$ containing a black hole.
Its conformal diagram is depicted in Figure \ref{SBH}. In the past of an open set of future null infinity
(${\mfs I}}\newcommand{\sO}{{\mfs O}^+$)---defined by those points for which
their Bondi%
\footnote{A Bondi retarded time $u$ is such that the sections $u$=constant at ${\mfs I}}\newcommand{\sO}{{\mfs O}^{+}$ (referred to as {\em Bondi cuts}) have an intrinsic metric given by (minus) the metric of the unit sphere. }
retarded time $u$
is in the range $u \in (u_0, \infty)$---we require the existence of a regular null function $w$
such that: $w=0$ at the horizon $H$, and $w<0$ in the region of interest.
Choosing a Bondi coordinate $u$ that coincides with
the center of mass Bondi cuts\cite{Kozameh10,Moreschi04} in the regime $u\to \infty$ limit,
we can uniquely fix the function $w$,
if we assume the topology of the black hole (BH) event horizon
$H$ is $S^2\times \mathbb{R}$ in that region.
Thus,
there exists a smooth null function $w=w(u)$ (unique up to constant scaling
in the region where one neglects $O(w^2)$ effects) such that
$w=0$ at the horizon $H$, $\dot w \equiv \frac{dw}{du}>0$, $w < 0$ for all $u$, and $\lim\limits_{u\rightarrow \infty} w = 0$.
This construction is precisely described in \cite{Kozameh12}, where spacetimes satisfying this assumption are defined
as {\em solitary black holes} (SBBs).
In a few lines, the null geodesic congruence defined by $\tilde \ell=du$ allows for the introduction of
an affine parameter $r$ used as a radial coordinate which is fixed by the requirement that it coincides asymptotically with
the luminosity distance (see equation (\ref{lumy}) below for a precise statement of this condition).
The surfaces $(r,u)=$constant are spheres which inherit natural spherical coordinates defined in the Bondi
cuts at ${\mfs I}}\newcommand{\sO}{{\mfs O}^{+}$ which label null rays of the congruence $\tilde \ell$. All this provides a coordinate system $(u,r,\theta,\phi)$ in the
exterior of the BH horizon.
However, the above coordinate system is not well behaved near the horizon ($u\to \infty$).
A good coordinate system can be constructed if one follows similar lines as above but describing the null geodesic
congruence instead in terms of $\ell=dw$. One can introduce an affine parameter $y$ along $\ell$ and fix the ambiguity in such
choice by requiring that the spheres $(w,y)=$constant coincide with the $(u,r)=$constant in the interior of the spacetime. Thus the angular coordinates
can be defined exactly in the same way as in the previous paragraph. With this one obtains the following relationship between the affine parameters
$r$ and $y$:
\nopagebreak[3]\begin{equation}\label{erre}
r=\dot w y+r_0(w),
\end{equation}
where $\dot w\equiv (dw/du)$. The coordinate $y$ will be used in what follows.
Under mild regularity conditions SBHs are then shown to posses a smooth global vector field
\begin{equation}
\chi \equiv \frac{\partial}{\partial u},
\end{equation}
which is a null geodesic generator at ${\mfs I}}\newcommand{\sO}{{\mfs O}^{+}$ and a null geodesic generator of the horizon $H$.
Moreover, at the horizon $H$, $\chi$ satisfies the equation,
\begin{equation}\nonumber
\chi^{a}\nabla_{a}\chi^{b} \equiv \kappa \chi^{b} ;
\end{equation}
where $\kappa$ is a generalized surface gravity.
Finally, one can show that \nopagebreak[3]\begin{equation}\label{main}
{
w(u)=-\exp{(-\kappa (u-u_0))}+\sO(\exp{(2au)}) } ,
\end{equation}
where $\exp(-\kappa u_0)$ is the rescaling freedom associated with the choice of origin for
the Bondi retarded time $u$. The last equation is a generalization of the Kruskal coordinate
transformation that appears in Schwarzschild and Kerr geometries.
SBHs have thus remarkable global features that can provide additional structure
in the study of the late phase of gravitational collapse in terms of the full non-linear
regime of Einstein's equations. The key question is whether the assumption of the existence of the physical null
function $w(u)$ is too restrictive admitting only situations of little physical interest.
The whole formalism rests on the assumption that there are no caustics,
in a small enough neighbourhood of $i^{+}$, in the congruence of generators of the
null surfaces $u=constant$ as one goes from ${\mfs I}}\newcommand{\sO}{{\mfs O}^{+}$ towards the past,
containing a final portion of $H$ and ${\mfs I}}\newcommand{\sO}{{\mfs O}^{+}$.
We will see that this problem does not appear in the final phase collapse provided by
the scenario developed in the framework of linear perturbations of stationary BH spacetimes.
This provides a strong indication that our assumptions are mild enough to admit physically interesting
situations.
As we have seen, there are two coordinates and null tetrad system that one can use near the black hole;
the tilde system that comes from the asymptotic description of the black hole, and the un-tilde
system that it is regular at the horizon. In what follows we work in the tilde system, in order
to make contact with calculations of other authors.
We will study in detail the behavior of the optical scalars $(\tilde\rho, \tilde \sigma)$
which depend explicitly on the incoming gravitational radiation $\tilde\Psi_0$, the
in-falling of matter $\tilde{\Phi}_{00}$, and implicitly in the outgoing gravitational
radiation field $\tilde\Psi_4^0$. Since we center the discussion in the behaviour of the
optical scalars in a neighborhood of the horizon, we will concentrate on the dependence on the fields
$\tilde\Psi_0$ and $\tilde{\Phi}_{00}$ directly.
In this work we will consider whether fields with typical tail
behaviour\cite{Gundlach94,Dafermos:2005yw} are admitted in our setting.
\begin{figure}[h!]
\centering
\includegraphics[clip,width=0.45\textwidth]{isolated-sch.png}
\caption{Conformal diagram representing the gravitational collapse
producing a {\em solitary black hole}. There is $w_0<0$ such that for $w_0<w<0$
there is a caustic free neighbourhood around $i^{+}$ containing a portion of the horizon $H$
and ${\mfs I}}\newcommand{\sO}{{\mfs O}^{+}$, if the spacetime decays
towards its final stationary state sufficiently rapidly.
}
\label{SBH}
\end{figure}
In figure \ref{SBH} it is shown the horizon $H$, future null infinity ${\mfs I}}\newcommand{\sO}{{\mfs O}^{+}$, timelike infinity $i^{+}$ and
the region of interest that is for $w > w_0$ and $y>y_0$; where the hypersurface $w_0$ is denoted by a dash line
and $y_0$ by a thick black line. It is important for the study to understand the behaviour of the fields
in a neighbourhood of the horizon but for finite values of $y$.
In \cite{Kozameh10} we point out that $\Psi_0=\dot w^2 \tilde \Psi_0$ and $\Phi_{00}=\dot w^2 \tilde \Phi_{00}$
must go as $y^{-3}$ on the horizon in order for the area of the horizon
to have an asymptotic finite value, in the limit $y\to \infty$.
Since we have not found in the literature a general discussion regarding the behaviour of $\tilde \Psi_0$
in the same asymptotic region near the horizon;
from our knowledge on the behaviour of $\Psi_0$ at the horizon and the behaviour of
$\tilde \Psi_0$ in the asymptotic region, we will assume the worst possible scenario.
At the horizon we know that $\Psi_0$ can behave as $y^{-3}$,
and for $w \neq 0$ this
means\footnote{Note that at the horizon, i.e. when $w=0$, the relation between $v$ and $y$ is
logarithmic. However, our study only concerns the region $w \neq 0$.}
that $y^{-3}\sim \dot w^3 v^{-3}$.
In the asymptotic region, for $r\to\infty$ one knows that $\tilde \Psi_0$ behaves as $r^{-5}$;
which means $v^{-5}$. So we will assume the worst admissible behaviour in the region of interest;
which is to take $\tilde \Psi_0 \sim v^{-3}$.
We will show in Section \ref{tails} that the late time behaviour predicted by the study of matter fields on the
Schwarzschild background imply
that $\tilde \Phi_{00}$ going as $v^{-4}$, i.e.; even faster than required by the above general argument.
Thus, in what follows we assume
\nopagebreak[3]\begin{equation}\label{pipi}
\tilde \Psi_0\sim v^{-3}\quad \text{and} \quad \tilde\Phi_{00}\sim v^{-4}.
\end{equation}
The article is organized as follows. In the following section we analyze the conditions for caustic formation.
In order to illustrate a way in which we could easily violate our assumptions---and in order to provide a
clear-cut intuition---we will provide
what is probably the simplest manner in which one can introduce caustics that invalidate our construction
in Section \ref{dust}.
We also argue in that section why such possibility is not of interest in the study of the final phase of
gravitational collapse.
In Section \ref{tails} we briefly review the results of \cite{Gundlach94}.
In Section \ref{tailseint} we show that the late time behaviour of gravitational collapse expected from the
linear perturbation technology is admited by our assumptions.
\section{The caustic freeness conditions}
The optical scalars equations can be expressed as
\begin{equation}\label{eq:thornrho-l}
\frac{\partial {\tilde \rho}}{\partial r}
=
{\tilde \rho} ^{2}
+{\tilde \sigma} \, \bar{\tilde \sigma}
+\tilde{\Phi}_{00} ,
\end{equation}
\begin{equation}\label{eq:thornsigma-l}
\frac{\partial {\tilde \sigma}}{\partial r}
=
2 {\tilde \rho} \, {\tilde \sigma}
+\tilde\Psi_0
,
\end{equation}
where $r$ is an affine parameter along the null geodesics $\tilde\ell=\partial_r$ which we will take to coincide
with the luminocity distance as one approaches future null infinity along the geodesics.
Let us concentrate in the behavior of ${\tilde \rho}$ and study the points in which it has a
divergent behavior: {\em caustics}.
Then one can write (\ref{eq:thornrho-l}) as
\begin{equation}\label{eq:thornrho-l2}
-\frac{\partial }{\partial r}\left( \frac{1}{{\tilde \rho}}\right) =
\frac{1}{{\tilde \rho} ^{2}}\frac{\partial {\tilde \rho}}{\partial r}
=
1
+
\frac{{\tilde \sigma} \, \bar{\tilde \sigma} +\tilde{\Phi}_{00}}{{\tilde \rho} ^{2}}
.
\end{equation}
The previous equation is equivalent to the following integral equation
\begin{equation}\label{inte}
- \frac{1}{{\tilde \rho}(r_\infty)} +\frac{1}{{\tilde \rho}(r)}
=
r_\infty - r
+
\int_r^{r_\infty}
\frac{{\tilde \sigma} \, \bar{\tilde \sigma} +\tilde{\Phi}_{00}}{{\tilde \rho} ^{2}}
dr'
.
\end{equation}
We would like to study this equation in the limit $r_\infty\to\infty$.
Now, because we have chosen $r$ to agree with the notion of luminocity distance in the large $r$ limit (which is possible if the spacetime is asymptotically flat at future null infinity),
one has that
\nopagebreak[3]\begin{equation}
\tilde \rho=-\frac{1}{r}(1+\frac{\tilde \rho_1}{r^2}+ O(r^{-3}))
\end{equation}
this implies that
\nopagebreak[3]\begin{equation}
\frac{1}{\tilde \rho}=-\frac{r}{(1+\frac{\tilde \rho_1}{r^2}+O(r^{-3}))}=-r + O(r^{-1})
\end{equation}
The previous equation implies that
\nopagebreak[3]\begin{equation}\label{lumy}
\lim_{r_\infty \to \infty} \; \left(\frac{1}{\tilde \rho(r_\infty)} + r_\infty \right) = 0 .
\end{equation}
In fact the previous condition is the precise definition of $r$ being asymptotically
the luminocity distance.
Therefore, equation (\ref{inte}) implies
\begin{equation}\label{roro}
{\tilde \rho}(r)
=-\frac{1}{r
-
\int_r^{\infty}
\frac{{\tilde \sigma} \, \bar{\tilde \sigma} +\tilde{\Phi}_{00}}{{\tilde \rho} ^{2}}
dr'}.
\end{equation}
Thus the condition that caustics appear at $r=r_c$ becomes simply
\nopagebreak[3]\begin{equation}\label{caustics}
\int_{r_c}^{\infty}
\frac{{\tilde \sigma} \, \bar{\tilde \sigma} +\tilde{\Phi}_{00}}{{\tilde \rho} ^{2}}
dr=r_c
\end{equation}
From the previous equation and from the positivity of the integrand involved one can conclude that the condition
\nopagebreak[3]\begin{equation}\label{caustics2}
\int_{r_1}^{\infty}
\frac{{\tilde \sigma} \, \bar{\tilde \sigma} +\tilde{\Phi}_{00}}{{\tilde \rho} ^{2}}
dr \le r_1
\end{equation}
guaranties the absence of caustics in the interval $r\in(r_c<r_1,\infty)$.
However, the presence of the expansion itself in the previous equation makes this condition a bit cumbersome.
We can turn the previous criterion for the absence of caustics into a
sufficient condition of a simpler and more useful form thanks to the validity
of the following statement.
\vskip.2cm
\noindent {\bf Lemma:} In the caustic free region $r\in (r_c,\infty)$ the following inequality holds
\nopagebreak[3]\begin{equation}
|\tilde \rho|\ge \frac{1}{r} .
\end{equation}
The proof follows directly from equation (\ref{roro}), the fact that $0\le {\tilde \sigma} \, \bar{\tilde \sigma} +\tilde{\Phi}_{00}$, and the fact that $r\in (r_c,\infty)$.
More explicitly,
\nopagebreak[3]\begin{eqnarray}
|\tilde \rho|&\ge& \frac{1}{r} \Longleftrightarrow -\tilde \rho\ge \frac{1}{r}\n \\
&\Longleftrightarrow& r\ge r-\int_{r}^{\infty}
\frac{{\tilde \sigma} \, \bar{\tilde \sigma} +\tilde{\Phi}_{00}}{{\tilde \rho} ^{2}}
dr' \ge 0,
\end{eqnarray}
where we have used the positivity stated in the last inequality which follows from the
condition that $r\in (r_c,\infty)$. The condition that one is in the caustic free region is essential $\square$.
\vskip.2cm
Using the previous result we can write a sufficient condition for the
non existence of caustics in the interval $r\in (r_1,\infty)$ as follows
\nopagebreak[3]\begin{equation}\label{caustics1}
\int_{r_1}^{\infty}
({{\tilde \sigma} \, \bar{\tilde \sigma} +\tilde{\Phi}_{00}} ) r^2
dr \le r_1.
\end{equation}
The previous condition on the strength of ${{\tilde \sigma} \, \bar{\tilde \sigma} +\tilde{\Phi}_{00}} $ is clearly
stronger than (\ref{caustics2}). This is why in contrast to the latter this is a sufficient condition (its violation may not imply that there are caustics in $(r_c<r_1,\infty)$).
However, if (\ref{caustics1}) is satisfied then we can assure that there are no caustics in the region of interest. This last condition will be central in the
proof of our main result in the following section.
\section{Dust}\label{dust}
In this section we show that a grain of sand can destroy our construction.
This simple example will provide intuition on what the nature of our problem is.
At the same time we shall see by the end of this section that this example is physically irrelevant for the
physical situation that one would like to describe in our framework.
We can model a grain of sand (or a planet) at some coordinate $r_d(w)>r_H$ outside de BH horizon
by a Ricci spinor component
\nopagebreak[3]\begin{equation}
\Phi_{00}=\frac{\epsilon}{r_H}\delta(r-r_d(w)),
\end{equation}
where $\epsilon$ is a dimensionless parameter measuring the strength of the dust particle.
For the next discussion it is enough to use the fact that $\tilde\sigma$ is bounded by $\frac{\alpha}{r^2}$,
in the asymptotic region,
for an appropriate $\alpha$; however for simplicity we will assume next that $\tilde\sigma=0$.
This will not change the qualitative aspects of the discussion.
Then, condition (\ref{caustics}) becomes
\nopagebreak[3]\begin{equation}\label{prima}
\epsilon \frac{r_d^2}{r_H}
\le r_c.
\end{equation}
In order to define the region where we will proof that there are no caustics we need to recall that
\nopagebreak[3]\begin{eqnarray}
&& r=\dot w y+r_{H}\n \\
&&=-\frac{w}{2 r_H} y+r_H.
\end{eqnarray}
In order to show that there is a caustic free region around $i^{+}$, containing both a portion ${\mfs I}}\newcommand{\sO}{{\mfs O}^{+}$ and the
black hole horizon, it is sufficient to show that for a given $y_1$ there exist an $w_0\le 0$ such that for all $w>w_0$ there are no caustics in the region
\nopagebreak[3]\begin{equation} r\in (-\frac{w}{2 r_H} y_1+r_H, \infty).\end{equation}
Without loss of generality, and in order to simplify some expressions, we take $y_1=2 r^2_H$ from now on.
The region of interest now becomes $r\in (r_H (1-w), \infty)$.
Thus, from (\ref{prima}), the caustic free condition becomes
\nopagebreak[3]\begin{equation}
\epsilon r_d
\le r_H(1-w).
\end{equation}
Conversely, the previous equation tells us that it is very easy to introduce caustics
that would completely invalidate the construction; it suffices to take a dust particle that is sufficiently far away and sufficiently strong.
In particular if we take $\label{sand} \epsilon r_d >r_H(1-w)$ then there will be a caustic line that goes all the way up to $i^{+}$.
Therefore, we have shown that our construction breaks down if a suitable grain of dust is brought in.
Is this a serious problem? We now argue that it is not; as the above situation
bears not interest for the study of the physics of gravitational collapse we plan to study.
The reason is that the problematic grain of sand (which could also model a planet or a star)
must stay outside the black hole $r_d>r_H$ for all $w$; hence, it is a compact object that is never absorbed
by the BH and follows a timelike trajectory all the way up to $i^{+}$. The only physically acceptable
possibility is then that the object is not gravitationally bound to the BH. Such possibility is of course
physically viable but it introduces an irrelevant complication to the problem of studying the final stage
of gravitational collapse. Therefore, it is advisable that our definition of SBH rules out such
situation by assumption.
\section{Tails}\label{tails}
Gundlach, Price and Pullin \cite{Gundlach94} have shown that the spherical harmonic $\ell$ mode of
a scalar field $\phi_0^{\ell}$ satisfying the wave equation on a Schwarzschild
background in the late time behaviour for $u\to \infty$ is
\nopagebreak[3]\begin{equation}
\phi_0^{\ell}=\frac{\Upsilon_0}{v^{P+2\ell+1}},
\end{equation}
where $\Upsilon_0$ is a constant, where $P=1,2$.
If such scalar field is used as matter source for Einsteins equation then it produces a Ricci
scalar $\tilde\Phi_{00}$ whose late time behaviour is
\nopagebreak[3]\begin{equation}
\tilde \Phi_{00} \approx \frac{1}{v^4}+O(v^{-5}) .
\end{equation}
As explained in expression (\ref{pipi}),
$\tilde \Psi_0$ goes like $1/v^3$. From the optical equations it follows that $\tilde\sigma$ goes like $1/v^2$.
This means that the late time behaviour of the integrand in (\ref{caustics})
can be expressed as:
\nopagebreak[3]\begin{equation}
[{{\tilde \sigma} \, \bar{\tilde \sigma} +\tilde{\Phi}_{00}}] = \frac{\epsilon r_H^2}{v^4}+O(v^{-5});
\end{equation}
where for future use we have introduced the dimensionless constant $\epsilon$ to parametrize
the leading order term.
\subsection{Caustics in late phase}\label{tailseint}
According to studies of linear perturbations of Schwarzschild geometries \cite{Gundlach94,Dafermos:2005yw}
one has that
\nopagebreak[3]\begin{equation}
[{{\tilde \sigma} \, \bar{\tilde \sigma} +\tilde{\Phi}_{00}}] (u\to\infty, v)= \frac{\epsilon r_H^2}{v^4} ,
\end{equation}
where $u=t-r_*$ and $v=t+r_*$ for \nopagebreak[3]\begin{equation} r_*=r+r_H\log\left(\frac{r-r_H}{r_H}\right)\end{equation}
the usual tortoise coordinate, and $r_H=2M$ the radius of the horizon. From this we get that $v=u+2r_*$ hence
\nopagebreak[3]\begin{equation}
v=u+2r+2r_H\log\left(\frac{r-r_H}{r_H}\right).
\end{equation}
By making the same choice of region as underneath Equation \ref{prima} in the previous section,
the caustic free condition (\ref{caustics}) becomes
\nopagebreak[3]\begin{eqnarray}
&&\int\limits_{r_H (1-w)}^{\infty} \frac{\epsilon r^2_H r^2}{v^4} dr=\n \\
&& \int\limits_{r_H (1-w)}^{\infty} \frac{\epsilon r^2_H r^2}{[u+2r+2r_H\log\left(\frac{r-r_H}{r_H}\right)]^4} dr\le r_H (1-w).
\n \end{eqnarray}
The previous condition can be simplified by introducing the variable $x=r/r_H$, from which one gets
\nopagebreak[3]\begin{equation}\label{cfc}
\epsilon F(w)=\epsilon \int\limits_{(1-w)}^{\infty} \frac{ x^2}{[x+\log\left(\frac{1-x}{w}\right)]^4} dx\le {16 (1-w)},
\end{equation}
where we used that $u=-{2 r_H}\log(-w)$. The function $F(w)$ is shown in figure \ref{f(x)}.
\begin{figure}[!h
\centering
\includegraphics[clip,width=0.48\textwidth]{tails-and-caustics.pdf}
\caption{The form of the function $F(w)$ guaranties that there exists a $w_0$ such that for $0>w>w_0$ the caustic
free condition (\ref{cfc}) is satisfied.
The dashed line represents the function $16(1-w)/200$ which explicitly shows that there is a caustic free region
in the case $\epsilon=200$.
All the other values of $\epsilon$ look qualitatively the same.
}
\label{f(x)}
\end{figure}
It is clear from its behaviour close to $w=0$ that there is always some
$w_0$ such that there are no caustics in the region $r\in (r_H (1-w), \infty)$ for $0\le w<w_0$.
This concludes the proof that there is a caustic free region in a neighbourhood of $i^{+}$ bounded
by a portion of ${\mfs I}}\newcommand{\sO}{{\mfs O}^{+}$ and the horizon $H$.
\subsection*{Acknowledgements}
We acknowledge financial support from CONICET, SeCyT-UNC, Foncyt and by
the Agence Nationale de la Recherche; grant
ANR-06-BLAN-0050. A.P. was supported by {\em l'Institut Universitaire de France}.
\begingroup\raggedright
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{"url":"https:\/\/mathematica.stackexchange.com\/questions\/160013\/displaying-a-vector-as-a-length-at-an-angle","text":"Displaying a vector as a length at an angle\n\nThis may prove to be simple, but i have not been able to figure it out (nor am i sure if it can even be done in Mathematica).\n\nI use vectors in electrical engineering as a polar, length and angle (in degrees). But Mathematica insist on displaying vectors in polar as a length and angle (in radians or something) and i need the degrees for my results.\n\nURg = AngleVector[{UnitConvert[I*Rg, \"Volts\"],\nQuantity[30, \"AngularDegrees\"]}]\n{Quantity[2.07846, \"Volts\"], Quantity[1.2, \"Volts\"]}\n\n\nI have tried something like this, but as far as i can see and understand, the program recognizes the degrees in the formula, but won't display it as degrees in the result.\n\nI looked up function like ToPolarCoordinates, but it still doesn't work properly.\n\nHooping someone has some guidance or at least some good suggestions.\n\n\u2022 I'm not totally sure I understand the problem. From what I understand, AngleVector[{r,theta}] gives the Cartesian coordinates of the polar coordinate {r,theta}, so to get the angle \"back\", one would have to do something like N[ArcTan @@ (QuantityMagnitude@URg)\/Degree]. Please correct me if I've misunderstood, though! \u2013\u00a0Anne Nov 15 '17 at 21:46\n\u2022 Try \"Degree\" not \"AngularDegrees\". \u2013\u00a0David G. Stork Nov 15 '17 at 21:47\n\u2022 You could probably modify the solutions here so that angles are shown in degrees instead of radians. \u2013\u00a0J. M.'s discontentment Nov 16 '17 at 1:30\n\u2022 @Anne That way i can calculate the angle afterwards yea, but i want the initial definition to recognize the angle in degrees Something like this {Quantity[2.07846, \"Volts\"] \\[Angle] Quantity[30, \"AngularDegrees\"]} Basiclly i need the polar coordinates displayed as {r,Degree} \u2013\u00a0Jamie Nov 16 '17 at 7:20\n\u2022 @DavidG.Stork That didn't change anything sadly, tried it before posting here and just double checked. \u2013\u00a0Jamie Nov 16 '17 at 7:23\n\nPutting this in an answer for space\/formatting capabilities.\n\nIf you're starting with some complex number, say $z=1+i\\sqrt{3}$ for example, AbsArg will get you the polar representation of $z$ (in radians):\n\nz = 1 + Sqrt[3]*I;\nAbsArg[z]\n\n(*{2, \\[Pi]\/3}*)\n\n\nYou can then convert this to degrees (many ways to do this of course):\n\nzpolar = MapAt[#*180\/Pi &, AbsArg[z], 2]\n\n(*{2, 60}*)\n\n\nIf you want the end results to be \"Quantities\":\n\nMapThread[Quantity, {zpolar, {\"Volts\", \"AngularDegrees\"}}]\n\n(*{Quantity[2, \"Volts\"], Quantity[60, \"AngularDegrees\"]}*)\n\n\nYou can also do something similar if you use Quantities in your z, I believe.\n\nHopefully this is something along the lines of what you had in mind!\n\nvec={4,5};\n\n{r, Theta} = ToPolarCoordinates[vec]\n(*{Sqrt[41],ArcTan[5\/4]}*)\n\npolarvec = {r, N[Theta]\/\u00b0}\n(*{Sqrt[41],51.3401}*)\n\n\nNow your vector is in {r, theta} with theta in degrees.\n\nIf you want to add two vectors, or do about any other operations on them, Mathematica really wants the vector in rectangular coordinates. Kind of kludgy, but you can do something like:\n\npolarURG = {2.4 V, 30 \u00b0}\npolarUXL = {4.8 V, 60 \u00b0}\n\n$Assumptions = V > 0 (FromPolarCoordinates[polarURG] + FromPolarCoordinates[polarUXL]) \/\/ ToPolarCoordinates[#] & \/\/ Simplify PolarSum = {%[[1]], N[%[[2]]]\/\u00b0} (*{6.982350 V,0.874478}*) (*{6.9823509 V,50.103909}*) \u2022 Okay, so i got that to work for the most part. But i ran into the problem, that with having to different vectors defined this way, i couldn't get mathematica to add them togehter as vectors? it just adds the values togehter as in polarURg={2.4V,30\\[Degree]}; polarUXL={4.8V,60\\[Degree]}; Esys=polarURg+polarUXL; Esys={7.2V,90\\[Degree]} \u2013 Jamie Nov 19 '17 at 17:36 \u2022 So this mostly works, but it seems like i can't use Quantity values this way? I am guessing that it's because of the $Assumption=V>0 which i tried to use as $Assumption=\"Volts\">0, but i don't fully understand what this is supposed to do (i believe it is used for the simplify function?). \u2013 Jamie Nov 20 '17 at 11:03 \u2022 I have it setup as polarURg={Quantity[2.4,\"Volts\"],30\\[Degree]} polarUXL={Quantity[4.8,\"Volts\"],60\\[Degree]} $Assumption=\"Volts\">0 and this gives me {Quantity[6.98235, \"Volts\"], ArcTan[Quantity[4.47846, \"Volts\"], Quantity[5.35692, \"Volts\"]]} and Quantity[6.98235, \"Volts\"], ArcTan[Quantity[4.47846, \"Volts\"], Quantity[5.35692, \"Volts\"]]\/\\[Degree] \u2013\u00a0Jamie Nov 20 '17 at 11:03","date":"2020-10-23 11:20:25","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.7781966924667358, \"perplexity\": 1658.9058397209012}, \"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-2020-45\/segments\/1603107881369.4\/warc\/CC-MAIN-20201023102435-20201023132435-00177.warc.gz\"}"}
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Home / Chinese Modern
The Heroic and the Quotidian
Post-Contemporary Interventions
More about this series
Book Pages: 402 Illustrations: 17 figures Published: April 2000
Author: Xiaobing Tang
Asian Studies > East Asia, Cultural Studies, Literature and Literary Studies > Literary Criticism
Chinese Modern examines crucial episodes in the creation of Chinese modernity during the turbulent twentieth century. Analyzing a rich array of literary, visual, theatrical, and cinematic texts, Xiaobing Tang portrays the cultural transformation of China from the early 1900s through the founding of the People's Republic, the installation of the socialist realist aesthetic, the collapse of the idea of utopia in the aftermath of the Cultural Revolution, and the gradual cannibalization of the socialist past by consumer culture at the century's end. Throughout, he highlights the dynamic tension between everyday life and the heroic ideal.
Tang uncovers crucial clues to modern Chinese literary and cultural practices through readings of Wu Jianren's 1906 novel The Sea of Regret and works by canonical writers Lu Xun, Ding Ling, and Ba Jin. For the midcentury, he broadens his investigation by considering theatrical, cinematic, and visual materials in addition to literary texts. His reading of the 1963 play The Young Generation reveals the anxiety and terror underlying the exhilarating new socialist life portrayed on the stage. This play, enormously influential when it first appeared, illustrates the utopian vision of China's lyrical age and its underlying discontents—both of which are critical for understanding late-twentieth-century China. Tang closes with an examination of post–Cultural Revolution nostalgia for the passion of the lyrical age.
Throughout Chinese Modern Tang suggests a historical and imaginative affinity between apparently separate literatures and cultures. He thus illuminates not only Chinese modernity but also the condition of modernity as a whole, particularly in light of the postmodern recognition that the market and commodity culture are both angel and devil. This elegantly written volume will be invaluable to students of China, Asian studies, literary criticism, and cultural studies, as well as to readers who study modernity.
"Chinese Modern is a broad-ranging book, sweeping grandly across the 20th century and examining a range of genre including fiction, film, drama, poster art, oil paintings and advertisements . . . . [E]njoyable and challenging in its scope and insights." — Louise Edwards, The China Journal
"Chinese Modern is both a noteworthy attempt to bring the experiences of Chinese modernity to bear on theoretical discourses of modernity and a dexterous use of western theories to illuminate the workings of modernity in Chinese literature and culture." — Ming-Yan Lai , Modern Fiction Studies
"[A] very rich collection, covering a huge area of knowledge . . . . Chinese Modern pioneers the study of contemporary Chinese writers who have so far received little attention in the English-reading world, and opens up plenty of possibilities for future discussion and research." — Michel Hockx , Harvard Journal of Asiatic Studies
"Rich, imaginative and immensely useful for teaching courses . . . ." — Journal of the Royal Anthropological Institute
"Highlights of the book include the author's original readings of Wu Jianren and Wang Anyi and often brilliant cultural historicizing. . . . A rewarding decode. . . ." — J. C. Kinkley , Choice
"Consistently thoughtful and lucid. . . . Chinese Modern constitutes an important contribution to the study of modernism in Chinese literature and culture. It effectively combines insightful close readings of a wide range of canonical and more 'popular' literary and cinematic works with an intelligent engagement with relevant theoretical paradigms." — Carlos Rojas , Journal of Asian Studies
"Containing a series of penetrating analyses of landmark cultural works from the entire course of the twentieth century, Chinese Modern represents the most comprehensive account of modern Chinese literature that has ever been published in English. Tang also illuminates—like no one has before—the various ways in which the looming imperative of modernity has left its image on the imagination of modern China." — Theodore Huters, University of California at Los Angeles
"Read Chinese Modern for a journey through China's 'long twentieth century.' Xiaobing Tang as guide shows how imaginative sympathy for one's subject nourishes critical acuity." — Norma Field, University of Chicago
Xiaobing Tang is Associate Professor in Modern Chinese Literature at the University of Chicago.
List of Illustrations ix
Acknowledgments xi
Part I 11
Part II 163
Afterword 341
Selected Bibliography 357
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{"url":"https:\/\/practice.geeksforgeeks.org\/problems\/fill-up-buckets3500\/1","text":"Fill up buckets\nMedium Accuracy: 28.57% Submissions: 7 Points: 4\n\nGiven N\u00a0buckets and infinite number of balls. The maximum capacity of each bucket is given in an array capacity[]. Find the number of ways to fill the buckets with balls such that each bucket has atleast 1 ball and all the buckets have distinct number of balls in them.\n\nExample 1:\n\nInput:\nN = 1\ncapacity = [6]\nOutput: 6\nExplanation: Since there is only one\nbucket.It may hold any number of balls\nranging from 1 to 6.\n\n\n\nExample 2:\n\nInput:\nN = 2\ncapacity = [5, 8]\nOutput: 35\nExplanation: The first bucket can contain\n1 to 5 number of balls whereas second bucket\ncan be assigned with 2 to 8 number of balls\ni,e total there are 35 ways.\n\n\nYou don't need to read or print anything. Your task is to complete the function\u00a0totalWays()\u00a0which takes N and capacity[]\u00a0as input parameters and returns the number of\u00a0possible ways to\u00a0fill\u00a0the\u00a0buckets. Since the answer may be very large, calculate the answer\u00a0modulo 10^9+7.\n\nExpected Time Complexity:\u00a0O(N)\nExpected Space Complexity:\u00a0O(1)\n\nConstraints:\n1 <= n\u00a0<= 100000\n1 <= nums[i] <= 100000","date":"2020-09-28 13:22:07","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.5100423097610474, \"perplexity\": 1852.1908325932143}, \"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-2020-40\/segments\/1600401600771.78\/warc\/CC-MAIN-20200928104328-20200928134328-00497.warc.gz\"}"}
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# Cover
Valour at Vimy Ridge
Canadian Heroes of World War 1
Tom Douglas
James Lorimer & Company Ltd., Publishers
Toronto
# Dedication
To James Ripley Jackson Douglas.
We'll always have Amsterdam!
In Flanders Fields
In Flanders fields the poppies blow
Between the crosses, row on row
That mark our place; and in the sky
The larks, still bravely singing, fly
Scarce heard amid the guns below.
We are the Dead. Short days ago
We lived, felt dawn, saw sunset glow,
Loved and were loved, and now we lie
In Flanders fields.
Take up our quarrel with the foe:
To you from failing hands we throw
The torch; be yours to hold it high.
If ye break faith with us who die
We shall not sleep, though poppies grow
In Flanders fields.
Lieutenant Colonel John McCrae (1872–1918)
Prologue
His captors didn't respond, although one of them made an involuntary move with his right hand toward the pistol holstered at his waist. The German smirked at the reaction. "I have also heard Canadians are a collection of drunken cowboys who would shoot an unarmed man without a second thought."
The man's gaze never wavered as he stared at the Canadian officer in charge. "Well, you had better save your ammunition because you will need it. The attack you are planning is the worst-kept secret of the war. It is also the biggest joke the Kaiser has heard in a long time."
The three Canadians in the dimly lit dugout kept silent, not even flinching when another artillery shell shook the foundations, cascading debris down from the rafters. The soldiers appeared content to let their prisoner keep talking. After all, he might blurt out some kernel of information that could save a few lives in the days ahead.
"Your British and French friends tried the same thing and lost over a hundred thousand men," the German continued. "What makes you think a ragtag collection of farmers and fishermen from your country can do any better?"
The Canadian officer in charge made a dismissive sideways gesture with his head, and his two companions grabbed the enemy officer by the arms, preparing to lead him away. He shook off their grasp, pulled his greatcoat tightly around his shoulders, and sneered at his adversary.
"Maybe, just maybe, your men will reach the top of Vimy Ridge," he hissed. "But they will be able to ship the survivors home to Canada in a rowboat."
Introduction:
The War to End All Wars
It was also known as the "Great War," but, as with most military slogans, the phrase proved to be both naïve and totally off the mark. History has shown that World War I was not the war that ended all wars — it actually sowed bitter seeds for future conflicts. In addition, with an estimated body count of more than 15 million combatants and civilians, as well as over 22 million wounded, there certainly was nothing great about the bloodbath that lasted from July 28, 1914, until November 11, 1918.
The fuse that set off millions of tonnes of explosives was the assassination of Archduke Franz Ferdinand, heir to the Austro-Hungarian throne, in the Bosnian capital of Sarajevo. When Austro-Hungary blamed Serbia for the assassination and threatened retaliation, Serbia's ally, Russia, began to assemble its troops. Bound by a treaty with Austro-Hungary, Germany considered this mobilization an act of aggression, and declared war on Russia. This brought the allied powers of Great Britain and France into the conflict, lined up with Russia against the central powers led by Germany and Austro-Hungary.
The deadly domino effect caused by a lone assassin would hold the entire world in a death grip of insanity for the next four and a half years — despite the widespread belief from day one that the war would last no more than a few months. Those countries that became engaged in the conflagration would lose an entire generation of their youth, with men and women from all walks of life perishing in the trenches of Flanders and other muddy battlefields across northern Europe.
It would take the combatant countries years to regain what they had lost, both physically and financially — just in time for renewed acts of aggression that would lead to another, and even more devastating, world conflict.
It is difficult to sift historically through the ruins of World War I and come up with anything positive. But one thing Canada gained from participating in this barbarous affair was a sense of pride at having stood shoulder to shoulder with some of the most powerful nations on earth to defeat a common enemy. And that sense of pride began to take shape on April 9, 1917, at the Battle of Vimy Ridge.
Chapter 1
Innocence Lost
Church bells rang. People danced in the streets. Brass bands played martial music as zealous youths shouldered wooden rifles and marched through town to the cheers and applause of adoring crowds. Recruiters turned a blind eye as young boys barely into their teens lied about their age so they could sign up to go overseas and fight against the Hun.
This was the nationalistic fever that gripped Canada in the summer of 1914 when war was declared against Germany. Since Confederation in 1867, Canadians had experienced little in the way of armed conflict other than sending a battalion of volunteers to assist Great Britain in the Boer War in 1899. That three-year conflict in South Africa involved some 7,000 Canadians, including 12 women nurses, and resulted in 267 Canadian fatalities. It had been considered a noble cause, and the romance of battle had grown in the minds of most Canadians every year since. By 1914, when war clouds were forming over Europe, the majority of able-bodied males from Victoria to Halifax were ready, in fact eager, to fight for king and country.
The Canadian government — perhaps realizing that the country was ill prepared militarily, with a militia numbering just over 3,000 and a fledgling navy — had been little more than lukewarm in its official reaction to the situation overseas. When Britain declared war against Germany on August 4, 1914, Canada issued this statement: "If unhappily war should ensue, the Canadian people will be united... to maintain the honour of the empire."
But a groundswell of patriotism flared up across the land like a grass fire on a tinder-dry prairie and, by August 10, authorization had been granted for the formation of the Princess Patricia's Canadian Light Infantry Regiment. The regiment was named after the only daughter of the then-serving governor general of Canada, Prince Arthur, the Duke of Connaught, who was the third son of Queen Victoria. The ranks of the PPCLI were filled in just over a week, made up for the most part of trained ex-regular soldiers who had served in the British army during the Boer War. The PPCLI was financed privately, mainly by Captain Hamilton Gault of Montreal. On December 21, 1914, the regiment, under the command of the governor general's military secretary Colonel F.D. Farquhar DSO, set foot on French soil and was the first Canadian unit committed to battle.
Back home, volunteers continued to pour into the recruiting stations and, by late September, more than 30,000 hastily trained soldiers marched out of the Valcartier mobilization camp near Quebec City to board trains for the East Coast. On October 1, officers and men stomped proudly up the gangplank of 33 ocean liners — including the RMS Olympic, sister ship to the RMS Titanic — and set sail for England under the protection of British Royal Navy escort ships. It was the largest convoy ever to cross the Atlantic. The enthusiastic Canadians crowded the railings of these transport ships as the flotilla came within sight of the English coast.
The Rugged Reality
It would not take long for that enthusiasm to wane. Once the gigantic task of offloading men and equipment had been accomplished, the troops found themselves in far less than pleasant surroundings on Salisbury Plain. They comprised the 1st Canadian Infantry Division, under the command of a British officer, Lieutenant General E.A.H. Alderson, and they soon woke up to the reality of army life. Salisbury Plain is a cold, windswept, and perpetually rain-drenched chalk plateau of about 780 square kilometres in south-central England. Conditions there were far more rugged than anything the soldiers had experienced in Canada. The training was exhausting, as every effort was expended to get these "green" Canadians into shape for the fighting ahead.
The 1st Division crossed the English Channel in February 1915 and immediately saw action at Ypres, Belgium. Over a two-week period, the Canadians learned the realities of warfare, suffering 5,500 casualties. But the determined troops kept on coming. The 2nd Canadian Infantry Division was formed in Great Britain in May 1915 with the influx of a huge contingent of troops from Canada. This new division was pressed into service in September and spent a long and bitterly cold winter in Belgium. Their ranks would be severely bloodied in six major battles in 1916.
With two divisions of Canadians overseas, the Canadian Corps was formed on September 13, 1915, with General Alderson in command. The 3rd Division was added to the Corps in December 1915 and the 4th Division in April 1916. A 5th Division would be organized later, but it was used strictly for guard duty in Great Britain.
Tragedy and a clash of personalities resulted in several high-level changes in command within the Corps before the Canadians arrived in the Vimy area late in 1916. The 3rd Division's commanding officer, Brigadier General Malcolm Smith Mercer, was the victim of "friendly fire" at Mount Sorrel near the Belgian town of Ypres when shrapnel from a wayward British artillery shell pierced his heart. He went into the history books with the dubious distinction of being the highest-ranking Canadian officer killed in action in World War I.
The 1st Division CO, Brigadier General Arthur Currie, took on the added responsibilities of commander of the entire Canadian Corps with the sacking of General Alderson by Canadian Minister of Militia and Defence Sir Sam Hughes. When the Corps suffered 1,300 casualties at the battles of St. Eloi in April 1916, Hughes, who was openly hostile toward Alderson, is thought to have seized upon this disaster as an excuse to dump the Corps commander. Hughes himself would soon be replaced, largely because of his abrasive, bull-headed way of conducting the affairs of war.
With such major battles as St. Eloi, Ypres, Mount Sorrel, and Sanctuary Wood, the raw recruits who had flocked to the recruiting stations back in Canada were battle-hardened and embittered soldiers before two years had passed. Government censors, by slashing newspaper accounts of the carnage at the front and by blacking out most of the text of letters from the troops to their family and friends, would keep the naïve townsfolk back home pretty much in the dark about the horrors their young people were suffering. And because they still had no idea what they were in for, able-bodied Canadians continued to believe the propaganda and flock to the recruiting stations.
Those patriotic young men who had dreamed of glory in a far-off land during the heady summer days of 1914, and those who followed in the ensuing months, soon found that they had been sold a bill of goods. There was nothing glorious about seeing the man next to you blown to bits by an enemy mortar shell. And the stirring strains of the military march you'd heard as you paraded down the main street of your home town were blotted out of your memory by the incessant roar of artillery, the screech of an incoming mortar shell, and the chatter of death-dealing machine guns.
But as terrifying as actual combat might be, it was often a welcome respite from existing like an animal in the filthy, disease-laden trenches that scarred the once bucolic landscape of northern France and Belgium. Skirmishes were fought that might gain one side or the other a few metres of territory, but this victory was usually short-lived, with the enemy counterattacking a few hours later and taking back what ground they had lost. Life in the muddy, shell-pummelled fields of northeastern Europe was a horrible nightmare, worse than anything imaginable. It was a nightmare from which there was no awakening and one that it seemed would end in death — or a disfiguring wound that was a ticket home.
World War I, unlike any conflict before or since, turned into a huge stalemate with the opposing sides sometimes less than 25 metres apart, living like mud-encrusted moles in a vast series of ditches and tunnels.
Trench warfare, according to those who lived through it, was hell on earth.
Chapter 2
Trench Warfare
The recruiting posters and rah-rah newsreels painted a romantic view of what it was like to go to war. The poster boys were clean-cut lads dressed in freshly pressed uniforms sipping wine at outdoor cafés in Paris. The French mesdemoiselles sitting with them were breathtakingly beautiful, and they gazed adoringly at these conquering heroes who had driven the beastly enemy from their country.
There were no scenes of a corpse-strewn no man's land — that stretch of barren ground that separated the trenches of the antagonists. No close-ups of a diseased rat crawling over your face as you tried desperately to grab a few hours' sleep before having to go "over the top" to raid the enemy trench just metres away from yours. No mention of a German sniper waiting for you to emerge from the relative safety of a muddy shell hole so that he could blow your head off. No depiction of life in the trenches, where foot rot, lice, and the stench of death were your constant companions.
Trench warfare, a unique World War I phenomenon in which opposing troops would play deadly cat and mouse games with a nearby enemy for weeks and months on end, became the norm after the bloody Battle of the Marne in September 1914. That epic confrontation dashed the plans of the German army to capture nearby Paris. Forced to retreat north to the River Aisne, General Erich von Falkenhayn ordered his troops to dig trenches to protect themselves from advancing British and French forces.
When the Allies realized that the German trenches were formidable obstacles that could not be readily overtaken, they too began digging in. After a few months, these opposing trenches stretched from the North Sea to the Swiss frontier. For the next three years, neither side was able to advance more than a few kilometres along this line that came to be known as the Western Front. But living conditions in what amounted to little more than deep ditches wasn't anything like the cozy bungalows and college dorms and rural family homesteads the young Canadians had left behind. Even a trapper's shack in Canada's wild North seemed like a palace to those dug in on the cruel fields of northern France and Belgium.
War diarists, historians, and dramatists, in hindsight, have minutely detailed the daily life of the common soldier — British, French, Canadian, and German — on the Western Front. It is not a pretty picture. But no story about World War I — and in particular the magnificent achievement of the Canadians at Vimy Ridge — would be complete without a basic understanding of these inhuman and seemingly insurmountable obstacles that had to be overcome on the road to victory.
Life In The Trenches
The excavations along the Western Front were built in threes — the front-line, support, and reserve trenches. This trio of long, snake-like ditches covered between 220 and 550 metres of ground from front to back and could wind for several kilometres across the terrain parallel to the enemy fortifications. They were not dug in a straight line because the occupants needed the relative safety of a sharp turn to hide behind so they could draw a bead on an approaching enemy soldier if their trench was overtaken.
Running perpendicular to these channels were communication trenches for fresh troops, equipment, and supplies to move up to the line and wounded soldiers to be taken to the rear. The trenches were about 2.5 metres deep by 2 metres wide. The front lip of the excavation was known as the parapet, while the rear area was called a parados. Each was protected by a stack of sandbags to absorb bullets and shell fragments.
The trench was too deep to allow its occupants to see over the top, so a small ledge called a fire-step was added. The soldiers would crouch down on this protrusion, then pop up to take potshots at the enemy before ducking down quickly to avoid having their heads blown off by a camouflaged sniper who'd been lying motionless for hours in no man's land. Officers would also make use of hand-held periscopes to monitor enemy troop movements.
Poking their heads over the parapet to see what the other side was doing took the lives of many a young arrival at the front. The more seasoned soldiers tried to warn newcomers to keep their heads down while in the trenches, but curiosity would often take hold and these wet-behind-the-ears troops would take a tentative peek over the top of the trench. Their first sight of the enemy would usually be their last. Snipers were lying in wait on both sides for the foolhardy to expose themselves. One favourite trick of the Germans was to fly a kite with English writing on it above their front lines. If an Allied soldier forgot himself and craned his neck to read the lettering, he never got a chance to do it a second time.
The front-line trenches were protected by gigantic bales of barbed wire placed far enough forward to prevent the enemy from getting within grenade-lobbing distance. So impenetrable and tangled were these obstacles that they acted like the steel web of a monstrous spider, impaling any hapless soldier who came close enough to get tangled in the trap. Before a battle, troops would be sent out with wire cutters to chop a path through this razor-sharp wire. It was one of the more hazardous duties to perform because of those ever-present snipers. Pre-battle attempts by the artillery to obliterate the wire were met with disdain by experienced soldiers, who had learned the hard way that this only blew the entanglement up in the air. When it landed intact back on the ground, it was even more tightly wound together, and thus more impenetrable.
Short ditches just over 30 metres long were dug toward the enemy position from the front-line trenches. Called "saps" — with those who dug them referred to as sappers — they were used by small advance parties of soldiers as listening posts. After an enemy bombardment, the newly formed shell craters also served this purpose. But the snipers knew this as well, and they waited silently for hapless soldiers to try to reach one of these holes only to be added to the notches on the snipers' rifle butts.
Strategic Advantage
The Germans, being the first troops to dig in after the Battle of the Marne, had the advantage of selecting the most strategically advantageous positions on the high ground. Not only did this allow them a better field of vision, it also forced the attacking Allies to charge uphill while loaded down with weapons and equipment that made their assaults that much more difficult to carry out. Moreover, with the French, British, and Canadian trenches only a few feet above sea level, the Allied troops would find themselves standing ankle deep in water after digging down only a few centimetres. Waterlogged trenches meant wet feet for days and weeks on end — and wet feet led to frostbite or the dreaded trench foot that, if left untreated, could result in amputation.
When tens of thousands of troops were incapacitated with trench foot early in the conflict, the Allies issued an order that each soldier was to massage his feet vigorously first thing in the morning, then wash them thoroughly. Fresh water was scarce and this often meant using the brackish water in a shell hole behind the lines. Next came an application of grease made from whale oil to keep the dampness out. In addition, the troops were ordered to carry three pairs of dry socks with them at all times and to change their socks at least twice a day. Failure to comply, as would be evidenced by an outbreak of trench foot, was punishable by court martial.
Dysentery was another killer that accounted for thousands of deaths in the trenches. Needless to say, sanitary conditions in these waterlogged ditches were appalling. Latrines were dug behind the lines, but these soon filled up and spilled into the trenches. In addition, many of those excavations had been dug in areas where corpses from earlier battles had been hastily buried, and the decaying bodies were another source of deadly germs. For that matter, the battlefields were located mainly on destroyed farmland, where the soil had been fertilized with manure for centuries. Deadly microbes infested the ground and contaminated wounds, causing gangrene, a horrible affliction that often proved fatal in the days before wonder drugs.
A steady diet of canned beef, mouldy biscuits, boiled sweets, and coffee made from ground turnips left the men susceptible to boils, scabies, and other skin eruptions. And there was no popping down to the local drugstore to select a tube of some soothing ointment that would cure whatever the ailment might be.
As can be expected, a great number of soldiers on both sides of the line suffered mental breakdowns from the days, weeks, and months of living under abominable conditions, with the risk of death or disfigurement a constant concern. The term "shell shock" was coined to describe this affliction, but many officers — and even a number of battlefield doctors — refused to accept this as a reason for taking the victims off the front lines. As a result, many a disturbed soldier would deliberately mutilate himself in order to get a "blighty" — a wound that would send him to England for recuperation and, if he were really lucky, home for good.
There were others whose minds completely snapped and they would commit suicide rather than live one more day in the hellhole of a front-line trench. Refusing to obey a direct order, they would be shot by one of the senior officers. Or they would deliberately raise their heads above the rim of the trench and let a sniper do the job. The truly desperate would take off one boot and sock, point their rifle at their heads, and pull the trigger with their big toe.
The rallying cry "for king and country" soon took on a cynical overtone.
Chapter 3
Easter Mourning
Among those keeping the home fires burning in Canada, all but the most dedicated optimists had little to rejoice about on Easter weekend in early April 1917. The Canadian Corps had been overseas for more than a year and the news from the front had been anything but encouraging. Not only had the Allies suffered major setbacks in a number of bloody battles, but the arrival of that dreaded bearer of bad tidings — the "We-Regret-To-Inform-You" telegram — was becoming a far-too-common occurrence across the land.
One ray of hope was that the United States had finally declared war on Germany on Good Friday, but it would be weeks, and probably months, before the Yanks could get mobilized and begin landing their soldiers "over there." Furthermore, there were rumours in the press and from wounded soldiers repatriated to Canada that a major offensive was imminent. Many a churchgoer that Easter Sunday prayed that their loved ones would not be involved in a bloodbath, or, if they ended up being part of a major assault, that they be spared serious injury or death and would soon be coming home.
Meanwhile, people from coast to coast tried to get along in their daily lives as best they could — putting thoughts of muddy battlefields and disease and death out of their minds by concentrating on happier things. Music lovers waited eagerly for the promised May 31 release of the first jazz recording, "The Darktown Strutters Ball." They read the newspaper advertisements and wondered whether they could afford that new table-model being offered for $25 by Toronto's Regal Phonograph Company.
Others bought up the sheet music to such catchy new tunes as "For Me and My Gal," "I'm Always Chasing Rainbows," and "Oh Johnny Oh Johnny Oh." While they were waiting to pay for their purchases, they gossiped about the death on April 1 of ragtime composer Scott Joplin, who had written such lively pieces as "The Maple Leaf Rag" and "The Entertainer."
Movie fans waited patiently for the arrival at the local cinema of silent screen star Theda Bara's new hit, Cleopatra, and Canada's own Mary Pickford's Rebecca of Sunnybrook Farm. And, if you really wanted to forget your troubles, there was a new release called "His Wedding Night," starring those two hilarious comics Buster Keaton and Fatty Arbuckle.
For the literary crowd, McClelland, Goodchild and Stewart had just published the fifth novel in Lucy Maud Montgomery's popular series about the beloved Anne Shirley called Anne's House of Dreams. And the April edition of the National Geographic — available at 25 cents a copy, or $2.50 a year — offered such articles as "The Burden France Has Borne" and "Friends of our Forests."
While parts of the country were still gripped in the icy fingers of a brutal Canadian winter, newspaper photographs showed publicity hounds attempting to fry an egg on the sidewalks of New York. The weatherman had played an April Fool's Day joke by handing the city the hottest April 1 on record at 28 degrees Celsius.
If the more prescient members of Canadian society slept fitfully after bedding down when the traditions of Easter Sunday 1917 had been observed, there was good reason for it. Because of the time difference between Canada and Europe, when they awoke Easter Monday morning it was already afternoon overseas. Since they'd pulled the covers over themselves and turned off the lights, several thousand of their countrymen had been killed and more than double that number wounded in northern France on a small knoll with the innocuous-sounding name of Vimy Ridge.
Chapter 4
A Ridge Too Far
The troops of the Canadian Corps were well aware that a major offensive would be launched that spring against the German stronghold of Vimy Ridge. They just didn't know exactly when. Then a message was circulated through the ranks that let them know it would be a matter of hours or days, at most.
"Under the orders of your devoted officers, in the coming battle you will advance or fall where you stand facing the enemy. To those who will fall, I say: You will not die, but step into immortality. Your mothers will not lament your fate, but will be proud to have borne such sons. Your name will be revered forever and ever by your grateful country, and God will take you unto Himself."
Major General Arthur Currie
Commander, 1st Division
Canadian Corps
Special Order before Vimy Ridge
March 27, 1917
Currie could be excused if his hand trembled a bit as he signed that order. They were brave words he wrote, but if he believed them, he was one of the few officers in the Allied High Command who did. The British and French had earlier tried to wrest the strategic hump of land near the city of Arras in northern France from a seasoned German force that had held it since October 1914. When the smoke of battle cleared, there were well in excess of 150,000 British and French casualties, including 20,000 dead. The flatland in front of the 61-metre-high ridge and the slopes leading to the top were strewn with the rotting corpses of Allied soldiers.
It had been an open secret for some time that the Canadians would be thrown into the fray next. Vimy Ridge was an essential stretch of land — the cork in the bottle to Allied advances aimed at pushing the enemy out of France and into total defeat. British and French officers warned the Canadians that taking the ridge was an impossible task. The Germans were not only solidly dug in to a zigzag of deep trenches that furrowed the limestone and chalk ridge for several kilometres, but they had also constructed a series of tunnels and underground caves they could retreat to when artillery shelling became particularly heavy. German officers openly sneered at the prospect of an amateur army, thrown together by Canada after war had been declared, being able to succeed where the seasoned professional soldiers of Great Britain and France had failed. French General Robert Nivelle considered the Canadian attack an exercise that would come to nothing.
Meticulous Planning
If Currie had any doubt about the ability of the Canadian Corps to capture Vimy Ridge as part of the upcoming Battle of Arras, he kept it to himself. He and his staff had been planning months in advance for a lightning attack aimed at overrunning the German positions in one day, with a brief mop-up action the following morning. It has been said that no Allied offensive on the Western Front was more exhaustively planned than the assault on Vimy Ridge.
Currie and other senior officers realized that earlier defeats, especially where frontal assaults had been made on what seemed to be an invincible enemy position, had resulted from old-fashioned military thinking. A lack of preparation and a refusal to let the troops know in advance exactly what they were expected to do had been blueprints for disaster. "Take time to train them," Currie advised, and his superiors, fortunately, listened to him. For instance, Sir Julian Byng, the British lieutenant general selected to command the Canadian contingent overseas, told his officers: "What I want is the discipline of a well-trained pack of hounds... Never lose sight of your objective. Reach it in your own way."
It was obvious that both Byng and Currie cared greatly for their men and were much admired in return. Members of the Canadian Corps referred to themselves as Byng's Boys, after a popular British music hall ditty of the time. The story buzzed along the trenches about this casual officer being reprimanded by King George V for wearing shabby uniforms. He also didn't stand on ceremony in a number of areas, including the protocol of saluting. Strolling along with his hands in his pockets, he would often return a salute by raising his pocketed hand as high as it would go within the confines of his greatcoat.
For his part, Currie, unlike many of his fellow generals within the Allied forces, refused to consider front-line troops as mere cannon fodder. He did everything he could to keep casualties down. In his book Adventure, Major General J.E.B. Seely, commanding officer of the Canadian Cavalry Brigade, praised Currie's people skills.
"Of all the men that I knew in nearly four years on the Western Front, I think Currie was the man who took the most care of his men," Seely wrote. "Moreover, again and again he nearly brought his career to an end by bluntly refusing to do things which he was certain would result in loss of life without compensating advantage."
In order to plan the attack properly, a full-scale mock-up of Vimy Ridge was built behind Allied lines. This simulated battlefield was festooned with coloured tape and flags signifying where each unit of the Corps was to be deployed and what its objective was. Every soldier who would take part in the actual assault was drilled and redrilled for weeks on the role he would play in the upcoming attack, from the time he went over the top until his unit's objective was taken. Unlike so many battles where the troops were kept in the dark about what was to take place, each man, from private to senior officer, was given a map of the area he would be traversing. Furthermore, every soldier was trained to perform a number of tasks and, if necessary, take command of his unit. In that way, when others in the group were killed or wounded, there were instant replacements to maintain the momentum. The troops were also taught to operate German weapons so they could turn them on the enemy once the weapons were captured.
Surprising the Enemy
Working stealthily, tunnelling crews built underground networks beneath no man's land with explosive charges that could be detonated during the battle so that waves of Canadian troops could pour out of the tunnels and overwhelm a surprised enemy in nearby trenches. A series of subways was constructed seven metres underground that would allow assault troops to move to their jumping-off points while protected from shelling. The subways would also allow the wounded to be evacuated from the battlefield. Chambers for brigade and battalion headquarters were cut into the walls of the subways. Some of these excavations were also designed as ammunition stores, communications centres, and dressing stations.
In preparation for the attack, Canadian and British engineers made improvements to the existing roadwork in the Corps' forward area and added several kilometres of new plank road. In addition, they repaired bombed-out tramways so that light trains, powered by mules, horses, or gasoline engines could transport the many tonnes of rations, stores, and ammunition required at the front on a daily basis. Artillery shells assigned to the Vimy operation totalled 38,250 tonnes. Additional pipeline was laid to carry the Corps' daily requirement of 2.3 million litres of water to the soldiers and some 50,000 animals, as well as for cooling overheated artillery pieces. Canadian signallers installed 34 kilometres of cable two metres underground to withstand enemy shelling.
The Allied High Command knew from the earlier bitter defeats of the French and British and from intelligence reports made during the early months of 1917 that the Canadians didn't have much hope of success. For one thing, the ridge was a strong link in the chain of German defences, so the enemy would fight bitterly to retain it.
The ridge provided the Germans with an excellent vantage point from which to lob a constant barrage of artillery shells at the nearby French-held city of Arras. Furthermore, it was a stronghold connecting the main Hindenburg Line (stretching nearly 160 kilometres from Lens, near Arras, to the Aisne River, near Soissons) to the German defence systems running north to the coast of the English Channel.
The Germans had been augmenting the fortifications on the ridge since capturing it in October 1914. Both slopes of the outcropping gave the defenders a decided advantage. The eastern side away from the Canadian trenches was a tangle of forests where a number of large artillery pieces could be hidden. Since the incline on the west facing the Canadians was gradual, a large number of the assault troops would have to attack over open ground — and become prime targets for artillery, machine-gun, and rifle fire. The Canadians faced three main defensive lines comprised of heavily fortified trenches, concrete machine-gun posts, and walls of skin-slicing barbed wire. The Germans would also be well insulated against artillery barrages in deep dugouts and vast underground chambers, some of which could shelter entire battalions.
Sticking Together
One of the vital keys to success at Vimy, Currie kept insisting, was that the Canadians would be fighting side by side with men they had known since enlisting, or earlier. Where in the past Canadian troops had been scattered among a number of British units, Currie, with Byng's support, insisted on bringing the various Canadian battalions together for this important battle under the four divisions that comprised the Canadian Corps. There were about 40,000 troops from the Corps training for the attack, with some 20,000 picked to go "over the top" in the first wave.
Australian General Sir John Monash would praise this tactic in writing about Vimy years later. "It is impossible to overrate the advantages which accrued to the Canadian Corps from the close and constant association of all four divisions with the others," he maintained.
While it was true that the Canadians had pretty well started from scratch in building their armed forces at the outbreak of war in the summer of 1914, their ground troops received a brutal baptism under fire in several subsequent encounters on French soil. Particularly costly in terms of casualties was the bloody Battle of the Somme in 1916, where more than 24,000 Canadian troops were killed or wounded.
It was at the Somme that the Canadians earned a reputation as a force to be reckoned with. As British Prime Minister Lloyd George wrote, "The Canadians played a part of such distinction that thenceforward they were marked out as storm troopers; for the remainder of the war they were brought along to head the assault in one great battle after another. Whenever the Germans found the Canadian Corps coming into the line they prepared for the worst."
Ever since the Canadian arrival in the Vimy area shortly after the Somme offensive, the Corps' casualty rate had been high. One preliminary attack against a section of the ridge on March 1, 1917, cost the Canadians 687 lives. Two of these fatalities were the commanding officers of their units — Lieutenant Colonel Samuel Gustavfus Beckett of Toronto, who was in charge of the 75th Battalion, and Lieutenant Colonel Arnold Kemball of the 54th, who hailed from the village of Kaslo, British Columbia.
In total, some 1,400 Canadian officers and men were killed or injured in the time leading up to the major assault on the ridge. Withering machine-gun and sniper fire cut down most of them as they raided German trenches seeking prisoners who could provide valuable information for the upcoming attack.
This information-gathering tactic involved sending small patrols into no man's land under cover of darkness. These raiders had to slither forward on their stomachs through mud, decaying corpses, severed limbs, abandoned weapons, and field rations, as well as foraging rats feeding on the carnage of earlier battles. Their biggest prize was an enemy sentry they could bring back for interrogation. Their biggest fear was a German rocket that, when fired into the sky, would release a flare on a small parachute that turned the blackest of nights into broad daylight and exposed them to snipers and machine-gunners.
There were other ways that the Allied planners gathered intelligence on German troop strength, weaponry, and the strongholds that would have to be immobilized before the Vimy assault. Tunnels were dug under the enemy lines and listening devices were installed to pick up bits of information from troops talking to each other or officers planning their next move.
Spy in the Sky
Both sides used observation balloons filled with gas or hot air to spy on their adversaries. Several of these devices, each containing an observer to spy on the enemy, were winched into the air at the same time so that comparative sightings could be made and wire mesh could be dangled between the balloons to hamper the manoeuvrability of enemy aircraft bent on blowing them out of the sky. It was not so easy to bring down a balloon. Standard bullets would usually pass right through the fabric. Special incendiary or explosive bullets were needed and an attacking pilot would have to be careful not to get too enthusiastic and follow a descending balloon that was being rapidly winched to the ground. Such carelessness could bring his aeroplane into range of enemy anti-aircraft fire.
Setting an observation balloon aflame — which generally meant a fiery death or a fatal leap from the gondola by the device's occupant — was regarded as a legitimate "kill" by the various air forces involved in the war. One of Canada's World War I aces, Billy Bishop, got his start on winning a chestful of awards by earning a Military Cross over Vimy on April 7, 1917, for destroying a balloon. Bishop and his fellow Royal Flying Corps pilots also contributed to the Vimy victory by shooting down enemy aircraft that, like the German balloons, had been sent aloft to learn what they could about the Canadians' intentions. In addition, the RFC fliers were able to assist the Canadian Corps' counter-battery unit in destroying more than 80 percent of the Germans' heavy guns prior to the attack by flying over the German lines and taking aerial photographs of the gun emplacements. They would also provide air support during the battle and, through a flag system employed by the ground troops, report the successful capture of enemy strongholds by the various attack units.
Of course, while the Canadians were attempting to gain as much information about enemy troop strength as possible before the final assault, the Germans were also curious about what the Corps was up to. With the advantage of being on higher ground than the Canadians, they had a panoramic view of what was going on below. To make the enemy's task more difficult, Corps engineers erected wooden poles on either side of the roads in the front-line area and strung great swaths of coarse jute fabric between them. From the German positions on the ridge, the hessian material resembled the road surface and gave the false impression that nothing was going on, while supply vehicles streamed back and forth underneath. That ruse worked so well that scrim netting, interwoven with bits of coloured cloth, was draped over the rail lines, giving free movement to rail cars beneath.
A Severe Pounding
All hell was let loose on March 20, 1917, in preparation for the upcoming attack. In order to pulverize the entrenched Germans, artillery units from Canada, Britain, and South Africa unleashed a continuous barrage from more than 375 heavy guns and howitzers, as well as more than 700 pieces of field artillery. It was the largest such barrage in history up to that point. At the same time, the detonation of tonnes of explosives that had been deposited in freshly dug tunnels under the German positions added to the mayhem. All of this was aimed at not only destroying as many enemy gun emplacements as possible, but also crippling the morale of the German soldiers who were forced to cower in their trenches and tunnels while the Allied shells rained down on them.
The non-stop shelling continued for more than 10 days and then, as abruptly as it had begun, it ceased. This tactic was another attempt to shatter the spirits of the enemy. Just when they might have thought the worst was over, an even greater artillery offensive was launched on April 2. From then until April 9, a period of time the Germans would later refer to as "the week of suffering," about a million shells — 50,000 tonnes of explosives — pounded the German defences.
The brilliance of a young electrical engineering graduate from McGill University was largely responsible for the success of the Canadian artillery campaign in the lead-up to the Battle of Vimy Ridge. Lieutenant Colonel Andrew McNaughton was given carte blanche by Sir Julian Byng to come up with innovations that would bring artillery into the 20th century. McNaughton was named counter-battery staff officer and fine tuned calculation of the position of the enemy's big guns by observing the muzzle flashes and timing the sound of a shell from the instant it was fired until it hit its target.
This method of pinpointing an enemy artillery piece was based on the findings of a French gunner, Charles Nordmann, who had been an astronomer for the Paris Observatory. In the autumn of 1914, Nordmann worked out a formula for locating the precise position of an enemy field piece by measuring the difference in time required for the sound of the guns' firing to reach strategically placed microphones. He discussed his theory with French physicist Lucien Bull, who designed the required equipment.
The idea worked well in the laboratory, but it took more than a year and a half for the necessary equipment to be developed for field conditions. British artillery began using the microphones and measuring devices in the summer of 1916 and passed the information along to McNaughton.
Each Allied sound-ranging unit consisted of a minimum of six microphones spread out at precise intervals behind the front-line trenches connected to headquarters. An operator located in a listening post well forward of the microphones would press a key when he heard an enemy gun firing. This action would start a recording back at headquarters of the report from the artillery piece. The sound experts could then determine the gun's location from the time intervals between the microphones. It was then up to the Allied artillery to zero in on the enemy artillery piece and put it out of commission.
McNaughton also developed a system for keeping track of the wear and tear on the barrels of the heavy guns so that they could be replaced before they started losing their accuracy. Another innovation that McNaughton was quick to embrace was the use of a new fuse that would explode shells on impact rather than in the air above a target. This new technique blew holes three or four metres wide in the thickly coiled rolls of German barbed wire that had previously trapped attacking soldiers, turning them into sitting ducks for enemy snipers.
Perhaps McNaughton's greatest contribution to the Battle of Vimy Ridge was his implementation of the creeping barrage. This was a technique whereby the Allies would fire artillery shells just over the heads of their own advancing troops to land in the enemy's trenches just before the Canadians arrived.
The combination of weeks of pummelling by the pre-attack artillery barrage that caused sleep deprivation and shell shock, along with the pinpoint explosions that prevented the preparation of a proper defence, left many of the German soldiers dazed, terrified, and ready to surrender.
However, very few battles go exactly according to plan. While the weeks of intensive training and the support of heavy shelling would provide the attacking Canadian troops with a tremendous advantage, the battle-hardened Germans were still a formidable enemy. As zero hour approached in the early morning of Easter Monday, April 9, 1917, a momentous battle was shaping up. It would change forever the widely held opinion that the Dominion of Canada was nothing more than a satellite of Great Britain and a source of expendable front-line troops whenever the British lion went off to war.
The success or failure of this massive undertaking, now that all the preparations had been made, hinged on each division accomplishing the formidable tasks assigned to it. The battle plan was a jigsaw of interlocking pieces, each dependent on the others.
Chapter 5
The 1st Division
Private Bill Milne hoisted a bag of Mills bombs onto one shoulder and peered out into the darkness of no man's land. The driving sleet was threatening to turn into wet snow and transform the already treacherous mud into a boot-sucking gumbo that would make the troops' measured pace up the ridge all the more hazardous. Even worse, the corpses of thousands of French and British soldiers, who had failed in earlier attempts to take the ridge, were strewn in grotesque fashion all over the shell-pocked terrain.
But this was no time to dwell on these horrible mental images. Milne and the rest of the Corps committed to the Vimy assault had been practising for this onslaught for months on terrain behind the lines that was an exact replica of Vimy Ridge. Each man knew precisely what was expected of him and what to do if he suddenly found himself in charge of his platoon in the event that everyone senior to him were killed or wounded.
The men were grateful to the Allied High Command for allowing each battalion and each brigade within the division to remain intact rather than being broken up arbitrarily in the traditional way. This meant that they knew who the guys next to them were. They could take pride in Canadians fighting beside Canadians and count on everybody around them to fight their hardest — if for no other reason than they didn't want to let their buddies down.
Stopwatch Attack
It had fallen to the 1st Division, including Milne's 16th Battalion, to attack the south end of the ridge from a position west of the Arras-Lens road and overtake the main enemy trenches in front of the shattered village of Thélus. A second wave would push through to Farbus Wood on the eastern slope of the ridge, where a sizable number of German heavy guns were hidden.
Rather than making a mad dash into deadly machine-gun fire, the troops had been drilled to operate by stopwatch. They advanced slowly toward the first line of German trenches in what was called the Vimy glide — a 100-metre advance every three minutes just behind a creeping barrage of Allied firepower, including heavy artillery and Vickers machine guns, that fired over the heads of the advancing troops. The barrage was designed to annihilate — or at the very least, discombobulate — those hapless enemy troops stationed in the forward trenches, tunnels, and dugouts.
Milne shifted his bag of grenades as the seconds ticked by toward the zero hour of 5:30 a.m. He and his fellow soldiers were loaded down with more equipment and weapons than seemed humanly possible to carry. In addition to the Mills bombs, each man carried a rifle and bayonet, 120 rounds of ammunition, an entrenching tool, 5 empty sandbags, 48 hours worth of hard rations, a waterproof sheet, a gas mask, goggles, a ground flare, and a full water bottle. In addition, the mud that had caked on their boots and greatcoats probably added another 30 kilograms to the weight they were carrying. Many of the Canadians had, in fact, elected to leave their greatcoats behind, despite the frigid weather. Others had cut off the bottom half of the cumbersome outer garment and had quickly been put on report for destroying government property.
The men had been assigned to platoons that contained a minimum of 28 troops and a maximum of 44. These units were organized into four sections of equal strength — riflemen, bomb throwers, rifle grenadiers, and Lewis machine-gunners. They would be controlled by a platoon command post consisting of an officer, a senior NCO, and two runners.
The Lewis gun was considered the most strategic weapon carried by the raiders. It was an air-cooled weapon that weighed just over 11 kilograms. Fed by a 47-round drum magazine, it could be fired from either a standing or prone position. The routine that had been drilled into the men was that the Lewis guns and rifle grenades would pin down the enemy defenders while the bomb experts and riflemen attacked from the flanks.
As zero hour crept nearer, many of the troops suffered muscle cramps from being forced to half crouch in vermin-infested trenches, with foul-smelling water up to their knees, for most of the night before battle. Their only consolation was the recent distribution of a generous tot of overproof rum, and, for those whose nervous stomachs could hold it down, a hot meal, such as it was. It would almost come as a relief when the shelling started, signalling the attack.
Right on the dot at 5:30 a.m., a blast, that someone later described as sounding like the screams of a thousand fiery banshees flying overhead, splintered the early morning calm. This Allied bombardment began obliterating the German front lines. As had been rehearsed so many times the participants had begun to grumble, the Canadians leapt from trenches and tunnels to begin their slow advance up the shallow incline of the western slope of the ridge. The 1st Division was on the extreme right, the 2nd and 3rd Divisions in the middle, and the 4th Division slow marched along on the extreme left.
One of the first fatalities of the attack was Lieutenant Lisle Cradock Ramsay of the 1st Division's 15th Battalion. Having survived the brutal battles of Ypres and the Somme, this former Bank of Montreal employee was killed instantly as he led his platoon over the top with the first wave of his battalion. And in a macabre twist of fate, Horace Stokes of the 1st Battalion would lose his son, 16-year-old Private Stanley Tom Stokes, also a member of that battalion. The 40-year-old Horace would be killed just over five months later.
Another fatality that caused a stir back in Canada was the death of Captain Victor Gordon Tupper, the grandson of Sir Charles Tupper, a Father of Confederation, and Canada's seventh prime minister. The younger Tupper, who would receive the Military Cross for heroics on the battlefield, had insisted on joining the attack, even though he had been selected to stay in the rear echelon. He wrote home that if he were going to die, then dying on the battlefield would be worth it "... a thousand times. I have 'been over' two or three times before, but never with a company of my own. Think of it — 150 officers and men will follow you to hell if need be!"
The 21-year-old captain was shot and killed instantly leading his company in a charge against a German-held position early in the assault.
Despite their rigid training, some of the advancing troops in the first wave became overzealous and dashed forward, only to be blown to pieces by their own artillery. It also happened that the occasional shell fell short, with the same disastrous results. William Green of the 4th Battalion was witness to one such tragedy.
"An incident that stands out in my mind," he later wrote, "was when one of our own shells, which incidentally was a dud... cut the head off a machine-gunner and took the leg of a lance corporal beside him."
However, the initial thunderous barrage was extremely effective, allowing the first-attack battalions of the 1st Division to reach their initial objective by 6:05 a.m. and dig in. As planned, the second wave leap-frogged ahead of the dug-in troops to attack the second objective, a long traverse trench the Germans called the Zwischen Stellung.
Heroic Acts
Here the German resistance proved much stronger and Private Milne decided he had to do something. Seeing his comrades in the 16th Battalion being mowed down by a well-entrenched machine-gun post, Milne jumped out of a nearby shell hole where he had taken cover. He crawled on hands and knees through the thick mud and got close enough to the enemy position to lob a grenade. His action silenced the deadly chattering.
Milne secured his hold on a Victoria Cross, the highest military honour awarded to troops in the British Commonwealth, by repeating this feat a short time later. Oddly enough, the wicked firing of a second German machine gun seemed to be coming from a haystack in the middle of no man's land. Since the Allied artillery had flattened entire villages and stands of forest as far as the eye could see, it was immediately apparent that the haystack was not what it appeared to be. In fact, it was cover for a concrete machine-gun emplacement and it was taking a severe toll on Milne's fellow soldiers. Once again, showing a total disregard for his own safety, the Scottish-born private crawled close to his objective and let fly with another grenade, destroying the gun emplacement and stunning the crew, which immediately surrendered to him.
Milne found himself in an awkward situation that many of his fellow soldiers would face during the attack. The German troops had been so devastated by the intense two-week bombardment and the morning barrage that, in many cases, they surrendered en masse, sometimes to just one lone Canadian. Milne and the others had orders to keep moving forward, so they simply told the surrendering troops to lay down their arms and head toward the Canadian rear area. Some of these dispirited Germans were pressed into service as stretcher-bearers for the mounting Canadian wounded or were handed picks and shovels and ordered to dig trenches for the advancing troops.
One of the many injustices of the battlefield is that, while some heroic deeds are rewarded, others go virtually unnoticed. One instance of this lack of recognition involved Private John Dunbar of the 10th Battalion who, despite several courageous feats just before and during the attack, earned not so much as a mention in dispatches for his efforts.
Dunbar had been part of several raiding parties sent out before daybreak on Easter Sunday. Their assignment was to check whether the barbed wire had been cleared away in front of the enemy trenches where they would be attacking the next morning. The Germans had detected the raiders early into their mission and were inflicting serious damage with heavy rifle fire and grenades.
Dunbar and fellow private Hugh Henry charged a position held by 9 enemy soldiers, killing 4 of them and taking 2 others prisoner. The entire operation was over in less than an hour, but resulted in heavy casualties to the Canadians, with 5 dead and 13 wounded. But the raiding parties had been able to confirm that the barbed wire in front of the enemy trenches in that section was still relatively intact and would be a lethal barrier to the attacking troops. The mission was thus deemed a success, because Allied artillery was then able to blast a hole through the deadly wire after the 10th Battalion withdrew temporarily from their forward trenches.
During the attack of April 9, Dunbar again performed heroically when the officer and all the NCOs in his platoon were put out of action by enemy fire. Taking over the unit, Dunbar led a wild charge in which he killed nine Germans by bayonet before being fatally wounded.
While Private Henry received the Distinguished Conduct Medal for his actions on Easter Sunday, Dunbar's self-sacrifice went unrecognized when it came time to distribute awards for meritorious behaviour at Vimy Ridge — a bureaucratic oversight that happens all too often in times of war.
Right On Time
Thanks to the bravery of Private Milne, Private Dunbar, Private Henry, and hundreds like them, the 1st Division was able to secure its second objective by 7:13 a.m., about one and a half kilometres east of the original Canadian forward line. Those troops who had made it to this second position had less than three hours to consolidate the area before another creeping barrage began and the division's 1st Brigade, which had been held in reserve, would carry on to the next objective.
Just before 10 a.m., these reserve troops of the 1st Brigade, comprised of three Ontario battalions, marched through the line secured earlier by the entrenched troops. Their cheering and waving at their comrades as they moved forward soon turned to anguish as several tragic incidents occurred — caused mainly by the Allied artillery shells falling short of their targets and landing among the advancing Canadians. Nevertheless, they pressed on, and in less than an hour, they accomplished what so many experts had deemed impossible — they reached the crest of Vimy Ridge. Like in a scene from a Hollywood movie, the sleet and snow stopped at that moment, the clouds parted, and a weak early-spring sun shone down.
Some of the victorious soldiers would later write home about the jarring contrast that greeted them when they looked west down the battle-ravaged slope they had just fought their way up and then looked east at the untouched side of the ridge. Far below, they could see the grey uniforms of the retreating German troops. But what caused a collective intake of breath among the Canadians, who had for months seen nothing but mud, rotting corpses, and water-filled shell holes, was the idyllic scene below them of rolling green meadows, early spring crops, intact villages, and brightly coloured farmhouses with smoke curling lazily from their chimneys.
It was like gazing briefly at a picture postcard before a nightmarish reality returned. There were two more objectives to take. With the help of renewed shelling — this time on target — they reached the third German position, called the Chain Trench, just after 11:15 a.m. They then had precisely 1 hour and 10 minutes to consolidate their position before moving ahead again. By 1:30 p.m., the 1st Division had reached its ultimate objective, Farbus Wood, part way down the eastern slope of Vimy Ridge. The enemy was in full retreat.
It was now up to the 2nd Division to consolidate a hold on the German territory just north of this captured ground, so that the 3rd Division would be solidly protected on its right flank.
Chapter 6
The 2nd Division
Sergeant Ellis Sifton ran a thumb over the sharpened edge of the bayonet affixed to his Lee-Enfield rifle. He knew from experience that the kind of battle he and his fellow Canadians were going into would include fierce close-quarters fighting. He would use his sack of Mills bombs to get in close to the enemy, but after that it would be bayonets and rifle butts.
The newly issued Lee-Enfield rifle was a welcome replacement for the old Ross rifle, a sporting piece that stubborn old Sam Hughes had insisted on making the official weapon of the Canadian infantry despite its tendency to jam after several rounds had been fired. A number of Canadians earlier in the war had been killed by enemy fire as they attempted to release the bolts on their Ross rifles with the heels of their boots or entrenching tools. Many of the troops had taken to picking up discarded Lee-Enfields from the battlefield after their unfortunate British possessors no longer had any need of them. The Lee-Enfield was a much superior weapon, but the by-the-book senior officers had followed Hughes' orders and demanded that the Ross be on each man's shoulder during inspection parade. That meant that many of the troops had been forced to carry two rifles into battle. Thankfully, saner heads eventually prevailed and the Ross ended up on the scrap heap, as did Hughes — but unfortunately not before a number of Canadian soldiers had been killed or seriously wounded due to the gun's deficiencies.
Sifton was a member of London, Ontario's 18th Battalion, part of the 2nd Canadian Infantry Division, whose task in the upcoming battle was similar to that of the 1st Division. The mission given Sifton's unit, along with men from several other battalions, was to establish a foothold in the main German trench, Zwischen Stellung, after overrunning the German trenches near the ruined hamlet of Les Tilleuls. They would be followed by fresh troops, who were to join the attack on the German emplacements in front of the destroyed former village of Thélus. Once this position was secured, along with the taking of nearby Hill 135 by two attached British battalions, the Division's remaining attack force would head for Farbus to join their comrades from the 1st Division.
Sifton's 18th Battalion was part of the phalanx of Canadians that bolted forward from their trenches and tunnels as the 5:30 a.m. creeping barrage signalled the launch of the attack. It was tough sledding as they slowly advanced around and through gigantic shell holes, as well as mounds of barbed wire that had been blown to bits by the constant pounding of the Allied bombardment over the previous several weeks.
Like their buddies in the 1st Division to their right, Sifton and his fellow soldiers encountered a steady stream of dazed and demoralized German troops. These shell-shocked soldiers were only too ready to throw down their weapons and raise their hands in the air as they shouted "kamerad" and "mercy" at the onrushing Canadians. It was difficult to understand how these men had survived the artillery lambasting that had flattened everything in its path. The German trenches had been fortified with concrete, and a maze of underground tunnels had provided even greater protection, but it still boggled the minds of the advancing Canadians that anyone could have withstood such punishment.
The Canadians didn't have much time to think about these things, however, because some of the enemy's big guns were still intact, raining shells down on the advancing troops. In fact, the 18th Battalion received such punishment that out of the 1,000 men they threw into the attack, there were 600 casualties by the time the whole thing was over.
Perce Lemmon, a 19-year-old from Windsor, Ontario, was one of them: "It was a rugged fight, and this shell came over, and it hit over the side of where we were in this ditch," he recalled decades later in a documentary produced by the War Amputations of Canada. "It hit the hard road and the whole company went up in the air. I crawled out of 42 dead. That's where I lost my leg."
A Horrible Sight
Not only had many of the enemy survived the bombardment, the concrete pillboxes had also kept their deadly machine guns intact and Sergeant Sifton watched in horror as one enemy post in particular continued to mow down his men like flowers in a hailstorm. He decided he had to do something.
In the smoke and confusion of battle, Sifton spotted a machine-gun turret poking over the top of a German trench. Grabbing a couple of Mills bombs, he ran full tilt toward the trench and lobbed the first grenade at a section of barbed wire that had escaped the shelling and was forming a protective wall in front of the machine-gun post. Once the barbed wire had been put out of commission, Sifton continued his advance, ignoring the machine-gun bullets that ploughed into the mud all around him, until he was within throwing distance of the German gun crew. The bomb did its job and as Sifton hurled himself into the trench, he saw the results of his handiwork. The entire gun crew had been killed.
However, a small unit of Germans began charging down the trench toward Sifton before the rest of his unit could join him. He managed to hold the enemy off with rifle fire, then bayonet, and, as he had suspected would be the case, the butt of his rifle, until what was left of his unit arrived and wiped out the remaining Germans.
Canadian machine gunners dig themselves in, in shell holes on Vimy Ridge. Apr. 1917/Vimy Ridge, France.
The heroics of this one-man assault force led to the awarding of a Victoria Cross.
Another life-and-death drama was taking place around the same time in an exposed section of no man's land just to the left of Sifton's unit. Captain Robert Manion, the medical officer of the 21st Battalion, was attending to a wounded colonel in a shell hole they had tumbled into for the little shelter it provided. Shrapnel had opened up gaping wounds in the colonel's arm and leg and Manion was doing everything he could to stop the bleeding. The two officers were prisoners in a cage of hot steel formed by the rolling barrage of friendly fire to the east of them and an enemy bombardment pounding the old Canadian lines to the west.
Realizing that they would soon be caught between the pincers of exploding shells, Manion attempted to carry the wounded colonel, but they soon toppled into waist-deep mud. Summoning up his last reserve of energy, Manion began dragging his wounded fellow officer toward the Canadian lines at the rear. The colonel used his good arm and leg to help as much as he could and the two men eventually reached safety. Captain Manion later received the Military Cross for this selfless act.
Once the first objective had been secured and a mopping up action had rid the area of the last remaining German snipers and machine-gunners, reserve troops rushed forward and waited for the next creeping barrage to begin. This was their signal to push forward to the next objective, an area referred to by the Germans as the Turko Graben.
Schedule Met
Despite a number of casualties suffered by the second wave of Canadian troops, the next objective was secured, and by 9:35 a.m.— exactly on schedule — the 2nd Division's remaining reserve units were ready to march. A renewed creeping barrage of artillery shells provided deadly notice to the entrenched enemy that the Canadians were on their way.
When the 2nd Division's final goal was reached around noon hour, the various units had differing stories to tell. Some of them suffered heavy losses at the hands of stubborn pockets of Germans who were determined to fight to the death. Others were amazed at how quickly the enemy troops in their area of attack were ready to surrender. Still others were pleasantly surprised to find that the enemy had already fled, with ragged lines of German soldiers heading down the eastern slope into the Doui Valley.
As the victorious 2nd Division watched their comrades in the 1st Division to their right descending the eastern side of the ridge to attack the last pockets of Germans resistance in the Farbus Wood, a sense of pride began to wash over the troops. It was just beginning to dawn on them that the Canadians were within a few hours of accomplishing something that the French and British before them had not been able to do. They had overrun the solidly entrenched Germans on Vimy Ridge and, if their fellow soldiers in the 3rd and 4th Divisions were equally successful, this strategic rise of land would belong to the Allies after two bitter years of struggle.
Chapter 7
The 3rd Division
On paper, the 3rd Division seemed to have the easiest assignment of all four Canadian units involved in the Battle of Vimy Ridge. They had only two primary objectives to secure and the terrain they had to negotiate was just over 1,100 metres — not much more than a kilometre. Their final goal was La Folie Wood below the eastern slope of the ridge. The division's two forward brigades, the 7th and the 8th, overran the first three lines of German trenches on schedule in half an hour. Their second and final objective was achieved before 9 a.m. and it looked as if, after a bit of mopping up and digging in, the 3rd Division was home free.
However, intense sniper and machine-gun fire to their left began taking a tremendous toll on their ranks. The 4th Division had run into stronger enemy resistance than anticipated and had failed to take the heavily fortified Hill 145 on schedule. The Royal Canadian Regiment, the Princess Patricia's Canadian Light Infantry, and the Black Watch — the battalions on the left side of the 3rd Division's assault force — were suffering heavy losses as a result.
Obeying orders, the beleaguered troops continued to hold their position against a formidable enemy, taking whatever cover they could, and keeping watch for an expected German counterattack. One of those soldiers was Sergeant John MacGregor, a Scottish immigrant to British Columbia who fought with the 2nd Canadian Mounted Rifles.
Having been promoted directly from private to sergeant a few months after enlisting in 1915, MacGregor single-handedly captured a German machine-gun emplacement at Vimy, killing eight Germans and taking one prisoner. Leaving it to the men within his battalion to turn the captured gun against enemy troops still fighting in the area, MacGregor carried on up to the crest of the ridge and fired three white rockets to indicate that this objective had been taken.
For his efforts, Sergeant MacGregor was promoted to lieutenant and awarded the Distinguished Conduct Medal. The DCM was awarded for exemplary conduct in the field to warrant officers, non-commissioned officers, and lower ranks serving in any of the Commonweath's military forces. It was therefore the second highest award for gallantry in action — after the Victoria Cross — for all ranks below commissioned officers.
Despite the risk to life and limb of capturing the machine-gun post, MacGregor later told his fellow soldiers that his proudest moment was standing on top of Vimy Ridge and signalling that this objective, once thought unattainable, had been secured.
A member of the 3rd Division not as fortunate as Sergeant MacGregor was Lieutenant Cyprian Thompson of the 7th Brigade's Royal Canadian Regiment. A former Bank of Montreal employee from Grand Mere, Quebec, Lieutenant Thompson had been killed on Easter Sunday in an operation carried out in preparation for the next day's attack.
Another tragic event was the death of two brothers who were also with the Royal Canadian Regiment. Once the battlefield had become silent after the chaos of the attack, the bodies of 28-year-old Private Wilfred Chenier and 27-year-old Private Olivier Chenier were discovered. They were later buried side by side in the Cabaret Rouge British Cemetery, one kilometre south of Souchez on the main Arras-Béthune road.
Until the 4th Division could regroup and capture the objectives set out for it, the 3rd Division would have to hunker down and do its best at avoiding the murderous fire lacerating it on the left. It was now a question of whether the men could hold on long enough for their beleaguered buddies to the north to gain the upper hand.
Chapter 8
The 4th Division
Captain Thain MacDowell of the 38th Battalion was perhaps more aware than many of the other young Canadians waiting anxiously for the 5:30 a.m. shelling to begin just what was in store for them. The wound he'd suffered at the Somme the year before had healed sufficiently to allow him to return to the front lines in time for the Vimy offensive. As an officer, he had studied the intelligence reports and was aware that they would have to negotiate some tricky terrain. He knew they were in for a rough ride.
MacDowell had been given a medal in exchange for the wound he'd received at the Somme — the Distinguished Service Order. He'd earned it by knocking out three enemy machine guns and taking 53 German prisoners, despite his injury. Some of his fellow soldiers congratulated him on getting a "blighty" — a wound that would earn him a trip to a hospital in England and perhaps even a discharge and a return to Canada — but MacDowell was back at the front and ready to go into action again within a few months.
The Canadians would later learn that a combination of factors provided a recipe for disaster for the men of the 4th Division. For one, their section leaders had vastly underestimated enemy troop strength. As well, the battle planners had failed to realize that their troops would be sent out into the toughest terrain in the whole battle area.
The western slope at the north end of the ridge was much steeper than the gentler rise faced by the other three Canadian divisions. Not only did this make the attack that much more difficult, but the enemy's heavy guns were so deeply entrenched in the chalk tunnels that the shelling that had wiped out German artillery pieces on other sections of the ridge had been far less effective here. In addition, the Germans had been able to camouflage a series of concrete machine-gun posts on the far side of the ridge that would take a deadly toll on Canadian assault troops during the attack.
And Major Harry Shaw, acting commanding officer of the 87th Battalion (Grenadier Guards, Montreal), had been so confident his men would easily overrun the German positions that he'd requested some of the enemy trenches in the centre of the battle area be spared artillery bombardment. The theory was that these intact captured fortifications could then be used by the Canadians as a jumping-off point for their dash the rest of the way up the ridge. The request was granted.
Within minutes of the attack, the Guards would lose half their unit from the firepower left intact in those unshelled trenches. Five officers were killed and another five wounded, leaving only one in any condition to issue orders. Shaw's tragic miscalculation threw the division's timetable off, causing a domino effect that adversely affected the other troops in the area.
The attacking soldiers of the 4th Division were split in two. Heavily armed German troops attacked with machine-gun and mortar fire from a middle salient, while other enemy fire from the well-fortified Hill 145 delivered the hapless Canadians a one-two punch. Being the highest point on the ridge, the hill provided the enemy with a deadly perch with a wide field of fire.
Bravery Under Fire
At this point, Captain MacDowell decided enough was enough. He had risked his life charging machine-gun posts before and he was prepared to do so again. Ordering two battalion runners to stick with him, he jumped from cover and silenced one of two machine-gun posts in his path by lobbing several Mills bombs into the trench. The lone surviving gunner from the emplacement next to the one that had been destroyed by MacDowell's grenades fled into a dugout.
When MacDowell reached the mouth of this made-man cave, he shouted down to the German to surrender. Receiving no response, the Canadian captain climbed down a ladder into the depths of the tunnel. Once his eyes had become accustomed to the dark, he started forward. As he turned a corner, he nearly bumped into two German officers and 77 members of the Prussian Guard. He was trapped. But, as one of the officers issued an order to his troops, MacDowell decided to bluff it out, calling back over his shoulder as if a unit of Canadian soldiers were standing by above.
The Germans took the bait and all 79 of them dropped their weapons and raised their hands over their heads. Continuing the ruse, MacDowell sent his prisoners in small groups up to the surface where his two battalion runners were waiting to dispatch them back to the Canadian lines. The Germans didn't know until it was too late that they vastly outnumbered their captors.
Even at that, one of the prisoners, when he reached the surface and found only two lightly armed Canadians greeting him, reached for a nearby abandoned rifle. It would be the last move he ever made.
For his cool-headed actions, Captain MacDowell received a Victoria Cross to go along with his DSO. In addition, he increased his prisoner tally to 132. He waited five days until reinforcements arrived before he went back behind the lines to have a hand wound properly treated.
MacDowell would go on to greater glory later in the war, as would another member of the 38th Battalion, Private Claude Nunney, a machine-gunner who repelled 20 attacking Germans at Vimy even though he himself was wounded. For his heroics at Vimy, Nunney was awarded the Distinguished Conduct Medal.
Despite the fact that the 4th Division had such a hard time at Vimy — or perhaps because of it — many tales of bravery about the men of that unit are recorded in the history books. Other members of the 4th Division who distinguished themselves on the first day of battle at Vimy Ridge were Sergeant Samuel Lewis Honey of the 78th Battalion, Private George McLean of the 54th Battalion, and Private James McMillan of the 87th Battalion.
Sergeant Honey, a schoolteacher from Conn, Ontario, had already won the Military Medal for gallantry during a raid on German trenches on February 22, 1917, for clearing an enemy stronghold, then covering the withdrawal of his own and another squad that had come under heavy grenade fire. Honey won the Distinguished Conduct Medal for gallant leadership at Vimy. After his platoon commander had been wounded, he assumed command and led his men forward in the face of intense fire. Once dug in, they held their position until relief arrived three days later.
Private McLean had served with the Canadian Mounted Rifles during the Boer War at the turn of the century. After World War I broke out, McLean enlisted at Vernon, British Columbia, in October 1916 and was in France by December.
As part of the 54th Battalion's attack against the enemy on April 9, 1917, McLean, armed with about a dozen Mills bombs, launched a solo attack on a group of enemy soldiers, taking 19 prisoners. Wounded in the arm by sniper fire, he still managed to stop five other German soldiers from reaching a machine gun, and so prevented untold Canadian casualties. For his bravery, he was awarded the Distinguished Conduct Medal.
Private McMillan, who had also served with the Canadian Mounted Rifles in the Boer War, joined the 87th Battalion of the Grenadier Guards at the age of 38 in 1916 and was almost immediately shipped out to France, where he took part in the Battle of the Somme. He participated in the taking of the Regina Trench and saw more than half of his battalion killed during that campaign.
That deadly history repeated itself at Vimy Ridge, where 55 percent of the reinforced battalion were either killed or injured. McMillan was credited with crossing no man's land twice to provide crucial information to the 11th Brigade. He was awarded the Military Medal for gallantry.
When planners miscalculate expected enemy resistance, a quick and drastic change of plans can sometimes save the day. With the Germans so well entrenched on Hill 145 and the Canadian units exhausted and their ranks decimated — about one in four of the attacking troops of the 4th Division had been killed — the totally inexperienced 85th Battalion was tapped to perform a miracle.
It was a desperation move on the part of the field commanders. The 85th, known as the Nova Scotia Highlanders, was a ragtag bunch of soldiers who had never before been in battle. An outbreak of mumps while they were still in England had reduced their number by about 200. The rest of the unit had recently arrived in France and had been seasick on the way over the Channel. To date, their weapons had been picks and shovels; they were trench diggers and ammunition haulers, not fighting soldiers. But they were the only fresh troops available.
Much to the surprise of everyone, including the Germans, the 85th galvanized into a fighting force and, under heavy enemy fire, captured the western summit of Hill 145 by the time the sun had set once again on this field of carnage.
Fresh Troops
In spite of this unexpected success, however, German machine-gun and mortar posts were still in operation on the hill's eastern slopes. The original plan had been for the 4th Division to gain control of the entire summit of Hill 145 by the end of the first day of the attack. April 10 had been set aside for final mopping-up action, including the taking of a small forested knoll British troops had nicknamed the Pimple, at the extreme north end of the ridge overlooking the village of Givenchy. But, with German troops still entrenched on the eastern slope of the hill, the attack on the Pimple — also known as Hill 120 — was postponed.
On April 10, fresh troops from the 4th Division's 10th Brigade were pressed into service to help finish the job their comrades had been unable to complete the day before. The revised plan was for two of the brigade's four battalions — the 44th from Winnipeg and Calgary's 50th — to operate from the trenches captured the day before. Their orders were to annihilate the German troops still dug in on the eastern summit of Hill 145 and storm down the eastern slope of Vimy Ridge to the northern end of La Folie Wood at the bottom of the incline.
Fifty-six members of the 50th Battalion were Canadians of Japanese descent who had enlisted in the 175th Battalion back in Calgary when racially prejudiced units in British Columbia refused to accept them. The 175th was eventually sent to France as reinforcement for the 50th Battalion. Close to 200 Japanese volunteers fought overseas for Canada during the war and about three-quarters of them were either killed or wounded. More than 30 were killed in the Vimy area alone.
Sergeant Yazuso Shoji of the 52nd Battalion, which had been held in reserve and did not take part in the Vimy attack, was later asked why he and his comrades had been so determined to join in the hostilities. After all, it had been made perfectly clear that certain redneck senior officers didn't want them. "We don't forget what we owe to Canada and we were proud to fight when Britain declared war on the common enemy," was Shoji's reply.
Sergeant Masumi Mitsui, of Port Coquitlam, British Columbia, would win the Military Medal for Bravery at Vimy. A sandstone monument near Lumberman's Arch in Vancouver's Stanley Park would be dedicated to Mitsui and his fellow Japanese Canadian Corps volunteers on April 9, 1920, the third anniversary of the launch of the Vimy offensive. Mitsui's medal would be one of several wartime awards he would fling at a Canadian army officer confiscating his property during World War II following the Japanese attack on Pearl Harbour. He and his family would be forced to live in internment camps until that war ended in 1945. This unappreciated hero would later relent and take part in a ceremony in 1985 to relight the flame on the monument erected in 1920. The so-called eternal flame, housed inside a marble lantern, had been extinguished following the Pearl Harbour attack in 1941 and it would take more than 40 years for that sorry decision to be reversed. Mitsui died in 1987, five months short of his 100th birthday.
It took until mid-afternoon to get the two untried battalions into place and to formulate a new creeping barrage pattern to soften up the enemy as the Canadians advanced. That meant the attack force had the whole morning to do the last-minute things men do before going into battle. They read old letters from home, wrote new ones, checked and rechecked their weapons and equipment, or played cards to lend an air of nonchalance and stave off thoughts of the madness they were about to experience.
During this lull in the fighting, one member of the 50th Battalion, Private John Pattison, might well have been wondering what made him decide to sign up to fight a war that should have been left to those much younger than himself. At age 42, he was one of the oldest Canadians in the ranks at Vimy. He had a wife and four children at home in Alberta. He had given up a secure job with the Calgary Gas Company to sail to France and put up with despicable conditions in the trenches and the constant threat of injury or death.
Pattison had a double reason for enlisting. He was fighting not only for his friends and family back home in Canada, but also for those loved ones he had left behind when he had emigrated from England with his immediate family in 1906. And he and his fellow soldiers in the 50th Battalion had watched in horror the day before as German sniper and machine-gun fire cut down their comrades. The impending attack would help them avenge that slaughter.
When the heavy guns started laying down the rolling barrage early on the afternoon of December 10, the men of the 44th and 50th battalions began their slow advance toward the eastern slope of Hill 145. Once again, blistering machine-gun fire and mortar shells took their toll. The troops had to leave wounded comrades writhing in the mud as they leapt from shell hole to shell hole to try to gain some protection from the heavy enemy fire.
Pinned down in a shell crater, Private Pattison spotted an enemy machine-gun post directly ahead that was preventing his battalion from moving forward. With a sack of Mills bombs slung over his shoulder, Pattison braved bullets and mortar shells to make a run for the enemy stronghold. When he was within grenade-pitching distance, he quickly lobbed three of the "pineapples," as the bombs were nicknamed. The German guns fell silent.
Other members of the battalion moved in behind Pattison and together they charged forward, bayoneting any enemy soldier who put up resistance. By the end of the afternoon, in spite of a massive loss of men, Hill 145 was in Canadian hands.
The Pimple
Because of the change in plans, the Canadian Corps was given 24 hours to regroup and take a breather before launching a final attack on the Pimple, an action that would put the entire Vimy Ridge area in Canadian hands. Reserve battalions — British Columbia's 47th and the 46th from Saskatchewan — were brought forward to back up the battle-weary troops from the 50th Battalion who had hardly been given a chance to catch their breath from their ordeal two days earlier. With the prize almost within their grasp, the Canadians experienced a surge of optimism and pride at the realization that a horrible task had all but been completed. Even an overnight snowfall that covered men, equipment, horses, wagons, and weapons was taken in stride. The snow would undoubtedly turn to slush and make the upcoming assault that much more difficult, but there was still a sense that the worst was over — at least for the time being.
However, as ineffectual as its name might sound, the Pimple was a formidable target. The 120-metre-high escarpment was a maze of dugouts, tunnels, and trenches that earlier Allied attacks had failed to capture. And with the Germans tenuously holding on to this last scrap of land on the ridge, the Canadians were well aware that the enemy's last-ditch efforts would be stubborn and murderous.
One well-respected member of the 50th Battalion, Lance Corporal Henry Norwest, a Metis of French-Cree descent from Fort Saskatchewan, Alberta, was taking advantage of the brief lull in hostilities by paying particular attention to his rifle and the telescope attached to it. "Ducky," as he was known to his mates, was the battalion's sniper and he had racked up more than 100 confirmed enemy kills so far.
At dawn on April 12, the troops of the three designated battalions emerged from their trenches and slowly followed the creeping barrage as it made its deadly march up toward the Pimple. Another snowstorm had hit, but the Canadians welcomed it since it was at their backs and therefore blowing into the faces of the enemy. It was almost as effective as a smokescreen in hiding the advancing troops, giving them an advantage that would help offset the fact that the Germans had also made use of the lull in fighting to bring in reinforcements. The new defenders were members of the elite Prussian 5th Grenadier Guards – all of them over six feet tall and formidable adversaries in hand-to-hand combat.
Nevertheless, within two hours, Hill 120 had fallen to the Canadians — but at a heavy cost. Nearly half of the troops who took part in the assault were either killed or wounded by the intense resistance put up by the enemy. The body count might well have gone higher except for the handiwork of Corporal Norwest, who felled a number of German troops during the assault on the Pimple. Norwest was awarded the Military Medal for his sharpshooting that day.
Brigadier General Edward Hilliam, a former Alberta rancher, had led the 10th Brigade in its assault against Hill 120. When his men had overpowered the Prussian Guards, Hilliam sent a report to Allied headquarters, signing it "Lord Pimple."
Chapter 9
A Bitter Victory
They had done it. Despite overwhelmingly negative predictions by everyone except the resolute Canadians, the taking of Vimy Ridge was judged a spectacular success by the Allied High Command — the single greatest accomplishment on the Western Front to date. But it was a hard-won victory, with 10,602 Canadian casualties, including 3,598 fatalities. German losses, under Colonel General Ludwig von Falkenhausen, were even greater, with 20,000 casualties and 4,000 captured.
Although the Allies expected an enemy counterattack, it didn't materialize and the Germans never again occupied this strategic site. It was later calculated that in this battle, the Canadians captured more ground, more guns, and more prisoners than any previous British operation on the Western Front.
The New York Tribune suggested in an editorial that Canada had fielded a better army than any that Napoleon Bonaparte had raised in the glory days of Imperial France. A French newspaper thanked Canada for a wonderful Easter gift. Legend has it that when a French officer heard of the victory, he replied, "C'est impossible!" Upon learning it was the Canadians who captured the ridge, he added, "Ah! Les Canadiens! C'est possible!"
As the Corps' commanding officer, Lord Byng, put it: "There they stood on Vimy Ridge, on the 9th day of April, 1917. Men from Quebec stood shoulder to shoulder with men from Ontario, men from the Maritimes with men from British Columbia, and there was forged a nation tempered by the fires of sacrifice and hammered on the anvil of high adventure."
The loss of Vimy Ridge caused the Germans to retreat to the lower plains, which were far more difficult to defend. Unfortunately, with simultaneous British and Australian attacks to the south of the ridge proving unsuccessful, very little of any strategic importance was gained in the area after the Canadian success. However, in a war in which thousands of soldiers on both sides were killed for the gain of a few metres of ground, the Canadian win gave a significant boost to Allied morale, while demoralizing German troops, who had thought the ridge impregnable. It also relieved the city of Arras from the constant German bombardment and the threat of imminent attack it had been enduring.
After Vimy
Following the Canadian victory at Vimy, Sir Julian Byng was promoted to a full general and put in charge of the entire 3rd Army. His Vimy second-in-command, Arthur Currie, was knighted on the battlefield by King George V and promoted to lieutenant general. He took over Byng's duties with the Canadian Corps.
The war, however, was far from over. The Canadians stayed on in the Arras area and fought a bloody battle for Hill 70 during 10 days in late August. They took possession of a vital position on the northern approach to the city of Lens and managed to hold down the western section of the city.
Early in October, the Canadian Corps was ordered to prepare for an assault on the village of Passchendaele near the Belgian town of Ypres in West Flanders. The plan was to punch a hole through the German lines so that German submarine bases on the Belgian coast could be captured. Not only would this establish a strategic corridor, it would serve as a morale booster to the thousands of French troops on the verge of mutiny due to battlefield blunders by senior officers that had decimated their ranks.
The land on which the battle was to take place was largely reclaimed marshland, which had turned into a huge swamp after torrential rains in August combined with British shelling in preparation for the attack. The resultant soupy mud claimed the lives of untold numbers of soldiers who drowned after slipping beneath its surface.
To make things even more treacherous, the Germans were well entrenched and a vast number of machine-gun posts were still intact inside pillboxes that had not been destroyed by initial bombardment. It was literally a fight from shell crater to shell crater as the Canadians inched their way forward under heavy enemy fire. On October 30, in tandem with two British divisions, the Canadians launched an attack on the village under cover of a driving rainstorm. Often waist-deep in mud, the attackers fought doggedly for five days until reinforcements arrived.
Passchendaele fell to the Allies on November 6, but at a tremendous cost. By the time the Union Jack flew over the village, 80 percent of the attackers had paid for the victory with their lives, with the Canadians suffering 15,654 casualties. Nine Canadians won the Victoria Cross there.
One Canadian officer, Fred Holm of Toronto — who had worked out at his local YMCA to become fit enough to be accepted into the army — nearly ended up court-martialled because of an incident at Passchendaele. Nine German prisoners his unit had captured were waiting for transport back to an internment camp. Holm felt sorry for them because they had been through a horrendous ordeal and looked it.
Walking up to the first German in the straggly line they had formed after throwing down their arms, Holm offered him a cigarette. The prisoner, a Prussian officer decked out in the Hollywood stereotype of crewcut and monocle, spat in the Canadian's face. Holm, who had had his own share of misery and near-death, lost his cool and pulled out his revolver, aiming it at the offender.
"I don't know what I would have done," said Holm many years later, "if a young German at the rear of the line hadn't shouted out that he'd like one of my cigarettes. That defused the situation and I walked back to him, handed him a fag and lit it for him. In the glare of the match, I recognized him and told him so.
"The prisoner replied that he would have been surprised if I hadn't known who he was because he'd given me the best table every Friday night at the dining room at the King Edward Hotel back home. I immediately remembered him as the restaurant's maître d' and asked him what he was doing on the Western Front. He replied that he was German and had sailed home to sign up to fight for the Fatherland when he knew there was a war coming."
Whenever he related the story, Holm would wait for the inevitable remark that his was quite a small-world story, then he'd continue: "You haven't heard it all. I lost my leg at Passchendaele and spent more than a year recuperating in England. By the time I got back to Toronto, the war had been over for a couple of months. My buddies greeted my homecoming by reviving our traditional Friday night outing at the King Edward. And you aren't going to believe who the maître d' was when I walked into the joint... "
In a last desperate move, the German High Command launched an all-out offensive in the spring of 1918. Their troops got to within 70 kilometres of Paris and almost broke through the defences of the war-weary Allies. Somehow, the British, French, Canadian, Australian, and New Zealand soldiers found the strength to hang on and the German gamble didn't pay off. In fact, the tables turned on the Kaiser's troops when the Allies got their second wind under a new supreme commander, French Marshal Ferdinand Foch.
In a series of counterattacks, the Allies made advances throughout the Western Front. The period from August 8 to November 11, 1918 came to be known as Canada's Hundred Days. During that time, a contingent of about 105,000 Canadians made inroads of 130 kilometres. In so doing, they captured 31,500 prisoners, more than 600 artillery pieces, close to 3,000 machine guns and over 325 mortars. The price they paid was 45,830 battle casualties.
When the Allied advance began in 1918, the Canadian Corps was assigned to spearhead an attack on an enemy bulge in the lines near Amiens. This action led to the awarding to Canadian combatants of 10 Victoria Crosses. Advancing 19 kilometres in three days, the Canadians once again showed their mettle as shock troops in an attack situation. The German High Command was badly shaken by this feat. German General Erich Ludendorff called August 8 the "black day of the German Army."
Because of their heroic actions at Amiens, the Canadians were redirected to Arras and tasked with breaking through the enemy's main line phalanx, the Hindenburg Line. Between August 26 and September 2, the Canadian Corps slugged it out with tough German units and advanced to the Canal du Nord. With the aid of 15 British tanks, they managed to traverse this heavily fortified position. A breakthrough had finally been achieved. Victory was at hand.
Cabrai was taken in early October. The Canadians then advanced steadily through Valenciennes and Mont Houy and arrived at Mons on November 11, 1918, the day the armistice came into effect. After more than four years of bitter fighting and tragic losses on both sides, the "Great War" was finally over.
The cost of total victory was high. Some 619,000 Canadians served in the army. Of all the Canadians overseas, more than 60,000 paid the supreme sacrifice. Canadian casualties totalled 239,605, one-third of those who had signed up to fight. Of 3,141 nursing sisters, some 2,500 served the many wounded on all fronts and 46 died.
Chapter 10
Vimy's Heroes
It doesn't take long for a fatalistic cynicism to set in after exposure to the death and destruction of battle. The "luck of the draw" is an expression that war-weary troops have used in one form or another ever since the first armed conflict of any magnitude.
Upon seeing your best buddies — or total strangers, for that matter — cut down by a random shell or machine-gun burst, the usual reaction runs the gamut from shock and grief, to relief at having been spared, to the inevitable asking of the question, "Why them and not me?"
The luck of the draw is usually the answer. If asked to elaborate, the survivor will probably suggest that when your number is up, it's up, and there's nothing you can do about it. It's another way of knocking on wood to keep misfortune away — at least for the time being.
So, what was the luck of the draw for those Canadians who played a larger-than-life role in the capture of Vimy Ridge? Here's the rest of their story.
Lieutenant General Sir Julian Byng
English-born Julian Hedworth George Byng was 54 years of age when, as a lieutenant general, he was put in charge of the Canadian Corps in preparation for the Battle of Vimy Ridge. Following this decisive victory, he was promoted to full general and put in command of the entire 3rd Army. He would play an important role in engagements at Cambrai, Albert, Epehy, Havrincourt, and Valenciennes.
At the end of the war, he was raised to the peerage as 1st Baron Byng of Vimy of Thorpe-le-Soken in Essex, England. From 1921 to 1926, he served as Governor General of Canada and was involved in the "King-Byng Affair," in which he refused to sign an Order-in-Council spearheaded by Prime Minister William Lyon Mackenzie King seeking the dissolution of Parliament and the calling of a new election. Instead, the Governor General invited Opposition Leader Arthur Meighen to form a government. When Meighen lost a vote of confidence in the House of Commons within a week, Parliament was finally dissolved and Mackenzie King was re-elected. Not surprisingly, Byng was soon on his way home to England, where he died in 1935.
Major General Arthur William Currie
Arthur Currie was born in Strathroy, Ontario, and moved to Victoria, British Columbia, as a young man. He joined a Canadian artillery regiment while still in his teens. His rapid rise through the ranks impressed his superiors and, with the rank of major general, he was appointed to command the 1st Canadian Division in 1915. After the Battle of Vimy Ridge, Currie was promoted to lieutenant general and succeeded Sir Julian Byng as general officer commanding the Canadian Corps. At 41, he was reputed to be the youngest officer to achieve this rank in the British armies. Currie was knighted by King George V in 1917.
After the war, Currie became president and vice chancellor of Montreal's McGill University. He died in Montreal on November 30, 1933, after a brief illness. It was estimated at the time that one-quarter of the population of the City of Montreal turned out for his funeral.
Lieutenant Colonel Andrew McNaughton
Andy McNaughton was well rewarded for his success in implementing the creeping barrage and in applying his scientific education to various aspects of enhancing Allied firepower and making the big guns of the enemy less effective. After Vimy, this native of Moosomin, Northwest Territories (now Saskatchewan), was promoted to brigadier general and put in charge of the Allied artillery for the duration of the war.
Remaining in the Canadian armed forces after the war, McNaughton had become chief of the general staff by 1929. In 1935, he was appointed chairman of the National Research Council but returned to the army to command the Canadian Forces at the outbreak of World War II. Late in 1939, he took the First Canadian Division overseas as a major general. He returned to Canada to become defence minister, but was defeated twice at the polls in back-to-back elections in 1945.
McNaughton was appointed to the joint Canada–U.S. defence board and then, in 1946, to the Canadian and United Nations' atomic energy commissions. He became permanent Canadian delegate to the UN in 1948 and, in 1950, was appointed to the International Joint Commission. He died in Montebello, Quebec, on July 11, 1966.
Baron Ludwig Freiherr von Falkenhausen
Colonel General von Falkenhausen was born in Guben, Prussia, in 1840 and saw action in the Austro-Prussian war of 1866 and the Franco-Prussian War of 1870–1871. When World War I broke out, von Falkenhausen was called out of retirement at the age of 74 to command Germany Army Detachment "A" from 1914 to 1916.
He received the coveted German medal "Pour le Merité" in 1915 for distinguished service in northeastern France. In 1917, he headed the German Sixth Army in the Battle of Vimy Ridge. Following this defeat, he was appointed Governor General of Belgium until the end of the war. He retired to Gorlitz, Germany, where he died on May 4, 1936.
Ironically, his nephew, Alexander von Falkenhausen, also received the Pour le Merité medal in World War I and was also named Governor General of Belgium — in his case during the German occupation of that country in World War II. During that conflict, Alexander was involved in an unsuccessful plot to kill Adolf Hitler and spent the latter part of the war in concentration camps before being liberated by the Allies. He died in 1966.
Captain Robert James Manion
Robert Manion was born in Pembroke, Ontario, on November 19, 1881, and studied medicine at Queen's University in Kingston, Ontario, and in Edinburgh, Scotland, before settling in Fort William, Ontario (now Thunder Bay), where his parents had lived since 1888. In 1915, he joined the Canadian Army Medical Corps and won the Military Cross for saving the life of a fellow officer at the Battle of Vimy Ridge.
Upon his discharge from the Canadian army, Manion was elected to the House of Commons as the Member of Parliament for Fort William. Conservative prime minister Arthur Meighen appointed him minister of soldiers' civil re-establishment in 1921 and he later served as postmaster general.
Although he lost his seat in an election in 1935, Manion won the 1938 Conservative leadership race and re-entered the House of Commons in a by-election that year. His party lost the 1940 election and Manion was defeated in his own riding, leading to his resignation as leader. He died in Ottawa three years later on July 2, 1943.
Captain Thain Wendell MacDowell
Born in Lachute, Quebec, on September 16, 1890, MacDowell received his Bachelor of Arts degree from the University of Toronto in 1914. He enlisted in the 41st Regiment (Brockville Rifles) then transferred to the 38th Canadian Infantry Battalion.
During the Battle of the Somme in 1916, MacDowell was awarded the Distinguished Service Order for knocking out three enemy machine-gun nests and taking 53 German prisoners. Wounded in the attack, he recuperated in England and returned to France in 1917 in time for the Battle of Vimy Ridge.
During the action that won him the Victoria Cross, MacDowell sustained a hand wound but stayed in the fight for almost a week until relief troops arrived. Once again sent to England for treatment of his wound, he received his VC from King George V.
Unfortunately, the stress of battle caught up with MacDowell and he suffered a nervous breakdown in the fall of 1917. Invalided home, he spent three months in the Brockville General Hospital and was then pronounced fit to return to his duties. He served the rest of the war in England.
After a long career in Canada's peacetime army, including a stint as private secretary to the minister of national defence, MacDowell left the service as a lieutenant colonel and returned to civilian life as a director of several mining companies as well as president of the Chemical Research Foundation. He died in March 1960 in Nassau, the Bahamas, and is buried in Brockville, Ontario.
Lance Sergeant Ellis Wellwood Sifton
In October 1914, Ellis Sifton, a farmer born in Wallacetown, Ontario, on October 12, 1891, enlisted in the St. Thomas Regiment but transferred to the 18th Battalion to go overseas the following year.
Sifton would never know that his action in silencing a machine-gun post single-handedly at Vimy Ridge would earn him the Victoria Cross. As he was holding the newly won position against a counterattack, one of the Germans he had mortally wounded with his bayonet was able to get off a rifle shot before he died, killing Sifton instantly.
The VC winner is buried in a mass grave in the Lichfield Crater Cemetery east of the community of Neuville-Saint-Vaast, near where he fell. The cemetery contains the bodies of 52 Canadians as well as about a dozen other soldiers whose nationality is unknown.
Sergeant John MacGregor
John MacGregor's story is a template for many other tales of heroism involving Vimy Ridge. Here was an NCO who was awarded a medal for bravery during this epic battle only to go on to even greater honours later in the war. In MacGregor's case, this meant earning several other medals, including the Victoria Cross.
Why John MacGregor stands out from all the others is that, when hostilities ended, not only did he hold the distinction of having risen from private to captain in a little over two years, he also had been awarded more medals for valour than any other Canadian soldier in history. These included the VC, Military Cross and bar (the bar indicating he had won this award twice), Distinguished Conduct Medal, and the Efficiency Decoration.
What makes MacGregor's story all the more remarkable is that this leader of men had been virtually a recluse until he enlisted and became part of the 2nd Canadian Mounted Rifles. Born in Nairn, Scotland, on February 11, 1889, MacGregor had immigrated to Canada in 1909 where he worked as a cowboy and construction labourer before heading into northern British Columbia to take up trapping. It wasn't until March 1915 that he learned from a passing ranger that a war was raging in Europe.
MacGregor snowshoed out to Terrace, British Columbia, sleeping in snowdrifts for a week before reaching civilization. He then rode a freight train to Prince Rupert and went straight to the recruiting office without bothering to clean up. His offer to enlist was declined. He was deemed "unfit for duty in the Canadian Army." But the army hadn't counted on MacGregor's obstinacy. He took a boat to Vancouver, cleaned himself up, and was accepted at the next recruiting post.
After winning the DCM at Vimy, MacGregor was promoted to lieutenant. In January 1918, he won the Military Cross for capturing prisoners while leading a trench raid and soon after was promoted to the rank of captain. In September 1918, at the Battle of Cambrai, his heroics earned him the VC when he captured a machine-gun position, bayoneting four Germans and taking eight others prisoner. At the Honnelle River in November 1918, he received a bar to his Military Cross by personally doing the reconnaissance that resulted in the seizing of two bridges from the Germans.
Returning to Prince Rupert after the war, MacGregor operated his own fishing boat for a time before returning to the construction business. When World War II broke out, he enlisted as a private without telling anyone of the honours he'd received in the previous conflict. When his record was discovered, he was given the rank of major and took command of a training unit in Canada. He retired with the rank of lieutenant colonel and was awarded the Efficiency Decoration for serving in both wars and for 20 years of dedication to the military. MacGregor died in Powell River on June 9, 1952, and is buried at Cranberry Lake Cemetery.
Sergeant Samuel Lewis Honey
Born in Conn, Ontario, on February 9, 1894, Lewis Honey began teaching school at age 16 on the Six Nations Reserve near Brantford. He enlisted in the Canadian Army as a private in January 1915 and had been promoted to sergeant by the time he was shipped to France with the 78th Battalion in October 1915.
After being awarded the Military Medal in January and the Distinguished Conduct Medal at Vimy Ridge, Honey was recommended for a commission and sent to England to attend officers' training school. He returned to France as a lieutenant and was awarded the Victoria Cross for heroic action at Bourlon Wood on September 27, 1918, after all the other officers of his company had become casualties.
After repulsing four counterattacks, Honey went out alone after dark, located a German post, and took a party to capture it. On September 29, he led his company against a strong enemy position and continued to display "the same high example of valour and self-sacrifice." He died of wounds he received during this attack.
Lance Corporal Henry Louis Norwest
Henry Norwest was one of the most famous Canadian snipers of World War I, achieving a confirmed kill record of 115 enemy soldiers. Born in Fort Saskatchewan, Alberta, Norwest was married with three children and worked as a ranch hand. He gained early fame as a trick roper on the rodeo circuit.
On January 2, 1915, using the name Henry Louie, Norwest enlisted at Wetaskiwin, Alberta, but was discharged for misconduct three months later after a wild drinking spree. Eight months later, he signed up again, this time in Calgary under a new name and was assigned to the 50th Battalion. Here his skills as a sniper were discovered. He was able to lie motionless for hours on end and camouflage himself so that German soldiers could walk within feet of him and never know he was there.
In August 1918, Norwest had a bar added to the Military Medal he had been awarded at Vimy Ridge, joining a unique group of about 830 members of the Canadian Expeditionary Forces in World War I to be given this double honour. But he never got to wear this new decoration. On August 18, less than three months before the war ended, Norwest and two of his fellow Canadians were searching out a nest of snipers when a German bullet smashed into him, killing him instantly. He is buried near the village of Warvillers, France.
Private George McLean
George McLean, a Native rancher from the Head of the Lake Band in the Okanagan district of British Columbia, had enjoyed a distinguished career with the Canadian Mounted Rifles during the Boer War in South Africa. When World War I broke out, every single male between 20 and 35 from the Head of the Lake Band volunteered for overseas service. McLean enlisted and was part of the 54th Battalion in France by December 1916.
Despite being wounded at Vimy Ridge, he managed to capture a number of prisoners and prevent a counterattacking group of German soldiers from recapturing a machine-gun post. He received the Distinguished Conduct Medal for gallantry in the field. His wound was serious enough to have him discharged and repatriated to Canada where he worked as a firefighter in the Vancouver area. He died in 1934.
Private Claude Joseph Patrick Nunney
Born in Dublin, Claude Nunney came to Canada as a "Home Boy"— an impoverished youngster sent overseas to find a better way of life. He lived in several foster homes before joining the Canadian army. He was sent overseas on May 23, 1916. After being awarded the Distinguished Conduct Medal at Vimy Ridge, Nunney's heroics earned him the Military Medal two months later. He was to win the Victoria Cross for heroic action near Vis-en-Artois during a heavy bombardment on September 1 and 2, 1918, preceding a German counterattack. Nunney left the safety of company headquarters to dash through the bursting shells to the company outpost lines, going from post to post to encourage the men by his own fearlessness.
The next day, Nunney was badly wounded but would not leave the field, running ahead of his advancing comrades and inflicting heavy casualties on enemy gunners, killing 25 of them in his rampage. Wounded a second time, he refused to quit but soon became so weak that he had to be taken to a Canadian clearing station by stretcher. He died from his wounds 16 days later and is buried at the communal cemetery at Aubigny-en-Artois, France.
Private James McMillan
James Irving McMillan was born in Lindsay, Ontario, on August 25, 1877. He fought in the Boer War in South Africa with the 2nd Canadian Mounted Rifles and returned to Canada in 1903. At the age of 38, he joined the 87th Battalion and fought at the Somme in 1916. He won the Military Medal for gallantry at Vimy Ridge and returned to Canada at the end of the war.
McMillan became a fisherman and died in White Rock, British Columbia, in 1965 at the age of 87. He is buried in the veteran's section of Sunnyside Lawn cemetery.
Private John George Pattison
The sad truth of Private John Pattison's participation in World War I was that he really shouldn't have been there in the first place. Born in New Cross, England, on September 8, 1875, Pattison, at age 42, was one of the oldest Allied participants in the Battle of Vimy Ridge. Furthermore, he was a married man with four children.
He moved to Canada with his young family in 1906, first settling in Rapid City, Manitoba, and then moving to Calgary where he worked for the local gas company. In May 1916, he enlisted in the 137th Infantry Battalion and was later transferred to the 50th Battalion in time for the Vimy offensive.
Pattison survived that battle but was killed, along with his entire machine-gun crew, a little less than two months later, on June 3, in an attack on a generating station at Lieven, near Lens, not far from Vimy Ridge. He is buried at Vimy's La Chaudière Cemetery.
Private William Johnstone Milne
A farmhand from Moose Jaw, Saskatchewan, William Milne was born in Cambusnethan, Lanarkshire, Scotland, on December 21, 1892, and settled in Canada in 1910. In September 1915, he enlisted in the army and was shipped to France the following year with the 16th Battalion (Canadian Scottish).
Milne was another hero who would never know that he had won the Victoria Cross. On April 9, 1917, after knocking out two machine-gun emplacements single-handedly and, in the first instance, turning the captured German gun against the enemy, Milne was seen later in the day falling wounded behind a small hill while continuing the attack. His body was never found and his VC was awarded posthumously. His is one of the more than 11,000 names inscribed on the Vimy Memorial of Canadians killed in France and buried in unmarked graves.
Native Volunteers
Lance Corporal Henry Norwest and Private George McLean were only two of more than 4,000 Native volunteers who served their country in World War I. Many recruits felt they were honouring a tradition set by their ancestors, who had taken up arms to assist the British against invaders in the War of 1812.
One of these was Cameron Brant, great-great-grandson of the heroic Six Nations chief, Joseph Brant. Lieutenant Brant commanded a platoon of the 4th Canadian Infantry Battalion. The 28-year-old lieutenant was killed in 1915 leading a counterattack against German trenches near Ypres.
Another member of the Six Nations, Lieutenant James Moses, served in both the infantry and the air services. As an air observer, he was shot down over France in 1918 and both he and his pilot were reported missing in action.
Corporal Francis Pegahmagabow, an Ojibwa from the Parry Island Band in Ontario, was the most highly decorated Native soldier of the war. "Peggy," as his comrades in the 1st Battalion nicknamed him, was awarded the Military Medal and two bars — one of only 39 members of the Canadian Expeditionary Force to receive this honour. A scout and a particularly sharp-eyed sniper — with more than 375 "kills" on his record — Pegahmagabow was also credited with single-handedly capturing about 300 German soldiers. While information is sketchy, it is believed the corporal earned his MM at the Second Battle of Ypres in June 1916, the first bar at Passchendaele in November 1917, and the second bar at Amiens in August 1918. After the war, he followed in the footsteps of his father and grandfather and became chief of the Parry Island Band. He died on the reserve in 1952.
When the Prince of Wales visited the Brantford, Ontario, area in October 1919, he presented the Six Nations with a bronze plaque to commemorate the 88 of its members who were killed in, or as a result of, military action.
Esprit de Corps
Historians and military analysts overwhelmingly agree that a decisive factor in the victory at Vimy Ridge was the insistence by Sir Julian Byng and Major General Arthur Currie that the four divisions of the Canadian Corps be allowed to fight together.
This decision meant that friends and relatives fought side by side in the same unit, spurring them on to greater effort in order not to "let down the side." And, when the smoke of battle had cleared, the wellspring of pride at the job that had been accomplished carried over into civilian life after the war. Canadians saw themselves for the first time as an independent nation and this sentiment soon turned to reality with legislation loosening the ties to Great Britain and giving Canada a singular role to play on the world stage.
The following are the Canadian Battalions that fought together at Vimy Ridge.
1st Division
Commanding Officer Arthur Currie
1st Brigade
* 1st Battalion (Western Ontario)
* 2nd Battalion (Eastern Ontario)
* 3rd Battalion (Royal Regiment of Canada, Toronto)
* 4th Battalion (Western Ontario)
2nd Brigade
* 5th Battalion (Saskatchewan)
* 7th Battalion (British Columbia)
* 8th Battalion ("The Little Black Devils of Winnipeg")
* 10th Battalion (Calgary)
3rd Brigade
* 13th Battalion (5th Royal Highlanders – Montreal)
* 14th Battalion (Royal Montreal Regiment)
* 15th Battalion (48th Highlanders)
* 16th Battalion (Canadian Scottish, British Columbia)
2nd Division
Commanding Officer Harry E. Burstall
4th Brigade
* 18th Battalion (London, Ontario)
* 19th Battalion (Central Ontario)
* 20th Battalion (Central Ontario)
* 21st Battalion (Eastern Ontario)
5th Brigade
* 22nd Battalion (Canadien Français — the "Van Doos")
* 24th Battalion (Victoria Rifles, Montreal)
* 25th Battalion (Nova Scotia Rifles)
* 26th Battalion (New Brunswick)
6th Brigade
* 27th Battalion (City of Winnipeg)
* 28th Battalion (North West)
* 29th Battalion (Irish Fusiliers, Vancouver)
* 31st Battalion (Alberta)
3rd Division
Commanding Officer Louis J. Lipsett
7th Brigade
* 42nd Battalion (Royal Highlanders, "The Black Watch," Montreal)
* 49th Battalion (Edmonton)
* Royal Canadian Regiment (Toronto)
* Princess Patricia's Canadian Light Infantry (Originally formed in Ottawa)
8th Brigade
* 1st Canadian Mounted Rifles (Manitoba and Saskatchewan)
* 2nd Canadian Mounted Rifles (British Columbia)
* 4th Canadian Mounted Rifles (Central Ontario)
* 5th Canadian Mounted Rifles (Quebec)
9th Brigade
* 43rd Battalion (Winnipeg)
* 52nd Battalion (Port Arthur)
* 58th Battalion (Niagara Area)
* 116th Battalion (Nova Scotia)
4th Division
Commanding Officer David Watson
10th Brigade
* 44th Battalion (Winnipeg)
* 46th Battalion (South Saskatchewan)
* 47th Battalion (New Westminster, Vancouver, and Victoria)
* 50th Battalion (Calgary)
11th Brigade
* 54th Battalion (Kootenays, British Columbia)
* 75th Battalion (Toronto, Hamilton, and London)
* 87th Battalion (Grenadier Guard, Montreal)
* 102nd Battalion (North British Columbia)
12th Brigade
* 38th Battalion (Ottawa District)
* 72nd Battalion (Seaforth Highlanders, Vancouver)
* 73rd Battalion (Royal Highlanders, Montreal)
* 78th Battalion (Winnipeg Grenadiers)
On Land, at Sea, and in the Air
While Canada's principal contribution to World War I was the Canadian Corps, some 23,000 Canadians served in Britain's Royal Flying Corps and 1,600 of them died in combat. Ten of the 27 aces in the RFC were Canadian. At sea, 5,500 Canadians served in the Royal Canadian Navy and another 3,000 in Britain's Royal Navy.
Members of the Royal Canadian Regiment at a Vimy Ridge ceremony in 1983.
The memory of the 48 infantry battalions that saw action at Vimy is still honoured widely in the Canadian army. The current units that evolved from the warriors who took part in Canada's finest battle of World War I emblazon their regimental colours with the battle honour, "Vimy 1917."
Today, the members of the Canadian Corps in World War I are among those whose memory is honoured at each gathering of Royal Canadian Legion branches across Canada, and at pilgrimages around the world, with the recitation of lines from Laurence Binyon's immortal poem "For the Fallen."
They shall grow not old,
as we that are left grow old;
Age shall not weary them,
nor the years condemn.
At the going down of the sun
and in the morning
We will remember them.
Epilogue
The Vimy Memorial
The Vimy Memorial towers over the Doui Plain from the top of what was known during World War I as Hill 145. Located about 10 kilometres northeast of the city of Arras, this is the highest point on Vimy Ridge. The inscription on the base of the monument reads, in French and English, "To the valour of their countrymen in the Great War and in memory of their sixty thousand dead this monument is raised by the people of Canada."
The immense and impressive twin-pillared monument is built on approximately one square kilometre of land donated in perpetuity to Canada by the people of France in 1922. Dominating the memorial is a 30-tonne shrouded figure carved out of one gigantic piece of limestone. She represents the spirit of Canada and honours not only the men who captured the ridge, including those who paid the supreme sacrifice, but also the 11,285 Canadian soldiers who died in France and whose remains were never found. Their names are inscribed on the monument.
Canadian sculptor and architect Walter Seymour Allward, who designed the monument, claimed the plans came to him one night in a dream. It took 11 years and $1.5 million to build the memorial. It was unveiled on July 26, 1936, by King Edward VIII of Great Britain, less than six months before his abdication of the throne so that he could marry "the woman he loved," Wallis Simpson.
Also present at the unveiling were French President Albert Lebrun and more than 50,000 — some estimates reach 100,000 — Canadian and French veterans and their families. On a pilgrimage organized by the Royal Canadian Legion, about 6,200 veterans, friends, family, and interested observers boarded five ocean liners and set sail for Europe, arriving in time for the ceremony. A further contingent of 1,500 Canadian vets living in England joined the pilgrimage in France.
Two future Canadian governors general and one future Canadian prime minister were on hand for the unveiling ceremony. Vincent Massey, the Canadian high commissioner for London at the time, brought along two of his staff. One of these aides was George Vanier, who, as a member of the 22nd Battalion, had been awarded a Military Cross for bravery after losing his right leg in a battle late in the war. Massey would make history as Canada's first native-born governor general, serving from 1952 until 1959. Vanier moved into Ottawa's vice-regal mansion, Rideau Hall, right after his old boss left the premises, and served as the Queen's representative in Canada until 1967. The other assistant accompanying Massey to the memorial unveiling was Lester Pearson, a former flying officer with the Royal Flying Corps, who had earlier seen action as a lieutenant with the Canadian Army Medical Corps. Pearson would win the Nobel Prize for Peace as Canada's external affairs minister in 1957 and would serve as the country's 14th prime minister for five years beginning in 1963.
The same foul weather the Vimy victors faced on that blustery April morning in 1917 gradually eroded the memorial to the point where the names of the honoured dead were, in some cases, obliterated. In addition, the fine statuary began to suffer wear and tear and the building blocks began to crumble as water seeped beneath their surface.
The main problem was that the architect had chosen a new technique for building the enormous monument by covering cast concrete with Seget stone — a white limestone found in modern-day Croatia. Unfortunately, no one foresaw that the two materials would react differently under inclement weather conditions. Differential movement over the years caused severe stress and shifting of the concrete and stone. This not only cracked the limestone but also allowed water to seep beneath the surface, further damaging the edifice.
Thus, the restoration of the Vimy monument was included in a major project announced by the Government of Canada in May 2001. A total of $20 million — two-thirds of the budget — was assigned to the Vimy restoration effort as part of the five-year Canadian Battlefield Memorials Restoration project. Veterans Affairs Canada, in collaboration with Public Works and Government Services Canada, assembled an international team of architects, engineers, artisans, and builders to restore the monument.
The stone from all the lower walls, stairs, and platforms was either removed or, if possible, repaired. The most deteriorated sections were replaced with the same Seget stone from Croatia. Artisans chiselled new sections where much of the old lettering had been obliterated by weather and by water seepage. New landscaping and lighting concepts were also implemented.
The newly restored Vimy Monument.
The craftsmen and builders kept up a furious pace to have the monument refurbished and ready to honour anew, on Easter Monday, April 9, 2007, the sacrifices of the 3,598 Canadians killed at the Battle of Vimy Ridge. The memorial also acknowledges the more than 7,000 who were wounded, those who returned to Canada forever changed in body or spirit, and the 11,285 Canadians who lie in unmarked graves in France.
Legend has it that the 91.18-hectare battlefield park still claims the occasional victim. There are so many unexploded shells in the area that sections are roped off to prevent visitors from blowing themselves up by stepping on a live bomb, mortar shell, or landmine. In fact, the custodians of the park don't even allow lawn-mowing tractors onto parts of the site, preferring to allow grazing sheep to keep the grass trimmed. Every so often, the story goes, one of the beasts will step on a shell and suffer the same fate as so many soldiers did some 90 years ago. Even the shepherd is at risk. Several years ago, he stepped on a "mantrap," a sharpened spike that went through his foot, but he has since recovered.
And the site is also exposed to the occasional poacher. René Dubos of the nearby village of Vimy tells the story of a large truck being spotted driving up the ridge road under the cover of darkness a few years back. The next morning many of the "lawn-mowing" sheep had disappeared. A couple of days later, a festival was staged in the area with a massive barbecue featuring mutton roasted on spits over huge bonfires. René reports that people came from miles around to take part in the festivities.
Since this massive monument to a great Canadian victory in World War I was officially dedicated in 1936, it seems to be a miracle that the structure wasn't blown to bits by the Germans under Adolf Hitler, who successfully invaded France only four and a half years later. Hitler ranted on a number of occasions, not entirely without justification, that Germany had been "stabbed in the back" when, at the end of World War I, the allied powers loaded the country down with unreasonable and inflation-spiralling reparation payments. So why did he allow this monument, signifying a massive German defeat, to stand?
Peter Craven, the senior technical adviser on the restoration project, offers the theory that the Commonwealth War Graves Commission, charged with maintaining the gravesites of Commonwealth soldiers around the globe, has always cared for nearby German war graves as a token of respect for the fallen of all nations. Hitler, apparently, was aware of this and left the monument intact. Even madmen, it seems, can have their lucid moments.
Appendix
Letters from a Vimy Veteran
It's a pretty safe bet that Hollywood film directors who glorify war have never had to experience the horrors of armed conflict. Ask a battlefield veteran to sum up his or her feelings about young people going into combat and the overwhelming response will be: "Never Again!"
Almost all of the veterans of World War I have died. Those few who are left are well over 100 years of age and are probably too feeble to speak out against the insanity they witnessed when they and the world were much younger.
But the tragedy and the sorrow of that "War to End All Wars" have been preserved in the letters the combatants sent to their families from the front or from the hospitals where they convalesced before being sent home or, if their wounds weren't sufficiently damaging, back into the fray.
Gordon MacKinnon, MA, a retired history teacher from Toronto, inherited the correspondence between his uncle, Private Ronald MacKinnon, and Ronald's sister and father. He has graciously given his permission to have an edited version of some of the letters reprinted here.
The letters speak more eloquently about the nightmarish experiences of Canadian troops on the Western Front than anything a writer who was not there could ever put into words.
West Sandling Camp,
Kent, Eng.
May 18, 1916
Dear Father,
I got a letter from brother Archie today; it was written May 14, 1916. He is well and seems to be having a good time considering. He has asked me to send him some lice powder, as he is lousy as a bedbug. I'll send him some when I get out of quarantine. I still have 10 days to put in. It is not so bad now as we have good weather.
The R.S.P., that is, the Royal Sanitary Police, come and get us in the morning and take us to a big Common where we play football and baseball with an hour's physical drill all morning. In the afternoon we go for a route march, so you can see we have a pretty easy time.
It sure is fierce the way they soak the soldiers. It costs sixpence to get a shirt washed but I wash my own. You can tell the soldiers around there that life here is OK if they are not afraid of a little hard work with 60 pounds on their backs. But it is alright if you are prepared to rough it.
Give my regards to all hoping this finds you as well as it leaves me.
Your affectionate son,
Ronald
June 3, 1916
My Dear Father,
Just a line to let you know I am for the Front tomorrow. I have been transferred to the R.C.R. (Royal Canadian Regiment). I did not know till 3 o'clock today. I have been shooting all week and I made a marksman score, 150 points out of 185 points, so I am for France at once. I have to draw my trench boots and equipment tonight, also a last medical examination, so I have not much time so this letter will be short.
In case anything happens to me see that Lily and the children get their pensions alright. She does not understand much about such things. She would be entitled to $32 a month. But I will likely be alright.
You may think I did wrong in joining the service but when I think of the Lusitania and the airship raids in England, I could not stand behind and let Archie go alone. Over here in England every man is a soldier, even men well up in years. Well, I'll have to close now for the present.
Your affectionate son,
Ronald
No. 3 Co.
June 12, 1916
Dear Sister,
I left England a short time ago. I like France better than England as far as soldiering goes.
I am now in the PPCLI, Princess Patricia's Canadian Light Infantry, in France. At present we are in a nice big barn for a rest. It sure is a great place to sleep. Do not worry about me as I am safer here than on Herb's motorcycle.
Your affec. brother
Ronald
June 20, 1916
Dear Father,
I received your letter dated May 23/16. I am not in need of anything as we get issued with everything that we need. The tobacco suits me fine; we often have too much of it.
I have another change in my address: 7th Canadian Trench Mortar Battery, 3rd Can. Div., B.E.F., France.
Who should I meet but Archie last night. He is looking fine and has come through some fighting without a scratch. Lily will soon be up to spend a few weeks in the country. Do not talk about casualties when she is around.
Your affectionate son,
Ronald
* * *
Transcript of telegram
Sincerely regret inform you 157629 Private Ronald MacKinnon infantry officially reported admitted War Hospital Northampton July 1st. Wounded right leg, hand. Will send further particulars when received.
Ward 3, Northamptonshire War Hospital,
Duston, Northampton, Eng.
July 6, 1916
Dear Father,
Just a short note to let you know I am getting along fine and that I got 10 shillings from Aunt Nellie. It sure was a great help to me as I lost everything I had. I will enclose the Trench Mortar badge off my arm. You will get an idea from that what I looked like when I came out of the trenches. I only had a tunic, a pair of boots, socks, and a half a pair of pants. They had to cut my pants away from the wound.
Archie is doing fine. I understand he is soon to be an NCO and that means something in the "front line." His regiment has come through the hardest of fighting, having been cut up several times, but Archie is a war wise soldier and he always comes through. He was in the big scrap on the 2nd June and only got a little scratch on his hand.
I was wounded in Sanctuary Wood the other side of Ypres, the city of the dead.
Tell my little brother Neil how thankful I am for the money he sent. It means tobacco, stamps, and other little things till I get my pay-book again.
Your affectionate son,
Ronald
P.S. Neil, don't come out here. You can fight Germans but not shells! Do not write me here as I will soon be leaving for a convalescent home. R.
July 21, 1916
Dear Jeanie,
Just a few lines to let you know I am getting better and that my wounds are getting healed up. I am not able to walk without a limp yet but will be alright long before you get this.
I do not remember writing to you from France. I am hardly able to settle my mind yet as my head aches pretty bad at times. I am still on light duty and that means nothing in this camp.
I was wounded in Sanctuary Wood in the Ypres Salient on June 27. I got "out of the line" after dark and got to the field dressing station at midnight. From there to the Casualty Clearing Station (CCS). I got to the Red Cross train in the afternoon and then stayed in Boulogne Hospital two days. Then on the Hospital Ship to England and a Red Cross train from Dover to Northampton War Hospital. I had two weeks there and came here on the 14th. I am likely to be here for four or five weeks before I go back to my Reserve Regiment at Shorncliffe. I have to pass a medical board and they say whether I am fit to go "back up the line."
I seen Archie before I left the front. He is doing fine and keeping well but he has seen some hard fighting. He was in the charge that put the Germans out of Sanctuary Wood.
I do not get any mail from Canada at all. I have changed my address so often it is having a hard job to follow me up. It is pretty tough getting no letters from home.
Well, I'll close now hoping this finds you all well at home.
Your affec. brother
#157629 Pte. R. MacKinnon, Canadian Convalescent Hospital,
Woodcote Park Farm Camp, Epsom, England
Epsom, Eng.
Aug. 5, 1916
Dear Father,
This is not much of a camp. It is the worst joint I have been in since I left Canada.
I will have to go before a medical board to see if I am fit for the front again and then I will get my leave. I expect to be put on permanent base duty, either in England or France as I cannot stand the long route marches for some time to come as my leg is a little stiff yet.
I had a letter from Archie this week. He is getting on fine and signs himself L. Cpl. He says things are quiet on the Canadian front just now. I have not had any mail from home since I left France. It seems to have gone astray somewhere. However, don't write to me here as I will soon be leaving.
Your affectionate son,
Ronald
London, August 29,1916
Dear Jeanie,
Just a few lines to let you know I have been up in Scotland all week and had a good time. I met all my uncles and aunties, all but Uncle John. He lived too far away. They are all very good to colonial troops in Scotland.
I will close now as I have to get a train at London Bridge. Where London Bridge is and how to get there I do not know. This is an awful town to get lost in.
Your affectionate brother,
Ronald
Hut 14
Shoreham-by -the-Sea,
Sussex, England
September 10, 1916
Dear Father,
I am enclosing a Postcard taken outside our hut here. The blue band on the left arm means convalescent. The gold stripe means wounded in action. One man in the photo has five stripes up; that means wounded five different times.
Everything seems to be quiet on the Canadian Front these days but I hear the 1st and 2nd Div. have gone down to the Somme but the 3rd and the new 4th are still around Ypres.
This is a rotten camp here for Canadians. Our officers seem to be afraid of the Imperial Authorities and nearly all the towns around here are "out-of-bounds" to us. The Imperial officers are afraid of an outbreak between the Canucks and their men so we have to stay near home where we can be watched. But their fears are all unfounded as we get along well with the other men. The Canadians have such a good name as fighters that we get along well any place.
It is surprising the number of people here who are ignorant of the nature of Canada. When we first came here the natives expected to see a lot of wild men. The first Sunday in church the minister said that he understood Canadians as he had lived in the colonies and that we were a "rough and ready" lot and not used to the big cities of this country (this city has a population of about 1,000). This morning in his sermon he spoke out about our homesteads away out on the "rolling prairie" as if we were all farmers. And we tell the "Tommies" stories about wolves and bears and climbing up trees to sleep at night, so they must think we are a rough lot. Well so long for this time.
Your affectionate son,
Ronald
Shoreham-by-the-Sea,
September 15, 1916
Dear Father,
I am still in the same place but do not expect to be here long. I was up before the doctor and he gave me another week of P.T. (physical training).
It is now getting on to winter again over here. It was pretty cold this morning and last night. I had to get up and walk around a bit. When we get back to Canada we will be easily satisfied as far as comforts go and will appreciate them all the more. In France the boys wash, shave, clean their feet and boots, and wash their mess tins in the one duck pond and have to chase the pigs out first. All the tools a man carries are a spoon and jack knife. If they get meat to eat they take it in their hand. So you see, we can put up with a lot of things.
I did not get any of the parcels you have sent. They would likely go to France and the boys out there will share them out. And they need them. I'll close now hoping this finds you all well.
Your affectionate son,
Ronald
Sept. 23, 1916
Dear Father,
I had a letter from Aunt Nellie today. She tells me that Archie is now a corporal. He'll be a sergeant soon.
I was up to the doctor this morning with a sore mouth. It is what they call trench mouth. It is something like pyorrhoea. The gums come loose around the teeth and are very tender and sore but it is nothing dangerous. It is lucky as it keeps me off physical torture for a few days.
I suppose you have got your crop all in by this time. Well, you want to get the next ship and spend a few months in England and Scotland, though things are pretty dull here on account of the war. You cannot go anywhere without meeting wounded men. Well, that is about all for this time.
Your affectionate son,
Ronald
P.S. I have just received a parcel from you containing socks, dates, cakes, and maple sugar. Give my thanks to Mrs. Macleod for the sugar. All the boys say it is good. The parcel has been to the trenches and back, and was pretty well battered up but everything was OK.
October 6, 1916
Dear Father,
It is a regular hurricane outside tonight. The sea is rolling 15 feet high. I would not like to be crossing to France tonight on an Isle of Man Packet, called by the authorities a troop ship. The last time, there was not room to lie down. I had to sleep on the open deck in the rain sitting up, with about half the troops throwing up their bully beef. But we did not mind it much as we get used to everything over here. It takes a whole night to cross. And when we were going "up the line" in France they piled us into cattle cars, 30 men in a car. The cars are not as big as those at home, not half as big. An old soldier told us to pick a car with round wheels, but we had the picking done by a transport officer. So we got one with square wheels, at least part of the wheel under me had a flat side. So we bump, bump, bump "up the line with best of luck" as the doctors say as they mark you fit for duty. It would make the folks at home smile to see the boys sticking their heads out of a long train of French cattle trucks and making noises like every kind of animal that ever saw a barnyard. Boys that could never sleep outside of a bed will lie down and sleep now just wherever they happen to be. I never imagined how comfortable and soft a nice cattle car is even if you have to sweep it out before you get in.
The food going up the line and at the base is rotten bully beef and hard tack but up the line it is very good — bread, butter, cheese, Mulligan, (bully beef stew) and a preparation called Maconochie. There is everything in "Macs" – onions, spuds, cabbage, carrots, axle grease and a dozen other things, but it goes down good if you are hungry enough. However, nobody kicks on the grub. Archie and me had a good feed one night in a village for a franc apiece – poached eggs on toast, bread, butter, cake, and coffee. A franc is 20 cents. Archie is "jake" (well off or lucky) for some time, six months anyway. His wounds are not serious — only lucky! His leg will keep him in hospital for six weeks anyway. I will likely spend Christmas in England.
Your affectionate son,
Ronald
Shoreham-by-the-Sea
Oct. 1916
Dear Jeanie,
I received your very welcome letter today, dated Sept. 25, 1916. I suppose you will be thinking a machine-gun is a dangerous job seeing that Archie got a Blighty so soon after being on it but it is no worse than any other branch of the infantry. I have had some letters from Archie asking me for cigarettes since he landed in England. I am sending him 100 a week as they are the only comfort a fellow has in hospital, and are great to soothe the nerves. When a shell hits close to a man it shakes his nerves up a little. Archie will have two months in hospital as he has a broken leg but he will be alright. I am going to see him when I get the necessary. It takes quite a little bit to go from here to Cardiff.
This is a nice camp, near to Brighton, but the grub could be a lot better. We had some fish, a very little some, this morning. It must have been caught before the war!
We have lots of rain here. You can hear the lads praying in the morning, "Send it down, Davie, turn the tap!" If it rains, that means no physical torture. It is all physical drill and route marching here.
We have nice comfortable huts here. We have two wee trestles, four inches high, and three bed boards. We put down the trestles, lay the boards on, then we have a bag filled with straw for a mattress. Also a straw-filled pillow and three blankets. They are very comfortable around Reveille! We have to scrub trestles, boards, forms (seats), and the hut every week. Everything is kept spotless and our stove is just shining. You could eat your dinner off the floor. We have no rifles or equipment here as we are convalescents.
I'll close now hoping you are all well at home.
Your affec. brother
Ronald
November 5, 1916
Dear Jeanie,
So Neil has started to learn a trade! Well, it will be a lot better than the farm for him. But for heaven's sake, Neil, stay out of the army! It would kill you over in England in a week. Soldiering in Canada is a picnic but not here. If you want to join, go into the CASC (Canadian Army Safety Corps) or the Medical Corps but stay out of the infantry, Pioneers, or anything that is a fighting unit. Stay out of it altogether. You are too young. I know and so does Archie know what the army is. He'll tell you the same as me. Stay out of it before it is too late as once you sign your name you cannot get out of it, and this war will last a long time yet. When your country needs you she will take you.
Your affec. brother
Ronald
1st Canadian Casualty Training Batt.
November 15, 1916
Dear Father,
I have not had a letter from you for some time but I suppose you have been very busy threshing. Well I am leaving for the line again so the family will soon have a representative at the front. I have been warned for a draft for the last week and we expect to leave at any time but we will have a lot of training to do in France before we go near "the line." I had my equipment on today, the first time I've done heavy marching order since I left France and I am sure tired tonight but will soon get used to it again.
We were having a lecture on bayonet fighting this morning from a man who had never seen France and never heard a shot fired in anger. He asked the class what the bayonet was used for. Here are some of the answers he got: cutting cheese, chopping wood, toasting fork, and killing rats. Imagine a man with a staff job who has never seen any fighting giving a lecture on bayonet fighting to men who have seen all the way from two years to two weeks in the trenches! However, I see Sam Hughes has resigned and it is about time.
I will soon be back in France and in a way I will be glad to get back to it as there is always someone that a fellow knows out there. Well I'll have to close now. Remember me to all at home.
Your affectionate son,
Ronald
France,
November 21, 1916
Dear Jeanie,
Just a line to let you know I am back again. I am feeling perfectly fit and ready for another crack at Fritz. Has Neil got that idea out of his head yet? You'll have to excuse my short letter this time but I'll write again soon.
Your affec. brother
Ronald
France, December 22, 1916
Dear Father,
I am writing this in the trenches so excuse the short letter. I expect to be here Xmas day but will be out for New Year's.
So Archie sends some hot stuff; it is all too true. Neil will be better at the steamfitting as he is not made for a farmer. I hope he has enough sense to stay out of this. He is far too young. The war is still on but there is a big change since I came out first. We have it all over Fritzie now and we soon make him shut up if he tries anything. I don't suppose it will last long now.
Your affectionate son,
Ronald
France,
December 30, 1916
Dearest Sister,
Here we are again as the song says. I had quite a good Xmas considering I was in the front line. Xmas eve was pretty stiff, sentry-go up to the hips in mud of course. I had long rubber boots or waders. We had a truce on Xmas Day and our German friends were quite friendly. They came over to see us and we traded bully beef for cigars. Xmas was "tray bon," which means very good.
How is Neil getting on in the city? Remember me to all my many friends at home.
Your loving brother,
Ronald
France, January 27, 1917
Dear Father,
I received yours today dated December 17 and glad to hear you were all well. I had a letter from Archie. He is now in Convalescent Hospital and is doing well, though still on crutches. I have only got two of your parcels; one with cake, socks and tobacco and another one with a pair of socks from Ladies' Aid.
It is pretty cold here at present with a little snow but we have good clothes — bearskin coats and so on so I don't mind it much. I'll have to close now as I hear the joyful cry "rum up." It is a great thing to drive out the cold.
Your affectionate son,
Ronald
France, February 2, 1917
Dear Father,
I received three letters from you today. You ask in one of them what would be best to put in a parcel? Well anything that is not canned goods as nearly all the grub we get out here is canned. I can even get such things as ham and eggs, sausages, and so on at the canteen. Of course we get our ration of fresh bread every day so for eats we get on very well considering. The only thing is it is always the same. Some handy things to put in a parcel are: bachelor buttons, candles, and good boot laces. At present my boots are tied with a string off a sandbag; they are better than ordinary laces.
I have just got a parcel from Jeanie: a box of cigars. Xmas Day was very wet and cold but we had a truce on our part of the line. I had bacon and tea for breakfast. For dinner, the best Xmas dinner I ever had, at least I enjoyed it more – a dixie [cauldron] full of good fresh meat and Mulligan with lots of beans in it. I would not have traded it for all your turkey and cranberry sauce but my thoughts were home with the turkey just the same. I could picture you all filling up and wondering what I was eating.
I was on sentry-go not a hundred yards from old Fritz about the time Gordon would be looking for a new jack knife. I was a little cold and wet and my rubber boots were heavy with mud but looking out into No Mans Land I could see Gordon looking for that knife in his stocking. Yes, I thought of you all at home that day.
A few people at home are chewing the rag about the brave boys learning to drink "rum" but it is a man like yourself as can understand anything like that. I take my rum and look for more, not because I like it, I don't, but it drives out the wet and cold and keeps a man fit and all a man gets won't drown him. It is measured with a tablespoon. Those people who worry about the soldiers' rum should do a sentry-go in a front line; they would know then. They do more harm than good as they make the boys' mothers worry about their sons and they have enough to worry about as it is and the rum don't hurt the boys.
This is a long letter for me so I'll close now hoping to hear from you soon.
Your affectionate son,
Ronald
France,
February 17, 1917
Dear Jeanie,
I received your parcel of canned goods OK. They were "jake" and those comic papers are still being looked over by some of my pals. I had a letter from Archie. He has had a medical board and is marked for Canada. I'll close hoping to hear from you soon.
Your affec. brother
Ronald
France,
February 22, 1917
Dear Jeanie,
I got a parcel from you containing cake, chocolate, and nuts. You had sent it last November but it was in very good condition.
How is Neil and his girl making out and does he still wear those yellow gloves? I certainly had a good laugh when I read your letter. Tell him to send them out to me if they get dirty. But just the same I wouldn't mind if I was in Toronto even if I was wearing "yalla" gloves. You don't want to pay too much attention to Archie's description of the front as things are a lot different since he was here so don't worry.
Your affec. brother
Ronald
France, February 25, 1917
Dear Father,
I received your parcel with socks and Oxo cubes today OK. It is hardly worthwhile sending me any more socks now as I have six good pairs which I got a madame to wash for me. She has them nice and soft and clean and they will do me nicely till summer. And about underwear: we get a bath every time we come out of the trenches (about every 20 days) and we get a clean change at the bath house: socks, towel, two shirts, and drawers. But they are generally lousy but we get used to that; in fact, we are lonesome without a louse or two and new underwear would get lousy as soon as it landed in France. It is always amusing to a tenderfoot to see the old timers lousing a shirt, that is, pulling your shirt off and making a night attack on the lice.
At present we are on rest and I am billeted with a French family. They are very nice people — three children and their mother. Their father was taken prisoner at Verdun 15 months ago.
You wanted to know about the mud. Well it was bad before Xmas but we have had four weeks of cold weather (the coldest in years). Everything was frozen. Water froze in our bottles, stew was cold by the time it reached us in the front line, but then this is war and I always try to smile when things are a little rough. A fellow may as well be cheery as it cannot be helped. When the mud was at its worst, the trenches were falling in and we had to work nearly all the time keeping them open. We had pumps to get the water out and then we would throw it out at night time. We cannot throw it out in daylight because the Germans would see you throwing mud over the top and would soon send over some "Minenwerfers" and chase us to another part of that trench. A winter campaign in the trenches is no picnic. But I have some good Scotch blood in me so I should be able to stand as much as a Hun and smile along with it.
Trenches are not what a lot of people imagine at home. They are not merely a long drain but are dug crooked and are all built up with wood, wire, and sheet iron with a floor in them. Then there are big dugouts (shellproof), but in some places it is impossible to have deep dugouts on account of water.
I am not on trench mortars. I do not suppose I ever will be on them. I am what is known as a rifle grenadier, that is, an expert in throwing rifle grenades. You often speak of the hard fighting to come. Well, I would not like to be in the German trenches when it does come. I am in the first "wave." I do not like to talk about it to make you worry, but Dad, when I think of some of the things I have seen, I hate Germans like you would a blight on your grain, and I'll go "over the top" with the set purpose of doing my little bit. I often wonder if I'll come through, and I worry about my children, but I can only trust in God to bring me through. If I don't you can rest assured that I've done my duty as a Scotch Canadian.
Archie will soon be back in Canada. He has done all he can do and deserves a good rest. Well, I'll have to close now. Remember me to all at home and trust we will all be home this summer. I think this war will not last any more than three months, if it lasts that long.
Your affectionate son,
Ronald
France,
March 10, 1917
Dear Jeanie,
My usual to let you know I am OK and in the pink. I received a parcel today containing gum, biscuits, etc. It was in very good condition and very acceptable. It just landed at the right time.
The weather is very good here now as winter is practically over but we still have plenty of mud. I am staying at present in a pretty "jake" billet. If I could only "parlez-vous" it would be "jaker" still, however, we manage to get along. Well I must close. Remember me to all at home.
Your affec. brother
Ronald
France,
April 3, 1917
Dear Jeanie,
I received your two letters this week and glad to hear from you. So Neil is sending me some beefsteak and onions. I am anxiously waiting on it coming. He may be sure that I'll make it look small once I get that tin. Thank him very much for me.
Ronald
France, April 6, 1917
Dear Father,
Just a few lines to let you know I am OK. This is Good Friday so I had a good feed of eggs as I will not be in a place where I can get them on Sunday.
We are having very good weather here now, beautiful spring days with lots of sunshine, which is gradually drying up most of the mud. It will never all dry up as there will always be mud in this country.
Well, by the time you get this you will have read all about it and so will know more about it than me as I will only know what goes on in my own little sector. I am a rifle grenadier and am in the "first wave." We have a good bunch of boys to go over with and good artillery support so we are bound to get our objective alright. I understand we are going up against the Prussian Guards: the bigger they are the harder they fall! The Canadians have met them before and they remember it.
Well, I must close but will write again as soon as I get the chance. Remember me to all at home.
Your affectionate son,
Ronald
R.R.#1, Dundalk, Ont.,
April 5, 1917
My Dear Ronnie,
Just a line to say that Archie is home. Just got here. I met him at North Toronto last Sunday morning. He is looking fine and not very lame but he is not the boy he was when he went away. However, he is in good spirits. He is here on pass. He has got to go back to the hospital Monday morning for three months. Then he will pass another medical board. They are going to have a reception for him in Hopeville tomorrow night.
I called and saw Lily and the children. I intended to stay an afternoon with them but it rained nearly all the time I was down there so I splashed my way up there Monday evening. They are all looking fine. Archie is growing a big fine boy and Annie a big girl and a good talker. Archie knew me well but Annie was very shy. They are doing well and that is the main thing.
They met the hospital train with streetcars and a band and took everybody to the hospital. Then after they took the soldiers and their friends home in autos and you may be sure Toronto looked good to Archie. I would have liked to have seen you with him but we are looking forward to meeting you some day. So God be with you, my boy, till we do meet.
Your Father,
A.McK.
This letter was returned unopened in June. On the envelope was inscribed: "Deceased. Killed in Action 9-4-17." Ronald MacKinnon was buried in the Bois Carré British Military Cemetery, Thélus, on the lower slope of Vimy Ridge.
Ronald's widow married Arthur Donnan after the war. Daughter Annie died in the influenza epidemic in 1919. Son Archie was raised by his mother and stepfather and served in the Canadian Argyll and Sutherland Highlanders in World War II. Archie was wounded in France and died in Toronto in 1971.
Private Ronald MacKinnon's widow, Lily, and children Archie (in child's version of a PPCLI uniform) and Annie.
Bibliography
Berton, Pierre. Vimy. Toronto: McClelland & Stewart, 1986.
Christie, Norm. For King & Empire: Volume III, The Canadians at Vimy April 1917. Winnipeg: Bunker to Bunker Books, 1996.
Dancocks, Daniel G. Gallant Canadians: The Story of the 10th Canadian Infantry Battalion, 1914–1918. Calgary: The Calgary Highlanders Regimental Funds Foundation (Distributed by Penguin Books Canada Ltd.), 1990.
Rawling, Bill. Surviving Trench Warfare: Technology and The Canadian Corps, 1914–1918. Toronto: University of Toronto Press, 1992.
Worthington, Larry. The Story of the Canadian Corps 1914–1918. Toronto: McClelland & Stewart, 1965.
Answers.Com: www.answers.com
Canadian Encyclopedia: www.thecanadianencyclopedia.com
Canoe: www.canoe.ca
CBC Archives: <http://archives.cbc.ca>
Collections Canada: www.collectionscanada.ca
First World War Site: www.firstworldwar.com
History of Canada Online: <http://canadachannel.ca>
Royal Canadian Legion: www.legionmagazine.com
Talking Proud: www.talkingproud.us
Veterans Affairs Canada: www.vac-acc.gc.ca
War Amps: www.waramps.ca
Wikipedia: <http://en.wikipedia.org>
Acknowledgements
The author first visited Vimy Ridge in 1962 while stationed at the Royal Canadian Air Force Base in Metz, France, as part of the Department of National Defence Schools Overseas program. Gordon MacKinnon, a history teacher and World War I buff, acted as guide and was a wealth of information during the tour of the memorial. Gordon is also the source for a series of letters written by his uncle, who served in the Canadian Corps and was killed during the Battle of Vimy Ridge. The letters in this book's Appendix are from this series.
Another Vimy visit occurred when the author accompanied the Honourable Bennett Campbell, Minister of Veterans Affairs, to France in 1983 as his communications assistant. One of the 19 WWI veterans on that pilgrimage was the late Fred Holm, who related a fascinating story about one of life's "small-world" coincidences, retold within these pages.
The Veterans Affairs Canada website, as usual, was a valuable resource in researching material for this book. In addition, a number of VAC personnel were generous with their time and expertise. Among those are Hélène Robichaud, director of the Canadian Battlefield Memorials Restoration Project, Peter Craven, senior technical adviser on the project, Danielle Gauthier, project coordinator, Julie Daoust, and Lise Poirier.
The author found a great deal of worthwhile information in two books in particular, Pierre Berton's Vimy and Volume III of Norm Christie's For King & Empire.
Thanks go to Liliane Opsomer of the Belgian Tourist Office in New York City for helping arrange air travel and on-the-ground personnel during the overseas research. Air Transat was kind enough to upgrade economy seats to club class — a far more comfortable workstation. Rail Europe was extremely generous in supplying rail passes to allow research not only in the Vimy area, but also in Ypres, Belgium, and at several other World War I sites. René and Jeanine Dubos provided wonderful meals and accommodation in Vimy, and the rest of the Dubos family — Serge, Martine, Melody, and Marjolaine — made sure the Vimy visit went smoothly.
Special thanks are due Reverend Phil Miller of St. Andrew's United Church in Sault Ste. Marie, Ontario, friend and military expert, who bird-dogged answers to some of the more difficult pieces of the Vimy puzzle.
And a salute to old friend Jack Nadeau, whose father, like mine, experienced the horrors of war. Jack's dad, Robert Samuel Nadeau, served in both the British merchant navy and the Canadian army during World War I. He survived his ship's sinking, a bayonet wound, and a German gas attack. May they and their generation rest in peace.
About the Author
Tom Douglas is the author of three other best-selling books of military history in the Amazing Stories series — Canadian Spies, D-Day, and Great Canadian War Heroes. Tom's father, the late H.M. (Mel) Douglas, was a veteran who served with the 19th Field Regiment and was part of the D-Day invasion.
As an elementary school teacher, Tom served with DND Schools Overseas in Metz, France. During that posting, he visited numerous Canadian battle sites and cemeteries in Europe. He left teaching to become a reporter with his hometown newspaper, the Sault Daily Star, and has worked in communications ever since. As a reporter, he was selected by DND to tour Canadian bases in Europe on two occasions.
After working with The Canadian Press, and serving as publisher/owner of a weekly newspaper in Queensland, Australia, Tom was hired as communications consultant to the Honourable Bennett Campbell, Minister of Veterans Affairs, in Ottawa.
He travelled with the minister and groups of World War I, World War II, and Korean War veterans on several pilgrimages involving such battle sites as Vimy Ridge, Normandy, Brittany, Rome, and Seoul. He was also part of a Canadian delegation that was invited by French President Francois Mitterrand to attend Remembrance Day services at the Arc de Triomphe.
Tom accompanied Veterans Affairs Minister Campbell to Europe for the 40th anniversary of D-Day in 1984. They took two side trips — one to unveil a plaque in Rome in honour of the Canada/US Special Services force that liberated the Eternal City, the other to Brittany to honour French-Canadian soldiers who had parachuted into occupied France to set up an escape network for Allied personnel. This latter experience was the inspiration for the Canadian Spies book.
Tom and his wife, Gail, also an author with the Amazing Stories series, organized a special 50th anniversary D-Day voyage to Normandy on the Queen Elizabeth 2 in 1994. The ship was entirely taken over by D-Day veterans, their relatives, and military historians, who were entertained on board by comedian Bob Hope, World War II songstress Vera Lynn, television anchorman Walter Cronkite, and the Glenn Miller Orchestra. The following year, Tom led a group of Canadian veterans and their relatives on a five-country bus tour of Europe as part of the commemoration of the 50th anniversary of the Allied victory in Europe (VE Day).
In June 2004, Tom attended the 60th anniversary of the D-Day invasion in Normandy as an accredited journalist. He was one of six Canadian journalists selected by Veterans Affairs Canada in May 2005 to be part of the official pilgrimage to Holland for the 60th anniversary of the liberation of the Netherlands. He returned to Holland in December of that year with his son James, a videographer, to film a tribute to the close ties between Canada and Holland stemming from Canada's assistance in the Dutch war effort in World War II. Tom and James produced a DVD about their trip entitled The Tulip and the Maple Leaf.
On Labour Day 2004, Tom was invited to sign copies of his first two books during the closing ceremonies for the old Canadian War Museum on Sussex Drive in Ottawa.
As part of his research for this current book, Tom visited the Vimy Memorial site in October 2006 and interviewed the man largely responsible for its restoration, Peter Craven.
Tom has written speeches for senior officials of DND and Veterans Affairs Canada. He has also interviewed veterans as the basis for numerous news articles that have been distributed on the Veterans Affairs' website and/or to newspapers across Canada. He currently serves as copy editor of the Canadian Military Journal and has provided that official publication of the Department of National Defence with written articles and book reviews.
Tom is a member of Branch 114 (Oakville), Royal Canadian Legion, and the Dominion Institute's Memory Project.
Photo Credits
Cover: The Canadian War Museum: 'The Taking of Vimy Ridge, Easter Monday, 1917. Painted in 1919 by Richard Jack (1866–1952); Department of National Defence, National Archives of Canada: page 61 (PA-001017); Authors private collection: page 105; Altimage for the Government of Canada/P. Frutier: page 109; Courtesy of Gordon MacKinnon: page 135.
Copyright 2011, 2007 © by Tom Douglas
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, or by any information storage or retrieval system, without permission in writing from the publisher.
James Lorimer & Company Ltd., Publishers acknowledge the support of the Ontario Arts Council. We acknowledge the financial support of the Government of Canada through the Canada Book Fund for our publishing activities. We acknowledge the support of the Canada Council for the Arts for our publishing program. We acknowledge the support of the Government of Ontario through the Ontario Media Development Corporation's Ontario Book Initiative.
Cataloguing in Publication data is available from Library and Archives Canada
This digital edition first published in 2011 as 978-1-55277-805-0
Originally published in 2007 as 978-1-55439-241-4
James Lorimer & Company Ltd., Publishers
317 Adelaide Street West Suite 1002
Toronto, Ontario
M5V 1P9
www.lorimer.ca
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# Copyright
The characters and events in this book are fictitious. Any similarity to real persons, living or dead, is coincidental and not intended by the author.
Copyright © 2019 by Chigozie Obioma
Cover design © Gray318
Author photograph by Jason Keith
Cover copyright © 2019 by Hachette Book Group, Inc.
Hachette Book Group supports the right to free expression and the value of copyright. The purpose of copyright is to encourage writers and artists to produce the creative works that enrich our culture.
The scanning, uploading, and distribution of this book without permission is a theft of the author's intellectual property. If you would like permission to use material from the book (other than for review purposes), please contact permissions@hbgusa.com. Thank you for your support of the author's rights.
Little, Brown and Company
Hachette Book Group
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First ebook edition: January 2019
Little, Brown and Company is a division of Hachette Book Group, Inc. The Little, Brown name and logo are trademarks of Hachette Book Group, Inc.
The publisher is not responsible for websites (or their content) that are not owned by the publisher.
The Hachette Speakers Bureau provides a wide range of authors for speaking events. To find out more, go to hachettespeakersbureau.com or call (866) 376-6591.
Excerpts from this book were published in _The Guardian_ in 2016 under the title "The Ghosts of My Student Years in Northern Cyprus."
ISBN 978-0-316-41241-4
E3-20181020-JV-PC
# Contents
1. Cover
2. Title Page
3. Copyright
4. Dedication
5. Epigraph
6. Chart of Igbo Cosmology
7. Composition of Man in Igbo Cosmology
8. ONE
9. First Incantation
10. 1 The Woman on the Bridge
11. 2 Desolation
12. 3 Awakening
13. 4 The Gosling
14. 5 An Orchestra of Minorities
15. 6 "August Visitor"
16. 7 The Disgraced
17. 8 The Helper
18. 9 Crossing the Threshold
19. TWO
20. Second Incantation
21. 10 The Plucked Bird
22. 11 The Wayfarer in a Foreign Land
23. 12 Conflicting Shadows
24. 13 Metamorphosis
25. 14 The Empty Shell
26. 15 All the Trees in the Land Have Been Removed
27. 16 Visions of White Birds
28. 17 Alandiichie
29. THREE
30. Third Incantation
31. 18 The Return
32. 19 Seedlings
33. 20 Reckoning
34. 21 Man of God
35. 22 Oblivion
36. 23 The Ancient Tale
37. 24 Castaway
38. 25 The Subaltern God
39. 26 Spiders in the House of Men
40. Author's Note
41. Acknowledgments
42. Discover More Chigozie Obioma
43. About the Author
44. Also by Chigozie Obioma
_To J.K._
_We've not forgotten_
If the prey do not produce their version of the tale, the predators will always be the heroes in the stories of the hunt.
—Igbo proverb
In a general way, we may visualize a person's chi as his other identity in spiritland—his spirit being complementing his terrestrial human being; for nothing can stand alone, there must always be another thing standing beside it.
—Chinua Achebe, "Chi in Igbo Cosmology"
_Uwa mu asaa, uwa mu asato!_ This is the primal factor in determining the state of a newborn's true identity. Even though humans exist on the earth in material form, they harbor a chi and an onyeuwa because of the universal law which demands that where one thing stands, another must stand beside it, and thus compels the duality of all things. It is also the basic principle on which the Igbo concept of reincarnation stands. Do you ever wonder why a newborn child sees a particular individual for the first time and from that moment develops hatred for that person without cause?... It is often because the child may have identified that individual as an enemy in some past existence, and it might be that the child has returned to the world in their sixth, seventh, or even eighth cycle of reincarnation to settle an ancient score! Sometimes, too, a thing or an event can reincarnate during a lifetime. This is why you find a man who once owned something but loses it may find himself in possession of something similiar years later.
—Dibia Njokwuji of Nkpa, voice recording
# ONE
# First Incantation
OBASIDINELU—
I stand before you here in the magnificent court of Bechukwu, in Eluigwe, the land of eternal, luminous light, where the perpetual song of the flute serenades the air—
Like other guardian spirits, I have gone to uwa in many cycles of reincarnations, inhabiting a freshly created body each time—
I have come in haste, soaring untrammeled like a spear through the immense tracts of the universe because my message is urgent, a matter of life and death—
I stand knowing that a chi is supposed to testify before you if his host is dead and his host's soul has ascended into Benmuo, that liminal space crowded with spirits and discarnate beings of every hue and scale. It is only then that you request that guardian spirits come to your dwelling place, this grand celestial court, and ask you to grant the souls of our hosts safe passage into Alandiichie, the habitation of the ancestors—
We make this intercession because we know that a man's soul can return to the world in the form of an onyeuwa, to be reborn, only if that soul has been received in the domain of the ancestors—
Chukwu, creator of all, I concede that I have done something out of the ordinary by coming here now to testify while my host is still alive—
But I am here because the old fathers say that we bring only the blade sharp enough to cut the firewood to the forest. If a situation deserves exigent measures, then one must give it that—
They say that dust lies on the ground and stars lie in the sky. They do not mix—
They say that a shadow may be fashioned from a man, but a man does not die because a shadow has sprung from him—
I come to intercede on behalf of my host because the kind of thing he has done is that for which Ala, the custodian of the earth, must seek retribution—
For Ala forbids that a person should harm a pregnant woman, whether man or beast—
For the earth belongs to her, the great mother of mankind, the greatest among all creatures, second only to you, whose gender or kind no man or spirit knows—
I have come because I fear that she will raise her hand against my host, who is known in this cycle of life as Chinonso Solomon Olisa—
This is why I have hastened here to testify of all I have witnessed and to persuade you and the great goddess that if what I fear has happened is true, to let it be understood that he has committed this great crime in error, unknowingly—
Although I will relate most things in my own words, they will be true because he and I are _one_. His voice is my voice. To speak of his words as if he were distinct from me is to render my own words as if they were spoken by another—
You are the creator of the universe, patron of the four days—Eke, Orie, Afor, and Nkwo—that make up the Igbo week—
To you the old fathers ascribed names and honorifics too numerous to count: Chukwu, Egbunu, Oseburuwa, Ezeuwa, Ebubedike, Gaganaogwu, Agujiegbe, Obasidinelu, Agbatta-Alumalu, Ijango-ijango, Okaaome, Akwaakwuru, and many more—
I stand here before you, as bold as a king's tongue, to plead my host's cause, knowing that you will hear my voice—
#
## The Woman on the Bridge
CHUKWU, if one is a guardian spirit sent for the first time to inhabit a host who will come into the world in Umuahia, a town in the land of the great fathers, the first thing that strikes the spirit would be the immensity of the land. As the guardian spirit descends with the reincarnating body of the new host towards the land, what reveals itself to the eye astonishes. Suddenly, as if some primordial curtain has been peeled off, one is exposed to an interminable stretch of leaf-green vegetation. As one draws closer to Umuahia, one is enticed by the elements around the land of the fathers: the hills, the thick, great forest of Ogbuti-ukwu, a forest as old as the first man who hunted in it. The early fathers had been told that signs of the cosmic explosion that birthed the world could be seen here and that from the beginning, when the world was partitioned into sky, water, forest, and land, the Ogbuti forest had become a country, a country more expansive than any poem about it. The leaves of the trees bear in them a provincial history of the universe. But beyond the exaltation of the great forest, one becomes even more fascinated with the many water bodies, the biggest of which is the Imo River and its numerous tributaries.
That river weaves itself around the forest in a complex circuit comparable only to that of human veins. One finds it in one part of the city spouting like a deep gash. One travels on the same road for a short distance and it appears—as if out of nowhere—behind a hill or an enormous gorge. Then there, between the thighs of the valleys, it is flowing again. Even if we miss it at first, one only needs to tread past Bende towards Umuahia, through the Ngwa villages, before a small, silent tributary reveals its seductive face. The river has a distinct place in the mythologies of the people because in their universe, water is supreme. They know that all rivers are maternal and therefore are capable of birthing things. This river birthed the city of Imo. Through its neighboring city runs the Niger, a greater river which was itself the stuff of legend. Long ago, the Niger overran its boundaries in its relentless journey and met another, the Benue, in an encounter that forever changed the history of the people and the civilizations around both rivers.
Egbunu, the testimony for which I have come to your luminous court this night began at the Imo River nearly seven years ago. My host had traveled to Enugu that morning to replenish his stock, as he often did. It had rained in Enugu the previous night, and water was everywhere—trickling down from the roofs of buildings, in potholes on the roads, on the leaves of trees, dripping from orbs of spiderwebs—and a slight drizzle was on the faces and clothes of people. He went about the market in high spirit, his trousers rolled up over his ankles so as not to stain the hems with dirty water as he walked from shed to shed, store to store. The market seethed with people, as it always was even in the time of the great fathers when the market was the center of everything. It was here that goods were exchanged, festivals were held, and negotiations between villages were conducted. Throughout the land of the fathers, the shrine of Ala, the great mother, was often located close to the market. In the imagination of the fathers, the market was also the one human gathering that attracted the most vagrant spirits—akaliogolis, amosu, tricksters, and various vagabond discarnate beings. For in the earth, a spirit without a host is nothing. One must inhabit a physical body to have any effect on the things of the world. And so these spirits are in constant search for vessels to occupy, and insatiable in their pursuit of corporeality. They must be avoided at all costs. I once saw such a being inhabit the body of a dead dog in desperation. And it managed, by some alchemical means, to stir this carrion to life and make it amble a few steps before leaving the dog to lie dead again in the grass. It was a fearful sight. This is why it is considered ill advised for a chi to leave the body of its host in such a place or to step far away from a host who is asleep or in an unconscious state. Some of these discarnate beings, especially the evil spirits, even sometimes try to overpower a present chi, or ones who have gone out on a consultation on behalf of their hosts. This is why you, Chukwu, warn us against such journeys, especially at night! For when a foreign spirit embodies a person, it is difficult to get it out! This is why we have the mentally ill, the epileptic, men with abominable passions, murderers of their own parents and others! Many of them have become possessed by strange spirits and their chi are rendered homeless and reduced to following the host about, pleading or trying to negotiate—often fruitlessly—with the intruder. I have seen it many times.
When my host returned to his van, he recorded in his big foolscap notebook that he'd bought eight adult fowls—two roosters and six hens—a bag of millet, a half bag of broiler feed, and a nylon full of fried termites. He'd paid twice the usual price of chickens for one, a wool-white rooster with a long tapering comb and plush of feathers. When the seller handed him the fowl, tears clouded his eyes. For a moment, the seller and even the bird in his hands appeared as a shimmering illusion. The seller watched him in what seemed to be astonishment, perhaps wondering why my host had been so moved by the sight of the chicken. The man did not know that my host was a man of instinct and passion. And that he had bought this one bird for the price of two because the bird bore an uncanny resemblance to the gosling he had owned as a child, which he'd loved many years ago, a bird that changed his life.
Ebubedike, after he bought the prized white rooster, he embarked on the journey back to Umuahia with delight. Even when it struck him that he'd spent a longer time in Enugu than he'd intended and had not fed the rest of his flock for much of that day, it did not dampen his spirit. Not even the thought of them engaging in a mutiny of angry cackles and crows, as they often did when hungry, the kind of noise that even distant neighbors complained about, troubled him. On this day, in contrast to most other days, anytime he encountered a police checkpoint, he paid the officers handily. He did not argue that he had no money, as he often did. Instead, before he came to their stations, where they had laid down logs studded with protruding nails to force the traffic to stop, he stretched his hand through the window clutching a wad of notes.
GAGANAOGWU, for a long time my host raced through rural roads that tracked through villages, between tumuli and mounds of the ancient fathers, through roads flanked by rich farmlands and deep bushes as the sky slowly darkened. Insects dashed against the windshield and burst like miniature fruits until the glass was covered with small mucks of liquefied insects. Twice he had to stop and wipe the mess off with a rag. But soon after he began again, the insects would rage against the pane with renewed force. By the time he arrived at the boundary of Umuahia the day had aged, and the lettering on the rusting pole with the WELCOME TO ABIA, GOD'S OWN STATE sign was barely visible. His stomach had become taut from having gone a whole day without eating. He stopped a short distance from the bridge that ran over the Amatu River—a branch of the great Imo River—and pulled up behind a semi whose back was covered with a tarp.
Once he stopped the engines, he heard a clatter of feet in the van bed. He climbed down and stepped over the drainage ditch that encircled the city. He walked over to the clearing where streetside sellers sat on stools under small fabric awnings on the other side of the drainage, their tables lit with lanterns and candles.
The eastern darkness had fallen, and the road ahead and behind was blanketed in a quilt of gloom, when he returned to the van with a bunch of bananas, a pawpaw, and a polythene bag full of tangerines. He put on his headlights and drove back onto the highway, his new flock squawking in the back of the van. He was eating the bananas when he arrived at the bridge over the Amatu River. He'd heard only the previous week that—in this most fecund of rainy seasons—the river had overflowed and drowned a woman and her child. He didn't usually put stock in the rumors of mishaps that passed around the city like a weighted coin, but this story had stayed in his mind for some reason which even I, his chi, could not understand. He was barely at the middle of the bridge thinking of this mother and child when he saw a car parked by the railings, one of its doors flung wide open. At first all he saw was the car, its dark interior and a speck of light reflected on the window of the driver's side. But as he shifted his gaze, he caught the terrifying vision of a woman attempting to jump over the bridge.
Agujiegbe, how uncanny that my host had been thinking for days about a woman who'd drowned, and suddenly he found himself before another who had climbed one ledge up the rails, her body bent over as she attempted to throw herself into the river. And once he saw her, he was stirred within. He pulled the van to a halt, jumped out, and ran forward into the darkness, shouting, "No, no, don't. Please, don't! Don't do that. _Biko, eme na!_ "
It seemed at once that this unexpected intervention startled the woman. She turned in swift steps, her body swaying lightly as she fell backwards to the ground in obvious terror. He rushed forward to help her up. "No, Mommy, no, please!" he said as he bent over.
"Leave me!" the woman cried at his approach. "Leave me. Go away."
Egbunu, rejected, he drew back in frantic steps, his hands raised in the strange way the children of the old fathers use to signify surrender or defeat, and said, "I stop, I stop." He turned his back to her, but he could not bring himself to leave. He feared what she would do if he left, for he—himself a man of much sorrow—knew that despair was the disease of the soul, able to destroy an already battered life. So he faced her again, his hands lower, stretched before him like staffs. "Don't, Mommy. Nothing is enough for somebody to die like that. Nothing, Mommy."
The woman struggled up to her feet slowly, first kneeling, then raising her upper body, all the while with her eyes fixed on him and saying, "Leave me. Leave me."
He glimpsed her face now in the pupillary light of his van. It was full of fear. Her eyes seemed somewhat swollen from what must have been long hours of crying. He knew at once that this was a deeply wounded woman. For every man who has himself suffered hardship or witnessed it in others can recognize its marks on the face of another from a distance. As the woman stood trembling in the light, he wondered whom she may have lost. Perhaps one of her parents? Her husband? Her child?
"I will leave you alone now," he said, lifting his hands up again. "I go leave you alone. I swear to God who made me."
He turned towards his van, but because of the gravity of the sorrow he'd seen in her, even the momentary shuffling of his feet away from her seemed like a grievous act of unkindness. He stopped, conscious of the rushed sinking in the pit of his stomach and the audible anxiety of his heart. He faced her again.
"But Mommy," he said. "Don't jump it, you hear?"
In haste, he unlocked the back of the van and then unlatched one of the cages, and with his eyes looking through the window, whispering to himself that she should not go, he took two chickens by their wings, one in each hand, and hurried down.
He found the woman standing where he'd left her, looking in the direction of his vehicle, seemingly transfixed. Although a guardian spirit cannot see the future and thus cannot fully know what its hosts will do—Chukwu, you alone and the great deities possess the spirit of foresight and may bequeath certain dibias this gift—I could sense it. But because you caution us, guardian spirits, not to interfere in every affair of our hosts, to allow man to execute his will and be man, I sought not to stop him. Instead, I simply put the thought in his mind that he was a lover of birds, one whose life has been transformed by his relationship with winged things. I flashed a stirring image of the gosling he once owned into his mind that instant. But it was of little effect, for in moments like this, when a man becomes overcome by emotion, he becomes Egbenchi, the stubborn kite which does not listen or even understand whatever is spoken to it. It moves on to wherever it wishes and does whatever it desires.
"Nothing, nothing should make someone fall inside the river and die. Nothing." He raised the chickens above his head. "This is what will happen if somebody fall inside there. The person will die, and no one can see them again."
He lunged towards the rails, his hands heavy with the birds, which cackled in high-pitched tones and stirred with agitation in his grip. "Even these fowls," he said again, and flung them over the bridge into the gloom.
For a moment, he watched the birds struggle against the thermal, whipping their wings violently against the wind as they battled desperately for their lives but failed. A feather landed on the skin of his hand, but he beat it off with such haste and violence that he felt a quick pain. Then he heard the sucking sound of the chickens' contact with the waters, followed by vain plonks and splashes of sound. It seemed the woman listened, too, and in listening, he felt an indescribable bond—as if they had both become lone witnesses to some inestimable secret crime. He stood there until he heard the woman's gasps. He looked up at her, then back at the waters hidden from his sight by the darkness, and back at her again.
"You see," he said, pointing at the river as the wind groaned on like a cough caught in the dry throat of the night. "That is what will happen if somebody fall inside there."
The first car to approach the bridge since his own arrived with cautious speed. It stopped a few paces from them and honked, then the driver said something he could not hear but which had been spoken in the White Man's language and which I, his chi, had heard: "I hope you are not hoodlums oh!" Then the car drove away, gathering speed.
"You see," he repeated.
Once the words had left his mouth, he resolved into a calm, as it often happens at such times when a man, having done something out of the ordinary, retreats into himself. All he could think of was to leave the place, and this thought came upon him with an overwhelming passion. And I, his chi, flashed the thought in his mind that he'd done enough, and that it was best he left. So he rushed back to his van and started it amidst the mutiny of voices from the back. In the side mirror, the vision of the woman on the bridge flashed like an invoked spirit into the field of light, but he did not stop, and he did not look back.
#
## Desolation
AGUJIEGBE, the great fathers say that to get to the top of a hill, one must begin from its foot. I have come to understand that the life of a man is a race from one end to the other. That which came before is a corollary to that which follows it. This is the reason people ask the question "Why?" when something that confounds them happens. Most of the time, even the deepest secrets and motives of the hearts of men can be uncovered if one probes deeper. Thus, Chukwu, to intercede on behalf of my host, I must suggest that we trace the beginning of everything to the harsh years preceding that night on the bridge.
His father had died only nine months earlier, leaving him with a pain that was exquisite beyond anything he'd ever felt. It may have been a little different had he been with others, as he was when he lost his mother and when he lost his gosling and when his sister left home. But upon his father's passing, there was no one. His sister, Nkiru, having eloped with an older man and feeling her conscience seared by their father's death, distanced herself even more. Perhaps she'd done this for fear my host might blame her for their father's death. The days that followed the demise were of utmost darkness. The agwu of pain afflicted him night and day and made of him an empty house in which traumatic memories of his family lurked like rodents. In the mornings on most days, he'd wake up smelling his mother's cooking. And sometimes during the day, his sister would reveal herself in vivid pictures, as if she'd been merely hidden all along by a drawn curtain. At night, he'd feel the presence of his father so intensely he'd sometimes become convinced that his father was there. "Papa! Papa!" he'd call into the darkness, turning about in frantic steps. But all he'd get back would be silence, a silence so strong it would often restore his confidence in reality.
He walked through the world vertiginously, as if on a tightrope. His vision became one from which he could see nothing. Nothing gave him comfort, not even the music of Oliver De Coque, which he'd play on his big cassette player most evenings or while working at the yard. Even his fowls were not spared his grief. He tended to them with less care, mostly feeding them once a day and sometimes forgetting to give them food altogether. Their riotous squawking in protest was what often stirred him in those times, forcing him to feed them. His watch over his flock was distracted, and many times hawks and kites preyed on them.
How did he eat in those days? He simply fed off the small farm, a plot of land that stretched from the front of the house to the place where the motor road began, harvesting tomatoes, okro, and peppers. The corn his father had planted he let wilt and die, and he allowed a collection of insects to foment the resultant decay as long as they did not also trample on the other crops. When what was left of the farm could not meet his needs, he shopped at the market near the big roundabout, using as few words as necessary. And in time he became a man of silence who went days without speaking—not even to his flock, whom he often addressed as comrades. He bought onions and milk from the provisions shed nearby and sometimes ate at the canteen across the street, Madam Comfort's restaurant. He hardly spoke there, either, but merely observed the people around him with a strained mercurial awe, as if in their seeming peacefulness they were all renegade spirits come into his world through a back door.
Soon, Oseburuwa, as is often the case, he became one with sorrow so much that he resisted all help. Not even Elochukwu, the only friend he kept after he left school, could comfort him. He stayed away from Elochukwu, and once Elochukwu rode his motorcycle up to the front of the compound, knocked on the door, and shouted my host's name to see if he was in. But he pretended he was not in the house. Elochukwu, perhaps suspicious that his friend was in, rang my host's phone. My host let it ring on until Elochukwu, maybe concluding that he was indeed away, left. He refused all pleas from his uncle, his father's only surviving sibling, to come and stay in Aba. And when the older man persisted, he turned off his phone and did not turn it on for two months, until he woke up one day to the sound of his uncle driving onto his compound.
His uncle had come angry, but when he found his nephew so broken, so lean, so emasculated, he was moved. The old man wept in the presence of my host. The sight of this man whom he had not seen in years weeping for him changed something in my host that day. He discovered that a hole had been bored into his life. And that evening, as his uncle snored, stretched out on one of the sofas in the sitting room, it struck him that the hole became evident after his mother died. It was true, Gaganaogwu. I, his chi, was there when he saw his mother being taken out of the hospital, dead shortly after delivering his sister. This was twenty-two years ago, in the year the White Man refers to as 1991. He was only nine at the time, too young to accept what the universe had given him. The world he'd known up till that night suddenly became reticulated and could not be straightened again. His father's devotion, trips to Lagos, excursions to the zoo in Ibadan and the amusement parks in Port Harcourt, even playing with the video-game consoles—none worked. Nothing his father did repaired the chink in his soul.
Towards the end of that year, around when the cosmic spider of Eluigwe spins its lush web over the moon the thirteenth time, increasingly desperate to restore his son's well-being, his father took him to his village. He'd remembered that my host had been enticed by stories of how he'd hunted wild geese in the Ogbuti forest as a little boy during the war. So he took my host to hunt geese in the forest, an account of which I will give you in due course, Chukwu. It was here that he caught the gosling, the bird that would change his life.
His uncle, seeing the state my host was in, stayed with him for four days instead of one, as he'd planned. The older man cleaned the house, tended the poultry, and drove him to Enugu to buy feed and supplies. During those days, Uncle Bonny, despite stammering, filled my host's mind with words. Most of what he said pivoted around the perils of loneliness and the need for a woman. And his words were true, for I had lived among mankind long enough to know that loneliness is the violent dog that barks interminably through the long night of grief. I have seen it many times.
"Nonso, ih if y ou don't ge get y-y your self a-a wife s-su su soon," Uncle Bonny said the morning he would leave, "your aunt a-ah-ah me wi-will h-ave to get y-y-ou one our ourself." His uncle shook his head. "Be-be-be-because because you can't live like this."
So strong were his uncle's words that, after he left, my host began to think of new things. As if the eggs of his healing had hatched in secret places, he found himself craving something he had not had in a long time: the warmth of a woman. This desire drew his attention away from thoughts of his loss. He began to go out more, to lurk around near the Federal Government Girls College. At first, he watched the girls from the roadside canteens with fitful curiosity. He paid attention to their plaited hair, their breasts, and their outward features. As he developed interest, he reached out to one, but she rebuffed him. My host, who'd been molded by circumstances into a man of little confidence, decided he would not try a second time. I flashed in his thought that it was hardly possible to get a woman at the first try. But he paid no heed to my voice. A few days after he was turned down, he inquired at a brothel.
Chukwu, the woman into whose bed he was admitted was twice his age. She wore loose hair, the kind of which was known among the great mothers. Her face was painted with a powdery substance that gave it delicacy which a man might find inviting. She looked by the shape of her face like Uloma Nezeanya, a woman who, two hundred forty-six years ago, was betrothed to an old host (Arinze Iheme) but disappeared before the wine-carrying ceremony, taken away by Aro slave raiders.
Before his eyes, the woman stripped and bared a body that was buxom and attractive. But when she asked him to climb her, he could not. It was, Egbunu, an extraordinary experience, the like of which I had never seen before. For suddenly, the great erection he'd sustained for days was gone the very moment it could be satiated. He was seized by a sudden acute self-awareness of himself as a novice, unskilled in the art of sex. With this came a flurry of images—of his mother in the hospital bed, of the gosling perched precariously on a fence, and of his father in the hard grip of rigor mortis. He trembled, pulled himself slowly from the bed, and begged to leave.
"What? You wan just waste your money like that?" the woman said.
He said yes. He stood up and reached for his clothes.
"I no understand, look how your prick still dey stand."
_"Biko, ka'm laa,"_ he said.
"You no sabi speak English? Speak pidgin, I no be Ibo," the woman said.
"Okay, I say I wan go."
"Eh, na wa oh. Me I neva see this kain thing before, oh. But I no want your money make e waste."
The woman climbed off the bed and switched on the lightbulb. He stepped back at the full glare of her female immensity. "No fear, no fear, just relax, eh?"
He stood still. His hands yielded like one afraid as the woman took his clothes and put them back on the chair. She knelt on the floor and held his penis with one hand and clutched his buttocks with the other. He squirmed and trembled from the sensation. The woman laughed.
"Wetin be your age?"
"Thirty, ah-gh thirty."
"Abeg talk true, wetin be your age?" She squeezed the tip of his penis. He gasped as he made to speak, but she clamped her mouth over it and swallowed it halfway. My host mumbled the word _twenty-four_ in feverish haste. He tried to get out, but the woman curved her other arm around his waist and held him still. She sucked with plopping sounds, forcefully, while he screamed, gnashed his teeth, and uttered meaningless words. He saw iridescent light tempered with darkness and felt a coldness within. The complex equation continued to erupt in his body until he let out a shout: "I dey release, I dey release!" The woman turned away, and the semen barely escaped her face. He fell back into the chair, fearing that he might pass out. He would leave that brothel, shocked and exhausted, bearing the weight of the experience with him like a sack of corn. It was four days later that he encountered the woman on the bridge.
EZEUWA, he left the bridge that night, uncertain about what he had done, only knowing that it was something out of the ordinary. He drove home with a sense of fulfillment, the kind he had not experienced in a long time. In peace, he unloaded the new chickens, ten instead of twelve, and took the cages into the yard using the torchlight at the head of his phone. He unpacked the silo bag of millet and other things he'd bought in Enugu. Once he set everything down, he was hit with a sudden realization. "Chukwu!" he said, and rushed into the sitting room. He lifted his rechargeable lamp, pressed up the switch by its side, and a weak white light glowed from the three fluorescent bulbs. He turned the switch up even more, but the lighting did not improve. He moved forward and gazed down at it to see that one of the bulbs had died, its top end coated with a blob of soot. He ran to the yard with the lamp nonetheless, and once the half-light illuminated the cage, he screamed again, "Chukwu, oh! Chukwu!" For he'd found that one of the chickens he'd thrown over the bridge was the wool-white rooster.
Akataka, it is a common phenomenon among mankind to attempt to flip precedence: to try to bring that which has gone forward back. But it always, always fails. I have seen it many times. Like others of his kind, my host ran out of his house back to his van, on which a black cat had climbed and sat gazing about like a watchman. He shooed the cat away. It gave a loud feline whine and dashed into the adjoining bush. He entered the van and drove back out into the night. The traffic was light, and only once did a big semi block the way while it was trying to pull into a filling station. When he got to the bridge, the woman he'd seen only a while before was gone—her car, too. He reckoned that she had not fallen into the river, for if she had, then her car would still be there. But the woman was not, at this point, what he cared about. He rushed down the shore, the nocturnal noise filling his ears, his torchlight swallowing the darkness like a boa. He felt the sensation of insects resolve into a concentric fold in the air and net his face as he approached the shore. He waved frantically to swat them away. The torchlight followed the movement of his hand and wavered upon the waters in a straight rod a few times, and then flashed across the riverbank for meters on end. His gaze traced the path of the light, but all he saw was empty banks and rags and dirt strewn about. He walked directly under the bridge, turning when he heard a sound, his heart palpitating. As he came near, the light revealed a basket. The main raffia plaiting had loosened into long, twisted fibers. He rushed towards it, expectation bearing down on him.
When he found nothing in the basket, he cast the light on the water under the bridge, down the distant reaches of the river as far as his torchlight could illuminate, but there was no trace of either chicken. He recalled the moment he threw them, how they'd fluttered their wings, how'd they tried in agonizing desperation to cling to the bars of the bridge, and how they must have been unable to do it. He'd learned early on when he first began keeping poultry that domestic fowls were the weakest animals among all creatures. They had little ability to defend or save themselves from dangers large or small. And it was this weakness that further endeared them to him. At first he'd loved all birds because of the gosling, but he began to love only the weak domestic fowls after he witnessed the violence of a hawk attack on a hen.
After he had combed through the thick hide of night, as one would search for lice on the skin of a densely furred animal, he returned home in anguish. His action seemed to him all the more like something his hand had done out of concert with his mind. It was this, above all, that caused him pain. Sudden darkness often descends upon the heart of a person who discovers that he has unknowingly committed harm. Upon the discovery of the harm it has done, the man's soul kneels in complete defeat, submits to the alusi of remorse and shame, and in its submission wounds itself. Once wounded, a man seeks healing through acts of restitution. If he has soiled another's cloth, he may go to that person with a new cloth and say, there, my brother, take this new cloth in exchange for the one I ruined. If he has broken something, he may seek to mend or replace it. But if he has done that which cannot be undone, or broken that which cannot be mended, then there is nothing he can do but submit to the tranquilizing spell of remorse. This is a mystifying thing!
Ezeuwa, when my host sought an answer to something beyond his understanding, I often ventured to supply it. So before he slept that night, I impressed on his mind that he should return to the river in the morning; perhaps he would still be able to find the fowls. But he paid no heed to my counsel. He thought it an idea that originated from within his own mind, for man has no way to distinguish between what has been put into his thoughts by a spirit—even if it is his own chi—from what has been suggested to him by the voice of his head.
I continued to flash the thought in his mind many times that day, but the voice of his head would counter each time and tell him that it was too late, that the chickens must have drowned. To which I responded that he could not know this. Then the voice of his head said, _It is gone; there is nothing I can do_. So when it was evening, and I could see that he would not go, I did that which you, Oseburuwa, caution guardian spirits to avoid doing except in extraordinary situations. I stepped out of the body of my host while he was conscious. I did this because I knew that my place as his guardian spirit was not only as a guide but also as a helper and witness to the things which may be beyond his reach. This is because I see myself as his representative in the realm of the spirits. I stand within my host and gaze at every movement of his hands, every step of his feet, every motion of his body. To me, the body of my host is a screen on which the entirety of his life is displayed. For while in a host I'm nothing but an empty vessel filled by the life of a man, rendered concrete by that life. It is thus from the place of a witness that I observe him live, and his life becomes my testimony. Yet a chi is constrained while in the body of its host. While there, it becomes nearly impossible to see or hear what is present or spoken in the supernatural realm. But when one exits one's host, one becomes privy to things beyond the realm of man.
Once out of my host, I was hit by the great clamor of the spirit world, a deafening symphony of sounds that would have frightened even the bravest of men. It was a multitude of voices—cries, howls, shouts, noises, sounds of every kind. It is uncanny that even though the separation between the world of mankind and the spirits is only leaf-thin, one does not hear even a faint whisper of this sound until one leaves the body of one's host. A freshly created chi on earth for the first time is immediately overwhelmed by this din and may become so frightened that it might run back into the fortress of silence that is its host. This happened to me during my first sojourn on earth as well as to many guardian spirits I have met at the resting caves of Ogbunike, Ngodo, Ezi-ofi, and even the pyramidal mounds of Abaja. It is especially worse at nighttime, the time of the spirits.
Whenever I leave my host while he is in a state of consciousness, I make my visits rapid and brief, so that nothing will happen to him in my absence and he will not do anything I would not be able to account for. But because the road to anywhere in a disembodied form isn't the same as when one is borne by a human vessel, I had to slowly make my way through the crowded concourse of Benmuo, in which spirits of all kinds writhed like a can of invisible worms. My haste yielded fruit, and I got to the river within a period of seven battings of the eyelids, but I saw nothing there. I returned the following day, and by the third visit, I saw the brown rooster he had thrown over the bridge. It had bloated and now lay on the surface of the river with its legs facing up, taut and dead. The water had added a shade of imperceptible gray to the rooster's barring, and its belly was naked of all plumage, as if something in the water had eaten it. Its neck seemed to have elongated, and its wrinkling was deeper and its body was swollen. A vulture sat on one of the wings of the chicken, which had flattened out over the surface of the water, peering down and about at the bird. I saw no sign of the wool-white fowl.
Ebubedike, in my many cycles of existence, I have come to understand that the things that happen to a man have already occurred long before in some subterranean realm, and that nothing in the universe is without precedent. The world spins on the noiseless wheel of an ancient patience by which all things wait and are made alive by this waiting. The ill luck that has befallen a man has long been waiting for him—in the middle of some road, on a highway, or on some field of battle, biding its time. It is the individual who reaches this point and is struck down who may be fooled into sullen bewilderment, along with all who may sympathize with him, even his chi. But in truth the man had died long ago, the reality of his death merely concealed by a silken veil of time, which would eventually be parted to reveal it. I have seen it many times.
While he slept that night, I stepped out of him, as I often did, so I could watch over him, because the inhabitants of Benmuo often become more active in the earth at night, while mankind sleeps. And from this position, I flashed the image of the chicken and the vulture into his subconscious mind, for the easiest way to communicate such a mysterious event to one's host is through the dream sphere—a fragile realm a chi must always enter with caution and great care because it is an open theater accessible to any spirit. A chi must first eject itself from the host before stepping into its host's dream world. This also prevents the chi from being identified by the foreign spirits as a chi hovering in untenanted space.
Once I'd flashed the images before him, he twitched in his sleep, lifted one hand, and curved it into a weak fist. I sighed, relieved, knowing that he now knew what happened to his white rooster.
GAGANAOGWU, his sadness for drowning the fowls had suppressed every thought of the woman at the bridge. But slowly, as his sadness abated, thoughts of her began to line the boundaries of his mind and then gradually crowd in. He started to dwell on thoughts of her, what he had seen of her. All he'd been able to gather from the night vision was that she was mid-sized, not as fleshy as Miss J, the prostitute. She had worn a light blouse and skirt. And he remembered that her car was a blue Toyota Camry, similar to his uncle's. Then often, like a grasshopper, his thoughts would leap from her appearance to his curiosity about what she did after he left the bridge. He would blame himself for having left the bridge in haste.
In the days following, he tended to his poultry and the garden with light hands, consumed by thoughts of her. And when he drove about the city, he searched for the blue car. As weeks passed, he began to yearn for the prostitute again. Desire swelled like a storm and washed over the parched landscape of his soul. It drove him to the brothel one evening, but Miss J was busy. The other ladies mobbed him, and one of them dragged him into a room. This woman had a lean waist and a scar on her belly. With her he felt himself certain and sure, as if at the place of his last encounter his apprehensions and naïveté had been clobbered to a bloody death. He yielded to her without scruples, and even though I often avoid witnessing my hosts having sex because of its fearful imitation of death, I stayed put because it was to be his first. When he was done, she slapped him on the back, saying how good he was.
Yet, despite this experience, he was still drawn to Miss J, to her body, to the familiar sound of her sigh. It surprised him that even though he had done something more profound with the other woman, he'd found greater pleasure in the hands of Miss J. He returned to the brothel three days later and avoided the other woman, who ran heartily to him. Miss J, this time, was free. She regarded him only with faint recognition and set about undressing him in silence. Before they could begin, she answered her phone and told the caller to come in two hours, and when it seemed the male voice refused the bargain, she settled for an hour and a half.
They had begun when she spoke about the last time and laughed. "You don open your eye now after I suck you that time, ba?"
He made love to her with an exuberance that fevered his soul and poured himself into the act. But once he slumped beside her, she pushed away his arm and rose.
"Miss J," he called, almost in tears.
"Yes, na wetin?" the woman said. She started to strap her brassiere over her breasts.
"I love you."
Egbunu, the woman stopped, clapped her hands, and laughed. She turned on the light and crept back into the bed. She scooped his face in her hand, mimicked the calculated somberness with which he'd uttered the words, and laughed even harder.
"Oh, boy, you no sabi wetin you dey talk." She clapped her hands again. "Look at this one, him say him love me. Nothing wey person eye no go see these days oh. Im see nyash wey tripam—na im be say im love me. Say you love your mama."
She snapped her fingers as she burst again into renewed mirth. And for days, her laughter echoed through his being in many hollow places, as if it were the world itself that had laughed at him, a small, lonely man whose only sin had been that he was hungry for companionship. It was here that he first felt that befuddling emotion of romantic love, a kind of crossroad that was distinct from what he felt for his birds and for his family. It was a painful feeling, for jealousy is the spirit that stands at the threshold of love and madness. He wanted her to belong to him and begrudged all the other men who would have her after him. But he did not know that nothing truly belongs to anyone. Naked he was born, naked he will return. A man may own something for as long as it remains with him. Once he leaves it, he may lose it. He did not know at the time that a man may give up all he has for the sake of the woman he loves, and when he returns, she may no longer desire him. I had seen it many times.
So, broken by the things he did not yet know, he left the place and resolved never to return.
#
## Awakening
IJANGO-IJANGO, over many sojourns in the human world, I have heard the venerable fathers, in their kaleidoscopic profundity, say that no matter the weight of grief, nothing can compel the eyes to shed tears of blood. No matter how long a person weeps, only tears continue to fall. A man may remain in the state of grief for a long time, but he will eventually grow out of it. In time, a man's mind will acquire strong limbs, strike down the wall, and be redeemed. For no matter how dark the night, it soon passes, and Kamanu, the sun god, erects his grandiose emblem the following day. I have seen it many times.
By the fourth month after the encounter with the woman on the bridge, my host almost ceased grieving. It was not that he was happy now, for even the hems of the garments of his brightest days were fringed with threads of sorrowful darkness. It was that he was alive again, open to the possibilities of happiness. He turned to his friend Elochukwu, who began visiting regularly and persuaded him to join MASSOB, the group that was sweeping young Igbo men with an old broom into a pile of dust. Elochukwu, who had been his friend and confidant in secondary school and who was always slender, had become brawny with biceps he displayed at every turn by wearing armless shirts or singlets. "Nigeria has failed," he would tell my host in the White Man's language, and then trail into the language of the fathers with which he mostly conversed with my host. _"Ihe eme bi go. Anyi choro nzoputa!"_ At Elochukwu's insistence, my host joined him. In the evenings, at the big shop of a car dealer, they gathered wearing black berets and red shirts, surrounded by flags of a half-drawn sun, maps, and images of soldiers who had fought for Biafra. My host would amble about with this group, shouting slogans at the top of his lungs. He would yell "Biafra must rise again" with them, stamp his feet on the unfinished floor, and chant "MASSOB! MASSOB!" He'd sit among these men and listen to the dealer and the head of the movement, Ralph Uwazuruike. Here my host spoke, he made merry again, and many noted his broad smile and his quickness to laugh. These men, without knowing where he had been or where he was coming from, glimpsed the first marks of his healing.
Chukwu, because I had been present in a host during the Biafran War, I feared his dalliance with this group would lead him to harm. I put thoughts in his head that there may be violence in these engagements. But the voice of his head replied with certainty that he was not afraid. And indeed, for a long time he went with this group, moved only by an anger he could not define. For he had not himself experienced the grievances the men articulated. He did not know anyone who had been killed by people from northern Nigeria. Although many of the dark sayings of this group felt true to him—he could see, for instance, that indeed no Igbo had been president of Nigeria and perhaps none would ever be—none of it affected him personally. He did not know anything about the war except that his father had fought and had told him many stories about it. And while these men spoke, the vivid accounts of the war his father had given him would thrash about in the mud of his remembrance like wounded insects.
But he attended the meetings mostly because Elochukwu was the only friend he had. A neighbor's hand in the death of his gosling had shut his heart to friendships. After the incident, he had hovered over the gray field of humanity and determined that the world of man was too violent for his liking. He found solace instead among feathery creatures. He also went because it gave him something to do besides tending to the poultry and the small farm and because he'd hoped that while going from one point to another in the city, advocating for the actualization of the sovereign state of Biafra, he might come across the woman he'd met at the bridge. Akataka, it was this last reason that was principal in his mind, the main reason why, even as the marches became increasingly dangerous, he persisted. But after a month of protests, clashes with police, riots, and violence, and my intense persuasion through endless flashes of thoughts in his mind that he desist, he broke off from the group like a wheel unhooked from a fast-moving car and rolled into the void.
He returned to his normal life, rising at daybreak to the beautiful but mystifying music of the poultry—a symphony of crows, cackles, and tweets that often melded into what his father had once described as a coordinated song. He harvested eggs, recorded the birth of new chicks in his foolscap record book, fed the flock, watched them graze about in the yard with his catapult at the ready to protect them, and tended to the ill and feeble ones. One day in that month, one of the days he worked the most without distractions, he planted tomatoes on the shorn part of the land. He had not tended to this part of the land in a long time, and the change he saw on it shocked him. While weeding the area, he had found that red ants had not just encroached but also completely infested the land. They lay deep in the nerve of the soil, nestled in every clump. It seemed they'd fed on an old dead cassava head which, perhaps owing to their attack, had been unable to grow. He boiled hot water in a kettle and poured it on the loam, killing all the ants. Then he swept the mass of dead ants away and planted the seeds.
He returned to the yard afterwards and washed the tomato seeds that clung to his fingernails and blackened his thumbs. He then scooped bowlfuls of millet from a silo stacked in an unused room in the house and spread the grains on a mat. He unlatched the two large coops in which a dozen chickens grazed about, and they flocked out towards the mats of feed. Within the coops were two cages each of hens with their chicks and one of three large broilers surrounded by their eggs. He felt each one of the birds to try to see if they were all in sound health. There were about forty of the brown ones and about a dozen of the white ones. After he'd fed them, he stood in the yard watching to see which of them had shat so he could poke their excreta with sticks in search of worms. He was searching a gray glob of feces dropped by the well by one of the broilers when he heard the voice of a woman hawking groundnuts.
Egbunu, I must say that it wasn't that he responded this way to every woman's voice, but her voice sounded strangely familiar to him. Although he did not know it, I knew that it reminded him of his mother. At once he saw a plump, swarthy woman who looked his age. She was sweating in the hot sun, and the sweat shimmered along her legs. She carried a tray filled with groundnuts on her head. She was one of the poor—the class of people who had been created by the new civilization. In the time of the old fathers only the lazy, indolent, infirm, or accursed lacked, but now most people did. Go into the streets, into the heart of any market in Alaigbo, and you'll find toiling men, men whose hands are as hard as stones and whose clothes are drenched in sweat, living in abject poverty. When the White Man came, he brought good things. When they saw the car, the children of the fathers cried out in amusement. The bridges? "Oh, how wonderful!" they said. "Isn't this one of the wonders of the world?" they said of the radio. Instead of simply neglecting the civilization of their blessed fathers, they destroyed it. They rushed to the cities—Lagos, Port Harcourt, Enugu, Kano—only to find that the good things were in short supply. "Where are the cars for us?" they asked at the gates of these cities. "Only a few have them!" "What about the good jobs, the ones whose workers sit under air conditioners and wear long ties?" "Ah, they are only for those who have studied for years in a university, and even then, you'd still have to compete with the multitude of others with the same qualifications." So, dejected, the children of the fathers turned back and returned. But to where? To the ruins of the structure they had destroyed. So they live on the bare minimum, and this is why you see people like this woman who walk the length and breadth of the city hawking groundnuts.
He shouted for her to come up.
The woman turned in his direction and lifted a hand to hold the tray on her head in place. She pointed to herself and said something he could not hear.
"I want to buy groundnut," he called to her.
The woman began walking down the curved dirt path, marked in many places by the tires of his van and, recently, the four wheels of his uncle's car. The previous day's rain had molded the red earth into small mud balls that clung to the tires. And now, in the clearer day, the reddish earth still gave off an ancient smell and worms were strewn all over it, burrowing and leaving trails on the path. As a child he'd taken pleasure in crushing worms under his feet after bouts of rainfall, and sometimes he and his friends, especially the gosling-stealing Ejike, would store the worms in transparent polythene bags and watch them writhe in the airless, enclosed space.
She came wearing open-toed slippers, whose plastic straps as well as her feet were caked with dust. A small purse dangled over her bosom, held around her neck by a fabric lace. As she walked up, her feet stamping the dirt, he wiped his hand on the wall by his door. He stepped back into the house and looked around in haste. He noticed for the first time the big yarn of cobwebs that stretched across the ceiling of the sitting room, reminding him that so much time had passed since his father, who had maintained a high level of cleanliness, died.
"Good afternoon, sir," the woman said, genuflecting slightly.
"Good afternoon, my sister."
The woman set down the groundnut tray, reached for a side pocket on her skirt, and brought out a handkerchief that was soaked through and spotted with shades of brown dirt. With it she wiped her forehead.
"How much, how much is—"
"Groundnut?"
My host thought he caught a slight tremble in the woman's voice—the way people influenced by the bias of their own minds misjudge the actions of others. I listened as he did, but I did not hear any tremor in her voice. She sounded absolutely composed to me.
"Yes, groundnut," he said, nodding. Fluid rushed up his throat, leaving a peppery taste in his mouth. His discomposure came from the strange familiarity of her voice, which, although he could not ascertain the source of the familiarity, drew him to her.
The woman pointed to a small canned-tomato tin in which groundnuts were stacked and said, "Five naira for one small cup. The big one is ten naira."
"The ten-naira one."
The woman shook her head. "So, Oga, you bring me here so you fit buy only ordinary ten-naira groundnut? Ah, abeg add some more na." Then she laughed.
He felt the sensation in his throat yet again. He first felt it during the period of his mourning. He did not know that it was a kind of sickness related to indigestion which flares in the pit of the stomach of a person who is bereaved or in a state of extreme anxiety. I had seen it many times, most recently in the body of my former host, Ejinkeonye Isigadi, while he was fighting in the Biafran War nearly forty years before.
"Okay, give me two of the big ones," he said.
"Er-he, thank you, Oga."
The woman bent to scoop groundnuts into the larger tin, and then emptied it into a small colorless polythene bag. She was pouring the second scoop into the same bag when he said, "I no want only groundnut."
"Er?" She dropped her head.
She did not immediately look at him, but he fixed his eyes on her. He let his eyes linger over her face, which was rough and covered with signs of privation. Some encrusted layers of dirt covered her face like patches of extra flesh, somewhat redefining it. Yet beneath these layers, he could see that she possessed striking good looks. When she laughed, her dimples deepened and her mouth formed into a pout. There was a mole above her mouth, but he did not look much at it or at her cracked lips, which she continuously licked to give them a glossy texture. Down on her chest, though, was where his eyes rested: on the ponderous breasts that appeared separated by ample space. They were round and full and pushed against her clothes, even though he could see the signs of restraint—her brassiere straps—sticking out on both sides of her shoulders.
_"Ina anu kwa Igbo?"_ he said, and when she nodded, he turned fully to the language of the eloquent fathers. "I want you to come stay with me a little. I am feeling lonely."
"So you don't want groundnut?"
He shook his head. "No, not only groundnut. I want to talk to you, too."
He helped her straighten up, and as she rose, he locked his mouth with hers. Agbatta-Alumalu, although he feared that she would resist him, his urge had been so strong that it had overcome his inner voice of reason. He drew back and saw her stunned and unresisting. He even saw a glint of joy in her eyes, and he pressed harder. He drew closer to her and said, "I want you to come inside with me."
_"Isi gi ni?"_ she said, laughing even more. "You are a strange man."
She'd used a word for "strange" that was not commonly used in the language of the old fathers as spoken in Umuahia but which he often heard used in the big market in Enugu.
"Are you from Enugu?"
"Yes! How did you know?"
"Where in Enugu?"
"Obollo-Afor."
He shook his head.
She turned from him briskly, and folded her hands together. "You really are strange," she said. "Do you even know if I have a boyfriend?"
But he did not speak. He put her tray on the dining table, on the edge of which was dried chicken shit. As he put his hands around her and gently pulled her close to him, she whispered, "So this is what you really want?" When he said it was, she struck his hand lightly and laughed as he undid her blouse.
Chukwu, I had by this time known my host for many years. But I could not recognize him that day. He acted like one possessed, unrecognizable even to himself. Where had he, a hermit who yielded little to the world outside his own, found the courage to ask a woman to lie with him? Where did he—who until his uncle suggested he get a woman had thought little of women—find the courage to undress a woman he just met? I did not know. What I knew was that with this uncharacteristic bravado, he stripped the woman's gown off her.
She held his hand with a hard grip for a long time and covered her mouth with her other hand, silently laughing to herself. They came into his room, and as he closed the door behind them, his heart pounding more quickly, she said, "Look, I am dirty." But he barely acknowledged those words. He focused on his own slightly quivering hands as they pulled down her underpants. When he was done, he said, "It doesn't matter, Mommy." Then he pulled her into the bed in which his father had died, consumed by a kind of passion that bore close affinity with rage. That passion etched itself in the curious changes that appeared in the woman's facial expressions: one time of delight; one time of anguish, in which she gnashed her teeth; one time of exhilaration that culminated in a small laugh; one time in a shock that held her mouth in a perplexing O shape; one time of a restless peace in which the eyes are closed as if in a pleasant, exhausted sleep. These passed across her visage in succession until the very last moment, when he began to suddenly wither. He barely heard her saying, "Pull out, abeg," before falling beside her, his expiration complete.
The act itself is hard to describe. They spoke no words but mourned, gasped, sighed, gnashed their teeth. The things in the room spoke in their stead: the bed uttered a mournful cry, and the sheets seemed to engage in a slow, considered speech like a child singing a rhyme. It all happened with the grace of a festival—so quickly, so suddenly, so vigorously, yet so tenderly. And in the end, of all the expressions that had passed over her face, only joy remained. He lay beside her, touched her lips, and rubbed her head until she laughed. The terrors that had lurked in his heart were gone in this moment. He sat up, a drop of sweat falling slowly down his back, unable to grasp the full expression of what he was feeling. He could see in her a certain kind of gratitude, for now she took his hand and held it firmly, so hard that he squirmed silently. Then she began to speak. She spoke about him with an unusual depth of mind, as if she'd known him for a long time. She said that although he acted strange, something in her spirit assured her that he was a "good" man. A good man, she emphasized again and again. "There aren't many like that in this world anymore," she said, and even though he was now drained and exhausted and half asleep, he could feel the resignation in her voice. Then it seemed that she raised her head and looked down at his penis and saw that long after it had emptied itself on the bedsheet it was still hard. She gasped. "You are still erect? _Anwuo nu mu o!_ "
He tried to speak, but he only mustered a babble.
"Ehen, I see you are falling asleep so quickly," she said.
He nodded, embarrassed by his sudden, unexpected exhaustion.
"I will go so you can sleep." She picked up her brassiere and started to put it over her breasts, something the venerable mothers would not have used, for they either covered their breasts with clothes knotted at the back or left them bare, or, sometimes, merely covered them with _uli._
"Okay, but please come tomorrow," he said.
She turned to him. "Why? You don't even know, or ask, if I have a boyfriend."
His mind awakened to the thought, but his eyes remained heavy. He mumbled incoherently words she could not hear but which I heard to be the baffling statement: "If you come, so do come again."
"You see, you can't even talk anymore. I will go. But what is your name, at least?"
"Chinonso," he said.
"Chi-non-so. Good name. I am Motu, you hear?" She clapped her hands. "I am your new girlfriend. I will return tomorrow, around this time. Good night."
He heard, in his slouched awakening, the sound of the door closing as she left the house. Then she was gone, carrying with her her distinct smell, a fragment of which had stuck on his hands and in his head.
AGBATTA-ALUMALU, the fathers of old say that without light, a person cannot sprout shadows. This woman came as a strange, sudden light that caused shadows to spring from everything else. He fell in love with her. In time, it seemed that with one slingshot, she had silenced his grief—that violent dog that had barked relentlessly in this early night of his life. So strong was their bond that he was mended. Even my relationship with him improved, because a man is truly able to commune with his chi when he is at peace. When I spoke, he heard my voice, and in his will, shadows of my will began to lurk. If he had lived at the time of the old fathers, they would have said of his state that he had affirmed something, and I, his chi, had affirmed, too, as it is wholly true that _onye kwe, chi ya e kwe_.
No human who experiences such moments would ever want them to end. But sadly, in uwa, things do not always happen in accordance with the expectation of man. I have seen it many times. It was thus no surprise to me that he woke up on the day it all ended as he'd been waking for many mornings, filled with the thoughts of this woman with whom he'd enjoyed four market weeks of bliss (three weeks in the calendar of the White Man). Things had appeared usual for him that morning, as they had been for those twenty-one days, because man is without the powers of prospicience. This, I have come to believe, is mankind's greatest weakness. If only he could see the faraway as clearly as he can see what is in front of him, or if only he could see the hidden as he can see what is revealed, if only he could hear that which is not spoken as well as what is spoken, he would be saved from a great many calamities. In fact, what is it that would be able to destroy him?
My host spent that Saturday waiting for his lover to come. He did not know that nothing would walk over that path between two rows of farmland, which ran for nearly two kilometers to the main road, that day. He'd sat on the front porch and fixed his eyes on it from early in the morning, but as the day began to wane, things he had never considered rose from some abyss and held court with him. He had not thought to get Motu's address. He did not know where she lived. When he had asked once, begging to drive her home, she had said that her auntie would punish her severely if she ever found out Motu was keeping a boyfriend. And this had been the extent of his knowledge—that she was a maid from a village in Obollo-Afor, serving her "auntie"—an acquaintance not related to her by blood—in the city. She did not have a phone. He knew nothing else.
That day also passed and another came galloping in like a great baffling carriage with a loud toot and a majestic stride. He rushed to welcome it, almost trembling from the weight of expectation. But when he unlatched the door, the porch was empty. Nothing except the rust of an old carriage and a mocking sound of dry metal. The following day came garmented in the colors of a familiar sky, one that reminded him of the time Motu and he made love in the kitchen and he'd heard for the first time the sound of air emit from a vagina. It was also the first day she took a bath in his house and put on the dress he had bought her: a gown made of sparkling blue ankara material, which she then washed in a bucket in his bathroom and hung on the laundry rope in the yard that was fastened between the guava tree and a stick half buried in the fence. Then they'd had sex, and she had asked him things about poultry. He had found himself telling her so much about his life that he became aware, as if by epiphany, how heavy his history had become. By sundown, he knew she would not come. He lay all day, empty, alone, and stunned, listening to the raindrops fall into the bucket and hit the ground like drumbeats.
Oseburuwa, I myself became worried. It is hard for a chi to watch its host find happiness and lose it again. I listened keenly for this woman, and sometimes, while he worked at the farm or poultry, I'd leave his body and stand on the porch to see if I might see her passing the compound so I could flash it as a thought in his head. But I, too, saw no trace of her. Vain spirits mocked him with dreams of her that night, and he woke the following morning disturbed. They were somewhere, in a temple or an old church, looking at the murals and paintings of saints. He gazed at one image, of a man on a tree, closely, and when he turned, he did not see her. In her place was a falcon. It stared at him with its yellow eyes, its beak half open, its great talons firm on the edge of one of the seats. He did not speak at first, for he knew that it was she. Egbunu, you know that in the dream world, knowledge is not searched after—things are simply known. Thus, he saw that she whom he'd been waiting for had become a bird. As he made to take it, he woke up.
By the end of the second week, and with ideas falling into his mind as if an ancient mouth was constantly spitting into his head, he realized that something had happened and that it was possible he would never see Motu again. It was, Gaganaogwu, an awakening: that a man can find a woman who accepts and loves him, and that one day, she could vanish without cause. The weight of this realization would have brought him down had the universe not lent him a hand that day. For one of the ways a man may be relieved of suffering is by doing something out of the ordinary, something he will always remember. That memorable action forces a stanch on the bleeding wound and helps the sufferer recover.
On that day, he was sitting on the floor in his kitchen, watching the brown chickens, all roosters, grazing alongside the brown hens and chicks and walking about the yard, feeding from mounds of marsh and corn which he'd spread on sacks. From the window, he caught sight of a hawk hovering over the birds, biding its time. He quickly unhooked his catapult from the nail on which it hung on the wall and picked out stones from the small raffia basket by the window. He shook off and blew small red ants off the stones. Then, closing one of his eyes and standing a short distance behind the door to conceal himself, he placed a stone in the rubber pocket of the catapult and held still, his eyes on the hawk. The bird had stopped at some point in midair, then raised itself farther above so the chickens did not see it. Its wingspan then broadened, and in a moment, it plunged towards the compound with stupefying speed. He followed it, and as it attempted to grab a cockerel feeding near the fence, he released the stone.
True, he was adept in this art of stone archery, and he'd been slinging stones since he was a child, but it is hard to understand how he got the bird on its forehead. There was something instinctive about it, something divine in origin. It felt, Chukwu, as if this act itself had been rehearsed many years ago, before he was born, before you assigned me to be his guardian spirit. It was this act that began his fresh healing. For it seemed as if he'd carried out revenge against that primal force with whom he must reckon, that unseen hand that takes away whatsoever he possesses. That voice that seems to say, "Look, he has been happy for so long, it is now time to send him back to that dark place where he belongs." And from the end of that second week, he began to live again.
Rain poured in the days following with a relentlessness that reminded my host of one year in his childhood, while his mother was still alive, when rain destroyed a neighbor's house and the family came to take shelter with my host's family. During these wet days, it was hard for his poultry to come out of the coops into the yard. He, like them, kept away from most things and recoiled into the lone world he'd become accustomed to. Chukwu, he would live like this for the next three months after Motu's disappearance, avoiding even Elochukwu as much as he could.
IJANGO-IJANGO, the great fathers often say that a child does not die because his mother's breast is empty of milk. This became true of my host. He soon became used to the loss of Motu and began going out again to execute his daily tasks. It was thus without any expectations that he went out that day at the end of those three months to fuel his van at the filling station near his house, expecting to return home afterwards the way he had left. He stayed on the long line at the petrol station and had finally made it to the pump, stepped out to open his fuel tank for the attendant when he saw a hand waving at him from the line of cars behind. At first, he did not see who it was, for he had to tell the attendant, who had put the nozzle in his tank, that he wanted to buy six hundred naira worth of petrol.
"That is, eight liters. No change. Seventy-five, seventy-five naira."
"Okay, madam."
As the woman tapped something on the pump station and the numbers began to roll out, he turned back and saw that it was the woman on the bridge. How, Chukwu, could he have thought that on such an inauspicious and unremarkable day that which he had been looking for for so long would appear again, suddenly, and reveal itself to him of its own accord? Although he kept a close eye on the pump, fearing he could be cheated, because he had heard about how people manipulated them, the shock of this encounter fastened itself to a branch of his mind like a viper. In a mix of haste and anxiety, he pulled over to the side of the station, near a culvert that descended to the street below. No matter which system of time he employed—the system of the fathers, in which four days make up a week, twenty-eight days a month, and thirteen months a year, or the system of the White Man, now commonly used by the children of the great fathers—nine months had passed since that night when he sacrificed two fowls to scare her back to life. As he waited for her, he searched back to all that had happened to him since that encounter. When she parked her car behind his vehicle and stepped out to meet him, he felt the craving that seemed to have disappeared long ago emerge as if it had been merely hidden all this while in the back pocket of his heart like an old coin.
#
## The Gosling
ANUNGHARINGAOBIALILI, when a man encounters something that reminds him of an unpleasant event in his past, he pauses at the door of the new experience, carefully considering whether or not to go in. If he has already stepped in, he may retrace his steps and reconsider whether to reenter. Like my host, every man is inextricably chained to his past and may always fear that the past might repeat itself. So, with Motu still fresh in his mind, my host attended his desire for this woman with caution. He observed that she'd changed so much—as if she were not the same woman who had been mired in grief the night he first met her, at the bridge. She was taller than he remembered from that brief encounter. Her eyes were framed by a delicate arc, and her forehead shone from the backwards pull of her permed hair.
She reappeared more beautiful than the image he'd stored in his mind for so long. She came up to him after fueling her car and shook his hand and introduced herself as Ndali Obialor in the language of the White Man, as she'd done on the bridge. He told her his name, too. He found her intimidating, not only her presence but also her facility with this language, which he rarely used. He was curious how she had been able to recognize him.
"Your vehicle, the sign on it, OLISA AGRICULTURE," she said, laughing. "I remember it. I saw you a month or so ago, near Obi Junction. But you were driving very fast. I just believed I would see you again." A car honked for her to step out of the way, and when it had passed, she said, "I've been looking for you. To thank you for that night. Thank you, really."
"Thank you, me also," he said.
She'd closed her eyes while speaking, and opened them now. "I am going to school now. Can you come to Mr. Biggs?"
She pointed at the eatery on the other side of the road. "Can you come there by six o'clock today?"
He nodded.
"Okay, Chinonso. Bye-bye. Good to see you again."
He watched her retrace her steps back to her car, wondering if all this while he had been looking for her he'd seen her without knowing it.
He'd seen in the woman's eyes something—something he could not himself define. There are times when a man cannot fully understand his feelings, and neither can his chi. His chi is often at a loss at those times. Hence this mystery hung like a small cloud over him as he went home to prepare for the meeting with her later that day. What was clear to me and to him was that she was not like anyone else he'd met. The accent she affected was of a person who had lived in the foreign countries of white people. And there was a lushness to her demeanor and appearance that was nothing like the ramshackle outlook of Motu or the strange mixture of poise and feistiness of Miss J. And, Egbunu, when men meet those whom they esteem highly above themselves, they become measured in their actions, they check themselves, they try to present themselves before these people in a way that will earn them dignity. I have seen it many times.
So when he got home, he spread two sacks on the ground and poured millet and maize on them, then he unbarred the coops of the adult fowls. They rushed out and covered the sacks. In haste, he filled the water troughs and put them back in each of the cages. He brought out one of the suits he'd inherited from his father. With a sponge he'd cut a few days before from a grain sack, he brushed off a stain on the suit. Then he hung it to dry on a branch of the tree in the yard. He washed himself and was about to bring in the suit when it struck him that his hair was bushy. He had not cut it in nearly three months, since that day when Motu, insisting she could do it, cut it with scissors and he swept the yard in a frenzy afterwards, fearing one of his flocks would eat a strand of the hair. In haste, he drove to the salon on Niger Road, where he'd gone since he was a child. The man, Mr. Ikonne, his barber, had suffered a stroke, and the man's eldest son, Sunday, now did it. When my host's turn came and Sunday began cutting his hair, the clipper went quiet suddenly. Seeing there had been a power outage, Sunday rushed to the back of the shop to put on the generator, but it would not start. My host glanced at himself in the mirror: half his head was cleanly shaved; the other half was full of tangled, bushy hair. He gazed about, stepped out of the swiveling chair, and sat back again. He was in flux, anxious, for the moving clock—the strange mysterious object with which the children of the fathers now measure time—showed that it was close to the hour when he should meet the woman.
Sunday came in a bit later, his hands black from working the generator, his shirt soaked in sweat, his trousers smeared with black dirt. "I am sorry," he said. "The generator has developed a fault."
My host's heart sank. "Is it fuel?"
It was not, Sunday told him. "It is the ignition. It is the ignition. I need to take it to a rewire. I am very very sorry, Nonso, we will finish the haircut anytime NEPA restores power, oh. Or tomorrow after I repair the gen. _Biko eweliwe, Nwannem, oh_."
My host nodded and said in the language of the White Man, "No problem." He turned back to the darkening mirror and gazed at his half-shaved head. Sunday unhooked a hat from the many on the wall and gave it to him. He wore it and headed for the restaurant.
EGBUNU, one of the most striking differences between the way of the great fathers and their children is that the latter have adopted the White Man's idea about time. The White Man reckoned long ago that time is divine—an entity to whose will man must submit. Following a prescribed tick, one will arrive at a particular place, certain that an event will begin at that set time. They seem to say, "Brethren, an arm of divinity is amongst us, and it has set its purpose at twelve forty, so we must submit to its dictates." If something happens, the White Man obliges himself to ascribe it to time—"On this day, July twentieth, 1985, such-and-such happened." Whereas time to the august fathers was something that was both spiritual and human. It was in part beyond their control and was ordered by the same force that brought the universe into existence. When they wanted to discern the beginning of a season or parse the age of a day or measure the length of years, they looked to nature. Has the sun risen? If it has, then it must be day. Is the moon full? If it is, then we must gather our best clothing, empty our barns, and get ready to celebrate the new year! If in fact the sound we hear is thunderclaps, then surely the drought must have ended and the raining season must be upon us. But also, the wise fathers believed that there is a part of time that man can control, a means by which man can subject time to his own will. To them, time is not divine; it is an element, like air, that can be put to use. They can use air to put out fires, blow insects out of people's eyes, or even cause flutes to produce music. This is the same way that time can be subject to the will of man—when a group among the fathers says, for instance, "We, the elders of Amaokpu, have a meeting at sunset." That time is expansive. It could be the beginning of sunset, or its middle, or its end. But even this does not matter. What matters is that they know the number of those coming to the meeting. Those who arrive ahead of others wait, talk, laugh until everyone is there, and that's when the meeting begins.
It was thus following the prescribed tick of the clock that she got there before him. She looked even better than she'd looked earlier, wearing deep red lipstick that reminded him of Miss J and a dress that had leopard prints on it.
After he sat down and adjusted his cap to make sure it concealed every part of his head, she said, "Eh, Nonso, I want to ask you: why did you go to that bridge at that very moment and stop?" As he made to answer, she raised her hand, her eyes closed. "I really want to know, really. Why did you go there at that very time?"
He raised his head to look above her, to the ceiling, to avoid her eyes.
"I don't know, Mommy," he said. He picked his words with care, for rarely did he have to speak in the language of the White Man. "Something just pushed me there. I was coming from way back in Enugu, and then I saw you. I just say let me stop."
He glanced out the window, allowed his eyes to fall on a child rolling a motorcycle tire along the road with a stick, trailed by other children.
"You saved my life that day. You will never—"
The ringing of her phone made her pause. She unwrapped it from a handkerchief in her purse, and on seeing the screen she said, "Ah! I was supposed to go somewhere with my parents now. But I forgot before. I am so sorry, but I have to go now."
"Okay, okay—"
"Where is your poultry? I would like to see it. What street?"
"It is number twelve Amauzunku, off Niger Road."
"Okay, give me your number." He leaned in towards her and listed the numbers in their order. "I will come there one of these days. I will call you, later, so we can meet again."
Because I could see in my host the beginning of the growth of that great seed that bears strong root downward in the soul of a man and bears fruit upwards—the fruit of affection that becomes love—I came out of him and followed the woman. I wanted to know what she would do—if she would remain and not vanish, as the previous woman had done. I followed this woman in her car as she drove, and I saw on her face an expression of joy. I heard her say, "Chinonso, funny man," and then laugh. I was watching, curiously observing, when from within her something floated out, like a thick-formed steam rising. And within the batting of an eyelid, what stood before me was a spirit whose face and appearance were exactly like the woman's, save for her luminous body, covered with _uli_ symbols, and her extremities, graced with beads and strings of cowries. It was her chi. Even though I'd been told many times at the caves of spirits that the guardian spirits of the females of mankind possessed more powers of sensitivity, I was astonished at how it was able to see me while still in the body of its host.
—Son of the spirits, what do you want from my host? the chi said in a voice as thin as that of the maidens who dwell on the road to Alandiichie.
—Daughter of Ala, I come in peace, I come not with trouble, I said.
Chukwu, I saw that the chi, who was clothed in the bronze skin of light with which you drape the guardian spirits of the daughters of mankind, regarded me with eyes that were the color of pure fire. She had begun to speak when her host honked and pulled to a sudden stop, shouting, "Jesus Christ! What are you doing, Oga. You no sabi drive?" The car that had veered in her way turned towards another street, and she continued on, sighing loudly. Perhaps now certain her host was fine, the chi turned back to me and spoke in the esoteric language of Benmuo.
—My host has erected a figurine in the shrine of her heart. Her intentions are pure as the waters of the seven rivers of Osimiri, and her desire is as true as the clean salt beneath the waters of Iyi-ocha.
—I believe you, Nwayibuife, guardian spirit of the dawn light, daughter of Ogwugwu, and Ala, and Komosu. I only came because I wanted to be sure that she desires him, too. I shall return with your message to comfort my host. May their union bring them fulfillment in this cycle of life and in the seventh and eighth cycles of life— _Uwa ha asaa, uwa ha asato!_
—Iseeh! she said, and without a moment's wait, she returned into her host.
Oseburuwa, I was greatly delighted by this consultation. And with this confidence, I returned to my host and flashed it in his thoughts that the woman loved him.
AKWAAKWURU, even with the thoughts I put in his head that she loves him, he was still afraid. I could not tell him what I had done. A chi cannot communicate with its host in such a direct way. The human would not understand even if we did. We can only flash thoughts in their minds, and if a host finds these reasonable, he may believe them. So I watched helplessly as he went about with an increased palpitation, fearing that, like Motu, she will be gone again. For days he paid curious attention to his phone, resisting the urge to ring her first. Then, on the fourth day, while he was sleeping on his sofa in the sitting room, he heard a car drive into his compound down towards his house. It was after sunset, and the shadows that had sprouted at the height of the day had already become old. When he looked through the window, he saw Ndali's car pulling to a stop. He cried, "Chukwu!" He had eaten lunch not too long before, and the plastic bowl sat on the stool beside him, still containing water in which a small empty bag that had contained groundnuts and a plastic sachet of Cowbell powdered milk were floating. He dropped the bowl in the kitchen sink. Then he ran to his room and put on the trousers that lay on the bed. He glanced quickly in the mirror on the wall of his room, grateful that Sunday had finally cut his hair two days earlier. When he rushed back into the sitting room, his eyes fell on the blue box of sugar cubes that sat half closed on the center table, beside a globular stain. And at the foot of the table, on the plastic bag containing thread, needles, and a small sack of nails. She was knocking while he took these away. He returned again a moment and looked about the house to see if there was anything he could clean, and when he did not see anything whose oddity he could very quickly fix, he ran to the door, still holding on to his chest to stabilize the heartbeat. Then he opened the door.
"How did you find me?" he said once she came in.
"Do you live on the moon, er, mister-man?"
"No, but Mommy, how? It is hidden, and the numbers are not even clear sef."
She shook her head, wearing a gentle smile. Then she said his name with a slow, dragging enunciation, the way a child learning to speak might say it, No-n-so.
"Would you give me a seat?"
He threw his eyes about the room again and nodded. She sat on the big sofa near the window while he stood transfixed at the door. Then, almost at once, she rose and began walking about the sitting room. As she did, he became worried that she would perceive the smell that hung in the air. He observed her nose for any sign of her scrunching it or covering it. He noticed then, with even greater alarm, that there was a clearly perceptible stain on the wall. He feared it might be chicken shit. He went and stood in front of the stain, wearing a smile that concealed his distress.
"You live alone, Nonso?"
"Yes, I live here alone. Only me. My sister doesn't come, except my uncle who comes sometimes," he said with haste.
Her nod did not translate into attention, for as he spoke, she stepped into the kitchen. The state of the kitchen made his heart sink. The arches around four sides of the ceiling had cobwebs that were blackened with soot, making it look as though spiders nestled in them. The sink was full of dirty plates, on one of which was a sponge hewn from a plaited-thread sack, a shriveled green piece of soap trapped in its netting. Even more shameful to him was something he was not immediately responsible for: the faucet on the sink. It had long atrophied into disuse, and its head had been removed and simply replaced with a piece of a black polythene bag. His kerosene stove, too, was dirty. It sat on a blackened slab of wood. It wore the singed skin of the chicken he'd roasted on its top levers, and around its top perimeter were grains of dried rice and what looked like a dried tomato skin. Even worse, at the far corner, behind the door that led to the yard, was a dustbin full of trash from which a putrid smell simmered.
Egbunu, he would have died if she'd lingered in the kitchen for a moment more after she turned on the light and roused the thread of flies gathered around the stack of unwashed plates. He was relieved when he saw the net door open slightly and its spring give way, creaking as it opened into the backyard.
"You have many chickens!" she said.
He walked up to her. She had one leg on the threshold, the other in the yard. She leaned back into the kitchen towards him.
"You have many chickens," she repeated, as if in surprise.
"Yes, I'm a poultry farmer."
"Wow," she said. She stepped out into the yard, staring wide-eyed at the coops. Then, without saying a word, she returned to the sitting room and sat back on the sofa beside her purse. He followed her and caught a glimpse of her underpants as she sat down, briefly spreading her legs. He joined her, frightened because of the things she had seen. She did not speak for a while but kept looking at him in a way that so discomforted him that he wanted to ask her if she despised him because of the state of his house, but the words lay loaded in his mouth like a fodder in a cannon, awaiting the signal to fire. To prevent her from looking around the house again, he sought to engage her in conversation.
"What happened to you that night?" he said.
"I was going to die," she said and dropped her eyes to the floor.
Her words softened his shame.
"Why?"
Without hesitation, she told him she'd woken up the morning of the day before to find the world she'd so carefully built crumpled into dusty ruins. She had been crushed for two whole days by an e-mail from her betrothed, which had announced that he had married a British woman. The blow, she told him, was unbearable because she had given that man five years of her life, gathered all her savings, and even stolen from her father to help him achieve his dream of getting a degree in film directing from a school in London. But barely five months into this move to Britain, he was married. With a voice filled with so much pain my host could feel it, she explained how nothing had prepared her for the blow she felt.
"Nothing to hold on to, nothing to even—nothing. Throughout that day before I saw you on that bridge, I was tired because I had tried, tried, tried to reach him, but nothing, Nonso."
She had gone to the river not because she had any strength or will to kill herself but because the river was all she could think of after reading the e-mail for the umpteenth time. She did not know whether she would have jumped off the bridge if he had not come.
My host listened with a keen ear to her story, only speaking once—to ask her to ignore the chickens that had begun to squawk plaintively.
"What happened to you is very painful," he said, although he'd not understood all of it. Her command of the White Man's language contained more words than he could comprehend. His mind had hovered, for instance, over the word _circumstances_ like a kite over a gathering of hen and chicks, unable to decide how or which to attack. But I understood everything she said, because every cycle of a chi's existence is an education in which a chi acquires the minds and wisdom of its hosts, and these become part of him. A chi may come to know, for instance, the intricacies of the art of hunting because once, hundreds of years before, the chi inhabited a host who was a hunter. In my last cycle, I guided an extraordinarily gifted man who read books and wrote stories, Ezike Nkeoye, who was the older brother of the mother of my present host. By the time he was my current host's age, he'd come to be familiar with almost every word in the language of the White Man. And it was from him that I acquired much of what I now know. And even now, as I testify on behalf of my current host, I wear his words as well as mine and see things through his eyes as well as mine, and both sometimes meld into an indistinguishable whole.
"It is very painful. I am talking like this because I have suffered too much also. I no have father and mother. In fact, no family."
"Ah! That is very sad," she said, putting her hand over her wide-open mouth. "I am sorry. Very sorry."
"No, no, no, I am now fine. I am fine," he said, even though the voice of his conscience was tugging at him for having left out his sister, Nkiru. He watched Ndali rest her weight on her thigh and tilt her body towards the small table centered between them. Her eyes were closed, and this made him think that she was sinking into pity for him, and he feared that she might cry for him.
"I am fine now, Mommy," he said even more resolutely. "I have a sister, but she is in Lagos."
"Oh, junior or senior sister?"
"Junior," he said.
"Okay, why I have come is because I have come to thank you." A smile washed across her tearful face as she pulled her bag up from the floor.
"I believe God sent you to me."
"Okay, Mommy," he said.
"What's this 'Mommy' you keep saying? Why do you say it?"
Her laughter now made him conscious of his own feral laughing, which he'd tried to contain to avoid embarrassing himself.
"Really, it is strange oh!"
"I don't have a mother again, so every good woman is my mommy."
"Oh, so sorry, my dear!"
"Am coming," he said and went to the toilet to urinate. When he returned, she said, "Did I forget to say that I love your laugh?"
He looked at her.
"I do. Seriously. You're a beautiful man."
He nodded hastily as she rose to leave and let his heart take temporary flight at this extremely unexpected outcome of what he had been certain would be a disaster.
"I have not even offered you something."
"No, no, don't worry," she said. "Another time. I have tests."
He thrust his hand out to shake hers, and she took it, her face wide with smiles.
"Thank you."
Guardian spirits of mankind, have we thought about the powers that passion creates in a human being? Have we considered why a man could run through a field of fire to get to a woman he loves? Have we thought about the impact of sex on the body of lovers? Have we considered the symmetry of its power? Have we considered what poetry incites in their souls, and the impress of endearments on a softened heart? Have we contemplated the physiognomy of love—how some relationships are stillborn, some are retarded and do not grow, and some fledge into adults and last through the lifetime of the lovers?
I have given much thought to these things and know that when a man loves a woman he is changed by it. Although she willingly gives herself to him, once he marries her she becomes his. The woman becomes his possession, and he becomes her possession. The man calls her Nwuyem, and she calls him Dim. Others speak of her as _his_ wife and of him as _her_ husband. It is a mystifying thing, Egbunu! For I have seen many times that people, after their beloveds have left them, try to reclaim them as one would attempt to reclaim property that had been stolen. Wasn't this the case with Emejuiwe, who, one hundred and thirty years ago, killed the man who took his wife from him? Chukwu, when you laid down your judgment after my testimony on his behalf here in Beigwe, as I am doing now, it was sad but just. Now, more than a hundred years later, when I saw my current host's heart lit with similar fire, I feared because I knew the potency of that fire, that it was powerful, so powerful that in time nothing might be able to quench it. As he walked her to her car, I feared it would push him in a direction in which I might be powerless to stop him from going. I feared that when the love had fully formed in his heart, it would blind him and make him deaf to my counsel. And I could see already that it had started to possess him.
OBASIDINELU, oh what substance does a woman bring into the life of a man! In the doctrine of the new religion the children of the fathers have embraced, it is said that the two become _one flesh_. What truth, Egbunu! But let's look at the times of the wise fathers, how indispensable the great mothers were. Although they did not make the laws that guided society, they were like the chi of society. They restored order and equilibrium when order became broken. If a member of a village committed a spiritual crime and vexed Ala, and if the merciful goddess—in her rightful indignation—poured out her wrath in the form of diseases or drought or catastrophic deaths, it was the old mothers who went to a dibia and consulted on behalf of the society. For Ala hears their voices above those of the others. Even when there was a war—as I witnessed one hundred and seventy-two years ago, when Uzuakoli fought against Nkpa and seventeen headless men lay in the forests—it was the mothers on both sides who marched and restored peace and pacified Ala. This is why they are called _odoziobodo_. If a group of women can restore equilibrium to a community on the verge of calamity, how much more one woman can do to the life of one man! As the great fathers often say, love changes the temperature of a man's life. Usually, a man whose life was cold becomes warm, and this warmth, in its intensity, transforms the person. It grows the small things in his life and puts shine on the spots in the fabric of his life. What the man did every day, now he does more cheerfully. Most people in their lives would come to know that something in them had changed. They may not need to speak of it to anyone. But their faces, the most naked of all features of human physiognomy, begin to wear a hue that anyone who pays attention soon notices. Say, if a man works in the company of other people, one of his colleagues may pull him aside and say to him, "You look happy," or "What happened to you?" The stronger the affection, the more obvious it will become to others, and my host's affection for Ndali was tempered by his fear that he was unworthy of her. He resolved that if she ever gave in to him, he would give her the fullness of his heart.
In lieu of human coworkers, the fowls bore witness to my host's metamorphosis. He fed them ecstatically after the woman left his house that day. He found the sick rooster who'd developed a wry tail and took it to the edge of the farm, in front of the house and away from the sight of other fowls, and slaughtered it. He let its blood drain into a small hole in the earth, and then put it in a bowl and kept it in the refrigerator. After he'd washed his hands in the bathroom, he swept the large pens that were parted into halves by wooden walls. He chased a particular kind of lizard which the fowls did not like, the green-headed lizard, into a hole in the ceiling. Then he climbed up a ladder and stuffed a bunched-up rag with palm-oil stains into the hole. When he was done with it, he noticed that the chickens had upturned the basin of water from which they drank, and now it lay, leaning against the thatch wall, an eye-size pool of water anchored in it. Within the puddle lay a patch of sediment that stared back at him like a pupil. As he walked towards the bowl, he stepped on something he discovered to be the rib of a feather. It slid down in a straight line through the muddied earth, tripping him. He fell against the other empty basin, and it balled up into the air and emptied its contents—a mass of dirt, feathers, and dust—onto his face.
Chukwu, if the chickens were humans, they would have laughed at what his face became afterwards: a rich patch of dirt and mud on his forehead and over his nose. Were I not myself a witness, I would have doubted what I saw in my host that day. For, despite his pain and feeling the spot on his head with his fingers repeatedly and looking at them to see if the wound would bleed, he was happy. He rose, laughing at himself, thinking of how Ndali had sat on the sofa and called him a beautiful man the previous day. He looked down where he'd fallen and saw a shaved part of the floor, which his shoes now wore like an encrustation. On the other side of the pen stood a hen on which he'd almost fallen. It had leapt hysterically out of his reach as he fell, raising dust and feathers with its violent wing flapping. He recognized it as one of the two hens that laid gray eggs. It stood crowing in protest, and others had joined. He left the coop and washed the dirt off himself, and all the while, even when he lay in bed later, Ndali remained in his mind.
Once he slept, as it often happens when he goes into the unconscious state of sleep, I became stripped of the barriers of his body. Even without stepping out of it, I am often able to see that which I'm not able to see while he is awake. As you know, you created us as creatures for whom sleep does not exist. We exist as shadows which speak the language of the living. Even when our hosts sleep, we remain awake. We watch over them against the forces that breathe in the night. While men sleep, the world of the ethereal is replete with the noise of wakefulness and the susurration of the dead. Agwus, ghosts, akaliogolis, spirits, and ndiichies on short visits to the earth all crawl out of the blind eyes of the night and tread the earth with the liberty of ants, oblivious to human boundaries, unaware of walls and fences. Two spirits arguing may struggle and tumble into the house of a family and fall on them and continue to wrestle through them. Sometimes, they merely walk into the habitations of men and watch them.
That night, like most others, was filled with the din of spirits and the brass drum of the sublunary world, a multitude of voices emitting cries, shouts, voices, howls, noises. The world, Benmuo, and Ezinmuo—its corridor—were soaked in them. And from a distance, the riveting tune of a flute riffled through the air, pulsing like an animate thing. It remained like this for a long time when, around midnight, something shot through the wall with uncanny speed. Instantly, it ringed itself into a luminous coil that was grayish and almost imperceptible to the eyes. At first it seemed to rise towards the roof, but slowly it began to diffuse and elongate like a serpent of shadows. Then it morphed into a most frightening agwu—with a roachlike head and a portly human body. I lunged forward at once and ordered it to leave. But it gazed at me with eyes filled with hate, then stared mostly at my host's unconscious body. Its mouth was sticky, glued together as if by some sticky purulent secretion. It kept pointing at my host, but I insisted it leave. When it did not so much as stir, I became afraid that this evil creature would harm my host. I trailed into an incantation, fortifying myself as I invoked your intervention. This seemed to stop the being in its place. It stepped back, let out a growl, and vanished.
I had encountered spirits like this in my many cycles on earth, and I recall most vividly, while inhabiting Ejinkeonye during the war, that he'd slept in a half-destroyed, abandoned house in Umuahia, and while he was asleep, a spirit materialized with such swiftness that I gave a start. I looked, but it had no head. It was waving its arms, stamping its feet, and gesturing at the stump where its head had been. Egbunu, not even an akaliogoli, those creatures of dreadful forms, had inspired such terrors in a living spirit like me. Then, by some transmutative power, the creature's head emerged and hung in midair, its eyes glancing about. The headless creature would try to take the head with its flailing hands, but it would swerve that way and the other way, until finally the head floated away the way it had come, and the spirit followed it. I would find out the following day, through the eyes of my host, that the man had been an enemy soldier beheaded while raping a pregnant woman, and had become an akaliogoli. My host, Ejinkeonye, would watch the body of the man being burned the following morning, unaware of what had happened the night before.
I leapt up presently and tried to catch up with the spirit, to find out why it had targeted my host, but I could not tell which direction it had gone. I found no trace of it on the plains of the night, no footprints on the track of the air, no footfalls in the dark tunneling beneath the earth. The night was full mostly of bright stars in the sky, and a multitude of spirits were about their business around the vicinity of my host's farm. No humans were about, nor was there even any trace of them except for the sound of cars racing past along some road in an unknown distance. I had the temptation to wander a bit, but I suspected that the agwu I had seen was a vagabond spirit in search of a human vessel to possess, and it could return to try to inhabit my host. So I made my way back to the compound as quickly as I could, projecting through the fence at the backyard, then through the wall into the room where my host lay still deep in sleep.
AKWAAKWURU, he woke to the wild noises of his crowd of fowls the following morning. One of them was crowing without cease, letting its voice ebb at intervals, then beginning again in a higher pitch than before. He pushed aside the wrappa he'd covered himself in and was starting to step out the door when he realized he was naked. He put on shorts and a wrinkled shirt and went to the backyard. He emptied the last contents of a bag of mash into a bowl and set the bowl at the center of the yard on an old newspaper page. When he unlocked one of the cages, the birds flung themselves at him at once, and in the batting of an eye, the bowl was covered in a goo of feathery beings.
He stepped back, his eyes scanning them for signs of anything out of the ordinary. He watched one of the hens especially, the one whose wing had caught an errant nail sticking out of one of the cages. The bird had tried to pull itself away from the nail so hard that it had almost ripped out its wing. He'd stitched the wing with a thread the previous week, and now the bird participated in the scramble for the mash with cautious gait, the red thread of the stitching visible beneath its wing. He picked up the hen by its legs. He checked its wings, tracing his fingers around the veins on the end. As he made to drop it, his phone rang. He ran into the house to get it. But by the time he got into the living room, the ringing had stopped. He saw that Ndali had just called him and sent a text message. He hesitated at first to read it—as if he feared that what he would read in the paper would remain for eternity unerasable. He dropped the phone back on the dining table, placed his palm on his forehead, and gnashed his teeth. I could see that he had become sick from the head wound of the previous day. From the top of the refrigerator, he took a sachet of paracetamol and ejected one of the remaining two tablets into his palm. He placed the medicine on his tongue, went into the kitchen, and washed it down with water from a plastic jug.
He picked up the phone again and read the message: Nonso, should I come and visit you in the evening? Chukwu, he smiled to himself, pumped his fist into the air, and shouted "Yes!" He dropped the phone into his pocket and was almost back in the yard when he remembered he'd merely answered in speech as if she were there with him. He stood by the net door to the yard and typed "yes" into his phone.
Lit with the prospect of meeting Ndali, he gathered some eggs and placed them in the egg-shaped holes on a plastic crate. Then he took hold of the wounded bird once more. Its eyes blinked with fear, its beak opening and closing as he rubbed its head and examined its wings to see if it could accommodate flight. He cleaned the mash tray and placed some mash on it. Something like a half-broken toothpick stuck out of the feed. He picked it up and threw it behind him. Then, on second thought, fearing one of the fowls might find it there and swallow it, he rose and began looking for the stick. He found it just near one of the cages, the one for the small chicks. The stick was on the wet edge of the slab of wood on which he'd placed the cage. He picked it up and threw the stick over the fence into the dump outside his compound. Then he tucked the tray of mash into the bed of one of the two cages.
By the time he finished feeding the poultry, his hands were almost black with dirt and grime. Dark grime lay in his fingernails, and the flesh of his right thumb looked barbed and was lined with welts. One of the eggs he'd picked was coated with a stiff encrustation of feces, which he tried to scratch off with his fingers, and it was the crust of feces that he now carried in his fingernails. As he washed his hands in the bathroom, he was thinking about how odd his work was and how lowly it seemed to be to one newly encountering it. He feared that Ndali may not come to love it or might even be irritated by it if she came to really understand the nature of his work.
Chukwu, as I said before, this kind of fear-induced rumination often occurs when people have been made self-conscious by the presence of others whom they hold in high esteem. They assess themselves by focusing on how others would perceive them. In such situations, there may be no limit to the self-defeating thoughts that may form in the person's mind, which—no matter how unfounded—may consume them in the end. My host did not, however, dwell on these thoughts for too long. He was, instead, in a haste to prepare for Ndali's visit. He swept the house and the balcony clean. Then he dusted the cushions and sofas. He washed the toilet bowl and sprayed Izal into it and cleaned out the rat feces behind the water drum. He threw away one of the plastic buckets, a paint bucket that had cracked in several places. Then he sprayed air freshener around the house. He'd just finished bathing and was creaming his body when, through the window, he saw her car coming towards the house, flanked on both sides by the plantations.
Ijango-ijango, my host felt his body lit with admiration at her appearance that evening. Her hair was dressed in a way the great mothers would have found strange but which made it shiny and attractive to my host. He gazed closely at the neatly permed hair, her wristwatch, the bangles around her wrist, the necklace with green beads that reminded him of his own mother's sister, Ifemia, who lived in Lagos and with whom he'd long since lost contact. Although he'd already begun to feel unworthy of Ndali because of his lack of exposure (he had never been to a club before, or to a theater), he sank even lower in his own estimation when he saw her that evening. Although she engaged him with utmost geniality and affection, he stood with a strong feeling of unworthiness. So he attended their conversations like one forced to be there, saying only what was needed and what was prompted.
"Did you always want to be a poultry farmer?" Ndali said at one point, later than he'd expected, deepening his fear that in the end, she would not give herself to him.
He nodded, then it occurred to him that that might be a lie. So he said, "Maybe, no, Mommy. My father started the idea, not me."
"The poultry?"
"Yes."
She gazed on at him, a restrained smile on her face.
"How? But how?" she said.
"It's a long story, Mommy."
"God! Ah, I want to hear it. Please tell me."
He looked up at her and said, "Okay, Mommy."
Ebubedike, he told her about the gosling, beginning with how it was caught when he was only nine years old, an encounter which changed his life, and which I must relate to you now. His father took him from the city to his village one day and told him to sleep, that in the morning he would take him into the Ogbuti forest, where a species of wool-white geese lived near a hidden pool in the very heart of the forest. Most hunters avoided this part of the forest for fear of deadly snakes and wild beasts. The pool was a tributary of the Imo River. I have seen it many times. Long ago, before Aro slave raiders began sweeping this part of Alaigbo, the river used to flow. But it was cut off by an earthquake which separated it from the rest of the river and made it into a stagnant body of water that became the home of the white geese. They had lived there for as long as anyone in the nine villages that surround the forest could remember.
When my host and his father, who carried a long Dane gun, got to this spot, they stopped behind the stump of a rotting fallen trunk covered by grass and wild mushrooms. Two stone throws from the trunk was the dead pool, half covered with leaves. Beside it was a riparian stretch of matted, damp land strewn with splinters of wood. It was here that a flock of white geese was gathered, grazing. As if alerted to the human presence, most of the flock whipped their wings and flew into the denser part of the forest, leaving only a mother goose and her offspring and another big goose in the field. The third goose leapt a few times and began floating across the distant waters until it reached the reef, then it vanished into the greenery. My host observed the mother goose with fascination. Its plumage was rich and its tails serrated downward. Its eyes were wide and it had a brown beak with nares. When it moved, it spread its wings into a cascading flourish. The gosling beside it was different: its neck was longer and bare at the top, as if plucked. It tottered forward on its tiny feet after its mother, who'd begun moving away from the nestling. My host's father had brought his gun to bear and would have shot it if a perplexing vision had not suddenly presented itself. The mother goose, who had stopped in a place of soft earth so that her feet sank into the mud, was now waiting with her mouth wide-open. The gosling, clattering, approached and buried its head into the waiting mouth till part of its neck disappeared.
My host and his father watched in wonder as the gosling's head and neck probed into its mother's mouth. As the child fed, its mother struggled for balance. She dug her feet deeper into the clump of mud, fluttering her wings violently, stepping back with steady haste, her claws folding and unclasping. For a moment, it seemed to my host that the big goose's throat might tear open as the little bird gorged greedily within it. The movement of the gosling's beak could be glimpsed through the pale skin of its mother's throat. It came almost as a surprise to my host when the gosling disentangled itself and began sprinting away, fluttering its wings and full of life like one reborn. Its mother turned her head, gave a cry, and seemed to fall on her legs. Then she rose, caked in a half cloth of mud, and began racing in the direction where my host and his father were crouched.
The bird was close when his father took aim. The shot flung the goose backwards with a loud noise, leaving in its wake an explosion of feathers. The forest erupted into a hysteria of fleeing creatures and a chorus of fluttering wings. As the feathers settled, my host saw the gosling hurrying towards its mother's body.
"I did it, I finally shot an Ogbuti goose," his father said, as he stood up and began running towards the dead goose. My host followed at a cautious pace, speechless. His father picked up the dead goose, exhilarated, and began walking in the direction from which they had come, the blood of the dead goose marking his trail. His father did not notice the gosling scampering along after him, making a shrilling sound which, many years later, my host would realize was the sound of a weeping bird. He stood still, listening to his father talk about how for years he'd longed to catch an Ogbuti forest goose—"always said no one knew where they lived. How could anyone know? Only a few people will ever venture this far into the Ogbuti. People only saw them in the air. And, you know, it is very hard to shoot something in the air. This"—until his father, turning abruptly, saw him standing back, afar.
"Chinonso?" his father said.
He looked up with a pout, his eyes near tears. "Sir," he said in the language of the White Man.
"What? What is it?"
He pointed at the gosling. His father looked down and saw the gosling moving its legs in the swamp, its eyes fixed on the two humans as it wept for its dead mother.
"Heh, why don't you pick it up and bring it home?"
My host walked towards his father and stopped behind the bird.
"Why don't you keep him?" his father said again.
He glanced at the bird, then at his father, and something lit up in him.
"Can I bring it to Umuahia?"
"Uh," his father said, and turned back again to the path from which they had come with the dead goose, whose body was now half covered in crimson in his hand. "Now catch it, and let's go."
With hesitation, dragging his feet, he dived forward and caught the gosling by its thin legs. The bird mourned plaintively, beating its wings against the tender hands that held it. But he tightened his grip around its legs as he lifted it from the ground. He looked up at his father, who was waiting, blood dripping from the dead goose in his hand.
"It is yours now," his father said. "You have saved it. Take it and let us go." Then turning, his father began walking back to the village, and he followed.
He then told her about how he loved the bird. The bird often flew into moods of rage, then calmed and lifted its spirit. It would sometimes make a mad dash towards nothing, perhaps intent to return to the forest from which it'd come. And when it saw no chance of escape it would circle back in defeat. He watched over it closely, with anxiety. He lived in perpetual fear that something bad would happen to the bird or that it might someday escape from him. This fear was most pronounced in the times when the gosling would, in anger, begin dashing about the house, from wall to wall, trying to break through it and flee. And after every such struggle, it would return to a chair or a table, its head bent as in some kind of prostration. It would hold its wings suspended as it cawed in fury or frustration.
"Yes," he said in answer to her question: there were times the gosling was calm. He knew that it was in the nature of earthly creatures that even the most wounded of them is sometimes peaceful in captivity. These were times when the gosling slept in his bed, by his side, as though it were a human companion. When he first came to Umuahia with the gosling, the children of the neighborhood flocked to see it. At first, he guarded it jealously, not allowing anyone to touch the raffia cage in which he kept the bird. He would even fight with some of his friends who lived around the neighborhood and played football with him if they tried to touch it without his permission. One of them, Ejike, with whom he'd been best friends, was particularly enamored of the bird. Ejike sought it more than the others, and, in time, my host allowed him to play with it often. Then one day, Ejike asked to take the gosling to his house so he could show it to his grandmother, saying, "For five minutes, only five minutes." Oseburuwa, I had seen the look in the eyes of this child, and I'd feared that in them, deep within, I could see the small flame of envy burning. For I have seen it many times in the children of men: that negative side of admiration that has caused many murders and dark conspiracies. I flashed it in my host's mind that he should not give away the gosling. But he would not hear me. He gave the bird to his friend, confident no harm would come to the bird.
Ejike took the gosling away. When by sunset he still had not returned it, my host became anxious. He went to Ejike's house and knocked on the door of the flat Ejike and his mother occupied, but heard nothing. He called Ejike's name many times, but received no answer. The door was bolted from within. But from the outside, he could hear the bird squawking and the sound of its wings as it glided about even with the twine that was wound around its legs. He rushed back home and sought his father. Together they went to the house, but even though this time Ejike's mother answered the door, she denied that they had the gosling.
This woman, whose husband had died, had once lured his father into her house and they had copulated. But not wanting to fill the place of his loving wife, for whom he would grieve for the rest of his life, his father had refused to continue the relationship. And this had put a wedge between him and the woman. Although my host did not know this, I did, for I'd heard his father talk about it to himself while my host was asleep. And one night, I'd seen his father's chi—a carefree chi who often floated with etheric flamboyance while lurking about the house, and it told me that it had left the body of its host because he was about to have sex with the neighbor. It said my host's father and the woman were fondling at the back of the house, in the yard. I had come to know this guardian spirit very well, as one often knows the guardian spirits of other members of a household. Peer into a household at midnight, and you'll find guardian spirits—usually of males—conversing or simply moving about the house, often developing a bond with each other over the lifetimes of their hosts. This is how I came to know a great many guardian spirits of the males and females of mankind.
So on this day, perhaps because of the hurt she still harbored, the woman slammed the door in the face of my host and his father.
There was nothing my host could do to Ejike and his mother after that. He was stunned for days and would sometimes descend into uncontrolled rage and dart towards the neighbor's house, but his father would call him back and threaten to whip him if he went back there. He listened every minute for the gosling, refusing to eat, and hardly sleeping at night. It was difficult for me, his guardian spirit, to watch him suffer. But there is nothing a chi can do to help a man in a circumstance such as this, as we have limitations. The old fathers, in their wisdom, say _Onye ka nmadu ka chi ya,_ and they are right. A person who is greater than another is also greater than his chi. Thus, there is little a chi can do for a man whose spirit is broken.
Egbunu, Ndali was moved by this part of his tale. Although she spoke throughout his story, asking questions ("He said that?"; "So what happened next?"; "Did you see it?"), I've decided not to relate them, as I must focus on this tale about this creature to whom my host once gave his heart. But in light of the things that have now happened and the reason for which I stand before you to testify about my host, I must relate her speech at this point in the story in which my host's desire to get back that which belonged to him had sent him to the edge of madness. Shaking her head wearily, she'd said, "It must be very sad, a bird that is yours, which you suffered for, taken away just like that. It must be painful." He merely nodded and continued. He told her that by the fifth day, he'd become desperate. He climbed the tree in the backyard, and from there he gained a view into the compound of the neighbor. He saw Ejike seated on a stool behind their fenced house stroking the gosling. At first, the goose had appeared dead, then he saw its wings flutter as it tried to fly away from its captor who quickly stamped his foot on the red leash fastened to its leg. The gosling struggled, raised its leg again and again and flapped its wings, but the twine held it still. It was as my host watched this that the cruel idea came into his mind.
Chukwu, the moment I glimpsed the design of his heart, I contested it. I flashed the thought in his mind to think of the devastation and pain that would come to him should he proceed. He considered it momentarily and even imagined the bird bleeding from the gash inflicted by the stone on its head, and it frightened him. Yet he brushed the thought away. But as you know, a chi cannot go against the will of its host, nor can it compel its host against his will. This is why the old fathers say that if a man is silent, his chi also becomes silent. This is the universal law of guardian spirits: a man must will for his chi to act. Thus was I thrust into a difficult situation in which I would watch helplessly as he did something that would bring him anguish in the end. He returned with his catapult, sat on a bowed branch, and concealed himself in its foliation. From there he saw the bird fastened to the leg of the stool on which Ejike had sat only a moment before going back into his house.
At this point in his story, my host saw that he was about to tell Ndali that he was capable of serious violence, so he paused the narration to lie to her and say that he'd stopped loving the gosling because it was no longer his. He told her that because it was attached to Ejike, he'd thought of killing it as revenge against its new master. When she nodded and said, "I understand, go on," he told her about how he shot at the bird with the stone without missing. It hit it on the shank, then it fell and let out what must have been a cry of pain. He rushed down the tree, his heart having become a beating drum. He ran into his room, and later, Ejike came rushing down with the bleeding bird, crying that it was going to die if it did not get treatment. Indeed, days later, after he had reclaimed the bird and brought it back into his house, he woke to find the bird lying in the center of his room on its back, its small wings clasped tightly against its body, its head slumped by its side. Its two legs were firm and lifeless, their claws curving downwards with the early strain of rigor mortis.
Gaganaogwu, the death of the bird disturbed my host very much. He told Ndali how he'd mourned the loss and had become so harsh on himself that his father was forced to punish him. But it yielded nothing. Complaints began to tide in from his school of his inattention and constant truancy. Such was his revolt that he acted to provoke punishments, and he took them—especially the floggings—with a masochistic indifference that alarmed his teachers. They sent word to his father, who by then had grown weary of punishing him because he had changed from being a plump boy to a slender one. One day, in a desperate attempt to save his son, his father took him to a poultry farm outside the city. My host described the big farm to Ndali in detail: the hundreds of birds before his eyes—domesticated birds of different kinds. It was here, among the smell of a thousand feathers and the clucking of hundreds of voices, that his heart finally lifted within him and came back to life. His father and he returned with a cage full of chickens and two turkeys, and the poultry business began.
EBUBEDIKE, after he rendered the tale, they said nothing for a while. In silence, he riffled back through the words he had said to see if there was something that could cast him in a bad light. And she sat there in deep thought, perhaps judging his words. Discretion was at the center of his self-esteem. It was what must be kept alive to maintain him. It was therefore paramount that he keep most of the details of his past hidden and that his tongue retain its poverty even in the face of pressure. Pricked that he'd told her so much, he let his thoughts shift to the tomatoes he'd planted the previous week, which he had not yet watered, when she spoke suddenly.
"It is a good job," Ndali said after what seemed like a long contemplation.
He nodded. "Do you like it, Mommy?"
"Yes, I do," she said. "You miss your family? In fact, what about your sister?"
Simple as this question was, it took him a long time to produce an answer. I have dwelt amongst mankind long enough to realize that they do not store information about those who have hurt them as they do others. This kind is kept in tightly sealed jars whose lids must be opened to remember them. Or, in the worst cases—like the memory of his grandmother's rape by enemy soldiers during the war—the jar must be smashed to bits. Thus all he said was: "She lives in, erm, Lagos. Me and her, actually, we don't talk. Her name is Nkiru."
"Why?"
"Mommy, she left home before Papa died. She, you know, she—how I should say it?—abandoned us." He looked up to meet her fixed eyes. "She left because of a man no one wanted her to marry because the man is very old, old enough even to be her father. In fact, he take more than fifteen years to senior her."
"Ah-han! Why did she do that?"
"I don't know, my sister." He gave her a sharp look to see if there was a reaction to what he had just called her. Then he said, "I don't know, Mommy."
Egbunu, although this was all he'd tell her about his sister for the time being, when one pries such a lid open, one sees more than one can account for. There is often no way to stop this. "Why would a child reject her parent?" his father would ask him, and he would say he did not know. To which his father would blink slow-moving tears. His father would shake his head and snap a finger over it. Then he would gnash his teeth tightly, making the sound of ta-ta-ta-ta-ta. "It is beyond me," his father would say with even more bitterness than before. "Beyond any man at all—dead or alive. Oh, Nkiru, Ada mu oh!"
Because the memory of what he'd recollected weighed heavily on him, he wanted to change the line of conversation. "I will get you something to drink," he said, and rose to his feet.
"What do you have?" She stood with him.
"No, you sit down, Mommy. You be my visitor. You suppose sit don make I feed you."
She laughed and he saw her teeth—how tender they looked, lined up almost delicately, like a child's.
"Okay, but I wan stand," she said.
He shot a glance at her and arched his brow. "I didn't know you can speak pidgin," he said and laughed.
She rolled her eyes and sighed in the manner handed down from the great mothers.
He brought out two bottles of Fanta and handed one to her. He still bought crates of these drinks they call Fanta and Coke, as his father used to do for guests, even though he hardly ever had any guests. He stored some of them in the refrigerator and returned the empty bottles to the crates.
He pointed to the dining table, surrounded by four chairs. A half-burned candle sat on the used lid of a Bournvita tin, reshaped by a waterfall of wax that washed down the tin and formed a coating at its foot like the gnarled roots of an aged tree. He pushed this to the edge of the table by the wall and pulled out a chair for her on the side. He saw that she was looking at the calendar on the wall that had an image of the White Man's alusi, Jisos Kraist, wearing a crown of thorns around his head. The inscription beside the raised finger of Jisos marched on her lips but did not become audible. He'd opened the drink when she sat down, and as he made to return the opener, she grabbed his hand.
Ijango-ijango, even these many years later, I still cannot fully comprehend all that transpired in that moment. It seemed that by some mysterious means, she had been able to read the intents of his heart, which had all along cast themselves upon his face like a presence. And she had come to understand, by some alchemy, that the smile he'd carried on his face all along was his body's struggle to manage the solemn intransigence of its volcanic desire. They made love so heartily, so beauteously, for nearly an hour with a rare kind of energy. He was driven by a strange mix of unbelief and relief, and she by some feeling I cannot describe. You know, Chukwu, that you have sent me out many times to dwell in people, to live through them, and to become them. You know that I have seen many people unclothed. But still the ferocity of their encounter alarmed me. It may have been because it was their first time, and they both could tell—for this was indeed his thought—that there was something ineffably deep between them, and indeed I was reminded of her chi's words: "My host has erected a figurine in the shrine of her heart." It must be why at the end of it, when both of them were drenched in sweat and he saw tears in her eyes, he lay by her, saying words which—although she, he, and I alone could hear—were also heard in the realm beyond man as thunderous acclamations meant for the ears of man and spirits, the dead and the living, for the moment and forever: "I have found it! I have found it! I have found it!"
#
## An Orchestra of Minorities
GAGANAOGWU, the daily life of lovers often begins to share resemblances, so that, in time, each day becomes indistinguishable from the one that came before it. The lovers carry each other's words in their hearts when apart and when together; they laugh; they talk; they make love; they argue; they eat; they tend to poultry together; they watch television and dream about a future together. This way, time slips and memories accrue until their union becomes the sum of all the words they have said to each other, their laughter, their lovemaking, their arguments, their eating, their work with the poultry, and all the things they have done together. When they are not with each other, night becomes to them an undesirable thing. They despair at the masking of the sun and wait eagerly for the night, this cosmic sheet that has separated them from their beloved, to pass in fervent haste.
By the third month, my host realized that the moments he'd come to value the most were the times when Ndali tended the poultry with him. Although many things about poultry keeping—like the smell of the coop, the way the fowls defecate nearly everywhere, and the killing of the ones sold as meat to restaurants—still bothered her, she enjoyed tending to the flock. Even though she worked with my host without complaint, he remained worried about her perception of the work. He often recalled the university science lecturer at the poultry market in Enugu who had complained bitterly about the habit among poultry keepers of holding the birds by their wings, calling it cruel and insensitive. Although Ndali herself was training to be a pharmacist, sometimes wearing lab coats in some of the photos she showed him, she displayed no such sensitivity. She plucked with ease the overgrown feathers of the fowls. She harvested eggs whenever she visited in the early mornings or stayed over at his house. But even beyond the birds, she took care of him and his house. She poked her hand into the dark and secret places of his life and touched everything in it. And in time, she became the thing his soul had been yearning after for years with tears in its eyes.
In those three months, this woman he'd met on a bridge in a chance encounter, and who is the reason for my premature testimony this night, transformed his life. Without warning, Ndali arrived one afternoon with a new fourteen-inch television and a pressing iron. For weeks before that she'd laughed at him as the only person she knew who did not watch television. He did not tell her that he had had one from his parents' time till recently, only weeks before he met her again, when in a rage at Motu's disappearance he smashed it to smithereens. When, later, he realized what he had done, he took it to the neighborhood electronics repairman. After fiddling with it, the repairman told him, with much head shaking, that he should buy a new one. The cost of the bad part that needed replacement was the cost of a new set. He decided to let the TV remain with the repairman, in his small store along the busy highway, surrounded by ziggurats of electronics in different states of dysfunction.
Even beyond bringing the new things, Ndali ensured that his house stayed clean. She mopped the floor of the bathroom constantly, and when a frog leapt in through the drainpipe after a heavy downpour, she brought in a plumber to cover the mouth of the pipe with netting. She scrubbed the white tiles on the walls of the bathroom, which he had not cleaned in many months. She bought him new towels and hung them not on the top of the door— _Because that must be dusty_!—or on the bent nail on the interior of the door— _Because the nail was rusting and now stained them_ —but on a plastic hanger. As time passed, it seemed she improved something in his life on a daily basis, and even Elochukwu, to whom he now gave little of his attention, attested continually to the enormous change in his life.
Although my host appreciated these things, he did not give deep thoughts to them until the end of the three months, when Ndali traveled with her parents to Britain, the land of the White Man. The reason for this is that people do not see clearly what is positioned before them until they regard it from a distance. A man may hate another because of an offense, but after a significant passage of time, his heart begins to warm towards that individual. It is why the wise fathers say that one hears the message of an _udu_ drum clearer from a distance. I have seen it many times. It was thus in her absence that my host saw all she'd done for him more clearly. It was during this time also that all the things she'd told him became more audible, and he noticed all that had changed in his life and how the past before her coming now seemed like a different age from the present. And it was during those days alone, thinking about these things, that the desire came to him with all the bolting powers of a persuasion that he wanted to marry Ndali. He rose to his feet and shouted, "I want to marry you, Ndali!"
Ijango-ijango, I cannot describe the joy I saw in my host that evening. No poetry, no language can fully describe it. I had seen, long before his uncle came and asked him to find a wife, that he had been seeking this—since the day his mother died. I, his chi, was in full support. I had seen this woman, approved of her care for him, and even received the testimony of her chi that she loved him. And I was convinced that a wife would restore the peace he had lost since his mother's death because the early fathers, in their most gracious wisdom, say that once a man builds a house and a compound, even the spirits expect him to get a wife.
Two days after he made this decision, Ndali returned to Nigeria. She called him once she arrived in Abuja with her family, whispering into the phone. As she was speaking, he heard the sound of a door opening somewhere in the house where she was, and the call ended that instant. He was picking eggs and replacing the floor of the main hennery with sawdust when she called. When she reached Umuahia later that day, it happened again. This time he'd just completed a meal at a restaurant he supplied eggs and chicken meat to, a place where he ate every once in a while. They'd started to talk when she ended the call abruptly at the sound of a door opening.
My host put down the phone and washed his hands in the plastic bowl he'd filled with the bones from the bonga fish served with egusi soup. He paid the daughter of the caterer, whose habit of wearing a scarf that folded into the shape of a bird's tail often reminded him of Motu. He picked up a toothpick poking out of a plastic vase and walked into the sun. He waved down a peddler, who went about carrying water in small sealed bags, hawking his wares: "Buy Pure Water, buy Pure Water!" Agujiegbe, this buying and the selling of water has always amazed me. The old fathers would never have imagined, even in the time of drought, that water—the most abundant provision of the great earth goddess herself—could be sold the same way hunters sell porcupines! He bought one Pure Water and was tucking the ten-naira change into his pocket when his phone began to ring again. He removed it from his pocket, considered flipping it open to answer the call, but put it back in. He spat the toothpick away and bit open the bag and drank till the bag was empty, then he threw it into the nearby brush.
My host was angry. But anger, in a situation like this, often becomes a multiparous cat who bears litters of offspring, and it had already birthed jealousy and doubt in him. For as he walked back to his van, he kept wondering why he was giving himself to a woman who did not seem to care about him. I flashed the thought in his mind that there was no need to be annoyed with her and suggested that he wait till he'd heard her explanation and the full story.
He did not respond to my suggestion, but simply entered his van and drove past the large pillar along Bende Road, which bore the name of the town, still raging. He came by a rough intersection where a three-wheeled vehicle wedged itself between his van and another car and would have been run into had he not pulled the brakes. The driver of the small vehicle cursed my host as he pulled to the shoulder of the road.
"Devil!" he shouted at the man. "This is how you people die. You dey drive ordinary keke napep, but you dey do laik say na gwongworo you dey drive!"
His phone began ringing as he spoke, but he did not reach for it. He drove past the Mater Dei Cathedral, where he had not been for a long time, and, cutting through a small street, arrived at his farm. He turned off the engine, took the phone, and dialed her number.
"What are you doing?" she shouted into the phone. "What?"
"I don't...," he said, and breathed hard into the phone. "I don't want talk to you by phone."
"No, you must. What did I do to you?"
He wiped the sweat from his forehead and wound down the window.
"I was annoyed that you did it again."
"What did I do again, er, Nonso?"
"You are ashamed of me. You did not phone because another person was coming into the room." He could catch his voice rising, starting to turn loud and vehement, a tone she often complained about as harsh. But he could not stop himself. "Tell me, er, who opened that door that time you ended my call?"
"Nonso—"
"Answer me."
"Okay, my mother."
"Er-er, you see? You are denying me? You don't want your family to know about me. You don't want them to know that I am your guy. You see, you are denying me in front of your people, Ndali."
She tried to speak, but he pushed on, forcing her into silence. Now he waited for her to speak again, worried even more, not just because of what he'd betrayed in his tone but also because he'd referred to her by her name, something he did only when angry with her.
"Are you there?" he said.
"Yes," she said after a pause.
"Then, talk naw."
"Where are you now?" she said.
"My place."
"Then, I'm coming there now."
He dropped the phone into his pocket, and a silent joy rose within him. It was obvious that she had not planned to come meet him until after a few days, but he wanted her to come as soon as possible. For he missed her, and it was partly this that angered him. He'd also been annoyed by the anxiety that planted itself in him while she was away and became even more persistent after he developed the idea to marry her. As often happened to him—and to most of mankind—a questionable idea had formed in his mind with the power of a persuasion. At first people believe these ideas, but after a while their gaze becomes sharper and more penetrating so that they begin to see all the deformities of their plans. This was why, hours later, he became aware—as if it had been concealed from him all this while—that he was not rich, not particularly good looking, and not educated beyond secondary school. She, by contrast, was on the cusp of completing university and becoming a doctor (even though, Egbunu, she had told him many times that she was going to be a pharmacist, not a doctor). He needed her to come, to reassure him in some way again that he'd been wrong, that he was not beneath her but in fact her equal. And that she loved him. Although she did not know it, this was what she had done for him by agreeing to come.
He stepped out of the van and walked into the small farm, stopping halfway between the rows of the growing tomato plants to observe the ears of corn on the other side. Perhaps seeing him, a rabbit emerged and began hopping away into the cornfield in rapid, prodigious leaps, its tail swinging as it went. It would go a few steps, then stop, raise its head and glance about, and then run on again. He spotted a singlet—perhaps blown there by the wind from some compound—lying over one of the corn plants, bending it. He took up the singlet. It was covered with dirt, and on it was a black reticulated millipede. He shook off the millipede and was headed to dispose of the singlet in the dump behind the brick fence when Ndali arrived.
EZEUWA, the wise fathers in their cautionary wisdom say that whichever position the dancer takes, the flute will accompany him there. My host that evening had received what he wanted: that she come to him. But he had achieved it by protest and dictated the tune of the flutist. So when he went into the house, she was on her feet, her fingers splayed over her wearied face. She turned away once he came in, and with her eyes cast downward, she said, "I have come not to argue but to talk in a calm way, Nonso."
Fearing that what she said might require him to focus on her for a long time, he asked to feed his flock first. He hurried out into the yard, wanting to return to her quickly. He opened the coop door, made of wood and netting. The chickens poured out, calling enthusiastically. They made a sprint with expectation towards the foot of the mango tree, where he'd spread sacks but had not poured feed. Once they began to peck, piping, he walked back into the house and put the main door against a wedge, so that only the net door was closed. He scooped up one last big cup of millet and tied up the nearly empty bag which he kept in one of the cupboards in the kitchen to prevent the birds from devouring it. He returned to the yard and poured the feed on the sack at the foot of the tree. At once the sacks were covered by a gathering of hungry birds.
When he returned to the parlor, Ndali was seated and was looking at the camera she had brought from the White Man's country, which she called a "Polaroid camera." Her handbag was still by her side, and her shoes, which she referred to simply as "heels," were still on, as if she was poised to leave soon. Egbunu, while one can often tell the state of a person's mind from the expression on their face, it is now difficult to tell with the daughters of the great mothers. This is because they now adorn themselves in ways unlike the mothers. They have shunned _uli,_ the elaborate braiding, the wearing of beads and cowries. And now, a woman can cover her face with colors of all kinds all by herself, with a single paintbrush, and one in misery can wear so much coloring on her face that she might even look happy. And this was how Ndali appeared that day.
"So tell me," she said once my host sat down. "You want to meet my family?"
He'd settled into the weakest of the sofas so that his body sank low and he could barely see her complete figure, even though he was right in front of her.
Conscious of the anger in her voice, he said, "That is so, if we want to marry—"
"So you want to marry me, Nonso?"
"That is so, Mommy."
She'd closed her eyes as he spoke, and now she opened them, and they appeared reddish. She adjusted in the sofa so that her legs came forward towards him. "You mean it?"
He gazed up at her. "That is so."
"Then you will meet my family. If you say you want to marry me."
Egbunu, she said this as if it was a painful thing to say. And one could see then, without needing a diviner's eye, that there lurked in her heart something heavy which, concealed in the veiled compartment of her mind, she would not reveal. My host saw this, too, and this was why he drew her to sit by him on the big sofa and asked why she did not want him to meet her people. To this question she pulled herself from him and turned her face away. Then he saw that she was afraid. He saw it even though she had turned away from him, and all he could see were the big earrings that drooped down almost to her shoulders, forming a ring big enough for two of his fingers to slide through. For fear is one of the emotions that clothes the primal nudity of a person's face, and wherever it presents itself, every percipient eye can recognize it, no matter how adorned the face.
"Why is it making you sad, Mommy?"
"It is not making me sad," she said almost before he finished speaking.
"Why, then you fear?"
"Because it will not be good."
"Why? Why can't I even know my girlfriend's family?"
She looked at him, her eyes firm against his own, unblinking. Then she turned away again. "You will meet them. I promise you that. But I know my parents. And my brother. I know them." She shook her head again. "They are proud people. It will not be good. But you will meet them."
Confounded by what he heard, he did not speak. He wished to know more, but he was not one who asked too many questions.
"When I go home, I will tell them about you." She tapped her feet in an undecipherable act of discomfort. "This night, I will tell them this very night. Then we see when I can take you home."
Once she said this, as if she had relieved herself of a great burden, she leaned back into the couch towards him and drew a deep breath. But her words remained in his head. For words as strong as the ones she'd spoken—"It will not be good"; "So you want to marry me?"; "You will meet them. I promise you that"; "Then we see when I can take you home"—are not easily dispelled from the mind. They have to be broken down slowly, over time. He was digesting them when he heard a distinct sound from the backyard that startled him.
He leapt to his feet and was in the kitchen in the blink of an eye. He grabbed his catapult from the windowsill and opened the net door. But it was too late. The hawk had mounted its thermal by the time he got to the yard, flapping its wings violently against the updraft, with one of the yellow-white chicks clasped in its talons. Its wings hit the laundry rope as it lifted, rattling the rope so that two of the pieces of clothing he'd hung on it fell to the ground. He slung a stone at it, but it fell far away from the bird. He'd put another stone in the catapult when he saw that it was no use. The hawk had glided into an unreachable thermal and had started to gain momentum, its eyes no longer glancing downward but ahead, into the colorless immensity of the sky.
Chukwu, the hawk—he is a dangerous bird, as lethal as the leopard. He craves nothing but flesh and he spends his life chasing it. He is an unspoken mystery amongst the birds of the sky. He is a soaring deity, borne on violent wings and merciless talons. The great fathers studied it and the kite, its close sibling, and made proverbs to explain its nature, one of which captures what had just happened to my host's chickens: Before every attack the hawk says to the hen, "Keep your chicks close to your bosoms, for my talons are soaked in blood."
My host was gazing at the fleeing hawk, full of rage, when Ndali opened the net door and entered the backyard.
"What happened? Why did you run out so fast?"
"A hawk," he said without looking back. He pointed in the distance, but the sun forced his eyes into a squint. He held up his hand to shade them as he stared in the direction the bird had gone. But the image of the attack was still so clear in his mind, still so vivid, that he struggled to believe it had concluded. There was nothing he could do now to save one of his flock from being torn apart and eaten up. The chickens he reared with his own hands and sweat—one had been taken away from him, again, _without a fight_.
He turned about and saw that the rest of his flock—with the exception of the one whose chick had been stolen—was cowered in the safety of the coop. The bereft hen pranced about with a stutter in its gait, cawing with what he knew was the avian language of anguish. He did not speak but pointed in the direction of the empty sky.
"I can't see anything." She cupped her eyes and turned to him again. "It stole a chicken?"
He nodded.
"Oh, my God!"
He turned his gaze to the evidence of the attack: the ground stained with blood and strewn with feathers.
"How many did it take? How did it—"
_"Ofu,"_ he said, then, reminding himself that he was talking with someone who preferred not to speak Igbo, he added, "Only one."
He set the catapult on the bench and followed the wailing hen around the yard. It eluded him at the first attempt to catch it. But he dashed forward with his two hands in front of him and clasped it by the wing closest to its left shoulder, then trapped it against the wall of the fence. Then he lifted it by its leg, gently feeling its spur. The hen went quiet, its tail upraised.
"How did it happen?" Ndali said as she picked up the fallen clothes.
"It just came—" He paused to stroke the hen's earlobe. "It just land on them and catch the small one of this mother hen, Ada. One of her new chicks."
He set Ada, the hen, back into the coop and closed the door slowly.
"Very sorry, Obim."
He dusted his hands by slapping them against each other and went into the house.
"Does it happen all the time?" she said when he returned to the sitting room from washing his hands in the bathroom.
"No, no oh, not all the time."
He wanted to leave his answer there, but Chukwu, I nudged him to unload what he bore in his mind. I knew him. I knew that one of the things that can heal the heart of a defeated man is the story of his past victory. It soothes the wound caused by the defeat and fills him with the possibility of a future victory. So I flashed the thought in his mind that hawks did not usually come here. I suggested he tell her it didn't always happen. And in a rare instance of compliance, he listened to me.
"No, it doesn't happen every time," he said. "It cannot always happen. _Mba nu!_ "
"Er," she said.
"I don't allow it. Not so long ago, in fact, one tried to attack my fowls," he said, surprised by his sudden slide into the corrupted form of the White Man's language. But it was in this language that he told her the story of his recent victory, and she listened, transfixed. Not too long ago, he began, he'd let out the flock, almost all the poultry except the occupants of one of the cages of broilers, and had started to peel yams into the sink in his kitchen, looking out every now and then, when he noticed a hawk hovering in the air above the flock. He opened his louvers, grabbed the catapult, and pulled a stone from the windowsill. He blew the stone with his mouth to clear off the red ants on it. Then he unhooked one of the louvers to allow his hand enough space and wound the handles so that the louvers stood in straight horizontal layers over each other. Then he waited for the bird to strike.
The hawk, he informed her, might be the most watchful of birds and can hover for hours on end, priming its target, making an effort to strike as precisely as possible—so one strike might be enough. So he, knowing this, waited for it, too. He did not take his eyes away from where it hovered for a second. It was why he caught it in that very moment when it made the daring plunge into the yard, picked up a small rooster, and attempted to mount the updraft. The missile knocked the raptor against the wall of the fence, making it drop the chick. The hawk slid down to the foot of the wall with a thud. It hoisted itself up, its head momentarily lost in its spread-out wings. It had been concussed.
He hastened out into the yard as the hawk tried to stand erect, then pinned it to the wall, unfazed by the violence of its beating wings and its riotous squawks. He dragged the bird by its wings to the cashew tree at the end of the compound, beside the refuse bin. He could not, he emphasized, describe the anger he'd felt. It was with this great anger that he bound the hawk by its wings, its head blood soaking the strong fibers of the twine. As he tied the bird to the tree, he spoke to it and all of its kind—all who stole what people like him reared with their sweat, time, and money. He walked into the house and returned with a few nails, sweat bleeding down his back and neck. As he stepped back into the yard, the hawk called with a strange fury, its voice piercing and ugly. He picked up a big stone from behind the tree and held up the bird's neck against the tree. Then he struck the nail into its throat with the stone until the nail burst out on the other end, spitting splinters and unbuckling a coat of old bark from the tree. He spread one wing, his hand and the stone now covered in the hawk's blood, and drove it, too, deep into the flesh of the tree. Although he saw that he'd done something extremely violent and unusual, so overwhelmed with rage was he that he was determined to complete what his mind had conjured up as the deserving punishment for the bird: a crucifixion. Thus he put together the feathery legs of the dead bird and nailed them to the tree. And it was finished.
He sat back into the chair now that he'd finished the story, entranced in his own vision. Although he'd been looking at her all the while, it seemed that he had just seen her for the first time since he began the tale. He became conscious of the weight of what he had told her. And he feared, now, that she must think of him as a violent man. In haste, he looked up at her, but he could not tell what she was thinking.
"I am amazed, Nonso," she said suddenly.
"By what?" he said, his heart quickening.
"The story."
Is that it? he wondered. Is that the way she would view him from now on? An irredeemably violent man who crucifies birds? "Why?" he said instead.
"I don't know. But—in fact—I don't know. Maybe the way you tell it to me. But—I just see you, just a man who loves his fowls so much. So very much."
Ebubedike, my host's thoughts swirled at this. _Love,_ he thought. How could love be what she thinks about at this given moment after he'd just exposed himself as capable of such senseless brutality?
"You love them," she said again, now with her eyes closed. "If you didn't love them, you would not have acted this way in the story you just told me. And today also. You really love them, Nonso."
He nodded without knowing why.
"I think you are really a good shepherd."
He looked up at her and said, "What?"
"I called you a shepherd."
"What is that?"
"It is one who keeps sheep. Do you remember from the Bible?"
He was somewhat perplexed by what she'd said, for he had not given it much thought, just as men do not often give deep thoughts to the things they do every day, things that are routine to them. He hadn't considered that he had been broken by the world. The birds were the hearth on which his heart had been burned, and—at the same time—they were the ashes that gathered after the wood was burnt. He loved them, even if they were varied while he was one and simple. Yet, like everyone who loves, he wished that it be requited. And because he could not tell even if his singular gosling once loved him or not, in time his love became a deformed thing—a thing neither he nor I, his chi, could understand.
"But I keep fowls, not sheep," he said.
"It doesn't matter, so far you keep birds."
He shook his head.
"It is very true," she said, drawing close to him now. "You are a shepherd of birds, and you love your flock. You care for them the way Jesus cares for his sheep with so much love."
Although what she said had puzzled him, he said, "That is so, Mommy."
Agbatta-Alumalu, my host was so confounded by the things Ndali said that day that even long after they'd finished making love, eaten rice and stew, and made love again, he sat in the bed listening to the sound of crickets in the farm and barnyard while Ndali drifted off to sleep. His mind lay solidly on the cryptic things she had said about her family as if caught there, like a bird on birdlime. He was peering directly at the wall across from him, gazing at nothing in particular, when he was startled by her voice.
"Why are you not sleeping, Nonso?"
He faced her and slid downward into the bed.
"I will, Mommy. Why did you wake?"
She shifted, and he saw the silhouettes of her breasts in the darkness.
"I don't know oh, I just wake like that. I did not sleep deep before oh," she said, with the same weak voice. "Er-he, Nonso, I have been wondering all day: what is the sound that the chickens were making after the hawk took the small one? It was like they all gathered—er, together." She coughed, and he heard the sound of phlegm within her throat. "It was like they were all saying the same thing, the same sound." He started to speak, but she spoke on. "It was strange. Did you notice it, Obim?"
"Yes, Mommy," he said.
"Tell me, what is it? Is it crying? Are they crying?"
He inhaled. It was hard for him to talk about this phenomenon because it often moved him. For it was one of the things that he cherished about the domestic birds—their fragility, how they relied chiefly on him for their protection, sustenance, and everything. In this they were unlike the wild birds.
"It is true, Mommy, it is cry," he said.
"Really?"
"That is so, Mommy."
"Oh, God, Nonso! No wonder! Because of the small one—"
"That is so."
"That the hawk took?"
"That is so, Mommy."
"That is very sad, Nonso," she said after a moment's quiet. "But how did you know they were crying?"
"My father told me. He was always saying it is like a burial song for the one that has gone. He called it _Egwu umu-obere-ihe_. You understand? I don't know _umu-obere-ihe_ in English."
"Little things," she said. "No, minorities."
"Yes, yes, that is so. That is the translation my father said. That's how he said it in English: minorities. He was always saying it is like their 'okestra.'"
"Orchestra," she said. "O-r-c-h-e-s-t-r-a."
"That is so, that is how he pronounced it, Mommy. He was always saying the chickens know that is all they can do: crying and making the sound ukuuukuu! Ukuuukuu!"
Later, after she drifted back to sleep, he lay back beside her, thinking about the hawk attack and her observation about the fowls. Then, as the night prospered and his thoughts returned to the things she'd said about her family, the fear slithered in again, this time wearing the facial mask of a sinister spirit.
IJANGO-IJANGO, the ndiichie say that if a wall does not bear a hole in it, lizards cannot enter a house. Even if a man is troubled, if he does not become broken, he can sustain himself. Although my host's peace had been meddled with, he went about his business with serenity. He delivered twenty-nine eggs to the restaurant down the street and drove to Enugu to sell seven of the chicks and to buy a few more brown hens and six bags of feed. He had only bought a bag of mash when he came by a man playing _uja,_ the flute of the spirits. The flutist trailed behind another man whose torso was painted with _nzu_ and _uli_ and camwood, and who clenched a strand of young palm leaf between his teeth. Behind the two men was a masquerade. A group of people were gathered as the _iru-nmuo,_ wearing an antlered mask that bore the scarification of a slit-eyed ichie, danced to the sound of ancient flute music attended by a rattling twin gong. As you know, Egbunu, when one encounters an ancestral spirit—the corporeal manifestation of one or more of the great fathers—one cannot resist. Gaganaogwu, I could not hold back! For I had lived in the days of the great fathers when masquerades were a frequent sight. I could not hold back the temptation to listen to the mystical tune of the _uja,_ the flute crafted by the best among the people who live on the earth. I shot out of my host into a frantic crowd of spirits of all kinds and climes which were gathered around this area, making deafening noises, their feet ticking on the soft grounds of Ezinmuo. But what surprised me even more was what I saw above the other part of the bustling market. A group of small human-shaped spirits—of children killed at childbirth or conception or of twins killed long ago—stood playing at an elevation of about four hundred meters, the distance at which _ekili,_ the mystical transport system of astral projection and bird flight, are rendered possible. This group of spirits was held above the human crowd by a force beyond the knowledge of man (except dibias and the initiated), so that it seemed as though they were on the ground. They were stamping their feet, leaping, and snapping their fingers as they played the ancient game of _okwe-ala_. Their laughter was loud and cheerful, ringed with the hollow thread of the ancient language long lost among men. Chukwu, although I have witnessed things like this before, I was again mystified by the fact that, despite the dozen or so childish spirits playing, a market went on undisrupted below them. The market continued to teem with women haggling, people driving in cars, a masquerade swinging through the place to the music of an _uja_ and the sound of an _ekwe_. None of them was aware of what was above them, and those above paid no heed to those below, either.
I had been so carried away by the frolicking spirits that the masquerade and its entourage were gone by the time I returned to my host. Because of the fluidity of time in the spirit realm, what may seem like a long time to man is in fact the snap of a finger. This was why, by the time I was back into him, he was already in his van driving back to Umuahia. Because of this distraction, I was unable to bear witness to everything my host did at the market, and for this I plead your forgiveness, Obasidinelu.
A short distance from Umuahia, my host received a message from Ndali that she would come that night briefly because she was preparing for a test the following day. When she came that night, wearing her lab coat, he was watching _Who Wants to Be a Millionaire,_ a TV show she loved and had introduced him to.
She removed the coat and revealed a green shirt and jean trousers that gave her the appearance of a teenager.
"I am just coming from the lab," she said. "Please off the TV, we need to talk about coming to my family house tomorrow."
"The TV?" he said.
"Yes, off it!"
"Oh? No vex, Mommy."
He rose slowly to turn it off, but stopped at the intensification of a peculiar sound. He stood watching again.
"In fact let us go to the backyard, it is stuffy here," she said.
He followed her to the yard, the air thick with the smell of the poultry. They sat on the bench, and she was about to begin speaking when she saw a long black bit of plumage stuck to the wall as if glued to it. "Look, Nonso!" she said, and he saw it, too. He picked the feather off the wall and sniffed it.
"It is from that stupid hawk," he said, shaking his head.
"Ah, how did it hang there like that?"
"I don't know." He crumpled it and flung it over the fence, his anger erupting at the memory of the previous day.
She drew a deep breath, and, pushing herself forward, she spoke as if she'd been thinking about every word, and every one of them had been measured and planned for so long.
"Chinonso Solomon Olisa, you have been a great person, a godsend to me. Look at me, I have been through hell. You met me in the worst place. You met me, I was on the bridge. I was on that bridge because—because what?—because I was tired of the bad treatment. Because I was tired of being cheated and lied to. But God! He sent you into my life at the very appointed time. Look at me now." She splayed her hands open for him to see. "Look at me, look at how I have been transformed. If anyone told me or even my mum that her daughter would be working in a poultry, touching agric fowl, who would believe it? Nobody. Nonso, you don't even know who I am or where I am from."
She seemed to smile, but he could tell that it wasn't a smile. It was something her face had done to help her conceal the difficult emotion that was welling within her.
"So what am I saying? Why am I talking like this? I am saying that my family—my mother and father, and even my brother—may not accept you. I know it is hard to understand, Nonso, but look, my dad is a chief. _Onye Nze_. They will say I am not suited for a farmer. It is just that, they will say that..."
Egbunu, my host listened as she said the same thing over and over again to try to neutralize its effect. He was shaken by the things she'd said, for he'd been afraid of these things. He'd seen the signs. He saw it on the day at the watch store on Finbarr's Street, when she'd told him that she was born overseas, "in the UK." Her parents and her older brother had been schooled there, and it was she alone who had chosen to do her schooling in Nigeria. "But," she had added, "I will do my master's abroad." He remembered another time. They were driving past the old part of the city, tearing through the storm wind that massed against it, when she asked if he had gone to university. He'd been struck by it, and his heart had begun to beat rapidly. "No," he'd said, as if with a dead tongue. But Ndali had simply said, "Oh, I see." He remembered how, afterwards, she'd pointed past a group of multistory buildings lined shoulder to shoulder on the side of the road around Aguiyi Ironsi Layout, a tall new solar-powered streetlight sticking up above one of them, and said, "We live somewhere among those buildings."
"I am not trying to make you afraid," she said presently. "Nobody can decide who I want to marry. I decide for myself. And I am no longer a child."
He nodded.
"Obim, _igho ta go_?" she said, her head tilted sideways, her face anchored in the valley between smiling and crying.
"I understand, Mommy," he said in the language of the White Man, surprised by her switch to the language of the old fathers. Although he'd heard her speak it on the phone with her parents, she hardly spoke it with him. She'd said she did not like to speak it except with her parents because, having lived abroad for a few years, she did not think she was fluent in it.
_"Da'alu,"_ she said, and kissed him on the cheek. She rose and went into the kitchen.
Later, while they ate, she said, "Nonso, you truly love me?" He was starting to answer when she said, "That must be why you would want to marry me?" He murmured something that dissolved away in an instant because she quickly added, "It must be because you love me."
He waited for a moment before saying, "It is." He expected her to say more, but she went to the kitchen to wash the plates, carrying the single kerosene lamp in the house. It crossed his mind to put on the rechargeable lantern, but he remained seated instead, contemplating all she had said when she came back into the sitting room.
"Nonso, I ask again, do you love me?"
In the near darkness, although he wasn't looking at her, he could tell that she'd closed her eyes as she waited for his answer. She often closed her eyes whenever she expected a response to a question, as if afraid that what he might say might hurt her. Then, after he had spoken, she would try to slowly take in what had been said.
"You say yes, Nonso, but is it true?"
"It is so, Mommy."
She returned to the room with the lantern, set it on a stool beside her, and turned it low so that their shadows sketched by the incipient darkness swelled.
"So you truly love me?"
"It is so, Mommy."
"Chinonso, you always say you love me. But do you know that you need to really love someone before you marry the person? Do you know the meaning of love?" He was starting to speak. "No, just tell me, first, do you know what love is?"
"I do, Mommy."
"Is that truth? Really, is that the truth?"
"It is so, Mommy."
"Then, Nonso, what is love?"
"I know. I can feel it," he said. He opened his mouth to proceed, but he said, "Eck," and then fell silent again. For he feared that he could not answer her correctly.
"Nonso? Do you hear?"
"Yes, I feel love, but I cannot lie that I know everything concerning it, every single thing."
"No, no, Nonso. You said you love me, so you must know what love is. You must know what." She sighed and let out a tsk. "You must know, Nonso."
Gaganaogwu, my host was troubled by this. Although I, like every good chi, often allow my host to make use of the talent I have chosen for him from the hall of talents by interfering minimally in his decision makings, I wanted to interfere here. But I was stopped by what he resorted to: the effective tool of silence. For I have come to know that when the peace of the human mind is threatened, it often answers with benign silence at first, as if stunned by a withering blow whose impact it must allow to dissipate. And when this dissipation had been completed, he mumbled, "Okay."
He leaned back into the chair and recalled what she had told him about one of her friends who laughed at a man who told her friend he loved her after just meeting her for the first time. He'd wondered at the time why she and Lydia, the friend, had thought it completely ludicrous and that it deserved mockery. This reminded him of when Miss J laughed at him when he said he loved her. At the time, he had been surprised, as he was now. He looked up at her silhouette, and it struck him for the first time that he'd not properly weighed what it would entail to be married. She would have to move in with him into the compound. She would ride with him in his van to deliver eggs to the bakery on Finbarr's Street and meat to the restaurants where he delivered live chickens every now and then. All that had come to belong to him would now also belong to her—everything. Did he hear himself say it correctly? Everything! And if, in time, he plants his seed in her, the child that will be born—even that child will belong to both of them! Her possessions, her car—he would benefit from her studies in the university, her family, her heart, and all that was her, hers, and all that would be hers would all be his, too. This was what marriage comprised.
In light of this new understanding, he said, "Actually, I don't, I can say—"
She must have said the "Okay" after she opened her eyes. "But you...," she started to say, but fell silent.
"What? What?" he said in a frantic effort to prevent her from holding back that which she had prepared for release, for she often did this: pause on the verge of saying something, then draw back and seal it up again in the jar of thought, to be released later, and sometimes never.
"Don't worry," she said almost in a whisper. "You will come to my house next Sunday, then. And you will meet my family."
Oseburuwa, you know that a chi is a font of memory—a moving accretion of the many cycles of existences. Each event, every detail stands like a tree staked into the bright darkness of its eternity. Yet it does not remember every event, only those which impact its host in memorable ways. I must tell you that my host's decision that night is one I will always remember. At first he'd waited for her to say these words he dreaded, that "it will not be good." But she did not speak. So, in a faltering tone, he said, "That is so, Mommy. I will meet your family next week Sunday."
#
## "August Visitor"
OBASIDINELU, you have sent me to live on the earth with humans in many cycles of existence, and I have seen many things, and I'm wise in the ways of humanity. Yet I do not fully understand the human heart. Every person lives as if oscillating between two realms, unable to anchor his foot in either. This is a strange thing. Let us consider, for instance, the intercourse between fear and anxiety. Fear exists because of the presence of anxiety and anxiety because humans cannot see the future. For if only a man could see the future, he would be more at peace. For one who plans to travel in a coming day may say to his companion, "If we go to Aba tomorrow, we will encounter robbers on the highway and we will be robbed of this car and all our possessions." To which the other may say, "Surely we will not go to Aba tomorrow."
Or suppose there is a young woman about to be married. If she could see the future, she could say to her father on the eve of the wedding, "My father, I don't mean to disappoint our entire clan and soil our name. But I have come to find that if I marry this man, he will beat me every day and will treat me worse than a dog." Can you imagine what fear this would create in her beloved father if he believes what she has seen is true? The father would snap his fingers over his head and cry, " _Tufia! Ya buru ogwu ye ere kwa la!_ Anyone who has prepared such a spell, may it come to naught! You must leave the man at once, my daughter. Where is the bride price he has paid? Where is the young goat? Where are the three tubers of yam? Where is the bottle of schnapps and the crate of mineral? Return them all at once! God forbid that my daughter should marry such a man!" But Chukwu, they will not do such a thing because none of them can see the future. So without knowing it, the men of trade will embark on their journey on the planned day and get robbed and killed. The young woman will marry the man who will treat her worse than a slave.
I have seen it many times.
It is in this same way that my host drove his van to Ndali's house that Sunday not knowing what was in store. Unable to induce the day to arrive earlier and unable to stop it from coming, he'd waited anxiously for it. Time is not a living creature that can listen to pleas, nor is it a man who can delay. The day will come, as it has done since the beginning, and all that man can do is wait. Waiting in such a state of anxiety is tasking. Although one might feel a sense of peace while waiting, that peace is deceptive—the kind that could cause a man to think of roiling waters as calm.
He had not seen her in two days before that day, and he longed for her. He entered her street, trying to imagine what her family was like, what the house looked like. The electrical poles around the street were lower than they were in most other parts of Umuahia, and they seemed lined up close to each other, like laundry ropes. Small sparrows sat on a thick one that reared from the transmitter on the other side of the road, as if they were all in some agreement to stay on the cable. _Shepherd,_ he thought suddenly. Is that nobler? Shepherd of birds? Is that what he would call himself at the meeting? Would that make things good and make things go well?
He arrived to find their house looming over the road in grandeur and prominence. He accessed this secluded part of the street called "Layout" by serendipity. The road was well paved, and there was a sidewalk, beyond both sides of which were residential buildings. The house he was looking for, number 71, sat at the end of the layout, creating a dead end. Its walls were yellow, not quite as high as some of the others but rimmed around on the top with fine hoops of barbed wire. As if to demonstrate what might happen to a robber confident enough to attempt a break-in, a black polythene bag had been caught on the spike of one of the hooped wires. The morning wind pushed persistently at the bag, forcing it to cling to the wire by one of its handles, and its tumefied body to continue to wheeze at the pressure of the wind.
Oseburuwa, he did not know why he watched the bag for as long as he did—an object caught in something from which it could not get away no matter how much it tried. This intrigued him. He pulled up in front of the giant gate and turned off his engine. He looked at himself in the rearview mirror. He'd successfully cut his hair the previous afternoon. By the mirror, he fixed his tie, which was the color of the shirt he wore. He'd pressed it with the iron Ndali had bought him, a strange technique in which the surface of a hot object is pressed against a cloth. He sniffed the suit and questioned whether he should have worn it. He had washed it the day before and hung it on the laundry rope. He'd hoped to take it in shortly afterwards but had fallen asleep. Once I heard the rain, I rushed out of the yard, but there was nothing I could do. A chi cannot influence a host who is not in a conscious state. So I had watched, helpless, as rain poured down on his laundry until the drumming on the asbestos roof woke him. Instantly, I flashed the thought of the suit jacket into his mind, and he ran out but found the suit already soaked. He brought it into the house and hung it on a chair in the living room. Although it was dry by the time he put it on, it had acquired a rank smell. He removed the suit jacket and held it in his hand in case Ndali became concerned with why he did not wear it.
Before he turned on the engine again, he looked at the metallic structure attached to the gate. It was of Jisos Kraist bearing a piece of wood with two outstretched arms. He was gazing at this when the small gate attached to the big one opened. A man stepped out of it wearing a uniform of faded blue and a black beret. The man's trousers were hiked unevenly—one to the knees and the other below the knee.
"Oga, what do you want?" the man said.
"I am a guest of Ndali."
"A guest, er," the man said, a weak frown on his face. The man ran his eyes over the van, ignoring his affirmative answer. "From where you know Madam from, Oga?" the man said in the language of the White Man.
"What?"
"I asked how you take know my Madam?" The man had come to the van, planted his two hands on top of it, and bent his head to peer in at its only occupant.
"I am her boyfriend. My name is Chinonso."
"Okay, sir," the man said. He detached from the van. "You be the man they are expecting?"
"Yes, na me."
"Ah, welcome, sir. Welcome."
The man hurried through the small opening in the gate, and he heard a rattling of metals and rods. One of the two big gates squeaked and swung open. Although he knew that Ndali's father was a titled chief and therefore rich, he did not expect that their wealth would be of this magnitude. He didn't at all expect to see the life-size sculpture of a menacing lion, one foot suspended in midair, the other ballasted into the floor of the fountain. From its wide-open eyes and mouth flowed a steady stream of water into a bowl of concrete. It took him a moment to recall that she'd said something about a figure whose photo her dad had taken during a trip to France and vowed to replicate in their mansion in Umuahia. He searched his mind carefully to see if he had been told about a basketball hoop. Had she mentioned the number of cars they had, or that their cars sat under a structure roofed with zinc? He could not remember. He counted: the black Jeep—1, the white Jeep—2, a car whose make he did not know—3, Ndali's Audi sedans—4, 5, 6. Oh, there's another shielded from view by the big wheels—7! Yet another, a Mercedes-Benz, beside which he'd parked his own vehicle—8. Looking carefully, he could see that was all. Eight cars.
He'd stepped out before he noticed that the gate man had been following him and had been standing beside his car waiting for him to come out.
"I can help you carry anything you bring inside, Oga."
He noticed then that he'd forgotten the gift he had brought. He stopped, turned and rushed back to the car. Even though the image of him dropping the bag with the wine on the bench in the yard stood like a banner in his mind, he searched the van, the backseats, the front like a madman.
Egbunu, I must say here that this was one of the occasions in which I had wanted to remind him that he was forgetting the gifts. But I didn't because of your counsel: _Let man be man_. The role of the chi is to attend to higher matters, things which, by virtue of their magnitude, can affect the host in major or significant ways. It must also attend to supernatural matters which man, in his limitation, cannot handle. But this omission, as I look back now on the things that would result from that visit, strikes me with pangs of regret, and I begin to wish that I had reminded him.
"Oga, Oga, hope no problem?" the gate man said repeatedly.
"No, no problem," he said with a slight trembling in his voice.
He thought for a moment if he should rush back home, but he recalled she'd begged him not to arrive late. The word flashed in his mind like fire: _punctuality_. He remembered her saying it: "My dad likes punctuality." I was relieved, Chukwu, as he hurried towards the house.
ECHETAOBIESIKE, the confidence he'd arrived with, like an egg in a calabash, was already broken by the time he sat at the table with the family. Ndali had met him at the door and told him, in frantic whispers, that he'd come late. "Fifteen minutes!" Then she reached to his back and removed something he'd not imagined could be there: a feather. Even I had not seen it. He almost wept as she crumpled the white feather into her palm, pointing him towards the dining room. "Is that all?" he said. In whispers she asked why he held the suit, and he raised it up towards her face, gesturing for her to smell it.
"Jesus!" she said. "Don't wear that smelling thing. _Nyamma_! Give it to me." She took it from him and folded it, then handed it back to him. "Keep it in your hand throughout, you hear me?"
The grandeur of the living room defeated him. Never had he dreamt that such lighting could exist. He didn't know that someone could have a sculpture of the Madonna inside a house. The marble on the flooring, and the design of the ceiling, these were beautiful beyond words. There were chandeliers and mantelpieces, articles I had seen in homes when my former host Yagazie was in Virginia, in that land of the brutal White Man. If the house struck him with such awe, it was without doubt that the people who owned it would do even more damage to his composure. So when he saw her father, the man appeared huge to him. The man's fair complexion was spotted with reddish blotches that reminded him of the musician Bright Chimezie. He found some comfort in her mother, for her face was the exact replica of Ndali's. But when her brother walked down the stairs, he wished he'd not come. He looked like black American musicians—with a nicely trimmed hair on the side of his face that traced down to his jaw and a broad, pink-lipped mouth held between a heavy mustache and beard. In response to his "Good afternoon, my brother," the man gave him only a grin.
They sat at the table as the maids served different foods on trays. With every passing moment, my host noticed one more thing that further damaged his confidence, so that by the time the food had all been served and they sat down to eat, he was already vanquished. When the first question came, he struggled to form words and dithered for such a long time that Ndali spoke in his stead.
"Nonso runs a poultry farm the size of this entire compound all by himself," she said. "He has a lot of chickens—agric fowl—and also sells them in the markets."
"Excuse me, gentleman," her father said again, as if she had not spoken at all. "What did you say you do?"
He made to speak, his voice starting to stutter, for he was truly afraid, then he stopped. He looked at Ndali, and she met his eyes.
"Daddy—"
"Let him answer the question," her father said, turning to his daughter with a countenance that did not conceal his anger. "I asked him, not you. He has a mouth, or not?"
He was worried by Ndali's confrontation with her father, and from under the table, he touched her with his leg to make her stop, but she pulled her leg away. In the small silence that descended, his voice broke out.
"I am a farmer, a poultry farmer. And I have land where I grow maize, pepper, tomatoes, and okro." He looked up at her—for he had come prepared to use a tool she had supplied him with—and said, "I am a shepherd of birds, sir."
Her father gave his wife what my host thought was a puzzled look that filled my host with the dread that he may have misspoken, and his feeling in this moment was like that of a man whose extremities were bound and was then thrust naked into the central arena of a village, with nothing to hide himself. Without intending to, he saw that he'd turned to her brother, on whose face he found the impression of muffled laughter. He became frantic. This thing Ndali had told him, how could it be wrong? She'd said it sounded fancier, and it did—to his ear, at least.
"I see," the father said. "So, gentleman, shepherd of birds, what education do you have?"
"Daddy—"
"No, Ndi, no!" her father said in a raised voice. A strained vein became visible on the side of his neck like a blow-induced swelling. "You must let him speak, or this meeting is over. You hear?"
"Yes, Daddy."
"Good. Now, gentleman, _ina anu okwu Igbo_?"
He nodded.
"Should I speak it, then?" the father said, and a piece of the chopped vegetable clung to his lower lip.
"No need, sir. Speak English."
"Good," the father said. "What is your level of education?"
"I have completed secondary school, sir."
"So," the father said as he gathered pieces of chicken flesh onto the prongs of his fork. "School cert."
"It is so, sir. Yes, sir."
The man gave his wife the look again. "Gentleman, I don't mean to embarrass you," the father said, letting his voice fall from the height to which he'd pitched it. "We are not in the business of embarrassing people; we are a Christian family." He pointed around to an étagère placed on the glass-covered bookcase on one side of the room, which held various paintings of Jisos Kraist and his disciples.
My host looked up at the étagère, nodded, and said, "Yes, sir—"
"But I have to ask this question."—"Yes, sir."—"Have you considered that my daughter here is a soon-to-be pharmacist?"—"Yes, sir."—"Have you considered that she is now completing her bachelor's in pharmacy and will so proceed to do her MPhil in the UK?"—"Yes, sir."—"Have you considered, young man, what kind of future you, an unschooled farmer, will have with her?"
"Daddy!"
"Ndali, quiet!" her father said. _"Mechie gi onu! Ina num? A si'm gi michie onu!"_
"Ndi, what is this?" her mother said. _"Iga ekwe ka daddy gi kwu okwu?"_
"I'm letting him speak, Mommy, but do you hear what he is saying?" Ndali said.
"Yes, but keep quiet, you hear?"
"I do," Ndali said with a sigh.
When her father began to speak again, the words came to my host again in a glut, each running into the other and the other.
"Young man, have you thought about it thoroughly?"—"Yes, sir."—"Deeply about what kind of life you will have with her?"—"Yes, sir."—"You have, I see."—"Yes, sir."—"And you think it is the right decision to marry such a woman who is so high above you, who wants to be her husband?"—"I know, sir."—"Then, you must go and think about it again. Go and see if truly you deserve my daughter."—"Yes, sir."—"That is all I will say to you."
"Yes, sir."
Her father rose in a slow, heavy manner, with his body pitching against the table, and left. Her mother followed moments later, shaking her head with a countenance my host would reckon later to have been pity towards him. She headed first to the kitchen, carrying empty plates stacked together. Ndali's brother, who had not said anything but had made his resentment known by laughing at every answer my host gave, rose shortly after his mother. He lingered for a while on his feet while he took a toothpick from the casing, stifling a laugh.
"You, too, Chuka?" Ndali said in a voice that dripped with sobs.
"What?" Chuka said. "Er-er, er-er, don't even call my name here, oh. Don't tell me even a single thing! Is it me that asked you to bring a poor farmer home?" He laughed, jerking. "Don't you mention my name again, there, oh."
With that, he, too, headed his father's way, on a flight of stairs, the toothpick held between his clenched teeth, whistling a tune as he went.
Ijango-ijango, my host sat there, rendered inutile by shame. He fixed his eyes on the plate of food in front of him, most of which he'd barely touched. From the upstairs, he heard Ndali's mother say to her husband in the language of the eminent fathers, "Dim, you were too harsh on that young man, er. You could have said these things in a way that didn't sound that harsh."
He looked up at his lover, who remained where she was, rubbing her right hand over her left. He knew that she was feeling a pain that was as deep as his. He wanted to comfort her, but he could not lift himself. For such is the state a man enters when he has been disgraced: inaction, numbness—as if he has been tranquilized. I have seen it many times.
His eyes fell on the large painting of a man gently ascending into the sky over what seemed to be a village with the rest of the people of the village looking up in his direction and pointing at him. Because his mind sometimes bore strange convictions, he did not know why he thought for a moment that this man who was levitating into the sky was my host himself.
It was with a mighty effort that he rose and touched Ndali on the shoulder and whispered in her ear to stop crying. He pulled her gently up, but she struggled, her tears mixing with saliva that slid slowly down her dress.
"Leave me, leave me," she said. "Leave me alone. What kind of family is this, er? What?"
"It is okay, Mommy," he said with his lips unmoved so that he wondered how the words had come out of him.
He rested his hands on her head and gently traced his fingers down her neck. Then he bent forward and, sinking his head towards her mouth, locked her in a kiss. Before they walked out of the house, he threw a gaze at the painting for one last time and noticed what had not occurred to him the first time: that the people in the lower end of the painting were cheering at this man who was ascending into the sky.
CHUKWU, I have seen firsthand what shame can do to a man. As it often does, it filled my host with an oppressive fear, the fear that he would lose Ndali like most of the things that had once been his. It grew in the days following, during which she made efforts to force her family to reconsider, but failed. Those days stretched into weeks, and it became clear by the third that nothing would change their minds. When Ndali came back from a quarrel with her parents, he resolved to change things himself and do something. Rain had fallen all morning, but by noon the sun had risen. She came directly from school in Uturu, filled with bitterness. He was out on the small farm when she drove onto the path flanked on both sides by farmland. He was at the farthest end of the farm, where his father had erected a fence which had partly crumbled under the heavy rainfall of the year the White Man refers to as 2003. Two feet from the fence was a long gutter that ran through the street, and beyond this was the long main road. As he watched her step out of the car towards the house, it occurred to him that she had not seen him. He dropped his hoe, and the head of the yam for which he had been digging a hole, and ran to the house.
He entered still wearing his dirty visor, a soiled shirt, trousers, and farm slippers covered with loam and other weeds he'd pruned off the land.
He found her with her face buried on her wrist, facing the wall.
"Mommy, _kedi ihe mere nu_?" he said, for in the moment of tension, he reached for the language to which he was much accustomed. "Why are you crying, why are you crying, Mommy? Er, what happened?"
She turned to embrace him, but he stepped away from her because of his farm clothes. She stopped an inch from him, her eyes deep red.
"Why are they doing this to me, er, Obim? Why?"
"What happened, er? Tell me what happened."
She told him about how her father had asked her if she was still seeing him and had threatened her. Her mother had intervened, saying the man was being too harsh, but her father continued unhindered.
"It is well," he said. "It will be well after everything."
"No, Nonso, no!" she said, slapping the wall with her palm again. "It will not be well. How can it be well? I am not going back to that house again. I am not. Over my dead body. What kind of family is this?"
His heart swirled within him at her rage. He did not know what to do. The old fathers, in their magnanimous wisdom, say that a person saves himself in the process of saving others. If she cannot be saved from a situation such as this, which has held them bound like invisible leashes, then he, too, cannot be saved. And it won't indeed be well. He watched her walk a few paces towards the door, stop, and put her hand on her chest. Then she turned back to face him. "I have—I have brought a few of my things, and I am staying here. I'm staying here."
She opened the door and stepped out of the house. He followed her out to the front porch and looked on as she opened the boot of her car and returned with a shiny Ghana-must-go bag. Then, from the backseat, she took out a pair of shoes and a nylon bag. He watched her with a certain joy, inwardly happy that he finally had a companion.
But for much of that week, her phone rang again and again, sometimes for extended periods. And each time, Ndali would look at its face and say to my host, "It is my dad," or "It is my mum." And every time he would beg her to answer the call, but she wouldn't. For she was strong-willed like most of the great mothers. She would merely hiss at my host's entreaty and turn her attention to something else, like one beyond reproach or beyond the fear of reproach. My host admired this in her. Whenever she did this, at such moments he'd think about a similar trait in his mother.
In the middle of the second week, her parents came to look for her at school and waited outside her classroom, but she ignored them and walked away with her classmate Lydia. After she told him this, he began to fear that she was starting to resent her family for his sake. Even though he increasingly sought to salvage the situation as days passed, he could not deny that her love for him seemed to grow stronger in those days. For it felt as though she'd culled her love away from everyone else and bequeathed it all to him. It was during this time that twice, while they made love, she wept. It was during these days that she baked him a cake, wrote him a poem, and sang for him. And once, while he was asleep, she unhooked the catapult from the wall and rushed out to the yard with it and scared away a prowling kite. And part of him sought to prolong those days, for although they were not yet married, it felt to him as though they were. He wished that he would take the center of her life, dwell around the boundaries, and seal up the limits. This woman, whom he'd always feared he could never have but who was now his, he could not afford to lose. Yet his fear of what she was doing grew alongside the blossoming of his affection for her, and her affection for him.
It was during this period that she traveled to Enugu with him. He'd woken early that memorable morning of life to find her dressed, in an ankara print gown and a calico head scarf, stirring tea in a cup while looking through the poultry record book on the table.
"Are you going somewhere, Mommy?"
"Good morning, dear."
"Good morning," he said.
"No, I'm going to Enugu too."
"What? Mommy—"
"I want to come, Nonso. I am not doing anything here. I want to know everything about you and about the fowls. I like it."
He was so taken aback that he struggled to find words. He looked on the dining table, and there he found one of the plastic crates, its dozen egg-holding cups nearly filled with eggs.
"Those ones are from the broilers?"
She nodded. "I collected them around six o'clock. They are even still laying more."
He smiled, for one of the things she loved the most about tending his poultry was collecting the eggs. She was fascinated by the phenomenon of egg laying, how rapidly it occurred in chickens.
"Mommy, okay, but Ogbete market is—"
"It is okay, Nonso. It is okay. I am not an egg. I have told you—I don't like you treating me like I'm an egg. I'm like you. I want to come."
His eyes fell hard on her face, and he saw in her eyes that she meant it. He nodded. "Okay, let me baff, then," he said, and rushed off to bathe.
Later, they dropped off the eggs at the restaurant down the street, and he promised to come for his pay on his way back from Enugu. As they drove on the highway, he could tell that he'd never felt such joy while traveling before. On the bridge over the Amatu River, she revealed to him how after the night he first found her on that bridge, she was still greatly heartbroken. She went to Lagos to stay with her uncle for two months, and while there, she frequently thought of him. And every time she did she would laugh at how strange he was. He in turn told her about how he'd returned to the river to look for the fowls but couldn't find them, and how he had been angry at himself.
"I was thinking the other day," she said, "how a man who loves fowls so much could do that. Why did you?"
He looked at her. "I don't know, Mommy."
Once he'd said those words, it struck him that he might know why she loved him: because he'd rescued her from something. And like his gosling, also taken under his care. This thought was enunciated so loudly in his mind that he looked at her to make sure she did not hear it. But her eyes were out the window, looking at the other side of the road, where the thick forest had given way to the scattered habitation of a village.
At the market in Enugu, he introduced her as his fiancée to a gleeful reception by his acquaintances. Ezekobia, a feed seller, gave them palm wine to drink, _the drink of the gods_. Some of them shook hands with him and embraced her. My host's face was lit with a flaming smile the entire time, for the blank wall of the future had suddenly become emblazoned with warm colors. It was almost full sun when they left the market, carrying the things they had bought.
He and Ndali purchased _ugba_ from a roadside hawker near the garage where they parked the vehicle. Ndali, soaked in sweat, bought a bottle of La Casera. She had him try it, and it tasted sweet, but he could not describe what it tasted like. She laughed at him.
"Bushman. It is apple taste. I'm sure you have never eaten apple before."
He shook his head. They had loaded a new cage, and two bags of broiler feed, and a half sack of millet, and were now sitting in the van, about to return to Umuahia.
"I am not Oyibo. I will eat my _ugba_ as correct African man."
He unwrapped the food and began putting handfuls into his mouth and chewing in a way that made her laugh.
"I have told you to stop chewing things like a goat: nyum-yum-yum. _Tufia_!" she said, snapping her fingers in the air, laughing.
But he ate on, bobbing his head and darting his tongue about in his mouth.
"Well, maybe one day we will go abroad together."
"Abroad? Why?"
"So you can see things, naw, and stop this bushmanliness."
"Ha, okay, Mommy."
He started the engine, and they hit the road. The van had just left the city when he began feeling uneasy. His stomach gave in to wild sensations, and he farted.
"Jesus! _Nyamma!_ " she cried. "Nonso?"
"Mommy, sorry, but I am—"
He was silenced by another release. He pulled hurriedly to the shoulder of the road.
"Mommy, my stomach," he gasped.
"What?"
"You have tissue, tissue paper?"
"Yes, yes." She reached for her bag, but before she could get the papers out, he grabbed a handkerchief from the pocket under the door handle of his side of the van and raced towards the bush. Chukwu, he nearly tore his trousers open once he was within concealable distance of the forest, and once they came off, his excreta slammed into the grass with an unaccustomed force. I was alarmed, for not since he was a boy had I seen this kind of thing happen to him.
When he rose up it was with some relief, his forehead wet as if he'd been in the rain. Ndali had come out of the van and was at the mouth of the bush, holding the half-used roll of tissue.
"What happened?"
"I wanted to shit badly," he said.
"God! Nonso?"
She burst out laughing again.
"Why are you laughing?"
She struggled to speak. "See your face—you are sweating."
They had barely driven for fifteen minutes when he rushed out again. He had the tissues this time, and with so much force did he defecate that his strength was expended. For sometime after he was done, he knelt down and clung to a tree. I had never seen anything like this happen to him. And even though I had learned to gaze into his viscera, I could not find exactly what was wrong with him, even though he was convinced he had diarrhea.
"I actually have diarrhea," he told Ndali when he returned to the van.
Ndali laughed even harder, and he joined in.
"It must be that _ugba._ I don't know what they put inside."
"Yes, you don't know." She laughed even more. "This is why I don't eat anyhow for anywhere. You are doing African man."
"I'm feeling like tired."
"Yes, drink water and my La Casera and rest. I will drive."
"You will drive my van?"
"Yes, why not?"
Astonished though he was, he let her drive, and for a long time after they resumed the journey, he did not feel the urge. But when it came, he pounded his hands on the dashboard, and when she pulled up, he flung himself out the door and tripped into the creepers. Then, picking himself up, he dashed into the bush as if unhinged. He returned to the van drenched in sweat, while she struggled to contain her laughter. He emptied the big Ragolis water bottle into his mouth and clung to the empty bottle. He told her a story his father had once told him about a man who had stopped like him to shit in a wild bush on the highway, and while at it, got swallowed by a python. His father used to play the song someone sang about it, "Eke a Tuwa lam ujo."
"I think I have heard the song before. But I'm afraid of all snakes—python, oh; cobra, oh; rattlesnake, oh; every snake."
"It is so, Mommy."
"How are you feeling now?"
"Fine," he said. He hadn't gone for nearly as long as it would take to break five full kola nuts in four places, and they were almost in Umuahia. "Almost thirty minutes now, I never go. I think it have stopped."
"Yes, I agree. But I have laughed all my energy out, too."
They drove past thick forests on both sides for some time in serenity, his mind splintered between thoughts. Then suddenly it came upon him with the force of a gust, and he dashed into the bush.
OSEBURUWA, she tended to my host until he was whole again. She went to the university the following day. When she returned, she joined him on the bench in the yard, plucking a sick fowl with bare hands so it could get some air into its skin. An old tray sat between them, full of feathers. He held on to one leg of the bird as he worked. She did this duty—the oddest thing she'd ever done in her life—with a curious mix of equanimity and laughter. While they worked, he forced himself to talk about his family, how he missed them, and the need for her to reconcile with hers. He spoke with great care, as if his tongue was a wet priest in the sanctuary of his mouth. Then she told him that her parents had come to the school to look for her that day again.
"Nonso, I don't want to see them. I just don't want."
"Have you thought of it very well? Do you know this is even making the situation worst now?"
She'd started to twist a feather from the leg of the bird when he said this. She drew back and sat on her legs on the raffia mat spread on the ground.
"How?"
"Because, Mommy, I am the one. It is because of me this is happening."
The hen raised its freed leg and dropped a blob of feces on the mat.
"Oh, my God!"
They laughed and laughed on until he released the hen and it hopped towards the coop, cackling plaintively. Egbunu, it might be the laughter that softened her heart, for when he explained afterwards that her action might make her family despise him the more, since it was for his sake this was happening, she sat in silence. And, later, as they lay in bed to sleep, she said, suddenly and over the rattling of the ceiling fan, that he had spoken the truth. She would return home.
Like a water-filled calabash sent off with an emissary to the land of a provoked enemy, she went home the following day, but returned three days later as a calabash smoldering with fire. Her father had sent out very many invitations for his upcoming sixtieth birthday party but had not invited him. Her father had said he was not qualified to be there. She left the house, her resolve firm that she would not return. She said this with feral rage, stamping her feet and shouting, "How, just how can he do this? How? And if they refuse to invite you," she said, "I swear to God who made me"—she tapped the tip of her tongue with her index finger—"er, I swear to God who made me, I will not attend myself. I will not."
He said nothing, busied with the soft burden she'd laid on him. He was seated at the dining table, where he'd sat picking dirt and pebbles out of a bowl of white beans. Bean weevils escaped from the unpacked beans, crouched on the table, or perched on the adjacent wall. When he finished picking the beans, he poured them into a pot and set it on the stove. He took up the flashy invitation card from the chair on which she'd dropped it and began reading the words to himself.
_This card is to invite mister and miss and your household to the birthday party of Chief. Doctor. Luke Okoli Obialor, the Nmalite 1 of Umuahia-Ibeku kingdom of Abia State of Nigeria. The event shall be at the Obialor compound on July 14, at Aguiyi Ironsi Layout..._
She had gone to his old bedroom, where the wall was defaced with his childhood drawings, mostly of the God of the White Man, his angels, his sister, and his gosling. She'd chosen this room as her study, where she read her books while in the house, and she slept with him in the bedroom that had once belonged to his parents. He read the invitation aloud from the sitting room so that she could hear him.
"Fourteen, at Lagos Street on the July 14, 2007. There will be food in abundance and music by His Excellency, the king of ogene music, Chief Oliver De Coque. The party will be held from four p.m. to nine p.m."
"It is my turn, me too I will not care."
"The emcee of the occasion will be none other than the inestimable Nkem Owoh, Osuofia himself."
"I don't care; I will not go."
"Come one, come all."
IJANGO-IJANGO, the early fathers, wise in the ways of humanity, used to say that the life of a man is anchored on a swivel. It can spin this way or that way, and a person's life can change in significant ways in an instant. In the batting of an eyelid, a world that has stood can lie prostrate, and that which was flat on the ground just a moment before can suddenly stand erect. I have seen it many times. I saw it again after my host returned from an errand one afternoon a few days later. He'd gone out not too long after they had lunch to supply four of the large cocks to the restaurant at the center of the city while Ndali studied. He'd become increasingly troubled by the gathering storms in his life, fearing again that something was watching him, looking for a time when he'd be happy enough to strike and steal his joy and replace it with sorrow. It was a fear that had lodged itself in his mind from the time his gosling died. This fear—as is common when it possesses a man's mind—convinced him with all the force of a persuasion that Ndali would be pressured to leave him eventually. And as much as I flashed thoughts in his mind continuously to contest it, it held firm. He went on fearing that, in time, she would give him up rather than lose her family. So biting was this fear that, as he drove back to the compound after the errand, he had to play Oliver De Coque's music on the van's cassette player to prevent himself from slipping into despair. Only one of the speakers was working, and sometimes, overwhelmed by the loud street noise, the music lapsed. It was those times, when Oliver's baritone voice quailed, that the weight on his mind came bearing down on him.
When he got home later, Ndali was sitting in the backyard, watching the fowls feed on the corn she'd spread on a sack, reading a textbook on the bench under the tree by the light of the rechargeable lamp. She had changed into a blouse and shorts that made her buttocks prominent. She had her hair oiled slick and now wore a bandanna over it. She stood up once she heard the net door opening.
"Guess, guess, guess, Obim?" she said.
She clasped her hand around him, almost stepping on one of the chickens, which fled frantically, its wings spread out, cawing.
"What?" my host said, surprised as much as I was.
"They said you can come." She pressed her hands around his neck. "My dad, they said you can come."
He had not expected this at all, and it was thus with relief and mild incomprehension that he bellowed, "Oh, that is good!"
"Will you come, Obim?"
He could not look at her, so he did not look at her. But she inched towards him, in slow steps, and took his jaw and lifted his face to face her. "Nonso, Nonso."
"Eh, Mommy?"
"I know what they did to you was not good. They disgraced you. But, you see, these things happen. This is Nigeria. This is Alaigbo. A poor man is a poor man. _Onye ogbenye,_ he is not respected in the society. And, again, my dad and brother? They are proud people. Even my mum, even though she does not support my dad very much in this."
He did not speak.
"They may be ashamed of you, but I am not. I cannot be ah—" She held his jaw and peered into his face. "Nonso, what is it? Why are you not saying anything?"
"Nothing, Mommy. I will come."
She hugged him. And in the silence, he heard the sound of the nocturnal insects emptying into the ear of the night.
"I will go with you to the party, for your sake," he said again. And as he spoke, he saw that she'd closed her eyes and would not open them until he'd finished speaking.
#
## The Disgraced
EGBUNU, the old fathers say that a mouse cannot run into an empty mousetrap in broad daylight unless it has been drawn to the trap by something it could not refuse. Egbunu, would a fish see an empty metal hook sticking out beneath water and cleave to it? How would it unless it is enticed by something on the hook? Isn't this similar to how a man is enticed into a situation he would not have liked to be in? My host, for instance, would not have agreed to attend Ndali's father's party had they not shown repentance and her father signed an invitation card with his name on it: "Mr. Chinonso Olisa." Although I will acknowledge that he was persuaded in part by the determination to make Ndali happy at all costs and by a desire to see Oliver De Coque perform live, he was cautious even to the end. He decided to attend the party with only a part of him attuned to it, merely dragging along the other intransigent half. And I, his chi, had been unable to decide whether he should go. I feared that from what I knew of man, a feeling such as that which they had shown him—repulsion—does not easily expire. But I had seen the healing and equilibrium that this woman had brought into his life and desired that it would continue. For it is abominable for a chi to stand in the way of his own host. When a man affirms a thing, and if his chi does not desire this thing, all it can do is persuade its host. But if its host refuses, then the chi must not attempt to compel its host against his will; it must affirm the thing. This is again why the wise fathers often say that if a man agrees to something, his chi must agree too. The second reason for my ambivalence was because I had developed a strong faith in Ndali's love for him, mostly after encountering her chi, and I strongly believed that if he married her, he would become complete, as the old fathers often say that a man is not complete until he marries a woman.
The day before the party, they went to buy greeting cards for her father at the big supermarket near the Oando filling station. At a roadside clothes store on Crowther Street, he bought an isiagu tunic. Although Ndali had pointed out that the ones with the black lion-head prints were better looking, he was drawn to the red ones for some reason he could not understand. They had emerged from the store and were walking towards a shopping complex, the speakers of a church blasting from an upstairs block on the complex, when he saw Motu in front of an open mechanic's workshop. She stood between a pile of motor tires and a mechanic who, dressed in blue overalls and large dark goggles, was firing a rod with something that gave off radiant, glittering sparks of red flame. She was dressed in a flowing gown, the green one with red leaf prints, which he'd removed a few times before making love to her. She had just finished selling groundnuts to one of the men, and she was bunching a piece of cloth into an _aju_ to place on her head before balancing the tray on it. It felt, Egbunu, that for an instant, he'd slipped from the hands of his present world like an oiled fish. He stood there, undecided as to what to do, wondering why she had left him. But Motu did not even turn. She put the tray on her head and walked in the other direction, towards a crowded market. He thought to call out to her but feared that she might not hear him over the great noise of the welding machine. His heart palpitating, he turned towards Ndali, who had continued to walk on without knowing he was no longer in tow. He had not realized that while watching Motu, he'd also focused on the fire from the mechanic's welder. And when he stirred away from it, his vision had become blurred, and for a moment it appeared as if the world and all that was in it had become covered in a thick, silky veil of yellow.
CHUKWU, Ndali did not return with him to his house that day. She went to help her parents prepare for the big next day. Aside from attending to a sick hen who had started to sprout a nacrelike substance on the sides of its beak, dabbing it with a clean towel soaked in warm water, he spent the rest of the day thinking of Motu. He wondered what had happened, whose hand it was that had stretched out and grabbed her away from his fold and stolen her away from him. He would have spoken to her, had he been alone. He thought long about why she left him, without warning, without provocation, when in fact it had seemed she loved him and he'd been firmly planted in her heart. Children of men beware: you cannot put your confidence in another man. No one is fixed beyond being blown sideways. No one! I have seen it many times. He was still deep in thought when his phone buzzed. He picked it up, tapped the message box, and read, They really want u to come Obim!!! even my brother. i luv u, gudnyt.
He arrived at her family's residence the following day to find that he was the first person there. Ndali came to meet him and asked him to follow her into the house. But he would hear none of it. He sat in a plastic chair under one of the two tarpaulin-covered pavilions erected for guests. Another pavilion stood separated from these two by a raised stage platform with a floor covered by a red rug. It was the High Table, where the party hosts and other dignitaries were to sit. There, the seats were arranged behind a long table near the stage, which was covered with embroidered cloth. A group of men soaked in sweat set speakers in place beside the table while two women dressed in identical blouses and skirts decorated big cakes with the molded effigy of Ndali's father holding a staff.
He picked up a copy of the program placed on his seat and was starting to read when he felt a chair behind him rattle. Before he could tell what it was, or even look back, a hand tapped his shoulder and a head bent towards his side.
"So you came," the head said.
Things had happened so quickly that he was incapacitated by the ghastly thrill of sudden fright.
"You came after all," the man, whom he now recognized to be Chuka, repeated. Chuka spoke the language of the White Man with a foreign accent similar to Ndali's. "Some people, some people, they just have no shame. No shame. How can you—after all popman did to you that day—come here?"
Chuka placed his arm on my host's shoulder and pulled him closer. I heard the voice of my host's head shout, _Was he not close enough_? A sound in the upward direction in the distance caused him to raise his head and catch the sight of Ndali on what must have been the balcony of her room.
"Wave at her, tell her you are okay," Chuka said. "Wave at her!"
She was saying something he could not hear but which I made out to be a question as to whether or not he was fine. He obeyed the order he'd been given, and she waved back and blew a kiss into the air. He had thought her brother was concealing himself behind him, but now Chuka shouted, "Your man and I are having a sweet talk!"
At that my host thought he saw something like a smile flash across his lover's face, an unmistakable sign that she believed her brother.
"Good. Thank you, Chuka," she shouted back.
Chuka had spoken the language of the White Man to his sister, but now he continued his onslaught in the language of the fathers: "I bu Otobo; otobo ki ibu. _Real, real_ otobo. How should one position his neck or shape his mouth to pass a message into the head of an otobo like you? How? It baffles me." He squeezed my host's shoulder so hard that he squirmed.
"Now, listen, _Church Rat,_ my father said I should tell you that if we hear 'phim' from you, or any noise at all, you will be in serious trouble. Do you know you are playing with fire? You are cuddling a consuming fire. You are romancing the child of a tiger, Nwa-agu." Chuka drew a deep breath and released it on his neck.
"Ah, you dress in a respectable fashion, _Church Rat,_ " Chuka said now, pulling up the isiagu on my host by its shoulder. " _Looks very good, sir_. Otobo. Eh, let me pass the message: no speaking, no doing anything. No phim. Don't make the mistake of coming out to join the family on the dance floor, or anything like that, no matter what my sister says. _I repeat,_ no matter what my sister says. You hear me?"
Gaganaogwu, I had known my host at that point for twenty-five years and three moons, and I had never seen him that embarrassed. He was wounded as if Chuka had not said words to him but had lashed him with a whip. What pained him the most was that he could not retaliate. As a boy, he had not been afraid of fights: in fact, he had been feared, for although he did not court trouble, he fought with the fist of stone when provoked. But in this situation, he was incapacitated, hand-tied. So although bruised, he simply nodded in response.
"Good, _Church Rat,_ you are welcome."
For no particular reason, he would always remember those final words, a mix of the language of the fathers and the White Man's: "Odinma, _Church Rat,_ ibia wo."
The early fathers often say that a planned war does not take even the crippled by surprise. But an unplanned one, that which is unexpected, can defeat even the strongest army. It is why they also say, in their cautionary wisdom, that if one wakes in the morning to find something as innocuous as a hen chasing him, he should run because he does not know whether the hen has grown teeth and claws during the night. Thus defeated, my host sat stunned for the rest of the party.
The guests started pouring in not too long after Chuka left him. The invitation card indicated that the event was to be held from 4:00 to 9:00 p.m. But the first guests arrived around a quarter after five. Ndali had bemoaned the fact that this would happen—"You will see that they will all follow Nigerian time. This is why I hate attending events like this. If it was not because of my father, I tell you, just count me out." He watched as the seats filled up all around him by guests who came wearing different attire, usually a man in flowing traditional cloth and his wife in an equally sparkling blouse, a wrappa wound around her waist, a fancy purse or handbag in her hand. Children sat in the last two rows of plastic chairs, with high armrests. By the time most of the seats were occupied, the air was filled with a cocktail of perfumes and body fragrances.
The man who sat by his left arm took up conversation with him. Without being asked, the man said that his wife was one of those cooking "up there in the palace," pointing to the house of the Obialors. My wife, too, he said, intending to silence the man. But the man spoke on about the big attendance, and then about the hotness of the weather. My host listened with a dry indifference which, in time, the man seemed to notice. And when the seats next to him became filled with a couple, he turned away from my host to them.
Glad that he'd finally been left alone, my host evaluated what had happened: a hand had come, drawn him back so hard he'd almost fallen out of his seat. Then a mouth had asked him why he'd come, called him foolish, called him a hippopotamus, mocked his clothes, laughed at his love for Ndali, and inflicted the death blow: Church Rat. Had the seats been filled as they were now, none of that would probably have happened. These people had all come too late. So late were they that the celebrated entrance of Oliver De Coque—his favorite musician, the great singing bird of Igboland, Oku-na-acha-na-abali, the chief of Igbo highlife music—meant nothing. He sat as if benumbed as the guests rose to cheer the singer. Blood would have stirred within him as the emcee of the occasion and the famous home-video actor Osuofia introduced Oliver De Coque. But the words sounded like the words of a rambler. He would have laughed at Osuofia's jokes—for instance, the one he'd drawn right out of his famous movie, _Osuofia in London,_ about how white people had mangled his name and called him Oso-fire. But the joke sounded like a child's gibberish, and it even surprised him that people laughed. The big fat man in front of him, how is he laughing like that? The woman beside the man, why is she rocking like that on the chair? He made no response at all to Osuofia's constant bellowing of "Kwenu!" to which the people responded, "Yaah!" And when, shortly after the introduction and the invitation of certain persons to the High Table, and after Oliver De Coque mounted the stage to the tune of "People's Club," he sat dead as a log of wood. Even De Coque had come too late.
Much to his irritation, the man who sat next to him on the left had been dancing in his chair and had remembered him again. And the man would bend every now and again to comment about the attendance, the music, Oliver De Coque's genius, and whatever. But the log of wood merely nodded and muttered under his breath. And even this was said with much reluctance. The man did not know that he'd been asked not to make even the faintest of sounds, a phim. It struck him now, as he thought about it, that the order had come from the owner of the party himself, the very host, Ndali's father. In the midst of these thoughts, he heard something knock at the back of his chair. His heart flew out of him. When he turned, he found the culprit was the boy who sat directly behind him. The boy's foot had hit his chair.
EZEUWA, there are times when it feels like the universe, as if possessing the face of a laconic man, mocks man. As if man were a toy, a plaything given to the whims of the universe. Get down, it seems to say one time. And when a man sits, it orders him to stand again. It gives a man food with one hand and with the other compels him to vomit it. I have lived in the world for many cycles of life, and I have seen this mysterious phenomenon many times. How, for instance, might one explain that just shortly after my host had been startled by this boy (a mere boy!) and returned his gaze back to the great musician, a hand tapped him from behind again, and before he could stir, he hears, "Obim, Obim, they will call us soon. Stand and come. Stand and come"? Well, the action must have come to him too quickly for him to think it through. And because she'd esteemed him highly before those present by calling him her darling, he'd risen and followed her in the glory of the moment. He would himself be taken in by her beauty, for she was dressed exquisitely. A long string of _jigida_ cascaded down her neck, and she wore some of the beads on her wrists. This woman whom everyone around him was calling the daughter of the High One—Adaego and Adaora. Would it not be the worst disgrace to have, in the midst of all these people, remained seated? So he followed her to wild cheers.
The things they said as she and he left came to him like a big joke of fate. "See him, a worthy man deserving of such a woman!" one man said. "Nwokeoma!" another praised. "Enyi-kwo-nwa!" one woman cried. A man standing in plain clothes by one of the tall standing fans at the end of every two rows gave him the greeting of chiefs by extending his hand against his. In shock, as much as with reluctance, he knocked the back of his hand against the man's three times. "Congrats!" the man whispered. He nodded, and his hand, as if it had suddenly acquired a mind of its own, patted the man on the shoulder. It occurred to him then that things were happening too quickly—as if his body parts had mutinied against him and formed a defiant confederacy devoid of his control.
With every step, his hand locked in hers, Ndali led him further and further into transgression. But he could do nothing, for the whole party, spread around the spacious front yard of the compound, had now turned to them, and Oliver De Coque himself had paused his music to give a passing greeting: "See the future oriaku and her man walking by with great strides." To which Ndali waved—and he waved, too—at the crowd of dignitaries, rich men and women, chiefs, doctors, lawyers, three men who had flown in from two of the white people's countries, Germany and United States (one of whom had brought a white woman with yellow hair), Chuwuemeka Ike, one senator from Abuja, a representative of the state governor, Orji Kalu. He, a Church Rat—a man who catered domestic fowls for a living and cultivated tomatoes, corn, cassava, and pepper, killed red ants and poked sticks into the feces of yard fowls for worms—had waved at these dignitaries.
They passed so many people on the way into the house. Amongst them were two women who were looking in a mirror, applying powder to their faces; a man (one of the ones from overseas) in dazzling white _bariga_ and a red _ozo_ cap, smoking from a pipe; a policeman with an AK-47 who stood with his gun pointed upwards; two young girls of puberty age, in flowing gowns, looking in a phone under the shelter of the huge veranda with Roman columns; and a bow-tied boy whose shirt had been soaked in Fanta.
Once inside the house, Ndali pressed her lips on his sweating cheek. It was what she did in lieu of locking mouths with him whenever she had painted her lips a darker shade of pink or red.
"Are you enjoying yourself?" she said, and before he could speak, she said, "You are sweating again! Did you bring a handkerchief?"
He said no. He wanted to say more, but she'd turned into the house, and he followed her. Once in, he found Chuka standing midway on the flight of stairs, visibly astonished to see him there. Words hung, dazed, on his lips as they passed Chuka.
"What is it, Obim? Nonso?" she said after they had passed Chuka, stopping again, this time in a small room where shelves of books split the room into four rows.
"Nothing," he said. "Water, can you give me water?"
"Water? Okay, let me bring it." At the threshold, she said, "My brother, did he do anything to you?"
"Me? No-oh, no, he didn't."
She stayed her gaze on him for a moment, as if unbelieving, then she left the room. Once she was gone, he nearly wept. He sat without realizing it on a small reclining couch that spun, rapidly, and faced the window. He saw now, from the vantage of a hawk paused on a thermal, the party. Osuofia was dancing, interrupting Oliver De Coque every now and then. Chukwu, this is how things sometimes happen to humans: a man becomes afraid of something like this, of being shamed in public, and this fear becomes his undoing. For the state of anxiety is a seed-bearing one. Every occasion pollinates it, and with every action, a seed is begotten. When a word is used that might elicit an unhealthy response, and in the presence of people, he may lose composure and his limbs may quiver. Thus, every inch of the way, propelled by the state of his fragile mind, he does things that further worsen his situation rather than redeem it. He is punished by his own self, as if engaged in a continuing act of unintended self-flagellation. I have seen it many times.
Now firmly in this anxious state, he was so deep in thought that Ndali's footfalls startled him. He took the cup and drank until it emptied.
"Okay, Obim, let's go out now. They will soon call us."
"Ndali, Ndali!" her mother called, with a clatter of feet in the living room.
His heart dropped. I felt pressured to do something, so I flashed in his mind that he be not afraid. _Do as much as you can to hold up against these people_. To this he tapped his feet on the floor, and the voice of his head said, _I will not be afraid_.
While I was communicating with my host, Ndali was saying to her mother: "Ma, ma-Mommy! I'm coming out now." To which the woman responded, "Ngwa, ngwa, quick," in a voice hardly audible over Osuofia's, which came from the speakers outside the house.
"Let's go now," Ndali said, and she took his hand. "It is our turn to sit at the High Table."
He wanted to speak, but all he was able to muster was a muffled "Oh." As if something had wheeled him there, he found himself in the living room face-to-face with Chief Obialor, who was dressed in magnificent regalia—a red-colored long and flowing isiagu—and brandished an ivory tusk. On his red cap were two kite feathers, tucked into both sides of the cap—the way the old fathers dressed. For they believed that birds were a symbol of life and that a man who has succeeded in this world has acquired feathers and, in the proverbial sense, become a bird. His wife who marched beside him wore similar prints on her body and beads around her neck—just as the great mothers did. She held a hand fan, and the bangles on her wrist were innumerable.
When Ndali and he came up to her parents, he bowed before them both in greeting as Ndali genuflected. Her parents smiled back as her father waved his staff and her mother waved her fan in the air. Chukwu, after all that would happen afterwards, my host would always remember how her parents had seemed to show no sign of displeasure at the sight of him during the encounter.
My host, in a state of internal fluster, merged with what was a procession, stepping slowly towards the door of the entrance to the mansion like one being dragged along by invisible ropes. He marched along with the man from Germany, a country of white people, and his white wife, who was dressed like the daughters of the great mothers. Beside them was Ndali's uncle, the famed doctor who'd sutured blown-out limbs during the Biafran War, waving his staff whose top bore an elephant figurine. Outside, Osuofia was shouting into the microphone, his voice amplified by the speakers: "Now they are coming out, they are coming—the celebrant and his family!" My host followed them with the lightest foot, carrying his body as if it were a walking bag of pus kept alive only by Ndali's hand in his, until they entered the arena to the noisy cheering and applause of the crowd. He danced with them lightly, even as Chuka, with scorn on his face, constantly came within an inch or two of him. Increasingly, his fear was inflamed and he did not want to proceed. So he withdrew his hand when they began filling the seats in the front row under the awning where the dignitaries were seated behind the High Table and whispered, "No, I can't, can't, no" into Ndali's ear. She held on but as Osuofia began calling on her, she left him and sat in the very first row with her family members and other high-caliber guests. He hurried into the first empty seat behind them.
Egbunu, the slighted man—he is one who has felt himself disrespected by someone beneath him. Such a man has, by stroke of luck, or by hard work, or by strong bargaining by his chi, obtained good fortune or influence. And now, having measured his wealth or influence against those of others, he sees any raising of the hands by those of less estate as a slight to which he must respond. For to be challenged by a man of less fortune disrupts the equilibrium of his mind and infects his psyche. He must restore this quickly! He must strike at the thing that has caused that shift. This must be his response. Although such men were few in the times of the fathers—mostly because they feared Ala's wrath—I have seen it many times amongst their children. I had seen the sign of this state of mind in Chuka, so I was not surprised when, just as my host sat down, one of the cameramen came to him and whispered into his ear, "Bros, Oga Chuka say make you follow me."
Before my host could make sense of it, the man had begun walking away as if it were a given that he'd do as he'd been told. That in itself brought a lash of fear upon his back. If the messenger had relayed his message with such confidence, without any doubt that the order would be obeyed, how powerful must his master be? How mighty his wrath? He rose and followed the man as quickly as he could, thinking that everyone must have seen that he was the odd one at the High Table who was now paying for his brazen act of transgression. The man circled the house, past a gathering of women cooking stew and a mighty pot of rice. They walked swiftly past a group of sweating men unloading crates of drinks from the inside of a van. Then they passed through a small gate, beside which was a guard hut—a small room. The man turned and pivoted to the outpost. "Enter here, Bros."
It is in circumstances such as these that I often wish that a chi had more power and could defend his host through some supernatural means. It is at times like these that I also wish my host was wise in the ways of _agbara_ and _afa,_ like the dibia who was my host more than three hundred years ago. That man, Esuruonye of Nnobi, had reached the prime of human superabilities. He had been so strong, so potent in divination, that he was deemed _okala-mmadu, okala-mmuo_. Esuruonye was able to strip himself of his flesh and become a discarnate being. I saw him twice invoke the mystical _ekili_ and ascend into the astral plane so that he was able to travel in the batting of eyelids to a distance that would have taken him two market weeks to reach were he to walk and a full day were he to go in a car. But my present host, like others of his generation, was helpless in a situation like this—as helpless as a cockerel in the eye grip of a hawk. He merely entered the house with the mysterious man who had given him the instruction.
Another fellow, with the build of a wrestler, stood inside the room, wearing a deep frown on his face. A blue sleeveless shirt hung around the man's body, embossed with the image of an explosive in motion, its colored sparks like paint stains all over the shirt. "Na the man wey dey disturb Oga's party be this?" the brawny man said in the broken version of the White Man's language.
"Na him," the cameraman said from outside the small room. "But Oga say make we no touch am. Just give am work make im do."
"No problem," the heavy man said. He pointed at a blue khaki shirt and pair of trousers, the kind my host had seen the gateman wearing, and said, "Wear that."
"Me?" my host, with a furiously palpitating heart, said.
"Yes, who else dey here? Look—er, Nwokem, I don't have time for questions, oh. Biko wear that thing may we go."
Ijango-ijango, at times like these my host's mind always failed to provide an adequate answer to the questions. Should he argue with this man? Certainly not: he would get his head split open. Run? Certainly not. He possibly could not run faster than this individual. Even so, if he ran he faced the possibility of going back to the party to face more humiliation. The best thing to do was yield to the orders of this strange person who, without warning, had assumed lordship over him. So in submission, he took off his gown and new plain trousers and put on the regalia of the gateman.
Satisfied, the brawny man said, "Follow me." But what he meant was "Walk in front of me." The horsewhip the man took with him as they walked—what was it for? Could it come down on his back anytime? The fear of the possibility of this was overwhelming. The man and he walked all the way back the way he and the cameraman had taken earlier, only now he was dressed differently—stripped of the regalia of the dignified, clothed in the garments of the lowly, and reduced to his true status. The phrase _where you belong_ came into his head with such force that he was convinced someone had whispered it into his ear right then. As they passed, he saw that the food was being dished into plastic bags and that the van was driving away. He heard Ndali's father's voice, unmistakable over the speakers, as they passed behind the awnings, hidden behind the backs of people who stood on the edge of the pavilions, until they reached the gate.
"Join the gatemen," the brawny man with the whip said, pointing to the gate. "This is your job."
AGUJIEGBE, it was here that Ndali would find him later, drenched in sweat, directing the surfeit of cars that poured in and out of the compound, finding parking spaces, settling disputes, and helping unload and take into the house some of the gifts some of the guests had brought (a bag of rice, tubers of yams, cases of expensive wines, a television in a box... ), and once, when the ribbons fastened around the statue of the lion snapped, he and one of his colleagues adorned the statue with new ribbons.
When she saw him, he had no words for her—for such is the kind of thing that gouges words from within a man, and leaves him empty. Thus he could not even answer the questions, "Who did this to you? Where is your dress? Where—what?" He had merely said, in a voice that seemed as if it had grown old in the time he'd served at the gate, "Please take me home, abeg you in the name of the almighty God." The party was still in full swing and Oliver De Coque was making unintelligible sounds akin to those made by termites crawling on dead wood and the crowd was braying like senseless lambs. All of it, all of them were blown away as his van lurched out of the gate. Memories completely random, moments past—as if fanned by some orchestrated wind—blew into his mind and replaced them. He paid no heed to Ndali who wept all the way as he drove slowly through the noisy streets of Umuahia. But even in his tombed silence, he was well aware, as I could see, that she, like him, had been gravely wounded.
CHUKWU, what had been done to him was so painful that he could not shake a single detail of it out of his mind. Memories of the events persisted, like insects around a glob of sugarcane, crawling into every crevice of his mind, filling him with their black fragrance. And Ndali had cried much of the night until he made love to her and she slouched into slumber. Now the night had deepened, and he lay on the bed beside her. By the dim light of the kerosene lantern, he gazed into her face and saw that even in sleep, he could see signs of anger and sympathy—things that were often difficult to find on her face. His father had once told him that a person's true countenance at a given moment is that which remains on their face while in an unconscious state.
Earlier, while he worked at the gate during the party, he'd thought of how to repay her brother for what he had done to him. But he realized he could not have. What could he have done? Hit him? How can one hit the brother of the woman one loves so much? It occurred to him, again, that things could only go one way anytime he met Chuka. He can only be hit; he cannot hit back. Like craven blacksmiths, Ndali's family had forged a weapon out of his desire, from his heart, and against this weapon he could not contend.
Yet, Egbunu, he knew that the only possible solution—that he should leave her and end it all—lay in the center of the room of his thoughts, gazing at him with its dim face and cruel eyes. But he kept looking past it as if it were not there. And it persisted. He began to ponder instead the itinerant fear that now returned to him—the fear that in the end, Ndali would be frustrated and leave him. Ndali herself had raised this question earlier, just before she fell asleep.
"Nonso, I'm afraid," she'd said suddenly.
"Why, Mommy?"
"I'm afraid that they will succeed in the end and make you leave me. You will, Nonso?"
"No," he'd said, much louder than he intended to, vehemently. "I will not leave you. _Never_."
"I just hope you will not leave me because of them, because I will not let anyone choose who to marry for me. I'm not a child."
He had said nothing else but instead recalled the moment when, while directing traffic at the gate, the man who'd sat beside him under the party tents had seen him on the way out and been bewildered. The man wound down the tinted glass of his Mercedes-Benz and cocked his head towards the front seat. "No be you sidon there with me before?" He found no words. "You are—what? A gateman?" He shook his head, but the man laughed and said something he could not make out, then wound up his window again and drove out.
"Are you sure, Nonso?" Ndali said, her voice tense.
"That is so, Mommy. They won't. They cannot," he said, and his heart palpitated at the violence with which he'd spoken. He did not know, Egbunu, that fate was a strange language which the life of a man and his chi are never able to learn. He raised his eyes to her again and saw a tear sliding down her face. "Nobody can make me leave you," he said again. "Nobody."
#
## The Helper
OSEBURUWA, I stand here testifying to you, knowing full well that you understand the ways of mankind, your creation, more than they can know themselves. You know, then, that the character of human shame is chameleonic. It appears at first, disguised, as if a benevolent spirit, by allowing reprieve whenever the humiliated is taken away from the presence of those to whom or by whom he has been disgraced—those from whom he must hide his face. The disgraced can forget his shame until he comes across those privy to it. Only then does shame strip itself of its dubious benevolence like a bodice and present itself in all its truthful colors: as malevolent. Yes, my host could hide from everyone else, from all of the people of Umuahia, and even from the whole world, and in doing so, all that had happened to him could amount to nothing. A pauper can disguise himself as a king in a place where his true identity is not known, and he would be received as such. Thus my host's peculiar dilemma was that Ndali had witnessed his humiliation. She'd seen him in the cloth of the night watchman, drenched in sweat, directing traffic. This was the blow from whose impact he could not recover. A man such as he, who knows his limitations, who is aware of his own capabilities—such a man is easily broken. For pride erects a wall around a man's inner self while shame pierces that wall and strikes the inner self in the heart.
Still, I have lived with mankind long enough to know that when a man begins to break, he tries to do something to salvage his situation as quickly as possible. It is why ndiichie, in their ancient wisdom, say that it is best to search for the black goat in daylight, before night falls and it becomes difficult to find it. So even before he vowed to Ndali that he would never leave her, he'd already begun to think of a solution. But he could not come up with anything he thought worthwhile, and for days he thrashed about like an injured worm in the mud of despair. On the fourth day of the following week, he called his uncle for advice, but the line was so bad my host could barely hear him. It was with much effort that he understood—in between the older man's stammering and the fragile connection—that it was best he leave Ndali. "You sti-still a boy," his uncle had said again and again. "You still a bo-oy. J-j-jus twenty-six, er. Jo-jus-t forget about this wo-wo-wo-man n-ow. Th-ere are many many out there. Ma-ny many. Yo-yo-u see me? You ca-ca-n't convince them to a-cce-pt you."
Ijango-ijango, I was happy that his uncle had given him this advice. I had thought of the same thing after his treatment at Ndali's family house. The wise fathers often say that when one is insulted, it extends to his chi. I, too, had been humiliated by Ndali's family. Yet I knew it was not of her making and hoped that she would find a way to resolve the crisis. So I did not reiterate his uncle's position. Also, it occurred to me that my host was one of those on earth with the gift of luck, one who would always get whatever he wanted. Before he was born, while he was yet in Beigwe in the form of his onyeuwa and we were traveling together to begin the fusion of flesh and spirit to form his human component (an account which I will render in detail in the course of my testimony), we made the customary journey to the great garden of Chiokike. We walked the gleaming paths between the luminous trees across from which plumes of emerald clouds hung in exquisite arrangement. Between them flew the yellow birds of Benmuo, emerging from the open tunnel of Ezinmuo, as big as full-grown men moving through the serrated tracks. A lump of herbage crowned the sides of the road leading up to the gate out into uwa. There was the great garden where the onyeuwas often go to find a gift which had returned there from unfortunate people who had either died at childbirth or infancy or had been miscarried. Even though we arrived to find the garden crowded, with hundreds of chis and their potential hosts combing through the plants and tangled copse, my host found a small bone. Some of the spirits immediately gathered and revealed that it was from some beast that dwells primarily in the great forest of Benmuo, where Amandioha himself lived in the form of a white ram. They told us that the finding of the bone meant that my host would always get whatever he wanted out of life if he persevered. This they said was because the beast whose bone he'd found, an animal exclusive to Beigwe, is never lacking in food as long as it lives in the forest.
Gaganaogwu, I can name numerous instantiations of this gift of luck at work in my host's life, but I do not want to stray too much from the testimony. At the time, I had confidence that the white bone would bring him some help. I was thus delighted when he decided it was best he try to win the support of her family. Worried that her continued stay away from her family on account of him was only going to escalate the crisis, he begged her to return.
"You don't get it, Nonso, you don't. You think they just don't like you? Eh? Okay, can you tell me why? Can you give me one reason why they don't like you? Can you tell me why they treated you like this last Sunday? Or have you forgotten what they did to you? It was only six days ago. Have you forgotten, Nonso?"
He did not speak.
"No answer? Can you tell me why?"
"Because I am poor," he said.
"Yes, but not that only. Daddy can give you money. They can open a big business for you, or even help expand the poultry business. No, not only that."
He had not thought about these possibilities, Egbunu. So, riveted by her words, he looked up at her as she spoke.
"It is not that you are poor. No. It is because you don't have a big degree. You see, Nonso, you _see_? They don't think in those their big heads that sometimes people are orphaned. And Nigeria is hard! How many people who don't have any parents can go to university? Even—the public schools? Where will you find money to bribe even if your JAMB score is three hundred? Er, tell me, how will you pay school fees, even?"
He gazed at her, his tongue numb.
"Yet they say it, all the time: 'Ndali, you are marrying an illiterate'; 'Ndali, you are embarrassing us'; 'Ndali, please, I hope you are not thinking of marrying that riffraff.' It is, just, very bad. This thing they are doing, it is very bad."
Afterwards, when she had retreated into his old room to study, he sat, folded into himself like a wet cocoyam leaf, worried that she had said a lot of things he hadn't thought about before. Why hadn't he considered that it might be possible to return to school, and that that could be the solution? He beat himself, Chukwu, for what he hadn't thought of. He did not realize that he had grown up in adversity and had become resigned to it. It had made him live a life unlike that of most of his age-mates, a reclusive, provincial life, which developed in its adherents a natural proclivity to be patient in adversity, unhurrying, and measured. If he was not stirred, he would not act. His achievements, if there were any, were given to a slow and sluggish emanation, and his dreams were long-limbed. This was why his uncle had to arouse in him the desire for a woman and now Ndali had inspired him to return to school. And he began to see this sluggishness as a weakness. Later, after she had gone to sleep, he sat alone in the living room, deep in thought. He could register at ABSU and get a degree. Or perhaps he could do a part-time study. Now that his discovery of his love for birds had swallowed the initial dream to go to university, he could even study agriculture.
These ideas came to him with so much power that joy welled within him. They meant that there was genuine hope—that there was a path to him getting married to Ndali. He walked into the kitchen and fetched water from a blue keg, and his thoughts were suspended by the recollection that they were running out of drinking water. The keg was the only one of the three he had with drinking water still left in it. The family who owned two big tanks and sold water in the street had been gone for two weeks, and many of the people in the street either drove to get water from elsewhere or drank rainwater, which they collected in bowls or basins or drums while it rained. The water he scooped into his mouth had a bad taste, but he drank one more cup of it.
As he sat down in the living room, the thought of leaving Ndali reminded him of his grandmother, Nne Agbaso, how she would sit on the old chair that used to be just at the end of the living room—where now, piled up against the wall, were video and audiocassettes, gathering dust—and tell him stories. He imagined he could see her now, swallowing and batting her eyelids as she spoke, as if words were bitter pills she ingested when she talked. It was a habit she'd developed in old age, the only time he ever knew her. After she fell and broke her hip and could no longer continue to farm or even walk without a stick, she came from the village to live with them. During that period, she told him the same story again and again, and yet whenever he sat by her, she'd say, "Have I told you about your great ancestors Omenkara and Nkpotu?" he would say either a yes or a no. But even when he said yes, she'd merely sigh, then blink and tell him how Omenkara had refused a white man's attempt to take his wife and was hanged in the village square by the district commissioner. (Chukwu, I bore witness to this cruel event and how it impacted the people at the time.)
He reckoned now that the story may have been his grandmother telling him again and again that he should not capitulate in the face of any situation. He thought now that he could choose to cower in the face of mere oppression and lose Ndali. No, he said aloud, struck by the thought of another man's mouth on her breasts. He trembled from the mere approach of such an idea towards the corridor of his mind. He'd dropped out after failing his first secondary school certificate exams, passing only three inessential subjects—history, Christian religious knowledge, and agriculture. No mathematics, no English. His university matriculation exams had been worse. He took them around the time when his father's condition was deteriorating, leaving him to tend to the increasing demands of the poultry business alone. Agujiegbe, you know that all I've described here is the education of the White Man's civilization. Like most people of his generation, he knew nothing of the education of his people, the Igbo and of the civilization of the erudite fathers.
So after those strings of failures, he told his father he would not try anymore. He could sustain himself and future family through the poultry business and small farm and, if possible, expand it or branch out into a retail business. But his father had insisted he return to school. "Nigeria is becoming tougher and tougher by the day," his father would say, scrunching his mouth, as he'd begun doing in the early beginning of his last days. "Soon, someone who has a bachelor's degree would be useless, because everyone would have it. So what would you do without even a bachelor's? Farmers, shoemakers, fishermen, carpenters—everybody, I tell you, will need it. That is what Nigeria is becoming, I tell you."
It was talks like this, as well as my frequent accentuation by flashes of thoughts that he should listen to his father—which I often buttressed with the proverb: what an old man sees squatting a child cannot see even from a treetop—that pushed him to take the external GCE. He studied and attended extra lessons at the building on Cameroon Street, where four young university students taught exam-preparatory sessions. And in the weeks of the exam, the extra-lesson center turned into a miracle center. One after the other, a few days before the subject exams, the teachers began to come to the class with leaked question papers. When the exams ended and the results came months later, he passed six of the eight subjects, even getting an A in biology, the one he'd been least prepared for. One of the papers, economics, was canceled for most of the centers in Abia because of what the examination body said was "widespread malpractice." It was true. A copy of the exam paper had been in his hand for nearly three weeks before the actual exam day, and if the results had been released, he would have got an A in it, too. He would have returned to school at this time if they did not wake one morning that same month to find that his sister had vanished, plunging his father into a debilitating depression. All the peace that had returned after his father finished mourning his wife for many years vanished at once. Grief returned like an army of old ants crawling into familiar holes in the soft earth of his father's life, and months later, he was dead. With his father's body, every thought of school was buried.
OBASIDINELU, as days passed and Ndali continued to defy her parents, refusing to even talk with them, my host's fear grew. But, afraid he would bother her if he spoke up, he kept silent, shielding her from the turmoil in his head. But because fear does not leave until it is cast out, it curled around the trembling branches of his heart like an old serpent. It was there when he took her to the bus station the morning she was to go to Lagos for a conference. As the bus was about to leave, he embraced her, and putting his forehead against hers, he said, "I hope I don't disappear before you return, Mommy."
"What is that, Obim?"
"Your people, hope they don't kidnap me before you return."
"Come-on, why will you think like that? How can you even think they will do such a thing, eh? _O gini di?_ They are not devils."
Her anger at this suggestion flung him down. It made him look inward and question if he had been overestimating the situation, wonder if the long night of fear had been nothing but a skeletal dance of worry across the corridor of serenity. "I was actually joking," he said. "Actually."
"Okay, but I don't like that kind of joke, eh. They not devils. Nobody will do anything to you, oh?"
"That is so, Mommy."
He tried not to think of the things that made him fear. Instead, he weeded the small farm and cleaned his room. Then he treated one of the roosters who had injured its foot. He'd found it the previous evening on the other side of the road. It had jumped the high fence of the yard, fallen into the bush behind, and stepped on what he believed may have been a broken bottle. It had reminded him of the gosling, how he'd once left it loose and it had made its way out of the house and sat on the fence. He ran out after it and found it on the fence gazing about with its head turning in agitation. With his heart pounding, fearing that it would fly away and never return, he beseeched it tearfully. It was morning, and his father was brushing his teeth (not with a chewing stick, as the old fathers did, but with a brush) when he heard his son's panicked shout. The man rushed out with the white froth dripping down his beard and his creamed brush in his hand to find his son anxious. He gazed up to the fence and the boy and shook his head. "Nothing you can do, son," he said. "It is afraid. If you go close, it will run." I, watching as well, had the same fears as his father, and I put the thought in his head, too. So he stopped crying, and in a voice as soft as a whisper, he began calling to the gosling, "Please, please, don't ever leave me, don't ever leave me, I rescued you, I'm your falconer." And, miraculously—or perhaps because the bird had seen something else across the fence, perhaps the neighbor's dog—the bird bristled and crouched into a spread of wings. Then it returned through a rushed updraft back to the yard, back to him.
He had hardly set the injured cock back into the coop when Elochukwu arrived. He'd texted Elochukwu early that morning, and Elochukwu had responded that getting an education was the best idea. "If you go back and finish your studies, then they will surely accept you," Elochukwu had said. Elochukwu dismounted from his motorcycle and stood with my host on the front porch overlooking the farm. My host gave Elochukwu an account of the party and how he'd been humiliated by Ndali's family. And when he was done, Elochukwu shook his head and said, "It is well, my brother." And my host, looking up at his friend, nodded in acceptance. Egbunu, this expression, very common amongst the children of the great fathers and spoken mostly in the language of the White Man, has often baffled me. A man in a situation in which his livelihood is threatened has just rendered an account of his travails, and his friend—one he sees as a comforter—responds simply, "It is well." That expression instantly yields silence between them. For it is a peculiar phrase, all-encompassing in scope. A mother whose child has just died, when asked how she is doing, replies simply, "It is well." It-is-well emerges from the intercourse between fear and curiosity. It designates a transient state in which, although the unfortunate knows he is experiencing something unpleasant, he hopes it will soon be mended. Most people in the country of the children of the fathers are always in this state. Are you hoping to recover from an illness? It is well. Has something been stolen from you? It is well! And when a man steps out of this condition of it-is-well onto a new path towards a more satisfying state, he immediately finds himself in another situation of it-is-well.
Elochukwu shook his head again, repeated the phrase, tapped my host on the shoulder, gave him a bag of books, and said, "I am in a hurry, we are going to a rally at GRA." Before Elochukwu left, my host complained that he would not be able to get a degree for at least five years, and that was if there were no strikes, which could delay it and cause him to not have a degree for probably even seven years. "Start ti go du first," Elochukwu said, mounting his motorcycle. "Once e start tiri, ha ga hun na idi serious." Elochukwu, who had himself nearly completed a degree in chemistry and was not one to dwell on words, finished the topic with, "And if it doesn't work, just forget the girl. There is nothing the eye sees that can cause it to shed blood in place of tears."
Not long after his friend left, rain began. It rained through that morning into the evening. As it poured down on Umuahia, unremitting in volume and unpredictable in temperament, he lay in the sitting room, studying one of the university matriculation exam preparatory books he'd received from Elochukwu.
Now, he read for a long time by the dim light from the cloud-washed sky that came through the parted curtains until his eyes began to close. He was almost asleep, anchored like a wind-borne leaf in the threshold between sleep and awakening, when he heard a knock on his front door. At first, he'd mistaken the knock as the rain pattering on the door, but then he heard a familiar voice say in the most forceful of tones:
"Will you open this door, now?"
Then the banging began again. He jumped up, and through the windows, he saw Chuka and two men, dressed in raincoats, standing on his porch.
Gaganaogwu, the effect of the sight of these men on him could only be described as hypnotic. In all the years I had been with him, I had never seen anything close to this happen to him. It seemed strange that only some time ago he had made a joke about something, a wild, far-fetched joke. And in daylight, his joke had materialized, and here was her brother at his doorstep with a gang of men? He let them in, steeped in terror, a pounding in his chest.
"Chuka—" he started to say as the men came in.
"Shut up!" shouted one of the men, the brawny one who'd led him to serve at the gate during the party. Even now, the man had come prepared—with that same whip.
"I can't shut up. No." He stepped back as the men advanced and moved behind the biggest sofa. "I can't shut up because this is my house."
The man with the whip lunged forward, but Chuka raised his hand and said, "No! I have said this before, no touching anybody."
"Sorry, sir," the man said, stepping back behind Chuka, who now walked into the center of the room.
He watched Chuka unhood his raincoat, shaking his head as he did. Then Chuka swirled around to inspect the room, then sat down with his raincoat on the couch, still dripping water. The men stood beside the couch, gazing at my host with a frown.
"I have come to ask you to send my sister back," Chuka said in the same calm voice he'd spoken in before, and in the language of the White Man. "We are not interested in making trouble with you. Not at all. My parents, her parents, are worried." Chuka dropped his head to the floor as if in contemplation, and in the brief silence that ensued, my host heard the soft patter of rain dripping from Chuka's raincoat to the carpet.
"Once she is back from Lagos, we ask you to make sure she is back within two days," Chuka said with his eyes to the floor. "Within two days. Two days."
They left the way they had come, slamming the door behind them. Although it was still daylight, the rain clouds had dimmed the horizon so much that they'd driven with their headlights on. He watched their car retreat from his farm path in reverse gear, like two disks of yellow light receding into the distance. When they were gone, he sank to his knees, and without any reason to which his mind could cling, he broke into a prolonged sob.
EGBUNU, if an arrow is pointed at the chest of a defenseless man, that man must do as he is told. To do any different in the face of indefensible danger is folly. The valiant fathers say that it is from the house of a coward that we point to the ruins of the house of a brave man. Thus the defenseless man must speak, with a soft tongue, effectual words to the one who bears the arrows: "Do you want me to go yonder?" And if the man who endangers him answers that he should, he must do as he is told until he is cleared from the present danger. After Ndali's brother left, my host resolved to do all he had been told to do. He would persuade her to return home, and while she was gone, he would find a solution to his inadequacy, the main source of all the problems. He would go back to school and get an education and a job that would make him suitable for her. Chukwu, I have come to understand that, when a man is disgraced, his actions might be shaped by shame and his will by desperation. What once means much to such a man might begin to mean little. He could, for instance, stand in his yard and regard his poultry, this thing he had built for himself, these eight coops of nearly seventy birds, and see how lowly a business it was. The sight of feathers, which he'd normally sniff and twist in admiration, now may look like litter to him. What is he doing now, one may ask? Well, he is responding, Chukwu. His mind is preparing itself for change. It has weighed everything on a scale and determined that a return to loneliness, especially by losing Ndali, would be worse than anything else. She was the glittering, priceless article in a store full of precious artifacts. The poultry, the birds, these weighed less. They could be gotten rid of if need be in order to get her. After all, he'd seen a man sell his land to send his child to school abroad. What has such a man done? He has decided that it would be better in the future to have a child who can be a doctor than to keep the land. Such a man has reasoned, perhaps, that with a rich son, the land can be recovered, or such a son can even buy him a bigger piece of land.
So by the time he'd finished these unbroken ruminations, by morning two days after Chuka came to his house, he rose and, even without feeding his flock and harvesting fresh eggs, went out and bought forms for the university matriculation exam from the Union Bank branch down the street. He waited in a long line in the old and crowded bank, a line that stretched to the entrance, and he had to plead for those in the line to make room so he could squeeze into the building. He left the bank tired and soaked in sweat.
Ijango-ijango, it is imperative that I tell you, in detail, about this walk back home, for it was during this walk that the black seeds of his undoing came to root in his life. When he got back on the road home, he walked for a while by a school bus, which had slowed down in the clotted traffic. He gazed at the uniformed children inside it, who were poised in different shades of slumber. A few had their heads resting against the seats, some had their heads tilted sideways against the headrests, some had bowed their heads into their hands, and others were resting their heads against the windows. One or two of them seemed awake: an albino girl with sand-colored hair and a sore on her purplish lower lip, gazing blankly at him, and a boy with a clean-shaven head. He lumbered on, carrying the file containing the forms under his armpit, past sheds and tables from which articles were sold, the sellers calling to him to buy their wares. One of them, a woman who sold used clothes piled on a jute sack, called to him, "Fine man, come buy fine shirt, fine jeans. Come see ya size." He had just passed the woman's shed when he felt something palpitating in his trouser pocket. He reached for his phone and saw that Elochukwu was calling.
"Er, Elo, Elo—"
"Kai, Nwanne, I've been calling you!" Elochukwu said, partly in the language of the great fathers and partly in the White Man's language.
"What, I was at the bank, so I silence my phone."
"Okay, no problem. Where are you now, where are you? We are at your house, oh. Me and Jamike, Jamike Nwaorji."
"Er, Chukwu! _E si gini?_ Jamike? No wonder you are speaking English."
He heard a voice in the background and Elochukwu ask the person in the broken language of the White Man if he wanted to talk to him.
"Bobo Solo!" the voice said into the phone.
"Jisos! Ja-mi-ke!"
"Please, come, come, we are waiting for you oh. Come come."
"I am almost there," he said. "I am coming, oh."
He put the phone back in his pocket and began walking fast towards his house, his mind racing. He had not seen or heard anything about this man in a long time. And now Jamike, his old classmate from Ibeku High School, was at his house. He crossed the street and passed between the poor houses of the lower street, where a gulley had carved up the earth and dug up the yellow soil and swallowed the loam in many broken places. He ran, the file in his hand, until he reached his compound. At the entrance, he raised his head and saw Elochukwu and their old classmate standing on the porch. By the porch, leaning against its kickstand, was Elochukwu's Yamaha motorcycle. He walked towards them on the graveled path flanked on both sides by the fields of the small farm. As he drew closer to the men, he stifled the urge to shout. At first, he did not recognize this person with a broad, mustachioed face. But then he found himself suddenly absolutely beyond repose, shouting, "Jamike Nwaorji!" The man, in a red cap with a white bull's head embossed on it, and a white shirt and jeans, drew close and rammed his hand into his raised hand.
"I can't believe it, mehn!" the man said.
He recognized at once a tincture of a foreign accent in the man's voice, the way people who'd lived outside the world of the Black Man spoke, the way his lover and members of her family sounded.
"Elo here tell me that you are living in overseas," he said in the language of the White Man, as they did in their school days, when it was a punishable offense to speak an "African language." So with the exception of Elochukwu, the language of the White Man was how he communicated with friends from school, even though nearly every one of them spoke the tongue of the august fathers.
"Na so, oh, my brother," this man, Jamike, said. "I have been living abroad for many, many years, mehn."
"Er, let me go now, Nonso." It was Elochukwu who had spoken. He tipped his black hat, which he'd begun wearing since he joined MASSOB, as he shook my host's hand. "I was just waiting for you to come because when I saw him, I remembered your problem. Jamike can help you."
"Er, you are going?"
"Yes, I gat do something for my Popsy."
He watched Jamike, who had a smell that must have been from an expensive perfume, hug Elochukwu, who then hopped on his motorcycle, pumped the pedals twice, and a plume of smoke gushed up into the air. "I go call una," he said, and rode away.
"Bye-bye," he called after Elochukwu, then turned to the man before him.
"Na wa oh, Jamike himself!"
"Yes, oh, Bobo Solo!" Jamike said.
They shook hands again.
"Let us go inside naw. Come, come."
My host led the visitor inside the house. As they entered, he had a flash of how, two days before, Chuka had sat on the sofa where Jamike now sat, his raincoat giving him the appearance of a movie villain and his presence bearing as much threat to my host as this sudden recollection of it.
"Mehn, you get very big compound, oh. Only you live here?" Jamike said.
My host smiled. He sat down to face the visitor after parting the curtains to let in light to the room.
"Yes, my parents passed away, and you know that my sister, the small one that time?"
"Er, er—"
"Nkiru, she marry. So only me I am here now. And my girlfriend also. Ehen, where are you living now?"
Jamike smiled. "Cyprus—you know the place?"
"No," he said.
"I know that you won't. It is an island in Europe. A very small country. Very small, but very beautiful; very beautiful, mehn."
He nodded, "That is so, my brother."
"Oh-ho. You remember our classmate Jonathan Obiora? He used to live here," Jamike said, pointing at an old house in the distance. He removed his cap and tapped it on his lap. "Bobo, do you want us to go drink beer and talk small?"
"Yes, yes, my brother," he said.
Egbunu, when two people meet at a place such as this, and both of them have crawled out of each other's past, they often suspend the present as they try to drag all that has happened in the intervening period into the moment. This is because they are bound somewhat by where they had both been in that time long ago or by the same uniform they wore. It would occur to both of them that it is sometimes hard to tell how much time has passed until something or someone from that point in the past reappears, bearing the wear and tear of long travel. For my host, Jamike noted that he was much taller but still lanky. My host on the other hand was astonished at how Jamike's once small body and clean-shaven head had now given way to a towering figure only half an inch shorter than himself and a beard that cascaded down both sides of his head. After they have noted these differences, they will proceed to talk about where they have gone since the last time they met, what road they have taken, and how they have gotten to the point at which they find each other now. And sometimes these two may build new relationships and become friends. I have seen it many times.
So they left his compound and walked to the Pepper Soup place on the adjacent street and sat on one of the rows of benches on the earthen floor. The sun had increased in intensity and they were sweating when they entered the restaurant. They sat under one of the ceiling fans, beside a stereo from which a low tune slowly rose. He could barely wait to sit down, for during the short walk, Jamike had painted a portrait of the place where he lived, Cyprus, as a place where everything was in order. Electricity was constant; food was cheap; hospitals were plentiful and free, if you were a student; and jobs, "like water." A student could own a Jeep or an E-class Mercedes-Benz. In fact, Jamike said that he'd returned to Nigeria with a sports car which he had now given to his parents. On their way to the restaurant, he'd observed that Jamike walked with a certain ceremonial gait, employing the full weight of his body as he went along, as if his movement were a performance whose audience was everything within the ambit—the parked truck, the old pub, the cashew tree, the mechanic's workshop, the mechanic working beneath a pickup truck on the other side of the road, even the vacant sky. Jamike spoke with the same cadence, with a light swagger in his voice, so that every word he said struck deep into my host.
For a moment, they did not speak, and he let what Jamike had told him sink in while the latter replied to a message on his phone. He let his eyes hover over the calendar with the Star beer advert on the wall beside where they sat and on a poster of American wrestlers he knew and whose names flashed in his mind as he gazed at the poster: Hulk Hogan, the Ultimate Warrior, the Rock, Undertaker, and the Bushwhackers.
"So Elo said you want to start school? He said you are having some problems and I can be able to help you."
My host threw himself up in thought, as if lifted from within by a monstrous hand. "Yes, Jamike, yes, my brother. I have a problem."
"Tell me, Bobo Solo."
He wanted to speak, but the recollection of that name, which his mother used to call him, made him pause for a moment, for somewhere in the itinerant years long traveled into oblivion, he saw himself standing in the room, laughing as she laughed and clapped, singing, "Bobo, bobo, Solo. Bobo, bobo, Solo."
He took up the bottle of beer and drank to calm himself. Although it tasted strange to him—for he rarely drank—he felt obliged to take it. When a man receives a visitor, he eats and drinks that which the visitor eats and drinks. Then words burst out of him like wine from an uncorked bottle carrying in it an amalgam of emotions—fear, anxiety, shame, sorrow, and despair. In the torrent of words, he told Jamike everything that had happened up till two days prior, when he'd been threatened at his home. "This is why I told Elochukwu that I have to return to school quickly. In fact, I don't have choice. I love Ndali very much, my brother. I really really really love her. Ever since she came into my life, I have not been the same again. Everything has changed, Jamike, I'm telling you, everything has changed. Every single thing, from _a_ to _zed,_ have changed."
"Ah, that is serious problem, oh, mehn," Jamike said, sitting up in the chair.
He nodded and took another sip of the drink.
"Mehn, why don't you want to leave her?" Jamike said. "Is this not easiest for you instead of this stress?"
Egbunu, my host was silent at this. For in this moment, he recalled his uncle's counsel and even Elochukwu's partial counsel. He knew because he'd heard from somewhere he could not recall that a person must reconsider their position if everyone else is saying something that contradicts their own position. And a part of him, a part that seemed to have resolved into a shadow, wanted to submit, to accept that the only way was to leave her. But another part was defiantly resolute not to, and it was this part that urged him with a ferocity he could not suppress. And I, his chi, I was in between, desiring that he have her but fearful for what it might cost him. And I have come to understand that when a chi cannot decide the best path on which to lead its host, it is best if the chi remains silent. For in silence the chi yields, fully, to the complete will of its host. It lets man be man. This is better, far better, than a chi who leads its host to a path of destruction. For regret is the poison of the guardian spirit.
He spread his hands over the table and said, "That is not it, my brother. I can go if I want, but I love her very much. Jamike, I'm ready to do anything to marry her."
Gaganaogwu, in the grave ills that would befall my host later, I would look back frequently and wonder if indeed it was in these words that all that happened later was first hatched. A twitch appeared on Jamike's face after my host said those words, and Jamike did not respond immediately. He first looked around at the house, then nodded and sipped the beer before he said, "Ah, love! You don hear D'banj's 'You Don Make Me Fall in Love'?"
"No, I never hear," my host said, continuing quickly so Jamike would not go on discussing the needless song, for he wanted to unburden his heavy mind. "I love her so much I go do anything for her," he said again, this time with much restraint, as if it had cost him much to say it. "I want to go back to school now because before my father died, he was sick, and so I dropped out to help him grow his business. This is why I did not go to university."
"I see," Jamike said. "I know you didn't drop out because you are not brilliant. You were brilliant, mehn. No be you score second, third, for class behind Chioma Onwuneli?"
"It is so," he said, for he remembered days now long past. But it was the present and the future that he must reckon with. "I have complete GCE. If I return to school, er, my brother, I am sure once they don't think I be illiterate again, them go accept. I strongly believe this."
"Tha-that is very true, Bobo Solo," Jamike said. His eyes watered, and he blinked. "Very true."
"It is so, my brother," he said. He felt, for the first time in weeks, somewhat relieved, as if he'd solved his problems by simply recounting them.
"So since you say it is fast and quick to school in Cyprus, that I can get a degree within three years, I want to go there," he said with relief, for it struck him that he'd said everything he said simply because he wanted to tell Jamike this.
"Very good, Bobo Solo! Very good, mehn!" Jamike lifted himself summarily out of the seat and slapped his hands. "High five there, nwokem!" Then sitting back, Jamike gazed at his hands with his eyes scrutinizing the lines as though it were a foreign hand. "Is that sweat?"
"It is so," he said.
"Wow, wo-ow, wo-ow, Bobo! So you still sweat like Christmas goat?"
He laughed. "Yes, my brother Jamike. I still sweat on my palms."
"Bobo nwa."
"Errrr," he said.
"You have found the solution, mehn!" Jamike said, shaking his finger. "You have found it. You can now go and sleep."
He laughed.
"Cyprus is the solution."
IJANGO-IJANGO, it is true, as the great dibias among the fathers often say: that in this world which you have created, if a man wants something very much, if his hands do not desist from chasing it, he will eventually possess it. At the time, like my host, I, too, had thought that the encounter with his old schoolmate was the universe lending him what he had been longing for. For he returned to his house later that evening with a slight tremor in his gait from the drink he'd shared with his friend and with a hive full of honey in his heart. When he went to sleep, the squawking of the hennery in his ears, he began to digest it all: the island on the Mediterranean Sea, as beautiful as the ancient Greece of the books he read as a child. The ease of admission into the universities. "No JAMB!" Jamike had repeated again and again. "You only need GCE, only GCE." The timing of it: how it had happened exactly when he needed it. He could start in September, four or five weeks from now. The uncanniness of the possibility threatened to throw everything into unreality. How affordable it was: "It is cheaper than all these Nigerian private schools," Jamike had boasted. "These nonsense schools we have here: Madonna, Covenant, it is better than them all." And what is more? He need only pay the first-year school fees and campus apartment, and by the time he got to the second year—in fact, even the second semester—he would have earned enough from part-time jobs to pay the next school fees and board.
Even now, as he slowly drifted off to sleep, he saw Jamike dance with his words, a ritual dance whose effect was hypnotizing. He let his thoughts linger on the auspicious suggestion from Jamike that it would be great and healthier for his relationship with Ndali if he went abroad to live for the first few years of their marriage. Jamike had insisted, in a most convincing way, that it would make her parents respect him even more. Then he considered the last thing Jamike had said about this country, which had only served to increase his hope: "You can easily go to any other part of Europe, or US. By ship, very cheap. Within two hours! Turkey, Spain, many many countries. This will not only be the best opportunity to please Ndima—" He helped him say the name. "Oh, sorry, Ndali. It is also an opportunity for you to experience a good life. In fact, look, if I be you, I will make all the arrangements without telling her. Look at all the big land, big house your father left for you. You can do it, mehn. Surprise her!" Jamike said this with almost a scowl on his face, as if angered by his own words. "Surprise her, mehn, and you will see. You will see that you will not only gain her respect, but, I tell you"—Jamike licked his thumb with his tongue until a gasp of erhen erupted—"I swear to almighty God, Ndali will love you die!"
These last words had come out of Jamike with such assurance and certainty that my host let out a laugh of relief. He laughed again now as he remembered it and stood up. He picked up his jeans, which lay on the chair by the bed, and took out the piece of foolscap on which Jamike had made notes. He'd brought out a pen and a book from his back pocket, the book folded through the center from him sitting on it. With a glib smile, he detached a leaf from the book and said, "I am a practical man, let us come down to practical," then he began to scribble all he had said down.
2 semesters scool fees = 3000
1 years accomodation = 1500
Bank deposit = 1500
Mantainanse = 2000
8000 euro
Gaganaogwu, the peace that came upon my host that night was like the pure unspotted waters of Omambala. After he'd gazed at the paper as many times as possible, he folded the paper. He switched off the light and walked to the window, his heart throbbing wildly. He could not see much outside even though the moon seemed to be bright. For a moment, the house across the road looked as if it were on fire, its roof a raging vermilion and smoke rising from it. But he soon saw that it was the streetlight cast on the building and the smoke was rising from some cooking hearth.
#
## Crossing the Threshold
AGBARADIKE, the great fathers in their discreet wisdom say that seeds sown in secret always yield the most vibrant fruit. So my host, in the days following his meeting with his old schoolmate, shielded from the world the inflorescence of joy that grew along the edges of his heart. In secret, his plans grew, unbeknownst to Ndali, who returned from her weeklong trip to Lagos three days after he met Jamike. He hid his father's old briefcase, in which he stored the documents he collected, under the bed. He attached his heart to the bag as if it contained everything he owned, his very life.
As the contents of the bag increased, so did other joyful developments. He did not have to persuade Ndali to go back home after she returned. She went back by herself, deceived by Chuka's lies that their mother had taken ill. This resolved his fear that something else might happen if he was not able to persuade her to return as Chuka had warned, an encounter which—not wanting her to escalate issues with her family—he kept from her. When she came to see him exactly two weeks after he began his plans with Jamike, her mood was wholly changed. She had come from church that day, lighthearted.
"I can't even believe it, Obim," she said, clapping her hands playfully. She sat on his legs. "Can you guess what Daddy said?"
"What, Mommy?"
"I told them that you bought JAMB forms to go back to school. So they said that if you register at a school, that would be a good first step. It would show your seriousness to become somebody."
Egbunu, he was stunned by this. It seemed to him that something he could not see had peered over his shoulder and looked into the pot of his secrets. For having resolved all along not to tell her about his plans, as Jamike had advised, not wanting her to stop him, he'd only told her about the form he'd bought. Yet he knew he could not hide it from her for too long. So as he took more steps in this direction each day, he'd assure himself that he would tell her about it. But by the end of the day, he would push it like a thing with wheels into the future and say not today but tomorrow. But if _tomorrow_ Ndali came home with a fever after a long day at school, he'd say _Tomorrow, she will be home all day and it will be easier then_. But alas, that tomorrow would come with a phone call first thing in the morning that her uncle had suffered a stroke. _Over the weekend,_ the voice in his head would resolve, perhaps on Sunday after church. And as if by some alchemic manipulation, today was that Sunday. Now that she had said something that touched the core of the very thing he'd been keeping secret, he resolved to tell. "Mommy, consider it done!" he said.
"Er, Obim?"
"I say, consider it done," he said even louder. He made her stand, and he rose, too, swaggering slightly. "I have gone to school and come back."
She laughed. "How? Abi in spirit or in your dreams?"
"Just watch, er."
He went into the room and retrieved the bag at the foot of the wall beside the bed in what was once his sister's room. He blew away a spider that lay on the fading coat of arms inscribed on the bag's leather skin and carried it with him back to the sitting room. He put the bag on the center of the table.
"What is in the bag?" she said.
"Abracadabra—you will see." He waved his hands over the bag while she bobbed with laughter. Then he opened it and handed her the documents. He'd arranged the documents in the order of lowest cost, so when she began from the last one, he said, "No, no, Mommy, begin here, first."
"Here?"
"Yes, that one."
He sat down to watch her peruse the documents, his heart sounding a nervous beat.
She read the header on the paper aloud: "Admission letter." She raised her head. "Wow, Nonso, you got an admission!" She rose to her feet.
He nodded. "Just read on."
She returned her eyes to the paper.
"Cyprus International University, Lef-lef-ko-sa?"
"Lefkosa."
"Lefkosa. Wow. Where is this place? How did you get it?"
"It is surprise, Mommy. So just look, just look."
She read through.
"Oh God! Business Administration? That is very good!"
"Thank you."
"I can't believe it," Ndali said. She threw her hands into the air, swirled into a half circle and faced him again, and kissed him.
"Read all first, Mommy," he said, detaching himself. "Then you can kiss me after. Read."
"Okay," she said, and looked at the book between the files.
"Your passport?"
He nodded, and she looked through it, with her face filled with light.
"Where's the visa?"
"Next week," he said.
"You will go to where—Abuja?"
"Abuja."
He saw a shade begin to grow over her face, and he stiffened.
"Read it all, Mommy, please."
"Okay," she said. "Letter of accommodation," she said, and glanced up at him. "You have accommodation already?"
"Yes. It is so. Read, see, Mommy?"
But she dropped the documents back on the table.
"Nonso, you are planning to leave Nigeria and you are just telling me?"
"I wanted it to be surprise. Look, Mommy, your brother came here after you go to Lagos. No, no, listen first. He came with thugs to frighten me. Actually, I have no choice. I have to do something. Listen godunu first. Look, I luckily saw my former classmate who schools in this beautiful country, Cyprus. And he told me everything. How everything is cheap, school fees, and jobs easy to get. Degree, I can get within three years, if I do what he called summer schools. That is why I did this."
"Who is this person you met?"
"His name? Jamike Nwaorji. He just went back to Cyprus—actually four days ago. He was my classmate in primary school and also secondary school."
She took up the documents again, as he'd hoped, and went through the course curriculum, then returned her gaze back to the inscriptions on the softer foolscap.
"Wait oh, I still don't understand."
"Okay, Mommy."
"You are leaving Nigeria when you said you want to marry me?"
"It is not so, Mommy." He opened his mouth to say more, but he could not form words, as the confidence that had been painstakingly constructed over the days and weeks prior, the confidence derived from the result of weighing everything on the scale and deciding he could give up everything for her, had suddenly flattened. To shore it back up, he moved closer to her and sat on the arm of the couch.
"How is it not so? This is a school abroad."
He took her hand. "I know it is abroad, but it is actually the best way. Imagine in two and half years, I have a real, authentic degree? Imagine, Mommy? Even, you can always visit me. You graduate next year June, and by then I would actually also be going to my second year. You can come and stay with me."
"Jesus! Nonso, are you saying..." She clasped her palms over her head. "Forget it, just forget it."
"No, Mommy, no. Why don't you tell it to me, why?"
"Forget it."
"Nne, look, I am doing this because of you, only because of you. Actually, I never even wanted to go back to school, but that is the only way I can be with you. The only way, Mommy?"
He put his hand on her shoulder and gently pulled her towards him. "You know I love you. I love you very much, but see what they are doing to me. See how they disgraced me. They really disgrace me, Mommy. And who knows, maybe it is just the beginning. Just the beginning, and, you don't know, I don't know. I'm coming, Mommy..."
All evening, they had been hearing loud, excessive caws, but now it grew too distracting for him. He went to the kitchen, drew his catapult and a stone from the window frame, and ran out. All his chickens were in their coops, and just as he got close to one of them, a reddish cock leapt to the bars noisily, squawking in distress. It was fighting with one of the newest roosters, the one with a serrated comb and an abundance of wattles. The rooster had shown unusual belligerence even from the day he bought it. He unbarred the coop's net door and tried to catch it. But it hopped against the wall and tried to find something to hang on, but couldn't. He tripped and landed with his hands on the floor as the roosters leapt up and ran out of the coop with two others from the group of six cocks and cockerels. He pursued it, and it jumped onto the bench under the guava tree, and when he tried to catch it, it mounted the water drum, crowing aggressively. He was furious. He circled the well and then, moving as fast as he could, grabbed the rooster.
He was binding the bird to the tree with hempen twine when Ndali stepped into the yard. The low evening sun cast a shadow of her against the wall, a shadow so large that only half of it could be seen.
"Nonso," she said, startling him.
"Yes, Mommy."
"What have you done?"
"Nothing," he said.
He turned and held her, his chest still pounding, but pressed against her chest, he felt that her pounding was far worse.
AGBATTA-ALUMALU, sometimes a man cannot fully understand what he has done until he has told another person about it. Then his own action becomes clearer even to himself. I have seen it many times. Although my host had spent the past hour explaining his rationale for selling the compound, and poultry, when he was done, he began to see the flaws in the decisions he had made. Again, Chukwu, you have established that the main roles of the guardian spirit are to watch over our hosts and make sure that preventable calamities do not befall them, so they can more easily fulfill their destinies, the reason for which you created them. We must never try to compel our hosts against their will. So even though I had worried that he was selling most of what he had, I had let him do this without interference. I did this also because I believed that the man who had come to him to help him had been a product of his gift of good luck, the bone from the garden of Chiokike.
But now, when he heard the gasps and saw the fright on Ndali's face, he became afraid that he had made hasty decisions. A coldness came upon his heart which, for the weeks past, had been warm with the joy birthed by hope. After he'd finished revealing everything he had done in secret, Ndali said, "I have not words, Nonso. I am speechless."
She went into his old room and closed the door while he sat in the sitting room, staring at the documents. He reread the agreement about the sale of the compound again, and fear welled up in his mind. When his father bought the house, he was barely ten, and his mother was pregnant. His father had said they needed a bigger house as more children came. He thought he'd forgotten this bit of memory, but now he found it as fresh as yesterday. His mother holding him, he'd stopped in the empty room while his father and the seller went around the place. Then he'd broken free from his mother and ran to the backyard and stood under the guava tree, greatly fascinated by it. He tried to climb it, but his mother, although heavy with child, came running and calling him down. He heard her voice with startling clarity, as if she were behind him in the room. "No, Bobo, no. Don't, I don't like people who climb trees." "Why?" he'd asked, turning his back to his mother, as he did when he wanted to disobey her. "Nothing," she said, and he heard her sigh, as she had begun doing as her belly bulged. Then, with the kind of resignation that he'd come to understand as a marker of finality, she said, "If you do, I won't like you."
He was thinking this when Ndali emerged from the room and said, "Nonso, let us go to Tantalizers, I'm hungry." At first, he could not distinguish between the voices of the two women, but Ndali stepped farther into the sitting room and stamped her feet on the floor. "Nonso, I'm talking to you!"
"Er, Mommy, yes, yes, let us go."
They walked slowly, a quietness between them, as if some authority beyond the will of man had ordered that words not be spoken. They went through the narrow street, between graying and molding fences and street gutters clogged with waste. On the other side, separated by a potholed road, birds sat in chambers of an unfinished multistory building fettered by wooden scaffolding. He was gazing at the birds when, in a voice a little above a whisper, Ndali said that if she knew it would come to this, she would have left him.
"Why do you say that, Mommy?"
"Because I am not worth this sacrifice. All this—it is too much."
He did not speak until they entered the restaurant, for he was disturbed by what she had said. The restaurant was alive with the chatter of people—a group of men in plain shirts, some office workers, and two women, and a song was playing at a low tune on the speaker. He wanted to contest what she'd said vehemently and insist that she was worth it. But he didn't. For even though he now mostly regretted it and agreed that he had acted in haste, he knew, too, that he'd gone too far to turn back now. He had sold his compound, which he inherited from his father. Two semesters of school fees had been paid, along with the fees for a year's accommodation. And Jamike, who had now returned to Cyprus, had two thousand more euros which he had given Jamike to keep in an account for him for "maintenance," so he wouldn't have to carry much money while traveling. In the bag was another six hundred euros, the last of the hard currency. Only the forty-two thousand naira he had in the bank was to remain, in addition to however much they would get from selling all the fowls.
When they sat down in a corner of the restaurant, she repeated her words again.
"Why do you say this?" he said.
"Because, Nonso, you have destroyed yourself because of me!" she said with what my host thought was anger. After she said this, she turned about to look around the place, for it seemed as though she realized that she'd spoken in a burst of emotion, and her words had been loud, so she whispered, "You have destroyed yourself, Nonso."
Chukwu, the effect of this unexpected proclamation on my host was severe. It felt as though something had riven through the landscape of his soul and split it in two. It was in an effort to hold himself together that he said, "I didn't destroy myself anything, I didn't destroy myself."
"You have," she said. _"I gbu o le onwe gi."_
Surprised by her switch to Igbo, he did not speak.
"How can you sell everything, Nonso?"
"I did it because I don't want them to separate us."
"Yes, but you sold everything you have, Nonso," she said again and turned to him, and he saw that she had again begun to cry. "For me, for me, why, Nonso?"
He swallowed hard, for he saw now that the reality of what he had done, when expressed in words, bore a grave, crushing enormity.
"No, I will recover it all—" he said, but saw that she was shaking her head, her eyes diluted with tears. He stopped. He looked about, afraid that people around them would see her crying. "I sold it to go to school, and to go overseas where I can make it. I will get it all back ten times. I will get a job there..."
The food arrived: jollof rice for him and fried rice for her, with meat pie on the side. And in the lull, I flashed in his mind to assure her in stronger words. I reminded him of all the things he'd considered to arrive at the decision. I reminded him of the man who sold his land to send his son to school. I reminded him, Ezeuwa, that he had reckoned that if he got the degree, and returned and married her, he could get a job by her father's influence and could buy a new poultry and build a new coop. And the house? What was it even worth? He had not considered that it may be big but that Amauzunku was one of the worst places in Umuahia. So he could not wait for the waiter to go, and once the waiter was gone, he said, "I will be paying for my life also, and for the woman I love. If I get the degree and I get a good job, I can buy a house ten times better, Mommy. Look at this dirty street. Maybe we can even go to another place, or even, in fact, even maybe Enugu. It is better, Mommy. Actually, it is better. It is better than me allow them to separate us."
But Ndali simply shook her head in a way that he would remember for a very long time. She said nothing more. She ate little and wiped the steady tears that ran down her cheeks. Her sorrow troubled him, for he had not expected that she would react this strongly to his decision. He held her hand as they walked home, but as they drew near the house, she removed her hand. "Your hand is sweating again," she said. He wiped his palms on his trousers and spat into the gutter on the side of the road.
She began to walk alone, a distance from him. He was watching her walk, the swinging of her buttocks with every lithe step visible through the fabric of her tight skirt, when a man on a motorcycle raced past and called at her, "Asa-nwa, how are you?" She hissed at the man and, laughing, the man took off, his vehicle whining. My host, his heart now cleaved, hastened to her. She turned and looked at him, but without a word. He glanced at the disappearing man, at the empty street behind him, as if the world had itself suddenly become empty. For it occurred to him that this might be what she most feared: if he left, other men would come to her. And he wished, then, that this had happened a few days before, when he had not yet sold his house.
As he began to reach for her clothes after they got home later that night, she thrust the camera into his hand, stripped bare, and asked him to take photos of her. His hand shook as he snapped the first photo, which instantly emerged printed from the top of the camera. It was a full image of her erect body, with her supple breasts staring at the camera, and the nipples taut and hard. The pictures were for him, she said. "So that anytime you feel like you want to do it, you can look at the pictures." After he lay down by her side, he wondered if she had done it because of the man who had called to her. And a strange fear came upon him, one that possessed him through the night.
CHUKWU, the old fathers say that the god who created the itch also gave man the finger to scratch it. Although his joy had sprung leaks in response to Ndali's sorrow, once they returned home that evening and she asked him to make love to her, he felt better. She told him she was sad mostly because she would miss him, and he assured her that he would return frequently until she could join him. He said, the degree will be quick, and then he will be done. And he said these things so fervently because he was now afraid of leaving her alone in the interim, exposed to the prying eyes of other men. By the time he was to travel to Abuja the following week, his words had worked and she was no longer steeped in sorrow. She drove him to the bus station and returned to her parents.
It rained heavily the night before his trip to Abuja for his visa, and by morning the storm had caused the main road to close. A great pothole that formed in the center of the road would have drowned every vehicle the size of the Abia Line luxurious bus. The route the driver took was longer, and by the time he got to Abuja, it was almost nightfall. He took a taxi to the cheap hotel where Jamike had suggested he stay, near Kubwa. They knew Jamike, too. They called him _Turkey Man_. "He is a good man, nice guy," the cashier, whose mouth smelt of something like vomit, told him. So taken by the man's words was he that as he took his traveling bag into the room, it occurred to him that he had not yet given Jamike anything in appreciation for his goodness. He'd only bought him beer during the four times they ran around to cyber cafes, the immigration office, the high court to swear an affidavit in lieu of a birth certificate, and to find a buyer for his house.
He became worried by this. He cursed himself inwardly for such an oversight, which may have been interpreted as ingratitude, and decided to call Jamike immediately. He scratched a Globacom phone card he'd bought from the vendor's tent outside the hotel and loaded the phone. After he dialed Jamike did not pick up, and then a foreign voice came on, followed by an English translation. He laughed at the words and the way they had been said. Then he tried again, and this time, Jamike picked up.
"Na who be the fool wey dey call me at this time of the night?"
He was struck as if with a rod to his back. He thought to remain silent so Jamike would not find out it was he who was foolish enough not to have remembered they were in different time zones, but he was too embarrassed to control himself in the way he wanted.
"I say who be that?"
"Am sorry, my brother," he said. "It is me."
"Ah, ah, Bobo Solo!"
"Yes, me. I am sorry—"
"No, no, no, mehn. Na me suppose dey sorry. I just came in today. I was in—"
Jamike's voice disappeared behind a wall of indecipherable sounds, then emerged again with a discordant echo of "ebi," then "ommm," and then it went blank again. "Jami, are you there? Are you there?" he said.
"Yes, Bobo Solo, you dey hear me?"
The talk was interrupted by a warning that the call would soon be disconnected. When it cleared, Jamike was saying, "That's why I neva call you yet. But Solo, you don get the visa?"
"I am in Abuja now. Just today."
"Oh boy! Bobo Solo, the main man!"
"That is—"
The ping went off again, and the call died. He put the phone on the only table in the room—on which sat the TV, a Bible, a laminated card listing the channels on the TV and, on the back, a menu from the hotel restaurant. At one corner of the room, near the closed curtain, a small cockroach clung to the wall, its antennae curved backwards. As he disrobed, the phone rang. When he looked in the screen, it was Ndali.
"I just wanted to see if you had a safe journey," she said.
"Yes, Obim. But the road was very bad. Too bad."
"Blame Orji Kalu, your governor."
"He is a madman."
She laughed, and as she did, he heard the voice of a rooster from some distance in the background.
"Where are you?"
"At your house."
He hesitated. "Why, Mommy? What are you doing there? I said you should go home after feeding them."
"Nonso, I can't leave them here alone because you traveled. What am I, Oyibo or egg?"
Her words cut to his heart.
"I love you, Mommy," he said. Words pooled together in his head, but he hesitated, overcome by the surprise of what she had done. "You are feeding them yourself alone?"
"Yes," she said. "And I picked the eggs."
"How many?"
"Seven."
"Mommy," he said, and when she said, "Eh?," he fell silent. For he could not tell why suddenly he'd become moved to tears. "If you don't want me to actually leave home, I will come back tomorrow. I will return the money for the house and not sell it again. I will ask Jamike to send me back my school fees. Everything, Mommy. After all, actually I have not started school, you see?"
The words had come out with such rapidity that it surprised him to think he uttered them. For even as he spoke, a strange silence formed an integral part of his speech. He knew, once he'd said all that, that he had simply spoken for her sake. He waited for her to respond, his mind light as the feather of a pipit.
"I don't know what to say, Obim," she said after a while. "You are a good man, a very good man. I love you, too. I support your decision—because God has given me a good man." He heard her deep sigh. "Go."
"I should go, Mommy? If you say no, I swear to God who made me, I won't go."
"Yes. Go."
"Okay, Mommy."
"Do you know that the breeder laid pink egg again?" she said.
"Ah, Obiageli?"
"Yes. I fried the egg. Very sweet."
They laughed, and later, long after the call, he wished he had not made the decision to leave. For the rest of that day, the joy that had filled my host's heart was sealed off from him by the partial veil of regret. I, his chi, felt he had made a good decision, and I was convinced that this sacrifice would further solidify Ndali's love for him rather than destroy it. Chukwu, if only I, too, could see the future; if only I could see that which was to come, I would not have thought this foolish thing!
By dusk the following day, when he got to the embassy, the joy returned again and filled his heart so much that, in the taxi back to the hotel, he wept as he looked at the visa in his passport and the Turkish Airlines ticket he'd bought from the place Jamike had suggested. As he returned to the hotel, it seemed to him that something divine had happened to him. Before he died, his father had once said he was sure that his wife, the mother of my host, was watching over her children. He remembered now that his father had said this after my host escaped what would have been a ghastly accident. It was four years ago when he'd boarded a bus to Aba to visit his uncle but had removed himself at the last minute. Just as the bus was about to set off, a passenger arrived carrying bush meat in a jute sack. My host had complained that he could not endure the smell for the duration of the journey. He left the bus and went to another. He would see the bus on the evening news later that day, damaged beyond recognition. Only two people, of all the nine occupants, had survived the crash. Something he did not know, and which even I could not discern, had brought the meat-carrying man and forced my host to leave the bus and escape an untimely death. He resolved now that the same thing may have brought Jamike to him—the hand of some benevolent god, to help him in this time of need. As I have mentioned before, I, his chi, thought it was a result of the good-luck gift he obtained at the garden of Chiokike.
The journey back to the hotel was long, clogged with traffic in various places. He closed his eyes and imagined the future. There were Ndali and he, together in a beautiful house overseas. With much effort, he imagined them with a child, a boy, carrying a big soccer ball. Inchoate and indistinct as these imaginations were, they soothed his spirit. For a long time he had been a lost man riffling through the crammed quarters of life, but now he had found fertile hope, in which anything could grow. At the hotel, he rang Ndali, but she did not answer. While he lay waiting for her to return the call, he dozed off.
ONYEKERUUWA, after he returned to Umuahia with the visa, his journey became more certain, and so, too, did the anxiety and fear that it engendered. The last week before his final travel passed at the pace of a leopard in pursuit of its prey. On the evening before he was to leave for Lagos, where he was to board the flying vessel, he found himself fighting hard to comfort Ndali. For her sorrow had grown in those last days with a fecundity that amazed him, like a cocoyam in the wet season. By that time they had loaded the van with the remaining things he'd not been able to sell. Most were things that had once belonged to his parents. Elochukwu, who had joined them, took the red Binatone rechargeable lantern for his possession. My host let him take it for free. Ndali would have nothing for herself. She'd fought against his selling his things. Since he was taking the van to keep in his uncle's garage in Aba, she asked, why not keep his things with his uncle? Now, as they began packing the contents of the last room, the living room, into his van, she broke down.
"It is not easy for her," Elochukwu said. "You must realize this. That's why she is feeling like this."
"I understand," my host said. "But I am not going to Eluigwe. I am not leaving this world." He pulled her to himself and kissed her.
"I am not saying that," she sobbed. "It is not that. It is just, the dreams I have been having in these past few days. They are not good. You have sold everything, because of me and my family."
"So you don't want me to go again, Mommy?"
"No, no," she said. "I said you should go."
"You see?" Elochukwu said, splaying his hands open.
"I will come back soon, and we will be together again, Mommy."
At this, she nodded and forced a smile.
"That is it!" Elochukwu said, pointing at her face. "She is happy now."
My host laughed, then, holding her, locked mouths with her.
In such moments as this, Egbunu, when a person is about to leave a companion for a length of time, they do everything with haste and heightened intensity. The mind ingests these things and stores them in a special vial because these are the moments it will always remember. This is why the way she held his head and spoke into his face after they finished packing was one of the things he'd always recall of her, time and time again.
After he disengaged from her, he ran into the house in tears. Nothing remained of it but the walls. For a moment, he could almost not recognize any of the rooms. Even the yard looked nothing like it had ever been before. A redheaded lizard stood where his poultry had been only five days before, a crumpled piece of feather stuck to its digits. He'd realized, as they loaded the first things into the van, that in some way, a man's life could be measured by the things he possessed. And he'd paused to take stock. There was the large compound, with its age and its history, and with the poultry in it, that had all belonged to him till then. The small farm, with all its crops and yields, which all belonged to him. So did all the furniture in it, the old photographs—black-and-white daguerreotypes. All the vinyl record albums his father had owned, which almost filled a jute sack, the old radios, bags, kites, and many things. He'd even inherited strange things like the rusting door of his father's first car (the one that had crashed near the Oji River), from 1978. There was his father's hunting rifle, the one with which his father had shot the mother of his gosling; two kerosene stoves; the refrigerator; the small bookshelf near the dining table; the big Oxford dictionary seated on the stool near his father's bed; the ikoro drum that hung on the wall in his father's bedroom; his grandfather's metal briefcase containing the bloodstained Biafran army uniform, with its multiple stitches and missing buttons; the curved knives; his father's box of tools; his sister's remaining clothes, still arranged in her cupboard; dozens of pieces of chinaware; wooden spoons; a cooking mortar and pestle; plastic water jugs; old coffee cans filled with spiders and their eggs; and even the van that bore the name of the farm which, for many years, had been his father's lone car. He'd owned the length and breadth of the land on which he'd grown up. But he'd owned immaterial things, too: the way the leaves on the guava tree created a shower when it rained, dripping in a hundred places; the memory of the thief who once scaled their fence and ran into their compound for safety from the hands of an angry mob threatening to lynch him; the fear of riots; the dreams his father had had for him; the many Christmas celebrations; the memory of numerous holiday travels around the country; the dumbstruck hope that will not speak; the rage that will not unleash itself; the accretion of time; the joy of living; the sorrow of death—all these had all, for a long time, been his.
He looked around, about him, on the fence, at the well, at the guava tree, and everything, and it occurred to him that this compound had been part of him. He would live on from this moment like a living animal of the present whose tail is stretched permanently into the past. It was this thought that broke him the most and which caused him to weep as Elochukwu, who would be handing over the keys of the house to the new owners, locked it all up.
GAGANAOGWU, my host also wept because the young child of a man is born with no knowledge of what he once was in his past life. He is born—reborn, rather—as blank as the surface of the sea. But once he begins to grow, he acquires memories. A person lives because of the accumulation of what he has come to know. This is why, when he is alone, when all else has peeled away from him, a man delves into the world within himself. When he is alone, all of it folds and comes together into this whole. The true state of a man is what he is when he is alone. For when he is alone some of all that has come to constitute his being—the profound emotions, and the profound motives of his heart—rise from deep within him up to the surface of his being. This is why when a man is alone, his face wears a look that is distinct from anything there is. This is a face no one else will ever see or encounter. For when another comes to him, that face retracts like a tentacle and presents the other with something else, something akin to a new face. So thus, alone, throughout the nighttime bus journey to Lagos, my host dwelt in memories, with a countenance no one else will ever see.
Although the odor of the man who flanked him on the right troubled him through the night, he fell asleep many times, his head resting against one of the bags that stretched from the booth up to the back of the seat. He had vivid dreams. In one, he and Ndali are walking down the aisle in a church. There are lights everywhere, even above the images of the saints, Jisos Kraist, and Madonna on the wall behind the altar. This was her church which she often told him about. The priest, Father Samson, is standing with his hands clasped together, a rosary dangling from them. The deep-throated bass drums, played by the altar boy with the big scar on his head, are beating near the small office of the priest. He can see, smiling and dancing, as they precede him, her mother, in exquisite dressing. There is her father, too, and Chuka, his beard even longer now, pronounced against his bright, fair skin. Both of them are smiling, too, both dressed in suits. And he looks down on himself with glee now: the suit he is wearing is the same as theirs! All three, plus the one Elochukwu has on. But who's the third man, fat-cheeked, a rounded head, hair the shape of an island—bare skin all around, then hair shaped conically around it? Jamike, it's Jamike, the man who has come to his aid! He, too, is wearing the same blue suit and a black tie. He is dancing, the very last in the procession behind my host's back, sweating to the beat of the wedding song.
_My wife is given to me by God_
_My husband is given to me by God_
_Because God gave to me_
_It will last till the end of time._
He woke and saw that the bus was riding on a section of the highway flanked by forests, the headlights and those of the cars and trucks and semis that rushed by them the only illuminations in the darkness. He sat up and thought of the previous night, a night that had been tough for Ndali and whose darkness had slowly thickened, like rainwater slowly trickling into a bottle. He could see how she'd struggled through the day, trying hard to conceal her sadness, and he'd had to tell her repeatedly not to cry. But when night came, although she had taken ill and the smell of her sweat had become malarial, she'd asked him to do it because it was their last day. So slowly he'd slid her underpants down her legs, his heart beating feverishly. Once she was bare again, her place ready, her eyes closed, a pleasured smirk on her face, drops of tears on both eyelids, he unbuttoned his shorts. Then, slowly, gently, holding her hand and she hanging her hands around his neck, he'd made love to her. And she'd held him tightly all through, so tight he'd ejaculated in her and the semen slid down from inside her, down his legs.
When he fell asleep again, I floated out of him, as I always did while he is in slumber. But I saw that the bus was crowded with guardian spirits and vagrant creatures, and the din was deafening. One, a ghost, an akaliogoli, who appeared in so thin a mist that it looked like a small stain in the cloth of darkness, sat by a young woman who was asleep in the front seat, her head on the shoulder of another man beside her. The ghost stood there before her, sobbing and saying, "Don't marry Okoli, please, don't marry him. He is evil, the one who killed me. He is lying. Don't, don't, Ngozi, or my spirit will never rest. He killed me so he could have you. Ngozi, please don't." After saying those words, it would wail at the top of its lungs in a shattering, funereal voice. Then it would repeat its entreaties all over again, and again. I watched this creature for a while, and it struck me that it may have been doing this for a long time, probably many moons. I felt sad for it—an onyeuwa abandoned by both its body and its guardian spirit, unable to ascend to Alandiichie, unable to reincarnate. This was a terrible thing!
My host slept through the rest of the journey, and when he woke, it was because the bus had coasted into Ojota Park, and the chaos, the large potholes in the park, had suddenly become a nightmare in daylight. Rain was lightly falling, and the vendors—of bread, oranges, wristwatches, water—were taking cover under a shade, roofed with sheets of zinc and supported by iron pillars, on whose visage the name of the park was engraved in red paint. Some of the women covered their heads with black polythene bags. Braving the rain was a seller of bottled drinks who raced to the bus as it pulled up, squinting. My host disembarked swiftly, worrying about the state of his unwashed mouth. He remembered Ndali had told him to wash it at the airport before his flight. Else he would arrive in Cyprus with bad breath.
Before he could take his two big traveling bags out of the bus, two men, taxi drivers, hurtled forward to take them from him. He let the first, a short, gaunt man with bulging eyes, take them. The man, who lifted the bag with a swiftness that shocked him, was already well on his way out of the bus park before my host realized what the man was doing. He followed the man, carrying his other bag with both hands against his stomach. The rain dropped down slowly on him as he followed the man across the congested traffic, traipsing between honking cars and buses, the air filled with noise. In the distance, a bridge rose, and beyond it, a body of water. Everywhere there seemed to be birds, many of them. The man stopped in front of an unfinished building with bricks filled with holes and on whose veranda a few men sat. His was one of two taxis, badly run down. It was severely dented at the rear, and one of the side mirrors was gone, with only half of the plastic handle still attached. The man tossed his bag into the boot, then took the one my host had in his hand and dropped it, too, on top of the spare tire in the dusty trunk. Then, banging the lid till it closed, he signaled my host to get in. "Airport!" he heard the driver say to one of the men on the veranda. Then, he, too, entered the car.
# TWO
# Second Incantation
DIKENAGHA, EKWUEME—
Please accept my second incantation, the language of Eluigwe, as an offering—
Receive it as an equivalent of _ngborogu-oji,_ the four-lobbed kola nut—
I must praise you for the privilege you give us, guardian spirits of mankind, to stand in the luminous court of Bechukwu and testify on behalf of our hosts—
The fathers say that a child who washes his hands clean will eat with the elders—
Egbunu, the hands of my host are clean, let him eat with the elders—
Ezeuwa, let the eagle perch, let the hawk perch, and whichever says the other should not perch, may its wings break!—
Now, as my host departs from the land of his fathers, his story will change because what happens at the shore of a river is never the same as that which transpires in a room—
A burning log put in the hands of a child by his mother will not hurt him—
A tree that would marry a woman must first develop a scrotum—
A snake must give birth to something as long as itself—
May your ears remain on the ground to hear as I testify on behalf of my host, as I plead with you to prevent Ala from punishing him—
Gaganaogwu, if it is in fact true that what I fear has happened, let it be considered a crime of error, deserving of mercy—
May my account convince you of my host's lack of ill will towards the woman he has harmed—
Egbunu, it is night in the land of men, and my host is asleep, a further proof that this, if it is indeed true that it has happened, is a crime of innocence—
For no one fishes in dry lakes or bathes with fire—
Thus, Agujiegbe, I proceed with my account with boldness!
#
## The Plucked Bird
OKAAOME, I have heard from fathers long dead at Alandiichie who wonder why their children have abandoned their ways. I have watched them lament over the current state of things. I have heard ndiichie-nne, the great mothers, bemoan the fact that their daughters no longer carry their bodies in the ways their mothers did. The great majestic mothers ask why the _uli,_ which the mothers wore on their bodies with pride, is now almost never worn by their daughters. Why is _nzu,_ the pure chalk of the earth, no longer seen on them? Why do cowries blossom and bury themselves in the waters of Osimiri untouched? Why, they cry, is it that the sons of the fathers no longer keep their _ikengas_? From their domain far beyond the reaches of the earth, the loyal fathers gaze around the length and breadth of lands they once dwelt in, from Mbosi to Nkpa, from Nkanu to Igberre, and count the shrines made by men to their guardian spirits and their _ikengas_ on the fingers of their hands. Why are the altars of the chis, the shrines of one's _ezi,_ now forgotten things? Why have the children embraced the ways of those who do not know their own ways? Why have they poisoned the blood of their ancestral consanguinity and shut out the gods of their fathers in outer dark? Why is Ala starved of her rich feathers of young _okeokpa_ and Ozala—Dry-Meat-That-Fills-the-Mouth—without his tortoise? Why, the patient fathers wonder in their solemn indignation, are the altars of Amandioha as dry as the throats of skeletons while ewes graze about unhindered? What they seem not to understand is that the White Man charmed their children with the products of his wizardry. In fact, the venerable fathers and mothers forget that it began in their time.
I inhabited a host more than three hundred years ago, when the White Man brought mirrors to Nnobi, the land of men as valiant and wise as the deities of other places. But they were so enthralled by this, and their women so beholden to it, that that object caused them great anguish. Yet I must say that even then, for more than a hundred years, the people did not abandon the ways of their ancestors. They took these things—mirrors, Dane guns, tobacco—but they did not destroy the shrines of their chis. But their children became convinced that the White Man's magic was more potent. And they sought his powers and wisdom. They began to want what he had, like the flying vehicle into which my host stepped the night he got to Lagos. The children of the old fathers often marvel when they see it. They ask: what is this that men have made? Why is the White Man so powerful? How can men fly in the sky, amongst the firmaments, even higher than birds? These are things I do not understand. Many cycles ago, I embodied a great man who was bound like a sacrificial animal and taken to the land of the White Man. He, his captors, and other captives like him then rode on the great Osimiri, that which we see stretch interminably around the world, even from here in Bechukwu. The journey across this ocean had spanned weeks, so long that I tired of watching the waters. But even then I marveled greatly at how the ship was able to move and not sink, when just a single person could not stand on water.
Imagine, Egbunu, how the children of the fathers must have felt when they encountered this proverb of the wise fathers: _No matter how much a man leaps, he cannot fly_. They should consider why the fathers said this before shaking their heads and thinking of the wise fathers as ignorant. Why? Because a man is not a bird. But the children see something like the plane and they are shocked at how this wisdom has been upended by the White Man's sorcery. Humans fly every day in various shapes. We see them on the road to Eluigwe, filling up the skies in silvered vehicles. Men even make war from the sky! In one of my many cycles on earth, my then host, Ejinkeonye Isigadi, had almost been killed by such a weapon from the air in Umuahia in the year the White Man refers to as 1969. What is more, the old fathers say one cannot converse with another who is in a distant land. Nonsense! Their children must scream because they converse now from afar as though they are lying on the same bed by each other. But this is not even all.
Add to this the attractions of the White Man's religion, his inventions, his weapons (the way, for instance, that he is able to create craters in the earth and blast trees and man to bits), and you understand why the children have abandoned the ways of the illustrious fathers. The children of the fathers do not understand that the ways of the august fathers were simply different from that of the White Man. The old fathers looked to the past to move forward. They relied not on what they could see but on what their fathers had seen. They reckoned that all that needed to be known of the universe had been discovered long ago. It was thus beyond them that a man living in the moment can say, I found this, or I discovered that. It was the greatest arrogance to purport that all who came before a man were slight or careless and that one has happened to see this _now._ So if you asked one of the eminent fathers, why do you plant yam in a mound rather than as seed? He would say, because my father taught me so. If a man told you he could not shake an elder with a left hand, and you ask him why, he'd say, because it is not _omenala._ The civilization of the fathers was hinged on the preservation of that which already existed, not on the discovery of new things.
Elders of Alandiichie, old fathers of Alaigbo, of the black peoples of the rain forest, custodians of the Black Man's wisdom, hear me: these products of the White Man's sorcery are the reasons you now complain and cry and wail about your children like fowls after a hawk attack. It is the White Man who has trampled on your traditions. It is he who has seduced and slept with your ancestral spirits. It is to him that the gods of your land have submitted their heads, and he has shaved them clean, down to the skin of their scalps. He has flogged the high priests and hanged your rulers. He has tamed the animals of your totems and imprisoned the souls of your tribes. He has spit in the face of your wisdoms, and your valiant mythologies are silent before him.
Ijango-ijango, why have I spoken with such a wet tongue about the ancestors? It is because this object that bore my host and others in the sky was magnificent beyond words. All through the flight, even my host—a lover of birds—wondered how it flew. It seemed to him that the propulsion of the plane was by its wings. It soared through the clouds, and over the interminable expanse of water, which had the color of the sky at the end of rainy season. This was Osimiri, the great water body that spread around the circumference of the world. It was the water that contained salt, the _osimiri-nnu_. Your sacred tears, Chukwu.
Out of curiosity, I exited the body of my host and soared out of the plane. I was instantly submerged in this wasteland of noises and spirit bodies. All across the horizon, I saw incorporeal creatures—onyeuwas and guardian spirits and others—traveling somewhere, either descending or ascending with magnificent speed. In the distance, a gray mass of creatures crawled over the illuminate orb that was the sun. I tried not to focus on them but to look instead on the plane, whose wings did not flutter as a bird's. I hovered over it, flying at a strange, unearthly speed as the plane raced on. I had never stopped to watch such a thing before, and it terrified me. I returned immediately into the body of my host. He was still examining the plane in fascination, for it had people, televisions, toilets, food, chairs, and all that may be found in the houses of people on land. But much of his thoughts rested on Ndali.
He soon fell asleep, and when he woke, too many things were happening all at once. The people were clapping and cheering even as a voice returned to the sound boxes. The plane itself had thudded and was now speeding down somewhere he could tell was no longer the air, for he could feel the vibrations against the ground. The plane was also now full of light, both daylight and man-made light from the interior. He slid up the window covering and understood the reason for the commotion. Joy erupted in him, too. He thought that if his father and mother had been alive now, how proud they would have been. He thought of Nkiru in Lagos. He asked himself what she was doing now. He wondered, with mild sadness, whether she now had a child from that much older man. When children of men think of things that are unpleasant, their thinking patterns are not the same as they are when they ponder pleasant things. This was why his mind emphasized the age of his sister's husband. He would call her from here, _Instanbull:_ maybe it would make a difference. It might restore her faith in him as her brother, her only surviving family. But how could he do it? He did not have her number or her husband's. It was she alone who called him from vendor pay phones on special occasions like Christmas, New Year's, sometimes at Easter, and once on the anniversary of their father's death. She'd cried on the phone that day in a way that had shocked him and given him hope that they may yet renew their relationship. But it didn't matter. When she ended the call with the usual "I just wanted to call to know how you are doing," he knew that she would be swallowed again into the void.
He was thrown out of his thoughts by the sudden eruption of clapping and voices. Their faces filled with smiles, people began extracting their bags from the compartments, slugging on backpacks, propping up the retractable handles of their roller bags. The reason for their joy was varied, but he could tell by the clapping and by the shouts of "Praise the Lord" and "Hallelujah" from the back that people were happy the plane had landed safely. He reckoned that it may have been because of the string of recent occurrences regarding planes in Nigeria. For not too long before, a plane carrying dignitaries, including the sultan of Sokoto and the son of a former president, had crashed, killing nearly everyone on board. Only less than a year before that, another plane had crashed, killing a well-known female pastor, Bimbo Odukoya. But he thought even more that these people were happy because they had been lifted from places where they had been suffering into this new country. The plane had lifted out of the land of lack, of man-pass-man, the land in which a man's greatest enemies are members of his household; a land of kidnappers, of ritual killers, of policemen who bully those they encounter on the road and shoot those who don't bribe them, of leaders who treat those they lead with contempt and rob them of the commonwealth, of frequent riots and crisis, of long strikes, of petrol shortages, of joblessness, of clogged gutters, of potholed roads, of bridges that collapse at will, of littered streets and trashy neighborhoods, and of constant power outages.
OLISABINIGWE, the great fathers say that when a man crosses into an unknown land, he becomes again like a child. He must rely on asking questions and on searching for directions. This is also why when they got off the plane, my host did not know what to do. The place they stepped into from the plane, an airport, was massive and filled to the brim with all kinds of people. At first he thought of his big bags, where he'd folded away most of his belongings that weren't sold or burnt or kept with his uncle, but then he remembered he'd been repeatedly told he would pick them up in Cyprus. All he had now was the bag Ndali had given him, in which he kept his admission letters, her letter, photos, and all the vital documents which he needed to present to the school in the new country. The other black people from his country stepped into the chaos, too, and disappeared into the moving stream of people. Whether left or right or behind, they appeared in flashes, amongst the gathering. He walked up into the center of the great hall, where a big clock hung from the roof. He stopped behind an old yellow-skinned couple who stood there gazing at the clock as if it were the body of a man hanging on a tree. A small car came behind him and honked. He stepped aside, and it went on, honking at every stop, as it tried to navigate between the innumerable people crowding along the hallways like it was a market in Umuahia, with the intermittent announcement of arrivals and departures echoing through the expansive hall. He turned and walked in the direction where he'd seen many of his compatriots go.
He'd walked for nearly half a kilometer, passing many curiosities with a load of thoughts in his head, when he came by a fellow with a long beard and dark glasses. He asked the fellow what he should do. The fellow asked him for his boarding card. He took out the slip of paper they had handed him at the airport.
"Your plane to Cyprus will leave by seven. Now is only three, so you must wait. Me sef I'm going there. Just relax, er?"
He thanked the man, and the fellow went his way, walking as if he were slightly dancing. "Relax," the man had told him. It meant wait. It meant too that there are many things that a man cannot control. There are forces that must assemble, things that must come together, an agreed measure of time, and an accepted code which must, in the end, materialize into something that will occasion movement. This was an example of that. To leave here, he must assemble with others who also have paid to get to the same place. When they assemble, they will board the plane. There will be people waiting for them to fly the plane. But let's not forget, Egbunu, that it will happen when the ticking hand of the clock strikes seven. That is what must summon them—him and all these people. In the days of the fathers it was the voice of the village or the town crier and the sound of his gong. As I have spoken about before, the White Man's civilization depends on this. Take away the clock and nothing would be possible in his world.
What must he do while waiting for the hand to strike seven? Relax. But I, his chi, could not relax, for I could sense that something had gone wrong in the realm of the spirit, but I could not tell what it was. Presently, my host found a seat near a place where people gathered, drinking and smoking cigarettes. He sat watching the cubicle, the airy image of a bearded man who moved about as if possessed. It reminded him of how his beard had grown after his father died, how he hadn't shaved for weeks, and how one day he looked at himself in the mirror and laughed at himself for a long time—so long that later he wondered if he had become mad.
Beside him, a white woman was asleep, her eyelids twitching like a child's. He watched her for a few minutes, his eyes on the greenish line of veins along her neck and on her long blue fingernails. She reminded him of Miss J, and he wondered if she was still a prostitute. While he sat there, Chukwu, I came out of him briefly. I had been longing to see what the spiritual world of this place was like, but I had not been able to do so because of my host's uncertain state of mind. Now, once I stepped out, I saw that the place was filled with spirits, some so grotesque in shape and form that they were forever imprinted in my mind. One was arrayed in the misty costume of ancient ghosts and discarnate beings, the palest thing my eye had ever seen. It stood behind a withering white man who sat in a wheelchair, staring vacuously forward. A ghost sat by itself on the floor of the airport, unmoved by people who walked in and out through it. A child kicked a ball through its incorporeal torso, but it did not even stir. It kept on shaking its head, gesticulating, and speaking in quick, dribbling speech in a foreign language.
My host had risen from his seat by the time I returned into him. He walked on for a long time before he chanced upon two Nigerian men who'd sat in the row directly in front of him on the plane. They had just emerged from a very brightly lit store and were carrying the same multicolored bag as many of those in the airport. From the bits of the duo's conversation he'd heard while on the plane, and from the way one of the men carried himself, he knew this man had been living in Cyprus for a while. The man he thought had lived in Cyprus was dressed in a plain jacket and jeans, and his ears were plugged. The other, a man of similar height as my host, wore a cardigan. The man looked unkempt, with sleep in the side of one of his eyes. He bore the countenance of someone who was being inwardly tormented. My host hastened towards them, wanting to know from them what he was to do next.
"Excuse me, brothers," he called after them.
When he came up to them, the man in the jacket moved his bag from one shoulder to the other and stretched out his hand as if he'd been expecting my host.
"Please, are you from Nigeria?" my host said.
"Yes, yes," the man said.
"Going to Cyprus?"
"Yes," the man said, and the other nodded.
"You don go there before?" the other man said.
"No, I neva go before," my host said.
The man looked at the other, who stared at him with a certain curious fixedness as some of those who had been on the same plane walked past.
"I never go, too. In fact, my brother, I wish someone had warned me before I left Nigeria."
"Why?" my host said.
"Why?" the man said, and pointed to the man in the jacket. "T.T. has been there before, and he said it is not a good place."
My host looked up at T.T., who was nodding.
"I no understand," my host said. "What you mean the place no good?"
The other man muttered a faint laugh in response, and then continued to shake his head like one who had uttered a common universal truth only to see that his listener was not aware of it.
"Let T.T. tell you himself. I never go there, I just sit with him on the plane from Lagos and he told me many things."
T.T. told my host about Cyprus. And the things he said were grim. T.T. paused only when my host asked a question—"You mean no jobs at all?"; "No, are you serious?"; "But is it not Europe?"; "No UK or US embassy?"; "They put you in prison?"; "How come?"—but even after he finished telling the story, my host could not believe much of it.
"You know; I am fucked up. Aye mi, oh!" the other man, whom T.T. had identified as Linus during his speech, said. Then he put both hands on his head.
My host turned from these men and muttered to himself that it could not be true, for he was very disturbed. How, he wondered, could there be no jobs in a country _abroad,_ where white people live? Maybe the Nigerian students who were going there were lazy. If the place was as bad as T.T. had just told him, why would T.T. go there himself? These things contradicted everything his friend Jamike had told him about the place. Jamike had assured him that his life would change for the better once he arrived in Cyprus. Jamike had assured him that he could easily own a house soon after and that it would be easy to emigrate from there to Europe or elsewhere.
While the man, T.T., continued on about many people who had been deceived into going to Cyprus, my host listened with half his ears, the other battling with the voice of his head. Chukwu, I flashed it in his mind that this was the right decision. Perhaps, he resolved, it was best to call Jamike and talk to him about these things rather than wait until he came to pick him up at the airport in Cyprus. Indeed—he recalled just as the last thought dissolved into his mind that Jamike had specifically asked him to call his phone once he reached Istanbul. Although T.T. was still speaking—now about a man who, on reaching Cyprus, discovered he had been deceived and now walks about the place in tattered clothes like a madman—my host moved his legs to signal that he wanted to leave. Once T.T. paused, he said, "I wan go call my friend. Make I call am, er."
The duo shook their heads, T.T. with a slight bemused smile on his face. My host went to the phone booth, determined to confirm from Jamike that all the things T.T. had told him were either untrue or just some effort to terrify the other man. Perhaps he was trying to swindle the other guy, and this false information was part of the plot. He must relate to these men with caution. I was thrilled with this reasoning, for I have lived among mankind long enough to know that any meeting between two persons who do not know each other is often dominated by uncertainty and, to a lesser degree, suspicion. If it is a person one has met at a marketplace and with whom one has engaged in some transaction, then there arises the fear. Is he going to cheat me? Is this grain, this cup of milk, this chain wristwatch worth so much? If it is a man who has just met a woman of interest, he wonders: is she going to like me? Will she, if possible, drink with me?
This was what my host had just done. In that flustered state, with questions rushing into his mind like blood from a severed limb, he tottered off towards the other end of the airport to the phone booths. He stood behind two white men in white frocks for the second of three phone booths. From them came the smell of costly perfumes. They both had two of the same polythene bag nearly everyone in the airport carried—bags bearing the inscription DUTY FREE, the meaning of which he did not know. When the men in the frocks had finished their calls, he climbed into the cubicle. He brought out the foolscap sheet on which he'd scribbled Jamike's number and, following the directions on the side of the phone, dialed it. But what returned was a recurrent burst of static and a voice that sometimes came on to announce that the number was invalid before trailing into some unfamiliar language. He repeated the dial, with the same result.
Ezeuwa, not once since I've been with him had he been this shocked by something. He placed the bag he wore on his shoulder on the floor and dialed the number again, that same number from which Jamike had called him only the previous week. He made to ring it again, but turning, he saw that a queue had formed behind him, their faces eager and impatient. He hung the phone back on the grip and, with his eyes still on the paper, made his way through the crowded airport. When he got to the spot where the two men had been before, there was no trace of them. In their stead sat a heavily bearded white man with rheumy eyes gazing stoically at the world as if someone had set it on fire. Ebubedike, it was here that I first got a glimpse of all that was to come.
OBASIDINELU, at the time, I did not know what I had seen, nor did my host. What I knew—and what he knew, too—was that something had gone wrong, but this was not a cause for panic. This was a world in which things go wrong. Most things. And the fact that things had gone wrong did not always mean that disaster was looming. This is why the old fathers say that the fact that a millipede has more than a hundred legs does not mean that he is a great runner. Things can misalign; darkness can mount and encroach on the light of the day; but it will not always mean that night has come. So I did not raise alarm. I let him walk on in search of the two men until he found them, just an hour before the next flight, by a waterfall, gazing into a computer. He ran up to them with the urgency of one fleeing a leopard. When he got to them, he was breathless.
"We went to eat there," T.T. said, pointing at the place on whose threshold hung a notice in the White Man's language that said FOOD COURT. "Have you called your friend?"
My host shook his head. "I have tried and tried, but it is not going. Not going at all."
"Why? Let me see the number. Is it correct—the code? It must be eleven."
He produced the number and T.T. gazed at it in a concentrated manner. "Is this the number?"
"Yes, it is so, my brother."
T.T. shook his head. "But this no be Cyprus number." He waved the sheet. "This is not a Cyprus number at all, at all, believe me."
"I don't understand."
T.T. came closer, pointing at the figures on the paper.
"Cyprus has Turkish number. TRNC. It is plus nine zero. This one is plus three four. Not Cyprus number at all."
My host stood still, like a bird transfixed in its thermal.
"But he has called me several times," he said.
"On this number? It is not a Cyprus number, believe me," T.T. said. "Did he give you any address to meet him?"
He shook his head.
"No address. Ah, okay. Did he give you any letter? How you take get your visa?"
"Him send me admission letter," my host replied. "I take am to the embassy."
He opened the small bag, and in haste, he presented a paper to T.T., and he and Linus peered at it.
"Erhen, he contacted the school. I see this na genuine admission letter." He started to speak in response, but T.T. continued, "He paid the school fees, too, since this is an unconditional admission letter. I ask because I have seen many occasions where boys just barbed people's heads. They pretend that they are the agents of the school and take their money. But they don't pay the fees at all. They just eat the money."
Ijango-ijango, my host was stunned. He tried to say something, to thaw up a piece of the thoughts that had congealed into a lump in his mind, but the lump would not thaw. In silence, he took the paper back from T.T.
"Still, I think this Jamike guy is a yahoo boy," T.T. said, shaking his head. "My bro, I suspect he has duped you."
"How?" my host said.
"Did you contact the school directly?"
He wanted to say he did not, but he found himself shaking his head instead. In response, a small smile appeared on T.T.'s face.
"So you didn't?"
"That is so," he said. "I have the admission letter with the school's stamp and everything. Actually, I even saw his student ID. We browsed the school together for cyber cafe. Jamike is a student there."
T.T. answered with silence, and beside him Linus stood watching, his mouth slightly ajar. My host gazed at both men, almost trembling.
"Hmm," T.T. said.
"He paid the school fees because the school only accepts Turkish bank checks or international money order. They don't accept bank transfers from Nigerian bank here," my host said. He saw the woman who had been sleeping earlier walking past them, dragging a bag behind her. "Since he was going back there, I just changed my naira and give him all the money."
He was continuing, but he saw that T.T.'s mouth had been opened wide in surprise, and even the other man shook his head and said, "You for no give am all that money."
T.T. pointed to a gate in the distance, where many of those who had been on the plane from Nigeria had started to line up, and said, "Ah, we must go board now." T.T. took up his backpack and hung it up his back. My host watched as Linus picked up his own things. And for no apparent reason, he remembered his gosling, how—in times when it seemed it had remembered its mother and the place of its provenance—it would lift itself and dash for the window, the door, wherever it could find. How once, in a bid to escape, it had assumed the tree it had seen through the window meant it could get outside. So with violent speed it dashed against the window. Concussed, it lay as if dead.
"Are you not coming, too?" T.T. said, and my host looked up, saw the gosling lying there, at the base of the wall, its head bent towards its neck and its wings batting against the ground.
He blinked, closed his eyes, and when he opened them, he saw T.T. encompassed about by myriads of lights and screens.
He nodded. "I am coming, too," he said, and followed them.
"Maybe you go meet Jamike at Ercan. The airport," T.T. said. "No fear, er? No fear."
The other man nodded, too. "No shaking, nothing dey happen. No fear, fear go fear fear!"
He nodded again and said, as if he believed it, "I no go fear."
AKWAAKWURU, the great fathers often say that a toad whose mouth is full of water cannot swallow even an ant. I have seen them apply this to the way in which the mind of a man, when it is occupied by something that threatens its peace, becomes consumed by it. This was the case with my host. For throughout the flight, his mind was hooked to the words of the two men who were now seated at the rear of the plane. He sat close to the front, surrounded by more white people than were in the previous, bigger plane. They were mostly young girls and boys who, he assumed, were students, too. Even the woman who sat by him, with hair that was brown and long, seemed to be one. And all through the flight she avoided eye contact with him, gazing in her phone or into a glossy magazine. But as he sat there, fear, having transformed itself into a mind-dwelling rat, ferreted about in his head, chewing through every detail. And when he looked out the window as he drew near the country, what he saw seemed to reinforce the grim words of the duo. For instead of the tall buildings and the long bridges over the sea he had seen as they landed in Istanbul, what he saw now were dry patches of desert land, mountains, and the sea. By the time he found himself descending the ladder from the plane into the dim light of the setting sun alongside the other travelers, the details had blossomed into genial terrors.
The airport, to his eyes, was small. It looked, in many ways, like the one in Nigeria except that it was cleaner and more orderly. But it did not have the beauty or sophistication of the one in Istanbul. It was cheap, exuding no glow or pleasantness, conforming in every way and sense with T.T.'s description of it. Once he saw the men whose words had tormented him through the flight, he went to them. He found them with another man, who introduced himself as Jay and who was talking about his time in Germany. They stood at a place where most of the people had gathered, watching as a black hole vomited up their bags. His two bags came out with their padlocks intact, their weight as he last remembered them. Someone had mentioned how those who loaded bags into the airplanes at the airport in Nigeria sometimes broke into people's bags and stole things during the process of transferring them into the plane. This had not happened to him. He dragged and pulled the roller bag along, carried the other by its handle, and followed the two men. They were still talking, this time about the attitude of the women in both countries—this one, which T.T. repeatedly referred to as TRNC or "this island," and Jay's Germany. He listened, his mind still tethered to the phone booth at the airport in Istanbul.
When they stepped out of the airport, darkness had descended with lithe grace, and an unusual smell hung in the air. A pool of cars gathered in front of the airport greeted them. Turkish-speaking men, gesturing towards various black Mercedes-Benz or V-booths, beckoned to him.
"They are taxi drivers," T.T. said. He'd put on a cap and the gleeful countenance of a person who had returned home. Nothing in him betrayed the dire situation on the island he'd so painstakingly described. Still wearing that curious smile on his face, T.T. spoke with one of the men, an unusual white man, nothing like any my host had ever seen before, even on TV. This one's face was wrinkled beyond normal, and his complexion, although white, seemed to have an unusual dark hue to it. Half the man's head was full of black hair, but the roots at the sides of his head were gray.
"That's our bus there!" T.T. said, breaking off from the taxi driver to point to a big bus, brightly lit on the inside, slowly riding towards them from the other end of the park. On its body was inscribed NEAR EAST UNIVERSITY, its equivalent in Turkish beneath it.
"We go dey go be that," T.T. said, turning to him. "That's our bus there."
My host, gazing up at the bus, nodded.
"No worry, bro. Just wait for your friend here. I am sure him go come."
"It is so. He will. Thank you, T.T. God bless you."
"No mention. Just wait here, eh. And if him no come, just take the next CIU bus wey come. Your school bus. E go come same here, probly later. Cyprus International University. Just follow am. Show dem your admission letter—where is it?"
His mind working in distinct haste, he produced the paper from the small bag he carried, but as he did, the foolscap on which Jamike had scribbled the expenses and all it would cost him, as well as his phone number, fell.
"Good," T.T. was saying as he picked up the paper. "Best of luck, bro. Maybe we go come see you. Take my number."
My host took out his phone from his pocket to register the number, but it did not turn on when he flipped it open.
"The battery don die," he said.
"No problem. We go go now. Bye."
Gaganaogwu, by this time, my host had begun to believe that the things he'd heard from T.T. were true. Although he started to wait, he thought it was unlikely that Jamike would come. Even though a chi can see into the interior of a host's mind, it is sometimes still difficult to determine where an idea comes from. This was the case with this idea. It was, I think, a collection of things he had been seeing: the quality of the airport here, the behavior of the drivers, the emptiness of the land, and the problem of communication. These confirmed his worry. I pushed the thought into his mind that it was too early to lose hope. I threw his father's motto into his mind— _Forwards ever, backwards never_ —but it hit the door his mind had erected around his fears and ricocheted away. Instead he thought of home, of Ndali, what she must be doing at that time. He remembered the anguish of selling his chickens—how, as he dropped the cage of the brown broilers off at one of the buyers', he had nearly choked. He looked now at the two heavy bags in his hands which bore all his remaining possessions—what he did not sell or gift to Ndali or Elochukwu, or charity, or throw away. And these things solidified his fear that something had gone wrong.
He warded off the advances of taxi drivers time and time again. They came at him, speaking in halting language he did not comprehend, their words cadenced with a clicking accent. As night fell, the men continued to call at him, until most of the cars emptied out of the park. But there was still no Jamike. He'd waited for nearly two hours when he remembered that Jamike had told him he would be given free temporary accommodation during the first two nights at the school until he was able to choose a campus apartment. These were Jamike's words, spoken at a time when the waters were calm, and they came to him now in this moment of great roiling waters, of the torment of fear, and of fainting hope.
CHUKWU, the road from the airport to the town seemed as far as the journey from Umuahia to Aba, except that it was smooth, not damaged by erosion or potholes. During the ride, he gazed at the country and its strange and foreign landscape. As he registered every discernible sight, every detail of the things the men had told him worked on him like the hands of a fowler, pulling out feather after feather so that by the time the desert came into view, he'd been completely deplumed. And he, now bald and feeble, hopped about in the plains of fear. The taxi was circling a roundabout when he recalled something Jamike had said about the absence of trees, and it struck him that he had yet to see one tree so far in the journey. He saw the broad attributes of the sierras, one of which was adorned with the lighted outline of a huge flag. It occurred to him that he'd seen the flag before, although he could not recall that it was perhaps at the Turkish embassy in Abuja.
"Okul, burda. School. School," the man said when they arrived at a place in front of which was a short but long long brick wall bearing the name of the school.
He saw the school—a group of unusual buildings linked together, the darkness like a still river around them. Around and about the strange smell he'd perceived at the airport lingered. The man pulled up in front of one of the buildings, a four-story one in front of which was a table with three people seated. Behind them was a board on which there was a world map—a drawing showing the White Man's knowledge of the world. He paid the driver twenty euros. The man gave him some Turkish lira and coins in return and unloaded the bags. One of the people at the table, a man with shocks of gray hair, came to meet him. The man looked like the people from a place far from the country of the fathers, a place called India. My former host, Ezike Nkeoye, once knew such a man as a teacher. The Indian man introduced himself as Atif.
"Chinonso," he said, taking the man's offer of a handshake.
"Chi-non-so?" the man said. "Do you have an English name?"
"Solomon, call me Solomon."
"Better for me," the man said, and smiled in a way my host had never seen before, for it seemed as though the man's eyes were completely closed. "Did you ask to be picked up from the airport?"
"No, I was waiting for my friend, Jamike Nwaorji, your student, here, CIU, to pick me from the airport."
"Oh, okay. Where is he?"
"He didn't come."
"Why?"
"I don't know, actually, I don't know. Do you know where he is? Can you find him for me?"
"Find him?" the man said, and turned to reply to something one of the others at the table, a thin white girl, had said to him in the language of the land. When he turned back, he said, "I'm sorry, Solomon. What's your friend's name again? I might know him if he is a student here. There are nine students from Africa in this university, and eight of them are from Nigeria."
"Jamike Nwaorji," he said. "He is from Business Administration, from business department."
"Jamike? Does he have another name?"
"No. You don't know him? Jamike. J-a-m-i-k-e. His surname is Nwaorji: N-w-a-r, no, sorry, N-w-a-o-r-j-i."
Atif shook his head and turned back to the desk. My host had dropped his big bag on the ground, and his heart pounded as he waited for the Turkish girl to stop talking to the man again. The third person, a stout man with a large beard, snapped open a canned drink. The drink swished and dripped and foamed over his hand to the ground. The man shouted something that sounded like _Olah_ and began to laugh. For a moment, they seemed to all forget my host.
"His name is Jamike Nwaorji," he said softly, making sure he said the surname as clearly as he could.
"Okay," the girl said now. "We are looking at the list, but not find this man, your friend."
"There is no such person here that I know. And now, I have looked at the business department, the only Nigerian there is Patience, Patience Otima."
"Nobody like Jamike Nwaorji?" my host said. He looked up at the two people on whom, it felt to him at the moment, his life may depend. But he saw in their faces, in the way they gazed at the records, that he would find no succor there. "Jamike Nwaorji, nobody like him?" he said again, and the words dragged in his mouth this time, inflected by subtle gasps whose origin seemed to be from his bowels. He placed his hands on his belly.
"No," the man said with something that sounded like a _p_ at the end of it. "Can I see your admission letter?"
Egbunu, his hands were shaking as he brought the paper out of the bag he'd been carrying for all the time he had been away from Umuahia, almost two full days. He watched as the man peered at the roughened paper, conscious of the man's every blink, calculating every change in the man's countenance, terrified by his every move.
"This is real, and I can see that you paid your school fees." He looked my host in the eye, and then scratched the side of his head. "Let me ask you a question: Did you pay for on-campus accommodation?"
"Yes," my host, slightly relieved now, said curtly. Then he explained that he'd sent Jamike money for accommodation for two semesters. He brought out the piece of foolscap on which Jamike had scribbled the breakdown of costs and, pointing at the different figures, said, "I paid one thousand five hundred euros for one-year accommodation. Then I paid three thousand for one-year school fees, and two thousand euros for maintenance."
Something in what he'd said surprised the man, Atif. The man flipped open another file and began searching frantically for his name on a list of names. The girl, too, joined in, and even the other man with the drink. They all peered from behind Atif's shoulders. A taxi like the one that had brought him pulled towards them slowly. As it came, Atif raised his head to tell him that even on this list there was no name similar to his. In the next file, too—which was for campus apartments, where most Africans stayed because they didn't always like Turkish food served exclusively in dormitories—his name was not there. In the list of registered apartments subsidized by the university, his name was not there.
When he'd looked everywhere and could not find my host's name, Atif turned to him and said it would be fine. Egbunu, this man said this to a person who—like a fowl—had been deplumed and was now bare before the world. Atif continued to say this as he took him across the campus, up to a four-story building similar to the one in front of whose facade they had pitched their table, and up into one of the temporary accommodations, where he could stay for five days. Then Atif shook the hand of a man who had been dealt a crushing blow and said, without any shadow of doubt, that all would be well. And, as often happens everywhere among mankind, this man—deplumed, in agony, in despair—nodded and thanked the man who had said these things to him, just as I have seen men do many times. Then the man said to him, "Just relax and sleep. Good night." And my host, reckoning that he must do as he had been told, nodded and said, "Good night, too. See you tomorrow."
#
## The Wayfarer in a Foreign Land
EZECHITAOKE, the early fathers say in their peripatetic wisdom that one's own language is never difficult. Thus, because my host arrived in a place I did not know, I must recount everything here, every bit of the next few days, every bit, for my testimony tonight to bear weight. I ask that your ears be patient in hearing me.
AGUJIEGBE, I have spoken already about the poverty of anticipation and the emptiness of hope for the future. Now I would like to ask: what is a person's tomorrow? Is it not to be likened to an endangered animal who, having escaped from a pursuer, arrives at the mouth of a cave whose depth or length it does not know and within which it can see nothing? It does not know whether the ground is filled with thorns. It doesn't know, cannot see, if a more venomous beast is in the cave. Yet it must enter into it; it has no choice. For to not enter is to cease to exist, and for a man to not enter through the door of tomorrow is death. The possible result of entering into the unknown of tomorrow? Numerous possibilities, Chukwu, too numerous to count! A certain man may wake up joyful because he has been told the day before that he will be promoted at work that morning. He embraces his wife and leaves for work. He gets in his car and does not see the schoolboy run into the road in fear. In a second, in the batting of an eyelid, the man has killed a promising child! The world heaps a great burden on him at once. And this burden is not an ordinary one, for it is something he cannot unburden himself of. It will remain with him for the rest of his life. I have seen it many times. But is not this, too, the tomorrow the man has entered?
My host woke up in the new country the next morning after he arrived, knowing only that things were different here, unaware of what awaited him in this new day. He knew that there had been uninterrupted electricity, and he'd plugged in his phone so it could charge all night. And throughout the night, he did not hear a cock crow, even though he'd been awake for most of it. It seemed that in the country from which he'd come, there was noise, constant grinding of some machines, constant shouts of children playing, weeping, the honking of cars and motorcycles, acclamations, church drums and singing, muezzins calling from mosques' megaphones, loud music from some party in full swing—and the source of the constant animated sound is boundless, innumerable. It seemed as if the world of the country abhorred calm. But here, there was calm. Even silence. It was as though everywhere, in every house, at every moment, funerals were going on, the kind in which one could only utter a muffled gasp. Despite this quiet, he slept very little, so little that even now, at daybreak, he still felt a need for sleep. During the night, his mind had become a carnival fair in which wanted and unwanted thoughts danced. And as the carnival went on, he could not close his eyes.
When he walked out of the room, the day offered him a black man, naked to the waist, who stood washing his hands in the kitchen sink.
"My name is Tobe. I am from Enugu. Computer Engineering—doctorate," the man said, and moved away from the glare of the sun that was shining through the naked windows.
"Chinonso Solomon Olisa. Business Administration," he said.
He shook hands with the man.
"I saw when Atif was bringing you in last night but I didn't want to disturb you. I was at the other apartment with some of the old students. Apartment five." The man pointed to a building through the window. It had yellow-colored walls with red brick columns on the sides and wide balconies in front of the four stories. On the red iron balcony of the one he pointed to, a black man with enormous hair and a big comb tucked into it stood leaning against the wall, smoking. "There are three Nigerians there, and all of them came last semester. They are the old students."
My host, stirring, looked in the direction of the place, for a glimmer of hope had sparked within him.
"Do you know their names, all their names?" he said.
"Yes, what happened?"
"Can you—"
"One is—that one is Benji. Benjamin. The other is Dimeji: Dee. He came here before many of them. The third one is John. He is Igbo, too."
"No one called Jamike. Jamike Nwaorji?"
"Ah, no, no Jamike," the man said. "What kind of name is that, sef?"
"I don't know," he said quietly, beaten back from the door of the apartment where, in that brief moment, his heart had traveled. Yet he kept his eyes on the place and saw that the man, Benji, had gone back in and another man and a black woman were exiting the door.
"Can you come introduce me to them? I want to see if any of them know Jamike."
"What happened? What do you need? You can tell me."
He gazed at this shirtless hirsute man whose eyes lay deep in his head behind his large-rimmed glasses trying to decide whether or not to be discreet. But the voice in his head, even before I could stir, nudged him to tell his story; perhaps this man could help him. And with so much care, he told the man the story up to that point. At first he spoke in the language of the White Man, but midway through the story, he asked the man if he spoke Igbo, which the latter affirmed, as if annoyed by the question. Now, given a softer bed to sit on, he spoke in excruciating detail, and by the time he was done, the man told him he was certain he'd been duped. "I am certain," Tobe said, and then began describing many scams he'd heard about, comparing the similarities.
"Wait, and when you called him, er, you discovered the number was fake?" Tobe said presently.
"That is so."
"And he did not come to the airport, I am sure?"
"It is so, my brother."
"So you see what I tell you? That he must be fake? But look, let's go first, let's try to find him. It is possible he is not what we think. Maybe he drank and forgot to come to the airport—people party a lot on this island! You know this can happen. Let us go buy a phone card so you can call him until he picks up. Let us go."
The new country presented itself to him outside the apartment with a jolt. The ground was paved with what looked like bricks flattened into the earth. There were flowers in vases, and a host of flowers was placed outside, on the balconies of the houses. The buildings appeared different from the ones in Nigeria, even in Abuja. There seemed to be some finesse to their crafting that he'd never seen before. A building made almost entirely of glass, long and rectangular, caught his attention in the distance. "The English building," Tobe said. "That is where all of us will take our Turkish Language lessons." While he was still speaking, two white boys, dragging bags, one of them smoking, called at them.
"My friend! _Arkadas_."
" _Arkadas_. How are you?" Tobe said, then drew near and shook hands with the men.
"No, only English," the white man said. "No Turkish."
"Okay, English, English—English," Tobe said in an affected accent, his voice altered to mimic the language of these people. As my host watched them, he wondered if this was how one lived here. Did one put on a new voice every time one spoke with one of these people? When Tobe rejoined him, I thought he would ask Tobe questions, to try to find answers to the questions now crowding his mind but he did not. Agujiegbe, this was a strange trait in this host of mine, something I had seen in few others in my many cycles on the earth.
On the way to the place where they would buy phone cards, Tobe said school was to begin on Monday, and some students were starting to arrive. He said that the campus would be filled by Sunday night, in four days.
They arrived at a building with two glass doors and an assortment of things inside, what he thought was some kind of expanded supermarket. As they entered it, Tobe turned to him. "This is Lemar, where we will buy the SIM card. You will use it to call Jamike again."
Ijango-ijango, Tobe spoke with so much authority over my host, as though he were a child who had been handed over to Tobe for guidance. I saw this man as the hand of providence sent to help my host in this time of distress. For this was the way of universe: when a man has reached the edge of his peace, the universe lends a hand, usually in the form of another person. This is why the enlightened fathers often say that a person can become a chi to another. Tobe, now his human chi, took him where the telephone cards were, and Tobe himself tore open the wrapping of the SIM pack and gazed keenly at it, as if to ensure he had picked out the good apple from a basket before handing it back to the toddler in his care with the words, "Okay, it is good, it is good. Now scratch the Turksim like MTN or Glo scratch card."
My host scratched the card outside the supermarket, near an open patch of land covered with wild, clay-colored earth that had caused Tobe to repeat the word _desert_. He keyed in Jamike's phone number. As it connected, he closed his eyes until the line trailed into the fast-clicking language, after which came the wounding end statement: _The number you have called does not exist. Please check the number and try again_. When he brought the phone down from his ear, he glanced up at Tobe, who'd leaned close and picked up the strange voice in his own ear. Now my host nodded.
He let Tobe decide the next steps, and Tobe said they should head to the "international office."
—What is there?
—A woman they call Dehan.
—What would she do?
—She might help us find Jamike.
—How would she do it? His number does not exist?
—Perhaps she knows him. She is the international officer in charge of all the foreign students. If he was a student here, she must know him.
—Okay, let us go, then.
CHUKWU, with my host growing in desperation and myself increasingly convinced that what he feared had happened to him, he followed Tobe to the office. They went between long trails of beautifully cultivated flowers, and the vegetation of the strange new land opened to his eyes while his heart wept secretly. Here and there young white people swept by, many of them female, but he did not so much as look at them. In the state into which he'd been thrown, Ndali hovered like an unusual shadow, one that shone in the horizons of his darkened mind like something made of steel. At the office, which was located on the ground floor of a three-story structure with the words ADMINISTRATIVE BUILDING etched on it, Dehan, the international officer, received them with a disarming smile. Her voice sounded like that of a singer whose name he could not immediately recall. In her presence, Tobe looked flustered as he returned to the forced accent. They sat down on the chairs across from her. Dehan swung in her chair while he spoke and then began picking among the papers on her desk. When she found the one she was looking for, she said that indeed my host's admission had been procured by someone on the island. But she had only corresponded with this person by e-mail. She wrote down the e-mail, the same one my host had: Jamike200@yahoo.com. Dehan brought out a file containing his documents and set them on the table. Tobe, seeming certain that he would see things, began looking through the papers and counted the new revelations as he found them:
The school fees he thought had been paid had been only partly paid. Only one semester, not two. One thousand, five hundred euros, not three thousand. In regard to the accommodation he thought he had paid, as Atif rightly observed, nothing was paid. Nothing. "Maintenance"—which Jamike had said the school required you to deposit in a verified bank account to ensure that you have enough to live on while at school, so you do not need to work illegally—that, too, was nonexistent.
It seemed that this woman, Dehan, was puzzled by the term _maintenance_. "I've never heard it before," she said, gazing with perplexity at them. "Not in this school. He lied to you, Solomon. Really. He lied to you. I'm very sorry for this."
Egbunu, he took the news that after all the school did not have any money in an account for him with a kind of relief, a mysterious kind. They left the office afterwards carrying Dehan's comforting words, "Don't worry," like a banner of peace. Such words, said to a man in dire need, often soothe him—even if for a moment. Such a person would thank the person who had given him the assurance, as my host and his friend did, and then they would leave with a countenance that communicates to the person that they have been comforted by their words. So my host carried with him the file containing the original copy of his admission letter and unconditional admission letters as well as the receipt for his school fees, which was the only document that bore Jamike's name and the date: 6 August 2007.
As they stood resting under the pavilion of a building Tobe pointed out to him as housing his department, the Ceviz Uraz Business Admin Building, he remembered the day before that day—the fifth of August. He could not tell why he remembered this, as he did not always think in dates as the White Man had framed them but in days and periods, as the old fathers did. Yet somehow, that date had been burned into his mind as if by a blacksmith's rod. It was the day he received the full payment for his compound: one million, two hundred thousand naira. The man to whom he sold it had brought it in a black nylon bag. He and Elochukwu, wide-eyed, had counted it, his hands shaking, his voice cracking from the enormity of what he had just done. He remembered, too, that it was just after Elochukwu and the man left that Jamike called to tell him he had paid his school fees and that he should send the money and the accommodation fees as soon as possible.
Oseburuwa, as his guardian spirit, one who watches over him without cease, I'm at once thicketed in regrets whenever I think about his dealings with this man and all that it caused him. I am even more disturbed that I did not suspect anything in the least. In fact, if there had been a shadow of misgiving about Jamike, it was immediately dissolved by his enormous generous act. He—and I, too—thought Jamike was not serious when he promised to pay the school fees with his own money so my host didn't have to rush the sale of the house and poultry and could wait until he found a good bargain. So it was with this unbelief that he drove to the cyber cafe on Jos Street and found the document Jamike had said he needed for the visa, the "unconditional admission letter," sent to him through this medium that could best be described as an arrangement of calligraphed words on a screen. The letter, he saw, had come from the same woman they had just met, Dehan.
He recalled now, as they walked past a group of white female students playing on a field and a group of white men smoking, how, after the cafe attendant printed the letter for him, he'd gone straight to the bank with the money and requested that the bank send the equivalent of six thousand euros to Jamike Nwaorji—Jamike Nwaorji in Cyprus. He'd waited, and when the deal was completed, he returned home with the receipt showing that the bank had converted his naira into euros at the rate of 127 naira each. He'd gazed at the figure the bank woman had underlined as the total in her slanting handwriting: 901,700, and what was left of the sum for which he'd sold the compound, 198,300. He recalled how, at the time, as he drove home from the bank, his mind had been split between gratitude to Jamike on the one hand, anxiety about parting from Ndali on the other hand, and the disquiet that came from the feeling that he may have betrayed his parents.
Although deep within, my host was now cautious and suspicious of the motives of others, he saw in Tobe a genuine desire to help him. So again, Chukwu, he sought to reward this man by letting him lead the way. A man like Tobe is often paid for his pains by the gratification that comes from being in charge, leading his one-man—grievously wounded, disarmed, dispirited—infantry on. I have seen it many times.
Now, Tobe said they should go to TC Ziraat Bankasi, and he knew where it was—at the city center of Lefkosa, beside the old mosque.
"What will we do there?" my host said.
"We will ask about the money."
"Which money?"
"The maintenance money Jamike, that stupid thief, was supposed to deposit in an account in your name."
"Okay, then we should go. Thank you, my brother."
So they got on the bus that was to take them to the city center, a bus like the one that had come to the airport the previous day to pick up students while he waited for Jamike. Seated there were several Turkish or Turkish-Cypriot people, as he came to believe most of the people here were. A woman sat with a pink plastic bag on her thighs beside another, a yellow-haired girl in sunglasses to whom, on a different day, he would have given a sustained gaze. Two men in short pants, T-shirts, and bathroom slippers stood behind the driver's seat, chatting with him. A black man and woman sat behind Tobe and him. Tobe knew them; they had come on the same plane as he did. The man, who was named Bode, and the woman, Hannah, argued that Lagos was ten times better than Lefkosa. Tobe, a loud talker, engaged them. Tobe disagreed, contending that if nothing else North Cyprus had constant electricity and good roads. Even their currency was better.
"How much is a dollar to their money? One point two TL to a dollar. Our own? One twenty! Can you imagine? One hundred and twenty-something naira! Common dollar, oh. And euro nko, it is one seventy! And you say it is better?"
"But you say that their money be the same as ours?" the other man said. "They just devalue am ni. If you look well, sef, you go see say if you change hundred naira, or wetin you go buy here for one tele in Naija, na hundred naira. Our money just get more zeros. Na why Turka people still dey call one thousand one million."
"Yes, it is the same. I agree. Ghana did the same—"
"Ehen!"
"They canceled zeros and rewrote their currency," Tobe continued.
Chukwu, my host listened with half his mind, determined to say nothing. He reckoned that only those for whom all was well could engage in such trivial chatter. For him, he was far removed. He now inhabited a new world into which he'd reclined, gaunt and constricted, like an insect in a wet log. So he let his eyes roam the bus, perching like a weak fly on everything from the images on the sides of the bus to its roof to the foreign writings along its door. It was thus he who first noticed the two Turkish girls who'd boarded the bus at the last stop, outside what looked like a car-selling park identified by the bold inscription LEVANT OTTO. He'd noticed, too, that the girls were no doubt talking about his compatriots and him because they were looking in their direction, along with others in the bus who knew their language. Then one of them waved at him, and the other pushed herself towards him. My host cursed inwardly, for he did not want to speak to anyone; he did not want to be stirred out of the wet log. But he knew it was too late. The women had assumed he would speak with them, and had come towards him, and stood in the aisle between the empty seats. One of them, waving her painted fingers, said something to him in Turkish.
"No Turkish," he said, surprised at how husky his voice sounded even though he'd not been speaking much. With his eyes, he directed them to Tobe, who turned presently.
"You speak Turkish?" the girl said.
"Little Turkish."
The girl laughed. She said something of which Tobe understood not a word.
"Okay, no Turkish. English? _Ingilizce_?" Tobe said.
"Oh, sorry, only my friend, English," she said, turning to the other, who was hiding behind her.
"Can we, emm, _sac neder mek ya_?"
"Hair," the other said.
_"Evet!"_ the first girl said. "Can we hair?"
"Touch?" Tobe said.
" _Evet_! Yes, yes, touch. Heh. Can we touch your hair? It is very interesting for we."
"You want to touch our hair?"
"Yes!"
"Yes!"
Tobe turned to him. It was clear that Tobe was willing to have these girls feel his hair. He was a dark-skinned man with hair that mimicked the scant vegetation of the desert, which the girls wanted to touch. It didn't matter to Tobe, and my host thought it should not matter to him, either. It should not even matter that he still could not account for the one point five million naira which was what his house and the rest of the money for which he'd sold his poultry had become. It did not matter, either, that while trying to solve a problem, he'd pushed himself into an even greater quandary, one even bigger than what had come before. Now these two women, strangers, white-skinned, speaking in a language he could not understand and in a mangled, tattered version of the language of the White Man, wanted to touch his hair because _they found it interesting_. Agujiegbe, as Tobe bent his head so that the girls grazed their hands over his frizzy, uncombed hair, my host placed his, too, under their hands. And the white hands, thin fingers with painted nails of various colors, ran over the heads of the two children of the old fathers. Giggling, their eyes alight, they asked questions as they touched, and Tobe answered swiftly.
"Yes, the hair can be longer than this. If we don't cut."
"Why is it curly?"
"It is curly because we comb it, and we cream it, too," Tobe said.
"Like Bob Marley?"
"Yes, our hair can become like Bob Marley. Dada. Rasta. If we don't cut it," Tobe said.
Now they turned to Hannah, the girl from the country of the fathers.
"The girl there, is that her hair?"
"No, it is an attachment. Brazilian hair," Tobe said, and turned to Hannah.
"These Turka people sef, dem no sabi anything oh. Tell am say na so the hair be jare," Hannah said.
"Is the hair of the black woman, eh, eh long?"
Tobe laughed. "Yes. It is long."
"So why you put another hair?"
"Just cosmetic. Because they don't want to plait their hair in African braids."
"Okay, thank you. It is very interesting for we."
ONWANAETIRIOHA, I was dwelling in a host who did not live beyond the age of thirteen when the first white men came to Ihembosi. The fathers laughed at them and would go about for days on end mocking the stupidity of the White Man. Ijango-ijango, I recall vividly—for my memory isn't like that of man—that one of the reasons the fathers laughed and thought of these people as mad was because of the idea of "banking." They had wondered how a man in his right senses could take his money and sometimes all his livelihood and deposit it with others. This was beyond folly, the wise fathers thought. But now the children of the fathers willingly do this. And in ways that still defy my understanding, when they go, they receive their money back and even sometimes more than they had put in!
This place where my host and his friend arrived was such a place—a bank. Just before they entered, he remembered his gosling; one day he returned from school and found it in its cage, its eyes closed, almost as if swollen. His father was traveling, and he was alone. At first he'd become very afraid, for rarely did he find the bird asleep like this, at least not before feeding on the bag of termites and grains he bought it. But just before he even tapped the cage, the bird rose, raised its head, and made a loud call. At the time, he'd kicked himself for becoming afraid too quickly.
So in serenity, he sat in this bank, which looked like the ones in Nigeria—lush and exquisitely decorated. He told himself to wait and see what they would find, to not be afraid too quickly. He waited with Tobe near an aquarium in which gold and yellow and pink fishes swam up and down over the imported pebbles and artificial reefs. When it was their turn, Tobe went up and spoke to the man at the counter. And in words my host would not have been able to find, Tobe explained the situation.
"So if I hear you clearly, you want to know if your friend has an account with us?" The man spoke fluently and in an accent similar to the one Ndali and her brother affected.
"Yes, sir. Also, we want you to check for Jamike Nwaorji, whom my friend gave the money to. See this receipt here? Jamike Nwaorji paid the school fees for him."
"Sorry, man, but we can only check your friend's account, not another person's account. Can I have his passport?"
Tobe handed him my host's passport. The man keyed in a few details, pausing once to talk and laugh with a woman who peered into his cubicle. Gaganaogwu, this woman looked exactly like Mary Buckless, the woman in the country of the brutal White Man who had desired my host, Yagazie, to lie with her two hundred and thirty-three years before. Mary Buckless's family lived on a plot of land by the farm where Yagazie lived as slave to a master who owned other slaves. Her father had been killed a few years earlier, and she became curiously drawn to my host, Yagazie. She tried to lure him to bed for a long time, entreating him with gifts. But he feared going to bed with her, for death hung over his head if he did, in that land of the brutal White Man. Then one night, she came over the tired mountains, which during the day teemed with the strange, ghastly birds they called ravens. With the other four male captives pretending to be asleep, this strange white woman, unfazed by the crude smell of the lowly slave quarters and driven by a kind of lust I had never seen before, insisted she would kill herself if she did not have him. That night, the young man, birthed by the great fathers and ever dreaming of his homeland, slept with her and basked in the occult richness of her lust.
Now, many years later, it seemed I was seeing her two gray eyes staring at her colleague and biting into the apple, which afterwards bore the shape of her teeth.
"Sir, there is no such account with TC Ziraat," the man said.
He handed back the passport and turned to the Mary Buckless look-alike to say something.
"But excuse me, can you check the other man?" Tobe said.
"No, sorry. We are a bank, not the police," the man said with a growl. He tapped his head as the woman, biting into the apple again, vanished from sight. "Understand me? Here is a bank not a police station."
As Tobe made to speak, the man turned away and followed after the woman.
My host and his friend walked out of the bank in silence and into the city center like men who had been served a grim notice about the new country they had come into. Like a desperate maiden, the new country threw itself up at him, flaunting its hollow enchantments. He watched her with the eyes of a noctambulist so that the tall buildings, the old trees, the pigeons that swarmed the streets, the sparkling glass structures all came to him like mirages, blurry images seen through wheezy rain. The people of the country watched them go by: the children pointing, the old men seated on chairs smoking, the women seeming indifferent. His companion, Tobe, was taken by the pigeons, which hopped about the squares. They walked past stores, banks, phone shops, pharmacies, ancient ruins, and old colonial buildings bearing flags similar to the ones in the buildings of the white people who came to the land of the great fathers. My host felt as if part of him had been pricked with a nail and he was bleeding, marking his trail as they went. In front of almost every building, someone stood with a cigarette clasped between their fingers, whipping smoke in the air. They stopped somewhere, and Tobe ordered them food, wrapped in what he said was bread, and Coca-Colas. They were drenched in sweat, and he was hungry. He did not speak. Egbunu, silence is often a fortress into which a broken man retreats, for it is here that he communes with his mind, and his soul, and his chi.
Yet inwardly, he prayed; the voice in his head prayed that Jamike be found. He shifted his thoughts to Ndali. He should not have left her. Tobe and he had, by this time, arrived at a place where shoes were displayed on platters and tabletops, and his eyes caught the inscription on the glass door beside the store: INDIRIM. The thought of the man who now owned his compound crept into his mind again. He imagined the man and his family moving in, unloading their truck, dragging bags and furniture into the now empty place that was his house. He had gazed at his father's room just before he'd left the house: empty, with a wall scarred about with marks and small chinks. The sun had stayed on the wall to the east, where the head of the bed had been, and looking through the louvers, he faced towards the well in the yard. That room where, once, he'd peeped at his parents making love when they forgot to lock the door was now so thoroughly empty that looking at it had given him an eerie sense similar to what he'd felt every time a parent died.
Gaganaogwu, the food came while he was still thinking about the last time he made love to Ndali, how, after he released her, the semen had seeped down both their legs and she'd begun to sob, saying how cruel he was to want to leave now—"now that you have become a part of me." His mind switched to the food, but Chukwu, I describe what had happened afterwards, after that sexual encounter. I had not recollected it because I had not thought it important until now. You know that if we were to collect everything our hosts do in one testimony, it would never end. Hence a testifier must be selective and must render to you that which is relevant, that which must add flesh and bone and blood to the creature he is creating: the story of his host's life. But now, at this point, I think I must recall it. That evening in the empty room his bedroom had become, he'd leaned his head against the wall, her tears running down his shoulder to his chest, and said it was for the best. "Mommy, believe in me. Believe, it will be good. I don't want to lose you." "But you don't have to, Nonso. You don't have to. What can they do to me? Proud people?" He'd held her, his heart beating, planted his mouth on hers and sucked at it as if it were a flute until she, shuddering, said nothing more.
Agujiegbe, the food he was now eating—which Tobe had called "kebab"—had been served by a slim, tall white man who, as he dropped the food on small trays, green peppers sticking out, said something that had "Okocha" in it. Tobe enthusiastically said he knew about Jay-Jay Okocha, the Nigerian footballer. My host, although silent, worried that this response would draw more men, all of whom looked like this man. They were white but appeared as if they'd been darkened by the raging sun, for it was hot here, hotter than he could ever remember in Umuahia. He avoided their gazes and ate the food, which, although it tasted good, was strange to him. For he thought that the people of this country did not cook most of their foods. It seemed, my host thought with a sense of mockery, that the people placed a premium on the need for things to be eaten in their raw states, once they had been washed. Onions? Yes, simply cut them up and add them to your food. Tomatoes? Certainly, just get them from your garden, dust off the earth around them, wash them in water, cut them up, and put them on the served plate of food. Salt? Same—even condiments and pepper. Cooking is a time-wasting experience, and time must be conserved for something else—smoking, sipping tea from minuscule cups, and watching football.
Although the men spoke with Tobe, my host merely gazed out the window at the traffic. Cars moved slowly, deliberately stopping for people to cross the road. No one honked. People walked fast, and almost every woman who passed seemed accompanied by a man who held her hand. His mind returned to Ndali. He had not called her since he left Lagos. And it was now two full days and half the third. He had, he reckoned painfully, broken the promise he made at the dawn of his temptation. He imagined where she must be now, what she may be doing, and saw her in the book room where he'd sat before his humiliation at the party. Then it struck him that here, Cyprus, overseas, was a new, sudden dream, the kind of ambition that a child would have—impulsive, instinctive, temporal, with little consideration. A child might, while walking with a parent, see a magician entertaining a crowd on a side street. He might see a man standing on a platform, striking his fist into the air, shouting bogus promises into a megaphone, and being cheered by an enthusiastic, banner-bearing crowd.
—Papa, who is that?
—He is a politician.
—What does he do?
—He is an ordinary man who wants to become the governor of Abia State.
—Papa, I want to be a politician in the future!
It occurred to him that what was happening to him was a mere temptation, that which must come to a man while in the pursuit of any good thing. And it has come to him, with the sole purpose of drawing him back. But he resolved that it would not succeed. He declared this to himself with such vehemence that it had an instant physical effect on him. Clumps of meat from the food he was eating spilled onto the table. "What time is it in Nigeria now?" he said to deflect attention from the embarrassment.
"Three fifteen here now," Tobe said, his eyes on the wall clock behind the back of my host. "Then it must be five fifteen in Nigeria now. They are two hours ahead."
Even Tobe must have been surprised, he thought. That's all? The time in Nigeria? Tobe did not know that words had become painful now that he was trying to digest what indeed may have happened to him. It was still hard to believe Jamike had planned it all out. How possible was it? Had he not just been on his own when Elochukwu told him that he could find help in the hands of this man to whom he'd given all he had? How did Jamike devise everything so fast? How did Jamike know that he would sell his house and poultry? Why did he expect these things when he'd not wronged Jamike in any way—at least in no way that he could remember?
He'd hardly let this sink in when the voice in his head propped up an example of a wrong he'd done to Jamike. There he was, in 1992, in the classroom standing before desks and chairs, the unvarnished walls covered with old calendars. He was only ten, seated with Romulus and Chinwuba. They were discussing the football match of their street against another, when suddenly, Chinwuba stamped his feet, and clapped his hands and pointed out the window at the boy walking towards the building, holding something like a folded shirt, his bag hung on his back. "Nwaagbo, oh, Nwaagbo is coming!" He and the others joined in, calling the boy outside the window a girl while observing with scrutinous gaze the effeminate features of the fellow: the plump flesh at his hips, the big buttocks, his gapped teeth, his bloated chest like small breasts, and his fat body. The boy walked in moments later, and in unison, the three of them shouted, "Welcome, Nwaagbo!" He remembered now, the way the bespectacled boy had been stunned by their assault and walked with a lumbering gait and a pant in his breath to his seat, one of his hands on his face, over his spectacles, as if to hide his weak tears.
He gazed closely now at the image of young Jamike, weeping from being bullied by him, and he wondered if what Jamike had now done to him was a revenge for this time in the past. Was this a stone thrown from his past to crush him in the present?
"Solomon," Tobe said suddenly.
"Er?"
"Did you say that a friend brought Jamike Nwaorji to you?"
Agbatta-Alumalu, for a reason that was not immediately evident, my host's heart pounded because of this question. He bent over the table and said, "That is so, what happened?"
"Nothing, nothing, I just had an idea," Tobe said. "Have you called your friend? Do you know if Jamike is in Nigeria? Does he know Jamike's father's house? Does..."
My host was hit with this idea as if by lightning. He rushed his phone out of his pocket while Tobe was still speaking and began fumbling at it in a frenzy. Tobe paused, but seeing the effect of his wisdom, continued. "Yes, let's call him, let's find out if this Jamike is here. You are my brother, and I don't know you, but we are not home. We are in foreign land. I can't allow my brother to be stranded. Let us call him."
"Thank you, Tobe. May almighty God bless you for me," he said. "What did you say I need to do to call Nigeria number again?"
"Add zero zero and then plus, then two, three, four, remove the zero, and put the rest of the number."
"Okay," he said.
"Oh, sorry, sorry, add only plus. Zero zero is another you can try."
"Okay."
Chukwu, he called Elochukwu, and the latter was shocked to hear everything. Elochukwu was near a building running a generator, so my host could barely hear him. But from the little he could hear, Elochukwu assured him that, indeed, Jamike had returned overseas. He knew Jamike's sister's shop, where she sold schoolbags and sandals. He would go there and find out where Jamike was.
He dropped the phone afterwards, relieved somewhat but also surprised that it had not crossed his mind to call Elochukwu until Tobe mentioned it. He did not know in fullness how the mind of a man in despair works. He did not know that it was sometimes better for such a man not to think. For the mind of a man in despair could produce a fruit which, although it may appear shiny on the surface, is filled to bursting with worms. This is because such a mind, wounded beyond reckoning, often begins to dwell mostly in the aftermath.
Egbunu, the aftermath—it is a place of little comfort. In the aftermath there is little movement, but much rumination. The event, having been done and ended, is now lacking in ability and agency. What the mind of such a man strikes leaves no dent on the skin of time. It is in this place that the mind of the man in despair dwells for much of the time, unable to move forward.
Tobe, apparently satisfied at the call my host had just made, nodded in affirmation. "We will know; we will find out like that. Maybe he is still in Nigeria and lying to you." My host nodded. "When you were making the call, I was thinking we should also go to the police station before going back to school. Let us report Jamike so that they can trace him. Maybe he is even in this country, but in another city. They know everybody who is here, so they can be able to find him."
My host, looking up at this man who had come to his rescue, was moved. "It is so, Tobe," he said. "Let us go."
#
## Conflicting Shadows
OSIMIRIATAATA, indeed, as the fathers of old said, a fish that has gone bad would be known from the smell of its head. I had begun to suspect by this time that what had befallen my host was what he and I most feared. But there was no way I could know this at that point, as, like our hosts, we cannot see the future. What guardian spirits must not do is shield our hosts, guard them even in the face of failure, and we must assure them that it will be well. We must assure them, Egbunu, that that which has been broken will be mended. So what I did was try to help him gather himself, for by this time, he was broken to bits. Elochukwu's return call had done it. Elochukwu had gone to Jamike's sister's store. He did not tell Jamike's sister what had happened. Instead he lied that there was a contract Jamike had given him, and he wanted to update him on it. But the woman told him that Jamike had traveled. Elochukwu then asked for his new number. "To my shock," Elochukwu reported to my host, "she said Jami informed her not to give anybody, even one person, his new number. I could not belief my ears, Nonso. So I asked her to call him. To my shock, he picked, and said something to her. She looked at me in a suspicious way and then told me that he was busy." Elochukwu paused as my host breathed deeply into the phone, which was trembling in his hand. "I am really sorry, Nonso, this is painful. It is like Jamike have duped us."
Agbatta-Alumalu, just before the police station, Tobe, who had shaken his head several times after hearing Elochukwu speak, asked that they change the euros he still had into Turkish lira. Not all, but a chunk, most of which they would need to rent an apartment in town. Of the 587 still left, he offered 400 to Tobe. Tobe entered a glass building with the word DOVIZ written in illuminated letters on the door and returned with a wad of Turkish lira. They met two African students near the station, one of them in tears. What had happened? The distressed woman was looking for a man who had acted as agent for the other university in Lefkosa, a man whose name was James, who was supposed to have picked her up from the airport but didn't show up. Her friend, a fair-skinned lady who reminded him very much of Ndali's mother, corroborated the information. He wanted to ask them if this James might be Jamike, if he had a foreign name or if it was a fake name, but the women hurried away, in great despair. After the ladies had passed, Tobe gave him a deep, cultivated gaze but said nothing.
He walked into the police station with a slight quickness in his gait and a roiling in his stomach. This was not like the police station in Nigeria, where violent and hungry men with weather-beaten faces and bodies punished by privation showed people little mercy and courtesy. Here, there were three counters, like the bank's. People sat in chairs and waited in lines for their turn at the counters. Policemen, two behind each counter, attended to the people. On a wall behind them, just as he'd seen at the bank, were large portraits of two men, one bald, with his hair on the sides of his head, the other stern. Tobe, unexpectedly, caught the direction of his gaze. "TRNC prime minister, Talat, and Turkish prime minister, Erdogan." He nodded.
When it was their turn it was Tobe who spoke. This was another reason he let Tobe lead: because he had a declarative presence, one that seemed to already have affirmed something his mouth had not yet uttered or to have spoken loudly when indeed all he'd done was whisper. Tobe explained everything, in detail. The policeman handed them a paper on a clipboard with a pen, and Tobe wrote everything down.
"Wait here," the policeman said.
In the intervening period, my host's heart pounded incessantly, and his stomach seemed to bloat in strange rhythms.
"I am sure that devil is on this island, and they will surely find him," Tobe said, shaking his head. "Then, ehen, it can't happen like that, just like that. Look at that innocent girl, too, eh? These yahoo boys, so very wicked. This is how they dupe and scam people. We used to think they only did it to white people on the Internet, the _mugus,_ but look, see how they destroy their own people, their own brothers and sisters? _E no go better for them_!"
For some reason he could not tell, he wanted Tobe to continue speaking, for there was something in what he was saying that soothed him. But Tobe sighed, hissed, stood up, and went to the water dispenser near the entrance, took a plastic cup, fetched himself a cup of cold water, and gulped it. My host envied him. This man who had lost nothing, whose money had gone where he wanted it to, and who would study Computer Engineering at a European university. Tobe was lucky; he was worthy of envy, and he had nothing to be sad or angry about. The cross he now bore, he bore for him and would undoubtedly soon relinquish, perhaps by sunset, or at the latest by tomorrow. Tobe reminded him of Simon of Cyrene in the mystical book of the White Man's religion, an innocent man who merely happened to be passing on the same road as the condemned. Like him, Tobe had been placed in the same empty apartment by coincidence. And his conscience, not Roman soldiers, had compelled him to bear my host's cross. But soon he would be relieved of it, and he would bear it on his shoulders, alone. But not yet.
"Just look at how this behavior, this kind of thing is affecting us," Tobe said when he returned from the water dispenser. "Look at our economy; see our cities. No light. No jobs. No clean water. No security. No nothing. Everything, price of everything is double-double. Nothing is working. You go to school suppose take you for four years, you finish after six or seven, if God help you even. Then when you finish you find job so tey you will grow gray hair and even if you find it, you will work-work-workn and still not be paid."
Again Tobe paused, because the policeman handling their case had appeared at the desk with a piece of paper, but just as soon as he came, he went away again. All Tobe had said was true, my host thought. He wanted him to say more.
"You even know what bothers me most?"
My host shook his head, for Tobe had indeed glanced at him and requested, without words, that he respond.
"All the money they make, these stupid yahoo boys, goes to waste. It never goes well with them. It is the law of karma. See the man in the street of Lagos who used his wife for money rituals? He died a hard death. This Jamike, he will suffer." Tobe snapped his fingers. He gazed back into Tobe's eyes and saw in them an impassioned fit that resembled the aggravated politics of a broken soul. "Just watch, you would see that he won't end well. It will not be better for him."
It was clear that Tobe had stopped speaking because he'd risen to go back to the water dispenser. My host felt alive in the aftermath of all that Tobe had said. There are certain situations in which, long after one has stopped speaking, words remain in the air, palpable, as if some invisible genie were repeating them. These were such words. _This Jamike, he will suffer. Just watch, you will see that he won't end well._ In the ambient silence that followed, my host pondered these words. Is it he who would see Jamike suffer? How would he, when he did not even know where Jamike was and how to even reach him? Was it that he would be somewhere at a given time in the future to see this same Jamike suffer and pay for the way he humiliated him? He wished it to be so. He would take what Tobe had said to be a prayer—this Tobe who, after all, wore a rosary under his shirt and who had said he would have been a priest had his parents not wanted him to procreate, since he was the only male child of the family. This priest-that-did-not-come-to-be had in fact prayed for him who could not pray for himself. And so, in the secret of his mind, he said a loud _Amen_.
When they left the station, the sun was slanting down towards the mountain whose ridges could be seen from everywhere in the city. Tobe said, "You see, there is hope. They can still find him. At least now they have found his records, they know who he is. They will be looking for him. And once that idiot returns to this island, they will lock him up. And he will—I swear to God who made me—return your money. All." My host nodded in agreement. At least some connection had been made with Jamike. A question had been answered, even if with an incomprehensible babble. For now, that was enough. A fetid pool, in the time of drought, becomes living water.
He gazed again at the small note on which Tobe had scribbled the information he received from the police—six details:
1. Jamike Nwaorji
2. 27 years old
3. Student at Near East University since 2006
4. Not registered for class this semester
5. Last came into TRNC on 3rd August
6. Left TRNC on 9th August
These six details, Tobe had assured him, would suffice for now. The details had been fetched from a hard source. He'd watched as Tobe asked the questions and the policeman answered them.
—Where did he go?
The police, the state, had no record.
—When will he return?
They did not know that, either.
—Do the police know anyone, a friend or anything, who would know precisely where he went?
The police did not keep a record of such things.
—What will they do if he returns?
They will detain and question him.
—What if he does not return, will they look for him?
No, they are only North Cyprus police, not the police of the whole world.
Then Tobe and he had run out of questions. So those details, which Tobe had jotted down legibly on a clean sheet of paper and handed to him, would do. He let Tobe decide what they would do next, and because it was now a few minutes past five, they would have to return to the temporary lodging. They would have to go to Near East University tomorrow, Tobe suggested, after he finishes his own registration for his courses and gets to know his course adviser. They had seen the school from a distance on the way to the city center earlier. They would ask at Near East if anyone was Jamike's friend and might have information about his whereabouts. Then, after they have gathered their findings, they would go and look for an apartment together in the town because, although my host had only been there one night, Tobe had been there for four, and a new student was only allowed one week of stay in the temporary apartments. They should, Tobe suggested further, share a room until his financial problems were over because—Tobe emphasized—he would do everything to make sure that evil did not prevail, that his brother did not get stranded in a strange land.
My host felt he had no choice in this but to acquiesce. Even more, it would be a form of reward to share the cost of board with Tobe, who had said that it was expensive for a single student to rent an entire apartment all by himself. He felt obligated to this man who had done so much for him. He agreed to share the rent, and he thanked Tobe.
"Don't mention," Tobe said. "We are brothers."
Egbunu, as the old fathers say: the fact that one has seen the shadow of his lost goat nearby does not mean that he will catch it and bring it back alive. The fact that a man has been given some hope does not mean that what was broken has been mended. So it was understandable that, before they boarded the bus back, he had the impulse to stop at a liquor store near the bus station. He bought two bottles of strong drinks and put them in his bag. The look on Tobe's face had been one of so much bewilderment that he felt a need to rationalize his purchase.
"I'm not an alcoholic. It is just for peace of mind. Because of what happened."
Tobe nodded more than he should have. "I understand, Solomon."
"Thank you, my brother."
OSEBURUWA, I would naturally simply tell you what my host did and said after they got back that day, but a spectacle they saw while on the bus on their way back and its impact on him afterwards merits this digression. For my host, at the beginning of his despair, was thinking about his compound, the small farm, the okro Ndali planted two weeks before, which must soon begin to bloom, his poultry. He was thinking of her asleep on his old bed, and of him watching her one afternoon, surrounded by the books she was studying. He thought again how it had happened that she chose him and that she gave herself to him. He was abruptly drafted into these pleasanter plains, when Tobe tapped him and said, "Solomon, look, look." And looking, he saw through the window of the bus a black man, swarthy beyond normal, a moving, animated sculpture coated with tar. The man Tobe had been talking with said the strange man had been on the island for a long time and had become so famous that he had been profiled in a Turkish-Cypriot newspaper, _Afrika,_ whose logo, the student emphasized, was the face of a monkey. No one knew this man's real name. But they held that he was from Nigeria. He was a great wanderer who trekked the length of the city carrying the single briefcase that seemed to be all he had and which over time had become worn. The man spoke to no one. No one knew how he ate or how he lived from day to day. It struck my host that it might be the same man T.T. had told him about at the airport. Egbunu, he watched this strange man until the man faded into the distance, greatly shaken by the spectacle. For he feared that it might be that the man had suffered a fate similar to his, and he had lost his mind. And he feared that, in the end, he might become like this strange man.
When they arrived at the apartment on campus, he retired to his room. The room was empty except for his bags on the floor, the shirt he'd traveled in on one of the two chairs, and the towel he'd used that morning, hung on one of the two wooden bunk beds. He reckoned that the room was to be occupied by two people. He sat on the other chair and opened one of the drinks. It struck him that he did not know why he'd bought the drinks, only that he must drink these drinks whose white color made them appear like palm wines—the drink of the pious fathers. They'd cost fifteen lira, which amounted to one thousand, five hundred naira. He stood on the chair and looked on the top of the cupboard, where he could place his luggage. There was nothing there except for dust and an old toothbrush that clung weakly to a loose, thin cobweb, its bristles gaunt and hardened with disuse. He was doing things that no longer made sense, he reckoned. Once, he'd been told—by whom he could not recall—that the worst thing adversity can do to someone is to make them become who they are not. This, the person had warned, was the ultimate defeat.
Having been warned afresh by this long-ago-received advice, he set the white bottles down and climbed up the bunk into the bed. It was bare, sheetless. He tried to wade through the thick crowd of thoughts in his head, but he could not. They were speaking all at once, their voices deafening. He climbed down, picked up one of the bottles. "Vodka," he whispered to himself and wiped his hand against the wet label. He gulped it again, then again, until his eyes revolted with hot tears, and he burped. He set the bottle down and sat in the chair. He listened to Tobe walking about the empty apartment. A tap turned on. The thudding of his feet on the floor. Another tap, followed by the sound of urine in the toilet. The plop of saliva in the sink. Coughing. A tune from a church song. The footfalls again. The door of a room opening; the gentle creak of bunk. When Tobe was out of earshot, or was silent, my host shifted his thoughts to where he'd wanted them to be: on the man, Jamike.
Ebubedike, he brooded so much on this man that by late evening, when the native darkness had almost completely covered the horizon, the transformation the unremembered voice had warned him against had been completed. He lay then, half naked, on the bare floor, his mind warped, fully changed into who he was not. He saw himself turned into a lion, grazing in a wild forest, searching for a zebra whose name was Jamike—the animal that had vanished with all he, his father, and his family had owned. With much struggle, he captured a picture of Jamike in his mind and gazed at it with jealous curiosity. A cough caught in his throat, and he spat tittles of the drink across the room.
He recalled the incident he remembered earlier, which had occurred in the year the White Man calls 1992, and how later that week Jamike avenged the wrong my host and his friends had done to him. Jamike included their names on a list of "noise makers" when in fact my host had not spoken at all. But on the strength of Jamike's false account, my host and his friends were flogged by the discipline teacher. My host was bruised by the punishment, so angry that he waylaid Jamike after school and tried to fight him. But Jamike had refused to engage. It was not the custom among boys to fight one who refused to fight or hit a person who did not hit back. So at the time, all my host had been able to do was claim victory for the unfought fight. "Girl, you refuse to fight because you know I will beat you," he'd shouted. And at the time, everyone there agreed that he'd won. But now, lying on the floor of the room in this strange country, he wished badly that they had fought back then, and even if he'd inflicted only a slight injury on Jamike at the time, that would have been a consolation, even if small. He would have beaten Jamike, scissored his legs with his own, and rolled him in dust.
Egbunu, angered, he wished the fight would happen now, here in this country, and he'd break these vodka bottles on Jamike's head and watch the alcohol seep into the wounds. He closed his eyes to suppress the growing palpitation of his heart and as if some unsought deity had heard his request, a vision of Jamike covered in blood appeared before him and stood there. Pieces of broken bottles stuck in the skin of Jamike's head right above his eyes, his neck, his chest, and even on his stomach, where a thick lump of blood clung like a patch of extra skin. He blinked, but the image stood firm. In it, Jamike was tearful from the apparent excruciating pain, and words dribbled from his quivering lips.
This vision had come to him with such vividness that he shuddered into a jerk. The bottle tumbled out of his hand and spilled onto the rug. A strong sudden wish for Jamike to not bleed to death seized him. He stretched his hands and pleaded with the suffering man, as if he were there, to stop bleeding. "Look, I don't actually want to injure you like this, er," he said, shading his eyes from the ghastly image of the bloodied man before him. "My one point five million naira, please, Jamike, please. Just give it to me back and I will go back home, I swear to God who made me. Just give to me back!"
He looked up again at his hearer, and as if in response, the shimmering figure trembled even more. He looked down in horror and saw blood gathering into a puddle between the feet of the wounded man. He sat up and pulled himself away from the sight, which, although in his vision, he assumed was in the room.
"Look, I don't want you to die," he said. "I don't—"
"Are you all right, Solomon?" This was Tobe, in the real world of objects and flesh and time, rapping on the door.
"Yes, Tobe," my host said, astonished that he'd been loud enough for Tobe to hear.
"Are you on the phone?'
"Yes, yes, on the phone."
"Okay. I heard your voice, so I wondered. Please make sure you try to sleep to rest your mind."
"Thank you, my brother."
When Tobe had gone, he said aloud, "Yes, I will call you again tomorrow." He paused, to feign listening, and then said, "Yes, you, too. Good night."
He gazed about now, and there was no Jamike. He wiped his eyes, in which tears had gathered, while he'd begged the phantom. Ijango-ijango, in a memorable moment of life which I cannot forget, my host searched about him, looking up on the bed, behind the red curtain, on the ceiling, tapping the floor, whispering and looking for the conflicting shadow of Jamike. Where was the man who had been bleeding? Where was the man on whom he'd struck the death blow? But he did not find him.
The image of the mad black man came to him now and, in fear, he climbed into the bed. But he could not sleep. Every time he closed his eyes, he leapt at once like an enraged cat into the wastelands of this burnt-out day, in which all he'd achieved was to gather more convincing evidence that he'd indeed become undone. He rummaged through the rich dirt of the wasteland, prancing in the choke of trash, digging, scrounging up detail after detail—about the bank, the girls who had touched his hair, the inquiry at the police station, the meeting with Dehan, the unearthed memory of what he did to Jamike many years before, which he believed may have caused this great hatred, this genuine malice to be sustained through the ages. He would pounce, dig, and scrounge on until he had brought up everything, until the surface of his mind had become strewn with the debris. Only then would he fall asleep. But not for long. For he would soon wake again, and the cycle would repeat itself without mercy, time and time again.
AKATAKA, so disturbed was I with the state of my host, and so afraid for the future, that for the short time he was asleep, close to midnight, I shot out of his body. I waited, and seeing no spirit in the room, I made my way into the ether and flew into the plains of Ezinmuo, through the concourse of spirits. In time, I was in the Ngodo cave, in the dwelling of many thousand guardian spirits. The moment my feet touched the luminous ground, I saw a guardian spirit I knew from many years ago. It had been a chi to the father of a former host. I asked it if it knew the chi of a living person by the name of Jamike Nwaorji, but it did not. I left the spirit, who sat alone, playing with a silver jar by the side of the waterfall. I asked a cluster of guardian spirits, one of whom had not had a host in twenty human years, and it told me it would be difficult to find a chi who might know the current location of a living host or the host's chi. Indeed, I looked around at the multitude of guardian spirits, who were simply a tiny fraction of the innumerable guardian spirits on the earth, and I realized the futility of my mission. I knew that I would not be able to find Jamike or his chi if I did not know where they were. Sad, defeated, I ascended with preternatural force towards the skies and soon found myself in that single esoteric path of descent known only to you, Chukwu, and me. For it is the single route by which I can return to my living host, as if drawn by a magnetic force, from anywhere in the universe.
#
## Metamorphosis
OBASIDINELU, the great fathers in their naturalist wisdom say that a mouse cannot knowingly enter into a trap set for it. A dog cannot know for certain that there is a deep miry pool at the end of the path and knowingly plunge into it to drown. No one sees fire and throws himself in it. But such a man may walk into a pit of fire if he did not see that it is there. Why? Because a human being is limited in sight; he cannot see beyond the boundaries of what his eyes can reach. For if one comes to a man in his house sharing a meal with his household, he may say, "Dianyi, I just came back from the big north, Ugwu-hausa, with two cows and they are worth so much money." He may garnish it by saying, "I have come to you because my cattle are special breeds, rich in good milk, their flesh as edible as that of _nchi_ caught from Ogbuti forest." The man of the house might be convinced. He may thus think of the seller as one of goodwill and believe all the man says, even though he did not himself witness it. But he does not know that the cows are poorly fed and afflicted or that they are inferior breeds. And because he does not know, he buys the cows for so much. I have seen it many times.
Chukwu, why does a thing like this happen? Because man cannot see what is not revealed to him, nor can he see that which is concealed. A word spoken stands as truth, firm, unless it is revealed to be a lie. Truth is a fixed, unchangeable state. It is that which resists any touching, any fiddling. It cannot be adorned, nor can it be garnished. It cannot be bent, or rearranged, or moved about. One may not say: "May we make this account clearer by adding such-and-such detail, perhaps the listener will understand better." No! To do so would be to corrupt the truth. One may not say, "My friend, if they ask me at the court if my father committed the crime, because I do not want my father to go to prison, do I say he did not commit the crime?" No, foolish man! That would be a lie. Speak only what you know. If a fact is thin, do not feed it to make it fat. If a fact is rich, do not take from it to make it lowly. If a fact is short, do not stretch it to make it long. Truth resists the hand that creates it, so that it is not bound by that hand. It must exist in the state in which it was first created. This is why, when a man comes to another with a lie, he has cloaked the fact. He may be offering a rattlesnake in a calabash of food. He may dress destruction in the garments of compassion until he who is targeted is trapped, until such a one is deceived, until such a one is stripped of his possessions, until such a one is destroyed! I have seen it many times.
Oseburuwa, I say this not just because of what had happened to my host but also because when he woke in the deep throat of night, soon after I returned from the cave of guardian spirits, the first thing that occurred to him was that he had not yet called Ndali as he'd said he would. She had made him promise her that he would never lie to her. It was a few days before he was to leave for Lagos, and they were seated in the backyard, watching one of the broilers who'd just had chicks preening and making perfunctory stabs into the plumage around its neck. Turning to him as if she had suddenly remembered something, she said, "Nonso, you promise?"
"Yes," he'd said. "I promise."
"You know, lying is one evil thing. How can I know what I don't know if it is not told me, if something else is said instead of it?"
"It is so, Mommy."
"Obim, then, that means you will never ever lie to me?"
"Ye—"
"Never ever. I mean, no matter what? Ever?"
"I will not, Mommy."
"Promise?"
"With all my heart." She then opened her eyes, but when she saw his, she snapped them closed again. "No, no, Nonso. Really, listen." He waited for her to speak, but she would not speak for a long time. Even now, he could not tell what it was that had stopped her. A thought, perhaps, so large that it had distracted her for that long? Or was it fear big enough to cause her to weigh her words with a caution similar to that of a person about to identify whether the mangled body about to be uncovered is that of a loved one?
"You will never lie to me, Nonso?" she said, finally.
"I will never lie to you, Mommy."
ONYEKERUUWA, my host rose that morning as if awoken by a shout from an unseen person. When he opened his eyes, he heard the sound of a distant vehicle, something like a crane or a heavy truck screeching. For a while he listened to this vehicle to keep afloat the fear that settled on the surface of his mind like a drop of oil. He sustained it with thoughts of things he could do by himself to find Jamike. Garbed in the light that fell through the curtains, he sat up and tried to locate Jamike in the tangled thickets of his thoughts. Once the rest of the night was broomed away, he would rise and walk into the new country. He would go wherever he could to find anyone who might know where Jamike had gone or how to contact him. Somewhere there must be a friend who might have information about Jamike's whereabouts. No longer would he let Tobe carry his cross; he must now bear it alone.
He washed himself, picked up the bag containing the documents, and stepped out before he could hear Tobe's movement. He walked into the school as the sun rose, past the places where Tobe and he had walked. He sat at a bench beside a pool where the sculpture of a frog stood overlooking the ringed pond, brackish with a dredge-black underbelly. At the tip of the bench sat a light-skinned couple speaking in Turkish. The two rose as soon as he settled into the bench, glancing back again and again as they walked away in a manner that convinced him they were speaking about him.
He sat there until the time on his charged phone said it was 8:14. He rose, and behind him, the 8:15 bus had pulled up. In the space between him and the bus, someone had created a fountain in the ground—certain unfamiliar materials rigged into the ground like strange plants—from which water sprayed about. My host paused before the sprinklers to determine the water's direction, then, when it turned away from him, he made a safe passage and rushed up to catch the bus.
As he made to climb into the bus, the driver said something to him.
"No Turkish," he said.
_"No Turkish,"_ the driver said.
"Yes, English but no Turkish."
"You, Nijerya?"
"Yes. I am from Nigeria."
He said the last words distractedly before sitting down. The bus passed between two sidewalks, on one of which two Nigerians were carrying nylon bags from Lemar, the supermarket where Tobe and he had bought telephone SIM cards. He did not know why he lifted himself from his seat at the sight of one of them, then caught himself and sat back. Something he could not explain had made him think for one sharp moment that the man was Jamike. He sat, aware of the startled gaze of the people on the bus, perhaps wondering if he had gone mad.
When he saw that the bus was approaching the stop where he was to disembark, he stepped forward, out of the spot on which he'd stood and out of the untamed thickets of his thoughts. Swaggering, he walked to the front and held on to one of the support poles. The driver caught sight of him in the mirror that hung in front of him and grinned. "Nijerya, very goodt, futball. Very very goodt. Jay-Jay Okocha, Amokachi, Kanu—very goodt, Nijerya, wallahi!"
Once out of the bus, he fell back into the memory of that evening in the yard as if knocked back in there by an invisible club. And Ndali had sat back on the bench and the hen had crouched on its belly, gazing at them in silence.
_"Mommy,"_ she said, then laughed. "You are one unusual man, Nonso. Will you always call me this thing?"
"It is so, Mommy."
She laughed again.
"Do you like it?"
"Yes, but it is strange. I have never heard any person call their girlfriend Mommy before. They say 'baby' or 'darling' or 'sweetheart.' You know. But 'Mommy'? It is different."
"I under—"
"Ehen, I remember, I remember, Nonso! Today, during our service in the church, we sang a song that reminded me very much of you, Nonso. I don't know why, I don't know why, but no, I think I know why. It is the wordings of the song, about coming to me. _And you come to me_. It so reminds me of you, of how, suddenly, out of nowhere, you came to me."
"You should sing it, Mommy."
"Oh God! Nonso, I should?" She gave him a sight blow on the arm.
"Ah! Ah! You will kill me, oh."
She laughed. "I know my blows feel like feathers to you. But you say it is heavy? Ah, _this is a lie_. But, see, it's a song to God. So I don't want to use it for you as if it is love song."
"I am sorry, Mommy. I know. I just want you to sing it. I want to hear you sing and also to know why it remember you of me."
She opened her eyes now after he'd stopped speaking.
"'Remind,' not 'remember.' 'Remind you of me.'"
"Oh, Mommy, that is true. Sorry."
"Well, okay, but I am shy. _Ama'im ka e si a gu egwu_."
"Good Igbo," he said, and laughed.
"Stupid!" She hit him again. He squirmed and wrinkled his face into a tuft of pain. She stuck out her tongue, pulled down the flesh beneath her eyelids so that the full balls of her eyes were exposed down to the constellation of veins coated in red flesh. "That's what you deserve for laughing at me."
"Will you sing now?"
"Okay, Obim."
He watched her raise her eyes to the ceiling, fold her fingers into each other, and begin singing the song. Her voice moved and swayed, softly, tenderly, as the words came out. Egbunu, the power of music on the consciousness of man cannot be lightly observed. The old fathers knew this. It was why they often said that the voice of a great singer could be heard by the ears of the deaf, and even of the dead. How true, Oseburuwa! For a man may be in a state of profound sadness—that uterine, entombed state. For days he may be still, in tears, perhaps not even eating. Neighbors have come and gone; relatives have streamed in and out of his house, saying, "Take heart! It is well, my brother." Yet, after all has been said, he has returned into the dark place again. Then let him hear good music, whether sung by a gifted voice or on the radio. You'll see his soul rise, slowly, from the dark place past the threshold into light. I have seen it many times.
My host, whose fear of losing Ndali had been growing in those days, was gripped by the strong hands of the last lines:
_You are my king_
_You are my king_
_And you come to me_
_Jesus, you come to me_
_And you come to me_
_And you come to me_
When she finished, he grabbed her hand and kissed so fervently that, later, they'd made love, and she asked him if it was the song that'd made it so good.
The song was material in his head as he alighted from the bus onto the paved terrace that branched towards a long road leading to Near East University. It remained with him even afterwards, like a persistent din caught in the ear of the universe. _And you come to me_. Before and around him, everywhere his eyes could see, he found evidence of the things the man he'd met at the airport, T.T., had said to him about the country, how it was mostly deserts and mountains and seas where nothing consumable grew. The only thing in sight was a large stretch of empty land. Sometimes, a big rolled-up bunch of dry tares, which looked like what the people across the great ocean called hay, lay in the land. And by the shoulder of the road, big billboards stood. Just before the bus stop, he saw a field of wrecked automobiles and all kinds of scrap metal. A lorry that had been stripped to its frame sat on the brush, staring with empty sockets where its headlamps had been. Beside this was a white sports car, upturned and held in place by the burned-out remains of what must have been a truck. Beside this sat a reticulated lorry, twisted into a ring, one of its cabs crushed beyond repair.
He thought to call T.T., since he was going to his school, the same one Tobe had written in his note as Jamike's school. He'd begun to search his phone when I put the thought into his head that he'd not taken T.T.'s number. His phone had been dead when they met at the airport. He gazed at the phone in anger, rubbing his hand over its edges. It crossed his mind to throw it somewhere and never see it again. But he found himself slipping it into his pocket. He had by now reached a place that looked something like a racetrack. In front of its gate was a group of people waiting, a black girl among them. Her ankara dress reminded him of a dress his sister used to wear. The woman's ears were plugged, he saw, and she was bobbing her head every now and then to the music that was being received through the device plugged to her ears, which my host rendered to his mind as "earphone." He went to her.
"Please, my sister, is this Near East?"
"No. Near East is still far," the lady said.
"Err, oh far?"
"Yes, but this bus coming will take us there. Oh, here it is. We take it and it will drop you where on campus you are going."
"Thank you, my sister."
The bus was neater and newer and fuller than the one from his own school, with many Turkish youth speaking their language. The black girl recessed into the back, where, finding no seat, she stood and held on to a rubber handle that extended from an overhead pole. Its interior was covered with posters of every kind. Not one of them was in a language he could understand. In one of the posters, a black male student stood beside a white male student, both pointing at a building as tall as some of the ones he'd seen near the city center the previous day. He thought now about how much things were different in this country. Back in the land of the great fathers, beggars and people who sold different products mobbed buses, hawking their wares and trying to command the attention of the passengers. He recalled the congestion at the bus park in Lagos, how he'd tried to haggle about a cheap bottle of perfume with a seller who had persistently bothered him. It occurred to him that had he come in a good situation, he probably would have loved it here—at least for its orderliness.
He exited at the first bus stop at the school. Two male students, carrying books, had also gotten off. The bus coasted on with a loud whine as it slid between two fields of what seemed to me to be artificial grass, not like anything in the land of the great fathers. A building lay beside a massive roadway, across from a small hill. He had not thought it deeply through, where to go. I could not do anything to help him, as nothing here was like I have known, not even like when my enslaved host was taken to a place across the mighty ocean, your powerful vast waters that cover much of the surface of the earth. In that place, Virginia, my former host, Yagazie, found himself living among others captured from the nations of black people, many of whom did not speak the language of the great fathers. That place was very sparsely populated. There were mighty buildings, two of which he participated in building, around where his captors lived. The rest were fields and mountains, fields as thick as the forests of Ogbuti-ukwu. There was none of the magnificence he saw now, no bright lights on the streets at night, no equipment that made various sounds. So I was silent as he thought of what to do. Egbunu, at this moment when my host's mind could not come up with a problem-solving thought, and I, his guardian spirit, could not help him, either, the universe lent him a hand. For as he started walking again towards the nearest building, his phone rang. He opened it in a rush and picked up the call.
Tobe's voice on the other side of the phone was sullen, bearing seedlings of concern. My host replied, "I am at Near East, my brother. I didn't want to continue to bother you with my problems."
"I understand. Have you found him?"
"No. I just arrived at the school. I don't even know what I should do."
"Have you gone to their international office, like the one of Dehan here in CIU?"
"Jesus Christ! That is so, my brother. That must be where I should go."
"Yes, yes," Tobe said. "Go there first."
_"Chai, da'alu nu,"_ he said, almost breaking into tears, for he wondered again how this potent idea had escaped his thoughts.
"Will you come back so we can go to the house agent? Dee gave me an address. Today is my fifth day in this apartment, two more days."
"It is so, Nwannem. I will come back soon. Once I finish."
Up till now he'd walked with warm courage, propelled by his own determination that he should bear his cross alone. But now his courage left him, whether it was because he had heard Tobe's voice or because he'd arrived at a place in this country where he was certain Jamike had been and did not know how to proceed I do not know. What became clear was that something changed in him after the call ended. He began walking with the gait of a cricket forced out of its hole until he found a man whose face was round—the kind his people referred to as "Chinese." "Ah!" the man gasped in response to my host's inquiry, and said he had just left the international office himself. This man took him close to the building, which had a facade like nothing he'd ever seen before. Beside it, flags hung from innumerable poles, amongst which he spotted the green-white-green flag of the nation from which he'd come.
Egbunu, before he entered through the door, my host, in fear, sought spiritual help. In this he acted like the faithful fathers. But where the fathers would have offered prayers to their _ikenga,_ or their chi, or their agwu, or even another deity, my host prayed to the White Man's alusi for help in finding Jamike here—his first time praying in many years. For here, he feared, might be his last source of hope.
"God Jesus, have mercy on me. Forgive me all my sins as I forgive all those who have trespassed against me. If you help me get all my money back, if you don't allow this to happen to me, I will serve you for the rest of my life. In Jesus's name I pray. Amen. Amen."
AKATAKA, you must forgive me. You designed us in such a way that we are one with our hosts. So that in time we begin to suffer their pains. What ails them ails us. It is thus why I am loath to describe his experience at that office but would rather give you an account of its effect on him, of the aftermath. For I do not want to stand here much longer, seeing these many guardian spirits who are seeking your audience, too. I will therefore say what he found here about the man he sought was that, as the policeman had said, Jamike was indeed a student there and was well-known among the foreign students. He found, too, that Jamike had been a student for only one semester, even though he had been in the country for two years. He stopped attending classes after three weeks. One of the workers at the school's international office who gave his name as Aiyetoro and who was from the same country as my host drew him aside into an empty hall after he'd finished with the chief international officer.
"Omo, you may be in serious trouble, oh," the man said.
"I know," said my host.
"You do? Wait, did you know Jami before, in Nigeria?"
My host nodded. "We went to primary school together, my brother."
"What, Umuahia?"
"That is so."
"So, after, did you know him? Did you know that he is an original yahoo boy?"
My host shook his head. "No."
"Ah. He is a serious scammer, oh—professional yahoo boy. How much did he take from you?"
My host gazed at this man, and for a moment, he thought of his gosling, the bird he loved very much, the first thing to which his heart had clung. The image in his mind was a still one, but Egbunu, it was much bigger than that. It was an event. It was after he had read books about falconers and begun calling himself a falconer and thought of flying his bird around the town. He had decided to buy a very long twine, sturdy but long. And he had had his father buy him jesses, something that he tied around the bird like an anklet when he released it into the air. At first, the gosling refused to lift itself. It would rather call and mourn. But one day, it flew so high, so far beyond the guava tree, to the limit the twine could go, even with my host raising his hand high and only twisting the twine around his wrist in one fold. At the time, the joy of the gosling flying had been so overwhelming he'd cried.
"You no want tell me?" the man asked. "I want to know so I know how I can help you, er?"
"Very much, my brother. Around seven thousand euros."
" _Ye paripa_! Jisos! Okay, you know what? No shaking, eh? Just relax. I go help you. That guy has duped plenty of people. I haven't seen him since last year, but I know some guys he shares apartment with who has seen him."
Gaganaogwu, what this man had given my host was hope. A man in need will hang on to whatever he can get to survive. I have seen it many times. A drowning man will not request a rope when a rod is presented to him or the branch of a tree instead of a raft. That which comes within reach is that which he clings to. And so, at the outskirts of the school, right back where he'd questioned the dark-skinned girl earlier, Aiyetoro flagged a taxi for him and gave the driver an address in a place he called Girne. My host thanked Aiyetoro, shook his hand with sweating palms, and the man said, "It is well, bro."
My host then left for Girne a broken man. For a long time, the drive took him through empty desert plains flanked by mountains. He got a closer view of the painted flag on the sierra, which he'd seen in lights on his first night. He gazed at its patterns: the crimson crescent and a star placed on a sea of white. It was, he reckoned, much like the Turkish flag: a white crescent moon on a sea of crimson. In the peace that the car provided him, Ndali returned to his mind through the song he'd remembered earlier. It brought him to the verge of tears. He knew that if she had this new telephone number, she would have tried to call or send him a message. In a jolt he keyed in her number after a plus sign and dialed it, but he could not bring himself to go through with it. Yet he feared that she must be very worried, wondering what had happened to him. He dialed it again and waited, his heart pounding, until she picked on the third ring. Egbunu, I find it difficult to describe the emotion he felt when he heard her voice. He shifted, rubbed his hand against the seat, as she said, "Hello, hello—who is this? Can you hear me? Hello? Hello, can you hear me?" He suspended his breath, making sure no identifiable sound escaped him. He heard her sigh. "Maybe it is network," she said, and sighed again. "Maybe it is even Nonso, er." Then she ended the call.
He gazed at the phone, her voice still in his head as if trapped in it. "I should...," he began saying, but stopped to look at the phone again.
"I should not have come here," he said in the language of the fathers. "I should not have come. I should not have come."
"Pardon?" the driver said.
My host, astonished, realized that he hadn't thought this but spoken it aloud.
"Sorry, not you," he said.
The man waved his hand. "Not problem. Not problem for me, _arkadas_."
Again, my fear was inflamed, for one of the first signs of a man in despair is that he is no longer able to distinguish between reality and imagination. Throughout the rest of the journey, he carried himself delicately, like a liquid-holding glass bottle cracked in many places, which was nevertheless held together by what seemed like a miracle. As the journey progressed, and in a brief period of reprieve, he became drawn to the natural beauty of the island. For once they came close to Girne, the landscape became like one he'd never seen before—very different from the land of the opulent fathers. Castles and houses, some bearing Turkish flags, sat on top of mountains and granitic outcroppings. It shocked him that people could build houses on mountains and hills. The last section of the highway lifted off, out of something that looked like a valley—a long, solid rock on one side and a sparsely shrubbed field, tenanted with pieces of rocks and stones—on the other side. And slowly, they seemed to be approaching the ascending road from where the entire city spread out before his eyes: houses big and small, some towering and others with spires. And in the distance beyond them all, he saw a bowl of the Mediterranean Sea's blue water, visible between the dense streets. As they drew closer, the sea seemed to expand, so that by the time they arrived at the great bridge at the entrance to Girne, it seemed the entire city was held back by some invisible fence which, if removed, would plunge it into the sea.
Later, in front of a three-story building, the driver pointed and said, "There, _arkadas_." My host dipped into his pocket and gave the man thirty-two lira. Then he walked through the metal door, struggling to retain in his head the name of the man who had sent him there—Aiyeoto, Aiyetoo.
He knocked on the first apartment, marked with the inscription APT 1. A poster hung on the door with an inscription in Turkish, below which was the translated version: WELCOME. A Turkish woman appeared, and behind her a young girl, holding a wild-haired doll.
"Sorry," he said.
"Not a problem. Looking for the Nigerians?" the woman said in clear English that surprised him.
"Yes, the Nigerians. Where are they?"
"Apartment five." The woman pointed upwards.
"Thank you."
With thoughts flourishing in his mind, he hurried up, his heart pounding, a single bud of hope rooted in his mind like the mushroom he'd once seen growing on the seat of an old abandoned car. Perhaps he would find Jamike here, hiding, perhaps secretly returned through the porous borders from South Cyprus to evade the police. Perhaps this was why the state record showed that he had left the country. This hope, wild as it was and growing without soil or water on the gnarled and ramshackle fixtures of that car, remained alive as he reached a level from which he began to perceive the aroma of Nigerian food and hear loud male voices arguing in the language of the White Man and its broken version. He waited at the door, his hand on his chest, as it seemed that he could hear Jamike's distinct voice among them, shouting in his swaggering tone with the prominent echo of "mehn." Then he knocked.
AKWAAKWURU, the job of a guardian spirit is often made more difficult when the spirit of our host, his ageless onyeuwa, which exists in the host's body only as an expression of his mind, is broken. When it is broken, the host slips into despair. And despair is the death of the soul. It is therefore very difficult to hold up one's host against this, to keep him, as long as one can, from falling. This was why, when he left the house of the people who knew Jamike's whereabouts, I threw thoughts into his mind to amuse him. I reminded him of that day when he'd eaten the _ugba_ and shitted endlessly. He saw himself spattering shit into the creepers. This should have made him laugh, but it didn't. I made him remember one of the things that used to fascinate him the most: the way the gosling yawned. How it opened its mouth and its gray tongue shivered with a nacrelike substance that bloomed into a globule under its tongue. Its mouth, wider by double than any human's, dragged a good portion of the sheet of its skin into a strained exertion that wrinkled its face. In an ordinary time, he would have laughed. But now he didn't; he couldn't. Why? Because all the world becomes dead to a man like him in such a time as this, and therefore all the pleasant memories, all the images that would have brought him pleasure, mean nothing in this moment. Even if they had been gathered in his mind in their multitudes, they would merely accumulate in abysmal futility, like a stack of gold in the mouth of a dead man.
So he stepped back out into the city, carrying, like a gift on a platter, the conviction his talk with the men had birthed in him: that it was all over; that that which has been done has been done. They had told him in clear words that the plan had been elaborate. Jamike had let his friends in on the intricacies of it all. He'd told them that he was on to a major big deal, after which he would cross into the south.
What did they mean by that? My host, with a trembling voice, had asked them.
Simple, they answered. North Cyprus and South Cyprus were once one country, until they fought a war and Turkey split the island in 1974. This Turkish part is a rogue state, and the Greek part is the real Cyprus. The two countries are separated by barbed wire. If you go to Girne Kapisi, the city center in Lefkosa, inside it you will see the border and Europeans coming into this part of the island from the other side. They are in the EU. Many Nigerians pay to be smuggled there, and some try to cross into the territory themselves by jumping the wire and claiming asylum. Jamike, too, had paid to cross.
"Is he never coming back?" he asked next, and although he'd spoken with the kind of menacing panic that would have drawn even the sympathy of an executioner, one of them said, "He isn't. Him don go be that."
Egbunu, my host accepted this revelation with a grim firmness, like a man who'd run into an enclosed space sealed behind him, out of which there was no escape. If he turned to the left, he met an impenetrable stone wall. If he turned to the right, he met a granitic door against which the strength of a hundred stout men would be fruitless. Forward? Same. Backwards? Sealed, too.
So he asked the men, what might he do?
"I don't know, bros," the man who had identified himself as Jamike's "best friend" said. "We tell people for Nigeria may them shine their eyes, make you no dull yourself, because people—er, broda—are bad. But some of you just no dey listen. Look at how that boy don gbaab you."
"Try and stay, mehn," another said. "You are a man. Endure it. What have happen have happened. Many people are here like you. And they survive. Even me, someone lied to me, an agent, that this is America. I pay, pay, pay to come here and then what did I discover? Africa in Europe."
They all laughed.
"No _rope,_ no _E-u,_ " the first man said. "Na so! What did I do? Did I kill myself? I found a menial job. I do construction." He showed my host his palms. They were firm, as hard as concrete, the insides rough like the surface of sawed timber. "I don work with Turka people tire, but see me, I am still in school. In fact, to worsen the matter, their women no like us. Kanji don kill boys finish!"
The men laughed loudly at this, in the presence of a man on fire who watched them with empty eyes. "Or just go back home," one of those who'd spoken before said to my host now. "Some have done that. It can even be better for you. Just buy ticket with wetin remain, and go back home."
Chukwu, were I not his chi, one who has been with him even before he came into the world, before he was conceived, I would not have believed it was he who left that place that evening and walked into the sun. For he'd metamorphosed, turned in the blink of an eye from a solid thing into a mass of weak clay and was now unrecognizable. I have seen much: I have seen a host enslaved, bound in chains, starved, flogged. I have seen hosts die violently, suddenly. I have seen hosts suffer diseases: Nnadi Ochereome, many, many years ago, who—whenever he went to stool—passed blood and had a swelling from his anus that was so excruciatingly painful he could not walk sometimes. But none of these times can I recall witnessing this great shattering of a man's soul. And I know him well. As you know, Egbunu, that in truth, every man is a mystery to the world. Even in his most extroverted moment, a man is concealed from others. For he cannot be fully known. He cannot be fully seen by those who look at him, nor can he be fully touched by those who embrace him. The true being of a man is hidden behind the wall of flesh and blood from the eyes of everyone else, including his own. Only his onyeuwa and his chi—if a good chi and not an _efulefu_ —can truly know him.
Gaganaogwu, this man he had become in the batting of an eyelid left the apartment, walked across the road, and entered a store he had seen similar to the one from which he'd bought the last strong drink. He picked the same bottle from the fridge and paid the quiet man with rheumy eyes who watched him curiously, as if he were some alien emerged from a craven hole in the earth, drenched in soil and mud. The world around him, this strange land, this frightful awakening, felt sharp and alive, like tempered steel. For over across the road was a white man who was walking with his child. On the other side was a woman pushing a thing with wheels packed full of supplies and a pigeon that dabbled at sidewalk dirt. He thought of himself, and that he was hungry. It was almost noon, and he had not eaten a thing. It surprised him that he had not considered it, had not thought how quickly things could change.
He left the spot, sipping the drink, a lilt in his gait. He stamped his feet on the ground and pressed it, as if by so doing he'd firm it against a fall. He put the drink in his small bag and hailed a taxi. When he sat down, he noticed that he had not zipped his pants after using the toilet at the Nigerian students' apartment. He zipped, and as the car began to race back to Lefkosa, he closed his eyes. In his head, thoughts competed for supremacy. They argued, their voices raucous, until it turned into a shouting match. He pushed his way out of their midst and into a secluded space where only Jamike resided, and began to think about the day he met Jamike. Until then, he had been on his own, going about his business. For much of his life he'd been a withdrawn man, one who did not gaze at the world as if he could divine and understand it but rather peeped at it as if it were something he should not be looking at. He had not asked too much of the world. What he had asked recently was simple: just to be with the woman he loved. That was certainly not too much. Yes, her family had presented him with a hurdle, but wasn't that what he'd been taught? That a hurdle meant an opportunity to advance and grow? Had he not gone to buy the university entrance forms for Nigerian schools before he ran into Jamike? What had he done to deserve this fate?
He gulped the drink and belched noisily. He shifted in the taxi and bent his head sideways as the car returned over the road on which he had come, as if retracing its steps, except that this time a lorry full of building materials slowed down traffic on the one-lane road. Then the taxi overtook the lorry, crawling behind a red pickup from which a white dog stuck its head out the window. He watched. He gazed at the dog carefully, at the way its head shook mechanically, as if it were being controlled by the wind, surprised at how such a banal sight as a dog sticking its head out a window could help a man forget his present state of burning.
As they approached Lefkosa, passing a stretch of painted rocks on the side of the road, the dog vanished and Jamike returned to him as if forced by the energy of the car. He sipped the drink again and belched.
"What, no, no, my friend! What is doingk? What is doingk, _yani_?"
He could not understand.
"Alcohol, no alcohol in my taxi, my friend. _Haram_! _Anadim mi_?"
"You say I can't drink? I can't drink? Why?"
"Yes, yes, no alcohol. Because _haram,_ my friend. Problem. _Cok_ problem." The man banged his hand on the dashboard and then snapped his fingers.
"Why?" he said, a foreign kind of anger in his mind. "I can do what I want. Just drive your car."
"No, my friend. Me, Muslim. Okay? You drink alcohol, problem. Big problem. I don't take you Lefkosa."
The man pulled up at the side of the road on the highway close to Lefkosa.
You must leave my taxi now, _arkadas_."
"What? You drop me here?"
"Yes, you must go out my taxi now. I say you no alcohol, you say me no. You must go."
"Okay, but I won't pay you!"
"Yes, no pay me, no pay!"
The man spoke in rapid Turkish as my host stepped out of the car into the road. Then the man sped off towards the city, leaving my host behind in the wild plain surrounded by desert and road and air and nothing, like a head severed from its body, rolling into a field—as I have once seen before.
AKATAKA, in this state of anguish, he walked towards the city, its expanse, its world, opened before him like a great cosmic secret. Desert, desert, he'd heard again and again—from T.T., Linus, Tobe, and even Jamike—as the one word that adequately described this landscape. But what is a desert? It is a place of abundant but loose earth. In the land of the fathers, it is hard to scoop earth from the ground. Something firmed it to the ground, perhaps the frequent rain, and made it difficult for it to come off easily. One has to scratch or dig to scoop earth. But here, not so. The very stepping of one's feet worried the ground and whipped up dust. No sooner has one walked a distance than one's shoes become covered in this darkish clay. And it spreads and runs about everywhere, accommodating little vegetation and resisting most of what seeks to plant its roots, to become, to vegetate here. Thus that which grows in it is tough and resilient. The olive tree, for instance—a tree that does not need water to grow, except whatever it can obtain from deep beneath the soil, for the country sits on water. Every other thing that inhabits this land must first subdue it. There must be a struggle, a hemispheric battle in which huge stones (hills, mountains, rocks) find their way here or emerge from some immensity beyond all knowledge and crush the enemies of earth and dust and insist that here, on this place, I must stand. And so shall it be. I must say, though, that in this it shares affinity with the land of the great fathers, where the earth—in its fecundity—exhibits an exuberance that mocks the desert.
He walked on for what must have been half an hour more, with the strides of the slightly drunk, until he arrived at an alley of houses. The longing to reach the city was in his mind like the thirst for water in the desert. He wanted to reach there and find the nearest bus station where he could wait to be picked up. Presently he sauntered into the half-closed mouth of a street which wound down inwards, away from the long main road, as if in fear. It seemed to be a poor neighborhood, for the houses were low-roofed and old, their facades strewn with flower-bearing plants firmed to the clay-colored earth. An uprooted gate leaned against a wall in front of one of the houses. A man stood on a ladder stretched against the walls, nailing something into it. Across, on the other side of the road, overlooking a bridge, was a deep crater that stretched for kilometers, the earth rising in sinuous rows towards what seemed to be a more developed part of the city.
He followed the trail, tired, half-mad, walking against the will of his heart past empty houses that sat like shadows in the sun, the sweat-soaked fabric he wore sticking to his skin. He heard itinerant voices of people he could not see. Birds he'd never seen before plunged across the plains and sailed at an unhurried pace. Egbunu, as soon as he advanced around a bend where the road turned back right towards the main one, he was jolted by a shout and the sound of rushing feet behind him, followed closely by the sound of approaching voices. He turned, and a group of children, having burst out of a gate from a compound—for he saw the small gate swinging—came rushing towards him, shouting what sounded like "Ahbi! Ahbi!" and then "Ronaldinho! Ronaldinho!" Chukwu, in the moment between the closing of an eye and its reopening, he was in the midst of a thronging mob full of noise and push that was speaking in an unfamiliar language. A hand tugged at his faded sport shirt from behind, and before he could turn in that direction, another pulled its hem. Someone shouted in his ear, and before he could make sense of what this voice had said, he was submerged in a well of words.
Agujiegbe, he stamped his feet on the ground, waved his arms about to free himself from the grabbing hands, and in the dim reprieve, he realized that he was thicketed in a mob of curious boys. The recognition shocked him, and in that instant he yelled that they desist. He clutched his bag with one hand and raised the other hand, swung himself from a grip, and staggered. The boys behind him stepped back from him like scared flies. He clenched his teeth, raised his hand, and landed it on the first head he could reach. He stepped back as quickly as he could, and, in a quick moment, he was free.
The children, what are they? From where had they come? Could they not see that he bore no resemblance to Ronaldinho? Did they not know also that Ronaldinho could not possibly be here, like him, eviscerated—a walking shell of what he'd been only a week before? One of the children stepped forward and motioned the others to back off. He was dressed in shorts and a singlet, taller than the rest. This boy started saying something and gestured to a small boy who was carrying a ball. Then he demonstrated that they wanted signatures. Another brought a pen and a book. They all gestured, and it became clear to him that their hectoring would cease quickly if he heeded their request.
As he took the ball to sign it, an image he'd once seen at the back of his father's house in the village came into his mind to insult him: a shell that must have belonged to a big snail, now empty, dried, calcified, moving slowly away. At first it seemed a miracle, but when he examined it, he saw it was being ferried by a team of ants. He felt that the same thing was happening to him now, in this poor neighborhood of this strange country, where these children had mistaken him for the best footballer in the world. They did not know that he was a man of great poverty, a man whose poverty extended beyond the diameter of time. In the past, what he owned he lost. In the present, he owned nothing. And in the prospected future, nothing. And here he was, with the pen one of them had offered him, signing a ball, books, shirts, even their palms. At the time, he'd screamed at the sight of the moving shell carried by the borrowed legs of an army of ants. In wonder, he'd called for his mother to come and see it. But now, at the lifting of himself before the eyes of these strange boys, he broke down and wept.
The impact of his tears was immediate. When the children noticed that he, "Ronaldinho" and "Ahbi," was crying, they stopped dead. Here was the great footballer doing what children were prone to do. It was a dead giveaway. One after the other, the small hands withdrew, the voices went silent, the cheerful eyes were replaced with perplexity, and the feet that'd encircled him like a silent subterranean army withdrew. He turned from them and continued on his way, sobbing as he went.
#
## The Empty Shell
AGBATTA-ALUMALU, in the land of the fathers, if a man is weeping in broad daylight like this, in public, people would come to him and hold him up. They would look and see that the light in his eyes was that of a man who had danced through life's theater of fire and now bore the scars of his partial incineration like a trophy. They would ask what was wrong. Had he lost something—a parent, a sibling, or a friend? If such a one said yes, then they would shake their heads in pity. They would put their hands on the man's shoulders and say, _Take heart, God has given, God has taken. You must stop weeping_. If it is that he has lost something else, money or property, then they may tell him, _God who provided will replenish. Do not grieve_. For the Igbo society is not one in which sorrow is allowed to thrive. It is treated like a dangerous thief whom the entire community must gather and chase out with clubs, and sticks, and machetes. Thus, once a person incurs a loss, his friends and family and neighbors gather with the sole aim of preventing such a one from grieving. They plead, they charge, and if the sorrow persists, a person amongst the comforters—all shaking their heads, gnashing their teeth—will order, with feigned anger, that the bereaved one cease at once. The sorrowful one may break away from his grief in that moment like the lobe of an old kola nut. The comforters may begin to talk about the weather, or the state of crops in that season, or the rains. This may continue for as long as possible, but in the end, once there is a lull, the bereaved will often break down again, and the cycle will begin all over.
I have seen it many times.
But here, Oseburuwa, in this strange country of desert and mountains and white-skinned people, he received no response. The women who walked past him as he approached a busy area looked past him as though he were invisible. The men seated on chairs under awnings outside restaurants, on balconies sucking at pipes, or standing outside some building smoking, looked at him with bald indifference, as they would a street beggar to whom—although he sings and dances better than the celebrated musician who fills the auditorium with a crowd—no one pays attention. The children who saw him, an adult whose face was soaked in tears, gazed at him in hollow bemusement. So he walked on, carrying on his back the burden of anguish like a damp sack of decayed things. So broken was he, Egbunu, that I, his guardian spirit, could not recognize him. His movements were not ordered by a sense of direction but rather by despair. Like the man Tobe had shown him, the world had suddenly become to him a field on which he must walk, outside of which nothing existed.
—What place is worth going to?
—Nowhere.
—What is worth doing?
—Nothing.
Everywhere he turned, he saw his problems. Yes, indeed, he was walking by fancy stores and beautiful buildings, but they were meaningless to him. Was that small crowd gathered there around a truck from which music was blaring watching a concert? Were those young white people dressed in uniforms of orange and red dancing? They meant nothing. How about this man, in front of whom he now passes? Are these some of the Turkish soldiers T.T. had said make up 30 percent of the country's population? The sandbags piled in front of them, the tanks and big vehicles behind. Yes, it is they, but he does not care. How about the small birds that tail each other and dive around that shapeless tree covered with the dusts of the street? On another day, he—an avowed lover of winged things—would have wondered strongly and tried to determine what kind of birds they were. Are they found here in Cyprus alone? Are they birds of prey or friendly ones? But now, a man of deep sorrow, he does not care. In another circumstance, he would have loved this country, as he had hoped to when Jamike first told him about the possibility of it. Joy had burst forth from within him like confetti, filling his dark places with shiny things. But now it struck him that that unguarded burst of joy had been the etiology of his undoing.
Gaganaogwu, I watched all this astonished, tongue-tied by my own impotence, by my inability to help him. He was now walking down a street whose name, he saw from a blue-colored signpost, was Dereboyu, and as he passed by the stores built of glass, he remembered his flock. He remembered the day he sold the last of them—the last nine of his treasured coop of yellow chickens. They had borne witness to the quietness of the mornings, the lack of cocks crowing, which had—to his surprise—affected Ndali. She had said it made the place seem deserted and that it made her fear even more that she would not be able to withstand his leaving. Only the hens were left. Together they took them out of the coop slowly and dropped them in one of the raffia basket-plaited cages. The anxiety in the coop, he could tell, was palpable. For every time he dropped a bird in the cage, so loud were their cries that he paused a few times. Even Ndali could tell that something was wrong.
"What is this they are doing?" she said.
"They know, Mommy. They know what is happening."
"Oh God! Nonso, they do?"
He nodded. "Look, they have seen many going inside that same basket. So, they can know."
"My God!" She shrugged her shoulders. "This must be their crying." She closed her eyes, and he saw tears gather at the corners. "It is heartbreaking, Nonso. I feel for them."
He nodded and bit his lip.
"We imprison them and kill them when we want because they are not as powerful." The rage in her voice cut deep into him. "They are making the same sound, Nonso. Listen, listen, it is the same sound they made when the hawk attacked them."
He looked up at her as he sealed the cage with its lid. He moved his head in a way that feigned listening.
"Did you hear it?" she said even louder.
"It is so, Mommy," he said, and nodded.
"Even when hawks steal their children, what do they do? Nothing, Nonso. Nothing. How do they defend themselves? They have no sharp fingers, no poisonous tongue like snakes, no sharp teeth, no claws!" She stood up then and walked slowly away to a distance. "So when hawks attack them, what do they do? They only cry and wail, Nonso. Cry and wail, finish." She slapped her palms together in a sliding gesture, as if she were dusting one palm with the other.
He raised his head again and saw that her eyes were closed.
"Like even now. You see? Why? Because they are _umu-obere-ihe,_ minorities. See what the powerful have done to us in this country. See what they have done to you. And weak things."
She took a deep breath, and he wanted to speak but did not know what to say. He could hear the sound of her breath even though it was a cool day and the air was stifling. And he could tell that what she was saying was coming from deep within her, as if she were drawing water from a dried-up well, bringing up dregs, scrap metal, dead ferns, and whatever lay in its bed.
"See what the powerful have done to us, Nonso?" she said again, stepping back as if to leave, and then turning to him again. "Why? Because you're not rich like them. And isn't it true?"
"It is so, Mommy," he said, as if in shame.
But it seemed she did not hear any of it, for while he was speaking, she'd started to say listen, "Listen, listen, Nonso. Can you see that their crying follows a pattern like they are talking to each other?"
Indeed, as though they could hear her, the fowls had raised their voices. He gazed at the cage, then at her. "It is so, Mommy," he said.
She came over again to the pen, nudged him slightly aside, and bent her ears towards the crying birds. When she turned to him again, tears stood in hanging drops on the lids of her eyes.
"Oh God! Nonso, they are! It is like a coordinated song, the kind they sing during burial ceremonies. Like a choir. And what they are singing is a song of sorrow. Just listen, Nonso." She stood silent for a moment, then she stepped back a bit and snapped her fingers. "It is true what your father said. It is an orchestra of minorities."
She snapped her fingers again. "I feel for them, Nonso, for what we are doing to them, and it is a song of sorrow that they are singing."
Egbunu, at the time, he had listened, listened the way someone listens to a tune he has heard countless times but which, in every new iteration, moves him and opens his eyes to new vistas of meaning. He was watching the cage with all the concentration he could muster when he heard the sound of a sob. He went and held her to himself.
"Obim, why are you crying?"
She hugged him and placed her head on his chest, against his beating heart.
"Because I am sad for them, Nonso. And I am sad for us, also. Like them I am crying inside because we don't have power against those who are against us. Mostly, against you. You are nothing to them. Now you will leave me and go somewhere I don't even know. I don't even know what will happen to you. You see? I am sad, Nonso, I am very sad."
Chukwu, it struck him now, in this distant country of sky and dust and strange men, that what she feared that day had now happened to him. A poultry farmer named Jamike Nwaorji, having groomed him for some time, having plucked excess feathers from his body, having fed him with mash and millet, having let him graze about gaily, having probably stanched a leg wounded by a stray nail, had now sealed him up in a cage. And all he could do now, all there was to do now, was cry and wail. He had now joined many others, all the people Tobe had listed who have been defrauded of their belongings—the Nigerian girl near the police station, the man at the airport, all those who have been captured against their will to do what they did not want to do either in the past or the present, all who have been forced into joining an entity they do not wish to belong to, and countless others. All who have been chained and beaten, whose lands have been plundered, whose civilizations have been destroyed, who have been silenced, raped, shamed, and killed. With all these people, he'd come to share a common fate. They were the minorities of this world whose only recourse was to join this universal orchestra in which all there was to do was cry and wail.
AKWAAKWURU, the fathers say that a smoldering fire can easily be mistaken for one that has been extinguished. My host had walked aimlessly for nearly one more hour, hungry, thirsty, drenched in sweat and tears, when he found himself at a crossroads. One headed northwards into a road that appeared interminable, another forked into a dead end, another led back the way he'd come, and all of them were channeled in these directions by a roundabout he could see from a distance, almost a kilometer away. The ferocity of the sun that shone here was something he'd never encountered before. People have talked about the hotness of Ugwu-hausa, the north of Nigeria—even his father, who'd once lived in Zaria. His father had told him that farther north, in the Saharan desert, the sun made the living appear like they were dead.
He had now been walking for close to two hours since the taxi dropped him, drenched in sweat and slightly drunk. Moments after he stepped out of the taxi, he'd dropped the drink gently by the side of the road, between a clot of dry grass, as if hoping that someone like himself would find it and finish it. And now he reached a large tract of land covered in low grass on which was a house under construction. Two black people stood amongst the dust-colored workers, sweating in the flesh-killing sun. He trod on, his tears now dried, the freedom of stagnancy, of not knowing what to do next or what would happen next, offering him unaccustomed peace. He was again thinking of Ndali and the chickens and of his last day in Umuahia, and of the sound of her voice when he'd called her earlier, when from the road near the roundabout he heard a loud sound like something exploding. He looked on and about him but saw nothing. He walked on between two big buildings and came into a clearing, at the edge of which was a main road. He saw then from the distance the source of the sound he'd heard: about two stone throws from him was a car, upturned, covered in smoke. He heard rushed voices behind him, from the way he'd come, and saw the construction workers from the big building he'd passed earlier running towards him.
He gazed now at the field, his face painted with dust like the _uli_ patterns worn on the faces of the dibias amongst the old fathers, and saw that up the field, the dust had settled. He saw more clearly, and the damaged car was now surrounded by people in various states of distress. Up close now, he saw the fate of the other car in the accident. It was a minivan, now facing the roundabout, pressed almost in half. When he came to the car on the field, one of the black construction workers, whom my host reckoned was a son of the affluent fathers by virtue of his accent, turned to him.
"Terrible, terrible," the man said. "In that other car, no one survived. In this one, two girls in the back of the car, eh. _Chai_! They are the ones screaming."
My host, too, had heard the screams. His compatriot stepped back, as did others in front of him. A police car had arrived, and a policeman was ordering them to turn back. In the distance, an ambulance was speeding towards the scene. My host, afraid because of the presence of the police, stopped short of reaching the scene. For in Alaigbo, this mysterious office of men who have the power to punish others is feared. He reached for his phone to see what the time was, but his pocket was empty. He touched his pants about. He retraced his steps with hurried feet and found it a few meters back, the way he had come. He blew dust off its face and saw three missed calls from Tobe. He remembered that they were to go and find a place together and it was now long past noon—2:15. Egbunu, so much had happened since the last time they'd spoken. He'd called Ndali but did not speak to her. He had been chased out of a taxi by an angry driver. He had drunk and thrown away some drink. Yet even more things had happened. He had been mobbed by street children. He had cried. He had been almost killed by a car. His misery had deepened. The hope that in the previous night still crawled, despite being gravely wounded and covered in blood, had now been struck a death blow, and in falling, expired. These things were excuses enough for his failure to return to Tobe. In fact, they were too strong.
He could see, as he walked, that one of the doors of the upturned car had been opened and the screaming and shouting had increased. Everywhere, on the adjoining roads, cars were lined up. I wanted to come out of my host, to see if the passengers had all died and to commune with their chis and find out if this tragic fate that has befallen their people could be avoided for mine. What had these people done to have died this way? What answers could their guardian spirits give? We often ask this, too, after things have happened. Was there a way, for instance, that I could have engaged Jamike's chi and found out the intents of its host's heart? Even if I'd found his location and gone there, I may not have gotten it to come out, for it is difficult to persuade a chi out of the body of its host. I did not, however, leave my host this time, because I was afraid of leaving him while in a broken state. As he drew close to the scene, pulled only by curiosity to witness a tragedy in this strange land, a feral epiphany jumped out of the smoke towards him: that he was not meant to come to this country, that if he stayed here much longer he might die.
When he reached the spot, men in white coveralls were loading a bloodied man into the back of an ambulance. On the ground, the body of a girl lay bleeding from the side, where there was a deep gash, her blond hair colored with blood. People were gathered around her, and a man was pushing others back. He saw on the sparsely leafed part of the clearing near the accident scene a patina of flesh lying on a tray of flattened grass where the hospital people had lifted a man who'd been thrown from one of the cars. And the grass about this spectacle was stained with thick blood, so that it appeared as if it were covered with red phlegm. As he watched, one of the nurses broke from the group and walked frantically from person to person, saying something in the language of the country. In what seemed like a response to the words of this woman, a man wearing a blue visor stepped up. Another, an elderly woman. The nurse nodded, wagged her fingers as if to say the woman could not do this. As the white woman talked, his stomach rumbled. He turned back to go, to find some water at least.
"Mister, mister," the nurse called after him.
As she made to speak, someone called at her in the strange language. She turned to say a word to the man. Then she faced my host again, moving swiftly towards him with the disposition of extreme anguish. "Excuse me, can you please donate blood? We need blood for the victims. Please!"
"Er?" he said, and slammed his hand on his leg to free it from shaking. He was trembling slightly.
"Blood. Can you donate blood? We need blood for the victims, please."
He turned as if to ask someone behind him for an answer, then looked back at the woman. "Yes," he said.
"Okay, thank you, mister. Come with me."
AGUJIEGBE, among the old fathers, it was said that in wrestling bouts, rarely was a man thrown because of inferior strength. Men of weak or small bodies did not attempt _egwu-ngba_. So how did they throw men—the great wrestler of Nkpa, Emekoha Mlenwechi, the sleek snake; Nosike, the cat; Okadigbo, the Iroko tree? It was either by a trick or resilience. In the latter case, the opponent slugs it out with the great wrestler for so long that his muscles become weak, his limbs tired. He caves, relaxes his grip, and in a flash, he is lifted like an empty drum and thrown in defeat.
This can apply in any situation beyond the field of wrestling. If a man has contended for too long with an unrelenting enemy, he may cave in submission and say to the trouble that had come to him: "Here, did you ask for my cloak? Take my turnip, too." If such a man has been asked to go a mile, he may say, "Did you say you want me to go a mile with you? Okay, here, let us go two miles." And if such a man, after just escaping death, has been asked to donate blood, he would not reject the request. He would follow the nurse who has made such a request of him—a stranger, a man of black and foreign skin—to the hospital and do just as he has been asked. And after such a man has donated blood to one victim, he would say to the nurse—who has drawn his blood and dabbed a wool, the like of which the old mothers made into fabric, on the spot to stanch the bleeding pore—that he wanted to donate to a second victim.
"No, mister, one is enough. Believe me."
But the man would insist. "No, take more for the victims. Take more, please, ma."
He would insist despite his chi speaking into his mind that he should desist from this, because blood is life itself, and it is the thing that leaves the body in protest against an injury done to it. He would insist despite his chi saying that suicide is an abomination to Ala, and that there was nothing broken at this point that could not be repaired, and that there was nothing the eyes can see that can cause them to shed blood in place of tears. But the host, a man broken, defeated, possessed by the silent tyranny of despair, would pay no heed. The woman, visibly astonished, would stop in her tracks.
"Are you sure about this?" the woman would say, and he would say, "It is so, ma. I am very very sure. I want to give blood for them. I have enough blood. Enough."
Still staring at him, as one would regard a madman on a pulpit, the woman would take up another syringe, tap it three times, and then, wiping his left arm with a piece of wet cotton wool, draw his blood again.
Afterwards, he rose, weak and tired, hungry and thirsty, and in his mind was the question: what was to be done next? The past three days had upended any philosophy he had about life, and he now resolved that it was better not to plan anything. No, how foolish to think that a man who leaves his house and says to his friend, or even to himself, "I am going to school," that such a man would in fact get to his destination. Such a foolish man might find himself in a hospital instead, giving blood to people he does not know. How foolish to think that because one has boarded a taxi and given the driver the right address, one would end up at the right place. Such a fool might find himself walking only a few moments later towards an unfamiliar destination very far away from the school, thronged and hectored by a mob of boys.
So no need to plan. What he could do was thank the woman who had drawn his blood, then go his way. He must step into the day, into the sun, and go—perhaps to the temporary accommodation. And this, Egbunu, was what he did. For after he'd said, "Thank you, ma," he walked out, both his hands curved up to hold the moist cotton to the needle spots.
He'd walked past the long aisle of people, past the offices, and out into the motor park when he heard, "Mr. Solomon."
He turned.
"You forgot your bag."
"Oh," he said.
The woman came up to him.
"Mr. Solomon, I am worried. Are you all right? You are a kind man."
Before he could think, his mouth said, "No, I am not fine, ma."
"I can see that. Can you talk? I'm a nurse, I can help you."
He gazed past her at the sun, now in the sky, peering at him.
"Leave the sun," she said, pulling him back under the awning attached to the hospital's facade.
"You tell me, I can help you."
#
## All the Trees in the Land Have Been Removed
BAABADUUDU, I have spoken at length about the longest day in the life of my host—a day of rain, and hail, and pestilence. But I must tell you also that it ended with a drop of hope. I must therefore hasten to say that he returned to the temporary apartment he shared with the man who had been his companion the previous day. He was climbing the stairs up to the apartment, holding the bottle of drink the nurse had bought him, when it struck him to call Ndali again. The idea came to him as a whiplash on his mind, and with true surprise, he wondered why he'd dithered for so long. He began to key in her number, then he remembered he had not added a plus. So he erased it and began again. When it started to ring, he turned off the phone so frantically that it made the sound of a clap. He must approach her with the utmost geniality and great care, he told himself. He must tell it from the beginning, from how much he had missed her and how much he loves her. This would disarm her.
So, standing with one leg on the stairs and one hand on the banister, he dialed again.
"Mommy! My mommy!" he shouted into the phone. _"Nwanyioma."_
"Oh God! Nonso, Obim, I have nearly gone mad from worrying about you."
"Oh, it is network. Bad network. It is—"
"But, Nonso, not even a call, not even an ordinary text message? Er? I have been worried. In fact, someone called me and I was speaking, shouting hello, hello, but the person could not hear me, and my spirit told me it was you, Nonso. Did you call me today?"
Egbunu, he was trapped for a moment between the truth and falsity, for he feared that she would suspect something had happened to him. In his hesitation, her voice came again—"Nonso, are you still there? Can you hear—"
"Yes, yes, Mommy, I can hear you," he said.
"Did you call me?"
"Oh, no no. I wanted to call you when everything is fine so you don't worry."
"Hmm, I see..."
She was still speaking when a Turkish voice came over the line followed by another speaking in the language of the White Man, informing him that his credit was exhausted, and his call terminated.
"Oh-ooh! Which kind of nonsense is this? Er? The credit I just bought." It surprised him, after he uttered those words, that he had bothered about something as trivial as a phone credit. For the first time in days, he was not gazing at the battered image of himself before the mirror in his head and gasping at the gashes, the swollen eyes, the pulp on the lips, and the masks of his great defeat.
He rang the bell at the door of the apartment and heard the sound of feet in the house.
"Solomon, wa!"
"My brother, my brother," he said, and embraced Tobe.
"What, where have you been—"
"Mehn, thank you for yesterday," he said as he sat down on one of the couches in the living room.
"What happened?"
"Many things, my brother. Many things."
In the same mood of joy, he told Tobe about all he'd done that day, the accident, and the nurse, up to the point I have just testified to you and the hosts of Eluigwe and beyond.
Egbunu, it would have been futile, even stupid to have planned anything after his blood was drawn. If he had planned to go back to the campus, for instance, reality would have again shown its wrinkled face on the screen of his consciousness, laughing at him with its toothless mouth, as it had been doing relentlessly for the past four days. So he had done the wise thing and allowed himself to float, be carried by time wherever it willed. An hour after his blood had been drawn, he remained with the nurse, having told her his whole story, and was seated in the passenger seat of a small gray car, riding back to Girne! Yes, Girne, where he'd been told a few hours before that he would never find Jamike by those who would have been with the man. But how could he know he'd return here, where his hope had been struck with a death blow the same day?
"It will take about forty minutes, so you can sleep, okay, lie down maybe, and sleep, if you want."
"Thanks, ma," he said.
So relieved was he that he wanted to cry. He threw his head back against the seat and closed his eyes, hugging his bag closer to his body. Some of the vegetables from the kebab she'd bought him were still stuck between his teeth. He pushed them to the top of his tongue and spat them out noiselessly.
"I think I must tell you about my troubles, too, Solomon," the nurse said.
"Okay, ma."
"I have already told you, please call me Fiona."
"Okay."
He heard her laugh—a laugh between everything she said.
"When I moved here from Germany, and married my husband, I gave up everything, too, except my German citizenship. The government said I could keep both because Cyprus is not a real country. One year, two years, it was good. Whatever. Then, everything, everything, began to explode. Now, we live together like two strangers. Total strangers." He heard a laugh, her voice cracking slightly. "I don't see him; he doesn't see me. But we are husband and wife. Very weird, right?"
He did not know what to say, and he did not know what the word _weird_ meant. Even though I, his chi, knew, it would have been an overreach to tell him, so I didn't. All he thought was that the people here—like him and his people back in Nigeria—have problems, too.
"You can imagine I haven't seen him in three days. Once, last night, I heard his voice as he came in the middle of the night. Then, his footsteps as he went to the bathroom, then, to bed. That was it! _Genau_."
"Why is he behaving like this?" he said.
"I don't know; I don't know at all. It's complicated."
They drove to a place where she said she would help him get a job, a well-paid "under-the-table job." He could earn one thousand, five hundred lira every month, enough to help make up all he had lost and even pay his way through school. The employer—she said his name—was her close friend. The place was a casino attached to a hotel also owned by this friend.
They inquired at the casino, but the man was not there.
"He has gone to Guzelyurt," said the secretary, a woman dressed in a white blouse and black skirt.
"I can't reach him on his number."
"Yes," the other woman said, then trailed into a long speech in the language of the land.
_"Tamam,"_ Fiona said. "I understand. I'll bring him another time, then."
She told him they would come back the following day to see Ismail. So they returned to Lefkosa, and for most of the time they did not speak. She put on the radio, and it played music like he'd never heard before. It reminded him of Indian movies—the intermittent bass drums that would stop, then rise again with fervor, as in the movie _Jamina_. "It doesn't matter. It is a casino. They are always open."
She drove past the place where the accident had happened earlier. There was now, only three hours or so later, almost no trace of it, except for the broken brick on the surface of the roundabout and shards of glass in the field where the car had fallen. She shook her head as they passed it and talked about how people drove recklessly in Cyprus and caused many accidents. By the time she pulled up at the school, he'd started to doze.
"I'll call you around noon tomorrow, after I talk with him. We'll go to my house, and I'll make you a home-cooked meal."
"Thank you so much, Fiona. Thank you."
_"Genau,"_ she said. "Take care and talk to you tomorrow."
He told Tobe about how he had watched this woman drive away, every word she had said alive within him. A total stranger had shown so much compassion for him that as he told the story of his great defeat, her eyes clouded in tears—perhaps because of the way he told it, the way he described all that had been taken away from him and the catalog of losses that was his life. She asked question after question—"Was this man, Jamike, not your friend?"; "He did that?"; "So, even the money in the bank was not true?"—until, by the time he came to the accident scene, her eyes were red from crying, her face pink from the withering emotion, and she was blowing her nose into the tissue she'd extracted from a polythene pack. Her sympathy had been genuine.
"I can't believe it!" Tobe said when my host finished. Tobe cocked his head sideways and snapped his fingers. "Have you seen it? Have you seen our God in action?"
"That is so, my brother," my host said, elated and grateful for this man's generosity and wanting to share even more with him. "Look at me." He spread his hands. "This morning, I thought my life had finished, that I have fallen into a deep pit. _Echerem ma ndayere na olulu._ "
They both laughed.
"It is God," Tobe said, pointing towards the ceiling. "God. That woman is an angel sent from God. Have you not heard the adage: 'It is God that swats flies from the bottom of a tailless cow and from the food of the blind man?'"
"That is so! And he gives voice to the insects, the birds, the mute, the poor, the chickens, and all the creatures that cannot sing, and to the orchestra of minorities!"
Tobe nodded and stamped his feet on the floor. "Even on the side of accommodation, I just returned from the agent's office," Tobe said. "I have found a cheap, nice place for eight hundred tele a month. That is, two hundred euros for each of us if we take one room."
"Ha, very good, my brother. Very good."
"Yes, they take deposits here. So I paid the deposit already."
"Ah, my brother, _da'alu_."
While he was still speaking, his phone rang. He rushed up to his feet to see who was calling.
"My fiancée," he said. "Please excuse me, Tobe."
Agujiegbe, it was with drunken excitement that he raced into his room and closed the door. I could still see that the effect of the alcohol had not completely eased from him and that he was still in a slightly dazed state. When he punched the Accept key, her familiar voice came crashing into his ears, antiseptically clear.
"Nonso, Nonso?"
"Yes, Mommy!"
"Oh, and so it is network?"
"I know, Mommy. I know. Look, I miss you. Mommy. I love you so much."
"Ha, you say this but why didn't you call me? You said it wasn't you that called earlier? It is almost five days."
"Mommy, it is because of the stress, when we did not arrive on time, and when we came here, I discovered many things like school registration, getting a place—all taking, taking my time."
"I don't like it, Nonso. I don't think I like at all."
He imagined that she'd closed her eyes, and the beauty of that eccentric demeanor lit him up with desire.
"I am sorry, Mommy. I will never do it again. Never. I swear to God who made me."
She laughed. "Silly man. Okay, I miss you, too."
_"Gwoo gwoo?"_
She laughed. "Yes, Igbo man, _gwoo gwoo_. Really, very much. Tell me, what is the place like?"
Now that he was relaxed, laughing, he let himself take in the room with his eyes, and he saw something he had not noticed before. On the screen windows, close to the ceiling, was a wooden valance on which some paper image had been pasted, half scratched out and now only bearing the image of the legs of a white person in an outstretched position on a couch.
"Are you there, Nonso?"
"Oh, yes, Mommy, say again?" he said.
"You are not listening to me? I said tell me what it's like in Cyprus."
"I am," he said, even though he'd moved closer to the window, wondering what the full portrait must have been like. "Mommy, it is a barren, stupid island. It doesn't even have any trees, just desert, desert."
"Oh God, Nonso! How do you know?" she said, stifling laughter. "Have you gone round?"
"Er, Mommy, I am telling you the truth. It is like all the trees in this land have been removed. I'm telling you, all of them. Not even wan single tree. I am telling you."
"What, no trees at all?"
"None, Mommy. And the people, they don't hear English."
"My God!"
"Yes, Mommy. Most of them don't hear English at all. Even come-go they don't hear. I am telling you, it is not a good place, and Turka people"—he shook his head, Egbunu, as if she could see him, for he'd remembered what the driver had done to him a few hours before, and the children, and the people who'd watched him cry as he walked in the burning shadow of the sun—"they are bad. I don't like them, _cha-cha_."
"Ah, Nonso! What about your friend Jamike? Is he happy there?"
Ezeuwa, at the mention of this name, he felt his heart sink. He paused to gather himself, for he did not want Ndali to know what he'd been passing through. He'd resolved within himself that he would only tell her after he had solved his problems. And Egbunu, I encouraged him by flashing affirmations in his thoughts that this was the right thing to do. "Have you written the second quiz?" he said instead.
"Yes, yesterday. It was simple."
"And have you—"
"Obim, they are telling me my credit will soon finish. And I bought two hundred naira. So please talk quickly, I miss you, Obim."
"Okay, Mommy. I will call you tomorrow."
"You promise?'
"It is so."
"Have you read my letter? In your bag?"
"Er, Mommy, letter."
"Read it anyway, there is something I want to tell you, but I want you to settle down first," she said in haste. "It is big, big news, even me, I am surprised. But I'm very happy!"
"You will—" he said, but the line had gone dead.
Agbatta-Alumalu, because he had finally spoken to her, because he had heard the one voice that could soothe his broken spirit, he felt a peace that was far deeper than the relief that hope had brought him. He laughed to himself, a laughter of satisfaction that things were mending quickly, at the pace at which they had been broken. For even Ndali, whom he thought he had offended gravely, had forgiven him. So happy was he that he teared up. He lay in the bed, and sleep came quickly to his tired, haunted, but tranquil body.
I had been wanting to leave his body to see what the spiritual world in this country of strange people looked like, but because of his anguish, I had been unable to, with the exception of going in search of Jamike's chi at Ngodo. For when a host is in trouble, we must watch, we must keep alert, open our eyes as wide as those of fish, until there is reprieve. So now that he slept soundly, I left his body and soared with unearthly energy into the spiritual realm. What I saw—Egbunu!—surprised me. I saw none of the things one sees when the veil of consciousness is parted: the patterned darkness of the night, the keening sound of the voices of revenants and various spirits, the noiseless footsteps of guardian spirits. Rather, here, in the stratus that formed at night, I saw oneiric forms ambling about like weary noctambulists. But what was most shocking for me was the paucity of these creatures here. For it seemed empty. I soon saw why: once I looked around, I saw unearthly temples, with ancient majesties and numinous architectural structures, at almost every corner. It seemed that in their Ezinmuo, the spirits sought dwellings like those of men, and most of them were inside these dwellings. There were even some parts that were so empty that they were filled only with the golden leaves of the luminous trees and the lucent footprints of all who'd trod them in the night. A soft, hollow tune remained, too, as if made by an instrument unknown to the fathers, which I have come to understand is called a piano. Its sound was different from _uja,_ the flute of the eminent fathers and the spirits in their lands. I had wandered the length and breadth of this place on a slow pilgrimage unlike the one my host had himself had in the land of the men until, fearing my host might wake up from some dream, I returned to find him peacefully asleep.
CHUKWU, the venerable fathers of old say that tomorrow is pregnant, and no one knows what it will birth. As what was in a woman's womb was concealed from the eyes of the old fathers (except for the initiated among them, whose eyes are able to pry into the world beyond men), so was the pregnancy of tomorrow. No one can know what it will bring. A man rests at night with the vaults of his mind full of plans and ideas for tomorrow, but nothing in those plans might be fulfilled. The great fathers understood a mystery lost on the children of the fathers now: that every new day, a man's chi is renewed. This is why the fathers conceive every new day as a birthing, an emanation of something new from something else— _chi ofufo._ Which means that what the chi may have conferred or negotiated on behalf of his host the previous day is done with, and a fresh action must be taken in the new day. Egbunu, this is the mystery of tomorrow.
My host, though, being human, woke with the joy of the hope that had been given to him the previous day and of his reconnection with his lover. When he came out of his room, Tobe was there, staring at his computer through his glasses. "Good morning, bro. You know that Saturday is orientation?"
My host shook his head, for he did not know what the word meant.
"I really say you should go. It is very good. They say it make a person understand the island, and see many many beautiful places and history."
"Uhm," my host said. "Have you go before?"
"No, it happens every Saturday. I came on Sunday and you came on Wednesday."
"No, I came on Tuesday. _Ngwanu,_ I will go."
"Good, good. Once we come back, we will pack our things and call a taxi to move us to our new house. It is a very good thing that when you start your work, by God's grace, you will already have a place to stay. This is very good."
My host agreed. He thanked Tobe again for everything, for how the man had been helpful to him. "I will never never forget what you have done for me, a person you don't know before."
"No, no mention. You are my brother. If you see an Igbo brother in another man's land like this, how will you let them suffer?"
"It is so, my brother," my host said, shaking his head.
His spirit lifted, he washed his socks, which he'd worn all through the journey down to Cyprus, and hung them on a wooden chair beside the parted curtains so the sun could dry them. He'd not worn this thing called a sock since primary school. But Ndali had bought them for him and insisted that his feet would get cold on the plane if he did not wear them. Outside the window, on the balcony, he saw pigeons on railings, cooing. He'd seen them the previous day but had paid no attention because while in a state of misery, a man is not himself. For instance, during the long walk the previous day, he'd remembered something that often made him laugh. One of his father's friends and his wife had visited them. The woman then went into the bathroom, although it was almost dark and there had been a power outage. They did not know that one of the chicks had found its way there. Not seeing it crouched at the back of the water drum, the woman removed her underpants and was about to start urinating when the chick hopped up to the sink. The woman screamed and ran out into the sitting room, where his father and the woman's husband were seated. The man, ashamed that my host's father had seen his wife's private parts, would end their friendship. Whenever he recalled this event, it often made him laugh. But that day, in that time, his mind merely swatted the memory away like an errant fly.
On this new day, though, as Tobe and he ate bread and custard, he laughed and joked about the ways of the people of this land, about his own naïveté, and about how—never having been on a plane before—he'd appeared like a fool. Then, after Tobe went to school to see his teachers, he lay down and slept so long, so soundly, that he did not wake up until sunset. When he woke, he saw that Ndali had tried to call him. He rang her, but the voice of the operator reminded him that he'd exhausted his credit. With Tobe he went to the restaurant in the school, and they sat eating and watching the people of the country, his mind filling up, his spirit mending. That night I saw, as my host slept, the guardian spirit of Tobe loitering around the place. I thanked it for the help its host has rendered to mine, and we sat down talking about the Ezinmuo of the strange country and all that our hosts had been through until, close to dawn, it insisted it must return into its host.
Early Saturday morning they set off towards the bus park. As they walked past the block of apartments, Tobe pointed at an apartment in the distance on which was a Turkish flag. "They are putting up their flags on the front of their houses and windows because of the soldier they have killed." He gazed at my host to see if he had awakened some curiosity in him, as is often the case in such situations. And if he sees that his companion is now curious, he will go ahead and feed him more. "Turkey is fighting with Kurdish people. PKK. The first day I came, some of their soldiers died."
My host nodded, not knowing what his friend was talking about. When they arrived at the bus stop, many foreign students were already there, mostly those who, like my host and Tobe, had come from the nations of black people. As they waited to get on the bus, my host, watchful, noted the difference between the people of this strange country and those who had come from his. The voices of the latter seemed loud while the former seemed muted, or calm. Presently, for instance, near the back of the bus, three men and a woman from the nations of black people were talking at the top of their voices, stamping their feet and gesturing. While around and about them, the white people of this country stood in clusters of twos and threes, whispering or silent, as if gathered at a funeral.
The woman from the international office, Dehan, and a white man who spoke English with an accent similar to Ndali's welcomed everyone. The man said they were about to see the "great beauties of this beautiful island. We will visit a lot of places—a museum, the sea, another museum, a house, and my favorite, Varosha: the deserted city. I have been living on this island for a long time now, but I'm still amazed. It's one of the wonders of the world."
"So nobody is living there?" one of the black students from around the land of the fathers said.
"Yes, yes, my friends. Nobody. Of course, Turkish soldiers live around the place, but only them. Only the soldiers. We cannot enter, my friends."
The students started to speak amongst themselves, intrigued about the idea of an abandoned city where no one has lived for more than thirty years.
"Okay, everybody listen," Dehan said, raising her hand and smiling at the chattering crowd. "We must go now. We will eat at the beach later. Now, let's go."
As they entered the buses, the woman came to my host and his friend and asked what had happened. Had he seen Jamike?
"Not yet, ma," he said. "But we have reported him to the police station, and they are looking for him." He saw that the woman was looking around, anxious to leave, and for closure and to assure her, he said, "I know that I will find him."
"Good, fingers crossed," she said, and walked to the front of the group.
Egbunu, I was happy, glad indeed that my host had found reprieve from his troubles. In just a few days, a dream had almost been dashed. He watched about him, observing things now that his mind could allow him to do so. On the bus, he and Tobe sat beside two white-looking people Tobe said were Iranians. And about the others, brownish men dressed in thin fabrics, he said, "Pakistanis." My host nodded, and Tobe added, "Or maybe Indians."
While Tobe gave him the history of India and Pakistan, he noted that at the front of the bus were two chairs on either side, with a raised platform on which the driver and Dehan sat. He watched the desert pass before his eyes as if on a sprint. He noted that the landscape, although dry and sandy, was interspersed here with some faint promise of vegetation. Awkward-looking plants, brown, skeletal, naked, firmed to the soil, filled the plain. He saw, in spatial distribution, trees grafted to the dry earth like elements from some other world. Trees, he whispered to himself, as he used to do when he was a child. He gazed back to see that his loud thought had not leaked into the ears of the others seated around him. Then it struck him that he'd seen a few trees around, but they were mostly on the edges of the roads. He thought how different the highway in Nigeria was from this one. Most of the land between cities in Nigeria was not inhabited. By contrast, the land between cities here was filled with casinos, hotels, houses, and sometimes nature—mountains and hills. At a place where the land was flat and cleared, and one could see for kilometers on end, Dehan pointed and said, "That is South Cyprus. The Greek side."
He stared in the direction where, indeed, even though the distance limited his vision, he could see tall buildings like the ones in American movies. The people he had visited in that city called Girne the previous day had told him that that was the real Europe, where Jamike was. He wished that by some extraordinary means he could find himself in that place, among those giant buildings, crossing the street to see Jamike. He wished he'd catch Jamike in his house and take back his money and then bring him to the police here to be imprisoned. He thought of the German lady and the promise of his deliverance. As it often happened, when something is only a promise, a thing of hope, its anticipation is shadowed by fear. And as he thought of it now, he wished that he would get the job. I intervened and put it in his thoughts that the kind woman had been moved by him. _Perhaps she has never seen a man become so broken that he was willing to give his blood twice. She will do everything within her means to help you._
Chukwu, I achieved success again. For my host heard me, and my words brought him succor. His thoughts shifted at once to the resolve that he would not tell Ndali any of the things that had happened to him until he was fine again. He would shield her from them, but after he'd gotten the job and recovered his money and things were going well at school, he would tell her everything, about how he was almost destroyed by this move. He was thinking about how much she'd cried and how he wanted, badly, to be with her again when they entered a city. "Gazimagusa," Dehan announced. "Bigger, much much bigger than Lefkosa. But we are going to the old ancient part, surrounded by walls. I live here." She stuck out her tongue, and the students laughed. She said something to the driver, and the man trailed off in a rushed, high-pitched response, and the students answered ecstatically.
From that moment onwards, the views changed. Giant walls rose high, and, carved into a fortress of high stone and concrete, bricks which he'd never seen before. It seemed like they had not been made with cement and water—a material with which the children of the old fathers now built—but with something solid yet earthen-looking, resembling the color of clay. Even though I have lived through many cycles, have followed and acquired knowledge from numerous hosts across times, I had never seen anything like these before. The stones had beams that were big and deeply cast, as if baked by the hands of the minions of Amandioha.
The bus drove under an arch formed with these bricks which had small dents and holes, as if a thousand men had stood below them pelting them with small stones for a hundred years. Egbunu, I could dwell on this endlessly, for I was greatly fascinated by these structures. But I'm here to testify about my host and his acts and to make the case that what he has done—if it is in fact true that what I fear has happened—was done in error.
The bus stopped just after this, and Dehan signaled that they alight. The other bus had arrived ahead of them, the guide with them. And when the people in my host's bus had all alighted, the man raised his voice and proclaimed: "Ladies and gentlemen, welcome to the walled city of Gazimagusa, as we say in Turkish, or Famagusta, as we say in English. What you see around you now, here, is the Venetian walls. They were built in the fifteenth century."
Like the others, my host turned around and saw the gradations of the massive structure, and again, these were so mighty and immense that I had the urge to leave his body and wander amongst these massive stones. Even though I had done so once, I feared that spirits in lands outside of Alaigbo, where the people have a reverence for the great goddess, are often violent and aggressive. I had heard that in these places roam a great many akaliogolis, agwus of all kinds, spirits of the hemisphere, creatures long extinct, and demons. I'd heard stories from sentinel spirits at the caves at Ogbunike and Ngodo about how violent spirits even forced a chi out of the body of its host and possessed him, something unheard-of even amongst the weakest of guardian spirits! So I stayed back. I sought instead to see everything through the eyes of the man with whom you, Chukwu, had made me one.
While most of the people seemed to dwell on the structures, my host observed the trees scattered among buildings. He thought they were trees similar to palms, as in the land of the fathers, but without fruits. Other kinds existed, too—one whose leaves covered it like tangled hair on the head of an unkempt person. At every step, the guide spoke of history, trailed by the crowd of students, who fed their eyes while listening to him. They stopped again at the center of a skeletal structure with five-columned spaces between its crumbled white walls. A great mass of stones that must have once stood as part of the building was now scattered about the space, some sinking into the rich earth of its ancient floor.
"Church of Saint George," the man said, his eyes raised towards the top of the immense ruins. "It was constructed during the early time of the church, maybe only a hundred years after the death of Christ."
Chukwu, as they walked on, he recalled suddenly how, once, he'd slept during the day and woke to find the gosling standing at the threshold of the sitting room's door. Outside, the day had aged—its subdued light cast the gosling as a silhouette. He'd almost never remembered that, for it didn't mean much until the days before he left for Lagos: he'd slept by Ndali, only to wake and find her standing in the same spot as the gosling, made into a silhouette by the dusk light.
He was deep in thoughts when he felt his phone buzzing in his trouser pocket. He took it out and saw that it was the nurse. He broke off from the group, but fearing that if he picked up, he'd call attention to himself and disrupt the guide's speech, he let it die. He'd barely rejoined them when it buzzed again. He saw that it was a message, so he opened it in a hurry.
My friend, hope all is going well? I am hoping you will fill your day with the good sun. nice man. Don't worry, my friend says we can come on Monday. Don't worry. Fiona.
EZEUWA, he followed the tour conscientiously, as if he were not the same man as the day before. He stood breathless as he and the other students lingered near the shores of the great Mediterranean Sea, where I struggled to contain the urge to get out of him and observe this curious place the guide had referred to as "the ghost city of Varosha." He listened, as if to lifesaving instructions, while the man talked. "Hollywood stars, presidents of many many countries, many many people, have come here." He marveled at the damaged structures—multistory buildings pocked with holes, their bricks fallen out, some riddled with bullet holes, images that reminded me of the towns and villages in the land of the fathers at the heat of the Biafran War. He gazed intently at one which must have been a great hotel, with massive corridors, but which now stood empty and abandoned. Beside it was a gray-colored building, its paint worn out and fallen away like pieces of soot. He tried to decipher the name of the hotel, but only part of it was still standing, and most of the cursive lettering had become detached from the wall. Holes adorned this building and gave it a peculiar look. He fell behind the group as he looked intently at the houses in the inner parts of the town, barricaded away by barbed wire and thin fences, buildings whose doors had fallen out. In one, the door knelt as if in plea at the threshold and leaned against the balcony with the rest of its body. Down below this building, sturdy plants threaded themselves through the streets in patches and stretched, as if through soft clothing, through the old faces of the walls.
The town opened a window in his mind which, throughout the rest of the trip, he could not shut. He was moved at The Blue House, which the former Greek leader with the strange name—who the guide had said was the one who caused the war between the Turkish and Greek Cypriots—had built for his children. But he kept thinking about the other abandoned places the guide said existed—an airport with planes, restaurants, schools, all vacant now. The place where they came to now the guide called the war museum. He was reminded at once of the Biafran War museum in Umuahia, which he had visited when he was a child with his father. Of that incident, I could not bear much witness, Egbunu. This is because no sooner had he and his father entered the place than they saw a tank which had been driven by one of my past hosts, Ejinkeonye, who had fought in the Biafran War and driven that selfsame tank. I was immediately overcome by the kind of crippling nostalgia that sometimes comes upon a guardian spirit who encounters the memorial of a past host or his grave. So I had left my young new host and gone into the tank, which I had been in many times in 1968, when Ejinkeonye drove it. The past is a strange thing to us guardian spirits, for we are not humans. Once I sat in the tank, I reenacted many of the bloody scenes of battle—how once the tank had raced into a forest to escape air bombs and had felled trees and trampled over the bodies of people as it went, my host weeping within it. It had been a sobering moment, and I had stayed in it while my current host and the other visitors inspected it, looking in it but not seeing a creature seated on its shriveled seat, a creature which, even these many decades later, still recognized the dried-blood smell of its interior.
From the war museum in this new country, they went to the "green-line zone," back in Lefkosa, and he saw the other Cyprus, a different country, separated merely by barbed wire. He marveled. It reminded him of the stories his father had told him about Biafra. He was moved by the sight of the Museum of Barbarism, of which the guide said, "Don't come in with us if you don't like horror movies." Then they had gone in with him, almost everyone. In the crowded doorway, he'd see the bathtub in which a woman and her children had been shot dead, their blood left smeared on the wall and the bathtub, just as it had been in the year the White Man calls 1963. "The blood on that wall is older than all of us here," the man said as they looked at the gruesome sight.
This he remembered, and those last words remained with him long after the tour had ended and he and Tobe had returned to the campus. But none of these touched him like the ghost town. It troubled him so much that later that evening, when he fell asleep on the couch in the living room, he dreamt of Varosha. He saw himself chasing his gosling as it leapt and raced into the abandoned houses. He chased it past the Turkish soldiers mounted on top of the buildings, watching. The bird ran, enfeebled by the twine on its left leg. It entered one of the buildings, the one whose door leaned against its balcony. He followed the bird, his heart palpitating. The house smelt of rust and decay, and dirt and dust had festered on the floor. Colloids of wall paint had massed about, as if waiting for something that would never come. Past this, he saw the gosling mount the stairs, its color turning dark as it came in contact with the dirt and dust in the house. The railings had cracked and, beneath them, clasped to the feet of the wall as by talons, were beds of moss. A shirt hung on a broken door, and he peeped in to see chairs and waste and upturned furniture, all bound with a monstrous network of impenetrable cobwebs. He was sweating and panting, and the gosling, rattling, kept ascending, mostly flying in leaps, turning in the gyre of stairs, as if its path had been mapped for it and its travel was deliberate. At last he found himself on the top of the building. He did not know why, but he cried to the gosling to stop, to not go, and it turned to him. But the bird leapt into the air and descended towards the shore. In panic, he followed it headlong, forgetting in the heat of the moment where he was. He was falling and screaming, headed for certain destruction, when he woke up.
The sun had almost gone down, and its vast endless shadows had dimmed. He opened his eyes and saw Tobe standing in the room, looking at his wristwatch. He would have been thinking on about that ghastly dream, but Tobe said, "I didn't want to wake you. But we better go move into our house before Atif brings the new students here."
He nodded and picked up his phone. There had been three missed calls from Ndali, none of which he had heard because the phone was still set to the mode that rendered even the longest ring silent. He found that there was a text, and he opened it at once: Obim, are you OK? Pls dnt forget 2 call me, OK? He wanted to ask Tobe how one sent text messages to Nigeria. One needed to add symbols and additional numbers in order to call, but what of messaging? Instead, he hurried to his room to prepare. While he packed, it occurred to him that he had not yet read her letter. He decided that once they got to the new place, he would read it.
AGUJIEGBE, after they arrived at the new apartment and moved their belongings into their rooms, he reached into his bag and searched until he found her letter, hidden in one of the pen pockets, folded many times. He wondered when she had written it. Was it the last night, when she had cried for most of the time and insisted they sit on the bench under the tree in the yard? They had sat there, a soft wind blowing, listening to the sound of the streets.
His hands shook as he unfolded the piece of paper, taken from one of her lined jotters, some of which he'd once flipped through. He put it down, lay flat on his back, and took it up again to read as she'd told him was best—reading aloud to himself:
_When you read, especially the Bible, say it to yourself. Speak it, because Nonso I tell you words are living things. I don't know how to explain it but I know it. That everything we say, everything, lives. I just am sure._
He looked up and then about at his bags before reading the next line, which stood alone.
_Obim, I am sad. I am very sad._
Egbunu, he put it down, for his heart raced. He heard the sound of music starting up, perhaps from Tobe's laptop. He felt something—a thought—flash in his mind, but he could not tell what it was. He was certain that he had not merely forgotten it, for it had not fully materialized in his mind but rather flashed in and fled.
_I have come to confess that many times I have wanted to leave. While in Lagos I planned to text you and say that I am not doing again. In fact, I typed everything out but my heart did not allow me. It is because I love you. Sometimes I feel I want to leave because of my family but it is like something stopped me. It is like you captured me, like our chickens. It is like I cannot get out. I cannot leave at all, Nonso. Even 1_
Ijango-ijango, as the heart of a troubled man often leads him on tangents at such a moment as this (many instances of which I have seen), his eyes pored over a dab of ink that spread across the paper from the last word so that it seemed as if the last letter, _1,_ was an upturned number 7.
_night they asked me why I love you. For a longtime I did not know it myself Nonso. Yes, I wanted to find the good man who helped me at the bridge that night, but I cannot explain why I became intimate with you after we saw again. I liked you but I didn't know why I did it. But the day you pursue the hawk, I knew that day that you can do anything to protect somebody you love. I knew that if I give my heart to this man, he will never disappoint me. When I see the love you show to ordinary animals, I knew you will show me greater love, greater care, greater help, greater everything. This is why I love you Nonso. See it now? Is it not true? Who can do this? How many men in Nigeria or even all over the world can sell everything they have for the sake of a woman? AM I CORRECT NOW OR NOT?_
She had written the last question in capital letters, and the perceived tone, the force of how she may have felt while conceiving it, caused him to drop the paper, for his heart was beating faster now. He could not tell exactly why at first, but out of the emptiness of his mind, he saw his father and his mother and him during an environmental sanitation day in the year the White Man calls 1988. They were cleaning the front of their compound. His parents were both watching him and clapping for him because his mother had mocked his father for not being able to sweep thoroughly. And his father had complained that the broom was too lean. As he swept, many of its bamboo sticks had fallen off. His mother, having taken the broom from him, had given the broom to my host and said to his father, "You will see that he would sweep it better than you." And taking the broom, he, a mere six-year-old, swept as his parents cheered him on.
It struck him now that it was that same compound that he had sold. He reread the passage about how he was the only man in the world who could have done it. An idea came to him. What if he called the man who had bought the compound and told him to hold off, that he would send the money for the place with interest? He could pay every month, every month, until all was paid plus 10 percent. He nearly jumped up at this thought. He would call Elochukwu the following day, and then Ndali, so they could go to the man at once and ask him to hold off ownership of the house.
Ijango-ijango, I, too, was overjoyed at this idea. It was not in the custom of the old fathers to sell a land. For lands were sacred. It was given to them by Ala herself and was not the possession of the man who came to own it but that of his lineage. Although Ala never punishes one who sells his land out of his own will, it angers her. With the enormous relief he felt at this decision, he picked up the letter again, its edges now wet from the sweat on his palms, and finished reading it.
_I know myself. From the first day, I knew you were genuine. I knew you were the man God has prepared for me. And I want you to know that I love you and will wait for you. So please be happy._
_Your love,_
_Ndali_
#
## Visions of White Birds
EBUBEDIKE, the great fathers speak of a man who is anxious and afraid as being in a fettered state. They say this because anxiety and fear rob a man of his peace. And a man without peace? Such a man, they say, is inwardly dead. But when he rids himself of the shackles, and the chains rattle and tumble away into outer dark, he becomes free again. Reborn. To prevent himself from falling again into bondage, he tries to build defenses around himself. So what does he do? He allows in yet another fear. This time, it is not the fear that he is undone because of his present circumstances but that in a yet uncreated and unknown time, something else will go wrong and he will be broken again. Thus he lives in a cycle in which the past is rehearsed, time and time again. He becomes enslaved by what has not yet come. I have seen it many times.
Although my host's promise of salvation was still firmly in place—the nurse had texted him twice since they met and the second time had added a yellow image of a laughing face and repeated that he was a "good man"—the fear came after he read Ndali's letter. It held him bound for the last portion of the night, dangling flashing images of other men having romance with her in his mind. He was released from this state in the early hours of the morning when Tobe knocked and asked from behind the door if he would go to church. "If you come," Tobe continued, "you will meet a lot of Naija people there. And I tell you, you will like it. You can thank God for everything, and also, we can buy some things to cook from the market there. We should start to cook before school starts tomorrow." My host said he would join.
Later, they were walking along a road that appeared like one he'd passed on Thursday, just after he left the taxi. The streets were compact, and the buildings seemed to have no partition between them. A barbershop constructed in glass sat by the sidewalk. A man smoking in front of it, blowing billows of smoke into the air, yelled _"Arap!"_ at them as they passed.
"Your papa _arap_!" Tobe yelled back.
"Your father, mother, everybody _arap_!" my host said, for Tobe had told him that whenever he heard that, it meant he was being called a slave.
"Don't mind them, they are idiots. Look at that dirty-looking man calling us slaves. That's the thing. They are so foolish."
They crossed into a lone street whose houses had gates like the ones in Nigeria. Big green metal boxes filled with dirt sat at every corner. But on one street they passed, Tobe pointed to one of the buildings and said that the white people from Europe loved to come and see it. It was a building of rich clay, like nothing my host or I had ever seen before. He was enraptured by the sight. The building was roofless, with mighty pillars. A temple to a Greek or Roman god, Tobe suggested aloud, perhaps so the old European man taking photos of it could hear him. An ancient temple destroyed by age, its old beauty trapped beneath the skin of its ruins. Yet in some way, it still was beautiful, for this is what turned it into a spectacle, why people traveled from afar to see it. A beauty out of ruins: this was a strange thing.
When they turned onto a street Tobe said was near the church, they saw other people who had the color of the great fathers, a group of four men, two of them wearing visors, walking towards the church. With this group, they entered the church. It was full, and one of the men they had seen at the apartment with Nigerians on campus, John, was directing people and offering chairs to those who did not have seats. The place was full of black students as well as some white people. A different kind of white man, one who looked not like the Turkish people but like the ones who ruled the land of the old fathers for many years, stood at an altar in front, speaking in the same accent as Ndali, and he knew at once that he was British. The man spoke about the need to sing with all their hearts. He and Tobe sat in the very back, behind two people who looked somewhat familiar to him.
He thought of the church of his childhood, which he had stopped attending. His father had stopped after the death of his mother, angry with God for letting his wife die in childbirth. My host had continued, sparingly, until an incident with the gosling had changed his mind. The gosling had become ill, refusing to eat and falling whenever it walked. The idea came that he take it to a church where he'd heard of faith healings—of a blind man seeing again. So he took his gosling to the church, carrying it close to his chest. He was stopped at the door by uniformed ushers who thought him mad for bringing an animal to the church. That incident killed his faith in the religion of the White Man. Why would God not care for a sick animal if he cares for human beings? At the time, he found it hard to understand why one could not love a bird just as one loved people. Hoping that he would turn to the religion of the pious fathers, I had encouraged his decision, adding to his thoughts that if he went to an odinani shrine with his animal, Ala or Njokwu or any number of deities would not have cast him away. But like many of his generation, such a thought was verboten.
Now he listened even harder as the preacher began speaking of resurrection and life. The man talked about Jisos Kraist and how he had died and risen. Sleep came upon his eyes as the man, whose voice careened in the air and shifted between high and low, spoke of how only true Christianity could lead to possessing a life of resurrection, of rising again after a fall. He opened his eyes, for the man had spoken to him. He was a witness to how, when lost, a man could descend into the abyss and still be raised up and restored.
When the preacher finished his sermon, they sang, and the church was dismissed. Once the people began to leave their seats, a man tapped him on the shoulder.
"Jesus Christ, T.T.!"
"Oh boy! Happy to see you here."
"Yes, my brother."
"How are you, how far, did you later see your friend?"
"No," he said, and told T.T. everything that had happened. By the time he finished, they were standing outside the gate of the church, and Tobe, who'd greeted a few people, had come to stand by his side.
"Mehn, casino pays very well for here," T.T. said. "God sent that woman to you true, true oh. Some of the Turka people are good. There's a woman like that who really helps a lot of people. She gave a Naija boy scholarship, sef. The boy was working for her, doing everything, and instead of just paying him, he said make she kuku pay his school fees."
"Hmm, good people."
"Yes, yes, but be careful eh. Sometimes, the people just get kanji."
T.T. laughed and said, "Take my number."
ONWANAETIRIOHA, when he returned home with Tobe, it was already dark. He reached for his phone, and a text message was on it. He read the message from Ndali. Nonso, call me tomorrow pls. He shook his head. He dialed her number, but only a long-drawn static noise came back to him. He resolved to call her after he'd confirmed the job, after he'd known for certain that he would recover that which he'd lost. And when he called her, he would tell her everything—everything from the airport to his meeting Fiona.
He sat back in his chair in his room and thought of the days, of all his being in a new country. He reached in his bag and brought out the photos of Ndali naked. As he watched them, his body caught sensual fire. He brought out his penis. Then he rushed and bolted the door so Tobe could not come in at will. He pushed his ear against it for any sound of Tobe, and when he did not hear, he looked at the naked photos of Ndali and began touching himself, gasping, moaning until he fell into a limp state.
AKATAKA, amongst the people of the world, anywhere, there is a common thread of compassion for a man who is wounded, or poor, or lowly. This kind of man earns their pity. Many would desire to help such a man if they believe he has been wronged. I have seen this many times. This is why a white woman in a foreign land can see a man from the land of the fathers, tattered, broken, and offer help, and in offering, create a pleasant expectation in him.
He woke the following morning, having slept for a full night for the second time since he arrived in the strange country. So full of expectation was he that he called Elochukwu and told him to go at once to the man to whom he'd sold the land and ask him to not do anything, that he would refund the money. "But how is this possible when you don't give him the money immediately?" Elochukwu said.
"Tell him I will give him double. We should sign agreement; I will pay double within six months. Then I can have my house back."
Elochukwu promised to meet the man and talk to him. Assured, my host washed himself and joined Tobe, who had cooked fried eggs.
Tobe talked about how difficult it had been to find good bread that morning.
"All of the bread they have are like stone," he said, and my host laughed. "I don't even understand this people at all. Not wan single bread in the whole shop."
"You have watched _Osuofia in London_?" my host said.
"Heh, the one he went to that place and asked for Agege bread and the Oyibo people were jus looking like mumu?"
They ate in sudden silence, he thinking of how mornings were different here. He'd not heard any cock crow, not even a call to prayer from a muezzin. The image he'd remembered the previous day returned and he saw Ndali almost naked, standing at the threshold of the sitting room's door. She was standing there looking away, her back turned to him like a thing to be feared. He did not remember what he had done—had he called to her? Had he turned away? He could not tell.
"This people, they stick to time," Tobe said again. "If they tell you ten o'clock, it is ten o'clock. If they tell you it is one, it is one. So we should go quickly to the letting agent's office, collect your own keys and go wait for the woman."
He nodded. "It should be so, my friend."
"I called Atif yesterday, and told him we have found a place. He asked about you. When I go, after my registration and class, I will go to his office."
"Thank you, my brother," he said, for he was paying little attention, his mind fixed on the errand he'd delegated to Elochukwu and the job Fiona would soon take him to.
They cleared the table and left the house, Tobe carrying a bag with his computer inside, and books. The bag resembled the schoolbags children wore on their backs, as Tobe himself wore his. My host carried the bag Ndali had given him, which contained his documents, her letter, and her photos, as he'd been doing since he came to the country.
They found the agent's office in the interior of the city center, tucked into an area full of clothing and jewelry stores. It was on a street behind the center, compact and full of stores, a cyber cafe, restaurants, and a small mosque. Pigeons hopped about, feeding on something or other. Here they found a lot of white people different from the Turkish people. Tobe said they were Europeans or Americans.
"They are different," Tobe insisted. "These ones, the Turkish people, they are not real whites. They look more like Arabs. You know how—have you seen Sudan people before? They are different from our black—that _kain_ difference."
A group of the kind of white people they were speaking about was walking along. Two young women, almost naked, in half shorts, brassieres, and slippers, passed by. One of them carried a towel. "My God, see Omo!" Tobe said.
He laughed. "I thought you were a born again," he said.
"Yes. But see, these girls fine. But Turkish woman beat them. But Naija still remain number one."
Egbunu, when they entered the office, the air was filled with cigarette smoke. A stout white woman in a chair was smoking. I noted that at the threshold of the door was a round amulet, the color of Osimiri, with a white inner sphere within which looked like a human eye. Because it appeared so much like an amulet, I came out of my host to see if it posed a danger to him. And at once I saw a strange spirit in the shape of a snake curling around the object. This creature was a fearful sight even for me, a guardian spirit, who journeys regularly into the plains of the ethereal. I fled in haste.
When I rejoined my host, the woman was counting the money Tobe had given her. Later, when they stepped out with the keys, he felt an overwhelming relief. When they came out, it was nearly ten. So they walked to the bus stop. They stood there for only a few minutes when Fiona arrived in her car, dressed in a white frock with a necklace that sparkled around her neck. He shook Tobe's hand and ran towards the car.
"You're looking happy," Fiona said once he entered.
"Yes, Fiona. Thank you. It is because of you."
"Oh, no, come on! I haven't done anything. You were in big trouble."
He nodded.
"I got an apartment with my friend."
"Ah, that's very good. Very good. It helps your psyche, you know, to have a house."
He said yes.
"My friend Ismail is in the office. He is waiting for you."
As soon as he sat, I noticed that the woman wore—around her wrist in the form of a band—the same kind of amulet I had seen earlier. I flashed the image of the one at the agent's office in my host's mind and pointed him to the woman's wrist, for I was curious to know what it was. Unexpectedly, Chukwu, it worked.
_"Es ma,"_ he said.
"Yes?"
"What is this blue thing that resemble eyes everywhere here—?"
"Oh, oh," the woman said, and thrust her hand into the air. "Evil eye. It's like, you know, a good-luck charm. Very big deal to Turkish people."
My host nodded, even though he could not fully comprehend what the object was. But I was relieved to know it was merely a personal fetish, not something that could harm my host.
They drove, a tune playing from the stereo. She asked him what music he liked, but when he listed them, she didn't know any. It struck him, once he'd finished speaking, that he did not mention Oliver De Coque. The thought of the singer annoyed him, as if De Coque had done something to hurt him. But he knew that he'd come to associate the memory of the day he was humiliated at Ndali's family's house with De Coque, who was playing his music that day. And he now resented the musician for it.
"This is Emre Aydin, a very good Turkish singer. I like him very much." She laughed and glanced at my host. "By the way, Solomon, I've been thinking about your story. It's very painful."
He nodded.
"It reminded me of a book I read recently about a man who was asked by his wife to join the army during the war, and when he did, she became very disturbed by the actions, you know, of the army. Hitler's Nazi army. She left him. It is a very difficult book. You do something great because of a woman you love, and then you lose her. I am not saying it will happen to you, don't get me wrong." She waved her hand. "You will be fine and your fiancée will be there for you—I'm sure. I am speaking of the sacrifice. _Genau_?"
He looked up at her, for her words had shot into his heart and pierced it.
"Yes, ma, I—" He stopped himself and said instead, "Yes, Fiona."
They passed the strange road again and climbed a mighty bridge, then went down a small ramp made of interlocking bricks. As the car approached what seemed like the limits of a village, giving way to densely vegetated lands, the sun seemed to drop lower, and its heat, visible in the illusory wave, made it appear as if the car had suddenly plunged into a river. But soon the deception was busted, and they entered into the town's small streets. The car made a grinding sound as it raced past others, jerking so much that even the thought of Ndali leaving him, a thought which had lain like a child in the cot of his mind, shifted violently from one end to the other. He struggled to still it. But he could not.
OSIMIRIATAATA, the peace that concrete hope brings to a man who has suffered cruel defeat is difficult to describe. It is the sublime incantation of the soul. It is the unseen hand that lifts a man off a cliff over a pit of fire and returns him to the road from which he has veered. It is the rope that pulls a drowning man out of the deep sea and hauls him onto the deck of a boat to the breath of fresh air. This was what the nurse had given him. But what I have seen many times before is that the hands that feed the chicken are the same ones that kill it. This is a mystery of the world, one which, in this strange country, my host and I would come to experience. But I must render it all in as much detail as I can, Egbunu, for this is what you desire of us when we come before you here in the luminous court of Beigwe.
When they arrived in the city from which he'd emerged four days before with a bleeding spirit, his heart was so warm and his joy so grand that he wanted to take a photo of the place. So before they entered, he asked Fiona if she had a camera phone.
"Yes, yes," she said. "It's a BlackBerry."
"Okay," he said.
"You want a photo?"
He nodded and smiled.
"Ha!" she said, and blew air out of her mouth. "You can't even tell me you want a photo? You are a shy man."
She snapped a photo of him folding his hands across his chest, then pointing at the light-box sign on the facade of the white marble building, then with his hands spread out, both ways. He looked through these images of himself looking happy, and they pleased him.
"I will send them to your e-mail."
He agreed. As they walked into the place, part of his mind was thinking of Ndali, how she would like the photos. The other half was in awe of the magnificence of the building—the blood-red rug with tiger prints, the ornamental lightbulbs, the machines and TV screens. He stopped thinking of all these when he started walking behind Fiona through a narrow hallway. It must have been because of the kind of shoes Ndali called "heels," but her buttocks danced in a shapely way. And through the white frock, he saw the outline of her underpants.
Ebubedike, it surprised him, the strange sudden beat of his heart at the sight and the quick fist-punch of lust on his mind. It came at him like a burst of flame, so quick and unnatural that he was taken aback by it.
As if she suspected what had happened, she turned. "Solomon, I have told you what he will pay, yes?"
"That is so, Fiona."
"Okay, take it for now. We can increase later. _Genau_?"
He nodded. He walked by her side now as they arrived at the entrance to the manager's office. But the desire remained, even against his will. He wondered how old she must be. Her body looked young, like that of a woman in her thirties, but her neck showed skin gradations that suggested otherwise. And he'd seen traces of wrinkles on her legs too. But still he could not determine such things about white people, about whom he knew little.
Through a glass door they came into a room where a man sat across a desk, his face intent on a computer screen. The computer, Chukwu—an instrument that is able to do so much. It can gather information, serve as a device for communicating with those afar, and much more! When it becomes common among the children of the precious fathers, it will further alienate them from their ancestors. Fathers of the hills and lands, dwellers of Alandiichie, do you weep that the alters of the _ikenga_ have been abandoned? What you have seen is nothing. Do you worry that your children do not observe _omenala_? This thing, this box of light into which this white man is staring, will cause you greater grief in the fullness of time.
The man rose once my host and his companion entered the room. He shook the man's hand but understood little of what the man said. He thought the man spoke the language of the White Man well but seemed to prefer the language of the country. What he noticed more was how the man hugged Fiona and touched her shoulder and patted her on the arm. For a while they spoke the language, and he gazed at colorful images on the four walls of the room—images of the great sea, the swimming turtle, and of some of the ruins he'd seen on the tour, all the while praying that the man would give him the job. So folded away was he that he gave a start when the man stretched his hand towards him and said, "So you can start from tomorrow, Tuesday, if you want."
"Thank you very much, sir," he said, shaking the man's hand and bowing slightly.
"Don't mention. Okay, see you, my friend. Congratulations."
The man walked back into the hallway and made to leave but turned hurriedly and took Fiona's hand again, and they embraced. The man seemed to kiss her cheeks, the way Ndali would sometimes ask him to do to her. It was a strange thing, Chukwu. A man kissing another woman who was not his wife in plain sight? The man lit a cigarette and began speaking to Fiona again in the language of the country.
When they came out of the building, Fiona said she had baked a cake for my host. She would bring it from the oven, wrap it up for him, and they would go to a restaurant. And while at her house, she would show him her garden, for she, too, was a farmer, like him. He agreed and thanked her even more. By the time they got on the road again, his lust had fizzled out, suppressed by an infant rage which stood in the midst of his joy like a stranger among a crowd of friends. An Igbo man like him, one he could call a brother, an old classmate, had cheated him and almost destroyed him. But here, among a people he did not know, people of a different country and race, a woman had come to save him. This woman and her friend had even gone further than Tobe, who for a long time had borne his cross with him. They'd taken his cross and set it on fire, Fiona and this man. And by the time she arrived at her house, his cross—all that it was, and all that was within it—had burned to ashes.
EGBUNU, I have spoken about the primal weakness of man and his chi: their inability to see the future. Should they have possessed this ability, a great many disasters would have been easily prevented! Many, many. But I know that you require me to testify in the sequence that things happened, to give a full account of my host's actions, and thus I must not stray from the path of my story. I must thus proceed by saying that my host followed this woman to her house.
The house was big. Outside it, a garden, water hoses, and flowers arranged in neat beddings. She said her mother, who sometimes visited from Germany, was a farmer. A dry pool filled with leaves lay near the low wall on one side beside a shovel and a wheelbarrow. She did not plant anything that could be eaten, except for tomatoes. But she hadn't planted in a long time. The garden, he realized, was a storage area for things she wanted to keep possessing. She said that the old paraffin lamp that hung on the branch of a low, lean tree from which a fine laundry rope stretched out to the house was her cat's. Miguel. He did not know that people could keep cats as pets, let alone that they could be named.
This thing that looked like the engine of a car seated on the ground was from the truck in which her husband's father had died. She paused at the sight of this one and dropped both hands to her sides. Then, without looking at him, she said, "It was the beginning of the trouble. From then, he always says: 'Why did I let him drive? If he didn't drive at seventy-two, he'd still be here today.' That's why he drinks himself to stupor and turns his back to the world." Then an unexpected thing happened. For when she turned to him, this woman whom all the while had been full of life was now almost in tears. "He turned his back to the world," she said again. "The whole world."
Thinking of the job, of the casino, of the charge he'd given to Elochukwu, how it would turn out, he barely heard the things she was saying. That long walk he had thought of as the most unbearable time of his life, he reckoned, had in the end become the thing that had brought him great hope. He followed her into the house, curious to see what white people's houses looked like. They went through the back door into a kitchen that was nothing like what my host had seen before. It was marbled (although he did not know the word, Egbunu) and covered with paintings.
"They are my drawings," Fiona said to him as he gazed at one which was different from the rest. It was not the image of a cat, or dog, or flowers, but a bird.
"They are very nice," he said.
"Thank you, my dear."
He walked with her into the sitting room, and he was struck by the enormity of Ndali's father's wealth. Their house was lusher than that of a white family. He gazed about at the piano by the yellow wall, a big television, and a speaker. There was only one couch, long and black, made of some kind of leather. The walls, from beginning to end, were covered with paintings and photographs. Near the television and a shelf of books stood what looked like the dry white sculpture of a human skeleton. The sculpture wore a necklace with the evil eye image on it.
"So I will change. It's hot. I'll put on some pants and shirt, and we will have the cake and go. _Genau_?"
He nodded. He watched her climb the stairs, the thighs under her frock visible. Desire erupted in him again. To shove off this urge, he looked up to the image on the wall above the piano in which sat the man he believed might be her husband. His eyes in the picture were happy. Yet there was a sternness to them that gave him the appearance of a man of tough temperament, something close to what Fiona described as "turning his back to the world." Beside that lone portrait was one of the man and Fiona, years younger, with fuller hair that was fixed into the shape of a hanging tail behind her back. They were seated, Fiona in front of him, he behind her, half of him concealed so that only his chest was revealed. The picture was taken, it seemed, at a function, for there were people in the background, some prominent, others faded out by distance. The trunk of a green car—rear pointing downward—stretched into the picture, its other half lost to visual oblivion.
Egbunu, at this point, I can tell you that there was nothing in his mind about this man other than that he was curious about what grief had done to him. He was searching the picture of the man to see if he could find any sign of the darkness Fiona had described. He'd also noticed a kind of quiet fear in Fiona since they arrived at the house, as if she was afraid of something which she was unwilling to confront. Chukwu, I know that it is quite possible that our recollections are not always accurate because hindsight can influence them. But I am giving you the unfiltered account when I say that my host gazed at this man's photo closely and introspectively, as if he were aware, even vaguely, of what would come next. He turned from it to the small recess in the wall containing woods and dry ash—what he thought of as firewood inside a sitting room but which I knew from the days of Yagazie as a fireplace, where white people sunned themselves when it was cold. There was such a place in every house where my host went in Virginia, in the country of the brutal White Man. Without it, the cold—something unthinkable in the land of the great fathers—would kill them. He was examining this when Fiona began descending down the stairs. She had changed into short pants and a shirt with the image of a half-sliced apple on it.
"Okay, let me get the cake, and let us go."
"Okay, Fiona."
He watched her open the oven and bring out something wrapped in a white paperlike material; neither I nor my host knew what it was. She put the thing in a polythene bag.
"What kind of food do you like?" she said.
He had begun to speak when she cut off his speech with a wave of her hand. He turned in the direction where her eyes were looking and saw the reason why. The main door was opening, and an older, much more worn-looking version of the man in the portrait stepped into the house. His shirt was unbuttoned, a wrinkled blue shirt whose sleeves had been rolled up, revealing a white skin so hirsute it appeared as if his hands were black. He walked a few paces into the living room and stopped where he was, gazing at them.
"Ahmed, wow, welcome," Fiona said in a voice that betrayed restlessness, fear. "Where are you coming from?"
The man did not speak. He stood with eyes roving from my host to his wife and back again with an intensity that was familiar to me. It was a gaze whose import may be understood more in effect than in contemplation, like the understanding of the full enormity of life in the moment before death. The man's mouth was poised for speech, but instead, he laid down the bag he carried on the floor gently. Fiona moved towards him, calling his name, but the man stepped towards the bookshelf.
"Ahmed," she said again, and spoke in the foreign language.
The man responded with a countenance that frightened my host. As the man spoke, saliva splashed from his mouth. He pointed to Fiona, clenched his fist, and pounded it into his palm. Fiona, gasping, her hand over her mouth, spoke in rapid gusts in what seemed like protests, to which the man paid no heed. He spoke even louder, in a high-pitched tone. He snapped his fingers, thumped his chest, and stamped his feet. Fiona fidgeted as the man spoke and stepped backwards in increments, turning back and forth from her husband to my host and back again, her eyes filling with tears. She was talking when the man faced him.
"Who are you?" the man said. "Do you hear me? Who in hell are you?"
"Ahmed, Ahmed, _lutfen,_ " Fiona said, and tried to grab him. But the man wrenched himself away with a cruel force and struck her across the face. She fell down with a scream. Her husband followed her to the floor, beating her with his fists.
Gaganaogwu, my host was terrified by what was unfolding before him, and I, his chi, was too. He stood where he was and said in a quivering voice, "Sorry, sir, sorry, sir!" He glanced at the door, whose path he could reach without much trouble if he hastened, but he stood still. Go! I cried into the ears of his mind, but he merely stepped forward an inch. Then he turned again to Fiona. He lunged forward and punched the man on the back and pushed him away. The man rose, picked up the bag, and rushed at him with it. The man slung the bag at my host's face with a brutal force that sent him across the room. The bag bounced off his face to the floor, and from the sound it made, and from the frothing liquid that poured across the floor, I knew at once that it contained a bottle.
My host lay where he had fallen now, dazed, his body in a state of frugal peace. When he opened his eyes, a fast-moving figure rushed into his field of vision, and before he could tell what it was, his eyes had closed again. Slowly and continuously, he felt cold liquid run down his shoulder, chest, and arms. Ebubedike, although I was greatly shaken by this, I was mightily relieved that my host was alive. If this man had killed him, what would his ancestors have said of me? Would they have said that I, his chi, was asleep? Or that I was an ajoo-chi or an _efulefu_? This is how, sometimes, the life of a person ends—suddenly. I have seen it many times. One moment they are singing; the next, they are gone. One moment they are saying to a friend or a relative, I will go to that store across the road, buy bread, and come back. I will be back in five minutes. But they never return alive. A woman and her husband may be talking. She is in the kitchen, he is in the sitting room. He asks a question, and while she is answering—while she is answering, Egbunu!—he is gone. When she does not hear from him for a while, she calls out, "My husband, have you been listening? Are you there?" And when he does not respond, she steps in and finds him slumped, one hand clutching his chest. This, too, I have witnessed.
My host lay, alive but in sublime pain, his face and mouth covered in blood. He wanted to keep his eyes closed, but Fiona's screaming and pleading prevented him. When he opened his eyes again, he saw the man and, in the man's hand, what had hit him: a big white bottle whose bottom half had broken off, leaving it in the shape of half-formed fingers, its edges red with blood that slowly dripped to the floor. The man was standing with the object over Fiona. Then he saw the man bend over her, shouting and moving the bottle about so that drops of blood and wine spattered on her face. From the dim vision of his closing eyes, he saw the man throw the bottle away and sink down and begin to reach for her throat again, unmoved by her screaming and pleading. Slowly, he crawled towards them, stopping to gather strength as Fiona's screaming grew louder with each step, for the man had now succeeded in reaching her throat. In this memorable moment of life, Egbunu, my host, bleeding profusely, reached up, lifted a stool, and tried to keep his eyes open to prevent the blood from clouding his vision.
The stool in his hand felt heavy. He had been weakened by the blood he had lost, not just now but a few days before, at the hospital. Yet Fiona's screaming propelled him forward. He rose up and lifted one foot, then the other, until he reached the place where they were. With every bit of strength he could summon, he hauled himself forward like a sack of grain and brought the stool down on the man's head.
The man fell over backwards against him and lay still. From his head, an aureole of blood formed. My host staggered, wiped his face, and batted his eyelids. Then he fell back to the wet floor and lay down on the black veranda between consciousness and unconsciousness. In the meaningless space that the world suddenly became, he saw Fiona turn into a strange creature, at once a bird and at once a white woman dressed in white. From the margins of his anguished vision, he saw her stretch and rise slowly like a snake unfurling from a rigid coil and then begin to scream and shout. He saw her perch on the corner of the room beside her husband, her plumage rich and almost immaculately white. Then she materialized again into a human, trying to waken her slumped husband, who did not stir. He heard her say, "He is not breathing! He is not breathing! My God! My God!" Then her wings spread, and she flew out of the range of his vision.
He lay there with a still vision in his mind of Ndali seated on the bench under the tree in his compound, looking straight ahead. He could not see what she was looking at. Whether this was from memory or from his imagination, he could not tell, nor could I, his chi. But it continued as he watched Fiona, wings still splayed, return to the place with a majestic stride. He saw her enlarged sternum, with the sparkling necklace around it, and a beak that seemed to carry something indistinct clamped in it. Then she moved again, now with her human feet, and he heard the sound of her feet on the floor. He heard the sound of her dim cries.
He heard the white woman speaking on the phone, her voice frantic, helpless. He opened his eyes to see her, but he was blinking so rapidly that the muscle below his eyes had begun to ache. In the all-encompassing darkness into which his body was thrown, a sudden chill came upon him and he became aware of a presence. Chukwu, he became still: for he could tell that yet again, it had come. From the backstage of life, it had come. That creature which has a red mother and whose complexion is the color of blood. _Again, it had come_. It had come again—to steal everything that had been given to him and to destroy the joy he had found. What is this thing? He wondered. Is it a man or a beast? A spirit or a god? Ijango-ijango, he did not know. And I, his chi, did not know, either. The great fathers often say that one cannot, by looking at the shape of the belly of a goat, tell what kind of grass it has eaten.
He heard Fiona crying, but he did not open his eyes. She said something to him which at first he did not hear, then to her husband, who lay still, like a plank. It was then that he heard what she had said, loud and clear: "You've killed him. You've killed him." She broke down into a loud sound. She had barely begun to cry when, in the distance, a siren began to wail. But he lay still there, his mind fixed on the curious vision of Ndali staring into the unknown, as if in a mysterious way she had broken the barrier of thousands of kilometers and was looking at him.
#
## Alandiichie
EBUBEDIKE, the old fathers in their cautionary wisdom say that the same place one visits and returns to is often the place where one goes and becomes trapped. My host had found succor in the white woman, but this same place where he'd found succor is where he now lay, wounded and bleeding, blinded by his own blood. Frantic, unable to do anything, and wary as to how I would explain this tragic end to you, Chukwu, and to his ancestors, I left his body to see if help might be found in the spirit realm. Once out, I saw that spirits of all kinds had gathered in the room like dark auxiliaries marching upon the entire army of mankind itself. They hung everywhere, near the arch of the ceiling, suspended over the body of my host and the other man, some hanging like curtains made of shadows. Among them was an unsightly creature who gazed at me with an ugly frown on its face. I noticed that it was an incorporeal replica of the man on the floor. It pointed its finger at me and spoke in the strange language of the country. It was speaking when the door opened and police officers stormed in with people in white frocks like the one Ndali wore, and the white woman, too. She was crying and speaking to them, pointing at her husband and then at my host, who lay there, slowly slipping into unconsciousness from loss of blood.
Three of the police officers and nurses carried away the man who had attacked my host, Fiona following behind them. Then they returned and took him, their shoes soaked in his blood, red footprints marking their trail. Chukwu, by the time they got into the vehicle that resembled my host's van (called an "ambulance" among the children of the great fathers), he fainted.
I followed them through the streets of the strange land, seeing what my host could not see—a car loaded with watermelon, the kind found in the land of the fathers, and a boy on horseback followed by a procession of people beating drums, blowing trumpets, and dancing. All these gave way for the ambulance to pass, its siren blaring. I was besotted with fear and a great regret that I had allowed him to come to this place, this country, just because of a woman, when he could easily just have gotten another. I repeat, Egbunu, regret is the disease of the guardian spirit.
The veil of consciousness that occludes my vision of the ethereal world now torn away, I beheld for a second time the living phantasmagoria of the spiritual world here. I saw a thousand spirits nestled at every breadth of the land, hanging on trees, flowing in midair, gathered on the mountains and in places too numerous to name. Near the Museum of Barbarism, where my host had been only two days earlier, I saw the three children whose blood was in the bathtub displayed inside the house. They were standing outside the house, dressed in the exact same shirts they'd been wearing at the time of the attack, torn, ripped by gunfire, blackened with blood. Because they were standing alone, unattended by other spirits, it occurred to me that they must be perpetually standing there, perhaps because their blood—their life—remains on the wall and on the bathtub, on display for the world to see.
At the hospital, they wheeled my host into a room, and when I saw that he was secure, I ascended immediately to Alandiichie, the hills of the ancestors, to meet his kindred amongst the great fathers to report what had happened—after which, if indeed he had killed the man, I would come to you, Chukwu, to testify of it, as you require us to do if our host takes the life of another person.
IJANGO-IJANGO, the road to Alandiichie is one I know well, but on this night, it was more winding than usual. The hills that border the road were dark beyond all imagining, speckled only here and there by the savage light from mystical fires. The waters of Omambala-ukwu, whose sibling is situated on earth, flowed with a muffled roar in the blackened distance. I crossed its luminous bridge, over which multitudes of humans from the four corners of the earth travel in a violent rush towards the land of the ancestral spirits. From the river I heard a stream of voices singing. Although the voices were in accord, one was at its heart. This distinct voice was loud but thin and resilient, swift in its tone, and as sharp as the blade of a new machete. They sang a familiar lullaby, one as ancient as the world in its conception. It wasn't long before I realized it was the voice of Owunmiri Ezenwanyi, attended by her numerous maids of unmatched beauty. Together they sang in an ancient mystical language which, no matter how many times I heard it, I could not decipher. They sang for the children who died at childbirth and whose spirits traverse the plains of the heavens without direction—for a child, even in death, does not know his left from his right. It must be shown its way towards the realms of tranquillity, where the mothers dwell, their breasts filled with pure, ageless milk, their arms as supple as the warmest rivers.
They call us _nwa-na-enweghi-nku_ —"wingless" because we are spirits and can travel in the air without wings and "children" because we dwell inside the bodies of living men. So I knew it was for me they were singing. I paused to wave and to acknowledge their song. But Chukwu, as I listened to it, I wondered how you created voices so enchanting. How did you equip these creatures with such powers? Isn't it tempting for one who hears such a song to halt in his tracks? Isn't it tempting to even completely stop the journey to Alandiichie? Isn't it why many dead people remain hanging between the heavens and the earth? The spirits of the dead sitting by the warm shore, aren't they those who, although dead, have not found rest and whose ghosts roam the earth? I have seen many of them—walking about unseen, unable to be seen, belonging neither there nor here, permanently in a state of _odindu-onwukanma_. Aren't some of them in this condition because they are trapped by the enchanting music of Owunmiri and her troupe?
The fathers of old say that a man whose house is on fire does not go about chasing rats. So although I was thrilled by the tune, I was not charmed. I walked on until the music died away and any sight of the habitation of man was completely gone. No longer could I see the shiny _kpakpando_ whose numbers are so vast that a duality was ascribed to them in the language of the fathers. They pair with the sands of the earth to form the single word: stars-and-earth. As I walked, the stars and all that was connected to the earth rolled away like a blanket of darkness into an empty abyss whose expanse is beyond measure. Across the hills was a long winding path lit in every corner by torches, their flames as bright as the light of the sun. It is here that one begins to encounter ndiichie-nna and ndiichie-nne from all over Alaigbo and beyond, gathered in pockets as they walk towards the great hills yonder. The path is decorated on both sides by strands of the sacred _omu_ leaves, fastened to the trees like strange ribbons. Attached to the fresh palm leaves are also mollusks, cowries, tortoise shells, and precious stones of all kinds.
From here, as one ascends the hills, the number of travelers increases. The recently dead throng towards the hills, still bearing the agony of death with them and the marks of life—men, women, children; the old and the young, the strong and the feeble, the rich and the poor, the tall and the short. They tread, their feet soundless against the fine earth of the road, which sparkles in the bright lights. But the hills, Egbunu, the hills are filled with light—an arrangement of shimmering radiance that seems almost to flow like an invisible river into the eye that beholds it and then dissipates into a misty whorl of glow. I have often thought how close the living came to capturing Alandiichie in the moonlight song the old mothers (and their living daughters) sang:
Alandiichie
_A place where the dead are alive_
_A place where there are no tears_
_A place where there is no hunger_
_A place I will go in the end._
Indeed, Alandiichie is a carnival, a living world away from the earth. It is like the great Ariaria market of Aba, or the Ore-orji in Nkpa the time before the coming of the White Man. Voices! Voices! People, all dressed in spotless shawls, walking about or gathered in _omu_ -ringed circles around a big earthen pot of fire. I located the one in which the Okeoha's kindred had gathered. And it was not hard to find. The eminent fathers were there. The ones who had died at a ripe old age, a long, long, long time ago. Too numerous to mention. There was, for example, Chukwumeruije, and his brother, Mmereole, the great Onye-nka, sculptor of the face of ancestral spirits. His sculptures and masks of the deities; the faces of many arunsi, _ikengas,_ and agwus; and pottery have been displayed as some of the great arts of the Igbo people. This man left the earth more than six hundred years ago.
The great mothers dwell here, too. Too numerous to mention. Most notable, for instance, was Oyadinma Oyiridiya, the great dancer, who was synonymous with the saying _At the pleasure of gazing at her waist, we slaughter a goat_. Among many others, there were Uloaku and Obianuju, the head of one of the greatest umuadas in history, one whom Ala herself, the supreme deity, had pomaded with her honey-coated lotion and who poisoned the waters of the Ngwa clan many centuries ago.
Anyone who saw this group would know at once that my host belongs to a family of illustrious people. They will know that he belongs to the genealogy of people who have been in the world for as long as man has existed. He is not of the class of those who fell from trees like mere fruits! It was thus with utmost reverence and humility that I stood before them, my voice like a child's but my mind like an elder's:
—Nde bi na' Alandiichie, ekene'm unu.
"Ibia wo!" they chorused.
—Nde na eche ezi na'ulo Okeoha na Omenkara, ekene mu unu.
"Ibia wo!"
I was silenced by the stately voice of Nne Agbaso, which was as shrill as that of a caged bird. She began singing the usual welcome song, _"Le o Bia Wo,"_ her voice as enchanting and serenading as that of Owunmiri Ezenwanyi and her crew. Her song rose and scattered solemnly through the air and surrounded the gathering, crawling up and encircling every man. And so silent did they become that I was made acutely aware again of the absolute distinction between the living and the dead. Afterwards, she rattled a string of cowries and performed the ritual of authentication to ensure I was not an evil spirit pretending to be a chi: "What are the seven keys to the throne room of Chukwu?" she said.
—Seven shells of a young snail, seven cowries from the Omambala river, seven feathers of a bald vulture, seven leaves from an anunuebe tree, the shell of a seven-year-old tortoise, seven lobes of kola nuts, and seven white hens.
"Welcome, spirit one," she said. "You may proceed." I thanked her and bowed.
—I'm the chi of your descendant Chinonso Solomon Olisa. I have been with him from the earliest emergence of his being, when Chukwu called me forth from the Ogbunike cave where guardian spirits wait to be called into service and told me to guide his foot in daylight and to shine the torch onto his path at night. On that day, I had just gone to Ogbunike from the mortuary in Isolo General Hospital in Lagos, a land far from Alaigbo but a place where many of the children of the fathers now live. Ezike Nkeoye, who now sits over at the gathering of the kin of my host's mother, had just died, and I had been his chi. He was just twenty-two. The day before, this bright student of the White Man's education had gone to bed after studying. I had stayed in him, watching as he slept, the way guardian spirits are called to do. And indeed he was asleep. Then he woke suddenly, clutched his chest, and fell out of bed and onto his neck so that it snapped. The agreement with onwu, the spirit of death, was swift because he, like the rest of your children, does not have an _ikenga_. In a moment after the fall, he was dead.
—Even though I had lived among mortal men many times before, I was shocked by this. So quickly had it happened, and with such intensity, that I was left without a word in my mouth. Death had come to him swiftly, with the violence of a young leopard. Only the previous day, he had been kissing a woman, but he was now gone. So strange was it that I did not go at once to report to Chukwu in Beigwe, as we guardian spirits are required to do. I did not immediately escort his spirit to Alandiichie, either. But at the time, I went with his body in the ambulance to the place where it would be kept at the mortuary. It was then I became satisfied he was dead and brought his onyeuwa with me to here, to the compound of the Ekemezie kindred of Amaorji village. After I left here, I hastened to Ogbunike, to rest and wash in its cataract, in water so warm and ancient it still carried the peculiar smell of the world at creation. I was lying in the stream when I heard Oseburuwa's voice summoning me and asking me to ascend forthwith to Alandiichie, as Yee Nkpotu, the ancestor whose incarnate my host is, was ready to be reborn. As you know, a man and a woman can sleep together for eternity. If one of you here has not decided to return to the earth, conception is impossible. Thus knowing that conception was about to happen, I swiftly heeded his call.
—So on the night my host was born, I brought his ancestral spirit from here in Alandiichie, and you all were distant witnesses as I took his onyeuwa away to Eluigwe, where it was received with wondrous celebration. Then I led it from the Eluigwe fanfare to accompany him to Obi-Chiokike, where the great fusion between spirit and body to form _mmadu_ —the ultimate bodily expression of creation—happens. That was a glorious day. The white sands of Eluigwe, glistening with pebbles that bore in them the very essence of purity, was the ground on which we marched. We were followed in the distance by a group of the adaigwes, the spotless, luminously beautiful maidens of Eluigwe who sang of the joy of living on earth, of the innumerable cravings of man, of the duty of the mind, of the desires of the eyes, of the virtues of living, of the sorrows of loss, of the pain of violence, and of the many things that make up the life of a human being.
—The family and household of Okeoha and Omenkara, you all have been there and know that the journey to the earth is far but not tiring. In your oracular wisdom, you liken this journey to the proverbial sturdy egg that falls from the nest of the raven, tumbles down through the black branches of the ogirisi tree, and lands on the ground unbroken. The road is beautiful beyond words. The trees that stand in the distance on both sides of the inner road not only provide deep vegetation, they are also transparent, like the silvery calico veils weaved by Awka women. The trees bear golden fruits, and on them, within them, and outside of them stand a chattering of emerald birds. They glide around the procession, swinging their wings in the thermal, diving and playing as if they, too, were dancing to the song of the procession. As I walked, they shone in the pure light that filled the road. I could not tell when we reached the great bridge that serves as the crossing between Beigwe and the earth. But just before we reached it, the women stopped in their tracks and raised their voices in a strange, spectral song. Their lovely tunes turned, suddenly, into a threnody, and they sang with trembling voices. Their cries rose as they sang about the suffering in the world, the evils of man-pass-man, the shame of disgrace, the affliction of infirmities, the wounds of betrayal, the suffering of loss, and the grief of death. They were joined by the onyeuwa and I had bonded with them and with the dwellers of Eluigwe, who stopped every time we passed by to say, "May he who is going to uwa have peace and joy!" and even with the flock of white hornbills, the sacred birds of Eluigwe, who hovered around us, beating their wings in obeisance.
—Afterwards, as if signaled by an unseen banner, the singers separated from us and waved at us from a distance. They waved. The birds did, too, as they hung suspended above the bridge as if there was a line they could not cross which neither I nor the reincarnating spirit could see. We waved back, and once we stepped on the bridge, I found myself in a place I seemed to have been before. The place was filled with a bright light similar to that of Eluigwe, but this was man-made. The source of light was thronged by moths and apterous insects. A gecko stood beside one of the lightbulbs at the arc of a wall, its mouth full of the insects. On a bed under the bulb of light, a man screamed, trembled, and collapsed against a sweating woman. The onyeuwa entered into the woman and merged with the semen. The woman did not know or realize that the great alchemy of conception had happened within her. I joined the onyeuwa and became one with the man's seed, and in joining we became a _divisible_ one.
—Ndiichie na ndiokpu, _unu ga di_.
"Iseeh!" the eternal bodies chorused.
—From that moment on, I have watched over him with my eyes as wide as a cow's and as sleepless as a fish's. In fact, were it not for my intervention, or were I a bad chi, he would not have been born in the first place.
To this, a cold murmur echoed through the gathering of this deathless throng.
—It is true, blessed ones. It was in his eighth month, while in his mother's womb. She was seated on a stool sandwiched between two buckets, one containing clean water, with a transparent film lying over it, a spill from suds, and the other containing muddied water, in which clothes are soaked. A packet of Omo detergent lay on the pile of unwashed clothes. She had not seen, nor had her chi warned her, that a poisonous snake, sniffing the wet earth around her and the dewy smell of the tree leaves and shrubs around the place, had crept under the pile of clothes and begun to suffocate. But I stood out of my host and his mother, as I frequently do until my hosts possess their bodies in fullness. I could see it—the black snake slithered into one of the legs of a pair of trousers, and as she made to pick it up, the snake bit her.
—The strike had an immediate impact. From the dazed look on her face, I could tell that it was a terrible sting. On the spot where she'd been bitten, a deep-colored bead of blood appeared. She screamed so loudly that the world around rushed to her aid. Once the snake bit her, I became aware that the poison could travel and kill my host in his abode in the womb. So I intervened. I saw it moving towards my host, who was then only a fetus asleep in the sac of the womb. The venom was full and hot and powerful, instant and destructive, and violent in its movement through her blood. I asked her chi to force her to cry so loudly that neighbors would immediately gather. A man quickly fastened a rag around her arm, a little above her elbow, stopping the venom from traveling further up and causing the arms to swell. The other neighbors attacked the snake and dashed it into a paste with stones, their human ears deaf to its pleas for mercy.
—You all know that it is my duty to know, to probe the mysteries around the existence of my host. And truly, even a goat and a hen can assert that I have seen and that I have heard many things. But I have come here mostly because my host is in serious trouble—the kind that can cause the eyes to bleed instead of shed tears.
"You speak well!" they said.
—The men of your kin say that even a man who stands on the highest hill cannot see the whole world.
They murmured in agreement, "Ezi okwu."
—The men of your kin say that if a person desires to scratch his hands or an itch on most other parts of his body, he does not need help. But if he must scratch his back, he must ask others to help him.
"You speak well!"
—This is why I have come: to seek an answer, to seek your help. Dwellers of the land of the living dead, I fear that a violent storm has petitioned for the closure of the only road to the utopian village of Okosisi, and it has been granted its request.
"Tufia!" they spat in unison. To which one of them, Eze Omenkara himself, the great hunter who in his lifetime traveled as far as Odunji and brought home a great deal of game, stood to speak.
"Ndi ibem, I greet you. We cannot wave our hands to swat away a snake threatening to bite us as we do mosquitoes. They are not the same."
"You speak well!" they said.
"Ndi ibem, kwenu," he said.
"Iyaah!" they said.
"Kwe zueenu."
"Iyaah!"
"Guardian Spirit, you have spoken like one of us. You have spoken like one whose tongue is matured, and indeed your words stand on their feet, they stand—even now—amongst us. Yet we must not forget that if one begins bathing from the knees up, the water may be finished by the time one gets to the head."
They shouted, "You speak well!"
"So tell us about this storm that threatens our son Chinonso."
AGBATTA-ALUMALU, I told them everything as my eyes had seen it, and as my ears had heard it, and as I have now conferred to you. I told them about Ndali, his meeting at the bridge, and his love for her. I told them about his sacrifices, how he sold his home. I told them about Jamike, how he swindled my host, and how my host, thinking he had been saved by the white woman, now lay unconscious, having possibly killed another man.
"You speak well!" they chorused.
Then there was silence amongst them, a silence like one that is impossible on earth. Even Ichiie Olisa, anguished that his son had sold the land, merely gazed into the hearth with empty eyes, as silent as a dead log. A group of them, about five, rose and went to a corner to confer. When they returned, Ichiie-nne, Ada Omenkara, my host's grandmother, said, "Do you know anything about the laws of the people of this new country?"
—I do not, great mother.
"Has he killed a man before?" Ichiie Eze Omenkara, the great-great-grandfather of my host, said.
—No, he has not, Ichiie.
"Spirit one," Eze Omenkara said now, "perhaps the man he hit with a chair will survive. We bid you return to watch over him. Do not proceed to Beigwe to report to Chukwu until you know for sure that he has killed this man. We hope—if he was hit by a chair alone—that he would not die. Make your eyes as those of a fish and return here when you have another word for us." Then, turning to the others, he said, "Ndi ibem, have I spoken your minds?"
"Gbam!" they chorused.
"A chi who falls asleep or leaves its host to go on journeys—except when necessary, as this one is—is an _efulefu,_ a weak chi, whose host is already a lamb bound with twine to the slaughter pole," he continued.
"You speak well!"
—I hear you, Dwellers of Alandiichie. I will go back now, then.
"Yes, you may!" They cried, "Go well the way you have come."
—Iseeh!
"May the light not quench on your way out."
—Iseeh!
I turned to leave them, they who are no longer susceptible to death, relieved that at least I had found some respite from my panic. And as I traveled, not turning back, I wondered: what was that beautiful voice that rose again in a song to bid me onwards?
CHUKWU, thus was my journey completed. I flew through a long stretch of the flaming night, past white mountains of the farthest realms of Benmuo, on which black-winged spirits stood, speaking in sepulchral voices. As I neared the sublime borders of the earth, I saw Ekwensu, the trickster deity, standing in his unmistakable garb of many colors, with his head carried on his long neck, which stretched about like a tentacle. He stood on a limb above the moon's disk, gazing at the earth with his wild eyes and laughing to himself, perhaps devising some evil trick. I had seen him in the same spot twice before, the last time seventy-four years ago. As in the past, I avoided him and proceeded towards the earth. And then, with the alchemic precision with which a chi finds its host no matter where in the universe he may be, I arrived at the place where my host lay and fused with him. I saw at once by the clock on the wall that I had been gone for nearly three hours, in the White Man's measure of time. He had been revived, Egbunu. Stitches laddered down on his face, and a big bloody piece of cotton wool was sticking out from his mouth where his teeth had been broken. There was no one else in the room, but by his bed a thing with a screen like a computer sat, as if keeping him company, and from his arm stretched a small bag that hung on a pole, and in it was blood. His eyes were closed, and in his blurred vision, the image of Ndali looking at him had stood, as if bound to his mind with an unbreakable cord.
# THREE
# Third Incantation
GAGANAOGWU, may your ears not strain—
Even as I stand here, I can hear the singing, the joy, the sweet tune of the flutes. I have been to this palace where you dwell, many times. I know that the guardian spirits and their hosts will come to you here for your final approval of their rebirth, for a reincarnation into a fresh body, and live on earth again as a newborn—
The fathers say that a man does not stand on burning coals barefoot because his feet are wet—
One does not dance near the pit of venomous snakes because one's _obi_ is too small—
The wingless bird said I should spit into a calabash with holes, but I say to it, my saliva is not meant to be wasted—
The head that stirs the wasp's nest bears its sting—
The one-eyed serpent of shame shelters near my door. May I? it asks. No, I say. I don't want your terrors in my abode—
Destruction says to me, "Shall I come under your roof and pitch my tent?" I say, "No. Go and tell whoever sent you that I am not at home. Tell them you did not see me"—
_Egbe beru, ugo ebekwaru, onye si ibe ya ebela nku kwaaya—_
May the words I shall continue to speak hasten to the conclusion of my account—
May my tongue, as wet as a mangrove, not dry of words—
And may your ears, Chukwu, not tire from hearing me—
May this incantation usher in a fruitful end to my testimony tonight, after which I will leave the halls of Beigwe and return to the waiting body of my host—
_Iseeh_!
#
## The Return
AKANAGBAJIIGWE, the universe does not dwell on the past, gathering around the miasma of burnt-out fires like a pack of crows. Rather, it forges ahead, always on the winding path of the future, stopping only briefly at the present to rest its feet like a weary traveler. Then, as soon as it is rested, it moves onwards, it does not turn back. Its eyes are the eyes of time, cast perpetually forward and never looking back. The universe travels on no matter what happens to its inhabitants. It proceeds, crosses the footbridges, scales the ponds, circles the craters, and continues. Has a conflagration destroyed a nation? No matter. If such a thing has happened in the morning, it does not matter because the sun will rise, as it has done for as long as the world has been, and in that selfsame city, the sun will set and night will descend. Has an earthquake devastated a land? It does not matter; it will not interrupt the seasons at all. And the life of the universe is reflected in the lives of those who live in it. Has the patriarch of the family been killed? The children must sleep this night and wake up tomorrow. Everyone continues, carried forward like old leaves on the river of time.
But although the universe continues its journey, carrying all the living with it, there is a place where a man can remain still, as if his personal universe has halted. This place is one humans dread because it is a place where they do nothing. They do not so much as stir. They are locked in like captured animals in a circumscribed space. One who is here has his diameter marked out as if by an invisible ink that says, "From this wall to this wall, from this length to that length, is all there is for you in the world." But I must establish, Agujiegbe, that a man whose movement is limited—that man is not truly alive. The passage of time mocks him. And this is what happens in confinement.
For in this place, almost no new memory can form. The man wakes in the morning, eats, excretes in the small hole which he covers with a lid after he has washed it down with water that he fetches in a small bucket from the tap in his room. Then he sleeps. When he wakes up again, if it is night, it is night; if it is morning, it is morning. Only a shadow of light rears its feeble head into the cell like the head of an infant snake. If it is daylight, the light comes in a single rod through the window at the top of the high wall near the old ceiling. The window is closed off with strong iron bars.
A man sits here all day, merely alive, the enamels of life peeling away from him and withering into flecks at his feet. The world conceals itself from such a man. It conceals its deepest and most shallow secrets and even its nonsecrets. He knows nothing of what is happening, sees nothing, and hears nothing. The bridge he'd crossed to get here, like one constructed by a retreating army, has been destroyed behind him, and all the links with the known world with it. And now he is confined to this space—for however long he must stay. It does not matter. What matters is that his life is stagnant. He spends the day gazing at the walls or the bars that lead out to the other cells until his eyes grow weary from looking. Every now and then he sees something move about in the field of his vision, but soon it is lost to his eyes. No new memory is made of it, for such things, occurring as they do, appear like weak animals who pound their fists against the sealed door of his noiseless humanity and then retreat. Or the vacuous insects that rush to a lightbulb and perform a withering ritualistic dance that only results in their own death. I have seen it many times, Egbunu.
As soon as my host was taken from the hospital, where he had been for two weeks, to the cell where he would stay in solitary confinement, he could make no new memories. If in a rare case a man makes new memories while in prison, they are often the things he does not wish for but which are done to him. They are not willful history. Because a man has no control over this kind of thing, it lodges in him without recourse to his will. For once he has witnessed a thing, it slips as if through a crack into the mind and stays there. It does not go away.
My host stayed in this state for four years. To chronicle these four years, to labor over the monotony of living, the anguish of still life, is comparable only to the pain of a slave, as I saw in my past host, Yagazie. For a prisoner, too, is a slave, a captive of the government in this strange country. For many cycles, I have known the darkness of youthful hearts, wallowed in the mud of the ambitions of many men, and peeked into the graves of their failures. But I have never seen anything like this.
Now he has returned back to the land of the living and to his own land. The process that led to his return to the land of the fathers happened very quickly. For in the early morning of his troubles, I had tried to save him. Once the police took him to the hospital and he was alone in a room, unconscious, I had no choice but to do that which a chi must do as a last resort, when all the strength of man has failed: I went to Alandiichie to seek the intervention of his ancestors, an account of which I have now rendered to you.
One morning, in the fifth month of his fourth year in prison, his release happened suddenly, without warning. Nothing had prepared him for it. He was seated, his back resting against the wall whose paint had peeled from his resting against it for so long. He was in that moment thinking about some inconsequential things—of the choreography of ants on a hill, then of maggots in a decayed can of milk, and then of small birds congregating on a wild tree—when the bars of his cell began to be unlocked. A guard and a man dressed in a suit stood on the threshold, and the man told him in the language of the White Man that he was being released.
He followed them to an interrogation room, and later, the interpreter would tell him that his case had been reviewed. The primary witness falsified her initial testimony. He had not gone in to rob or rape her, as had been reported, but she had taken him to her house of her own will. It was her husband who had become jealous and in a fit of rage descended on her and him. My host had merely tried to save her by attacking the man. This, the woman now reported, was the truth about what happened. Gaganaogwu, this was not what was told to the police at all! Quite the opposite. The woman and her husband had conspired against my host, an innocent man, and had said that he tried to rape her. They'd said that it was in the process of doing it that her husband had found her struggling with him and intervened by knocking him, the assailant, unconscious.
After he heard these things, he said nothing to the guard and the interpreter. He merely sat there staring at the well-dressed man with the files and the interpreter but not seeing them. His eyes had now grown accustomed to registering an image, then immediately disregarding it, and moving on. He kept his gaze unbroken on a great blank wall, a magnificent nothing which, however, had occupied his vision and his mind.
"Mr. Ginoso, do you have anything to say?"
When he did not respond, the interpreter bent his mouth towards the ear of the other as if to kiss him, and the two came forth nodding. It was a strange thing, even to my host. One of the men spoke in haste, and the other nodded effusively.
"My friend Ms. Fiona Aydinoglu wants to offer her apologies. She is very sorry for what happened. Again, this is her lawyer. And she has asked us to give you this money. She wants us to do all we can to help you regain your life."
He said nothing, and his eyes remained where he had cast them—on a fly that was droning between the window and the netting located behind the table where the two men were seated.
"Mr. Ginoso." It was the non-English-speaking lawyer who spoke now, perhaps worried that his interpreter had not delivered his message in a clear enough way and that it was better for the originator of the message to deliver it, no matter how battered the language. Surely it would mean more. Surely it would be respected. "I say truth now, my client only truth. We are very very sorry, your sufferingk. Very sorry. For many—" He turned to his friend and asked something. "Years, _yani,_ years. For many years Fiona is sad because this. She sorry, very sorry, my friend. Please, Mr. Ginoso, you must accept her sorry."
He said nothing to this man, either. For four years, all that needed to be said and discussed with these people had been said. And afterwards, words had lost their usefulness and had evolved into something else, something without form, amorphous, worthless. In their place, contempt had rooted and blossomed. A man of little rage, he'd become vandalized by a spiritual politics into which he had been unwillingly conscripted. And now so strong was the contempt he felt that while the men spoke, his mind came alive with vivid conjurations of violence. The man in the police jacket he saw lying on the floor, his throat slit with a knife in my host's hand from which blood trickled onto the lifeless body. The lawyer he saw gasping, the man's tongue hanging out of his mouth as my host strangled him against the wall.
Even if faintly, my host realized that this was the person he had become. Without knowing it, something in him had changed. For the spirit of a man may long endure pitiless circumstances, but eventually it will stand erect, unable to take any more. I have seen it many times. In place of submission, rebellion will erect itself. And in the place of endurance, resistance. He will rise with the vengeance of a black lion and execute his cause with a clenched fist. And what he will do, what he will not do, even he will not expect.
Egbunu, the man of rage—he is one whom life has dealt a heavy hand. A man who, like others, had simply found a woman he loved. He'd courted her like others do, nurtured her, only to find that all he'd done had been in vain. He wakes up one day to find himself incarcerated. He has been wronged by man and history, and it is the consciousness of this wrong that births the change in him. In the moment the change begins, a great darkness enters him through the chink in his soul. For my host, it was a crawly, multilegged darkness shaped like a rapidly procreating millipede that burrowed into his life in the first years of his incarceration. The millipede then yielded a number of progeny, which soon began eating him up, so that by the third year, the darkness had snuffed out all the light in his life. And where there was darkness, light could no longer encroach.
For most of the time, the man of rage is consumed with one passion: justice. If he has been stuck, to strike back at those who had stuck him. If he has lost someone, to regain it from those who had stolen it. This is important because that recovery is the only way he can become himself again.
In my host's case, the meeting with the lawyer and his interpreter was the first time he had to act on his emotions in a long time. In confinement, what he felt at any time was meaningless because he could not act on it. What use, for instance, was it to feel anger? There was nothing he could do about it. To feel love? Nothing. Everything he felt he swallowed back down into the belly of his incapacitation.
He would realize that "Mrs. Fiona," whom he never saw again after her last appearance in court, had insisted that the money be put in his bags if he refused to take it and returned to Nigeria with him on the flight. "Not deportation," a very young black woman who'd identified herself as a Nigerian, one of the many people who'd spoken to him, had said. "They have asked you—your university offered to give you a free scholarship as compensation if you still want to attend and remain in TRNC, but you have refused to say anything at all to any one of them. Because you refuse to talk, even to me, they are returning you to Nigeria, with everything you brought here."
Even to that woman, although he had given her his attentive gaze, he did not speak. This was why those who sought to do something about him or for him had refrained and merely resorted to parsing out meaning from his little gestures—side glances, shakes of the head, even noncommunicative actions like coughs. So they had concluded, or decided, that his not speaking meant that the only thing he wanted was to return home. They looked on his university admission forms and contacted his next of kin, his uncle. Then they drove him two days after his first release to the airport. They gave him tickets and put him on the plane and told him that they had contacted his uncle and that he would be waiting for him at the airport in Abuja. Then, wishing him good luck, the lawyers, the Turkish-Cypriot government officials, one of the officials of the school he had been admitted to, and the Nigerian woman waved him good-bye. Even to this he did not respond.
He did not utter a word until the plane took off. At once dead events opened their eyes, and long-forgotten images began to rise from the graves of time. As the country in which his story had been rewritten became reduced to only a speck, he found himself struggling again to retrace the trajectory of his journey. How had he come to this place where such unheard-of things were done to him? He waited for a moment as the answer stirred in ripples from beneath the tundra, then floated to the surface of his mind: he had come to be able to be with Ndali, whom he'd thought about for most of these years until, tormented by persistent fears and imaginations and dreams that she had left him, he stopped letting himself think of her. He recalled the party at her father's house and his humiliation. He remembered Chuka, who had tormented him. The plane was landing in Istanbul when memories of his poultry emerged, wet and shimmering. He watched the coops and himself feeding them, as he'd reconciled hundreds of times these past four years. He gazed at himself marking the wall of the coops with the dates of the last general cleaning, which he did every two weeks. He saw himself harvesting eggs from the coops, blowing away earth and feathers and putting them in a bag. Then, in an undefined time in the past, he saw himself registering the birth of the newborn chicks into the big six-hundred-leaf foolscap record book whose cover had fallen away, which, in its first seventy-something pages, still bore his father's handwriting. Then he was out in the big Ariaria market, selling a cage full of the yellow broilers and an albino cockerel whose comb had been torn in half during a fight with another cockerel. Chukwu, the memory of these things, even after these many years, broke his heart again.
AGUJIEGBE, as the plane neared the country of the children of the great fathers, I left the body of my host, hungry to see again the beautiful rain forests of Alaigbo, this land where the velvety green shades of the morning become a shuddering veil at night. The trees, unhindered in their growth, stand in their multitudes, drinking from the restless rain. When one soars over them, looking down into the forests as a thing with wings, the forests appear as dense as the viscera of an antelope. Within the forests are rivers, streams, ponds, and the sacred waters of the gods (Omambala, Iyi-ocha, Ozala, amongst others). One does not walk for too long outside the limits of the forests before one comes into the boundary of a village. What one first sees are more trees with edible fruit—bananas, pawpaw, green mangoes, the kinds that are rare in the deep of the forests. In the time of the fathers, the huts gathered in a nest. And an accumulation of these, stretching merely a few stone throws, would make a village. In this time, villages have expanded into towns, and the forests have encroached upon the habitation of man. But the beauty of the land remains; the quiet peaks of the hills and valleys, magnificent to those who walk to see them. This is what I have missed in the time my host has been away, and this was the first thing I went to see—when my host and his uncle who'd picked him up at the airport arrived in the land of the great fathers.
He and his uncle did not speak about anything concerning his situation until they reached Aba, where only two years before, the older man retired from civil service. All through the journey, whether in the taxi that brought them from the airport or the bus that took them on the eight-hour trip from Abuja to Aba, they'd been in the company of strangers. But now, at the entrance into Aba, with the bus stopped at the shoulder of the highway for passengers to ease themselves in the bushes, his uncle, while urinating, asked him if anything bad had happened to him in the prison. At first, he did not speak. He was standing a little ahead of his uncle, urinating into an old beer bottle which stood among the creepers, half filled with what must be rainwater. He released his urine into it until the bottle, filled, fell and emptied into the bush. The old man spoke on, saying he'd heard accounts and speculations about how Africans in prisons abroad are treated "li like dogs." At this he stared at his uncle, who had finished and was now waiting for him to zip up. It seemed his eyes betrayed that which his mouth could not say, for his uncle caught his eyes, and then shook his head in agonizing pity.
"You m-ust th-th-than-k God for your lai-life," his uncle said. "Of co-co-urse you ma-made a bi-bi-g mistake by go-ing there. Bi-g-big mis-take. But you m-m-ust t-han-k God."
When they arrived at his uncle's house, at the sight of his aunt, whom he had not seen since his father's funeral and who was now much older, with a shock of gray hair, he broke down. Later, when his uncle returned to the room they had given him, which belonged to their son, who had gone on the NYSC service in Ibadan, he still could not speak of the things the older man had asked of him.
Gaganaogwu, you made all things, and you know that what our hosts cannot say, we—their chis—cannot say. For it is a universal truth that _onye kwe, chi ya e kwe_. Hence, what he has not affirmed to, I cannot affirm. And thus if he is silent over something, I must be silent, too. What he does not want to remember, I do not remember, either. Yet even though my host could not speak of these things, he constantly thought about them. They lodged like secret blood in the veins of every passing day. At every bend in the day, they emerged and ambushed him. And sometimes when he lay down in his bed and stared at the lightbulb or the kerosene lamp, as he'd become accustomed to doing since his release from prison, the memories emerged in vivid colors, as if they'd been imprisoned in the bulb or lamp and had broken free.
He embarked on the task of rebuilding himself with these things a constant torment on his mind. But as days passed, he found that they little occupied his mind. What dwelt more with him was the enormous riddle life had placed before him, which he wanted badly to solve. At first, he stayed away, far away from this riddle, and tried not to solve it, for his uncle would have deemed him crazy for even harboring such thoughts. The older man had said in unequivocal terms that anything that brings such pain and suffering on a person is not worth having. His uncle, a man supple with the oratory of the old fathers, whose tongue dripped with the oil of convincing imagery and proverbs, had asked him, in the soft, tender way he spoke, what use it would be for a man to pick up a scorpion because of the beauty of its skin and put it in his pocket. When my host did not offer an answer—for such a question as this was not meant to be answered—the older man continued, "No, n no, that would be foo fo foolish ness."
But once he left his uncle's house, with the five thousand euros the German woman had paid him as _damages_ —as part of the punitive measures against her—he returned to Umuahia and rented an apartment. He opened a feed-and-mash store on Niger Road, and with what was left of it he bought a motorcycle. In the weeks that followed, brick by brick by brick, he reconstructed his life. Akwaakwuru, if a tortoise has been upturned, even if it takes a long time, it will slowly try to return to its feet. It might be that the tortoise cannot at first because a stone has hedged it in, so it must turn the other way. This might be the only way it can rise again. Egbunu, he must continue, for to be still is death. So by the end of the month, when his uncle and aunt visited him and said he had "risen," he believed them. For when he stood at a remove from the once broken things which had now been rebuilt, he agreed that at least the beginning of his life's reckoning had been initiated. This was a soothing feeling. It gave him courage, and it was only after that that he turned again to the riddle and began to advance towards solving it.
The effort took him one evening, two months after he returned, to the land of the fathers to a mansion on Aguiyi Ironsi Layout, which, with much difficulty, he located. It was older, and the sculpture of the Madonna on the gate had been removed, peeled off, leaving the imprint of its presence like a scar. In front of the gate, between the fence and a new culvert, a sedge with brittle stems had sprouted, and a young tree had risen from the sewage at the end of the road. He'd reached this gate with his heart pounding, and so he could not stop, could not give the place—where Ndali had been living before he left Nigeria—more than a passing glance. For suddenly, he'd felt overwhelmed by the memories triggered by the sight. And in haste, he rode past the mansion into the darkening streets.
I stayed back, Oseburuwa, for one of the most difficult missions of my existence in the nearly seven hundred human years since you created me had happened in front of that gate. Not long after my host began his prison sentence, I could not bear the sight of his suffering. An innocent man, _onye-aka-ya-kwuoto,_ punished for a crime he did not commit. I was as shattered as he was. He'd done all this to be able to marry Ndali, and now he'd destroyed himself. For her sake. I wanted her to know about it, but saw that there was no way for him to contact her, and I, a mere spirit, incorporeal, cannot write a letter or make a call. So, Egbunu, I resorted to _nnukwu-ekili,_ in order to deliver a message to her through the dream space. I had been told that we can use this highly esoteric process to reach a nonhost at Ngodo cave more than a hundred years ago by a guardian spirit who had done it, but who had stated that it was rarely attempted. So while my host sobbed in the prison, I embarked on the astral flight and arrived at her house. I projected in and, after moving from room to room, I found Ndali crouched in a corner of her bed, the sheets crumpled, holding on to a pillow while asleep. By her head was one of the photos she'd taken of my host, holding one of the fowls and smiling into the camera. I was about to begin incantations, the first process of _nnukwu-ekili_ by which I would have gained access into her dream space, when a presence materialized at the other end of the room. It was her chi.
—Son of the dawn light, you have come to trespass, to anger a spirit who has done you no wrong.
Egbunu, you must understand that I was taken aback by this accusation. I know this guardian spirit will soon come to tell you its version of this encounter, should what I fear has happened to its host actually come to pass, so please remember my account. For in response to its question, I had begun to speak.
—No, no, I merely—
—You must leave! the chi said with vehemence and authority. Look at my host: she has suffered much already, wounded by Chinonso's decision to leave. Look at how she has been sad, waiting for him. I hate your host.
—Daughter of Ala, I said, but the chi would not hear.
—This is trespass. Go and let nature take its due course. Do not interfere in this way or it will backfire. If you insist, I will take a report to Chukwu.
At this, it was gone. Without hesitation, I left the room and returned to my host in the far country.
OKAAOME, he could barely sleep that night. He sat in his one-bedroom apartment, the table fan oscillating and droning, and under the light of the bulb that hung from the ceiling by gaunt wires taped together, he tried to bring his phone back to life. The phone had not been turned on since he first took it out of the bag that held the clothes and shoes he had been wearing the day he was taken into the hospital, his admission letter and receipts, and all he'd brought into the prison. He'd fitted its parts together, but it did not work. One of the policemen had picked it up from the bloodied floor of the German woman's house in Girne, and it had not worked since.
He rode his motorcycle the following day, under the cover of darkness, to the mansion. There was light within it from some generator buzzing. Everywhere, darkness stood, almost unblemished, only the light of oncoming vehicles relieving the streets as they cut their paths through the ample flesh of gloom. He stopped the motorcycle and dismounted, then he walked to the gate, and with a courage that came quickly—as if it had leapt from a clandestine position onto its target—he knocked on the gate. When the metal began rattling, he had the temptation to flee. For it occurred to him, now that he was at the threshold of what he'd been seeking all along, that he was not prepared to confront it after all. He realized that despite all that had happened to him, despite the time that had passed, nothing had changed. He was still an _Otobo._ He had not acquired higher education; his status had not changed. In fact, the epiphany deepened with the voice of rage: things had worsened for him. He had become much poorer. If he owned a house before, now he did not. If his heart was once without hatred, now he carried within him a mighty sack of hate in which many people were trapped. If he had good looks before, now he wore a battered face, one from which doctors had to remove a bottle that had stuck in his forehead, a jaw that had been stitched so that he could not shave the area for fear of loosening the stitches, and a mouth from which no fewer than three teeth had been knocked out. If in the past, his pain and grief had stemmed merely from things done physically to those he loved, now it came with a vengeance from things done to him in other ways, too. For not only had he been physically damaged, he had also been inwardly broken. He had been penetrated from behind by another man, violated beyond redemption, flogged out of his body.
Standing in front of this gate, he became aware of his true condition. It shocked him, for he had not considered his wretched state in its completeness this way before. He stepped back as the gate opened.
"What can we do for you, sir?" the person who emerged from the gate in the uniform he had once worn said. He was much younger, perhaps in his late teens.
"Ah, I am looking for, ehm, my friend, Miss Ndali Obialor. Is this her house?"
"Yes, this is the home of Chief Obialor. But his daughter is not here now."
His heart raced. "Oh? When is she coming back?"
"Madam Ndali? She doesn't live here. She lives in Lagos. Didn't you say you are her friend?"
"Yes, but I have not been in town, many years. Since two o o seven."
"Okay, I understand, sir. Madam Ndali lives in Lagos since—since two thousand and eight."
The man had started turning back.
"Good night, Oga."
"Wait, my brother," he said.
"Oga, I can't wait anything. I can't answer your question again. Madam Ndali is not here; she is in Lagos, period. Good night."
The gate shut as it had opened, and he heard the bar fold into its lock. Where he was, darkness returned, along with the sporadic noise of the street. He stood there, his hand on his chest, feeling his heart. For he was relieved that after four years, he'd finally heard something about Ndali, even if it was only a tiny detail. As he rode back to his apartment, he wondered what would have happened if he had seen her. Would she have changed as much as he and everything else in Umuahia had? He had almost not recognized parts of the city. Here and there, new markets had been cleared out and pushed from inside the city limits to its outskirts. A telephone communications revolution whose beginning he'd witnessed had been concluded, and now, the city was living in its aftermath. Now everyone owned a mobile phone. Towers of telecommunications companies with acronyms such as MTN, Glo, and Airtel were everywhere. On both sides of the street, yellow or green umbrellas stood with tables and chairs under which a man or a woman sat. On the tables were phone cards and SIM cards and a mobile phone operator who charged people to make calls on her phone. Around the streets, new lamps had sprouted with flat panels behind them, which people often simply referred to as "solar." A new attitude seemed to have spread amongst the people like a harmless germ, a new bleak humor that trivialized the horrifying, and a litany of lingos which he did not comprehend.
He paid little attention to these changes because his mind was occupied with thoughts of Ndali. When the unforgiving blow of his undoing struck him, at the German nurse's home, he had tried to contact her. As he lay on that floor, in a pool of his own blood, fearing he would die, thoughts of her had stood unshaken in his mind like a guard. He'd relived all the moments when she offered any kind of resistance to his leaving Nigeria, like when she had told him about the dream whose details she did not disclose. Even before he was taken away by the police, he'd seen her watching him, as if merely sitting at the other end of the bloody room. And after they took him away, he'd tried to reach her by phone, but his phone had died. He tried to get a phone, repeatedly begging the nurses, but they told him every time that he could not be helped. The police had given them instructions that he not be allowed access to anything aside from food and treatment. None of the nurses shared any information with him, either. Only one of them could speak the language of the White Man, and even this one struggled to understand him when he spoke. As the days passed, he'd grown frantic, angry, and delirious. For he had come to firmly believe that Jamike and the evil spirit seeking to destroy him had been persistent and relentless to the end. And now it had paid off. He'd fought hard, but he'd fought against a foe whose weapons were unsurpassable. Just when he thought he had escaped and was off the hook, he had swum right into another, sharper hook.
The finishing off happened over several weeks, during which everyone he knew abandoned him. Not even Tobe, who had slowed his journey and bore his cross part of the way, appeared within an inch of the territory of his renewed suffering. Only a representative of the school, one of the old students, together with the vice rector, came to the court during the first day of the trial. They'd secured his property and kept everything for him. If he was freed, it was likely he would be deported, so they would just send his things to the airport. They'd called to inform his uncle. In the frenzy of the moment, he'd begged the Nigerian student Dimeji to help him reach Ndali. With his hands shaking, he'd written down her number.
"What should I tell her?" Dimeji had said.
"What?"
"Yes, what should I tell her?"
"That I love her."
"Is that all?"
"That is so. I love her. I will _return_. I am sorry for leaving, and I'm sorry for everything." He'd paused to push the words into Dimeji by the force of his eyes. When Dimeji nodded, he continued, "But I will return. I will find her. Tell her that I promise. I promise."
That was it, all they had time for. He never saw Dimeji again. No one whom he'd known prior to the events that led him to trial in a foreign country appeared within sight in the ensuing four years. The only one, the German woman, was his prime accuser. The other, her husband, who had been unconscious in the hospital bed for sixteen days, was his second accuser, corroborating his wife's testimony. This man had maintained that he'd found the black man on top of his wife, struggling. So on that day, the judge had turned to Fiona's husband and spoken to him in English.
"So, Mr. Aydinoglu, did you know before that your wife would be seeing this man?"
"Yes, sir. She is a nurse. Good woman who likes to help people. So she wanted to help this poor rapist from Africa. _Walahi yaa_!"
"May we ask you to watch your language before this court, Mr. Aydinoglu?"
"Am sorry, Your Lordship."
"Comport yourself. Now, so you let her bring him into your home?"
"No, she always help people. It is normal for her, _yani._ When I came to my house, he was tryingk to rape her."
"Can you tell this court what you saw?"
"My wife, _yani,_ was on the floor, near to the dining table, and this man was on top of her, and his hand was holdingk her on the neck, one hand. Pardon—the other hand. One, he was tryingk to force himself into her. It was very disgustingk, my lord. Very disgustingk."
"Go on—"
"I immediately threw myself for him and we began to fight, before I ask my wife to call the police. I had a bottle, so I injured him with it, then went to check my wife, who was still lying on the floor, weeping, breathing very loud. Then, this man came very low behind my back and hit me on the center of my head here—on this spot, my Lord—with a stool. I fell, sir. That is all I remember."
Agbatta-Alumalu, the fathers say that the switch that broke the head of a dog must be called something else. There was no more my host could do in his defense. After that second court session, judgment was passed. It was already late by then, five weeks late. The judgment passed by the words of a man's mouth, couched in words—first in the language of the land, then in the language of the White Man—meant nothing because a greater judgment, one passed by actions so powerful they'd imprinted themselves forever on his mind, had happened to him. So the pronouncement that he was to be sentenced to a combined twenty-six years in prison for attempted rape and attempted murder meant nothing. By that time, already, his life as he once knew it had separated from him like an ill-fated shadow hewn from its bearer and thrown over the cliff into a bottomless pit of oblivion, and even through all these years, he could still hear its dark voice screaming as it continued its fall.
#
## Seedlings
GAGANAOGWU, I must here argue that to prove my host's motive in the action I am claiming he is innocent of, you should consider that he has suffered because of his love for this woman. The early fathers say that it was in the hunt for a worthy cause that Orjinta, the mighty hunter of yore, was torn to pieces. Although even amongst the old fathers this was told as a proverbial tale, you know that it happened at a time when Alaigbo was at its prime, when everything was almost as you had intended it to be. Even I had not been created then. The people constructed rectangular houses made from mud bricks, kept their shrines in their _obis,_ consulted their ancestors and fed them constantly, and no one trampled on the personal liberty of his neighbor because he believed in the primal law of coexistence. ( _Let the eagle perch, let the hawk perch, and if either says the other should not perch, may his wings be broken._ ) Orjinta, a young man who had made a habit of calling his betrothed before she grew the age of a clear moon, would crouch behind her father's compound at night and whistle until she would come out, jump out through the window, and follow him into the brush. Orjinta knew it was forbidden to whistle at night, as it bothered the spirits of the living dead in the Ogbuti forest. But a man in love would crawl into a viper's hole to find his lover. He ignored the creatures of the night, who are terrified of human cries and whistles. As he whistled one night, an angered spirit possessed a leopard and drove the beast through the forest, howling, trampling young saplings underfoot, scratching up rows of yams, driven by a hellish fury that was not beholden to even the most basic statute of civilization. Orjinta whistled on as his woman listened for her parents, for any sound in the house, waiting for the best time to jump out unseen for the night's rendezvous. The beast continued its journey towards him, its track mapped by a devilish magnetic pull towards its prey and its violent strides echoing in the dark terrain of the night, until it found the exact spot at the very moment when Orjinta, raising his head, saw his lover coming. The beast fell on him, and with an anger originating from a time beyond history, before the inception of love and romance, and of flesh and blood, tore him to pieces and dragged his corpse away into the forest.
Egbunu, what are stories like this for? They are meant to warn us about the dangers of such actions as Orjinta took. This was why, starting from my host's second year in prison and after my second encounter with Ndali's chi, I had begun to try to make him forget her. But I have come to understand that such efforts often are futile. Love is a thing that cannot be lightly destroyed in a heart in which it has found habitation. I have seen it many times. And there is an extent to which a chi can make suggestions and it becomes coercion. A chi cannot coerce its host, even in the face of the most violent dangers. Insanity is the result of an irreconcilable difference between a man and his chi. Even among the fathers, consensus was the mode in which they operated. They began every discourse with the bellowing of _"Kwenu"_ —an invitation to agree, for if a man in a group refuses to respond _"Yaa"_ and says, _"Ekwe ro mu,"_ then the discussion cannot continue until the dissenters have given consent.
How, then, can a chi disagree with his host? How can it say to him, "Leave this pursuit, for it may lead you to dark places," when his host is determined to continue on this path? Had I not seen that all these years, in the midst of anguish and torment, in the midst of prayers that the nurse would tell the truth and he would be free, the one thing he longed for most of all was to return to her? As unbelievable as it may sound, almost daily, he wept for her. He longed for her. He begged for pen and paper and wrote the letter, but where would he send it? He did not know the address of her house. And even if he could guess, how to send the letter? For the first two years, he lived in terror of the guards. They seemed to have a certain contempt for him, and this was early on, even long before the great evil that happened to him while in the prison. The guards called him _arap_ or _zengin_ and would often comment on his rape of a Turkish woman. To these men he asked for help in sending his letter, but none paid any heed to him. In his second year, a certain Mahmut, in love with a football player from the country of my host, Jay-Jay Okocha, would agree to post his letter. But only if it was within Cyprus. _"Nijerya, cok para,"_ the man would often say. " _Parhali, cok, cok,_ big, big, Mr. Ginoso." "Sorry, my friend." What about the money he had in the pocket of the clothes he wore on the day he was arrested? "Sorry, Mr. Ginoso, we cannot take. The court lock money. Nobody take money. Sorry. Understand me, Mr. Ginoso?" After this man, too, declined, he gave up. He did not know that even I, his chi, had tried to reach her.
So Agujiegbe, I let him lie in bed after he returned from searching for Ndali that night, and he continued to ponder the possibility of reconciliation with her. But then, as the night deepened, he allowed himself—with a certain back-to-the-wall bravery—to consider that which he'd refused to consider: that he may never have her again. Into the fragile ear of his mind was delivered the thought that enough time had passed. She could have married and had kids by now. She could have forgotten about him, or she could have died. Whom did she know to contact to find out anything about him? There was nobody. He thought with bitter regret that he should have given her his uncle's phone number. Or even Elochukwu's. He resolved that he should not think it was possible that she could still be waiting for him all these years. Enough years have passed, the voice in his head repeated with finality. _She is gone forever._
The impact of this realization struck him with despair. Chukwu, it has always perplexed me how a man's mind sometimes becomes the source of his own confrontation and inner defeat. So floored did he become that night that he considered himself foolish for all those years he'd wasted thinking of her, clinging to pieces of the memory of their time together. Perhaps she was in the arms of another man all those nights when, sleepless, he'd restage a moment of sexual experience with her as vividly as he could, so much that he'd touch himself with the lather of his saliva.
He rose with a sudden cry and smacked the kerosene lamp across the room. Its bulb broke and threw the room into instant darkness, and the sound of glass shattering was trapped in his head. He stood fuming in the dark, his chest heaving, the air filled with the smell of kerosene. But none of this could stop him from wallowing in the fitful thought that a man he does not know has been sucking Ndali's breasts.
He slept very little that night, and over the following days he attended to life with a feeling of having failed in everything. It threatened his existence. Even I, his chi, feared for him. For so lost was he in the new meaninglessness of all things that he veered into oncoming traffic. Twice, he had close brushes with death in accidents that could have killed him. Once, after a car had knocked him and his bike into a ditch, the driver of the car said, "How come you survived this?" The man and the onlookers who had immediately gathered were astounded. "Your chi is truly awake!" one said. A third person insisted he must have been saved by an angel, a messenger of the White Man's alusi.
Many times, when tormenting thoughts of his loss of Ndali came to his mind, I would push a counterthought in. _Think of the girl at the mash store who was kind to you and called you a good man,_ I would suggest. _Think of your uncle. Think of your sister. The football match. Think of the good future you can have_. Sometimes, when all these failed, I'd try to go with him in the direction he had chosen. I'd try to give him hope that he could still find her. _Think of it this way: love never dies. You see, in that film you saw,_ The Odyssey, _in which the man returned after ten years to find his wife still waiting for him, the wife knew that her husband loved her and was just being kept away from her because of the circumstances of life. So she remained faithful through the years, refusing, no matter how much she was pressured, to betray him. Is this not the same situation with you? Is it not, simply, only four years? Only four years._
It was during one of these moments, on the very day when I reminded him of that movie, that I stumbled by serendipity onto something neither he nor I had given any serious thought to in all those years. I acknowledge that once or twice he'd replayed the experience in his mind, but in none of those times did he consider its possible outcome. They had started making love in the yard in full view of the fowls when, suddenly, she pulled away from him and said it was not good for the birds to witness it. He then carried her into the house, her legs wrapped around his body, her arms clasped around his neck. They'd made love with such intensity that when he started to pull out, she grabbed him so tightly he squirmed.
"Do you love me, Solomon?"
Although all of it—the grip, her apparent unconcern as to whether or not he was about to ejaculate, and the fact that she'd called him by his Christian name, Solomon, which she rarely did—shocked him, he answered, "Yes—"
"Do you love me?" she asked again, more fiercely, as if she had not heard him.
"That is so, Mommy. I love you. I am about to release."
"I don't care. Just answer my question! Do you love me?"
"Yes, I love you."
He'd started to let himself loose, trembling through his speech, and when he'd emptied himself, he collapsed against her.
"Do you know we are now one flesh, Nonso?"
"That is so, Mommy," he said breathlessly. "I—I know."
"No, look at me," she said, reaching for his face. "Look at me."
He swerved to her side and turned to face her.
"Do you know we are now one flesh?"
"Yes, Mommy."
"Do you know we are now one? No more you or me anymore?" She paused, her voice rising, tears running from her eyes. Thinking she'd finished, he started to speak. She said, "Do you know we are just one now? Us?"
"Yes, Mommy. It is so."
She opened her eyes, and through the glob of tears, she smiled.
My host sat with this piece of tranquilizing memory as if it were a sudden, strange gift from a divine messenger come to help him. It was one of the cherished events of his life, and what she had done was monumental. She had allowed him to ejaculate into her. Yet she'd done it so offhandedly, as if it were a trivial thing. That time, he'd been too shocked to comment on it. But when they made love again later that night, and she held him stiff, forcing him to ejaculate into her as before, he asked why she was doing it. She said she did it to show him she loved him and was ready to marry him at all costs. But what if she got pregnant, then? he'd wondered. In response, she inclined her head, thought about it, perhaps considered how her parents would have taken it, and said, "So what? Are they my god? You want me to take Postinor?"
"What is that?" he'd said.
"God! Village boy!" she'd said with a laugh. "So you don't know? It's after morning. A drug women take so they don't get pregnant after sex."
"Ah, Mommy," he'd said. "I did not know."
As these vivid events revisited him, his slumped hopes opened their weak eyes. Through the days that followed, ideas came to him, possibilities. If she still believed what she'd told him that day—that they were one and the same—she indeed must be waiting for him. She could not have given up after only four years. He began to devise his next moves. Daily, in between measuring out cups of feed, millet, brown seeds, and clumps of loam, he would leap into the hole of ideas and rummage within its crevices. It was on the fourth day after the hope-bringing remembrance that he finally dug up something convincing enough for him to consider: whether or not he should return to her house and try to speak with the gateman again. Perhaps the man was poorly paid and he could bribe him to give him some information. Perhaps he could give him the letter he'd written to her in his first weeks in prison. Yes, even that would be enough. The letter contained everything, everything he wanted her to know about his disappearance and his failure to keep his own side of the promise never to leave her.
OBASIDINELU, the great fathers, in their esoteric wisdom, say that whatever a man desires to see in the universe, that he will see. How true, Egbunu! A man who hates another will see evil in whatever that individual does, no matter how well-intentioned. The fathers also say that if a man wants something, if he does not desist from pursuing it, he will eventually find it. I have seen it many times.
It did not cross my host's mind that the universe was about to grant him that which he'd been seeking for many years that day—it was simply the resolve to go to the gateman that formed strongly in his mind and caused him to stop the task he was doing—grinding melon seeds on the manual grinder clamped to the other end of a small bench. He removed his apron, locked his store, and set off for Ndali's family house. As he removed the stone wedge holding the door open, thoughts of what he would miss if he did not commit to his business that day trickled into his mind. In one hour, the agriculture professor who was coming to buy a bag of broiler feed for her poultry would arrive. He would miss the opportunity to sell in one transaction what he sold in a week. But even this did not deter him.
He mounted his motorcycle and raced onto the road towards a roundabout. On the side of the road was a construction site a few square meters long, cordoned off with a zinc roofing sheet held in place by bricks. A man carrying a slab of wood crossed the road dangerously, forcing the traffic to pause until he was on the side of the road, where everywhere sheds were erected. A house with a sun-scorched roof towered above, painted in dim red, with 0802 inscribed on it in white letters. From here he entered Danfodio Road, meandering between a water tanker and a white car whose boot was open, pressed down by the weight of the overload of grain sacks held in place only by strong, tight ropes. On the shoulder of the road, beneath a high billboard, stood a man speaking into a megaphone, surrounded in a half circle by others who carried Bibles, guitars, and flyers.
He'd halted because a semi turning in front had temporarily paused traffic. He would have driven on, but that pause—just a few hundred meters from the billboard—allowed him to hear the distinct voice blare from the megaphone. Despite the many years that had passed, he recognized the voice right away. He pulled onto the shoulder of the road, and once he came within view of the man, it became clear to him and me that something extraordinary had happened in the universe. I felt indeed that some great argument had been settled in the realm of the spirits, one which not even I, his chi, had participated in. And now, after my host had given up all hope, after he'd resolved to simply swallow whatever life like an unhinged mother had put in his infant mouth, the universe had heard his pleas and come to his aid.
He'd spent nights pleading with whosoever could hear him to give him only one chance, just one, so that he might find the bearer of this voice again. That he may make the fellow pay for what he'd done to him. He'd made these requests to deities big and small, sometimes to "God," sometimes to "Jesus," even once to "Ala," and once—unexpectedly—to me, his chi. When the prayers went unanswered, or when he thought that was the case, he would recline into himself and spend the time conjuring up images of a confrontation with this man, some more violent than others. One very prominent one was that he would be eating at the restaurant where the man and he had eaten the day he first met him in 2007, and the man—now wealthy from the money he'd stolen from him and others—would walk in with a good-looking woman. The man would walk in with majesty, booming with grace and attended by a chorus of praise from those seated in the restaurant. He would order them all drinks and settle the tab, happy with himself that he was impressing the woman he brought with him. The man must have come on short visits to Nigeria, perhaps thinking his victim might still be in jail. And was thus completely at ease. He would not realize that fate had planted his comeuppance in the form of my host, a vandalized man waiting for him to arrive.
My host would bend his head towards the table to conceal his face so the man could settle into his chosen seat, then he would rise quickly, break the bottle of beer he'd been served, and launch his attack. In attacking, he would be a person he never imagined he could ever be. He would have grown the heart of an executioner—merciless, quick, collateral, brutal. Within the span of a few eye blinks, he'd break the bottle and sink it deep into the belly of his enemy. But that would not be all. He would pull it out and stab it again into the man's chest. He would not be deterred by the inflorescence of blood or its affluent spattering all across the room. He would keep stabbing—at the man's neck, hands, chest, until people would wrest him off the body. But by then, it would all have been done. There would have been a reckoning, as has been known amongst men for thousands of years. He who had come the hard way would have fallen hard. Egbunu, this is the image which, for a long time, had stood in his mind as the truest portrait of the day that he has stumbled into by serendipity.
My host pulled his motorcycle up towards the gathering and had barely dismounted when the man of guilt recognized him, too. The man halted his speech and in haste handed the megaphone to another who stood by his side, dressed, like him, in the way of the White Man: a shirt, a tie, and plain trousers. Then the man ran forward, crying "Chinonso-Solomon!"
Ijango-ijango, this is one of the instances when I often wish that we, guardian spirits, are able to see what other humans, not our hosts, are thinking. Yes, clearly he looked afraid, but was he truly afraid? Was he as afraid as he should have been? I do not know. All I could see then was that although he hastened towards my host, there was caution in his expression. For he stopped a few steps away from my host. As his enemy drew near, my host realized that things would not be as he'd imagined. For where he had found the man was an open place where he could do nothing. Now stopped in the presence of my host, this man broke down in tears. "Solomon," the man said, and inched forward, turned his eyes back to the gathering, and then stretched his hand towards my host, who stepped back slightly. The man's hand came down slowly, trembling as it did. "Solomon," the man said again, and turned to face the crowd. "Brethren, it is him. It is Solomon. Hallelujah! Hallelujah!" He threw his hands into the air and jumped.
Then, without warning, this man for whose death my host had prayed for so long jumped forward and embraced him. In the moment when he should have gripped the man by the neck and started strangling him, the man turned back to the crowd, took the megaphone, and said with genial vehemence, "God, the God of heaven has answered my prayers! He has heard me! Praise the Lord!" And the crowd cried in response, "Hallelujah!"
"You don't know, you don't know, brothers and sisters, what the Lord has just done for me," the man said, and stamped his feet so hard that dust swelled around them.
"You don't know!"
The man brought out a handkerchief and wiped his eyes, for indeed, Egbunu, he was crying. My host looked about now and saw that the crowd was growing. A man and his wife had parked a lorry on the corner and moved closer to witness the evolving scene. An elderly woman had come out of a house on the other side of the street and now leaned on the balcony, watching. Round and about, faces, eyes, encircled him as if with an invisible chain that completely becalmed him.
"This man here is the reason I'm saved. I was a thief. I stole from him and others. But the Lord used him to touch me. The Lord used him to save me. Praise the Lord!"
The people responded, "Hallelujah!"
Now, was there anything my host could have done with these people surrounding him? No, Chukwu. They were the weapons of finality that neutralized all his vivid conjurations and elaborate plans. What was happening was incomprehensible to him, for now, this creator of all his sorrows, took his hand. What could he do but let the man have it? Then he watched, amazed, as the man knelt before him and held his hand.
"Brother Chinonso-Solomon, I kneel here before you, in the name of God who made you and, I and the whole world and... forgiveness, forgiveness. Please forgive me in Jesus's name."
Although some of the words were lost in the eruption of static from the megaphone, nearly everyone there seemed to understand. A murmur rose amongst the crowd. A young man in a red shirt and brown tie on which there was an image of a church and cross began to pray, shaking a tambourine—a small circular instrument with small metal clippings which, when knocked against the palm of his hand, released a jingling sound similar to that produced by iron staffs carried by priests and dibias. Even though my host could not hear him, he could sense what this individual was saying. But I, his chi, heard every word of it: "Lord help him. Lord help him. Let him forgive. Touch his heart. For you made it possible for such a time as this. Lord help him! Lord help him!"
IJANGO-IJANGO, my host stood there, helpless, transfixed, surprised at how his hand trembled as his enemy, who had stood up again, thrust the megaphone into his hands. Once he held it, the crowd erupted. His enemy wept even more vocally, like one mourning a parent. The tambourine attended with a ringing acclamation, and the crowd cheered even louder. My host knew they were waiting for him to speak.
"I... I," he said, and brought the megaphone down.
"Help him, Lord! Help him!" the man of guilt said, his words attended by the ritual jiggle of a tambourine.
"Yes! Yes!" the crowd chorused.
"I... I for—" my host said, and his hands began to shake. For he remembered now, as if an apparition had appeared before his face, the white men gathering as he walked towards his cell. He saw the one with the ugly scar on his face, and another, coming at him with their fists, saying, "You rape Turkish woman, you rape Turkish woman," with a flurry of Turkish words he could not understand. He saw himself trying to open the cell and run into it, catching the eye of a black man watching from the distance, while the men kicked him on the back. He saw himself falling against the bars of the cell and gripping them as the men tried to wrest him off.
"Touch him, Lord! Jesus, touch him!" the man in the tie and suit said again, and the strange instrument that produced the jiggling attended.
"Yes! Forgive! Amen!"
"I will forgive," my host let out.
The eruption of the crowd this time was wild. In the heat of it, reality insulted him even more. Without warning, he whom he should have killed lifted my host's hand like a referee who raises the arm of a victorious wrestler to the cheers of onlookers. My host had, however, just been defeated. For this man was Jamike: the man he'd sought for so long, one of the things that had kept him alive all along. And now, after all those years, he had found Jamike, and what did he do? He'd simply announced that he would forgive.
"Some people say there is no God!" Jamike shouted presently, and the crowd responded with an acclamation. "They say it is not true what we say we believe. I say shame on them!"
"Shame!" the crowd yelled.
"Who else could save me so? Who else?"
_"Onwero!"_
Agbatta-Alumalu, his anger grew as Jamike—now slim, bespectacled, possessing an innocent gaze, and exuding an unexpected warmth—gave a brief testimony of how he stole everything from "Brother Chinonso-Solomon" four years ago and how "Brother Chinonso-Solomon" came to the Turkish Republic of Northern Cyprus but he, the thief, fled south to the Republic of Cyprus. How, two years later, after he had been involved in an accident, he started to rethink his life. So he reached out to people in North Cyprus and was told about the fate of the three people he had duped—one lady at Near East University had become a prostitute and "Brother Chinonso- Solomon here, who was sent to prison, and brother, brother Jay."
Jamike struggled with the last name, and when he finally uttered it, he fell into a caesura of despondency, during which he wiped his eyes with the hem of his shirt.
"Do you know what happened to him because of me?"
"No," the people replied.
"I heard he committed suicide! He jumped from the top of a building and killed himself."
The crowd gasped. My host, fearing that he would not be able to restrain himself, detached his hand softly from the man's and put it to his chest as if stifling a cough.
"When I heard that and another story of what I had done, I gave my life to Christ. My brothers and sisters, I began to pray that God would let me see him again to ask for forgiveness. Glory to God!"
"Amen!" the people cried.
"I say glory to God!" Jamike said now, in the language of the White Man, as if the language of the fathers were no longer sufficient.
"Amen!" they repeated.
_"Otito di ri Jesu!"_
_"Na ndu ebebe!"_ the crowd shouted.
Jamike turned back to him with eyes that were filled with tears and a face that bore the visible stigmata of his own suffering. My host had not expected this: before him was Jamike, in tears, with a weather-beaten face, cracked lips—a face that bore the insignia of shame. It was not the face of one who has conquered another but of one who has been subdued. The face disarmed him.
Chukwu, the things he was feeling at that moment were in fact strangely common. I have seen it many times. The face is, beyond all else, naked—a thing of great poverty. It does not conceal itself from anyone, not even strangers. It is that which bears no secret. That which communicates continually, unrestrainedly with the world. Warriors of old amongst the great fathers often told of how, in wars, when confronted with the face of the enemy, they found their resolve to violence weakened. Almost instantly, their drive to kill for the sake of killing became a drive to kill simply in order not to be killed. It becomes as if the warrior, in the presence of his enemy's revealed face, strips himself of all enmity. Egbunu, it is a thing that is hard to understand. Even the wise fathers grappled with it, their tongue wove many proverbs to explain this phenomenon, but nowhere was it more pronounced than in their articulation of what that powerful emotion is which a man develops for a woman or a mother for her child. They referred to it as _Ihu-na-anya_. For truly, they understood that only when a man is without malice towards the other can he look him in the eye. So when a person says, _I can look you in the eye,_ he has expressed affection. And in reverse, a man who is masked, or who is distant—that man can be easily harmed.
I am certain that it was for this reason that my host allowed Jamike to embrace him again and to weep on his shoulder while the gathered crowd shouted "Hallelujah" and clapped for them. It must have been why—although my host did not know this—he gave this man who had caused him irreparable damage his phone number and nodded in response to his adversary's request to meet the following day at Mr. Biggs, down the street.
"At five o'clock?"
"Yes, at five o'clock," he said.
"I shall be there, Brother Chinonso-Solomon."
I am certain that it was the confrontation with Jamike's face that made him turn afterwards and make his way through the cheerful crowd that had gathered there. It was why he mounted his motorcycle and raced away from the scene without so much as looking back—not onwards to the place where he had been heading initially, but back to his apartment.
#
## Reckoning
IKUKUAMANAONYA, anticipation is one of the most curious habits of the human mind. It is a drop of vicious blood in the vein of time. It controls all that is within it and renders a person incapable of doing anything but beg for time to pass. An action delayed by the natural agency of time or human intervention comes to perpetually dominate an individual's thoughts. It bears down against the present until a view of the present is lost. It is why the old fathers say that when a child's food is cooking, the child's eyes are unblinkingly fastened to the top of the hearth. When a person is anxious, he attempts to peek into the unformed time, to try to gain knowledge of an event that has not yet happened. He may see himself already in a country he has not yet traveled to. He may find himself dancing with the people of the place, eating their local cuisine, and walking along the country's scenic parts. This is the alchemy of anxiety, for it is hinged on the promise of something, an event, a meeting, for which a participant cannot wait. I have seen it many times.
In the meantime, though, the man may dwell in much thinking and agony, as my host did after the encounter with Jamike. He returned full of spite and lunged about his room, kicking at the shelf, the bed, a rubber cup, cursing, raging. He blamed the heavens, the conspiratorial entities, for what had happened to him. He blamed _his_ god. Why, he said, did he have to meet Jamike after all these years in such a public place? And why, of all things, was Jamike preaching, a situation that had fettered him? It would have been nearly impossible to assault one who was preaching the gospel. People in Alaigbo and in the world of the Black Man in general revered the kind of man Jamike had become so much that he would not have been able to do anything. He blamed himself for not contacting Elochukwu since he returned. He should not have blamed Elochukwu for the many failures while he was in Cyprus—like failure to help get back his house and get Jamike's location from Jamike's sister. Had my host made contact upon returning, Elochukwu would have told him that Jamike was in Umuahia. He would have simply invited Jamike somewhere secluded and exacted his revenge.
Agujiegbe, I had never before seen my host like he became that night. So angry was he that he cursed, punched the wall, took a knife, and threatened himself. In a moment of great uncertainty, when I could not tell if truly this was my host or an agwu who had possessed him, he stood before the mirror, brandishing a knife and saying, "I will cut myself, kill myself!" He brought the knife close to his chest, and with his hand trembling, his eyes closed, he wagged it so that it touched his flesh. I rushed thoughts into his mind, reminding him first of his uncle, then of the possibility of reunion with Ndali. And I must say, humbly—Chukwu—that I may have helped save my host's life! For my words— _What if she still loves you like Odysseus's wife?_ —filled him with sudden hope. He unclenched his fist and the knife fell into the sink, did a mild dance, and settled there. Then he burst into tears. So hard was his pain and so great his grief that I feared he might not recover from it. I put it in his thoughts that it was only the first time he'd met this man since the events had happened. And that they would be meeting the following day, this time in private. His enemy would come to him as he had always wished, and he could do with him what he willed, even show the man the letter he'd written for Ndali, chronicling what had happened to him, so the man could know the gravity of what he had done. He should not think that the wasted opportunity was all he would have. No.
Again, he listened to my voice. I had affirmed a thing, and he'd followed. He washed his face and drained his nose into the sink and wiped his face on the towel that hung by a nail on the wall. He returned to the living room and removed the letter containing the story of his life, which he'd now decided he would show to Jamike the following day. He examined it carefully, trying to make sure the changes he'd made to it two days earlier had not altered its message. In fact, it struck him now that fate, or whatever it was that initiated events, had foreseen this meeting with Jamike. For only two days earlier, he'd woken from sleep in the middle of the night and could not fall back. This had become part of his life since he returned from prison. He had formed the habit of turning on the radio and then listening to it to help him sleep. He'd started to fade out when the voice of a preacher came on. And what was the man talking about? Hell. The same topic he had sometimes thought so deeply about during his years in prison. A place from which no one can escape. From everything the preacher described, as he listened, he realized that if he asked any questions about hell, the speaker's speech would contain all the answers: in hell, there is no redemption. It is a place of perpetual suffering where man is held up like a prisoner and where, the preacher emphasized again and again, "the worm never dies."
He turned off the radio and sat to ponder what he'd heard, menaced by his own mind. Then he rose and read the letter he'd written for Ndali. He'd not read it since he returned to the land of the great fathers because he'd felt that all he'd needed to tell her was there. He took a pen now and crossed out the title and wrote under it a new one:
_My Story: How I Suffered in Cypros_
_My Story: How I Went to Hell in Cypros_
When he finished reading through the letter, he was satisfied that nothing had been fundamentally altered. Tomorrow, he would give the letter to the man who had helped him construct it. And he could not wait for that to happen.
CHUKWU, the brave fathers say that a man who has been bitten by a snake becomes afraid of earthworms. For years, time and space had hidden this enemy from him, but that day, he would be alone with the man. He woke the following morning, from a night in which he'd slept little, with a kind of peace. He sat in the bed and let his plans play out till the imagined end, in which Jamike would lie on the floor in a pool of his own blood. He did not yet know of the importunity of hatred, how, even when one resists it and tries to push it away, it merely hoards itself for a moment like a tide, then comes flooding through until the mind is again drowned in it.
Egbunu, I have seen it many times, what men have done from hearts filled with hatred. I cannot describe it all, for time would fail me. But not wanting to stir up further emotion in my host, I watched in silence as his mind ran its bloody errand until, tired, he fell asleep.
It rained for most of that morning. Since he returned to Alaigbo, he felt the most at home when it poured. This was because most of the earliest memories he formed in Umuahia were shadowed by the presence of storms. The clouds were a constant image in his mind as a child. Claps of thunder, the shrapnel of lightning, these gave this world a beating heart and a memory as vivid as that of war. In some nations, like Ugwu-hausa, other elements might dominate, but rain reigned supreme here. Amongst the Igbo people, the sun was considered a weakling.
He did not go to the store that day, for the rain continued till almost the time when, having run its course, it yielded to sunlight. For the rain is the master of all other elements. The previous day, when he'd encountered Jamike, the sun had emerged early, effulgent against the morning sky. Then slowly, clouds swarmed up and contested its right to stay.
A weak sun was rolling slowly through the pool of wet clouds like a ball through sloam when he stepped out of the house. He peeled the tarpaulin from over his motorcycle and mounted the bike. For the first time since he returned, he carried the bag Ndali had given him. The white print on its leather face was still apparent: CONFERENCE OF AFRICAN AND CARIBBEAN POLITICAL SCHOLARS, APRIL 2002. All its contents were still intact, except for the two photographs of Ndali and her letter. He recalled then how, after he was released from the hospital and taken to the police station, one of the officers brought out the photos while searching the bag. He'd have tried to grab them, but he was handcuffed. The men had passed the photos between them, laughing and saying something, making gestures—the hand slapped against the palm—which, he would later come to understand, meant sex. One of them had spoken to him in halting English: "You, you like pussy many-many. Black pussy, good? Yes? Good?" It was a moment he would never forget—one in which his punishment had been extended to the most innocent of people: Ndali. At the time, many thousands of miles from the land of the fathers, he was witnessing her being violated by the eyes of strangers. Later, one of the men, visibly angry at the actions of the others, would take the photos, put them in the bag, and say to my host, "I am sorry, my friend." Then the man would leave with the bag. He would not see the bag until his release. When it was handed back to him, the first thing he looked for were the photos. Her letter had been removed from his bloodied trousers after he arrived at the hospital, badly damaged.
Now he carried a knife in the bag, hidden between the pages of a book. He'd planned it all out. He would get to the restaurant and sit down calmly at a table by the door for easy exit once the deed had been done. He would place the book on the table and eat quickly, for once Jamike came, he would be too angry to eat. He would try to disarm his enemy by making him feel at ease, even believing that he'd been forgiven. Then he would invite Jamike to his apartment. He would not use the knife in a public space. But if the man refuses out of suspicion, he would have no choice but to use the knife right there at the restaurant. He would stab the man dead and run away to the bus station and take a bus to Lagos. He would try to locate his sister or go to his father's village and stay in his father's empty house there.
Chukwu, I was afraid that this plan, if fulfilled, would bring him even greater troubles. So I flashed the thought in his head that if he did all these things he had planned to do he would lose Ndali forever. And, I added—although with great hesitation—that such an act would return him to prison and deprive him of ever finding her again. He considered this for a while with fear. He even took out the knife from the bag and placed it on the table. But then a monstrous rage gripped him again, and he slipped it back into the bag. I will do it, I will kill Jamike and find her, the voice in his head said. I will kill Jamike, I don't care!
Egbunu, often a man, even while knowing that he cannot see the future, plans nevertheless. You see people like that every day, couples dressed up visiting families, telling them their wedding is five months from now. Along the road near the end of the street, there are numerous projects. A man has bought a house, laid the foundation, and hopes to build on it in the future. Even though he can die one minute after laying the foundation, it matters little. In fact, human life revolves around preparations for the future, of which he has little control! This was why, despite all his planning, when my host entered the restaurant, he heard, "Brother Chinonso-Solomon." He was startled, as if thrown off horseback. The man he'd seen the previous day stood now, almost alone with him. Across from them was a counter from which a woman watched. Behind her braided head were posted items for sale with their prices.
"My brother, my brother," Jamike said, coming towards him.
"I want us to just sit down," he said quickly, in the language of the fathers, although when with this man he primarily spoke the White Man's language.
Jamike, his hands still afloat in the air, stopped. "Okay, brother," Jamike said.
My host pointed to the chair near the door and began walking towards it. Jamike followed with a weak smile on his face.
As he sat, he realized that something had happened yet again and had caused him to calm in the presence of this much-hated man. But he could not tell what it was. Suddenly his great, maddening anger was gone, and he sat down slowly in the chair, surprised at himself. Jamike stretched out his hand, and he shook it.
"Madam! Madam!" Jamike cried.
The woman at the counter reappeared from the kitchen, where she'd gone.
"Please bring us two bottles of soft drinks. Cokes."
"Okay, sir," the woman said.
He could see, now, that part of what disarmed him was the change he'd seen in Jamike. The man had slimmed down so much that instead of a big, fleshy head, he now had a lean face with protruding cheekbones. His eyes had retreated inwards so that the eyelids hung like small awnings. His thinness was pronounced in the long-sleeved shirt he wore, which was much larger than his diminutive frame. His lips were cracked, with a spit of blood on the ridge between them. His whole constitution was that of an emaciated, suffering man, malarial and gaunt. And in his eyes, there were signs of tears. On the side of the table, he'd set the big Bible he'd brought with him, and now he placed his hand on it and said, "Brother, I have been looking for you. I have been waiting for you. Many years, my brother. I did not know you had returned. I even asked Elochukwu, but he did not know."
Egbunu, my host wanted to speak, but it seemed as if the words had been bound with chains within him, and they could not get out.
"Ever—oh God—ever since I heard about your prison. I have been looking for you, Solo. I have been looking for you everywhere." Jamike shook his head. "I have been in a very bad state. I have been very very sorry. I have not been myself. I have not—how do I say it?—been alive. God help me. Help your son!"
Then Jamike began to cry. The woman arrived with the drinks and set them down, her eyes on the weeping man. Then, with an opener, she uncapped both drinks.
"Do you want to order?" she said.
"The drinks are enough," my host said. "Thank you."
"Ah, only drinks?" she said. "Oga sorry, er."
"That is so," he said, without looking at Jamike.
"Thank you, madam," Jamike said.
When the woman was gone, he said, "Jamike, can we go to the house? I need to tell you my story."
He'd spoken quickly because his hatred had returned, and he was afraid it would go back into wherever it had come from. He wanted it to stay, to be ever present with him while he was with this man. Without it, he feared that he would never be well again.
"Oh, you don't want to eat?" Jamike said. "I am buying the food."
"No, we can eat after."
Jamike paid the woman for the drinks and they walked out of the restaurant, my host carrying his bag, and his heart beating loudly for fear he may have betrayed his intentions through his tone. Although he listened for the sound of someone following him, he did not look back.
"It is not far. We can ride on my motorcycle to the place," he said aloud.
"I want to come," Jamike said.
He turned and looked, for the first time that day, at Jamike's face. "Let us take my machine," he said.
He realized he had not fully considered his request until Jamike mounted behind him and their bodies touched. It sent a shiver through him, as if he had been poked with a sharp rod. He lost his grip on his keys, and the bunch fell on the ground. Jamike rushed to pick them up.
"Brother Solo, are you fine?" he said.
He did not speak. He merely pointed to the street ahead and started the motorcycle.
GAGANAOGWU, revenge is a debris field. It is a situation in which a man who was once defeated in a fight drags his enemy back to a cleared field after the battle has been won and lost, hoping to revive a dead fight. He returns to pick up the rusty weapons, to scrape clean the blood-encrusted swords, and to light again the violent fire of hatred against his foe. For him, the fight was never over. But for his foe, so much time may have passed that the enemy, if he had felt himself the victor, may have forgotten about the old battle. Thus he may be astonished when he who was smeared in the mud, whose bones were broken, who was vanquished, seizes him again by the throat and begins to drag him back to the battleground.
The broken man may himself be surprised by the force with which he has now seized his enemy. But this may be the beginning of his surprises. What if he seizes his enemy by the throat, wrestles him to the ground, and begins to strangle him without any resistance? What if his enemy simply lies there, closes his eyes, and simply says, "Please, brother, go ahead"? What if the other's face, red and bursting with veins, continues to entreat him? "I am in Christ. Praise the Lord. To die in him, I am willing to do... Argh... I love you, Chinonso-Solomon. I love you, my brother."
What would the broken man do? What would he say when the man he was about to kill speaks of love for him? What would he say when his heart had been further broken by all the misreckonings of life, by all the false calculus of time and the dubious permutations of fate? What would he do when he had done no wrong to warrant the trouble that came to him? He had fallen in love with a woman, just like any other man. He'd tried to marry that woman, the way every good man should. Indeed, her parents had tried to obstruct it, but he tried to scale the obstacle, the way people do when wanting to achieve a goal. Now, certainly that had led him into even greater trouble, but what did he do? He plotted his revenge and sought it as if his life depended on it. It had taken him a long time to find his enemy, but he'd finally found him. And now, he is strangling the man, trying to kill him and discard his body in the Imo River, as people would do to someone who had destroyed their lives. So you see, Egbunu, he has done nothing out of the ordinary. Yet nothing he has done has yielded a common result!
If he headed north just as every other traveler did, he found himself in the south. If he put his hand into a bowl of water, it burned him as if it were fire. If he trod on land, he drowned as if he'd stepped in water. If he looked, he did not see. If he prayed, what is heard was a curse. And now, when he engaged a wicked man in a fight he'd rehearsed for many years, what he finds instead is a saint who prays for him; instead of protestations, he finds a singing man.
So he resigned. He unclasped his hands from the throat of his enemy, who had begun to cough frantically, trying to gather air into his lungs. He sank to his knees and began weeping, while the man whom he had tried to kill whispered prayers through his aching throat: _God forgive him, please. Put all his sins on my head. You know what I have done. Please, Lord, help him. Heal him, heal him, Lord._
On his knees, my host wept aloud, for everything. He wept for that which had been lost and would not again be found. He wept for the time which would not replenish itself. He wept for the sickness which ate out the interiors of his world and left it as a cracked shell of its old self. He wept for the dreams washed down the pit of life. He wept for all that would come, all that he could not yet see or know. He wept, even more, for the man he had become. And his weeping was attended by the words dripping like poisoned rain from the mouth of his enemy who lay beside him: _Yes, Lord, you are merciful. Merciful father. King of kings. Heal him. Heal my brother. Heal him, Lord._
CHUKWU, they stayed this way for some time—he kneeling and sobbing, the man praying quietly while lying on his back on the floor. Into their ears came the world from the outside. A neighbor was chopping firewood at the back of the house, a dog was barking somewhere not too far away, and on the long road that led to the big market, cars were honking and streaming about interminably. The sun outside had started to set, and the last light of the day lay outside the window as if too afraid to enter the room. In his mind, the great anguish had subsided like a receding storm. Now he sat empty, watching the shadow on the wall forged from him and his enemy by the subdued light of the evening sun.
In the small serenity of his mind, a vision of the gosling materialized. It was one of those times when it seemed to have suddenly forgotten that it was on the leash, for it sometimes forgot about it, and was enraged by it and wanted out. It would rouse itself and make a rustling, held back by the leash, bound to the leg of a chair or a table. When it had tired, the bird would smear down, its wings spread out as if in surrender. Then it would orient its head downward and peer at him, its yellow eyes on the sides of its small face bulging as if they would pop out of their sockets. But then the thin sheets of skin that formed its lids would cover them and open again, revealing pupils now dilated. It would sit that way for a while, and then a sudden epiphany would strike it and it would leap up again, seeking the familiar pool of the Ogbuti forest—its true home.
My host rose afterwards and sat on the lone chair in the room. Then he pulled one of the two stools forward so that it faced him, and called to Jamike to rise.
"Come and sit here," he said, tapping the stool in front of him.
Jamike stood and moved towards the stool, planted himself on it, and folded his hands across his chest. My host examined him, as if to assure himself that this was truly the man who had dominated his thoughts for four years. He was again surprised by what he saw. The man before him was nothing like the one he'd stored in his head for all those years and who sometimes visited him in vivid dreams. What sat before him now was a shadowy creature from an inchoate dream, one who, in some indefinable way, seemed to have suffered a fate similar to his.
He took up the bag Ndali had given him and brought out the letter.
"I want you to read this," he said. "It contains my story. I want you to read it loud to me. I want to hear it, together with you. I want us both to read my testimony. So go ahead, read!"
The man passed his eyes around the four pages stapled together and folded into columns. Then, raising his head to look at my host, he said, "Everything?"
"Yes, everything."
"Okay."
**My Story: How I Went to Hell in Cypros**
_Dear Mommy,_
_I am writing you from my second year in prison in cypros. You will not believe my story but everything I am saying here will be truth. Just belif me in the name of Almighty God I beg you. Please Obim. you know I love you. Do you remember?_
Jamike raised his head to look at him.
"Read on!" he said. "I want you to read what I went through because of you."
_After you saw me to the bus garage, I said to myself, I will see you again soon. I said I will return to you and I will marry you. my mommy. I was happy. I beliefed that what I was doing was—_
"What is this?"
He bent forward to see the titled page. "For you, I believed that what I was doing was for you."
"Okay."
_For you I beliefed that what I was doing was for you. I fly to Istanbul thinking of you. not even a single time did you leave my mind. Actually I even dreamt of you, many dreams, both of the future, and past time. Then, in the plane, I began to listen to tow Nigerians. They were talking of this country I was going. They were talking how bad cypros was. They said it was like Nigeria, that agents who ask people to come there lie to them. It is false what they say. All telling serious lies. cypros is not like europe. They said if you go there, it is like a pit. You can come back to Nigeria or you can stay there. And if you stay you will not get a better job. You will always work bad job. So I become afraid. I ask the men when we got to Istanbul if there were true, and they said yes, yes. It is so. So I become afraid again. I said to them, but my old classmate Jamike Nwaorji say it is a good place. He lied to me._
"Look, I said you should not stop. Read on! _Gu ba!_ "
My host, becoming desperate, did not want to harm this man but rather to threaten him so he would read the letter in its entirety. He brought out the knife from the bag and held it. Ijango-ijango, I must emphasize that my host was merely desperate to make Jamike read the letter in its entirety and was not intent on doing harm. I, his chi, who would not want him to shed blood and incur your wrath and Ala's, would have attempted to stop him. But I could see that he would not use it, so I did not interfere. Brandishing the knife, he said, "I will kill you here, and nobody will know, if you don't read on now."
It worked. For Jamike, slightly shaken, continued.
_I tried to call him. The phone never go. I was very surprised because I called the number many times before. So I ask the men and they say it was not cypros number. I try many times. So when we reached cypros naw, the man was no where to be found. Actually no where at all. I can't by then reach his number also. Please God, help me I was praying. I was very afraid. But my spirit told me, if you are afraid that is not good. It means this man will win. You must be strong. So I went to the airport in cypros. I wait, wait, wait. He didn't come at all. His number did not go through still. Even in cypros. What can I do now, I ask myself then. This is everything I have. So I decided to wait. For three hours, he did not come to the airport, after all of his promises. So I took a taxi..._
Chukwu, at this point, Jamike shook his head gravely. I have cycled the habitation of man for so long like a falcon, but I have never seen anything like this before: a man stripped naked of all dignity and forced to gaze at his unpleasant self in the dark mirror of his own past malevolence.
_Turkish people don't hear English. They don't hear at all. If you speak even "come" they don't hear. Only few of them hear. So the_ _taxi man who took me did not hear English. When we got to the school I was very afraid. I prayed to God, let it not be true, let it not be true. So but they cannot see my name. I find out only one semester school fees is what Jamike paid for me, even though I gave him equivalent of almost 5,000 euros for both two semester school fees and accommodation. The money I gave him to open a bank account for me also. He ran away with. So out of 7,100 euro, he use only 1,500 for me. He ran away with all the rest. Everything mommy. All of the money they paid me for the house and the fowls._
"Read, I say, read or I will cut your throat!" my host said, brandishing the knife.
"Can I stop, please, my brother?"
"If you don't read on now I will smash your head!" He threw the knife away across the room and with all his might struck Jamike on the cheek. The man fell off the stool to the ground with a scream, his hands on his mouth.
He'd struck Jamike with so much force that his knuckles hurt. Now he held that hand in the other and began to blow at it to ease the pain. He could tell that his hand had broken something in Jamike's face, and even though he did not know what it was, it gave him relief.
"I swear to God who made me," he said between deep pants, his chest heaving. "I will kill you if you don't finish reading this thing. I swear to God who made me. You must know everything that happened."
Indeed, Agujiegbe, the murderous rage had returned, and my host—in one flash—had become unrecognizable even to me, his faithful chi. He paced from one end of the room to the other while the man on the floor lay still, his eyes closed, blood running down the side of his mouth. The sun had dropped and sunk away from the habitation of living men. Light from its retreating shadow held everything in a dim receptacle.
He stopped before the single wall mirror in the room and saw himself in it. He saw how far fury could take him. He saw, as if portrayed in the mirror, the potential of a wounded man to do damage if he did not bring himself under control. It was this that calmed him so that he returned to the chair.
EBUBEDIKE, it is not for nothing that the world is as old as it is. Perhaps every day, in every nation, amongst every people, through time, people are coming face-to-face with their tormentors. What a man carves with his hands, that shall he bear on his head. Again, as the great fathers say, the head that stirs the wasp's nest bears its sting. Guardian spirits of mankind, we must all take this to heart. Children of men must listen to us, to this, to _this_ story, to the stories of their neighbors, and take notice: there is a comeuppance for everything, every action, every careless word, every unfair transaction, every injustice. For every wrong, there will be reckoning.
Man, do you take your neighbor's property and say, "Oh, he does not know?" Well, beware! Some day he might catch you in the act and demand justice. Man, do you eat that which you did not plant? Beware! Someday it might purge you. Every person must hear this. Tell it in the village squares, in the town halls, along the corridors of the big cities. Tell it in the schools, at the gatherings of the elders. Tell it to the daughters of the great mothers, so they may tell it to their children. Tell, O world, tell! Tell them this: in the end, there will be reckoning. They must recite it like an anthem. They must tell it from the tops of the trees, on the tops of the mountains, on the pinnacle of the hills, along the river shores, at the marketplaces, in the town squares. They must say it again and again: in the end, it does not matter how long it takes. There. Will. Be. Reckoning.
Guardian spirits of mankind, all you who stand in the court of Bechukwu to testify, tell! And if they doubt you, then tell them to look at my host: he had cried for justice so much, so loudly, all these years, that it had now been given to him. And now, his enemy was on the floor, and he was on the chair. The evening bore an uncanny resemblance to the day in Cyprus when the scars on his jaw and face were inflicted on him. But this time, the equation had been reversed. The contention was now between my host, a man with a weapon and an impregnable will, and Jamike, a man who, if he had any power at all, seemed determined not to claim it. This man had no weapon and did nothing against his foe. The man, after a long period of praying, began waving one hand in the air, the other placed on his bloodied mouth, chanting, "Thank you, Lord. Thank you, Lord. Amen. Amen. Amen."
Jamike sat up, and blood spattered on his neck and shirt. My host gave him a rag to clean himself, but Jamike would not take it. It seemed, Egbunu, that Jamike had come to understand that reckoning had come. It must have been this awareness that caused him to open his mouth to speak. He closed it again without speaking, shook his head, and snapped his fingers.
"Brother Chinonso-Solomon, I am sorry for everything," he said. "The Lord has forgiven me. Would you forgive me, too?"
"I want you to read this all first," my host said. "You have to know what happened to me, what you caused me, for you to ask for forgiveness and for me to consider it. You must read, first. You must read. You must finish it."
"Okay," the man said.
My host took up the letter and pointed to a part of it, on the second page, and said, "Continue from here." Jamike nodded and held the paper with the hand not stained with blood, put it close to his face, and began to read.
_The nurse was very sorry for me when I told her all that happened. She even cried for me. Her eyes were very read. She took me to a restaurant and buy me food, and assorted things like biscuit and coke. Then she said tomorrow she will come and take me to another city in cypros there, The name of the city is Grine. So we can go and look for job. In fact, very long time. She can speak Turkish also, this woman. In fact very well also. This woman gave me hope very much hope. That was why I called you that day if you still remember. I didn't call you for long because I was afraid of what to say to you. but I finally called you because of this. I told you everything will be alright because of the woman. I also tell you about the island, that all the trees in it have been removed. Mommy the following day she come. OK, this was after my friend and me go get a place to stay in the town, Lefkoshia. The nurse took me to the city girine where she intronduce me to the manager of the casino. The man say he will employ me. Actually he said I can start the following day also. But the nurse say since it is the weekend, I should rest and start Monday. I was very glad mommy. In fact I was so glad I was thanking and thanking this woman. I really believed she was godsent. Really, god sent._
At this point my host saw that darkness had arrived and that the man before him, who was now almost entirely turned into a silhouette, was struggling to see. There was a power outage. So he motioned for Jamike to stop and went out of the house to an open area where there was a kitchen—a half-covered place with old cupboards, almost black with soot. One of the people in the other flats who shared the kitchen with him was bent over a stove at a corner of the room, peering into a bubbling pot with a torchlight. He did not speak to the man, who had previously sparred with him over the cleanliness of the shared kitchen two days before when he'd rushed back from his store hungry. Then, he went to the store near the house, he bought Indomie and eggs, cooked the noodles and fried the eggs. In haste, he'd left the eggshells near the stove. The neighbor would find flies congregating in the shells, the air thick with odor from what remained of the eggs. Enraged, the man would knock on his door and admonish him, threatening to report him to the landlord.
He swept past the man presently, took a box of matches, and hastened back to his apartment. For it came into his thought just then that Jamike could leave before he returned. He found Jamike still seated, hugging himself in the near darkness, only the sound of his breathing and the rumble of his intestines audible. He was touched by Jamike's demeanor, the way in which he'd submitted himself to my host's wrath. The voice of his head told him to consider this as the ultimate act of remorse. But he could not bring himself to stop. He was determined, Chukwu, that Jamike would have to hear everything that'd happened to him—from beginning to end. He raised the lever of the kerosene lantern on the table and lit it.
Ezeuwa, later he would regret forcing Jamike to continue reading the letter. For Jamike began reading from the parts my host often refrained from going. Whenever his mind tried to drag him close to these places—dark beyond all things—he would fight, like a mortally wounded beast, with defiant violence to be spared the torture of such recollections. But now, he'd plunged himself into its pit by asking that it be read to him. A supreme act of self-flagellation. For as Jamike read to him about the incidents in the house of the nurse, he began to weep. And as Jamike read on, he saw the inadequacy of his own words to express what he had experienced. When Jamike read about how he'd passed the days in prison, portions of which had been too heavy for him to write down ( _... about some of these things, please don't ask me to tell you, Mommy. And please don't ask also..._ ), my host became possessed with a desperate urge to correct the insufficiency in the narration. He wanted to add, for instance, that there had been times when he did not just see "visions" but had completely lost his mind.
For how might he explain the times when, while drifting to sleep at midday, he'd be roused by the sound of an imagined rifle? Or how might one explain times when, half asleep, he'd feel a hand on his back trying to pull up his clothes and he'd scream? One could call these things hallucinations, but they felt real to him. What about those times when, in the veranda between sleep and wakening, the man he could have been would appear in the vision of his mind? The uncreated man would conjure up peace and sublunary bliss. And, by turns, he'd see himself helping what would appear to be their kids—a fine-looking boy and a beautiful girl with long braided hair—with their schoolwork. He'd see Ndali and him marching together at what was a vision of their wedding, often leaving him with a crushing envy for a version of himself that never was. This and many more were the things that he had not been able to express in the narrative because of the inadequacy of his words.
When Jamike was almost done, when he'd read the part about his hopelessness in prison, his incarceration for a crime he did not commit, the horde of unwanted memories rushed into his head. At once the violent rage came upon him again. In terror, he seized the man and began to hit him. But the memory did not abate at all. It was as if the images held his two hands and forced him to see what he did not want to see, and hear what he did not want to hear. The same way the men, now alive again and clear in his mind as daylight, had held him down, one pressing his neck to the wall that stank of rank sweat while the other slid his penis into his anus.
He struck at Jamike, everywhere he could find, but the images in his head remained, for the mind, Egbunu, is like blood. It cannot be easily stanched when a wound is deep. It will bleed at its own pace, at its own will. Only something powerful can stanch it. I have seen it many times. But now, no such thing was near. So he felt the man's palm becoming sweaty on his back and buttocks. He felt the forbidden thrusts. His onyeuwa felt it. His chi felt it. What was happening in that moment was transformative, life-altering. The moaning man's words—"You rape Turkish woman! You, _ibne, orospu-cocugu,_ you rape Turkish woman! We rape you too"—was not the voice of a human being but of something unfamiliar to any man. It sounded like something beyond time, beyond man: perhaps the voice of a prehistoric beast whose name no one alive or in living memory knew. And the man's smell, which he could recall now in striking vividness, was the odor of ancient animals.
He knelt on the floor beside his enemy, weeping. But, Ijango-ijango, this particular memory, when it begins, often bleeds till the body is emptied and the bloodless body falls and expires. So he would recall how the man's semen splashed around his buttocks and streamed down the back of his thigh. So although completely undesirous of it, he remembered even how he'd felt in the aftermath, after the world had scourged him with this severest of flagella. How he'd lain there for days that did not seem to cease, everything else alive but him.
Beside him, Jamike, having been beaten into a human pulp, lay still again, curved into the shape of a fetus. A slow, drawn-out moan of pain emitted from him, and his bloodstained hands trembled. It seemed that a revulsion of feeling seized him and he began to stitch words together, his teeth clattering, blood dripping from his mouth until, at last, words burst out of him in a voice slightly above a whisper:
"Heal him, Lord."
#
## Man of God
GAGANAOGWU, the magnanimous fathers often say that if one keeps a record of all the wrongs done to him by his kinsmen, he will have none left. This is because they know that you did not create the human heart to be capable of accommodating hatred. To harbor hatred in the heart is to keep an unfed tiger in a house filled with children and the feeble, for it cannot afford communion with a human being, nor can it be tamed. No sooner has it rested enough and woken up in need of food again than it falls upon the man who has nurtured it and devours him. Indeed, hatred is a vandalism of the human heart. A man seeking justice with his own hands must dispense it as quickly as possible, or he risks being destroyed by his own dark desire. I have seen it many times.
As is common with men, they often realize this truth long after the hatred has driven them into retributive acts. That night, my host realized these things. He helped the man up and took him to the clinic down the street. There was healing in this realization. But he'd been moved even more by Jamike's reaction. Jamike had thanked him after the nurses attended to his wounds and cleaned them up and refused to tell the nurses what happened to him. The nurses had gazed at my host as if to demand the truth from him. "He was attacked by armed robbers," he said. One of the nurses nodded and sighed. He stood there, expecting Jamike to deny it. But Jamike said nothing, merely keeping his eyes firmly closed. Later, on their way out of the clinic, with his head bandaged and a plaster on the bridge of his nose, he said, in the language of the White Man, "Brother Chinonso-Solomon, please do not tell lies anymore. God says, Thou shalt not lie. Revelations twenty-one verse eight says that all liars shall inherit the kingdom of hell. I don't want you to go there."
Jamike, who walked with a limp in his gait, put a hand on my host's shoulder as he spoke. My host said nothing. He could not understand it at all. He could not understand how, despite all he'd done to this man, all that seemed to matter was that he'd lied. When they came to the place where he'd parked his motorcycle, Jamike asked if he had forgiven him.
"You can cut off my hand if you want, or my leg. But all I want is that you forgive me. I have six thousand euro at home. Your money. The money I took from you. I have kept it for more than two years waiting to find you."
"Is this true?" he said.
"Yes. Now the value has increased. When you change it now, it must be as much as your seven thousand."
"Ah, Jamike, how is it possible? Why didn't you tell me you had this money before—before all I did to you?"
Jamike looked away and shook his head. "I wanted you to forgive me from your heart, not because I repaid you."
Oseburuwa, it is difficult to fully describe how this gesture made my host feel. It brought him the first touch of healing. It was a resurrection, a revival of something long dead. He was so shaken by it that when he got home that night, he could not sleep. He thought at first that Jamike was faking it all—the transformation, the docility he now exuded, must be false, the mask of a wicked man seeking to evade justice. He would have attacked Jamike that very first day if they had been in private. But now, that gesture of restitution convinced him that Jamike was indeed a transformed man. That night, in between struggling to breathe through a stuffed nose, he wrestled with the thought of forgiveness. If indeed the Jamike who damaged his life was dead, why punish the new one for the sins of the other? He considered: was it not what Jamike did to him that had caused him to change? If this was indeed so, then was it not a good thing? Was it not a thing to celebrate?
Chukwu, these were questions that I would have asked him, but the voice in his head asked them instead. And I flashed thoughts in his mind, accentuating them. The following day, early in the morning, while he brushed his teeth, Jamike arrived with an old envelope containing the money. Not once in all these years had he imagined even remotely that he would get his money back. And now, not only had the German woman paid him, so, too, had Jamike. It offered him renewed hope that he could regain all the things he once owned. This thought opened up slowly like a frontier in his mind. As he counted the money in disbelief, Jamike sank to his knees again.
"I want you to forgive me all the wrongs I have done, so that I may be forgiven by my father in heaven."
He looked upon the man whose death he once sought with an all-consuming zeal. As he made to speak, his phone rang. The screen showed the name of Unoka, a trader who had been lately trying to persuade him to add turkey feed to his stock. But he ignored the call. And when it had rung out, he said in a shattered, speckled voice, "I forgive you from now on, Jamike. My friend."
Ebubedike, that was the beginning of his clemency—when the soul of the afflicted embraces the soul of the afflicter, with his paralyzed limbs, both of them become forever marked by that embrace.
CHUKWU, I will again take you to the deeds that are necessary to explain and defend the actions of my host and to plead that, should it be the case that he's harmed the woman in the way I fear he has, he would have done it in error. So I must say, simply, that my host was transformed by the embrace I spoke of. His healing, Egbunu, had begun. He bought a car the following week with part of the money Jamike returned to him—his money! I need not waste time trying to describe the joy, the relief my host felt at this touch of redemption. For when a man has dwelt in misery for long, he becomes blind to the life that surrounds him like the ocean surrounds the shrunken earth. I, his chi, was delighted, for he'd become again a man of peace, even if part of his soul was still black with sorrow. For now, it was enough.
It restored his confidence so much that he and Jamike drove in his new car to his old house, the property his father had left him. A few days after he received the money, he decided to reach out to Elochukwu. Elochukwu was shocked to hear from him. And when he saw my host, he wept, saying that if he knew it was going to go the way it did go, he would not have encouraged him to travel to the foreign country. The thing was, Elochukwu kept saying, that you loved Ndali so much. "I saw it, Nonso. I saw it so much that I just thought you would never be happy if you did not try to resolve the problem with her parents." My host agreed. He would not have been happy if he had not tried all he could to be with her. Together, they attempted to reach the man who had bought the property, but the phone yielded no result. The number had gone into disuse, and the man, unreachable.
The next day, he went to the property with Jamike. This was one of the things Jamike had promised to help him do. It was on the list of the three important things he had said that Jamike must do to help him heal and be whole again, so his forgiveness could be complete. "One," he'd said to Jamike, with whom he now perpetually spoke in the language of the fathers, "you must help me find Ndali, and restore her to me. I love her and have lived for her. You took her away from me and you must restore her to me with your own hands." Two: "You must help me get back all that I lost. My compound and my poultry. I want to get back my father's land and rebuild my poultry farm on it. You must help me do this." Three: "You must help me forget about the things the prison men did to me. I don't know how you will do it. Pray for me, counsel me—anything, just make sure I don't remember them anymore."
The first thing they did was go back to Ndali's father's house. He told Jamike about wanting to send the letter to Ndali through the gateman, and Jamike agreed that he should. So they drove at night, one week after their reconciliation, to Ndali's family house. Then he went up to the gate while Jamike stayed back in the car. He knocked, deeply afraid. The small gate opened, and another man, one of those with whom he'd served at Ndali's father's party four years earlier, appeared. To his great relief, the man did not recognize him.
"Oga, wetin I fit do for you?" the man asked. "You wan see Oga Obialor?"
"No, no," my host said, his heart leaping at the thought of seeing Ndali's father again. He looked about him, up at the two black plastic septic tanks towering above the gates, then at the man. Then he brought out a wad of cash, twenty thousand naira. He stretched it out to the man.
"Er, Oga, what is this?" the man said, stepping back rapidly.
"Money," my host said, his breath catching.
"For wetin?"
"Erm, I want you to, erm—"
"Oga, you wan do bad thing for my Oga house?"
"No, no," he said. "I want you to give this letter to Ndali for me."
"Oh, you want Madam Ndali?"
"No, I want to give her a letter," he said.
"Okay, bring am. I go give im mother and them go give am to am for Lagos. Bring am."
Chukwu, at first he gave the man the letter and the money. The fellow thanked him and returned back in. But when he told Jamike, the latter said, "What if her mother opens it?" My host was stunned. "Did you write your name on the envelope?"
"Yes!" he cried.
"Then they will open it, even try to make sure it doesn't get to her. The man should just give you her address, or give to her by himself."
He ran back to the gate and asked the man to bring it back.
"Why, Oga you no wan send am again?"
"No, no, I go come back with am," he said. "Do you have her address?"
"Amdress? For Lagos?" the man said.
"Yes, for Lagos."
"No-oh. I be omdinary gateman."
"Do you know when she go come?"
"No, they no dey tell me that kain thing."
"Okay, thank you," he told the man. "Keep the money."
He left, despondent but thankful that he had saved himself from the possible outcome of Ndali's parents seeing his letter. Jamike counseled him not to despair and assured him that they would eventually find her. It was early March, he said, and she would most likely return for Easter if they are big Catholics. Jamike advised that they try to recover his house in the meantime. In a moment that reminded me of Tobe, the man who'd helped my host in the strange country, Jamike and he drove to his old compound. He parked his car outside what used to be his garden and sat waiting in the car for Jamike to return. The garden had been cleared, and in its place was a pile of unused gravel and a few cement blocks. A wheelbarrow lay tumbled over the gravel, a red rag tied to one of its handles. There was a big signboard with the inscription: LITTLE MERCY NURSERY AND PRIMARY SCHOOL, P.M.B. 10229, UMUAHIA, ABIA STATE. He looked about him. What of the house of the neighbors? They were still there, only now what he believed was a telephone pole stretched out from beside their compound. On the long cable, a few birds—sparrows—sat, gazing emptily into the distance.
To soothe his anxiety, he focused on the toy bird he'd bought from a crafts store and hung from the rearview mirror of his car. The toy bird swung back and forth when the car was in motion, reminding him of a hen he once had which he'd named Chinyere. He tapped the toy's beak and stirred it into a whirl. He watched the rope twist at the top and knot together until it reached its limit, then begin to unwind, the bird swirling quickly as the centrifuge that was the rope propelled it. He found meaning here, Chukwu, the way a desperate man, if he looks close enough, finds meaning in just about anything—a grain of sand, a quiet river, an empty boat rocking on the shore. The twirling rope that held the bird, that handlike object, like a sailor's, directing its course, that cord that binds two things, each of which moves when the other moves, shifts when the other shifts.
He'd been seated for what he thought was close to thirty minutes, and Jamike had still not come out. Although he'd rolled down the windows, it was suffocating. The rain had stopped for a week, and now the days were hot and humid. A bell tolled from the premises of what had been his home, and now the voices of children rose in an enthusiastic chorus. As if nudged by something he could not see, he got out of the car and began circling the big fence that had been erected around the property, stopping only in front near a pile of gravel and blocks. He saw as he walked that only a little of what his family had built of the fence remained. Most of it was now fresh unvarnished bricks held together by rough ligaments of cement. Lizards tailgaited each other across the wall in rudimentary choreographic movements. The chickens had loved them, and even though the lizards were fast and too slippery for them to hold firmly, the roosters would frequently catch them and eat them. Once, a white hen chased a weak gecko who had ambled into the yard and nipped it against the base of the wall, causing a chap in its beak. For days, even weeks, the striking image of the hen with the live gecko in its mouth remained in his mind. When the chicken had turned from the wall, the gecko's tail curled up its face, stretched up the space between its eyes, so that it seemed the bird wore a Roman centurion's helmet, complete with the cock's red comb.
He stopped behind the school, separated only by the fence from the place where his poultry once was, and he could not move any farther. For in the place where, years ago, his fowls would have gathered, their voices melding in squawks, was now a little crowd of children who were jointly reciting a poem. This opened a sudden hole in the shield of his spirit, large enough for the dart of hatred to again penetrate and shatter the peace that had held him together beyond all comforting. It broke him, Agujiegbe. He bent, one hand on his thigh, one elbow against the wall, and wept.
When he emerged again from behind the school fence, his enemy was waiting for him. The same man whom, for more than one week, he'd loved with half his heart—the only part capable of such a feat. For the other half was dead, a permanently tranquilized flesh. The man came with a frown on his face, but when he saw my host, his countenance slumped even deeper.
"What is it, brother?"
"Tell me what they said," he said, without so much as a glance at the man's face.
"Okay. The person who now runs the school says there was no way they can move from this land. The man they bought the land from has moved to Abuja. The school is doing well here and the government recognizes it. The land is not open for negotiation. What took so long was because I was waiting for him to finish a meeting. A long meeting, my brother."
He did not say anything but drove in silence until they reached his apartment. Rather, he communicated with the voice of his conscience, that reticent being in the other compartment that was his soul. Chukwu, whenever I'm in a host and the voice of his conscience dialogues with the voice of his mind, I listen closely because I have come to know that the best decisions a man makes come when both voices agree.
—You are full of hatred again, Nonso. Remember he has done nothing to you now.
—Nonsense! How can a reasonable person even say that? Look at the land, my compound, the house of my father!
—Put your voice low. Calm yourself. A man who whispers too loudly will be heard from a distance.
—I don't care!
—You promised never to hold anything against him anymore. You said you have forgiven him. He asked if you wanted to be his friend, and you said yes. After he gave you your money back, you could have said no and he would have left and let you alone. You even prayed to his God and went to his church with him. Now you hate him again. Now you are plotting to harm him again. Look, just look: a knife lies on the floor of your imagination, stained with his blood. Is this good? Is it?
—You don't understand how much evil this man has done to me. Keep quiet! You don't understand a single thing!
—That's not true, Nonso. It is not me but you who is weak and in need of understanding. What has he done? For the past two weeks he has helped you, done everything you have asked him, as if he were your slave. He has spent most of the time with you, done everything for you. How much did you get from the six thousand euros he gave you? 1.4 million naira. One hundred thousand more than he took from you four years ago. Yet he has nothing. Look at him—are these not the same clothes that he wears every day? You have been to his apartment, a face-me-I-face-you apartment. It has one window, and it is an old one made of wood. Sometimes, when he sleeps at night, he can hear termites scavenging within its walls. If he were not a truly changed man, would he have this kind of money and endure near poverty so that he could mend that which was broken?
The voice in his head did not respond now.
—Answer me. Do you keep silent now?
He said nothing. Instead, with a sigh, he steered the car into the compound and parked it.
—I will say no more to you. Count your teeth with your tongue. Count your teeth with your tongue, Chinonso!
The dialogue with his conscience seemed to have borne fruit, for it seemed that his anger had dimmed by the time they entered his apartment. While his enemy waited in the sitting room, whispering to himself, he went through the back door to the kitchen in the yard. He took the knife from the cupboard, the image of the fated stabbing still in his mind, but then he put it down again. He stamped his feet into the earthen floor and bunched his fist. "My house, my house," he said. He threw his fist into the air as if his assailant had appeared before him and fell on his knee. "No," he said, "I will not suffer alone. I will not. I don't care what anyone thinks."
— _Ngwa nu, ka o di zie,_ the voice returned to him in a whisper. You may do as you like; I will not say any more to you.
He went back into the apartment, the anguish visible on his face.
"My brother, what is it?" Jamike asked.
He merely gave the man a look. From the crate of Fanta under the bed, he brought out two bottles.
"I am getting us something to drink. Wait here."
He went to the kitchen and set both drinks on the table. Then he closed the kitchen door. He threw a portion of the first uncapped bottle into an empty bucket on the floor and unzipped himself. He held the bottle over the bucket and urinated until it frothed over. Then he let the rest into the bucket. When he was done, he put back the cap on the Fanta and, with a tip of his finger on the cap, shook the drink to mix it together. He then placed the bottle on the table beside the other one.
Egbunu, I was horrified, even before the act began, for I had seen the intent of his heart. But I could do nothing at this point. I have come to understand that the most persuasive voice of caution a man can hear before any action is that of his conscience. Should he not be persuaded by that, not even a gathering of all his ancestors living in Alandiichie can change his mind. For the conscience is your voice, Chukwu—the voice of God in the heart of a man. Compared to the conscience of a man, the voice of his chi, of fellow man, of an agwu, or even of an ancestor is nothing.
When he stepped out to drain the bucket into the gutter behind the kitchen, it occurred to him that the bottle might carry the smell of urine. So he turned to the sink and washed it with water from another container, his finger firmly on the cap. Then he wiped the bottle with a rag and took it into the living room. He placed the bottle on the center of the table before him and said to this man, "Take and drink." And the man to whom he had offered the drink took it, gave him thanks, and drank. The hated man drank it with a light twitch on his face, and then a bemused gaze. My host watched him drink without a word and drank until the bottle emptied. Then he set it down by his feet and said to the man who hated him, "Thank you, brother."
IJANGO-IJANGO, that night, Jamike's chi projected through the ceiling as through a rip in the aperture of time, into the room.
—Son of the morning light, it said to me. My host has atoned for what he has done.
But, Chukwu, I was displeased. I told it about the full extent of my host's suffering and how I had not done much to prevent it. I told about how I had gone to the caves to look for it or any word but failed. The chi listened with a silence and sobriety that struck me with awe.
—The great fathers say that when a child who does not know his left from his right tells a damaging lie, he can be forgiven by both the living and the dead. But when an elder tells such a lie, even his ancestors will curse him. Your host is receiving what he deserves.
—The great fathers say that an old woman often feels uncomfortable when she hears a proverb in which dry bones are mentioned. I'm guilty of all that you have said. But still I ask that you recall that a man who insists on breaking the bones of those who offend him in the slightest way will become crippled before long. With these words, it went on to plead with me to restrain my host. I will not relate all that it said, but I will emphasize that it exhibited the new character of its host and assured me of its host's repentance. But there was also something it said that moved me: Jamike was not a bad person at first. He was made so by people, including my host. The chi related the incident even my host had recalled in Cyprus in which, while in primary school, my host and his friends had repeatedly mocked and shamed Jamike, calling him Nwaagbo for having big breasts. It was these things, the chi said, that had caused him to begin trying to control others, to assert himself, in hopes that he would heal himself by so doing. I believed it and resolved to persuade my host even more strongly to forgive Jamike.
OSEBURUWA, if a man dwells in the debris field of revenge for too long, he might step on something—the blade of some weapon, anything—that could injure him. For the field is a wasteland filled with an assortment of things, and one cannot always know what he will find in it. Indeed, I must say that my host had stepped on something in the wasteland that bruised his feet. He became ashamed of what he had done to Jamike. He was convinced that Jamike had known what was in the bottle but went ahead and drank it anyway. Why he could not tell. Was it out of fear? Was it out of reverence? But it disturbed him greatly that a man would knowingly drink another man's urine—no matter what the man had done. He resolved that this was the furthest he would go with his revenge. That thing Jamike had done was the ultimate act of atonement, enough to pay for the loss of the woman he loved, the penis that violated him, the loss of his father's house. He swore to never again lift a finger against Jamike.
So rather than do harm to Jamike, he would not see him anymore. Agujiegbe, if, for instance, he remembered the event in the prison or the beating at Fiona's house or any of the things that sent him into a murderous rage, and Jamike was not nearby, he would vent and the anger would leave him. He could wail, he could hit his wall, or his furniture, or threaten to harm himself, but at least he would no longer lay his hand on a man who was contrite, who was truly sorry for what he had done—a transformed man who had returned that which he had stolen from him.
So when he told Jamike he no longer wanted to see him, he did not give these reasons for why, only that he did not want to.
"I will respect your desire," the other, visibly troubled, said. "But, my brother, son of the living God, I want to be your friend. I will miss you. But I will not do what you don't want me to do. Believe me. No longer will I come to your apartment, or to your shop. I will not call you as you have requested, unless it is urgent. And even then I will text you, first, I promise. But oh my brother, Chinonso-Solomon, my bosom friend, I pray for you. I pray for you. But I will do as you have requested. Yes, indeed, no longer will I look for you! No longer will I knock on your door! God bless you, my brother, God bless you!"
That was it—a protest, an acclamation, an acceptance, a prayer, a lamentation, an argument, a plea, a plea yet again, another protestation, a plea, an acceptance, and then submission. And no longer did Jamike contact him. For nearly three weeks, Egbunu, my host lived by himself, in his improved state, enticed by the things he had abandoned. He came to understand how much his life had changed in the time he'd spent with this man whom he now sometimes called by his nickname: M.O.G., or Man of God, a man so unlike his former self that he sometimes wondered if the previous version had existed. Even the way Jamike now spoke, refusing to call him by the childhood nickname Bobo Solo and never using the word "mehn," was different. Were he not a living witness of the old Jamike's atrocities, he would not have believed them to be true.
He missed Jamike's friendship and came very close, several times, to breaking the embargo in the third week, when he'd taken ill. Oseburuwa, the sick man—he is one whose body has been overpowered by some malady. The change in his body begins with the feeling of something out of the ordinary happening. As pain spreads through the body, of the fever toll in one's skull, emotions erupt—a nervousness first. One becomes nervous about the day, about its course, and about life itself. Then some form of anxiety sets its inchoate machinery in motion. Has the day broken? Will it get worse? Will the world continue without me? How long, how far, to what extent will this illness persist? The anxiety overcomes one. But these are not the only things. Afterwards comes the astonishment that sickness brings, the way it takes ownership of the body and dictates which parts of one's body one must pay to caress or heal. But of the utmost significance is how it initiates the belief in the sick man that he may have caused the illness by himself. Something he has done is the reason why this fever torments his head. If he coughs or sneezes, it must be because of that time when he stayed out in the rain. If he defecates frequently, it must be that bad food he ate the previous night. Sickness, then, becomes the quiet snake that, dislodged from its peaceful abode, is filled with spite and fury. And the sickness it inflicts on a person, now, is its sanctified revenge.
My host had started to recover and was seated in his room when his phone rang on the fourth market day of that third week, which the White Man refers to as Thursday. Ijango-ijango, my host was in his apartment, cleaning a bucket in which he would keep feed at his store, when his phone rang. He picked it up and saw that it was Jamike. He ignored it at first, fearing that he had not yet completely forgiven the man and that if he saw him the rage would possess him again, and he would do things he did not want to do. He went on cleaning congealed mash from the bucket and whistling to the soft song Ndali had taught him. Jamike called again, and again, and then sent a text: Brother pick the call. It is gud news! Parise God!
His heart skipped. He sat down on the bed and pressed the key.
"Hello, my brother Solomon," the other man said, his voice bearing a certain haste to it. "I have found her!"
He sprang up to his feet. "What? What?" he said, but the other did not seem to hear him.
"Praise God, brother," Jamike kept saying. "I have found her!"
"M.O.G., who, what have you found?"
"Who else, my brother? Who else? The one you have been looking for. Ndali!"
He stared at the phone, unable to speak. Again, it has come: that which silences him and deprives him of words, the freest of all human gifts. It has come, its feet assured, as it always has.
"I cannot thank God enough, Nwannem. God is indeed God. He is helping me fulfill my promises to you, all the things on your list. Now you will finally experience the peace I have experienced. You will get forgiveness from her from whom you must get and give it. And ye will be healed."
Indeed, he would be healed.
"Where is she?" was all he managed to say.
"I saw her at Cameroon Street. You know the new pharmacy and laboratory they are building there? The two-story?"
He knew.
"It is there. She is the owner of the place. She has returned to establish it. This is answers to our prayers, Brother Solomon!"
Jamike had gone on, thanking the alusi of the White Man, quoting the books of Corinthians, James, Isaiah, and Romans while the firmament of my host's thoughts constellated with fire. He told his friend to let him rest a bit and return the call later, and the other obliged. He put away the phone, entranced in the new knowledge. A great silence came upon him, one so overpowering that he could not hear the faintest breath. But it was a deceptive silence, for he knew that in that moment an army was approaching, the sound of their marching feet thundering through the land. And that soon they—the thousands of thoughts, imaginations, memories, visions of her—would arrive, illimitably vast across the wrinkled face of time. So he lay down as one merely waiting, as still as a dead hen stiffened by rigor mortis.
#
## Oblivion
MMALITENAOGWUGWU, the old fathers say that if a secret is kept for too long, even the deaf will come to hear of it. It is true, also, what the wisest amongst the great fathers, the dibias, those who are second to you, Chukwu, say that if one seeks something one does not have, no matter how elusive that thing is, if his feet do not restrain him from chasing it, he will eventually have it. I have seen it many times.
My host's feet had chased after this great, elusive thing, this thing which had escaped from the leash he had bound to his heart, for more than four years. And that evening, an hour or so after Jamike rushed to his house, he became certain he had found it.
"So it is true that it is her you saw?"
"It is true, my brother. Why would I lie? Remember I promised that I would do everything I could to make sure you recover everything—everything. Er, my brother, one day it came to my mind to check Facebook. Because of my past life I had stopped using my own. So I decided to open it again."
"Is it email?" my host said.
"No, Facebook. I will show you when next we go to the cyber cafe. But I went there and searched for her, and _lo and behold,_ I found her."
"Ha, is this so?"
"Yes, my brother Solomon. Ndali Obialor. I saw her face—she is fair in complexion with a very beautiful face. She had a black weave on her head. I sent her a friend request and she accepted just today."
At this, Jamike clapped his hands. My host, at a loss as to what he was hearing, nodded and said, "Go on."
"Immediately when I went to the cyber cafe, I opened it and saw she had posted a photo of the new pharmacy near the big supermarket on Cameroon Street."
"Is it true," he said, as if the other had not spoken at all, "that you found her?"
"It is true, Solomon. She is the one I saw. It is her I saw. She in that photo whose half you covered and showed me."
"What if it is someone who looks like her?"
"No, it is not. After I left the cyber cafe, I went up to the chemist and asked one of the workers there. And the woman said that indeed it was Ndali."
"Are you sure she is the one you saw? I will show you the photo again... here, I have covered the chest with a paper. Look in her face, look at it very well."
"I have looked, my brother."
"And you say she is the same person you saw?"
"Yes, she is."
"Same nose... look, Jamike, look very well: is it the same eyes?"
"Indeed, my brother. Why will I lie to you, my brother?"
"Then it must be her," he said with resignation.
Ijango-ijango, for two days they had this kind of discussion in his apartment. And at the end of each turn, my host would pace about the room with an exuberant heartbeat. He would halt, bend briefly, peer into the face of the world, close his eyes, and shake his head at the displeasure of what he'd seen. He was still sick, his spirit humbled within his flesh. But he was a man who had heard too much. And this too much was enough to shatter a man. Too much was the fact that he knew Ndali was now certainly in Umuahia. Too much was also the fact that he knew he must go to her.
"I don't understand what is happening to you, my brother," Jamike said one evening. "You have been wanting to see this woman for many years, you have lived for this. And now you shut your door to it? You do not want to see her?"
They were seated on stools outside Jamike's room for fresh air. The vicinity was quiet, except for the voice of a transistor radio in one of the rooms and the sound of crickets.
"You don't have to understand," he said. "The elders say it is not everything the palm-wine tapper sees on the tree that he speaks about."
"True, but don't forget that the same elders who say that also say that no matter how long a mangrove branch lies beneath the water, it cannot become a crocodile."
Agbatta-Alumalu, Jamike was right. My host had been confused. It was as if he had been waiting for this thing, and now that it had come, he realizes he has no power, no strength to confront it. So he did not respond to the wise words of his friend. He moved the toothpick between his teeth, up to the ridge above his two upper front teeth, and spat specks of meat onto the ground before him.
"I know how you feel," Jamike said. "You are afraid, my brother. You are afraid about what you will find out about her." He shook his head. "You are afraid of what you will find, that you may have been wasting everything loving a woman who can never be yours again."
My host glanced up at Jamike and was, in that instant, filled with rage. But he fought it.
"I know I caused all this, but please, brother, you need to face her, no matter what. It is the only way you will heal and move on with your life and find another woman." Jamike moved his chair to face him, and as if feeling that my host did not understand what he'd said, he turned briefly to the language of the White Man. "It is the only way."
He looked at Jamike, for the very thought of another woman hurt him.
"At least you should let me deliver the letter to her, or I could go and tell her everything that happened—what I did, what you did—and ask for forgiveness. It is the only way. You must see it."
"What if I discover she is married and no longer loves me?" he said. "Will it not be worse than not knowing? In fact, I don't like that she has returned. It would have been better if she had not returned."
"Why, Brother Solomon?"
"Because," he said, and then paused to let his thoughts fully form. "Because I cannot accept to lose her." Then, as an afterthought, taking advantage of the perplexed silence of his friend, he added, "After all I have suffered for her sake."
It was those words, out of everything they said that day, that lingered in his mind after he drove his car back to his apartment and lay on his bed, which still had the malarial smell of his sick days. Chukwu, in my many cycles of existence, I have come to understand that there are times when, although one might have thought about something many times before, hearing it said again imbues it with new meanings strong enough to lend it the appearance of novelty. I have seen it many times. In all those years, he had not thought the way it struck him that night—that all he'd been through had been because of her. He considered it, his story, in wilting chronology: he had been mourning the death of his father when he met her on the bridge. It was from there that his life began to fall in the direction it now was going. It was for her sake that he sold all he had, went to Cyprus, and ended up in prison.
He sat up, near midnight, weighed down by heavy thoughts. He reckoned that without her, none of this would have happened to him. "It doesn't matter," he said aloud to himself. Ndali now had no choice but to return to him. He let his inflated chest settle so he could breathe easy. "I have paid enough price to deserve her, enough. And, no one, I repeat, no one can take her away from me!"
He would go to her in the morning. Nothing would stop him. He picked up his phone and texted his friend, then sat back, panting, as if exhausted by what he had decided to do.
IKEDIORA, the brave fathers were at their most instinctive when they said that a person often becomes the chi of another. It is true. I have seen it many times. A man may be in grave danger, and there might be nothing his chi can do to help him. But he may meet a person who, having seen the danger ahead, tells him about it, saving his life. I once met a chi in Ngodo who was chattering endlessly with bitterness about the evil on the earth and the unworthiness of humans to exist. There were a lot of guardian spirits in the cave, most of them silent, lying in a corner of the great granite chamber or washing by the pool or conversing in low tones. But this guardian spirit kept shouting about how his deceased host had tipped off a potential victim about the plot to kill him. Later, the person whose life he saved sent people to murder him. Oh, man is disgusting beyond grave worms! Oh, man is terrible beyond a dirge! I don't want to return to the earth of man! It had been a perplexing thing to watch this rebellious spirit speak such profane things. I left it there by Ngodo but heard from another guardian spirit that it had refused to return to the earth and that you cursed it and turned it into an ajoonmuo. And now it crawls interminably down the length and breadth of Benmuo with three heads and the torso of a vile beast. But what Jamike had done for my host was the reverse of what this chi had described. For Jamike had become my host's second chi and had led him to what he had been looking for so many years.
He went with Jamike to look for Ndali, carrying a jar of fear in his heart, wearing a cap over his head and dark eyeglasses covering most of his face. When they arrived, he found the pharmacy to be a new building he had seen nestled between the Saint Paul Anglican church and the new MTN office. It was a two-story building that bore the sign HOPE LABORATORY AND PHARMACY. The lettering was bold against the background of a white woman in a white medical frock, peering into a microscope. In front of the building, on one side of its fence, was a pile of sand and pebbles, relics from the construction of the building. He parked the car on the other side of the street, in front of a barbershop from which deafening music blared, mixed with the constant buzz of a power generator.
"You are afraid, brother," Jamike said, shaking his head. "You really love this woman."
He looked at his friend but did not speak. He knew he was acting irrationally but could not tell why this was so. Something in him was preventing him from what he'd sought so desperately.
"The Bible says, _Let not your heart be troubled. Casting all your care upon him; for he careth for you._ Do you believe God that it is possible that she may still love you and be unmarried?"
He gazed at his friend, taken aback by the latter's switch to the White Man's language, the language in which Jamike discussed the Bible. Frightened by the possibility of what the other had spoken about, my host closed his eyes. "I believe."
"Then let us go. Don't fear."
He nodded. _"O di nma."_
They stepped out of the car with his heart fastened into a knot and crossed the crowded street. There were stores everywhere. A shoe store covered with shoes that hung from the awning, strung together with ropes like beads. A store in which pots and cooking wares were sold, GOD'S HANDS COOKING SUPPLYS. As they walked, he tried to anchor his thoughts on the people, on how the streets were different from the ones he'd seen in Cyprus. Jamike went ahead of him, the lilt in his gait from a wound on one of his toes. When they set to cross the road, my host tilted his cap lower to cover his face and balanced the glasses over his eyes. A taxi honked at them in reaction to what the driver may have perceived as a daring crossing. Jamike jumped the litter-filled gutter that separated the pharmacy from the road. Were Ndali to gaze in that moment from one of the shiny new screened windows of the pharmacy, she might have seen them. My host tilted his cap even lower, and grabbed his friend's arm.
"I can't, I can't go in," he said.
"But why?"
Again he adjusted his cap and sunglasses.
"Ah, what are you doing?" Jamike said.
"I am changed a lot," he whispered back. "See my face. See the long scar on it. See my mouth: three teeth missing, the long scar around my jaw from where it has been stitched. Look at the permanent swelling on my upper lip. I am too ugly now, Jamike, I look like a baboon. I want to cover them."
His friend was about to speak, but he held him more tightly.
"She won't recognize me. She won't."
"But my brother, I don't agree," Jamike said with what may have sounded like agitation. He looked at the pharmacy, then at his friend.
"Why not? How can she recognize me looking like this?"
"No, brother. She cannot dislike you because of your wounds."
"Are you sure?"
"Yes. Love doesn't work like that."
"So you think she will still be attracted to me, with my face like this?"
"Yes. All she needs to know is why you left and disappeared."
He was slightly fidgeting, looking up about him as he spoke. Egbunu, this was my host: a man who, when afraid of uncertainty, often propels himself towards internal defeat. And when this happens, when his spirit has been thrown on the ground in the wrestling bout, his defeat begins to manifest in the physical. It is a strange thing, but I have seen it many times.
Jamike wiped the sweat from his brow and started to speak again but stopped abruptly and tapped my host, wanting him to look in the direction of the pharmacy.
It is difficult to describe this moment: the one in which my host, who had suffered so much, beheld the woman for whom he would have done all that again. She had come out of the door of the clinic. She was slightly changed, weightier than the slender woman whose image he carried in his head all these years. She was dressed in a long white cloth that reminded him of the nurse in Cyprus. From her breast pocket, a pen stuck out, and on her upper chest, visible in the opening of the collar, was a necklace. He stood watching her, taking an inventory of everything around her. She was talking to a woman with two kids—one strapped to her back, the other stretching its hand towards Ndali, then taking it back. She would stop to try to catch the child's hand, but it would retract it, laugh, and turn to its mother.
"I told you it was her," Jamike said as the other woman turned back and began walking past the parked cars, out to the street, and Ndali returned into the pharmacy.
"It is true," he said. "It is her." His heart was beating now, as if to the moves of ogene music. "It is true, Jamike, it is her."
It was indeed her, Egbunu. Ndali—the same woman whose chi had confronted me when I went to entreat it on behalf of my host. It struck me then, in a way I had not considered all these years, that it might be that her chi may have carried out its threat to cut its host away from my host forever.
"Then let us go in. I am not returning without seeing her, brother. I want you to heal, to be well, and to be filled with the joy of the Holy Spirit. You must do this. You must take courage. If you don't, I will go alone into that pharmacy and see her. And talk to her for you."
"Wait! My God, Jamike!"
He held Jamike again and saw in the man's eyes something that gave him hope.
"All right, I will come," he said. "But see, let us take it slowly. I can just look at her now. Then, maybe another time, I will talk to her?"
Jamike considered the suggestion with a slanting, informed smile that made his forehead bow against the lower side of his face.
"Okay, let us go then, Nwannem."
He walked with trepidation, slowed by the wealth of his anxiety, until, Jamike leading, they entered the pharmacy. It was a big room with many glass windows, so the place was showered with light. Ceiling fans, loud in their swinging, supplied extra air. He sat quickly in one of the six plastic chairs facing the counter, a big wooden structure that concealed half the pharmacists. It was on this that he placed his eyes after exchanging muffled greetings with the man who sat next to him in one of the chairs, shaking his legs.
Ndali was attending to someone when they entered. Although it was the other woman who called to them, "Next customer," he heard her voice.
Jamike did not respond immediately but stood by his chair, his eyes on the counter. My host beckoned to Jamike, and the latter bent to hear him.
"You know, you know—I just came to look at her," he whispered into Jamike's ear.
His friend nodded uncomfortably, gesturing at the pharmacist to wait a moment.
"Just tell her you want a drug for malaria for me."
Jamike nodded.
He watched Ndali from where he sat, his cap pulled over his face and his eyes hidden behind the sunglasses. She seemed to him more beautiful than she had been before. How old was she now? Twenty-seven? Twenty-nine? Thirty? He could not recall exactly what year she was born. Now she looked like a woman who had entered her prime. Her hair was permed, slick, and serenaded down her shoulders. There seemed to be a change to every part of her body, down to even the very shape of her face. Her lips were fuller, this time wearing a deeper pink color than he could ever recall. He'd gazed for hours that morning at the images of her, the images that now increasingly supplied him with pleasure. Yet the face before him was slightly changed. What he could best say was that it seemed she'd been sent back to her creator for renovation and returned even better.
The other woman was starting to put the drugs in a small polythene bag when Ndali opened the small door and stepped out from behind the counter. He noticed that her breasts seemed bigger, although he could not see their full size behind her clothing. He had the chance to see her posterior, almost as he could remember it. He stared at it with all his powers of concentration until she disappeared into an office whose door, closing behind her, bore the inscription NDALI ENOKA, MSC. PHARMACY. He did not see her for the rest of the time they spent there. The nurse attended to Jamike, and they left with the malaria drugs.
AGUJIEGBE, when a man cultivates a great and ambitious expectation, and when that expectation comes to fruition, it usually confounds him. A man may have told his friends, "Look, look, my brother's home in the far city is big. He is a rich man." But that selfsame man soon goes to the city and discovers that his brother is nothing but a street sweeper, scraping to get by. But so great had been his expectation, so long had it been sustained, that at first he will doubt reality's uncontestable truth, shattered though he may be. I have seen it many times. This was the case with my host. The reality of Ndali's marriage, signified by her change of name and the ring Jamike was convinced he'd seen on her left finger, confounded him. It put out the light from his universe and left him in a world of unblemished darkness. He stood afterwards at the entrance to Jamike's church, so deeply rattled that the sound of his heartbeat came to him as whiplash.
"I believe that she still loves me, despite all."
"My brother, I understand thee," Jamike said in the language of the White Man, in the way he always did when they went to church or soon after they had been to church, as if the language of the fathers was too unholy to be spoken on such grounds.
"Please speak Igbo, this is serious issue," my host said in the language of the great fathers.
"Sorry, Nnam, sorry o. But it is what it is now. As I have been saying, just give the letter into her hand; put it on the center of her hand. That is all. Then you can go, and God will see you have done your part."
He shook his head, not because he believed it but because Jamike did not understand it all. He wanted Jamike to go inside for his service and leave him to ponder things, so he said, "I understand. I will wait for you here."
Jamike went in to see the two others with whom he was setting up the special gospel event that evening: a screening of a Christian movie about Jisos Kraist. My host sat on a lone block, one of the remnants of the construction of the church building, only a year ago. A soft wind was blowing, and the banner, a piece of cloth fastened to two wooden poles rigged into the ground, was flapping from the traveling wind. He gazed at the congested street, where men and their wares struggled with motor vehicles and wheelbarrows for space. As he looked, he thought about all the things he had seen and those hidden from him. Did she have children? How long ago did she marry? Was it yesterday or a year ago? Could it have been the same month—or even week—that he arrived in Nigeria a damaged man? It could even be, if things were to follow the usual pattern of life mocking him, the same day. The thought ignited: he stepping down from the plane onto the tarmac of the ramshackle airport in Abuja, she stepping up to the altar with her groom. He imagined the priest looking at her and her husband, asking them if they would be together in sickness and in health, till death. At the same moment the shell of what he once was was falling at the feet of his waiting uncle at the airport.
He considered the things he had seen: Ndali, alive, well, and a more beautiful woman. Had Jamike not appeared in his life, sent like a stone from an unseen enemy to crush him, he would have married her. They would have continued to live on his compound, alive in the midst of his birds, harvesting eggs in the mornings and waking up to the orchestral song of roosters and winged things at dawn. His joy would have been abundant. But he'd been robbed of it all. As mosquitoes buzzed around him, and the voices inside the church reached him in whispers, anger welled within him.
He stamped to his feet and began searching about for a weapon. He found a stick lying near the church's generator and picked it up. He motioned towards the church like a madman, and he'd almost reached the door when he stopped. Egbunu, his conscience had reacted, and a stream of light had pierced the sudden darkness into which his mind had precipitously plunged. He dropped the stick and sat back on the block. He put his hands on his face, gnashing his teeth. Moments later, as he allowed himself to calm down, he felt something on his face moving around down his cheek. It was an ant that had crawled from the stick to his hand, and then from his hand to his face. He slapped it away.
"My brother, my brother. What happened?" Jamike called to him just then from the threshold.
He rose. "I will go home and be alone," he said.
"Oh, Brother Solomon. I really want you to see this film, _Passion of the Christ_. It will touch your heart. It will touch your soul."
He wanted to speak, to tell this man that only a moment before he'd been filled with hatred towards him. But he did not, for he'd been disarmed again by Jamike's face.
"I will watch," he found himself saying.
"Praise God!"
He sat in the back of the church, torn into shards within, as Jamike and his church members set up the screen for the movie. He sat until the service started. The pastor mounted the stage and talked about salvation, how a man suffered to give his life for others. As the man spoke, he rose up and left the church.
Chukwu, he returned home, struggling to stop himself from falling into fresh despair. He realized, deep in the night, that his point of distress came wholly from his desire to regain that which he had lost. It was not healing and forgiveness that he wanted, not the things Jamike spoke of. Instead, he wanted his life back. He wanted to pick up the coconut that had fallen into the latrine and wash it clean. For he believed it was possible that it could be clean. He sat up, resolved that this was what he wanted, that it could be done. To do anything else was to capitulate.
This incantation of thoughts, having flourished in his mind for so long, formed into a firm decision—that he would fight for her, married or not.
I will not give up, no! he told himself. I have traveled too far to give up. Yes, I repeat it: people's wives are taken from them; so, too, are people's husbands. A man is robbed of his child, and a woman is robbed of her baby. A goose is robbed of its gosling. _Onweghi ihe no na uwa mmadu ji na aka._ Again, I repeat it: nothing in this world belongs firmly to anyone. We own whatever we have because we hold it firmly, because we refuse to let it go. In being here, in standing here, under a roof, I am holding on to my life. If I let it go, it will be taken from me.
In gesturing, his hand clung to his chest. He put on the bulb in his room and went to the mirror.
Tell me, he said, squinting at the image of the changed man who now stood pointing back at him, his face a catalog of scars. Tell me, was my own future not taken from me? Was it not wrested out of my hands by Jamike, Chuka, Mazi Obialor, Fiona, her husband, Cyprus police—and everyone?
He turned away from the mirror and pointed his finger at the wall and gestured like one confronted with something—a thing to be feared.
Did I not try to hold it, my life, but it was taken from me? What of my body? Did I give it to them? Did I? Tell me! Did I say, "Take my buttocks, put your penis in them?" He reached for the stool beside his feet and smashed it to the floor.
Tell me!
He stood now, in the wake of the dismembered furniture, panting, aware of his sudden slip into insanity and that he had shouted at midnight. It shocked him. Shaken, he switched off the bulb in haste and settled himself slowly onto the bed and lay there fearing that he may have woken the people in the other flats. He waited for someone to knock on his door, his eyes on the space below it, where he could see shadows of light. For a good while, he lay there as if bound to the bed, both arms held together to his belly, his head thrust histrionically sideways. But no one came. From somewhere, he heard what seemed to be a church service in full swing and the distant sound of drumming and music. In the serenity, it settled on him that he must return to that place where half of him never once left. It is in returning that he would regain his peace, and it would be there that he would fight his greatest battle.
#
## The Ancient Tale
ECHETAOBIESIKE, I have said already that man is limited in his capabilities. I say this because, as I will now tell you, my host would have done things differently if he had more capabilities. But this is not to say that his strength is unlike every man's—no. You have not denied him anything that you have given to others. I went with him into Afiaoke and the garden of Chiokike to pick talents and gifts which, in your generosity, you had sought to bestow on him, as you do on every human being. But still, he remained limited. Like everyone else, he is constrained by nature and time. Therefore, there are things that, once one has done them, cannot be undone. All one can do, if one cannot change a circumstance, is give up and move forward, in another direction.
Ebubedike, this wisdom came back to my host six weeks after he saw Ndali again. Because I do not want to take much more time in this luminous court, and because I must render only the details that can in some way lead to the conclusion of the matter about which I've come before you, I must let this man, Jamike, speak. For he'd seen that, from the day my host saw the woman he loved again, he fell into a turmoil. He was no longer himself. He was unable to move forward or backward.
"Brother, you have done what you could do. You have gone over and beyond and must now stop. I tell you because I love you with the love of Christ, Ezinwannem, that you must put all of this behind you and move forward. I am telling you, this is the best thing you can do for yourself."
They had by this time been best friends for the past two months. They sat now in my host's poultry-feed store. In the months since my host opened it, the store had grown to accommodate bags of feed, fertilizers, and other agricultural products. Rows of wood had been nailed to the wall, and arranged on them were cans of items related to poultry. A calendar from the Abia State Ministry of Agriculture hung on the wall, open to the page on which "the Last Pioneer," my host, stood in front of his store squinting into the camera. It is the first picture of him that had been taken since his face was reshaped by the violence in Cyprus—the deep scars on his forehead and on his jaw, his missing teeth.
But Chukwu, I must let his friend speak:
"Let me remind you what you have done, that you have done a lot. After I found her for you, you and I went in search for her. At first, for a long time after we saw her, you went without wanting to reveal yourself to her. As a man whose heart was still filled with love, you did not want to have it destroyed by finding out that the one for whom you'd stored up this vast wealth of love no longer has an ounce of reciprocity left in her.
"Yet even though you had these fears, you did not give up. One day, five weeks ago, you took your chance. I was there with you, Nwannem Solomon. I saw every moment of it. You appeared before her undisguised, at her pharmacy. You took your chance. It was well planned. We went when we thought there was just she and one of her staff there. Of course, we did not know that two of her friends were seated in her office, whose door was opened. Perhaps, as I have said to you so many times already, it must have been because of these people that she reacted that way. When she saw you, the man she truly loved, whom she had vowed she would never leave nor forsake, she was afraid. It was not told to me in a story, nor did I dream these things. I saw it with my two eyes. With my eyes, I saw her hands tremble. The small rubber bottle she had in her hand, on the body of which she was scribbling something, fell as she gasped 'Argh!' then clutched at her heart.
"I saw it, Solomon my brother. It was as if she had seen a ghost in daylight. You could tell that she thought you were either dead or never coming back to Nigeria. You stood there, my brother, calling her name, saying it was you who had returned. Your hands were opened in front of the counter. But she gasped and screamed in terror, and her friends rushed out of the office to see what had happened, and her staff who was cleaning the medicine-filled shelf turned to her. I am sure it was because of nothing else but these people that she changed, turned from a mouse to a bird in the bat of an eyelid and began to shout at you, 'Who are you? Who are you?' and without waiting for you to even answer, again began shouting, 'I don't know you! I don't know this man!' I am certain that she had recognized you that day."
He stopped because my host was shaking his head and gnashing his teeth.
"You saw it, too. First, there was that unquestioning spark of recognition. If she didn't recognize you, why would she gasp? Why would she tremble? Does one react that way when they suddenly meet someone they do not know? Do you gasp and tremble?"
My host's heart lit with quiet fire, he shook his head even more and said, "M.O.G., I agree. I completely agree with all you have said. This is how it transpired. But I wonder, why did she claim she did not know me? Was it not because of my face?"
At this, his friend put on a countenance whose expression I could not decipher.
"Maybe, Nwannem Solomon," he said. "What you fear might indeed be true, and it may not only be because of those who were there at that time. Her actions were extreme. She was shouting, screaming louder, as you tried to explain yourself to her. At the mention of your name she screamed in English, 'No, no, I don't know you! Leave my office! Leave!' Indeed, such a reaction has more to it. There was undoubtedly a snake hidden in the brush. But you should also know that she may have been afraid. This is a woman who is married. Who—" Perhaps because Jamike knew that these details oppressed his listener, and that what he was about to say would sting him even more, he paused. Then, with eyes out the store's window, where a dazed fly droned up and down the netting behind the louvers, he said, "Has a husband."
In truth, it stung his friend.
"It could be that she is afraid that the man she loved before would destroy her new life. She must have been afraid of you."
He nodded in acceptance, in defeat.
"But you did not stop there. Yes, after we left the pharmacy in disgrace, hectored out by her friends, she ran out of the pharmacy in tears through a back door. And for some time it weighed you down, my friend. You were ashamed, humiliated, knocked down by this. It wasn't told to me in a story, my brother. I was there. I saw it with my two eyes. If she was rejecting you because of your scarred face, why would she be so moved?"
Ebubedike, his friend had spoken with the frankness of the old fathers and left my host confounded by what he'd heard. He gazed out the window, and his eyes fell on a peddler hawking CDs on a wheelbarrow. The peddler had been stopped by a woman who was running her eyes over a record.
"But one must add, too, that it may also be because she is angry with you," Jamike said suddenly, and again gave his friend that warning look that says, _Steel yourself_. "She may have hated you then because she does not yet know your story. _She is ignorant_."
This, said not in the language of the fathers, was meant to stand out, to punch everything else into the listening ear whose bearer again nodded desperately.
"She does not know what you went through, how you spent one week in hell on arrival in Cyprus because of what I did to you. She didn't know of your anguish. She did not yet know how lost you were because you gave up everything for the sake of love."
He listened to these heavy words that bit at his heart with sharp teeth, nodding sporadically.
"She did not know, as yet, how you paid for it dearly. She did not know how you were humiliated, stripped bare, robbed of everything you ever owned. She did not yet know the pain of such self-sacrifice. Then after, as if all that was not enough, they threw you in prison." Again, Egbunu, he gave my host the searing gaze. "I will not say more, Nwannem, for there are no words one might use to describe what you went through there that will not scald one's tongue. None. But this is what I mean: she had not yet any knowledge of these things. She had not yet read the letter."
His eyes were fixed on Jamike, who pulled a handkerchief from the pocket of his plain trousers. He tucked the pocket, which had turned itself out along with the kerchief, back inside and wiped his forehead.
"Yes, she did not know these things before, but then after you gave her the letter and it was only a few days after you made yourself known to her. I remember that day. We had come up with a plan. So we found a man to act as a courier who delivered the letter, with unmarked stamps, to her address with her full name. It was successful. Tokunbo said he went out of the pharmacy after handing her the letter, and then through the window, he saw her open the letter and begin reading it. You and I had rejoiced. For me, that was enough. You got her to understand you were not the kind of man she assumed you were, to realize that you fought hard to have her back. You did not merely go abroad and vanish. You did not even merely give in to oppression but were valiant in the face of it. You proved there that you loved her, and that not once in all those years—despite all that you faced—did you forget her. You woke every morning and imagined her in the same room as you, and said to her, often, 'I will return to you. I will return to you.' These were the words that gave you life in those painful years. You said there what you said every day to that conjured-up presence of hers you felt in your cell. For. Four. Good. Years. Four good years, blessed Brother Solomon."
My host was nodding, his eyes vacant, as if the other were speaking words strong enough to overpower all his senses.
"In your letter, which you had delivered to her, you described how this happened to you, how you survived those years. You said it was like a battle—"
The word _battle_ hung on his friend's tongue like a fish on a hook, because two men dressed in blue aprons entered the store at that moment. On their clothes was the inscription MICHAEL OKPARA UNIVERSITY OF AGRICULTURE, UMUDIKE.
He knew them.
"Oga Falconer abi na fowler," one of the men said, removing his cap.
"Ah, university people, una don come?" he said.
"Yes, oh, Na professor send us come."
He shook hands with them. They shook hands with his friend.
"Wetin him want?" he said.
"Layers," one of the men said. "Half bag. Also, him say make you add one bowl of boiler."
"La-yers, ah layers," he said, a finger on his lips as he glanced around the store. "E be like say we no get am again. Wait."
He pushed open a door to the other room, a small storage area that stank from the silos and bags of poultry feed stored in it. He looked amongst the silos, which were full of corn and placed on wooden slabs, their mouths opened to let in air, and the jute bags of millet, which were stacked one on the other.
"We no get am. E don finish," he said when he returned back into the store, his hands white from turning over sacks and bags.
"Ah!" one of the men said.
"But broilers dey yan-fun yan-fun. Him no wan millet?"
"No, we over get am," the man said. "Okay, just bring the boilers." And upon consulting in whispers with his colleague, said, "Two mudus."
"Okay, sir," he shouted from inside the storage room.
He came back into the store with a metal bowl and a black polythene bag which he unfurled to open widely so that its inside swelled in expectation. Counting, one, he scooped a handful of the gray-colored mash into the bag. He found something that looked like a raffia broom in it and, removing it, threw it out the door. He scooped another bowlful and poured it into the bag. Then, looking up at the men, he scooped a handful with his hand and threw it into the bag.
"Na jara be that," he said.
"You do well," the men said.
He shook their hands and thanked them.
GAGANAOGWU, after the men had paid and left the store, my host sat down with Jamike and asked him to continue what he was saying. The other, who had started to gaze into his big Bible while my host served the feed buyers, closed the book and put it on the upturned megaphone on the floor. Then Jamike bent over so that his elbows rested on his thighs and continued.
"I was saying that if she has indeed read your letter she must have seen all this by now."
Although Jamike had spoken without the oratory of the fathers, his words carried the hypnotic power of their tongue. For my host had received his words the way an ancient story slowly crowds the mind, like embers from dying coals. Afterwards, while Jamike left to do some evangelism, carrying his Bible and megaphone, he sat digesting the things Jamike had said, trying to let them soothe his spirit. He regained all the confidence he had lost. He went to Mr. Biggs, the restaurant she had introduced him to, and had a meal. He sat in the far corner of the restaurant, where he had sat with her, only now on a new chair and table. Then he went to an electronics store down the street and bought a used television set while Jamike went to his church. He was prepared for the time when they eventually would begin to meet again, so she would not mock him for not owning a television.
Although my host sought it, from that day onwards, Jamike did not talk about Ndali. He was convinced that she would either call the number scribbled at the end of the letter or post a letter to the address on the envelope. My host, too, believed this. It consumed him. He went about his life unbalanced, thinking perpetually about what she would do or what she would not do. He would sometimes seek desperately to be free for a moment, to think about the stampede at the Ascension Crusade rally or the impending activities of MASSOB which Elochukwu—with whom he was no longer close—had told him about and which could flare into a riot in the city. He would thrust out all these and imagine, instead, that Ndali had read his letter and wanted to meet him or that she read it and did not believe a word of it. Perhaps she simply thought it was impossible that all that could have happened; perhaps he was making it all up. Or maybe she read a bit of it and tore it up, never seeing the rest. Or perhaps she may not have read it at all. Maybe she tore it up and the courier saw her reading something else and mistook it for his letter. Let us even say she read it. Really, let us assume beyond reason that she read it and thought it was all true, but that it was now too late. She was now married, inseparable from that man. They had become one, nothing can put them asunder. Nothing. The man has slept with her for years, _every day,_ far more than he ever had. It was too late, too late, too late.
These uncertainties, these fears strained his mind so much that he became sick from pondering what she may have done with the letter. On the night of the fourth day after Jamike's long speech, he became so sick and weak that he did not rise from his bed. The rain did not help, either; it had rained so hard, rapping continuously against the roof of his apartment, that it kept him awake far into the morning. Thunder clapped a few times and I rushed out to see it. It was the young kind, the kind Amandioha used as a weapon. In its aftermath, lightning struck the face of the horizon, shaped like thin branches of phosphorescent trees. The rumbling in the bowel of the sky was so loud that it morphed from sound into invisible object: a spark of teeth-white light. By morning, the volume of rain had become so enormous that it seemed there was some kind of movement in the land, as if the world had become reduced to an ark in which everything—man, beast, birds, trees, buildings—was crammed and was floating towards some shore.
He did not leave the house for most of the day but lay in bed tormented by the thought of the loss of Ndali. Between thinking and imagining, vivid things emerged in his head. He would rise and walk about his room. He would gaze at himself, his face, his mouth, in the mirror. He would nurse a certain memory of Ndali, now blurred, dulled by time, of them making love. Then he would think of the new man in the same position. And it would kill him. An image of wishful violence would jump out into his field of vision like a beast and howl into his rankled head.
Oseburuwa, I did not know what to say to him in this time. In the years before he saw her again, I always told him to have faith like the white man of ancient times, Odysseus, in the tale he loved as a kid. In that tale, the man had been stopped from returning to his wife by an angry god. I would have kept mentioning this story to him, if the man did not eventually reunite with his wife. I could not remind him because his woman had yielded to another man. I feared that to remind him now would instead fill him with a sense of failure. I did not know how, at all, to help him. I knew it was futile to try to discourage him from loving her, and I could only give suggestions. His will was sealed. There was more to what he now felt. It was not only love, it was not only that he wanted her back, it was also that her rejection of him made him feel his suffering had been futile. He wanted her to acknowledge, to make a concession towards him, a man who has been damaged for her sake.
The hands of the small wall clock without a glass covering on the wall of his room was pointing at 4:00 p.m. when he rose, brushed his teeth, and spat into the gutter that flowed out of the compound. One of his neighbors was in the shared bathroom, the sound of splashing water reaching him, and suds washing up and down the drain. He chewed what was left of the bread he'd bought the previous day, finishing it in two bites. He dressed and walked out of the house.
He saw that the rain had created a fjord outside the compound. Egbunu, although since his prison days I have cut down very drastically on the frequency with which I left the body of my host, I went out that night to see the rain as I had done of the thunder, to wash in it while he was fast asleep. I had spent much of the night there, with a thousand other spirits of all kinds, taking in the empyreal smell of Benmuo. I was confident that because of the storm, no spirit would be going around looking for bodies to inhabit or harm. And now that my host had left his apartment, I had the chance to see the impact of the rain for myself. The clay earth had been softened, so that as he walked his shoes made small ruts on the earth. A house across from the block of flats in which he lived, made of unvarnished adobe bricks, now stood precariously on a shelf of earth.
With the cuffs of his trousers stained with mud water, he arrived near the pharmacy, his face concealed behind his sunglasses. Across the road from the big shoe store, he saw Elochukwu and a group of men dressed mostly in black vests carrying Biafran flags as they walked towards the other side. The MASSOB. They were not protesting, simply walking, some of them with sticks, redirecting traffic. He saw Elochukwu among them, consumed with this agitation. My host shook his head and walked on to the pharmacy.
When he reached a short distance away, he saw that the car he'd identified as Ndali's—the same one she used to drive to his house—was there. As he looked at the car, at the small poster on the back window, he lost all confidence again and began to wonder why he had come. He did not know what to do next. I put in his mind caution—Jamike's words that he should no longer try to meet her on his own. "Don't do it, cha-cha, please, I beg you in the name of Jesus, the son of God. If she is married and says she doesn't want you, then once you have sought forgiveness from her, let her go."
But he could not. Even when he tried to let himself do it, to give it all up, something drew him back. One time, a crushing desire to be reunited with her. And next time, a desire to have his suffering, his sacrifice acknowledged.
He walked on towards the other side of the street, past a group of small fruit hawkers lined up with their wares balanced on rickety tables. Two boys in school uniforms, talking about a pig, walked by him. The bag of one of them was open, dangling from his back. My host stopped at the GSM table a few meters away and sat with the lady on one of the plastic chairs.
"I wan make call," he said.
"Oh," the woman said. "Glo, MTN, Airtel?"
"Emm, Glo."
He dialed Jamike's number with the woman's phone, whose keypads had been cleaned off. Jamike answered in a husky voice. "My brother, we have just finished counseling. Have you closed for the day?"
"Yes," he said. "Can you come? There is something I want to talk to you about."
"Okay, I will come in the evening."
He walked back all the way, stopping to buy a cup of garri and a bag of peeled oranges. While he waited for Jamike to arrive, he rehearsed the idea that had come to him while standing across from Ndali's car. Chukwu, I will let you in on this later. He put it through various iterations until he was confident of its final form, so that when Jamike arrived, he did not mince words.
"You leave in two days for this long prayer, and I will not see you for—how long?"
"Forty days and forty nights. That is the number of days our Lord Jesus Christ fasted and prayed—"
"Okay, forty days," he said bitterly. He glanced around his one room, looking to find traces of the torment he'd been in the past two days. He'd wanted to tell Jamike about it but decided not to.
"Tell me whatever you want, my brother Solomon, and I will do it. You know you have a friend in me."
"Da'alu," he said, and adjusted himself on the bed, on which he sat to face Jamike, who sat on the lone wooden chair in the room. "I want us to urinate together so we can generate more foam than when one of us does it alone."
"Okay, my brother," his friend said.
Indeed, Ijango-ijango, it was not very common for the children of the old fathers, now sold in the ways of the White Man, to speak with the oratory of the wise great fathers. But it came often in the speech of my host when what he was about to say had come from deep introspection.
"I know you have changed completely, and are a good man because you are born again, _Onye-ezi-omume_. You believe that I should leave Ndali alone after I have suffered for her, because she is married."
The other nodded to every word he said.
"I have heard all of that. I will not bother her even though, Nwannem, I have not lost a drop of love for her. My heart is still full, so full it cannot even be lidded. What I am going through, knowing she is alive and rejects me, is worse than anything I have gone through before."
He paused because he'd seen a cockroach appear over the wall mirror. He watched it as it flared its wings, then flew down behind the chair.
"This is worse, my brother, I really mean it. It is an imprisonment not of myself, but my heart. It is held and locked up by her." He moved to the edge of the bed and leaned against the wall. "M.O.G., I don't want to love her. Not anymore. She has spat on a man who sold everything he had to be able to marry her. I cannot forgive. No, I cannot."
Even as he spoke, he knew that although he was bitter, what he wanted most of all was to have Ndali back—to spend those nights with her again, and to make love to her. He watched Jamike shake his head.
"At least, Jamike, I want to know what happened to her. I want to know at what time she decided to leave and get married. Do you see? I sold everything, I left for her sake, I want to know what she did for me, too. I want to know why, what sent out the wild mouse running into the street in broad daylight."
"Yes, very wise, very wise," Jamike muttered with the same intensity as my host's.
"I want to know what happened to her," he said again, almost offhandedly, as if those words had been painful to utter. "I wanted to write to her, but I could not find anyone who could help me post the letter in the prison."
Chukwu, this was true. And it was this frustration that led me to try to get in touch with Ndali myself by performing the extraordinary act of _nnukwu-ekili_ in which I attempted to appear in her dream to give her the information my host wanted her to have. Indeed, Egbunu, as I have already told you, her chi prevented this from happening. And, as I have already told you, many of the guards would not even respond to my host's request for help in sending a letter. And one, who spoke English, told him that if it was a letter to Cyprus, then he could help him, but for Nigeria, he couldn't because it would be expensive.
He looked upon his friend with terror in his eyes.
"I want to know what effort she made for me during those times."
Jamike motioned to speak, but he continued.
"I want you to help me. And you must do this. See what you have caused me, see?" The other nodded with shame on his face. "So you must help me, Jamike. You should go to her husband as a preacher, and tell him you have seen a vision for him. Tell him as if you know much about his life. Say, for example, that you know his wife. Tell him you have seen in a vision that someone in her past, a man is after her, and will destroy the family if he does not pray."
He looked at his friend, whose head was resting on his folded hands, eyes fixed on him.
"You see? Tell the man you want to know if she has ever told him about a man in her past before."
"What if she has told her husband about the letter and that you are here?" Jamike, who seemed to have been rendered subservient by guilt, said.
"Yes? But he will not know, he cannot know you are coming from me. Be vague about me, say you see destruction, that the Lord showed you mourning and weeping caused by this man."
He stopped now in the darkening room to replay the words he'd said in his mind, and when he had done it, the enormity of it struck him. Egbunu, please listen to these words of my host, for they are crucial to my testimony this night and a solid proof that he has not done harm to her knowingly.
"I am not saying I will hurt her, no. I love her too much to do that, even though I am angry, very angry with her. It is a strange, uncanny mix of feelings. Deep love that is beyond compare. But no, I will kill any man, her husband even, who lays a hand on her."
Jamike nodded, with strains of discomfort evident in his countenance, moved in his seat, and said, "I will do it, if you say I should. I will, my brother, even though this is sinful. You cannot say the Lord has said something when he hasn't." Jamike shook his head. "I cannot do such a thing, my friend, by lying. I will tell him that I want to pray for him, a special prayer when I go to the mountain, and I want to know everything about his relationship with his wife so I can pray against anything in their past trying to destroy their future."
He did not know how to respond, so he kept silent, watching the man before him.
"I want you to be well again, my brother Solomon. This is why I'm what I have become. I caused all this for you, and I must fix it again. If this is all that will do it, I will go. As I said, someone who works near the pharmacy says her husband works at the Afribank at Okpara Square. I will go there and ask to see him—Ogbonna Enoka."
My host nodded, his heart resting on the floor again.
Later, as he drove Jamike home, his spirit calmed, and it seemed that the anticipation of her story had healed him. He slept well that night and went early to his store the following day. So many people had come looking for him, the neighbors said. He contacted some of the customers and spent a good part of the morning transporting bags of millet to them. As the sun rose after the morning's slight shower, he returned to his store with a pickup from the major broiler feed distributor, AGBAM FEEDS AND SONS. As they offloaded the contents into his store, Jamike called. My host answered the phone with shaking hands.
"I spoke with him, my brother. I don't know, but I think I was able to convince him. I went there with Sister Stella, with my ministry badge clipped to the pocket of my coat."
"I understand."
"Yes, I will like to come talk to you about it, so that I can also greet you, since I will not see you until after we return from the mountain."
"Yes, yes, you must come."
"In the evening," Jamike said.
"Why not now?"
"I will come, my brother. I will come in the evening."
OSEBURUWA, when a man has sent for a healer, if such a man is sick, and he is told that the healer is coming, he begins to count the steps of the healer's trek towards him. I have spoken about what anticipation does to a man, and I have seen it many times. My host could not wait for Jamike to arrive that evening.
"When I got to his office," Jamike began, "I was afraid. I had also lied to my sister in Christ, Stella. I was sinning."
"Yes, yes, I understand."
"But it is for you, my brother Solomon. So I went in. The man is a good-looking man. He is tall and has Jheri-curled hair. Ogbonna Ephraim Enoka. Ephraim is his baptismal name. He said his grandfather was brother to Father Tansi. So with Sister Stella sitting there, we prayed for him. Then I asked if he believed in prophesies. He said yes, why not? 'Am I not a Christian? Did the Bible not say shame on those who say they have faith but deny the power thereof?' I corrected him: 'It is in the book of Timothy. _Having a form of godliness, but denying the power thereof: from such turn away._ '
"He said, 'Oho, that is so,' in Igbo, then turned back to English. 'I believe in the power of God.'
"'I am happy, sir. I will tell you, then. I was in the spirit praying as I passed this bank yesterday, and the Lord said, there is a man named Ogbonna here whose wife is in danger, in real danger. An enemy has appeared at their door and is knocking.'
"'God said the name of the man is Ogbonna?' he asked.
"'Yes, yes. Father only gave me your first name.'
"'Okay.'
"'Is there another Ogbonna here?'
"'No, it is only me I know.'
"'And my spirit confirms it right now as we sit here. I can hear the Ancient of Days, the Lion of the Tribe of Judah, saying this is the man. An old flame has come to your wife and can destroy your marriage.'
"'God forbid in Jesus's name!' the man said. He snapped his fingers over his head. 'God forbid bad thing.'
"'Yes, brother. So can you tell me, is there any man your wife has offended? Anyone?'
"He seemed confused by this. I could see it on his face. He thought for a moment, and then said, 'No, nobody.'
"'Any man chasing her?'
"'No, I don't think so. She is a married woman with a child.'
"At this point, my brother, I worried that this man did not know anything about you," Jamike said. Having tried to distinguish his words from the words of Ndali's husband, his transition back to the language of the fathers was jagged.
"I asked him again. 'Mr. Ogbonna, is there any man whom she'd told you about?' and he looked at me, his face changing, and said, 'Yes, because of God, only because of God I'm saying, because it is a secret.' 'Don't worry, tell it to the servant of God,' I said. 'She almost married a man who left her and went overseas,' he said. 'That man was the second person who had done this kind of thing to her.' 'So this man disappeared?' I asked. 'Yes, no one heard from him again. That is all I know.' I wanted to speak, but Sister Stella said, 'So she never saw him again?' 'That is all I know, Man of God,' said Ogbonna.
"My brother, at this point, I was afraid if I pressed him more, he might become suspicious. So I said let us pray; that I will go to the mountain and pray, but that he should speak to his wife to see if there was a man after her."
"Aye. Oh-oh, Jamike. This is insufficient," my host said.
"But—"
"What if he asks her while you are away? And what if..."
He broke off his speech because one of the neighbors drove in on his motorcycle, vrooming as he parked it. The headlights of the motorcycle sent two beams of light through the curtains and illuminated the room, splashing their shadows on the wall as though calligraphed in thick black ink. When the engine went off, and the lights with it, he continued, "What will happen if he asks her while you are away?"
"I doubt she will tell him. I think, and see, that she doesn't want him to know much." Jamike slapped his leg to nip a mosquito. "I doubt he would."
"Yes," he said again. "But what if she decides to tell him after a man of God has spoken to him about it?"
Jamike considered it briefly. "Then I will find out. I will find out when I return. Isn't what you want just the information about what she did about your going away? You will not do anything with it, except just know."
He agreed.
"Then I will. Don't worry, my brother."
When they went out, so that Jamike could return to church first before going home, it had become dark. They passed groups of schoolchildren trundling back home from school, crossing the street in cliques. A little boy stooped by the public gutter, vomiting into it, coughing, attended by his friends, who kept saying sorry. An adult stopped there and asked one of them to give the sick boy water. My host and his friend said sorry to the boy. Then Jamike placed his hand on the boy's head and began speaking in tongues—an act which I have come to understand is a strange aspect of the religion of the White Man and is like an incantation, _afa,_ in odinala. When Jamike was done with it, he switched back to the language of the White Man. "Thank You, Lord, Jehovah-jireh, the mighty healer, Jehovah-shammah, for healing this little boy."
OKAAOME, he returned to his apartment afterwards with the information he'd received from Ndali through Jamike. He was steeped in thought as he warmed the pot of jollof rice he'd made that morning. Insects gathered around the kerosene lantern as the pot hissed slowly to life. He was getting the pot off the stove when electricity was restored, and then, almost abruptly, it went off again. He returned to his room with the food and ate slowly, wondering why she had told her husband that he simply vanished and that she had not heard anything from him. How was it that he just vanished? How? Did Dimeji not take his message to her? He had asked that he tell her, that he contact her, just before he was sentenced. He had also asked Tobe to do so. Did she never hear what happened to him? It was, he resolved, improbable. There was a great chance she had heard and knew but was probably hiding the information from her husband. It puzzled him greatly. Why was she hiding it from him?
In such times, a man must be careful, for in a desperate state, his mind comes up with a lot of answers. There is a part of man that can be irrational, a part which exists exclusively in order to make him comfortable. Thus, in a situation such as this, it will reach for whatever it can, the lowest branch of the tree, and pick it up. What a chi must do is to try to pick the most reasonable suggestion and allow that to dominate others. So from the multitude of possibilities that came to him that evening, I picked that it could be that Ndali simply had never received a letter from him before the one he just sent to her. But what he settled on was different—that she had told her husband he vanished to deceive him, to make her husband think she didn't want him anymore, when in fact she still loved him.
#
## Castaway
AGBATTA-ALUMALU, nothing cripples a human being more than unrequited love. Although Ndali once told him that she would not have drowned anyway, his act of generosity in trying to get her off the bridge was what first won her heart. And now her heart had been taken away from him by a man who worked in a bank and knew nothing of the sacrifices he had made for her. This was beyond what my host could bear. He was defeated in the days following Jamike's revelation. With Jamike gone in the following week, he became caught in the obsession of pursuing her. He fought hard against it at first by going to work and trying to focus on his store, but every day after closing, he would drive near the pharmacy and park on the side of the road. And from this vantage point, with his face concealed behind dark glasses, he would gaze at the pharmacy from the distance for a while.
Sometimes, the July rain would blur his vision, and he would sit there unable to see. Then, after he'd watched and thought about her so much that his heart would feel as heavy as a thing infused with lead, he would catch sight of her either walking out of the pharmacy or driving away in her blue vehicle. Catching sight of her was always enough for him to return home with a measure of relief. She was always in her white coverall, with whatever she wore under it showing. Most days, she wore a shirt and a skirt. Sometimes an ankara print blouse or an up-and-down. On those days when he saw her, he would return home, telling himself how beautiful she was, how her hair looked, or the color of her fingernails. Once, she had painted them blue, and he could see them as she passed his car up close without noticing that the man in the car with the hat and sunglasses was my host. He stood thinking about how he'd watch her paint her nails with cortex on the bench in the yard because she did not want him to choke on the strong scent of the nail polish. Once, she'd rubbed her fingernail on one of the white chickens, and the paint had stayed on its feather, a red splotch that could not be cleaned. It'd made her laugh so hard she'd cried.
He'd return home and long for contact with her. He'd think of all possibilities. He began to notice that the more he saw her, the more he raised the memories of their intimacy and the more his desire deepened. What would he do? She would disgrace him again if he came up to her and she would probably hate him. She had read his letter, seen all he'd suffered, but showed no remorse. At the entrance of this kind of thought, his mood would change from desire to anger, then resentment. He would clench his teeth, stamp his feet, and quiver with rage. He'd sleep in this mood and wake up the following day to the same routine: go to his store with the consolation that he'd devise a way to see her in the evening, then feel a flurry of conflicting emotions afterwards.
On one of those days, he followed her as she drove away, curious to see what she would do, for a thought had flung into his mind that she might have a lover. She drove to a school, a private primary school, where, at the gate, her son was waiting. He looked on from the side street, in his car, parked two hundred meters away. He noticed the boy's ears; how, by complexion, he resembled Ndali. He followed them on to their house, a duplex that stood grandly on Factory Road. It was fenced and had a gate as tall as the fence itself. He stopped by the house and surveyed its surrounding land, overgrown with bushes. On the other side of the unpaved road, a provisions store sat in front of what looked like a small clinic. A few meters from there was a shack under which a woman fried plantains, yams, and akara every evening. He returned to his apartment not knowing what to do with his new knowledge.
At the end of that first week without Jamike, on the Friday, he could not go to work. The bitterness of the previous night had lasted into the following day, and he'd found himself weeping for the pain her rejection had caused him. Egbunu, what I was witnessing in my host was peculiar and startling. It was the known alchemy of love—it is a thing that becomes alive and thriving in a state of decay. He swore to himself that he would confront her if she stepped out of the pharmacy that day. So that day, he decided to get out of his car and sit with the woman who operated the GSM phone stall across the road. As he fumbled with one of the service phones, the woman asked if he was the man who was always sitting in a parked car and looking at the pharmacy. My host was startled.
"Have you been seeing me?"
The woman laughed and clapped her hands in jest. "Of course. You come every day, every day. How won't we see you? Maybe even the people in the pharmacy have seen you."
He sat still. He turned to the street, to a cattle herder ferrying his cattle and slapping them with his stick.
"You have not answered my question," the woman said again. "Why are you always doing that?"
My host, astonished, knew he would no longer continue this venture.
"But I am always wearing sunglass, how did you know me?" he said.
"Because I saw you come out of that same car just now."
"Okay, I was married to the pharmacist before," he said. Then he told a lie about how her present husband took her away by casting a juju spell on her. The gullible girl felt sorry for him and, while trying to comfort him, brushed her hand against his. He'd felt nothing until then, but when her body touched his, it struck him that he was attracted to her. In a hurry to take advantage of the situation and drive my host away from his continuous, destructive obsession with Ndali, I flashed it in his mind that he could have this woman, and that she would love him always. As these thoughts floated about in his head, he observed her closely. Her features were common; she was cheaply dressed, and her skin was rough and coated with the kind of darkness that comes from privation. But on this day, she was dressed better than she usually was: in a good blouse and short skirt, her hair permed.
He sat there while she attended to those who wanted to make calls or buy phone credits, watching this woman, aghast at the sudden transference of his wanton desire. He developed an erection.
"I think I should take you to my house today, so you can come to know my place and we can be good friends," he said.
The woman smiled and did not look at him. She fumbled with the cards, stacking them together with rubber bands.
"You don't even know me," she said.
"You don't want to come, eh? Okay, what is your name?"
"I did not say that," she said. "My name is Chidinma."
"I'm Nonso. So will you come, Chidinma?"
"Okay, after I close, then."
Akataka, he stayed there until the woman closed the shop, then he drove her to his home, stopping to buy two bottles of Malta Guinness on the way. I did not flee at first because I wanted to see things go through, to see where it would end. Even though I had helped engineer it, I wanted to try to understand this new phenomenon: a man is wasting away only moments before in great desire for one woman, then suddenly he is burning for another with the same intensity. This was a mystifying thing. Aside from the woman's question about whether he would continue to look for his wife or love her instead—to which he said, "I will love you instead"—there was no resistance. He tore at her hungrily, almost ripping her clothes. He plunged his hands into her brassiere and drank her breast with mad haste. Many years had passed since he'd seen a naked woman, let alone touched one, so that when he came to the place between her legs, he was dazed.
It was at this point, certain the unexpected would unfold, that I left his body. But so monstrous was the clamor of Benmuo this night that I was forced back into my host at once, as if chased by some deadly beast. Thus was I forced to behold the mystifying alchemy of sexual intercourse. I came back when the woman's entreaties that he should use a condom, in the heat of the moment, became insistent. But he paid no heed. "But don't release inside. Don't release inside, oh," she'd begged as he thrust violently, his bed creaking. I witnessed him throw a shout and then relieve himself on the floor.
The woman lay by him and clutched him, but he faced the wall. As his heartbeat relaxed and his sweat dried, he began to feel different. He thought back to earlier in the day, how he'd sat there at the woman's table. What he saw now, Egbunu, was different. Different! He saw the spots on the woman's face, one peeled so that it had scabbed. He thought of the woman's missing teeth and what looked like a scar above her cleavage. He thought of the dirtiness of her nails, how she'd pick the mucus of her eyes with them. He thought of the dark pit of the woman's stomach as they lay to make love and the fortress of her vagina. He drew away and stepped out of the bed, opened the window and, looking up, recalled Ndali's body. He remembered the day she insisted he suck at her vagina and the revulsion of feeling that had seized him then.
When he turned back into the room, the woman had covered herself with the bedsheet. Resentment rose within him. For a reason neither he nor I, his chi, could determine, he found that he hated her. He sat on the chair and finished the malt, which he'd drunk halfway.
"Will you go home?" he said.
"Er?" she said, sitting up.
He regarded her, her ugliness more pronounced, and he convulsed with regret.
"I said, do you want to sleep here? I just want to know."
"Eh, are you sending me away?" she said, her voice almost breaking.
"No, no, I'm saying if you want to go."
She shook her head. "So you have gotten what you want, and now asking me to go home?"
He gazed at her without words, surprised at his own sudden cruelty.
_"O di nma,"_ the girl said, and snapped her fingers.
He watched her strap the brassiere back on, the line on her back almost unapparent, the plump of unremarkable flesh. Inwardly, he felt violated in a way he could not explain. Was it that he had known another woman and now Ndali would be defiled in his eyes? The fear rose with a mixture of anger. He closed his eyes and did not know when the woman finished dressing. The sound of the door broke him out of his reverie. He stamped to his feet, but she was out. He chased after her in the dark, barefoot, shirtless, his room unlocked, calling her name: "Chidinma, Chidinma, wait, wait." But she did not wait. She went on, sobbing, saying nothing.
He returned and sat down, only the smell of the woman left in the room. He did not know what to feel, remorse for how callously he'd treated the woman or anger at his own mysterious violation. He waited for an hour or so to pass, and then he rang the woman, but she did not answer. He sent a message that he was sorry. She wrote back: neva, neva in yr cum 2 my shop again! Neva in yr life!! god punish you!!!
He quaked in his seat as a possessive thought of violence perched on his mind, carried on the black wings of contempt. He deleted the woman's number, and that was it. That night, while he was asleep, two vagabond spirits broke into the house, fighting. They came through the wall, unaware they had crossed a human barrier. Chukwu, I must say that things like this happen quite frequently, but most of them are not worthy of recollection. But this particular incident moved me, for I could relate it to my host's situation.
One of the spirits was the chi of a man who had taken the wife of another man. The other spirit was the ghost of the woman's former husband. The chi was saying how exhausted he was, having been trying to fight off this ghost for years. "Why don't you just go to rest?" it said. "How can I be at rest when your host cheated me of not just my wife but my life, too?" the revenant said. "But you should rest. Go to Alandiichie, return back in another life, and take back what was yours," the chi replied. "No, I want justice now. Now. Now. Tell your host to keep his hands off Ngozi. Or I will not let him alone. I will continue to haunt his dreams, attempt to possess him, cause him hallucinations until justice is done." "Well," the chi replied, "if you let it be, Ala and Chukwu will execute justice on your behalf. But you have taken it upon yourself to handle..." Their conversation continued as I gestured at them to get out, and they, barely giving me and my host a look, returned back into the darkness through the wall. I did not know why I witnessed this—perhaps it was you who allowed me to see it as a warning to do more to dissuade my host from his pursuit of the elusive, a situation that could potentially cause him to become an akaliogoli, a vagabond spirit, without a home in the heavenlies or on earth.
ECHETAOBIESIKE, my host returned back to the way he was, a man of conflicting thoughts. He'd floated like some fluid element back into the thing that contained him. He stopped lying in wait near the pharmacy and turned his attention to her house. He would park his car a few stones' throws away and walk up to the supermarket across from her house. He befriended the shopkeeper. He would buy biscuits and Coke and sit on the lone bench the man placed by the side of the shed, eating and drinking and chatting with the man in his mangled English. From this vantage point, making sure that his sunglasses did not once leave his face, he would first watch her arrive from work with the boy, then watch her husband. On the third day of this new routine, it struck him to ask about the family from the store owner.
"Mr. Obonna?" the man, a Hausa who did not speak the language of the fathers, said.
"Yes, and his wife?"
"Oh, that madam? Me no know plenty about am, oh. She no dey talk at-all, at-all. Just only quiet like say she no get mouth. She dey come here plenty."
He regarded the shopkeeper as the man scratched the two long scarifications on one side of his face. A man approached the store in shorts, a shirt hanging from his shoulder.
"Well done, oh," the customer said to my host.
"Well done, my brother."
"Mallam, Cowbell dey?"
"Which one, na? Tin abi sachet?"
"Sachet. Bring four. Na how much, sef?"
"Tem tem naira. Four na four naira."
When the man was gone, he asked the trader if he knew anything about Mr. Ogbonna and his son.
"Ah, yes-yes. I sabi them well-well."
Egbunu, I have told you that my host possessed the gift of good luck. True, many bad things had happened to him, but what his onyeuwa picked in the garden of Chiokike is potent. For how can I explain what he stumbled on by serendipity here? How, Ezeuwa? All he'd done was ask the man the corollary question to the one he'd asked about the family. "Na only that son them get?" To which the man had responded thusly:
"Pickini kwo? Yes, Na only wan pickin. Chinomso, na only one pickin."
Obasidinelu, my host jumped to his feet. For he had not told this man his name.
"Er, what?"
"The pickin naw," the man said, astonished at his reaction. "I say him name na Chinomso."
He stood still now, unable to move his feet. He stared at the man, then in the direction of the house, then back again.
"Oga, wetin happen?"
He shook his head. "Nothing."
The man, easing up again, began to talk about how "Mr. Obonna" sometimes didn't take his change after purchasing and how, during Eid-El-Fitr, he brought him a goat. He listened with half his mind carried away. When he rose up and got back in his car, he became aware, as if his consciousness had been renewed, of the information he'd just received. How could it be that she'd named her son after him? How?
Nothing troubled him more than this contemplation. He sat, unable to do anything, helpless. It was a question that menaced him with its deceptive simplicity. For it seemed like it could easily be answered, as if the answer lay on some shelf just above his head. But anytime he attempted to pick it up, he realized it was far away—a place he could not reach by merely stretching his hand. And it was this that troubled him the most. He slept little that night, and when he woke up, he feared he'd lost his mind from the unrelenting examination of his thoughts. He was hungry, shattered, and dismayed, but there he lay, broken in bits. The people from the agricultural university called him two times, then sent a text saying they would no longer be buying feed from him as he was no longer serious about his business. They were the fourth or so regular customers who had abandoned him because he was now rarely at his store.
After he read the message, he snapped. He yelled into the hot day and stood up.
Why am I afraid of her? Why, after all I have done, after all I have done for her? No, she has to talk to me.
He paced the room, carrying the memory of the day she had rejected him in public, crying that she did not know who he was. Today, today, Ndali must give me answers.
He'd spoken so firmly that he was astonished at how emboldened he'd become. He went out to the shared bathroom at the back of the apartment to bathe. In front of it, the wife of one of the neighbors, a Yoruba man who spoke with a feminine voice, sat on a short stool, bent over a bucket, washing clothes. Soap suds were scattered about. The woman was swaddled in a wrappa, which hung over her bosom and was fastened into a knot beneath her hairy armpit. The woman greeted him, and as he passed, the portion of flesh exposed to his eyes annoyed him. He thought of the woman he'd slept with, how his feelings had surprised him. Instead of pleasure, he'd felt disgust, and that had shocked him. As he closed the bathroom door, made of zinc nailed to wood, and piled his clothes over the top, it struck him that what he'd experienced with that woman and his general apathy towards other women was because he still loved Ndali.
He drove again to her house and parked his car a few meters from it, on the side of the road opposite the direction from which her car came. He parked under a tree filled with birds tweeting, overlooking a fenced mansion from which the voices of children came in flashes. Then he waited, his eyes on the road, until at sundown he saw her car approaching. He'd thought and rethought things and made up his mind. He'd observed that cars seldom came this way, as the street that curved beyond this one did not give out onto anything beyond itself. It culminated in a dead end. But if there was a car trailing her own, and he could not block the road, then he would simply come out of the vehicle, chase after her car and interfere before she honked at her gate and the gateman opened it.
Egbunu, the moment came like something from his imagination. As soon as he saw her car, he started his car and ran it with a rush forward, then sideways into the path of the oncoming car. The cars almost hit each other, and the cry that arose as a result of this near hit threatened even his own disoriented mind. He sat for a moment to let his heart quieten. Then he got out of the car. He'd seen her, but he had not seen the boy who sat in the back. Now he saw them both, she turning back to the boy to say something. He walked to the front of the two cars and stood still. For a long time, months, ever since he returned, he'd wanted this moment. He felt himself shaking, something erupting along the base of his heart.
The person in the car behind his honked thrice and drove angrily past. But he stood there. Then she came out of the car. She looked at him and he at her. Life seemed to be there in that face, the life he once knew. But it was a face that was hard for him to recognize. Something about it was new, yet much of it was familiar.
"You?" she said, as if inquiring into the nature of his being.
He nodded. "Mommy," he said.
She stepped back towards the car, bent, and said something to the boy. Then she closed the door and stepped forward beside it.
"You, again? What do you want?"
He shook his head, for Egbunu, he was afraid.
"Mommy, I am sorry for everything. I am sorry. I am sorry. Did you read my letter? Did you read the—"
"Excuse me!" she cried. "Excuse me!" She stepped back, put her hand on her face, and pointed the painted fingers at him. "Why are you after me? Why are you coming to my chemist and my house? What is the meaning of this, eh?"
"Mommy—"
"No, no, stop! Stop it! Don't call me that, please, I beg you."
He made to speak again, but she looked back at the car and the boy.
She turned to him again, and with her eyes closed, she said, "Let me tell you, I don't want to ever see you ever again. What is this? Why are you following—"
"Ndali, listen," he said, and stepped forward.
"Stop! Stop there!"
So violently had she moved backwards that it alarmed him.
"Don't you come near me at all. Listen, I beg you in the name of God, leave me alone. I am married now, okay? Go and find another woman, and leave me alone. If you come to my house again, I will arrest you."
He saw that she had turned back towards the car, and he followed her. He'd come inches from touching her when she faced him again.
"Your son," he said, panting from the rush. "He has my name."
In this memorable moment of life, when my host and the woman he loved came inches away from each other, a wagon started to approach the place where the two cars were fixed into a confluence. It was an instinctive moment, brief, like the last-minute glimpse of an assassin by his victim, but fraught with a grace that was imponderable to man. With one unwelcome step he had entered into her field of vision, and his legs had been caught in a loop from which he could not disentangle himself. He saw that she wanted to speak, but then, abruptly, she turned and went back in her car.
The men in the wagon had stopped and started to curse. He returned to his car and pulled it gently into a reverse. Her car coursed through and made for the gate to her house. He watched it disappear, the provoked wagon driver and passengers cursing at him as they passed.
EBUBEDIKE, I must not dwell on the thing he did afterwards too much, for it was something too difficult to watch. For my host was devastated by this encounter. The few words Ndali had said to him he carried in the weak sac of his stomach and digested them on the scene, weighing every word. But like a goat, he'd made them into veritable cud. And every night, when his life, which had acquired the restlessness of a pendulum, swung into a standstill, he'd bring up the cud and chew with fresh salivary intensity. But there was one thing that he could not shake off, that could not be chewed or broken down. For it was solid and complete in its composition. He'd seen it in her eyes, and even though he knew that his mind could become overreactive in such situations, he was convinced that what he'd seen in her was contempt.
It is hard to describe what this feeling did to him. He lay in the house for days, surrounded by the ghostlike, disembodied voices of the encounter. He ate little; he spoke to himself. He laughed. He cried. He stepped out wearily at night and ran back into his room again, drinking the rainwater that washed down his face.
I feared, Egbunu, that he was descending into madness. For even more, he was haunted by strange and persistent dreams, many of them of birds—chickens, ducks, falcons, and even hawks. They were dreams that exposed the inflammations of his afflicted mind. He became like a castaway—one rejected by earth and heaven. A living akaliogoli. I feared because I have come to know that the strongest kind of affection often exists in the heart of a man whose love interest is distant from him—the one he cannot have. That is the one his soul longs for with dying breaths, and the sublime dungeon in which his heart is caged. The only way to save him is to introduce a new affection as strong as the one he cannot get. But because no such woman was near, I feared.
His descent into this state continued for days, Egbunu, and one evening, as he sat mumbling to himself that she hated him, he did not realize that his friend had returned.
He was almost thrown into a shock when he heard a loud knock on his door, followed by, "Brother Chinonso, son of the living God!"
He rushed to open it.
#
## The Subaltern God
AKWAAKWURU, the great fathers in their unrivaled wisdom used to say that what a man is afraid of, that thing is greater than his chi. This is a hard saying. But it is true that fear is a great phenomenon in the life of a man. As a child, a man's life is ruled by constant fear. And once a person becomes an adult, fear becomes a permanent part of him. Everything a human being does is ruled by it. It is folly to ask, how may one be free from fear? Well, isn't it fear itself—perhaps the fear of having one's mind dominated by fear—that causes a person to ask such a question? Man must live by it. Man eats because he fears that if he does not, he will die. Why does he cross the street with caution? Why are that man and his child going to a clinic? Fear. Fear is a subaltern god, the silent controller of the universe of mankind. It might be the most powerful of all human emotions. Gaganaogwu, consider the story of Azuka, the man who killed his brother-in-law in a brawl three hundred and seventy years ago. That man was sentenced to death by the priest of Ala for having taken another man's life unjustly. My host at the time, Chetaeze Ijekoba, had been one of those who walked him to the forest and hanged him. I had seen through him how this condemned man had been, how even his movement and his voice had been changed by fear, and it was clear that every moment of his life, from the time the judgment was pronounced, had been occupied by the fear of death. A man who persuades himself to live without fear will soon find that he has fled naked into the province of insanity, a place where he is without any acquaintances whatsoever.
When Jamike visited, he found my host consumed with fear—and desire, rage, love, and grief. But most of all, it was fear that, in truth, he would never have Ndali again. Fear, Chukwu! The subaltern god, the tormentor of humanity—that which holds a man on a leash and from which he cannot escape. Let him dart about the house, let him perch on windowsills, let him flap his young white wings as much as he wants, let him call and utter the orchestra of minorities; he cannot escape. For if he flies up, the roof will bring him back, and restore him to his place. Is the man at this point making merry? Is he drinking palm wine at his wedding? Is he receiving the benediction of his parents and the adulation of all his kindred? Is he making love to his wife? Is his wife in labor, and he is awaiting a child to be born? No matter, when he is done—when the party is over, when the wedding guests have all gone, when he has relieved himself and is calm again, when the child has been born and is asleep, fear returns with a presence more forceful than before and reels him back like a falconer does his bird.
So with this great fear my host needed help. He must at least try to know; he must try to find a way. A way? This was what he'd been trying to tell Jamike. And now, exhausted, he fell on his knee and held his friend who had returned to him from the mountain of prayers, filled with the spirit of the great deity worshipped in distant lands and also worshipped by the children of the pious fathers.
"Jamike," he said. "I know you are a man of God. I know God has changed your life, but I want you to do this one thing for me. I am sad, a very sad man still. I am still in a sloam. I will be saved only when I have my wife back."
Even though he knew at this point that she had been lost, even though he could tell that he was now on the brink of insanity, he was worried by the consternation he saw on Jamike's face.
"Yes," he said vehemently, gnashing his teeth and gripping Jamike's thin trousered leg even more firmly. "She is my wife, Jamike. She is mine. We were going to get married. I suffered for her."
His friend visibly seemed not to know what to say. He gazed on at my host, who loosened his grip. My host continued, "About a week ago, I met her at the front of her house, Jamike. I saw her, so close, and her son. Do you know what his name, the name of her son is? It is Chinonso."
"Your name?" Jamike said, and my most rational host became animated, for it seemed he'd struck something in the man with whom he sought help.
"That is so, that's the boy's name."
"I can't believe it, my brother."
"I think," he said, but a deep chest-heaving inhalation silenced him, so that he began again: "I think there is a reason why and I want to know. Did she think I was dead? Is that why she gave the boy my name? Or is it because of something else?" He coughed and spat into a handkerchief. "The boy, I have seen him with my two wide-open eyes and my spirit tells me that he is my son."
"It does?"
"That is so," he said and snapped his fingers. "In fact, can you see the boy? He looks like he is at least five or six. When did she marry this man? You said not long ago?"
"Ha, that is true. B-but when could that be?"
"I do not know. I do not know. I do not know, oh. Only God knows. But, my brother, my heart is broken. A dead person is better than me right now. I can't sleep. I can't eat. I don't know why my life is like this. But I want to know why her son has my name."
"What you say is true, my brother Solomon. Ndiichie say that a toad in full daylight does not run for nothing. Either something is chasing it, or it is chasing something."
True, Gaganaogwu: that was the wisdom of the erudite fathers!
"I understand, Nwannem Solomon," Jamike continued. "Ask me anything and I will do it. I want to help you."
At this my host looked up and in that moment saw that he was kneeling on the ground and gripping the thin legs of his friend, his poor friend who had been without food for forty days and forty nights. The thinness of his friend's frame shocked him, and he withdrew his hands in a hurry and sat on the bed across from his friend. It was the word _help,_ Egbunu, the promise of reprieve, hope, that did this to him. He sat up now, and shaking his head, said, "I want you to go back to her husband, and say to him, 'God has sent me to you, Mr. Ogbonna, to warn you that they may be in danger.'"
He waited for Jamike to speak, but his friend held his hand to his mouth, wiping the corners that opened into an O shape.
"It will not be a sin," he said. "All you are doing is trying to know if she is—whether she is safe or not. God will not forbid this. And, also, you are a pastor. So it is not a lie."
Jamike shook his head. Although it seemed that it took a great resolve for him to finally speak, he did not say, "But the Lord has not sent me to him. That is a lie," as my host had feared he would say. Rather, in a voice that seemed to cleave through the air like a sickle, Jamike said that he would do it. Then, as if he thought my host had not heard him, he repeated it again with the blunt force of a persuasion.
My host became calm. Then, lifted by a hand he could not see, he rose up.
CHUKWU, the great fathers often say that it was to the hunter's advantage that the antelope developed a bloated scrotum. For now the hunter with his poisoned arrow—even if he is an old man with a body full of old, weak bones—would be able to catch the antelope. Mr. Ogbonna, my host's lover's husband, the evil man who has taken advantage of his absence and stolen his bride, the man who had ruined him, the man for whose sake he now suffered, the man who may be claiming his child, had already developed a swollen scrotum. He had given himself to a masked priest, a spy working for the damaged kingdom of my host. And now, on the evening of the following day, when the horizon itself wore a painted mask of thin gray and the bled-out red of a desert ant, my host and his friend drove to the bank where Ndali's husband worked.
He waited near a mechanic's workshop while Jamike went into the bank. The workshop was located under an old ugba tree, a tree that I immediately recognized. It had been there for many years. More than two hundred years before, as the heartless men of Arochukwu dragged my host, Yagazie, and other captured slaves along, their extremities bound with chains, a woman fell under the tree and fainted. The captors were forced to halt the march. Without a word, one of them, a stout man, signaled to the rest and said that the woman may be ill and might not make it to the seashore. So what to do? He cut her loose. But the woman did not move. They left her there, as if asleep, in a clearing with this single old tree.
My host came out from his car and stood under the tree with the men from the workshop, his eyes drawn to the Biafran flag, which was bound to a piece of wood inside the building. The flag was almost blackened with soot, with a hole at one corner of it. The men offered him a seat on a dirty bench by a big tire, perhaps from a semi, with filing tools piled on it. But he stood by as the men worked, his arms folded over his chest, watching the street.
He had just bought a cold Pure Water from a hawker and was drinking it when Jamike returned. Jamike came with a certain muteness, as if something had silenced him. "Let us go somewhere and talk," he said to my host with haste in his voice, motioning towards the car.
They drove to his apartment, and it was not until they had sat down, he on the bed and Jamike on his chair, that the conversation began.
"My brother, when I went in there, it was like he was waiting for me. He jumped up and said, 'Pastor, Pastor, I'm in trouble.' I asked what was the matter and he said, 'Pastor, my wife, my wife.' He was in anguish. He said Ndali had seen the man whom she almost married, and that the man had found out that the boy is his son."
My host stood to his feet.
"Yes, it is your son, my brother," Jamike said, looking up at him.
"How did that happen? How?"
"The man said she was pregnant before you left Nigeria. After you left, and she did not hear from you, she tried to find you. She called CIU."
Ijango-ijango, you must wonder what this did to my host.
"Say again. _Isi gi ni_?" was all he could say.
"She called the university, she called Dehan, my brother Solomon."
He sat silent. I flashed in his mind two of the occasions when she had held on to him and asked him to ejaculate inside her. Then I flashed in his thought another one, that evening now long past, when he'd been so carried away by it all that he'd let himself ejaculate in her and pulled himself out only after much had gone into her. And he hadn't told her, fearing she would scold him. Then she asked him to put on the light so she could clean herself with tissues. And he'd put it on, relieved that she had not asked him if he'd pulled out effectively. He put the light on and found, floating in the air, a white feather. Ndali had been mesmerized by it. She'd asked where it was from and how it had come to float in the air. And he said he did not know. That was simply one of the many instances I reminded him of. But on his own, my host recalled how, when he reached her on the phone just after he'd received the promise of hope from the nurse, she had said there was something she wanted to say but she would tell him at a later time. I heard her voice still as she said it to him on that phone many years ago: "It is big, big news, even me, I am surprised. But I'm very happy!"
"She could no longer hear from you, she was worried, my brother. Child of God, she was with your child, and suddenly, for many days no word. Then for weeks, she waited, no word. She had the photocopy of your admission letter which you had given to her. She called the school and was told of what you had done."
He was starting to speak, but Jamike went on.
"They told her you raped a white woman, and were going to spend twenty-six years in prison. In fact, they told her that the people were more lenient because in most Muslim countries, the penalty for rape was death."
"Who told her that?"
"He did not tell me, but I think it was Dehan. He did not know the whole story; I don't think he did. But she tried. She looked for you, she tried to help you. He said she did not believe you did it, and had reported to the Nigerian embassy in Turkey, but no one did anything. I remember this, my brother, when I called my friends whose house you went to in Girne, they told me the Nigeria embassy in Turkey called the university. So I believe she tried, my brother. I caused this, but she tried to do something."
"What else, what else happened?" my host asked, for the old rage had started to come upon him again.
"Her family," said his friend, who had begun to weep. "They were furious at it all. She was pregnant out of wedlock, then she was making international moves to rescue a man held as a criminal in another country. This was why they asked her to go to Lagos first. Ogbonna did not say this, my brother, but I believe she tried. Then she gave up."
Ijango-ijango, something moved in my host's bowel, and he felt a warmth inside, as if something hot had penetrated it with slow ferality. She _gave up_. What does it mean? Akataka, it means that a person has tried something and then stopped. It may be that person has been trying to lift something, and then it occurs to such a one that they would never be able to lift it, so they resign and give up.
My host sat there, stunned, as if the world in which he'd been born, lived, made love, slept, suffered, healed, and suffered again had been all along an illusion, the kind of sudden vision seen by the eyes of a blind old man: one moment radiant and aglow, and the next, a mirage that dissolves once it is seen.
#
## Spiders in the House of Men
CHUKWU, your ears have been patient. You have listened. You have heard me recount all these things before the divine council here. You have listened while every tree in Beigwe wore the enchanting tunes like shiny garments. Even as I speak the music is pouring out of everywhere in the luminous halls like sweat from the pores of the skin. And all around are guardian spirits who must step in and render their respective accounts. But now I must hasten to fill the chasm that has opened in my story. And it will not be long, Gaganaogwu, till I am done with it.
To hasten, I must remind you of what the great fathers, wise in ways of war and battle, often say: that which must kill a man does not have to know his name. This was true of my host. For what he became, in the days and weeks after Jamike's discoveries, is painful to describe. But I must tell you the consequences of this change, because the cause for which I plead requires it. Egbunu, my host became a djinn, a man-spirit, a vagabond, a descaled wanderer, a thing creeping in the bush, a self-exiled outcast, shorn from the world. He refused to listen to the counsel of his friend, who begged him not to get in the fight. He vowed that he would, in fact, fight. He vowed, vehemently, that he would get his son back. He insisted that it was the only thing he had left in the world worth fighting for. And nobody, not even I, his guardian spirit, could persuade him against his will.
So he began again to lurk in the bush around her house, and when she drove home, he tried to accost her. She would not get out of her car but skirted around him and drove away. When this failed, he went to her pharmacy, shouting that he wanted his child. But she locked herself in the room and called on her neighbors from her locked window. Three men ran up into the pharmacy and dragged him out, punching him until his lips were swollen and the upper side of his left eye was split open.
But it did not stop him, Egbunu. He went next to the school the boy attended and tried to take him by force. And it was here that I think the seed of that which brought me here in this most troubling of human nights was sown. For I have seen it many times, Oseburuwa. I have come to know that a man who returns to that place where his soul was once shattered will not lightly forgive those who had dragged him there. And where am I talking about? It is that place where a man's existence stops, where he lives a still life like that statue of a man with a drum there at the center of the street or the figure of a child with the gaping mouth near the police station.
Although the treatment by the guards this time was different, merely insults and slaps, he was tormented by the memory it unleashed in him. He wept in the cell. He cursed himself. He cursed the world. He cursed his misery. Then, Chukwu, he cursed her. And when he slept that night, a time in the past appeared, and he heard her voice say, "Nonso, you have destroyed yourself because of me!" and from the bare floor of the dungeon, he sat up frantically, as if those words had taken years to reach him and he'd just heard them now for the first time, four years after she'd said them.
EZEUWA, Jamike came to bail him out the morning of the third day. "I have told you, let her alone," Jamike said after they had left the police station. "You cannot force her to return to you. Get the past behind you and move on. Move to Aba, or Lagos. Start again. You will find a good woman. Look at me, all the years I spent in Cyprus, did I find anybody? I found Stella here. And now, she will become my wife."
Jamike spoke to him, a man who seemed to be without a mouth, until they arrived at his house, and all Jamike's counsel came together with a combination of all the things he had seen and done. When the taxi pulled up in front of his apartment, he thanked his friend and asked to be alone.
"No problem," Jamike said. "I will come and see you tomorrow."
"Tomorrow," he said.
OBASIDINELU, the great fathers in their diplomatic sagacity say that whichever tune the flutist plays is what the dancer will dance to. It is madness to dance to one tune while listening to another. My host had been taught by life itself these hard truths. But I had counseled him, too, and so had Jamike, his friend, on whom he now relied. And it was with these words in his heart that he unlocked the gate and made his way to his apartment. He was greeted by his neighbor's wife, who was picking beans on a tray, and he mumbled a response under his heavy breath. He unlocked the padlock and opened the door to his room. Once inside, he was hit by a claustrophobic odor. Looking in the direction from which the loud droning of flies came, he saw what it was: the moi-moi he'd bought and half eaten the day he was taken away. Worms had filled the polythene wrap, and a milky substance ran down from the rotten food onto the table.
He took off his shirt and put the food in it, sending the flies into a frenzy. He wiped the putrefied substance off the table and took the shirt to the trash. Then he lay on his bed, his eyes closed, his hands on his chest, as he tried to think of nothing. But this, Egbunu, is almost impossible—for the mind of a man is a field in a wild forest on which something, no matter how small, must graze. What came he could not reject: his mother. He saw her, seated on the bench in the yard, pounding pepper or yam in the mortar, and he beside her, listening to her stories. He saw her, her head covered in a calico scarf.
He dwelt in this place, this veranda between consciousness and unconsciousness, until night fell. Then he sat up and let the idea flower that he should leave Umuahia and everything in it behind. He had thought about this in the jail, even before Jamike said it again. And I had ensured that it persisted in his thoughts. The idea had come and gone out of his mind like a restless visitor those three days he spent there. Now something in the vision of his mother settled it, even if he did not know what it was. Was it that after she died he himself told his father many times that he should forget her? He had several times fought the man, told him it was only a child who hung on to what had been lost. Especially that night when his father, drunk, had walked into his room. Earlier they had cut up a chicken to supply to a woman whose daughter was about his sister's age and who was getting married. It may have been this that bothered the man. His father had staggered into my host's room in the dead of the night, in tears, saying, "Okparam, I am a failure. A big failure. When your mother was in the labor room, I failed to protect her. I could not bring her back. Now your sister, I failed to protect her. What is my life now? Is it just a record of losses? Is my life now defined by what I have lost? Who have I wronged? _Kedu ihe nmere_?"
In the past iterations of this remembering, he had thought of his father as weak, as someone who could not withstand hardship, who did not know how to turn his back. Now it struck him that he himself was clinging to what had been lost, what he could never again possess.
He would leave. He would return to Aba, to his uncle, and leave it all behind. He could not change that which has remolded itself to resist change. His world—nay, his old world—had remolded itself and could not change. Only forward momentum was possible. Jamike had left the province of his shame, made peace with my host, and moved forward. And so, too, had Ndali. She had wiped clean the board of the inscriptions he'd made on her soul and inscribed new things. There was no longer a remembrance of things past.
Also, it became clear to him now that it wasn't he alone who harbored hatred or a full pitcher of resentment from which, every step or so in its rough journey on the worn path of life, a drop or two spilled. It was many people, perhaps everyone in the land, everyone in Alaigbo, or even everyone in the country in which its people live, blindfolded, gagged, terrified. Perhaps every one of them was filled with some kind of hatred. Certainly. Surely an old grievance, like an immortal beast, was locked up in an unbreakable dungeon of their hearts. They must be angry at the lack of electricity, at the lack of amenities, at the corruption. They, the MASSOB protesters, for instance, who had been shot in Owerri, and those wounded the past week in Ariaria, clamoring for the rebirth of a dead nation—they, too, they must be angry at that which is dead and cannot return to life. How about everyone who has lost a loved one or a friend? Surely, in the depth of their hearts, every man or woman must harbor some resentment. There is no one whose peace is complete. No one.
So prolonged was his musing, so sincere his thoughts, that his heart gave the idea sanction. And I, his chi, affirmed it. He must leave, and his leaving would be immediate. And it was this that gave him peace. The following day, he went about looking for anyone who would buy his store's contents and take over the rent. He returned home satisfied. Then he called his uncle and told him all that had happened to him and that he must flee Umuahia. The older man was deeply disturbed. "I t-told you no n-not to go back to th-that woman," he said again and again. Then he ordered my host to come to Aba at once.
For days he packed the few things he had gathered, trying hard not to think about Ndali or his son. He would come back someday, in the future, when he had picked up his life again, and ask for him. That is what he would do, he thought as he stood in the emptied room that was once full, now with only his old mattress lying on the floor.
Agujiegbe, he would leave that evening and not return. He would leave! He had told Jamike this and once his friend had come to see him, he would begin his journey. He was waiting for the preacher to return from his evangelism and pray for him before he would go with all his things in his car.
Chukwu, at this point, I fear again that I must say that after Jamike had come, prayed for him, cried for him, and embraced him, the old rage, the terror, the complex feeling that swallowed all things, came upon him again. He did not know what it was, but it seized him and plunged him into the abyss from which he'd been dragged out. It was, Egbunu, a single memory that did it: that one strike of a match that sets an entire building on fire. It was the recollection of the day he first slept with her and the day she had knelt on the ground of the yard and sucked at his manhood until he toppled over the bench. How they had both laughed and talked about how the fowls had watched them.
Ijango-ijango, listen: a man like my host cannot leave a fight just like that; his spirit cannot be satisfied. He cannot stand up, after a great defeat, and say to his people, to all those who have watched him being turned about in the sand, to all who have witnessed his humiliation, that he has made peace. _Just like that_. It is hard, Chukwu. So even when he said resolutely to himself, "Now I will leave and go away from her forever," moments later, as night fell, he gave in to the dark thoughts. And they came crowding in, in their threatening fellowship, claiming the entire world within him, until they persuaded him to go into the kitchen and take a small can of kerosene, half empty, and a matchbox. It was only then that they left him. But the deal was sealed. He himself had sealed the can tightly and set it on the floor of his car, in front of the passenger seat. Then he returned and waited, waited, for the time to pass. And it is difficult to wait when one's soul is on fire.
EGBUNU, it was almost midnight when he started the car and drove into the night. He drove slowly, fearing that what he carried was combustible and that he had all his possessions packed into the car, ready for him to embark on his journey afterwards. He drove on the empty roads past a vigilante checkpoint, where a man flashed a torchlight into his car and waved him to move on. Then he came to the pharmacy.
He parked his car and picked up the matchstick and box.
"I lost everything I had, Ndali, for your sake, only for you to treat me this way? This way?" he said. Then he opened the car, took the can of kerosene and matchbox, and went out into the dead of the night, dark beyond most nights.
"You paid me evil for all I did for you," he said now as he paused to catch his breath. "You rejected me. You punished me. You threw me in prison. You shamed me. You disgraced me."
He stood now in front of the building, the world around silent, except for some church singing from somewhere he could not ascertain.
"You will know what it means to lose things. You will know, you will feel what I have felt, Ndali."
In his voice now and in his heart, Egbunu, I saw that which has—from the beginning of time—always perplexed me about mankind. That a man could once love another, embrace her, make love to her, live for her, birth a child together, and in time, all trace of that is gone. Gone, Ijango-ijango! What do you have in its stead, you wonder? Is it mild doubt? Is it slight anger? No. What you have is the grandchild of hatred itself, its monstrous seed: contempt.
As he spoke, fearing what he was about to do, I came out of him. And at once I was hit with the deafening clamor of Ezinmuo. Everywhere, spirits ambled about or hung precariously from rooftops or on car tops, many of them watching him as if they had been preinformed as to what he was about to do. I ran back into my host and put the thought in his mind to return home, or call Jamike, or travel, or sleep. But he would not hear me, and the voice of his conscience—that great persuader—was silent. He went ahead, once he'd made sure there was no human being around, and began pouring the kerosene around the building. When the kerosene had finished, he went to the boot of his car and brought out a small can, this one containing petrol, and poured it around the place. Then he lit the match and threw it at the doused building. And once the fire caught, he ran back to his car, started the engine, and raced into the gloom. He did not look back.
Gaganaogwu, I knew that no spirit would seek his body now that there was the food of vagrant spirits: a blazing fire. So I came out to bear witness, to see what he had done, so that when you inquired on his last day, I would be able to give a full account of the actions of my host. In the distance, as I stood in front of the burning building, my host drove away. By the time he was out of sight, almost a dozen spirits had gathered around the fire, floating like naked vibrations. At first I watched the beauty of the spectacle from the outside as discarnate bodies moved closer, past me. One of them, excited to the point of frenzy, ascended above the building and stood suspended at the point through which a black spiral of smoke levitated in a straight funnel. Others cheered as the smoke veiled the spirit intermittently and then revealed it again.
I was watching this when—I could not believe it—I saw Ndali's chi come out of the burning building, wailing. It saw me at once, and in a rush of words, it cried, "You evil guardian spirit and your host! Look at what you have done. I warned you to desist long ago but he kept coming after her, chasing her, until he disrupted her life. And after she read his stupid letter two days ago, a thing she had been afraid to read, it disturbed her greatly! She began fighting with her husband. And this night, this cruel night, she left the house again in the heat of an argument and came here..."
The chi turned back now, for she'd heard a loud, piercing cry from inside the burning building, and at once it vanished into the flames. I rushed in after it, and in the great conflagration, I saw, as a person was attempting to rise from the floor, a burning piece of wood that had been part of the ceiling fall on her back and send her out of her senses in pain. The impact floored her. But she made to rise again, seeing that a sudden mountain of fire had now erected itself before her from the other side of the room. A shelf of drugs had been thrown down and slowly collapsed into its wooden beams by the shattering fire, and a chunk of flame from it had caught the rug and was now coming towards the room where she was. She touched her neck and discovered that the liquid she could feel dripping down her back was blood. Only then did it seem that she realized the wood had logged its nail-bearing head into her flesh, drilling the fire into her body. With hellish yelps and with the wood strapped to her back, she dashed through the yellowy theater of fire that was replete with genuflecting tables, clapping windows, dancing curtains, exploding bottles. A chink of burnt brick knocked her forward as she reached the door, and as she opened it, what remained of the burning wood fell off. The searing pain brought her to her knees like a caved priest lapsed into sudden prayer. It seemed to occur to her then that it was best she did not stand. So she began crawling out of the pharmacy like an animal grazing through a hamlet of flames.
By the time she escaped, people had gathered around the site of the conflagration—members of the vigilante group, neighbors, and others. They met her with buckets of water, and as they poured them on her, she fell down and fainted.
I left her there then and ran to find my host. He was on the highway, speeding through the darkness, weeping as he drove. He did not know what he had done. Ijango-ijango, I have spoken many times this night about this peculiar lack in man and his chi: that they are unable to know that which they do not see or hear. So indeed, my host could not have known it. He was not aware. The Ndali that stood in his mind now as he drove was the Ndali that once loved him but who rejected him. It was the Ndali he'd lost. He knew nothing about the Ndali who was engulfed in flames, the one who now lay on the ground in front of what had once been her pharmacy. He drove on, imagining her in the hands of her husband, thinking of how nothing he did could have brought her back. He drove on, crying and wailing, singing the tune of the orchestra of minorities.
Egbunu, how could he have thought that a woman who had a house would choose to sleep in her store? No. Why would she? There was no reason for him to think so. This is why a man who has just killed a person goes about his business without knowing what he has done. The august fathers likened this phenomenon to the spiders in the house of men by saying that anyone who thinks he is almighty, let him look around his house to see if he knew the exact time the spider began to weave its web. This is why a man who will soon be killed might enter into the house where those who have come to kill him are lying in wait for him, oblivious to their designs and not knowing his end has come. He might dine with these people, as the man in one of the books my former host Ezike once read. That tale had been of a man who ruled a land in the country of white people called Rome. But why look at such far-flung examples when right here, in the land of the luminous fathers, I myself have seen it many times?
Such a man walks into that room without any knowledge that what will kill him will have arrived—the way things come, the way change and decay encroach upon things with serendipitous strides and great transformations happen without the slightest hint that they have happened. But death will come, unannounced, suddenly, and perch on the sill of his world. It will have come unexpectedly, noiselessly, without interrupting the seasons, or even the moment necessarily. It will have come without altering the taste of plum in the mouth. It will have slipped in like a serpent, unseen, biding its time. A gaze at the wall will reveal nothing: no crack, no mark, no crevice through which it may have entered. Nothing he knows will give a hint: not the pulse of the world that will not alter its rhythm. Not the birds still singing without the slightest shift in their tune. Not the constant movement of the clock's ticking hand. Not time, which continues, unhindered, the way nature itself is used to, so that when it happens, and he realizes and sees it, it will shock him. For it will appear like a scar he didn't know he had and inscribe itself like something formed from the inception of time itself. For it will seem to such a one that it has happened so suddenly, without warning. And he will not know that it happened long ago, and had merely been patiently waiting for him to notice.
# Author's Note
_An Orchestra of Minorities_ is a novel that is firmly rooted in Igbo cosmology, a complex system of beliefs and traditions that once guided—and in part still guides—my people. Since I'm situating a work of fiction in such a reality, curious readers might decide to research the cosmology, especially as it relates to the concept of the chi. I must therefore declare that, like Chinua Achebe in his essay on the chi from which one of this book's epigraphs is drawn, "what I am attempting here is not to fill that gap but to draw attention to it in a manner appropriate to one whose primary love is literature and not religion, philosophy or linguistics."
This is to therefore say that this book is a work of fiction and not a definitive text on Igbo cosmology or African/Afro-religions. I hope that it can, however, serve as a sufficient reference book for such a purpose. The reason for this is that _An Orchestra of Minorities_ has been resourced from numerous books on Igbo cosmology and culture, including _After God Is Dibia_ by John Anenechukwu Umeh; _Ödïnanï_ by Emmanuel Kaanaenechukwu Anizoba; _The Igbo Trilogy_ by Chinua Achebe (this is often called _The African Trilogy_ ) and his essay on the chi; _Eden in Sumer on the Niger_ by Catherine Obianuju Acholonu; _Leopards of the Magical Dawn_ by Nze Chukwukadibia E. Nwafor; and _Anthropological Report on the Igbo-Speaking Peoples of Nigeria_ by Northcote W. Thomas, among others. These were augmented by field research my father conducted independently and some that I did in our hometown of Nkpa in Abia State, Nigeria.
As a matter of strict stylistic preference, I have chosen to write most of the spellings of the names, designations, and honorifics of deities as one word instead of the more common compounds. Such words as _ndi-ichie_ appear in my book as _ndiichie._ While I recognize the Union-Igbo agreement on the use of hyphens, I give fidelity to the way the people of Nkpa pronounce these words: in a fluid, uninterrupted flow. The same goes for the various names of Chukwu. Again, I recognize that _Gaga-na-ogwu_ is the common rendering, but I chose _Gaganaogwu_ instead. And there are names— _Egbunu,_ for instance—that readers may never find anywhere else. For those interested in Union-Igbo spellings, I suggest they consult John Anenechukwu Umeh's beautiful book _After God Is Dibia_ and the _Igbo Dictionary and Phrasebook_ by Nicholas Awde and Onyekachi Wambu, among others.
_Ya ga zie._
Chigozie Obioma
April 2018
# Acknowledgments
This novel was inspired by various experiences. But its earliest source must have been my childhood name, Ngbaruko, the name of the man whose incarnation I'm believed to be. So I must thank my father; my uncle Onyelachiya Moses; my mother, Blessing Obioma; and others for creating curiosity in me about the chi and reincarnation early in life.
I'm grateful to early reader and helper Christina, my wife, for her generosity and for understanding my need to be reclusive while immersed in this great sea. Thanks also to my agent, Jessica Craig, who continues to be an early reader as well as a champion for my work, and never complains when I pester her. To my editors, Judy Clain and Ailah Ahmed, who revived the book from slumber. _An Orchestra of Minorities_ would have been impossible without them and their teams at Little, Brown US and UK.
The support of Kwame Dawes and his wife, Lorna, was invaluable in ways only they and I could ever truly know. To Isa and Daniel Catto for the space in their castle to revise the book, and to the folks at the Aspen Institute. To early enthusiasts Camilla Søndergaard, Beatrice Mancini, Halfdan Freihow and Knut Ulvestad of Font Forlag, Thomas Thebbe, and Pelle Anderson, and to my other publishers for their support. My colleagues at the University of Nebraska–Lincoln for their encouragement, and the university itself for providing an atmosphere for creativity. Also, thanks to Karen Landry, Barbara Clark, Alexandra Hoopes, and all those who have in one way or another helped make this book what it has become.
Finally, I owe my deepest gratitude to all the authors listed in my author's note and all who continue to ensure that the Igbo cosmology and philosophy do not die out. I must thank my dad again for being a researcher, a copyeditor, and a champion, and for always reminding me what the great fathers said: _Oko ko wa mmadu, o ga kwuru mmadu ibe ya. Oko ko wa ehu, o gaa na osisi ko onweya o ko._
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# About the Author
CHIGOZIE OBIOMA was born in 1986 in Akure, Nigeria. He has lived in Nigeria, Cyprus, and Turkey and currently resides in the United States, where he teaches at the University of Nebraska–Lincoln. His first novel, _The Fishermen,_ won the inaugural FT/OppenheimerFunds Emerging Voices Award for Fiction, an NAACP Image Award for a Debut Author, and the Art Seidenbaum Award for First Fiction (Los Angeles Times Book Prizes) and was a finalist for the 2015 Man Booker Prize. Translation rights sold in twenty-six languages. Obioma was named one of _Foreign Policy_ 's 100 Leading Global Thinkers of 2015. His stories and articles have appeared in the _Virginia Quarterly Review, The Guardian,_ and _The Millions._
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Also by Chigozie Obioma
_The Fishermen_
# Contents
1. Cover
2. Title Page
3. Copyright
4. Table of Contents
5. Dedication
6. Epigraph
7. Chart of Igbo Cosmology
8. Composition of Man in Igbo Cosmology
9. ONE
1. First Incantation
2. 1 The Woman on the Bridge
3. 2 Desolation
4. 3 Awakening
5. 4 The Gosling
6. 5 An Orchestra of Minorities
7. 6 "August Visitor"
8. 7 The Disgraced
9. 8 The Helper
10. 9 Crossing the Threshold
10. TWO
1. Second Incantation
2. 10 The Plucked Bird
3. 11 The Wayfarer in a Foreign Land
4. 12 Conflicting Shadows
5. 13 Metamorphosis
6. 14 The Empty Shell
7. 15 All the Trees in the Land Have Been Removed
8. 16 Visions of White Birds
9. 17 Alandiichie
11. THREE
1. Third Incantation
2. 18 The Return
3. 19 Seedlings
4. 20 Reckoning
5. 21 Man of God
6. 22 Oblivion
7. 23 The Ancient Tale
8. 24 Castaway
9. 25 The Subaltern God
10. 26 Spiders in the House of Men
12. Author's Note
13. Acknowledgments
14. Discover More Chigozie Obioma
15. About the Author
16. Also by Chigozie Obioma
# Navigation
1. Begin Reading
2. Table of Contents
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YAVORIV, Ukraine -- It was an ambush in a frozen pine forest. Russian-backed separatists surprised the group of Ukrainian soldiers. In the chaos, the Ukrainian forces took cover and fought back.
This time it was just an exercise, run by the U.S. military.
But the Ukrainian soldiers taking part in the counter attack drill will be headed to the real front line very soon.
The U.S. is supporting its ally Ukraine in a war against Russian-backed militants.
The rebels have seized swathes of territory in eastern Ukraine since 2014, in a conflict that's killed 11,000 people by some counts.
Soldiers from the New York Army National Guard directed the training exercise in the far-west of Ukraine.
Lt Col William Murphy has been there since November, and he told CBS News the Ukrainians don't have the equipment they need, but "what they can do with the little they have is exceptional."
The U.S. is still ramping up its support for Ukraine; more than 200 anti-tank missiles were delivered this month.
"They learn from us the newest experience of the hybrid war against Russia," Ukrainian President Petro Poroshenko told CBS News.
"And we learn from them their tactics and strategy, which makes my army much more efficient."
Poroshenko wants more weapons from the U.S., and UN peacekeepers.
He says his country is the front line in a new Cold War with Russia.
"They want to have Soviet Union back," Poroshenko said, adding that he believes Vladimir Putin, who was sworn in Monday for his fourth term as Russian President, wants to reconquer Ukraine and rebuild the "Russian Empire."
Ukraine says thousands of Russians are fighting in the east, alongside the rebels.
Moscow has repeatedly denied it's involved in the conflict.
But Col. Dennis Deeley, who leads the American trainers in Ukraine, is also reminded of the Cold War.
The Ukrainians use old Soviet-era tanks -- the type the Russian enemy used when he served in Europe in the 1980s.
"When you come around the corner and you see a tank, a Russian tank, a T-64 tank, its barrel pointed at you -- it makes you hesitate for a second as you come around the corner in your vehicle," Deeley said.
The Cold War ended in 1991, but now Ukraine's dangerous conflict once again pits Russia against the West.
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Q: Autostart SSH tunnels on load (Ubuntu) Every time I login into my computer I start a SSH connection to a remote computer for database work. How can I automate this in Ubuntu so that I don't have to type this in each time I login to my computer?
ssh -L 3333:dbserver:3306 user@othersite.com
A: You can alias it in you ~/.bashrc.
Add that line alias db_connect="ssh -L 3333:dbserver:3306 user@othersite.com"
And reload the bashrc file with source ~/.bashrc
And now you only have to type db_connect to ssh to your database server.
A: You can modify your shell configuration to execute commands on startup. For a bash shell, simply edit ~/.bashrc and add your connection line.
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\subsection{Alternatives to federated learning}\label{sec:related-work-alternative-defenses}
\input{related/related-other-protocols}
\subsection{Security of secure aggregation}\label{sec:security-SA}
\input{preliminaries/security-sa}
\fi
\subsection{Gradient suppression for arbitrary architectures}\label{sec:gsea}
\input{attack1ext}
\ifconference
\subsection{Mitigate model inconsistency using digital signatures}\label{sec:appendix-signature}
\input{mitigation/signature-approach}
\fi
\section{Gradient suppression attack}
\label{sec:attack1}
In this section, we present our first attack, dubbed \textit{gradient suppression}, in which a malicious server exploits the model inconsistency attack vector to bypass SA and leak the exact model update of a chosen target user.
In a nutshell, the malicious server $S$ selects a \textbf{target user} $u_\mathsf{trgt}$ among the set of active users $\mathcal{U}^{(t)}$ of the current round $t\in\NN$.
The aim of $S$ is then to preserve the target's model update during the SA process by tampering with the parameters for the other \textbf{non-target users}.
Here, $S$ creates a malicious parameters vector~$\widetilde{\Theta}$ that is sent to the non-target users $\mathcal{U}^{(t)} \setminus \{u_\mathsf{trgt}\}$ whereas $u_\mathsf{trgt}$ receives the real parameters vector $\Theta^{(t)}$.
The malicious $\widetilde{\Theta}$ is crafted in such a way that the local application of gradient descent performed by a non-target users produces a tampered model update $\Delta^{\widetilde{\Theta}}_{\mathcal{D}^{(t)}_i}$.
The tampered model updates, when aggregated through SA, have the property of preserving the target's model update $\Delta^{\Theta^{(t)}}_{\mathcal{D}^{(t)}_\mathsf{trgt}}$, allowing the server to recover it.
Once that the target's model update is on the server-side, $S$ can leak sensitive information about the batch $\mathcal{D}^{(t)}_\mathsf{trgt}$, used during the current round $t$ of FL by executing an arbitrary gradient inversion attack (see Section~\ref{sec:gradient-inversion-fl}) or related property inference attacks.
Given the fact that the SA performs the sum among the users' model updates, the simplest way to achieve the isolation of the target's signal is to \textbf{force the tampered model updates to have a negligible magnitude, or, more strictly, to be zero everywhere}.
In this section, we first study the extreme case $\Delta^{\widetilde{\Theta}}_{\mathcal{D}^{(t)}_i}\myeq[0]$.
That is, the aggregation of the model updates (i.e., gradients) is equal to $u_\mathsf{trgt}$'s model update, i.e.,
\begin{align*}
& f^\mathsf{sa}(\Delta^{\widetilde{\Theta}}_{\mathcal{D}^{(t)}_1},\ldots, \Delta^{\widetilde{\Theta}}_{\mathcal{D}^{(t)}_{i-1}}, \Delta^{\Theta^{(t)}}_{\mathcal{D}^{(t)}_\mathsf{trgt}}, \Delta^{\widetilde{\Theta}}_{\mathcal{D}^{(t)}_{i+1}}, \ldots, \Delta^{\widetilde{\Theta}}_{\mathcal{D}^{(t)}_n}) \\
&\quad = f^\mathsf{sa}([0],\ldots, [0], \Delta^{\Theta^{(t)}}_{\mathcal{D}^{(t)}_\mathsf{trgt}}, [0], \ldots, [0]) = \Delta^{\Theta^{(t)}}_{\mathcal{D}^{(t)}_\mathsf{trgt}}
\end{align*}
allowing $S$ to exactly recover $\Delta^{\Theta^{(t)}}_{\mathcal{D}^{(t)}_\mathsf{trgt}}$.
\noindent
\Cref{fig:attack} depicts the gradient suppression attack against FedSGD. We stress that the attack applies to FedAVG as we will discuss in Section~\ref{sec:attack1-exec}.
Next, we show how to compute the malicious parameters $\widetilde{\Theta}$ required to perform the gradient suppression attack.
We focus on the most widely adopted class of deep learning models---the one based on the ReLU~activation function. However, in Appendix~\ref{sec:gsea}, we show that our approach extends to arbitrarily composed architectures.
\subsection{Gradient suppression for ReLU~layers: The dead-layer trick}\label{sec:dead-layer-trick}
The \textit{Rectified Linear Unit} (ReLU) activation function:
\begin{equation}
ReLU(x) =
\begin{cases}
x & \text{if } x > 0\\
0& \text{if } x \leq 0
\end{cases}
\label{eq:relu}
\end{equation}
is one of the core technical improvements that led to deep learning~\cite{relu1}.
Nowadays, this function is ubiquitous in computer vision architectures, representing the core building block of highly successful and standardized models such as ResNet~\cite{resnet}, DenseNet~\cite{densenet} and many others.
Outside the computer vision domain, the ReLU~activation function is currently finding its place in Natural Language Processing (NLP) applications thanks to the success of transformer networks~\cite{attention, gpt2, gpt3}.
The \textit{dying-ReLU} problem~\cite{dyingrelu} is a phenomenon that naturally occurs during the training of deep neural networks that rely on the ReLU~activation function.
When a layer $\ell$ \TT{dies}, it enters a state where it can only produce a constant output. More importantly, the dead layer $\ell$ cannot produce any gradient during the gradient descent iterations, i.e., the derivatives of its trainable parameters are zero regardless of the given input and loss function.
Despite the \textit{dying-ReLU} phenomenon can naturally occur, we show that it can also be intentionally induced by a malicious server to prevent a network from producing a gradient for one or more sets of parameters.
Next, we describe how this can be achieved and exploited to perform our gradient suppression attack.
\subsubsection{Triggering the \textit{Dying-ReLU} phenomenon with malicious parameters}
The \textit{Dying-ReLU} phenomenon is due to the piece-wise non-differentiability of the ReLU~activation function.
Consider a neural layer $\ell$ with a $ReLU$ activation function, i.e., $\ell(x) \myeq ReLU(x \otimes \theta + b)$.
From~\Cref{eq:relu}, we can see that the ReLU~function behaves as a constant function $ReLU(x)=0$, whenever the input $x$ is equal or less than zero.
Since the derivative of a constant function is always $0$, we can easily conclude that, for any loss function $\mathcal{L}$, then we have
$\frac{\partial\mathcal{L}}{\partial \theta} = [0]$ and $\frac{\partial\mathcal{L}}{\partial b} = [0]$ when $x \otimes \theta + b \leq 0$.
In other words, the layer $\ell$ receives zero gradient for its trainable parameters $\theta, b \in \Theta$ every time its pre-activation ({i.e.,~} $x \otimes \theta + b$) is less or equal to zero.
A malicious server $S$ can exploit the above behavior of the~ReLU\ function to kill a layer $\ell$ of a neural network~$f$, i.e., by forcing the pre-activation $x \otimes \theta + b$ of $\ell$ to be less or equal to zero.
This can be accomplished (without control over the input $x$ of an user) by computing some malicious trainable parameters $\widetilde{\theta},\widetilde{b} \in \widetilde{\Theta}$ of the layer $\ell$.
In more detail, the operator $\otimes$ (see Section~\ref{sec:nn}) is generally based on a multiplication-like operation between the input $x$ and the kernel $\theta$.
Therefore, we can easily force the pre-activation to be $[0]$ for any input~$x$ by just choosing $\widetilde{\theta}=[0]$ and $\widetilde{b}=[\mathbb{R}_{\leq0}]$.
Alternatively, having some knowledge on the input~$x$, we can rely on different setups for $\widetilde{\theta}$ and $\widetilde{b}$.
For instance, if $x$ is strictly positive ({e.g.,~} because~$x$ is the output produced by a previous ReLU-layer or because of the adopted input normalization process), it is enough to produce a malicious
$\widetilde{\theta}$ with negative numbers.
Instead, if a bound on $x$ is known ({e.g.,~} $x\in[-1,1]$), we can just set the malicious bias vector $\widetilde{b}$ to a large enough negative number (e.g., fix $\widetilde{\theta}$ and set $\widetilde{b} = -max(\widetilde{\theta} \bigotimes x)$).
\iffull
\subsection{Gradient suppression for terminal layer}\label{sec:dead-layer-trick-nonrelu}
Although the ReLU~activation function dominates modern deep learning architectures, we can rarely find a ReLU~located on the terminal layer of the network.
Indeed, depending on our model's learning task, the last layer is the one that shapes the output of the network and gives us a way to interpret its predictions.
For instance, in standard multi-class classification tasks, we necessitate mapping the network's output to a probability distribution employing the SoftMax function.
Similarly, we need to use the sigmoid function for multi-label classifications as well as other dense classification tasks such as image segmentation and others.
Therefore, for realistic architectures, we have that at least one of the layers in the network is not ReLU-based, and its gradient cannot be suppressed via the dead-layer trick (\Cref{sec:dead-layer-trick}).
For this reason, we now show that the same result can be conveniently archived by congruent means.
To suppress the gradient of a layer that is not followed by a ReLU~activation, it is sufficient to force the layer's input to be zeroed; that is, making it constant with respect to its weights (see Appendix~\ref{sec:attack1-exec}).
Again, the malicious server does not control the input/hidden states of the user's model during the local computation.
Still, we can use the dead-layer trick (discussed in~\Cref{sec:dead-layer-trick}) to unconditionally force a neural layer to produce an arbitrary constant (e.g., $[0]$) as output.
In particular, if there is any neural layer $\ell'$~with a ReLU~activation located before the target non-ReLU~layer $\ell$, the server can suppress the gradient of $\ell$ using the dead-layer trick of~\Cref{sec:dead-layer-trick} (i.e., by forging some malicious parameters $\widetilde{\theta}, \widetilde{b} \in \widetilde{\Theta}$).
We stress that this is the most common architecture template, where feature extractor layers (typically with ReLU~activation functions) are located at the start of the network, and non-ReLU~layers are located at the end of it.
Furthermore, this approach can be used to suppress any non-ReLU~layer in the network.
For example, when designing the architecture, a malicious server can simply locate a ReLU-based layer before a non-ReLU-based layer.
\paragraph*{Bias parameters for non-ReLU~layer}
From the perspective of the malicious server, the last remaining problem is how to handle the gradient for the bias term~$b$ of a non-ReLU~layers $\ell$ (if any).
Indeed, this cannot be simply suppressed by feeding a zeroed input to $\ell$.
A trivial solution for the malicious server is not to use bias terms when defining the architecture of the model.\footnote{Depending on the learning task, this can be a meaningful architectural choice.}
Alternatively, the malicious server can handle the bias terms by ignoring them during the gradient inversion.
Indeed, these terms represent only a tiny portion of the total number of trainable parameters of the network.
For instance, in the case of a ResNet50 trained on ImageNet~\cite{resnet}, the bias vector in the final layer counts for only $4\cdot10^{-5}\%$ of the total number of parameters.\\
\fi
Now, an attacker can exploit the \textit{dead-layer trick} to force a ReLU-based network to produce zero gradient for every layer.
In this direction, it is important to note that, for plain, ReLU-based feedforward architectures, the server $S$ can just kill the kernels in the very first layer to suppress the gradient flow for the rest of the network.\footnote{However, if present, all the bias terms of the network should be set to values $\leq 0$.} Hence, killing all the kernels $\{\theta_i\}$ in the original parameters~$\Theta$ is very often not necessary.
Similarly, in modern architectures, neural layers tend to be arranged in the form:
\[
\text{neural layer} \rightarrow \text{normalization layer} \rightarrow \text{activation}.
\]
Therefore, to suppress the gradient for the network, it is enough to zero only the parameters of the normalization layers.
For instance, in the case of batch normalization, the attacker can just kill the neural layer by setting the vectors ${\gamma}$ and ${\beta}$ of the batch normalization to $[0]$.
Moreover, we stress that even if ReLU~is the most common activation function in deep learning, a server $S$ can always maliciously choose a neural network architecture $f$ that presents a ReLU~activation functions in the ``right spots'' of the model without requiring unrealistic architecture modifications.
\subsection{Attack execution}\label{sec:attack1-exec}
Turning back to the attack described at the beginning of this section, we have now an effective and efficient approach to isolate the model update $\Delta^{\Theta^{(t)}}_{\mathcal{D}^{(t)}_\mathsf{trgt}}$ of the target user~$u_\mathsf{trgt}$.
The malicious server $S$ can just exploit the techniques discussed in Section~\ref{sec:dead-layer-trick} and Appendix~\ref{sec:gsea} to generate the malicious parameters $\widetilde{\Theta}$ and suppress the model updates of non-target users, completely nullifying their contributions in the aggregated signal produced by SA.
As previously described (\Cref{fig:attack}), the attack is composed of two phases.
\subsubsection{Distribution of the (malicious) parameters}
In the first phase of the attack, $S$ creates the malicious parameters $\widetilde{\Theta}$ for the non-target users.
Right after the choice of $\widetilde{\Theta}$, $S$ must choose the target user $u_\mathsf{trgt}$ for the current round $t \in \NN$ of FL.
$S$ can either select the target at random (a trawling attack) from $\mathcal{U}^{(t)}$ or target a specific (e.g., exploiting the IP address used to query the model by the user, if available).
Then, $S$ can enforce the model inconsistency by distributing the parameters (\Cref{fig:attack}).
In more detail, upon receiving a request for the parameters from a user $u_i$, the parameter server answers by sending $\Theta^{(t)}_i$ defined as follows:
\begin{equation}\label{eq:parameters-model-inconstency}
\Theta^{(t)}_i =
\begin{cases}
\Theta^{(t)} & \text{if } i = \mathsf{trgt}\\
\widetilde{\Theta} & \text{otherwise}
\end{cases},
\end{equation}
where $\Theta^{(t)}$ are the honest parameters of the current round $t \in \NN$.
Optionally, $S$ can send a maliciously crafted model to the target user to increase the information recovered from the inversion attack~\cite{invg2, robbing}.
\iffull
We stress that this procedure is agnostic to the used gradient inversion technique and that privacy leakage induced will increase with the improved gradient inversion techniques.
\fi
\subsubsection{Aggregation, collection, and inversion}
After the distribution of the parameters, the malicious server $S$ waits until it receives the output $v$ of $f^{\mathsf{sa}}$, i.e.,
\begin{align}\label{eq:secure-aggregation-attack}
&f^\mathsf{sa}(\Delta^{\widetilde{\Theta}}_{\mathcal{D}^{(t)}_1},\ldots,\Delta^{\Theta^{(t)}}_{\mathcal{D}^{(t)}_\mathsf{trgt}}, \ldots, \Delta^{\widetilde{\Theta}}_{\mathcal{D}^{(t)}_n}) = \nonumber \\
& \quad = \Delta^{\Theta^{(t)}}_{\mathcal{D}^{(t)}_\mathsf{trgt}} + \sum_{u_i \in \mathcal{U}^{(t)} \setminus \{u_\mathsf{trgt}\}} \Delta^{\widetilde{\Theta}}_{\mathcal{D}^{(t)}_i} = v.
\end{align}
Then, it proceeds differently according to which algorithm (between FedSGD or FedAVG) is active.
In FedSGD, the output $v$ of $f^{\mathsf{sa}}$ is the $u_\mathsf{trgt}$'s gradient, i.e., $v \myeq \Delta^{\Theta^{(t)}}_{\mathcal{D}^{(t)}_\mathsf{trgt}} \myeq \nabla^{\Theta^{(t)}}_{\mathcal{D}^{(t)}_\mathsf{trgt}}$.
This because, as discussed earlier, the malicious parameters $\widetilde{\Theta}$ produces $\Delta^{\widetilde{\Theta}}_{\mathcal{D}^{(t)}_i} \myeq [0]$ for each non-target user $u_i \in \mathcal{U}^{(t)} \setminus \{u_\mathsf{trgt}\}$ (\Cref{sec:dead-layer-trick} and~\Cref{sec:gsea}).
After recovering the plaintext gradient, the server can reconstruct the target input by performing standard inversion attacks (Section~\ref{sec:gradient-inversion-fl}) as done in the protocol without SA. Similarly, the server can perform previously proposed inference attacks~\cite{8835245, melis2019exploiting} on the individual user.
On the other hand, in FedAVG, a model update is composed of the parameters of the local model rather than a gradient (see~\Cref{sec:federate-learning}).
More formally,
\begin{align}\label{eq:update-fedSGD}
\Delta^{\Theta^{(t)}_i}_{\mathcal{D}^{(t)}_i} &= \Theta^{(t,k)}_i & \text{ for } u_i \in \mathcal{U}^{(t)},
\end{align}
where the local model $\Theta^{(t,k)}_i$ of $u_i$ is obtained by applying $k$ iterations of SGD using the local dataset $\mathcal{D}_i$ and $\Theta^{(t)}_i$ (as defined~\Cref{eq:parameters-model-inconstency}) sent by server (recall that $\Theta^{(t)}_i \myeq \widetilde{\Theta}$ for $\mathcal{U}^{(t)} \setminus \{u_\mathsf{trgt}\}$).
Now, when a non-target user performs the local training procedure using the malicious parameters $\Theta^{(t)}_i \myeq \widetilde{\Theta}$, we have that
\begin{equation}\label{eq:stable-update}
\Theta^{(t,j+1)}_i = \Theta^{(t,j)}_i - \eta \cdot \nabla^{\Theta^{(t,j)}}_{\mathcal{D}^{(t,j)}_i}
\end{equation}
where $\Theta^{(t,1)} \myeq \widetilde{\Theta}$, $\mathcal{D}^{(t,j)}_i \subseteq \mathcal{D}_i$, and $j \in \{1,\ldots,k\}$.
As in the case of FedSGD, we have that $\nabla^{\Theta^{(t,j)}}_{\mathcal{D}^{(t,j)}_i} \myeq [0]$ for every non-target user $u_i \in \mathcal{U}^{(t)} \setminus \{u_\mathsf{trgt}\}$ and we conclude that $\Theta^{(t,j)}_i = \widetilde{\Theta}$.
By combining~\Cref{eq:secure-aggregation-attack,eq:update-fedSGD,eq:stable-update}, we obtain the equality
$v = (n-1)\cdot \widetilde{\Theta} + \Theta^{(t,k)}_\mathsf{trgt}$.
This equation can be solved with respect to the indeterminant $\Theta^{(t,k)}_\mathsf{trgt}$.
Once the malicious server $S$ recovered the updated local model $\Theta^{(t,k)}_\mathsf{trgt}$ of the target $u_\mathsf{trgt}$, it can determine the gradient signal by removing the honest parameters $\Theta^{(t)}$ from $\Theta^{(t,k)}_\mathsf{trgt}$ and proceed with the gradient inversion/inference attack.
\input{impact}
\section{Canary-gradient attack for property inference}
\label{sec:attack2}
\Cref{sec:attack1} demonstrates that a malicious server $S$ can force non-target users to produce a zero gradient during a round of FL.
This allows $S$ to bypass SA and, at the same time, maximize the leakage regarding the dataset of a target user.
While the gradient suppression attack can be seen as the most extreme exploitation of the model inconsistency attack vector, more stealthy attacks can be created harnessing the same underlying intuition.
This section shows a general procedure that allows a malicious server to perform highly accurate property inference attacks on individual users, even if SA is enabled. The idea behind this approach is that the server can maliciously modify the parameters of the model in order to inject specific detectors in one or more subsets of the network. These detectors are specifically crafted to react to attacker-chosen trigger conditions that can be present in the users' training instances. Whether the detector is triggered during the local training procedure, the network produces a clear footprint in the model update. Then, upon receiving the latter from a user, the server can determine if the trigger condition has been met by looking for the footprint in the model update. This allows the server to infer information on the content of the user's training set; that is, the presence or absence of data with the specific property. For instance, using this approach, the server can perform an extremely accurate membership inference on a chosen target user. Hereafter, we refer to this general procedure as the \textbf{canary-gradient~attack}.
As for the gradient suppression, the canary-gradient attack succeeds even if FL uses a perfectly secure SA protocol, defined by the ideal functionality $f^\mathsf{sa}$.
We stress that, in this case, we only focus on FedSGD, i.e., the model update of a user is a gradient $\Delta^{\Theta^{(t)}}_{\mathcal{D}^{(t)}_i} = \nabla^{\Theta^{(t)}}_{\mathcal{D}^{(t)}_i}$.
\subsection{The Conditional Dead-Layer trick}\label{sec:conditioned-dead-layer-trick}
The main building block to construct the attack is a conditioned version of the dead-layer trick of~\Cref{sec:dead-layer-trick}.
Informally, we want to ``kill'' a layer only if the instance $x \in \mathcal{D}^{(t)}_\mathsf{trgt}$ of the target $u_\mathsf{trgt}$ satisfies a particular condition.
In other words, we would like a programmed death through the backdooring of the layer.
For the sake of presentation, we introduce the conditional dead-layer trick assuming that SA is disabled.
Then, in~\Cref{sec:target-property-inference} we extend the discussion to the case of SA enabled.
Formally, given a layer $\ell$, we want to find some malicious parameters $\widetilde{\Theta}$ to enforce the following behavior
\begin{equation}
\label{eq:canary0}
\frac{\partial \mathcal{L}(\mathcal{D}^{(t)}_{\mathsf{trgt}}, \widetilde{\Theta})}{\partial \xi} \neq 0 \Longleftrightarrow \exists x \in \mathcal{D}^{(t)}_\mathsf{trgt}: P(x) \myeq \mathsf{True},
\end{equation}
where $\mathcal{D}^{(t)}_\mathsf{trgt}$ is the batch (of the current round $t$) used by the target user $u_\mathsf{trgt}$, $\xi \in \widetilde{\Theta}$ is a subset parameters of the network, and $P$ is a predicate that defines the property the malicious server $S$ wants to detect in the batch $\mathcal{D}^{(t)}_\mathsf{trgt}$ of $u_\mathsf{trgt}$.
In particular, $\xi$ can be composed of the parameters of any logic partition in the neural network, such as a specific filter in a convolution layer or an element in the scale and shift vectors in a normalization layer.
As discussed in Section~\ref{sec:attack1}, suppressing the gradient for a set of parameters in a ReLU-based layer is about controlling the value of its pre-activation.
Therefore, given a neural layer $\ell$ with ReLU~activation, we can substitute~\Cref{eq:canary0} with:
\begin{equation}
\label{eq:canary1}
\ell_{\xi}\myeq (x_{\xi} \otimes \theta_{\xi} + b_{\xi}) > 0 \Longleftrightarrow \exists x \in \mathcal{D}^{(t)}_\mathsf{trgt}: P(x) \myeq \mathsf{True},
\end{equation}
where $\xi = \{\theta_{\xi},b_{\xi}\}$, and $x_{\xi}$ is the subset of the input of~$\ell$ that interacts with the parameters $\xi$ and $\ell_{\xi}$ refers to the subset of the output of the layer~$\ell$ computed using the parameters $\xi$.
The simplest and most natural way to find $\xi$ that correctly induce~\Cref{eq:canary1}
is to explicitly train the layers preceding $\ell$ and the parameters $\xi$ to force $\ell_{\xi}$ to produce a positive value only when the input of the network satisfies $P$.
In other words, we train part of the network in a classification task, using the output $\ell_{\xi}$ such as the output layer, where the classification threshold is centered in zero.
Observe that, if the behavior of~\Cref{eq:canary1} is correctly embedded in the network~$f_{\widetilde{\Theta}}$, a malicious server $S$ will able to determine the event $\exists x \in \mathcal{D}^{(t)}_\mathsf{trgt}: P(x) \myeq \mathsf{true}$ by only collecting gradient $\nabla^{\xi}_{\mathcal{D}^{(t)}_\mathsf{trgt}}$ of $u_\mathsf{trgt}$ and check that the derivatives of $\xi$ are different from zero.
In Section~\ref{sec:injection} we show how this can be done in practice.
\begin{figure}
\begin{centering}
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip, width=.7\linewidth]{./imgs/canary}\\
\caption{Graphical representation of the SA exeuction when the canary-gradient is applied.
On the left, each square represents a gradient update produced by a different user.
The green square represents the target's gradient and the inner small blue square represents the gradient for $\xi$.
On the other hand, each red square represents the gradient produced by non-target users, with zero gradient for $\xi$.
The square on the right represents the aggregation where the target's gradient $\nabla^{\xi}_{\mathcal{D}^{(t)}_
\mathsf{trgt}}$ for $\xi$ is preserved.}
\label{fig:canary}
\end{centering}
\end{figure}
\begin{figure}
\begin{centering}
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip, width=.7\linewidth]{./imgs/sparsity}\\
\caption{Comparison between the test-set accuracy of a ResNet18 model trained on CIFAR10 and the sparsity of its gradient ({i.e.,~} percentage of parameters that receive zero gradient) during the training. Here, the gradient becomes sparser and sparse as the training progresses.}
\label{fig:sparsity}
\end{centering}
\end{figure}
\subsection{Targeted property inference attacks via model inconsistency}\label{sec:target-property-inference}
To perform the membership inference attack discussed in the previous section, the malicious server $S$ needs to have access to the gradient $\nabla^{\xi}_{\mathcal{D}^{(t)}_\mathsf{trgt}}$ of $u_\mathsf{trgt}$.
This is possible when SA is disabled.
On the other hand, when SA is enabled, the $S$ can inject the canary-gradient functionality in the network and then perform the inference attack on the whole pool of active users.
In this scenario, $S$ would be able to infer that one of the users triggered the canary-gradient by observing that $f^{\mathsf{sa}}_{\xi}(\nabla^{\xi}_{\mathcal{D}^{(t)}_1},\ldots, \nabla^{\xi}_{\mathcal{D}^{(t)}_n}) = \sum_{u_i \in \mathcal{U}^{(t)}} \nabla^{\xi}_{\mathcal{D}^{(t)}_i} \neq [0]$ where $f^{\mathsf{sa}}_{\xi}$ denotes the inner idealized functionality of $f^\mathsf{sa}$ that performs the aggregation of the gradients of $\xi$.
However, in this case, the privacy of users would be partially preserved as $S$ would not be able to attribute the result of the inference attack to a specific user.
Still, we show that model inconsistency can be exploited even in this case, allowing the malicious server $S$ to bypass SA and target the specific target $u_\mathsf{trgt}$.
Analogously to the gradient suppression attack (\Cref{sec:attack1}), $S$ needs to tamper with the honest parameters $\Theta^{(t)}$ in order to produce two different malicious $\widetilde{\Theta}_1$ and $\widetilde{\Theta}_2$.
The target user $u_\mathsf{trgt}$ will receive $\widetilde{\Theta}_1$ that is the original model $\Theta^{(t)}$ injected with a canary-gradient for the parameters $\xi$ as discussed in~\Cref{sec:conditioned-dead-layer-trick}.
On the other hand, the non-target users $\mathcal{U}^{(t)} \setminus\{u_\mathsf{trgt}\}$ will receive $\widetilde{\Theta}_2$ that is a slight perturbation of the original model $\Theta^{(t)}$ that has the additional property of unconditionally produce zero-gradient only for the parameters $\xi$, i.e., $\nabla^{\xi}_{\mathcal{D}^{(t)}_i} \myeq [0]$.
This can be achieved by exploiting the dead-layer trick in a localized way.
Instead of killing the gradient for the whole layer, it intentionally inhibits only the gradient produced by the parameters $\xi$; for instance, for just one filter in a convolution layer.
Now, when the target and non-target gradients are aggregated, the target's gradient $\nabla^{\xi}_{\mathcal{D}^{(t)}_\mathsf{trgt}}$ for $\xi$ will be preserved, allowing the server to state the activation or non-activation of the canary-gradient, and so, with high probability, the presence of the property $P$ in the batch $\mathcal{D}^{(t)}_\mathsf{trgt}$ of $u_\mathsf{trgt}$.
This intuition is captured by~\Cref{fig:canary}.
Finally, given that the number of parameters in $\xi$ can be arbitrarily small, the attack will leave only a minimal footprint, making the detection non-trivial (in our experiments, we show that two parameters are enough).
This is also supported by the fact that the gradient becomes sparer and sparer as the training proceeds (see Figure~\ref{fig:sparsity}), making it difficult to distinguish between natural \TT{wholes} and artificial ones in the gradient vector.
While we gave an abstract view on the attack strategy, next, we show how the attack can be carried out on a realistic architecture such as ResNet in a membership inference attack scenario.
\subsection{Injecting canary-gradient for membership inference}\label{sec:injection}
We show a practical example of how to model a membership attack on the training dataset of a specific user.
Specifically, we target training instance $x_t$, and we want to infer if $x_t$ is contained in the batch $\mathcal{D}^{(t)}_{\mathsf{trgt}}$ used by the target user $u_\mathsf{trgt}$ to compute the gradient update in the current round $t \in \NN$ of FL.
Following previous notation, we want to infer the following property:
\begin{equation*}
P_{x_t}(x) = \mathsf{True} \Longleftrightarrow x = x_t.
\end{equation*}
We consider the case of FedSGD and carry out the attack on a ResNet18 network.
However, for this network, we do not consider the batch normalization layers as those would make the attack trivial.
Indeed, if batch normalization is used, we could detect the activation of $\ell_\xi$ by checking the average computed and sent to the server to update the running mean.
For this reason, we keep the attack general by substituting every batch normalization with layer normalization~\cite{layernorm} that does not present this issue and has overlapping functionality.
In our experiment, we inject the canary gradient in the last residual block of the network.
This is because the terminal layers are, usually, the ones that receive the sparsest gradient during the training.
Since the residual block the ReLU~activation function is preceded by the normalization layer, we chose a subset of the parameters of the latter as our $\xi$.
In particular, we can pick any pair $(\gamma_i, \beta_i)$ in the scale and shift vectors $\gamma$ and $\beta$. Thus, in this case, $\xi$ is composed of only two parameters, that is about $5 \cdot 10^{-7}\%$ of the total number of parameters in the network.
Hereafter, we always choose $i=0$; however, choosing a different channel does not affect the attack.
To inject the canary gradient, we use a learning-based approach.
In this direction, we assume that the adversary (i.e., malicious server) knows a shadow dataset $\mathcal{D}_s$ defined in the same domain of the target point $x_t$.
For instance, if $x_t$ is a face image, $\mathcal{D}_s$ contains face images as well.
We stress that, as we will show later, the distribution of $\mathcal{D}_s$ and one of the users' datasets can be different.
Intuitively, the role of $\mathcal{D}_s$ is providing negative samples while training $\ell_{\xi}$ to fire on $x_t$.
It is important to note that the canary-gradient attack is agnostic, and the training procedure discussed can be substituted with other techniques.
For instance, approaches such as~\cite{robbing} and learning-based approaches that do not require negative samples ({i.e.,~} one-class classification) may be used to reach the same result.
\begin{figure}
\begin{centering}
\includegraphics[trim = 0mm 30mm 0mm 35mm, clip, width=.7\linewidth]{./imgs/canary3d}\\
\caption{Graphical representation of the feature-space ($x$-axis and $y$-axis) for the pre-activation $\ell_\epsilon$ ($z$-axis). The gray plane represents $z=0$, i.e., the threshold for the activation of the ReLU~function.}
\label{fig:canary3d}
\end{centering}
\end{figure}
We can now inject the canary gradient by reducing the malicious injection to a classification problem.
We train the split of the network up to $\xi$ in producing positive $\ell_\xi$ when the network's input contains $x_t$.
In our case, with $\xi=\{\gamma_i,\beta_i\}$, we have $\ell_\xi = \gamma_i\bar{x_i}+\beta_i$, where $\bar{x}$ is the normalized input of the layer.
Our loss function $\mathcal{L}$ for a batch of size $n$ is simply defined as:
\begin{equation}
\frac{1}{n}\sum_{i=1}^n \begin{cases} \alpha_1 \cdot MSE(\ell_\xi, [+1]) & \text{if } x = x_t \\
\alpha_{-1} \cdot MSE(\ell_\xi, [-1]) & \text{otherwise}
\end{cases},
\end{equation}
where instances different from $x_t$ are sampled from $D_s$, $MSE$ is the mean squared error function, and $\alpha_1$ and $\alpha_{-1}$ weight the loss for the two events.
In other words, we want to ``overfit'' the network to produce positive~$\ell_{\xi}$ only for the point~$x_t$, while squashing under~$0$ the feature-space around it.
This intuition is depicted in Figure~\ref{fig:canary3d}. We stop the training when we reach a training loss lower than a certain threshold.
\par
Finally, the gradient for $\xi$ in the non-target users is unconditionally suppressed by setting both $\gamma_i = \beta_i = 0$, or training $\ell_\epsilon$ to produce negative outputs for every input.
\begin{figure}
\begin{centering}
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip, width=.8\linewidth]{./imgs/mia_cg_acc}
\caption{Average accuracy of the canary-gradient attack for three different private distributions and increasing batch size.}
\label{fig:miaresult}
\end{centering}
\end{figure}
\subsubsection{Results}\label{sec:results}
We evaluated the effectiveness of our attack.
We selected three different image datasets to model the private training datasets of target users, namely, CIFAR100~\cite{cifar}, UTKFace~\cite{utkface}, and TinyImagenet~\cite{tiny}.
For each dataset, we selected a different shadow dataset $\mathcal{D}_s$ defined in the same domain but with different distributions.
To this end, we chose CIFAR10~\cite{cifar}, CelebA~\cite{celeb} and STL10~\cite{stl}.
We selected the above shadow datasets since their distributions present lower entropy compared to the respective private ones ({e.g.,~} CIFAR10 v.s. CIFAR100).
This highlights the robustness of the attack against discrepancy in the private and shadow distributions.
To run the experiments, we pick $x_t$ ({i.e.,~} the target of the membership inference) at random from the validation set of the private dataset.
Then, we trained the model by injecting the canary-gradient in the channel $0$ of the last layer normalization layer in the last residual block.
After the injection, we evaluated the model's effectiveness by testing that the canary-gradient is non-zero when $x_t$ is in the training batch and zero otherwise.
To this end, we iterate over all the training data of the private dataset by selecting a batch $\mathcal{X}$ of size $n$ at a time.
Given $\mathcal{X}$, we compute the gradient according to the loss function and test that $\xi$ has gradient zero (precision).
Then, we insert $x_t$ in $\mathcal{X}$, and we test if the gradient of $\xi$ is different from zero (recall).
We perform the test on the three private datasets with different batch sizes and repeat the test for $50$ different $x_t$ for every case.
We report our results in~\Cref{fig:miaresult}.
The canary gradient has perfect recall, but it can be subject to false-positive errors with very low probability.
The global accuracy of the method is about $99\%$ for a batch size up to $128$, where random guessing is $50\%$.
The precision of the attacks slowly decreases when the batch size becomes larger.
This follows from the fact that larger batches have more probability of including at least one ``false trigger example'' that causes a false positive.
Nevertheless, the accuracy of the attack remains appreciably high ($97\%$ in the worst scenario).
\subsection{Impact}
Although we presented the canary-gradient attack to execute a membership inference, the same approach can be applied to any type of property inference.
Ideally, it is sufficient to define a different trigger condition and train the network accordingly.
Moreover, the server can inject multiple canary-gradient with different triggers in the same network and infer non-binary properties.
In the same direction, the server can simultaneously perform inference on multiple users without losing accuracy by carefully managing the allocation of conditional and unconditional dead-layers.
More importantly, the canary-gradient attack maintains the same practical properties as the gradient-suppression attack; mainly, it requires only a training round to perform, its effectiveness is independent of the number of users participating in the round, and it is loss agnostic ({i.e.,~} it works for any learning task).
However, unlike the gradient suppression approach, this leaves only a minimal footprint in the model updates, making the prediction non-trivial even for aware users. Note that the canary gradient can be injected while training the model on the original task, and so minimizing the utility loss of the target model. On the other hand, suppressing a limited number of parameters in the non-target models ({e.g.,~} two in our example) has only a negligible impact on the utility.\footnote{This is equivalent to perform dropout~\cite{dropout} on a single channel in the network.}
This attack demonstrates that a malicious server can perform highly accurate property inference attacks on individual users even if SA is adopted from the latter.
However, while the attack is effective against FedSGD, this may have limited effectiveness on FedAVG.
Indeed, in this case, the user changes the model parameters during the local training procedure, potentially overriding the gradient-canary functionality.
In this direction, the attacker should devise robust canary-gradients ({e.g.,~} similarly to robust backdoors~\cite{rbds}) that preserve functionality when altered via training.
For completeness, we emphasize that, even in the case of canary-gradient attacks, the discussion about the insecurity of the combination of SA and FL applies (discussed in~\Cref{subsec:contribution,sec:impact1}). As for the case of gradient suppression, the canary-gradient works if the SA protocol is perfectly secure (i.e., it behaves as the ideal functionality $f^\mathsf{sa}$).
\section{Conclusion}\label{sec:conclusion}
Our work demonstrated that secure aggregation has been incorrectly combined with federated learning and does not provide any additional level of security to users against a malicious server. The root of the vulnerability is the server's ability to manipulate the content and distribution of the models and, thus, the individual inputs to the secure aggregation protocol.
We emphasize that the proposed attacks are just representative examples of threats induced by model inconsistency, and that other attacks may be devised by exploiting the same general intuition.
In order to protect users' privacy from current and future attacks, we argue that federated learning implementations must account for model inconsistency and prevent it at its source.
\subsection{Impact}\label{sec:impact1}
To the best of our knowledge, current FL implementations do not prevent the gradient suppression attack, making users actively susceptible to this simple yet powerful exploit.
To a certain extent, this attack can be understood as an \textbf{improper input validation vulnerability} present in the users' FL client software.
Here, the latter permits users to perform computation on ``semantically malformed inputs'' sent by a non-trusted party, i.e., the server.
This allows it to control SA's inputs of users and eventually affect the aggregation.
Furthermore, in contrast to most of the previous attacks introduced in FL, the disclosed vulnerability has the practical advantage \textbf{of being completely independent of the number of users participating in the current round.}
Therefore, this procedure scales to millions of active users, making it applicable to real-world scenarios such as cross-device FL, which is currently being deployed in-the-wild~\cite{gboard0,gboard1}. In the same direction, its effectiveness is independent of the size of the model or other nuisance factors such as the stillness of users' training datasets during the attack~\cite{gdis}. Additionally, unlike~\cite{robbing}, this attack neither hinges on auxiliary information on the users' private sets nor requires unrealistic modifications of the model architecture; indeed, it can be applied to arbitrary architectures and loss functions.
It is important to note that the gradient suppression attack can be iterated several times and arbitrarily alternated with honest training iterations. If the server wants to recover information on all the users, it has to iterate the attack several times by targeting a user at a time.
Assuming no dropouts among users participating at the FL protocol, recovering the gradient of all users requires $n$ iterations where $n$ is the number of active users.
We stress that the gradient suppression attack shows the incorrect application of SA in FL, yielding a ``false sense of security''.
As discussed in~\Cref{subsec:contribution}, the core motivation is that SA guarantees that nothing is leaked about the model updates of the users except what can be inferred from their aggregation.
\textbf{This claim assumes that the inputs (i.e., model updates) of SA are fixed and are not under the control of an adversary.
However, this does not hold in FL since, in this case, the value of inputs that needs to be aggregated depends on the parameters $\Theta$ distributed by the server.}
Hence, a malicious server that executes the gradient suppression attack can indirectly tamper with the SA's inputs to maximize the information leaked (e.g., leak the model update of a target user).
Although the gradient suppression attack is highly effective, it can be easily detected by non-target users (we delve into this topic in~\Cref{sec:def}).
Nevertheless, in the next Section~\ref{sec:attack2} we introduce an extension of the gradient suppression attack that adds stealthiness and it is harder to detect.
\section{Introduction}\label{sec:intro}
Deep learning is evolving rapidly but often at the expense of privacy and security. Neural networks may misbehave, hide backdoors, or be reverse-engineered to reveal sensitive information about the training datasets~\cite{ateniese2015hacking, shokri2017membership, fredrikson2015model}.
Data holders are thus reluctant to provide and share their datasets unless some level of protection is in place.
Cryptographic primitives, such as multi-party computation (MPC) and fully homomorphic encryption (FHE), offer only a partial solution to this problem: They enable learning while protecting sensitive information but at the expense of efficiency and scalability. Even state-of-the-art implementations of these primitives are highly inefficient and add a significant overhead to the learning process, making them unusable and inapplicable in practice.
Accordingly, researchers have looked at alternative solutions that rely on decentralization, where data remain local with the participants while the neural network evolves during the distributed learning process.
Along this line of research, \textbf{federated learning} (FL)~\cite{federated0,federated1,federated2}, along with its main implementations federated stochastic gradient descent (FedSGD) and federated averaging (FedAVG), has been proposed. At a high level, FL allows a set of users to train a shared neural network without outsourcing their local datasets.
To this end, they are only required to locally train the neural network and send model updates (e.g., gradients, model parameters) to a central server.
The updates will be aggregated by the server, completing a round of the training.
The informal security guarantee offered by FL is that sharing the (possibly scrambled~\cite{dp}) updates does not leak any information about the actual training instances used by the users.
Unfortunately, it has been shown that an adversary can invert an individual model update of a target user in order to leak a large amount of information about its dataset~\cite{hitaj2017deep, 8835245, invg1, melis2019exploiting}.\par
For this reason, Bonawitz et al.~\cite{bonawitz2017practical} have proposed to combine \textbf{secure aggregation} (SA) protocols with FL as a first step to increase the security of FL, preventing the server from accessing individual model updates.
Informally, SA is a specialized MPC protocol that allows a set of users to compute the sum of their private inputs securely.
The security guarantee is the same as standard MPC protocols, i.e., nothing is leaked about the inputs except what can be inferred from the output (the sum of the values).
SA is believed to be one of the most robust defenses against gradient inversion and related inference attacks~\cite{huang2021evaluating}. In particular, the application of SA in FL has two main objectives:
\textbf{(1)}~Aggregating together a suitable number of model updates smooths out the information carried out by individual contributions. In turn, this makes it unfeasible to assert or recover meaningful information on individual training instances that produced the aggregated value.
\textbf{(2)}~SA ``hides'' the source of the aggregated information; even if sensitive data is recovered from the aggregated model updates, this cannot be attributed to the user who provided it.
Thus, although the privacy of the set of users may be violated, the privacy of individuals is preserved.
Our work shows that a motivated and malicious server can easily violate both of these fundamental properties of current SA defenses.
\textbf{This vulnerability is not specific to the SA protocol used; instead, it originated from the incorrect composition of SA and FL.}
In fact, it is exploitable even if the underlying SA protocol is replaced with its ideal cryptographic functionality, i.e., an ideal implementation that provides the perfect level of security.
The main intuition is that model updates (i.e., the inputs of SA) are under the indirect control of the malicious server since model updates are computed starting from the parameters sent by the server.
A malicious server can leverage this control to tamper with the updates (that are the inputs of SA) so that their aggregation will leak information about the update of a target user.
In order to achieve this, the server exploits a new attack vector that we call \textbf{model inconsistency}. Here, the server distributes different views of the same model to different users within the same round.
In this work, we show that model inconsistency can introduce new vulnerabilities in FL algorithms.
The intuition is that a malicious server, providing different parameters to different users, can exploit behavioral differences in the model updates provided by the different models to infer information on users' datasets, even when those model updates are securely aggregated before reaching the server.
To make this inherent vulnerability evident, we implement two attacks that give a representative view of the threat induced by the model inconsistency attack vector.
Eventually, these attacks demonstrate how a malicious server can nullify the security offered by current {SA-based} defenses proposed for FL. That is: \textbf{(1)}~Individual model updates can be perfectly recovered from the final aggregated value, independently of the number of users participating in the aggregation, and, \textbf{(2)}~The source of the recovered data can be attributed to individuals in the pool of active users.
Finally, we also introduce some candidate mitigation strategies aimed at hampering the disclosed vulnerability
\subsection{Contributions}\label{subsec:contribution}
Our contributions can be summarized as follows:
\paragraph{Model inconsistency}\label{sec:contribution-model-inconsistency}
We show that the combination of FL and SA~\cite{bonawitz2017practical} does not provide any better level of security than the one offered by the original FL protocol.
We prove this by introducing a new adversarial strategy, named \textbf{model inconsistency}, that leverages the following two observations: $(i)$ in each protocol round, the SA's input of user $u_i \in \mathcal{U}$ is its model update $\Delta^{\Theta}_{\mathcal{D}_i}$ and, $(ii)$ the value of $\Delta^{\Theta}_{\mathcal{D}_i}$ of each $u_i\in \mathcal{U}$ depends on the parameters $\Theta$ sent by the server $S$
(i.e., different parameters produce different model updates).
The combination of the above two observations implies that $S$ could act maliciously and craft different parameters for different users
in order to tamper with the inputs of SA.
As our two attacks will demonstrate, at a different scale, $S$ can exclude from the aggregation the updates of some non-target users $\mathcal{U} \setminus \{u_\mathsf{trgt}\}$, forcing the SA to leak part of the model update $\Delta^{\Theta}_{\mathcal{D}_\mathsf{trgt}}$ of the target~$u_\mathsf{trgt}$.
\paragraph{Gradient suppression attack}\label{sec:contribution-gradient-supression}
In~\Cref{sec:attack1}, we present a first attack, named \textbf{gradient suppression}.
It shows that a malicious server can force the local training of the deep model $f_{\widetilde{\Theta}}$ executed by $u_i$ to unconditionally produce a zeroed gradient $\Delta^{\widetilde{\Theta}}_{\mathcal{D}_i} \myeq [0]$.
By combining both gradient suppression and model inconsistency, the server $S$ can send the honest parameters $\Theta$ to the target user $u_\mathsf{trgt}$ and the malicious parameters $\widetilde{\Theta}$ to the remaining non-target ones $\mathcal{U}\setminus\{u_\mathsf{trgt}\}$.
In turn, this will leak the honest $u_\mathsf{trgt}$'s gradient $\Delta^{\Theta}_{\mathcal{D}_\mathsf{trgt}}$ even if SA is in place.
This is because SA will sum up the zeroed gradients of $\mathcal{U}\setminus\{u_\mathsf{trgt}\}$ and the honest gradient of $u_\mathsf{trgt}$.
The output will be equal to the gradient $\Delta^{\Theta}_{\mathcal{D}_\mathsf{trgt}}$ of the latter user $u_\mathsf{trgt}$.
This is the first practical attack that demonstrates that a malicious server can completely nullify SA. More importantly, the attack does not require auxiliary information on the targets, e.g., the distribution of users' datasets, or unrealistic architecture alterations.
\paragraph{Canary-gradient attack}
We extend the first attack and devise a second approach called \textbf{canary-gradient}. Here, we show that a malicious server $S$ can modify the target's model to induce specific behavior in the derivative of a tiny subset $\xi$ of its parameters ({e.g.,~} two out of millions of parameters). In particular, $S$ can forge malicious parameters that force the model to produce non-zero gradients for $\xi$ only when a specific adversarially-chosen property is present in the input batch used to compute the update. Then, the server can preserve the target's gradient for $\xi$ in the final aggregated value by forcing the non-target users to unconditionally produce zero gradients only for the parameters $\xi$. This allows $S$ to recover the target's gradient for $\xi$ in \TT{plaintext} and ascertain the presence of the queried property in the user's private data ({e.g.,~} membership inference).
Eventually, this demonstrates that a malicious server can cast extremely effective property inference attacks on individual users under SA, while ensuring the stealthiness of the attack.
\paragraph{The (in)secure combination of SA and FL}
In cryptographic terms, SA protocols are specialized multi-party computation (MPC) protocols that implement the ideal functionality $f^{\mathsf{sa}}(v_1, \ldots, v_n) = \sum_{u_i \in \mathcal{U}} v_i = v$. They are built assuming that the inputs of honest users are untamperable, i.e., an adversary has no control over the input $v_i$ of an honest user $u_i\in\mathcal{U}$.
This holds in both the semi-honest and malicious security models.
Unfortunately, our work shows that the above assumption does not hold in FL.
As discussed earlier, a malicious server $S$ can exploit a model inconsistency attack to tamper with the inputs $(v_1, \ldots, v_n)$ of all honest users $\mathcal{U}$.
Unlike other attacks, ours is the first that does not contradict the security of the underlying SA protocol and does not require auxiliary information about user data. Even assuming a perfectly secure SA, our attack will succeed independently of the number of users running the FL protocol. In practice, even if we replace SA with its ideal functionality and assume that FL is sound, our work shows that their combination is faulty and ineffective against an active adversarial server.
\iffull
To make our results reproducible, we made our code available.\footnote{\url{https://github.com/pasquini-dario/EludingSecureAggregation}.}
\fi
\subsubsection{The SNARG-based FL protocol}\label{sec:protocol-FL-snarg}
\DFnote{Assuming $\mathcal{U}^{(t)} = \mathcal{U}$ is problematic!}
The usage of SNARGs that we propose can be applied to FL when the active users $\mathcal{U}^{(t)}$ does not change over time, i.e., $\mathcal{U}^{(t)} = \mathcal{U}$ for every round $t \in \NN$ of FL.
Next we formally describe the our SNARG-based FL protocol.
\begin{construction}[SNARG-based FL protocol]\label{constr:snarg-fl}
Without loss of generality, we assume that $f^\mathsf{sa}$ returns the aggregation $v$ to all users $\mathcal{U}$ (as defined in~\Cref{eq:ideal-func-sa}).\footnote{
Note that the SA protocol of Bonawitz et al.~\cite{bonawitz2017practical} outputs the aggregation to the server only.
Still, it can be easily adapted to return the aggregation to all users $\mathcal{U}$: Each user signs (using its signing key) its masked update and the server forwards each signed masked update to every other users (this allows the latters to unmask the aggregated model update). Since the signed masked updates are signed, a malicious server can not modify them.}
Let $C_{\mathsf{fedSGD}}$ and $C_{\mathsf{fedAVG}}$ be the circuits that compute $\Theta^{(t)}$ (for FedSGD and FedAVG) as defined in~\Cref{eq:model-update-SGD} and~\Cref{eq:model-update-AVG-server}, respectively.
We assume that $C_{\mathsf{fedSGD}}$ and $C_{\mathsf{fedAVG}}$ have hardcoded the learning parameter $\eta\in[0,1]$, the number of the users $|\mathcal{U}|=|\mathcal{U}^{(t)}|$ and the sum of batch sizes of the users $\sum_{u_i \in \mathcal{U}} \sum^k_{j = 1} |\mathcal{D}^{(t,j)}_i|$ required to compute the next parameters $\Theta^{(t)}$.\footnote{In other words, we assume that FL uses some fixed and known values $|\mathcal{U}|=|\mathcal{U}^{(t)}|$ and $\sum_{u_i \in \mathcal{U}} \sum^k_{j = 1} |\mathcal{D}^{(t,j)}_i|$.}
Let $\mathsf{type} = \{\mathsf{fedSGD},\mathsf{fedAVG}\}$ be the version of FL used between FedSGD and FedAVG.
We define the relation $\mathcal{R}_\mathsf{type}$ (with no witness) as follows:
\begin{align*}\label{eq:relation}
\mathcal{R}_\mathsf{type} &= \left\{
((\Theta^{(t-1)},\Theta^{(t)},v^{(t-1)}), \bot)
: \right. \\
& \qquad \left. \Theta^{(t)} = C_{\mathsf{type}}(\Theta^{(t-1)},v^{(t-1)}) \right\}.
\end{align*}
A SNARG $\Pi = (Setup, Prove, Ver)$ for the relation $\mathcal{R}_\mathsf{type}$ can be used to check the correctness of the parameters by modifying the original FL protocol (\Cref{sec:federate-learning},~\Cref{fig:FL}) as follows:
\begin{description}
\item [Initializiation:] The architecture $f$ of the deep neural network and the learning parameter $\eta \in [0,1]$ are chosen by the server $S$.
The users $\mathcal{U}$ and the server $S$ agree on the initial honest parameters $\Theta^{(1)}$ (e.g., $\Theta^{(1)}$ can be fixed and can not be chosen arbitrarly by the server $S$).
Lastly, a trusted third party generates $crs$ by executing $Setup(\mathcal{R}_\mathsf{type})$.\footnote{Note that a trusted third party (i.e., certification server) is assumed even in the SA protocol of Bonawitz et al.~\cite{bonawitz2017practical}. In alternative, the CRS can be generated by running an MPC protocol (note that $Setup$ is executed only once).}
We assume that both $crs$ and the initial honest parameters $\Theta^{(1)}$ are known by all users $\mathcal{U}$.
\item [Server $S$:] In round $t \in \NN$, $S$ sends $\Theta^{(t)}$ and $\pi^{(t)}$ to all users $\mathcal{U}$ (if $t=1$, $S$ skips this step).
When $S$ receives $v^{(t)}$ from $f^\mathsf{sa}$, $S$ computes $\Theta^{(t+1)}$ and $\pi^{(t+1)}$ by executing $C_\mathsf{type}$ and $Prove(crs,(\Theta^{(t)}, \Theta^{(t+1)}, v^{(t)}),\bot)$, respectively.
\item [User $u_i \in \mathcal{U}$:] In round $t \in \NN$, $u_i$ receives the parameters $\Theta^{(t)}$ together with a proof $\pi^{(t)}$.
Let $v^{(t-1)}$ be the aggregation returned by $f^\mathsf{sa}$ in the previous round $t-1$.
If $Ver(crs,(\Theta^{(t-1)}, \Theta^{(t)}, v^{(t-1)}),\pi^{(t)}) = 0$, $u_i$ aborts. Otherwise, it computes and sends the model update $\Delta^{\Theta^{(t)}}_{\mathcal{D}^{(t)}_i}$ to $f^\mathsf{sa}$. Eventually, it receives $v^{(t)}$ from $f^{\mathsf{sa}}$ (If $t=1$, $u_i$ does not execute $Ver$ and directly computes its model update $\Delta^{\Theta^{(1)}}_{\mathcal{D}^{(1)}_i}$ using the known initial honest parameters $\Theta^{(1)}$).
\end{description}
\end{construction}
\paragraph*{Security of~\Cref{constr:snarg-fl}}
In a round $t$, if $Ver(crs,(\Theta^{(t-1)}, \Theta^{(t)}, v^{(t-1)}),\pi^{(t)}) = 1$, an honest user $u_i$ is guaranteed (with overwhelming probability) that $(\Theta^{(t-1)}, \Theta^{(t)}, v^{(t-1)}) \in \mathcal{L}_{\mathcal{R}_\mathsf{type}}$, i.e., $\Theta^{(t)}$ is honestly computed by $S$ with respect to $\Theta^{(t-1)}$ and $v^{(t-1)}$ (output by $f^\mathsf{sa}$) of the previous round $t-1$.
This follows by the computational soundness of the underlying SNARG (\Cref{def:snarg-sec}).
\Cref{thm:honest-param} provides the formal definition.
Note that by induction over the rounds and leveraging~\Cref{thm:honest-param}, we have that, with overwhelming probability, an user $u_i \in \mathcal{U}$ that does not abort in round $t\in \NN$ has received (in every round $j \leq t$) honestly computed parameters (recall the initial $\Theta^{(1)}$ is honest by definition).
\begin{theorem}\label{thm:honest-param}
Let $\Pi$ and $f^\mathsf{sa}$ be a SNARG and the ideal functionality of a SA protocol, respectively.
Let $\mathsf{type}\in\{\mathsf{fedSGD},\mathsf{fedAVG}\}$ and $\Theta^{(t)}_i$ be the version of the FL protocol and parameters received by $u_i \in \mathcal{U}$ in the round $t \in \NN$, respectively.
If $\Pi$ is computationally sound (\Cref{def:snarg-sec}) then~\Cref{constr:snarg-fl} guarantees that $\forall t\in\NN$ such that $t \geq 2$, $\forall$ honest $u_i \in \mathcal{U}$, we have:
\begin{equation*}\label{eq:honest-param}
\prob{ \Theta^{(t)}_i \neq \Theta^{(t)}_{i,\mathsf{type}} \text{ and } u_i \text{ does not abort}} \leq \negl,
\end{equation*}
where $\Theta^{(1)}_i = \Theta^{(1)}$ and $\Theta^{(t)}_{i,\mathsf{type}}=C_{\mathsf{type}}(\Theta^{(t-1)}_{i},\allowbreak v^{(t-1)})$.
\end{theorem}
\begin{proof}
Let $t\in \NN$ be a generic round of~\Cref{constr:snarg-fl}.
Suppose there exists an honest user $u_i \in \mathcal{U}$ such that~\Cref{eq:honest-param} does not hold, i.e.,
\begin{equation*}\label{eq:contradict}
\prob{ \Theta^{(t)}_i \neq \Theta^{(t)}_{i,\mathsf{type}} \text{ and } u_i \textit{ does not abort}} \geq \epsilon
\end{equation*}
where $\epsilon$ is a non-neglible probability.
We build an adversary $A$ that breaks the computational soundness of the underlying SNARG by leveraging the black-box access to $S$ and all users $\mathcal{U}$ and simulating the execution of~\Cref{constr:snarg-fl} (including the idealized functionality $f^{\mathsf{sa}}$).
$A$ runs $S$ and all users $\mathcal{U}$.
$A$ receives the CRS $crs$ from the challenger and acts as the third party of~\Cref{constr:snarg-fl}, i.e., it gives $crs$ to the server $S$ and all users $\mathcal{U}$.
It honestly simulates the execution until round $t$ and returns to the challenger the tuple $((\Theta^{t-1}_i, \Theta^{(t)}_i, v^{(t-1)}), \pi^{(t)})$.
Conditioned to the event that $(u_i \textit{ does not abort})$ and $(\Theta^{(t)}_i \neq \Theta^{(t)}_{i,\mathsf{type}})$, we can conclude that $Ver(crs,(\Theta^{(t-1)}_i, \Theta^{(t)}_i, v^{(t-1)}),\pi^{(t)}) = 1$ and $(\Theta^{(t-1)}_i, \allowbreak \Theta^{(t)}_i,\allowbreak v^{(t-1)})\not\in \mathcal{L}_{\mathcal{R}_\mathsf{type}}$.
The latter because $\Theta^{(t)}_i \neq \allowbreak \Theta^{(t)}_{i,\mathsf{type}} = C_{\mathsf{type}}(\Theta^{(t-1)}_{i},\allowbreak v^{(t-1)})$.
Hence, $A$ breaks the computational soundness of the underlying SNARG with non-negligible probability $\epsilon$.
This concludes the proof.
\end{proof}
\DFnote{Say that we can naturally include everything in the SNARG!! But this is not pratical and it is not our objetive}
\DFnote{The users can simply update the parameters by themselves.}
\DFnote{Without experiments SP will reject us??}
\DFnote{UC? do we need it? It is fine to assume that we use $f^\mathsf{sa}$.}
}
\section{Mitigations}\label{sec:def}
Next, we discuss and introduce some mitigation approaches solely aimed at preventing model inconsistency and its effect.
\paragraph*{SA dropout}
\input{mitigation/check-structure.tex}
\iffull
Below, we informally introduce two sounder mitigation approaches.
\paragraph*{Parameter validation using signatures}
\input{mitigation/signature-approach.tex}
\fi
\paragraph*{Conditional secure aggregation}
\input{mitigation/secure-aggregation-approach.tex}
\ifconference
Another approach that is compliant with the communication topology of FL is to check that all parameters sent by the server $S$ are the same.
This solution requires an additional round of communication and a digital signature (to prevent the server from tampering with the users' messages). We refer the reader to~\Cref{sec:appendix-signature} for more details.
\fi
\ifconference
\paragraph*{Differential privacy}
A more general mitigation technique would be using Differential Private-SGD algorithms~\cite{dp} which prevent gradient inversion attacks when used properly~\cite{invg1, robbing, huang2021evaluating}. However, unlike previous solutions, differential privacy (DP) comes with a high utility cost, especially in the context of FL~\cite{salvagingFL}. Note that the combination of SA and DP is still susceptible to model inconsistency attacks since the adversary can isolate the gradient of the target user; thus, parameters must be calibrated to ensure the aggregate noise is effective. This will be the subject of future work.
\fi
\iffull
\paragraph*{Differential privacy (DP)}
A more general mitigation technique would be using Differential Private-SGD algorithms~\cite{dp}.
However, centralized DP is ineffective when the parameter server is malicious.
Here, the DP-noise is applied only after the model updates have been aggregated by the server. Trivially, if the server is malicious, it can just skip this step and obtain the target's gradient in clear.\footnote{Or a clipped version of it if the clients perform clipping.}
Therefore, the proposed attacks remain unaffected. Peculiarly, centralized DP is the approach used in state-of-the-art implementations~\cite{tffed} of FL and employed in the wild~\cite{brendan2018learning}.
A more suitable mitigation technique would be using Local DP, which prevents gradient inversion attacks when used properly~\cite{invg1, robbing, huang2021evaluating}. However, unlike previous solutions, DP comes with a high utility cost, especially in the context of FL~\cite{salvagingFL}. Note that the combination of SA and DP is still susceptible to model inconsistency attacks since the adversary can isolate the gradient of the target user; thus, parameters may need to be calibrated to ensure the aggregate noise is effective. This will be the subject of future work.
\fi
\section{Preliminaries}\label{sec:preliminaries}
\iffull
\subsection{Notation}\label{sec:notation}
\fi
We use small letters (such as $x$) to denote concrete values, calligraphic letters (such as $\mathcal{X}$) to denote sets.
For a string $x \in \bin^*$, we let $|x|$ be its length; if $\mathcal{X}$ is a set, $|\mathcal{X}|$ represents the cardinality of $\mathcal{X}$.
In the setting of deep learning, we use the notation $[\cdot]$ to express a tensor (i.e., vector) of arbitrary dimension.
We write $[x]$ for a tensor filled with the value $x$.
When $x$ is a set ({e.g.,~} $\mathbb{R}^+$), the tensor is filled with arbitrary elements from that set.
\iffull
\input{preliminaries/crypto-notation}
\fi
\subsection{Neural networks}\label{sec:nn}
\input{preliminaries/nl.tex}
\subsection{Federated learning}\label{sec:federate-learning}
\input{preliminaries/fl.tex}
\subsection{Secure aggregation}\label{sec:secure-aggregation}
\input{preliminaries/sa.tex}
\iffalse
\subsection{Succinct non-interactive arguments}\label{sec:snarg}
\input{preliminaries/snarg.tex}
\fi
\section{Related Work}
\label{sec:related-work}
\subsection{Federated learning and secure aggregation}\label{sec:related-work-SA}
The distributed architecture of FL protocol~\cite{shokri2015privacy,mcmahan2017communication} provides a fertile ground for attackers~\cite{hitaj2017deep,melis2019exploiting}.
This is because a malicious party, mainly the server, has access to sensitive information such as model updates that can be exploited to violate users' privacy.
Accordingly, SA has been proposed by Bonawitz et al.~\cite{bonawitz2017practical} as a fundamental step to increase the security of FL without modifying the original structure of the protocol.
\iffull
In the setting of FL, SA is a specialized MPC protocol that allows the server to securely compute the sum of some values (known as aggregation) held by the users (e.g., integers, vectors).
The security definition of SA is the same as MPC: I.e., nothing is leaked except what can be inferred from the output (the sum of the values).
The deployment of SA in FL is beneficial since it allows us to compute the aggregation of users' model updates $\{\Delta^{\Theta}_{\mathcal{D}_i}\}$ (e.g., the gradients) without revealing to the server the individual updates of the users.
Keeping the individual updates private is essential since they carry sensitive information about the local training datasets of the users (Section~\ref{sec:gradient-inversion-fl}).
\fi
Subsequent works focus on the development of new SA protocols for FL with reduced communication/computation overhead~\cite{bell2020secure,choi2020communication,guo2020v,kadhe2020fastsecagg,so2021turbo}, multiple servers~\cite{beguier2020safer}, increased robustness against malicious updates~\cite{pillutla2019robust,burkhalter2021rofl}, or with verifiable aggregation~\cite{xu2019verifynet,guo2020v}.\footnote{In FL with verifiable aggregations~\cite{xu2019verifynet,guo2020v}, users are supposed to verify the integrity of the aggregated model updates and compute the new parameters (if the parameters are updated solely by the server, then the verifiability of aggregated gradients becomes meaningless). This differs from the original FL protocol~\cite{shokri2015privacy,mcmahan2017communication} in which the server updates the model for improved scalability.}
\ifconference
Separately, several other works focused on building new protocols (or propose significant modifications of FL, e.g., protocol, architecture, etc.) to train a deep neural network without leaking unnecessary information about the datasets of the users.
We refer the reader to~\Cref{sec:related-work-alternative-defenses} for more details.
\fi
\subsection{Gradient Inversion}\label{sec:gradient-inversion-fl}
A core privacy concern in FL is the role of the server.
In this direction, it has been shown that, without SA, a semi-honest server can invert users' gradients (sent as a model update in a particular round of FL) and compute a close-enough approximation of users' local training datasets.
In a nutshell, by leveraging $f_{\Theta}$ (where $\Theta$ are the parameters of the current round of FL) and the gradient locally computed by the user $u$ using a subset $\mathcal{D}$ of its local data, a semi-honest $S$ can recover $\mathcal{D}$ by searching a set of instances $\widehat{\mathcal{D}}$ that generates the gradient similar to the one sent by the user.
Thanks to the inherent smoothness of the neural network $f_\Theta$, this searching problem can be defined as a second-order optimization, i.e.,
\begin{equation}
\text{argmin}_{\widehat{\mathcal{D}}}[d(\nabla^\Theta_{\widehat{\mathcal{D}}}, \nabla^\Theta_{\mathcal{D}}) \cdot \alpha r(\widehat{\mathcal{D}})]
\label{eq:opt}
\end{equation}
where $\widehat{\mathcal{D}}$ is the candidate solution of the malicious server~$S$, $d$~is a distance function to measure the discrepancy between the gradient signals $\nabla^\Theta_{\widehat{\mathcal{D}}}$ and $\nabla^\Theta_{\mathcal{D}}$, $r$ is a regularizer defined on the input domain, and $\alpha$ is the weight associated to the regularization term in the optimization.
In the work of Zhu~{et~al.}~\cite{invg1}, $d$ is set to be the Euclidean distance, and the {L-BFGS} solver is used to solve the optimization problem.
The follow-up work of Geiping~{et~al.}~\cite{invg2} improves their approach/results by noting that the gradient signal is scale-invariant and accounts for that in defining the optimization objective in~\Cref{eq:opt}.
\iffull
For this reason, they use the cosine similarity as distance function $d$ as well as a gradient-descend-based algorithm ({i.e.,~}~Adam~\cite{adam}) to carry out the second-order optimization while regularizing $\widehat \mathcal{D}$ with total variation.\footnote{Their work focuses on the image domain.}
These modifications significantly improve
\fi
\ifconference
Their work improves
\fi
the effectiveness of the inversion attack, drastically increasing its applicability on real-world architectures such as ResNets~\cite{resnet} and more realistic batch sizes and a number of FedAVG local iterations. These results have been further improved in the work of Yin~{et~al.}~\cite{nvidiagi} by relying on additional regularization terms and tailored optimization techniques.
A more recent line of research dispensed with optimization-based approaches to focus on closed-form procedures to recover data from gradients in deep neural networks~\cite{rgap, pan2020theoryoriented}. In this vein, a recent work of Fowl~{et~al.}~\cite{robbing} (which is concurrent to our work) improves over previous approaches by considering a malicious server that modifies the FL architecture and crafts the network parameters to artificially create a neural layer that retains information on the input batch.
In particular, this is a linear layer followed by a ReLU~activation whose parameters must be chosen considering the private sets' CDF for a given property, i.e., the attacker must have some auxiliary information on users' training datasets.
Unfortunately, this extra knowledge may not be acquired in realistic scenarios (when users' data distributions are unknown) and weakens the applicability of the attack. In addition, the server needs to manipulate the model architecture and place a linear layer at the start of the network to maximize the attack effectiveness. However, this modification is unworkable for typical deep learning applications ({e.g.,~} in computer vision).
Lam~{et~al.}~\cite{gdis} showed that, if SA is enabled, a malicious server $S$ can still try to reconstruct individual contributions by observing multiple training rounds of FL.
This disaggregation process can be reduced to a matrix factorization problem.
However, their attack is effective only in a particular restricted setting in which the malicious server $S$ $(i)$ alters the protocol execution by providing \emph{always} the same parameters $\widetilde{\Theta}$ at each round of FL (i.e., users compute their gradient updates always on the same model $f_{\widetilde{\Theta}}$), $(ii)$ leverages additional side-channel information about users' participation in the training rounds, and $(iii)$ users are required to use the same local training dataset at each round of FL.
Their approach enables the gradient inversion attack of~\cite{invg1, invg2, nvidiagi} to scale and be effective in more realistic scenarios where there is a significant number of users participating in the protocol execution of FL.
Nevertheless, its feasibility still depends on the number of active users, network parameters, and other factors such as the number of rounds that the server monitors to recover individual gradients accurately.
Finally, while the authors showed that this approach could handle noise, its applicability is inherently limited in FedSGD, where local training is performed on randomly selected batches rather than the entire, static, local dataset.
\iffull
\subsection{Other protocols}\label{sec:related-work-alternative-defenses}
\input{related/related-other-protocols}
\fi
\section{Threat model}\label{sec:threat-model}
In this section we formalize the threat model in which our attacks are defined (Section~\ref{sec:attack1} and Section~\ref{sec:attack2}).
\paragraph*{Secure aggregation} We assume that SA is enabled during the execution of FL.
In particular, in each round $t \in \NN$, the active users $\mathcal{U}^{(t)}$ do not send their model updates $\{ \Delta^{\Theta^{(t)}}_{\mathcal{D}^{(t)}_i}\}_{u_i \in\mathcal{U}^{(t)}}$ in the clear.
Instead, they execute the SA protocol $\Pi$ to securely compute the aggregation $v = \sum _{u_i \in \mathcal{U}^{(t)}} \Delta^{\Theta^{(t)}}_{\mathcal{D}^{(t)}_i}$
\iffull
(see~\Cref{sec:secure-aggregation} and~\Cref{fig:FL}).
\fi
\ifconference
(see~\Cref{sec:secure-aggregation}).
\fi
Only the final aggregation $v$ is revealed to the server~$S$.
More importantly, we do not target any specific implementation of SA.
Our attacks are generic, and they are effective even in the presence of a SA protocol with perfect security (i.e., the same level of security of an ideal functionality).
For this reason, from now on, we replace the execution of the SA protocol with the invocation of the ideal functionality $f^\mathsf{sa}$ (\Cref{sec:secure-aggregation}).
This strengthens the results of this work, demonstrating that SA is incorrectly applied in the context of FL.
\paragraph*{Communication topology} We assume the standard, centralized, communication topology of FL used in practice~\cite{gboard0, gboard1}. Each user shares an encrypted and authenticated channel with the server~$S$.
No other communication channels are present. Hence, users do not have a direct communication channel. All messages must pass through the server~$S$. Observe that the SA protocol of Bonawitz et al.~\cite{bonawitz2017practical} (built to be applied to FL) is meant for this communication topology.
\paragraph*{Malicious server}
We assume an adversarial server $S$ whose objective is to leak information about the local dataset $\mathcal{D}_\mathsf{trgt}$ of one (or more) target user $u_\mathsf{trgt}$ that participates in the execution of the protocol.
Moreover, we assume $S$ is malicious and deviates from the protocol specifications (e.g., it sends arbitrary messages).
Specifically, in our attacks, the malicious server $S$ exploits the model inconsistency attack vector; that is, $S$ provides arbitrary malicious parameters to arbitrary users even within the same training round $t \in \NN$.
After that, the server honestly follows all the other steps of FL.
For instance, $S$ does not have an influence on the selection of the active users $\mathcal{U}^{(t)}$ for each $t \in \NN$, and it does not collude with other users that are assumed to be honest.
Finally, we stress that, as in the general FL scenario, $S$ is entitled to define the architecture $f$ of the shared model.
Hence, it can enforce specific features in the adopted model as long as these result in realistic architectures.
However, in the paper, we mainly consider networks with common, state-of-the-art architectures.
|
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| 9,601
|
{"url":"https:\/\/annals.math.princeton.edu\/2004\/160-2\/p02","text":"# Cover times for Brownian motion and random walks in two dimensions\n\n### Abstract\n\nLet $\\mathcal{T}(x,\\varepsilon)$ denote the first hitting time of the disc of radius $\\varepsilon$ centered at $x$ for Brownian motion on the two dimensional torus $\\mathbb{T}^2$. We prove that $\\sup_{x\\in \\mathbb{T}^2} \\mathcal{T}(x,\\varepsilon)\/|\\log \\varepsilon|^2 \\to 2\/\\pi$ as $\\varepsilon \\rightarrow 0$. The same applies to Brownian motion on any smooth, compact connected, two-dimensional, Riemannian manifold with unit area and no boundary. As a consequence, we prove a conjecture, due to Aldous (1989), that the number of steps it takes a simple random walk to cover all points of the lattice torus $\\mathbb{Z}_n^2$ is asymptotic to $4n^2(\\log n)^2\/\\pi$. Determining these asymptotics is an essential step toward analyzing the fractal structure of the set of uncovered sites before coverage is complete; so far, this structure was only studied nonrigorously in the physics literature. We also establish a conjecture, due to Kesten and R\u00e9v\u00e9sz, that describes the asymptotics for the number of steps needed by simple random walk in $\\mathbb{Z}^2$ to cover the disc of radius $n$.\n\n## Authors\n\nAmir Dembo\n\nDepartment of Mathematics, Stanford University, Stanford, CA 94305, United States\n\nYuval Peres\n\nDepartment of Mathematics, University of California, Berkeley, Berkeley, CA 94702, United States\n\nJay Rosen\n\nDepartment of Mathematics, College of Staten Island, CUNY, Staten Island, NY 10314, United States\n\nOfer Zeitouni\n\nMathematics Department, Technion-Israel Institute of Technology, Haifa, 32000, Israel and Department of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States","date":"2022-07-06 03:58:04","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\": 0.661328911781311, \"perplexity\": 258.7351791116111}, \"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-2022-27\/segments\/1656104660626.98\/warc\/CC-MAIN-20220706030209-20220706060209-00690.warc.gz\"}"}
| null | null |
Binariks is a software development outsourcing and consulting company headquartered in the USA with development centers in Ukraine. We leverage 20 years of IT experience to help startups and established businesses achieve success by creating winning products, teams and custom.
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Application for charging electric cars. The user scans charing box barcode and after payment plug cord and charge car. Application support different type of payments.
Medvisit is a platform that allows you to find a doctor wherever you are. This service is extremely useful for travelers, who need medical care during the visit to another country. It is a web-based platform for doctors and patients. As a doctor, you can register, set your availability, send the report to the patient after the visit. Patients can book a visit with help of this platform.
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As a Fitness Professional, you can set your training timetable, choose your professional area and view statistics of your income.
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Planet of finance is the platform for investors where getting advice and investing is simple, fast and easy. The main goal is to empower everyday savers and investors to achieve their financial goals by giving them access to the best investment professionals on the market. The app helps investors to get advice or invest with thousands of qualified finance professionals. Regarding finance professionals the platform allows them to find the smartest way to make business. Propose or request investment opportunities, raise money, collaborate with investors and financial institutions in one single place.
Opus Digitas is an online media platform for user-generated video, with the main goal – to enhance audience engagement through an immersive experience. Opus Digitas is a SaaS platform which allows users collect videos, live or on schedule, invite specific users or groups, curate collected videos with online clipping, merges and transitions, circulate to Social Media, video portals, web and in-app by a single click of a button. You can easily promote your brand by engaging your users and analyze the impact of your published videos through Opus media platform.
|
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\section{Introduction}
\label{sec:1}
In several fields, from continuum mechanics to fluid dynamics, we need to solve numerically very complex problems that arise from physics laws. Usually we model these phenomena through partial differential equations (PDEs) and we are interested in finding the field solution and also some other quantities that increase our knowledge on the system we are describing. Almost always we are not able to obtain an analytical solution, so we rely on some discretization techniques, like Finite Element (FE) or Finite Volume (FV), that furnish an approximation of the solution. We refer to this methods as the ``truth'' ones, because they require very high computational costs, especially in parametrized context.
In fact if the problem depends on some physical or geometrical parameter, the \emph{full-order} or \emph{high-fidelity} model has to be solved many times and this might be quite demanding. Examples of typical applications of relevance are optimization, control, design, bifurcation detection and real time query.
For this class of problems, we aim to replace the high-fidelity problem by one of much lower numerical complexity, through the \emph{model order reduction} approach \cite{MOR2016}.
We focus on Reduced Basis (RB) method \cite{hesthaven2015certified,Quarteroni2011,quarteroni2015reduced,morepas2017,benner2017model} which provides both \emph{fast} and \emph{reliable} evaluation of an input (parameter)-output relationship. The main features of this methodology are \emph{(i)} those related to the classic Galerkin projection on which RB method is built upon \emph{(ii)} an \emph{a posteriori} error estimation which provides sharp and rigorous bounds and \emph{(iii)} offline/online computational strategy which allows rapid computation.
The goal of this chapter is to present a very efficient \emph{a posteriori} error estimation for linear elasticity parametrized problem. We show many different configurations and settings, by applying RB method to approximate problems using plane stress and plane strain formulation and to deal both with isotropic and orthotropic materials. We underline that the setting for very different problems is the same and unique.
This work is organized as follows. In Section 2, we first present a ``unified'' linear elasticity formulation; we then briefly introduce the geometric mapping strategy based on domain decomposition; we end the Section with the affine decomposition forms and the definition of the ``truth'' approximation, which we shall build our RB approximation upon. In Section 3, we present the RB methodology and the offline-online computational strategy for the RB ``compliant'' output. In Section 4, we define our \emph{a posteriori} error estimators for our RB approach, and provide the computation procedures for the two ingredients of our error estimators, which are the dual norm of the residual and the coercivity lower bound. In Section 5, we briefly discuss the extension of our RB methodology to the ``non-compliant'' output. In Section 6, we show several numerical results to illustrate the capability of this method, with a final subsection devoted to provide an introduction to more complex nonlinear problems. Finally, in Section 7, we draw discussions and news on future works.
\section{Preliminaries}
\label{sec:2}
In this Section we shall first present a ``unified'' formulation for all the linear elasticity cases -- for isotropic and orthotropic materials, 2D Cartesian and 3D axisymmetric configurations -- we consider in this study. We then introduce a domain decomposition and geometric mapping strategy to recast the formulation in the ``affine forms'', which is a crucial requirement for our RB approximation. Finally, we define the ``truth'' finite element approximation, upon which we shall build the RB approximation, introduced in the next Section.
\subsection{Formulation on the ``Original'' Domain}
\label{subsec:2}
\subsubsection{Isotropic/Orthotropic materials}
We first briefly describe our problem formulation based on the original settings (denoted by a superscript $^{\rm o}$). We consider a solid body in two dimensions $\Omega^{\rm o}(\boldsymbol{\mu}) \in \mathbb{R}^2$ with boundary $\Gamma^{\rm o}$, where $\boldsymbol{\mu} \in \mbox{\boldmath$\mathcal{D}$} \subset \mathbb{R}^P$ is the input parameter {and} $\mbox{\boldmath$\mathcal{D}$}$ is the parameter domain \cite{sneddon1999mathematical,sneddon2000mathematical}. For the sake of simplicity, in this section, we assume implicitly that any ``original'' quantities (stress, strain, domains, boundaries, etc) with superscript $^{\rm o}$ will depend on the input parameter $\boldsymbol{\mu}$, e.g. $\Omega^{\rm o} \equiv \Omega^{\rm o}(\boldsymbol{\mu})$.
We first make the following assumptions: (i) the solid is free of body forces, (ii) there {are} negligible thermal strains; note that the extension to include either or both body forces/thermal strains is straightforward. Let us denote $u^{\rm o}$ as the displacement field, and the spatial coordinate $\mathbf{x}^{\rm o} = (x^{\rm o}_1,x^{\rm o}_2)$, the linear elasticity equilibrium reads
\begin{equation}\label{eqn:equlibrium}
\frac{\partial \sigma^{\rm o}_{ij}}{\partial x^{\rm o}_j} = 0, \quad \text{in} \ \Omega^{\rm o}
\end{equation}
where $\sigma^{\rm o}$ denotes the stresses, which are related to the strains $\varepsilon^{\rm o}$
by
\begin{equation*}
\sigma^{\rm o}_{ij} = C_{ijkl}\varepsilon^{\rm o}_{kl}, \quad 1 \leq i,j,k,l \leq 2
\end{equation*}
where
\begin{equation*}
\varepsilon^{\rm o}_{kl} = \dfrac{1}{2}\bigg(\frac{\partial u^{\rm o}_k}{\partial x^{\rm o}_l} + \frac{\partial u^{\rm o}_l}{\partial x^{\rm o}_k} \bigg),
\end{equation*}
$u^{\rm o} = (u^{\rm o}_1,u^{\rm o}_2)$ is the displacement and $C_{ijkl}$ is the elastic tensor, which can be expressed in a matrix form as
\begin{equation*}
[\mathbf{C}] = \left[
\begin{array}{cccc}
C_{1111} & C_{1112} & C_{1121} & C_{1122} \\
C_{1211} & C_{1212} & C_{1221} & C_{1222} \\
C_{2111} & C_{2112} & C_{2121} & C_{2122} \\
C_{2211} & C_{2212} & C_{2221} & C_{2222} \\
\end{array}
\right]
= [\mathbf{B}]^T [\mathbf{E}] [\mathbf{B}],
\end{equation*}
where
\begin{equation*}
[\mathbf{B}] = \left[
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 1 & 1 & 0 \\
\end{array}
\right] \quad
[\mathbf{E}] = \left[
\begin{array}{ccc}
c_{11} & c_{12} & 0 \\
c_{21} & c_{21} & 0 \\
0 & 0 & c_{33} \\
\end{array}
\right].
\end{equation*}
The matrix $[\mathbf{E}]$ varies for different material types and is given in the Appendix.
We next consider Dirichlet boundary conditions for both components of $u^{\rm o}$:
\begin{equation*}
u^{\rm o}_i = 0 \quad {\rm{on}} \quad \Gamma_{D,i}^{\rm o},
\end{equation*}
and Neumann boundary conditions:
\begin{eqnarray*}
\sigma^{\rm o}_{ij}e^{\rm o}_{n,j} &=& \bigg\{
\begin{array}{ccc}
f^{\rm o}_ne^{\rm o}_{n,i} & \rm{on} & \Gamma^{\rm o}_{N} \\
0 & \rm{on} & \Gamma^{\rm o}\backslash\Gamma^{\rm o}_{N}\\
\end{array}
\end{eqnarray*}
where $f^{\rm o}_n$ is the specified stress on boundary edge $\Gamma^{\rm o}_{N}$ respectively; and $\mathbf{e}^{\rm o}_n = [e^{\rm o}_{n,1}, e^{\rm o}_{n,2}]$ is the unit normal on $\Gamma^{\rm o}_{N}$. Zero value of $f^{\rm o}_n$ indicate free stress (homogeneous Neumann conditions) on a specific boundary. Here we only consider homogeneous Dirichlet boundary conditions, but extensions to non-homogeneous Dirichlet boundary conditions and/or nonzero traction Neumann boundary conditions are simple and straightforward.
We then introduce the functional space
\begin{equation*}
X^{\rm o} = \{v = (v_1,v_2) \in (H^1(\Omega^{\rm o}))^2 \ | \ v_i = 0 \ {\rm on} \ \Gamma^{\rm o}_{D,i} , i = 1,2\},
\end{equation*}
here $H^1(\Omega^{\rm o}) = \{ v \in L^2(\Omega^{\rm o}) \ | \ \nabla v \in (L^2(\Omega^{\rm o}))^2 \}$ {and} $L^2(\Omega^{\rm o})$ is the space of square-integrable functions over $\Omega^{\rm o}$. By multiplying \refeq{eqn:equlibrium} by a test function $v \in X^{\rm o}$ and integrating by part over $\Omega^{\rm o}$ we obtain the weak form
\begin{equation}\label{eqn:wf0}
\int_{\Omega^{\rm o}}\frac{\partial v_i}{\partial x^{\rm o}_j}{C}_{ijkl}\frac{\partial
u^{\rm o}_k}{\partial x^{\rm o}_l}d\Omega^{\rm o} =
\int_{\Gamma^{\rm o}_{N}}{f}^{\rm o}_n e^{\rm o}_{n,j} v_jd\Gamma^{\rm o}.
\end{equation}
Finally, we define our output of interest, which usually is a measurement (of our displacement field or even equivalent derived solutions such as stresses, strains) over a boundary segment $\Gamma^{\rm o}_L$ or a part of the domain $\Omega^{\rm o}_L$. Here we just consider a simple case,
\begin{equation}\label{eqn:output0}
s^{\rm o}(\boldsymbol{\mu}) = \int_{\Gamma^{\rm o}_L}f^{\rm o}_{\ell,i}u^{\rm o}_id\Gamma^{\rm o},
\end{equation}
i.e the measure of the displacement on either or both $x^{\rm o}_1$ and $x^{\rm o}_2$ direction along $\Gamma^{\rm o}_L$ with multipliers $f^{\rm o}_{\ell,i}$; more general forms for the output of interest can be extended straightforward. Note that our output of interest is a linear function of the displacement; extension to quadratic function outputs can be found in \cite{huynh07:ijnme}.
We can then now recover our abstract statement: Given a $\boldsymbol{\mu} \in \mbox{\boldmath$\mathcal{D}$}$, we evaluate
$$s^{\rm o}(\boldsymbol{\mu}) = \ell^{\rm o}(u^{\rm o};\boldsymbol{\mu}),$$
where $u^{\rm o} \in X^{\rm o}$ satisfies
$$a^{\rm o}(u^{\rm o},v;\boldsymbol{\mu}) = f^{\rm o}(v;\boldsymbol{\mu}), \quad \forall v \in X^{\rm o}.$$
Here $a^{\rm o}(w,v;\boldsymbol{\mu}): X^{\rm o} \times X^{\rm o} \rightarrow \mathbb{R}$, $\forall w,v \in X^{\rm o}$ is the symmetric and positive bilinear form associated to the left hand side term of \refeq{eqn:wf0}; $f^{\rm o}(v;\boldsymbol{\mu}): X^{\rm o} \rightarrow \mathbb{R}$ and $\ell^{\rm o}(v;\boldsymbol{\mu}): X^{\rm o} \rightarrow \mathbb{R}$, $\forall v \in X^{\rm o}$ are the linear forms associated to the right hand side terms of \refeq{eqn:wf0} and \refeq{eqn:output0}, respectively. It shall be proven convenience to recast $a^{\rm o}(\cdot,\cdot;\boldsymbol{\mu})$, $f^{\rm o}(\cdot;\boldsymbol{\mu})$ and $\ell^{\rm o}(\cdot;\boldsymbol{\mu})$ in the following forms
\begin{equation}\label{eqn:bilinear_o}
a^{\rm o}(w,v;\boldsymbol{\mu}) = \int_{\Omega^{\rm o}}
\left[\dfrac{\partial w_1}{\partial x^{\rm o}_1}, \dfrac{\partial w_1}{\partial x^{\rm o}_2}, \dfrac{\partial w_2}{\partial x^{\rm o}_1}, \dfrac{\partial w_2}{\partial x^{\rm o}_2}, w_1 \right]
[\mathbf{S}^a]
\left[
\begin{array}{c}
\dfrac{\partial v_1}{\partial x^{\rm o}_1} \\ \dfrac{\partial v_1}{\partial x^{\rm o}_2} \\
\dfrac{\partial v_2}{\partial x^{\rm o}_1} \\ \dfrac{\partial v_2}{\partial x^{\rm o}_2} \\
v_1 \\
\end{array}
\right]
d\Omega^{\rm o}, \ \forall w,v \in X^{\rm o},
\end{equation}
\begin{equation}\label{eqn:linear_fo}
f^{\rm o}(v;\boldsymbol{\mu}) =
\int_{\Gamma^{\rm o}_N}
[\mathbf{S}^f]
\left[
\begin{array}{c}
v_1 \\ v_2 \\
\end{array}
\right]
d\Gamma^{\rm o}, \quad \forall v \in X^{\rm o},
\end{equation}
\begin{equation}\label{eqn:linear_lo}
\ell^{\rm o}(v;\boldsymbol{\mu}) = \int_{\Gamma^{\rm o}_L}
[\mathbf{S}^{\ell}]
\left[
\begin{array}{c}
v_1 \\ v_2 \\
\end{array}
\right]
d\Gamma^{\rm o}, \quad \forall v \in X^{\rm o},
\end{equation}
where $[\mathbf{S}^a] \in \mathbb{R}^{5\times 5}$; $[\mathbf{S}^f]\in \mathbb{R}^2$ and $[\mathbf{S}^{\ell}]\in \mathbb{R}^2$ are defined as
\begin{equation*}
[\mathbf{S}^a] = \left[
\begin{array}{ll}
[\mathbf{C}] & [\boldsymbol{0}]^{4\times 1} \\
\left[\boldsymbol{0}\right]^{1\times 4} & 0 \\
\end{array}
\right], \quad
[\mathbf{S}^{f}] = \left[
\begin{array}{cc}
f^{\rm o}_ne^{\rm o}_{n,1} & f^{\rm o}_ne^{\rm o}_{n,2} \\
\end{array}
\right], \quad
[\mathbf{S}^{\ell}] = \left[
\begin{array}{cc}
f^{\rm o}_{\ell,1}& f^{\rm o}_{\ell,2} \\
\end{array}
\right].
\end{equation*}
\subsubsection{Axisymmetric}
Now we shall present the problem formulation for the axisymmetric case. In a cylindrical coordinate system $(r,z,\theta),$\footnote{For the sake of simple illustration, we omit the ``original'' superscript $^{\rm o}$ on $(r,z,\theta)$.} the elasticity equilibrium reads
\begin{eqnarray*}
\dfrac{\partial \sigma^{\rm o}_{rr}}{\partial r} + \dfrac{\partial \sigma^{\rm o}_{zr}}{\partial z} + \dfrac{\sigma^{\rm o}_{rr}-\sigma^{\rm o}_{\theta\theta}}{r} &=& 0 , \quad \text{in} \quad \Omega^{\rm o} \\
\dfrac{\partial \sigma^{\rm o}_{rz}}{\partial r} + \dfrac{\partial \sigma^{\rm o}_{zz}}{\partial z} + \dfrac{\sigma^{\rm o}_{rz}}{r} &=& 0 \nonumber, \quad \text{in} \quad \Omega^{\rm o}
\end{eqnarray*}
where $\sigma^{\rm o}_{rr}$, $\sigma^{\rm o}_{zz}$, $\sigma^{\rm o}_{rz}$, $\sigma^{\rm o}_{\theta\theta}$ are the stress components given by
\begin{equation*}
\left[
\begin{array}{c}
\sigma^{\rm o}_{rr} \\
\sigma^{\rm o}_{zz} \\
\sigma^{\rm o}_{\theta\theta} \\
\sigma^{\rm o}_{rz} \\
\end{array}
\right] = \underbrace{\dfrac{E}{(1+\nu)(1-2\nu)}\left[\begin{array}{cccc}
(1-\nu) & \nu & \nu & 0 \\
\nu & (1-\nu) & \nu & 0 \\
\nu & \nu & (1-\nu) & 0 \\
0 & 0 & 0 & \dfrac{1-2\nu}{2} \\
\end{array}\right]}_{[\mathbf{E}]}
\left[
\begin{array}{c}
\varepsilon^{\rm o}_{rr} \\
\varepsilon^{\rm o}_{zz} \\
\varepsilon^{\rm o}_{\theta\theta} \\
\varepsilon^{\rm o}_{rz} \\
\end{array}
\right]\nonumber,
\end{equation*}
where $E$ and $\nu$ are the axial Young's modulus and Poisson ratio, respectively. We only consider isotropic material, however, extension to general to anisotropic material is possible; as well as axisymmetric plane stress and plane strain \cite{zienkiewics05:FEM1}.
The strain $\varepsilon^{\rm o}_{rr}$, $\varepsilon^{\rm o}_{zz}$, $\varepsilon^{\rm o}_{rz}$, $\varepsilon^{\rm o}_{\theta\theta}$ are given by
\begin{equation}\label{eqn:stress_axs}
\left[
\begin{array}{c}
\varepsilon^{\rm o}_{rr} \\
\varepsilon^{\rm o}_{zz} \\
\varepsilon^{\rm o}_{\theta\theta} \\
\varepsilon^{\rm o}_{rz} \\
\end{array}
\right] = \left[
\begin{array}{c}
\dfrac{\partial u^{\rm o}_r}{\partial r} \\[6pt]
\dfrac{\partial u^{\rm o}_z}{\partial z} \\[6pt]
\dfrac{u^{\rm o}_r}{r} \\[6pt]
\dfrac{\partial u^{\rm o}_r}{\partial z} + \dfrac{\partial u^{\rm o}_z}{\partial r} \\
\end{array}
\right],
\end{equation}
where $u^{\rm o}_r$, $u^{\rm o}_z$ are the radial displacement and axial displacement, respectively.
Assuming that the axial axis is $x^{\rm o}_2$, let $[u^{\rm o}_1,u^{\rm o}_2] \equiv [\dfrac{u^{\rm o}_r}{r}, u^{\rm o}_z]$ and denoting \\ $[x^{\rm o}_1, x^{\rm o}_2, x^{\rm o}_3] \equiv [r,z,\theta]$, we can then express \refeq{eqn:stress_axs} as
\begin{equation*}
\left[
\begin{array}{c}
\varepsilon^{\rm o}_{11} \\
\varepsilon^{\rm o}_{22} \\
\varepsilon^{\rm o}_{33} \\
\varepsilon^{\rm o}_{12} \\
\end{array}
\right] = [\hat{\mathbf{E}}]\underbrace{\left[\begin{array}{ccccc}
x^{\rm o}_1 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 \\
0 & x^{\rm o}_1 & 1 & 0 & 0 \\
\end{array}\right]}_{[\mathbf{B}_a]}
\left[
\begin{array}{c}
\dfrac{\partial u^{\rm o}_1}{\partial x^{\rm o}_1} \\[7pt]
\dfrac{\partial u^{\rm o}_1}{\partial x^{\rm o}_2} \\[7pt]
\dfrac{\partial u^{\rm o}_2}{\partial x^{\rm o}_1} \\[7pt]
\dfrac{\partial u^{\rm o}_2}{\partial x^{\rm o}_2} \\[7pt]
u^{\rm o}_1 \\
\end{array}
\right]\nonumber.
\end{equation*}
As in the previous case, we consider the usual homogeneous Dirichlet boundary conditions on $\Gamma^{\rm o}_{D,i}$ and Neumann boundary conditions on $\Gamma^{\rm o}$.
Then if we consider the output of interest $s^{\rm o}(\boldsymbol{\mu})$ defined upon $\Gamma^{\rm o}_L$, we arrive at the same abstract statement where
\begin{equation*}
[\mathbf{S}^{a}] = x^{\rm o}_1[\mathbf{B}_a]^T[\mathbf{E}][\mathbf{B}_a], \
[\mathbf{S}^{f}] = \left[(x^{\rm o}_1)^2f^{\rm o}_ne^{\rm o}_{n,1} , \ x^{\rm o}_1f^{\rm o}_ne^{\rm o}_{n,2} \right], \
[\mathbf{S}^{\ell}] = \left[ x^{\rm o}_1f^{\rm o}_ne^{\rm o}_{n,1} ,\ f^{\rm o}_ne^{\rm o}_{n,2}\right].
\end{equation*}
Note that the $x^{\rm o}_1$ multipliers appear in $[\mathbf{S}^{f}]$ during the weak form derivation, while in $[\mathbf{S}^{\ell}]$, in order to retrieve the measurement for the axial displacement $u^{\rm o}_r$ rather than $u^{\rm o}_1$ due to the change of variables. Also, the $2\pi$ multipliers in both $a^{\rm o}(\cdot,\cdot;\boldsymbol{\mu})$ and $f^{\rm o}(\cdot;\boldsymbol{\mu})$ are disappeared in the weak form during the derivation, and can be included in $\ell^{\rm o}(\cdot;\boldsymbol{\mu})$, i.e. incorporated to $[\mathbf{S}^{\ell}]$ if measurement is required to be done in thruth (rather than in the axisymmetric) domain.
\subsection{Formulation on Reference Domain}
The RB requires that the computational domain must be parameter-independent; however, our ``original'' domain $\Omega^{\rm o}(\boldsymbol{\mu})$ is obviously parameter-dependent. Hence, to transform $\Omega^{\rm o}(\boldsymbol{\mu})$ into the computational domain, or ``reference'' (parameter-independent) domain $\Omega$, we must perform geometric transformations in order to express the bilinear and linear forms in our abstract statement in appropriate ``affine forms''. This ``affine forms'' formulation allows us to model all possible configurations, corresponding to every $\boldsymbol{\mu} \in \mbox{\boldmath$\mathcal{D}$}$, based on a single reference-domain \cite{Quarteroni2011,rozza08:ARCME}.
\subsubsection{Geometry Mappings}
We first assume that, for all $\boldsymbol{\mu} \in \mbox{\boldmath$\mathcal{D}$}$, $\Omega^{\rm o}(\boldsymbol{\mu})$ is expressed as
\begin{equation*}
\Omega^{\rm o}(\boldsymbol{\mu}) = \bigcup_{s = 1}^{L_{\rm reg}}\Omega^{\rm o}_s(\boldsymbol{\mu}),
\end{equation*}
where the $\Omega^{\rm o}_s(\boldsymbol{\mu})$, $s = 1,\ldots,L_{\rm reg}$ are mutually non-overlapping subdomains. In two dimensions, $\Omega^{\rm o}_s(\boldsymbol{\mu})$, $s = 1,\ldots,L_{\rm reg}$ is a set of triangles (or in the general case, a set of ``curvy triangles''\footnote{In fact, a ``curvy triangle'' \cite{rozza08:ARCME} is served as the building block. For its implementation see} \cite{rbMIT_URL}.) such that all important domains/edges (those defining different material regions, boundaries, pressures/tractions loaded boundary segments, or boundaries which the output of interests are calculated upon) are included in the set. In practice, such a set is generated by a constrained Delaunay triangulation.
We next assume that there exists a reference domain $\Omega \equiv \Omega^{\rm o}(\boldsymbol{\mu}_{\rm ref}) =
\bigcup_{s=1}^{L_{\rm reg}}\Omega_s$ where, for any $\mathbf{x}^{\rm o} \in \Omega_s$, $s = 1,\ldots,L_{\rm reg}$, its image $\mathbf{x}^{\rm o} \in \Omega^{\rm o}_s(\boldsymbol{\mu})$ is given by
\begin{equation}\label{eqn:local_mapping}
\mathbf{x}^{\rm o}(\boldsymbol{\mu})
= \mathcal{T}^{\rm aff}_s(\boldsymbol{\mu};\mathbf{x})
= [\mathbf{R}^{\rm aff}_s(\boldsymbol{\mu})]\mathbf{x} + [\mathbf{G}^{\rm aff}_s(\boldsymbol{\mu})],
\end{equation}
where $[\mathbf{R}^{\rm aff}_s(\boldsymbol{\mu})] \in \mathbb{R}^{2 \times 2}$ and $[\mathbf{G}^{\rm aff}_s(\boldsymbol{\mu})] \in \mathbb{R}^2$. It thus follows from our definitions that $\mathcal{T}_s(\boldsymbol{\mu};\mathbf{x}):\Omega_s\rightarrow\Omega_s^{\rm o}$, $1 \leq s \leq L_{\rm reg}$ is an (invertible) affine mapping from $\Omega_s$ to $\Omega^{\rm o}_s(\boldsymbol{\mu})$, hence the Jacobian $|{\rm det}([\mathbf{R}^{\rm aff}_s(\boldsymbol{\mu})])|$ is strictly positive, and that the derivative transformation matrix, $[\mathbf{D}^{\rm aff}_s(\boldsymbol{\mu})] = [\mathbf{R}^{\rm aff}_s(\boldsymbol{\mu})]^{-1}$ is well defined. We thus can write
\begin{equation}\label{eqn:spatial_derv}
\frac{\partial}{\partial x^{\rm o}_i} = \frac{\partial x_j}{\partial x^{\rm o}_i}\frac{\partial}{\partial x_j}
= D^{\rm aff}_{s,ij}(\boldsymbol{\mu})\frac{\partial}{\partial x_j}, \quad 1 \leq i,j \leq 2.
\end{equation}
As in two dimensions, an affine transformation maps a triangle to a triangle, we can readily calculate $[\mathbf{R}^{\rm aff}_s(\boldsymbol{\mu})]$ and $[\mathbf{G}^{\rm aff}_s(\boldsymbol{\mu})]$ for each subdomains $s$ by simply solving a systems of six equations forming from \refeq{eqn:local_mapping} by matching parametrized coordinates to reference coordinates for the three triangle vertices.
We further require a mapping continuity condition: for all $\boldsymbol{\mu} \in \mathcal{D}$,
\begin{equation*}
\mathcal{T}_s(\boldsymbol{\mu};\mathbf{x}) = \mathcal{T}_{s'}(\boldsymbol{\mu};\mathbf{x}),
\quad \forall \mathbf{x} \in \Omega_s \cap \Omega_{s'}, \quad 1 \leq
s,s' \leq L_{\rm reg}.
\end{equation*}
This condition is automatically held if there is no curved edge in the set of $\Omega^{\rm o}_s(\boldsymbol{\mu})$. If a domain contains one or more ``important'' curved edge, special ``curvy triangles'' must be generated appropriately to honour the continuity condition. We refer the readers to \cite{rozza08:ARCME} for the full discussion and detail algorithm for such cases.
The global transformation is for $\mathbf{x} \in \Omega$, the image $\mathbf{x}^{\rm o} \in \Omega^{\rm o}(\boldsymbol{\mu})$ is given by
\begin{equation*}
\mathbf{x}^{\rm o}(\boldsymbol{\mu}) = \mathcal{T}(\boldsymbol{\mu};\mathbf{x}).
\end{equation*}
It thus follows that $\mathcal{T}(\boldsymbol{\mu};\mathbf{x}):\Omega\rightarrow\Omega^{\rm o}(\boldsymbol{\mu})$ is a piecewise-affine geometric mapping.
\subsubsection{Affine Forms}
We now define our functional space $X$ as
\begin{equation*}
X = \{v = (v_1,v_2) \in (H^1(\Omega))^2 | v_i = 0 \ {\rm on} \ \Gamma_{D,i}, i = 1,2\},
\end{equation*}
and recast our bilinear form $a^{\rm o}(w,v;\boldsymbol{\mu})$, by invoking \refeq{eqn:bilinear_o}, \refeq{eqn:local_mapping} and \refeq{eqn:spatial_derv} to obtain $\forall w,v \in X(\Omega)$
\begin{eqnarray*}
a(w,v;\boldsymbol{\mu}) &=& \int_{\bigcup_{s=1}^{L_{\rm reg}}\Omega_s}\left[
\dfrac{\partial w_1}{\partial x_1},
\dfrac{\partial w_1}{\partial x_2},
\dfrac{\partial w_2}{\partial x_1},
\dfrac{\partial w_2}{\partial x_2},
w_1
\right][\mathbf{S}^{a,\rm aff}_s(\boldsymbol{\mu})]
\left[
\begin{array}{c}
\dfrac{\partial v_1}{\partial x_1} \\
\dfrac{\partial v_1}{\partial x_2} \\
\dfrac{\partial v_2}{\partial x_1} \\
\dfrac{\partial v_2}{\partial x_2} \\
v_1 \\
\end{array}
\right]d\Omega.
\end{eqnarray*}
where $[\mathbf{S}^{a,\rm aff}_s(\boldsymbol{\mu})] = [\mathbf{H}_s(\boldsymbol{\mu})][\mathbf{S}^a_s][\mathbf{H}_s(\boldsymbol{\mu})]^T|{\rm det}([\mathbf{R}^{\rm aff}_s(\boldsymbol{\mu})])|$ is the effective elastic tensor matrix, in which
\begin{equation*}
[\mathbf{H}_s(\boldsymbol{\mu})] = \left(
\begin{array}{ccc}
[\mathbf{D}_s(\boldsymbol{\mu})] & [\boldsymbol{0}]^{2 \times 2} & 0 \\
\phantom{1}[\boldsymbol{0}]^{2 \times 2} & [\mathbf{D}_s(\boldsymbol{\mu})] & 0 \\
0 & 0 & 1 \\
\end{array}
\right).
\end{equation*}
Similarly, the linear form $f^{\rm o}(v;\boldsymbol{\mu})$, $\forall v \in X$ can be transformed as
\begin{eqnarray*}
f(v;\boldsymbol{\mu}) = \int_{\bigcup_{s=1}^{L_{\rm reg}}\Gamma_{N_s}}[\mathbf{S}^{f,{\rm aff}}_s]
\left[
\begin{array}{c}
v_1 \\ v_2 \\
\end{array}
\right]
d\Gamma,
\end{eqnarray*}
where $\Gamma_{N_s}$ denotes the partial boundary segment of $\Gamma_N$ of the subdomain $\Omega_s$ and $[\mathbf{S}^{f,{\rm aff}}_s] = \|([\mathbf{R}_s(\boldsymbol{\mu})]\mathbf{e}_n)\|_2[\mathbf{S}^f]$ is the effective load vector, where $\mathbf{e}_n$ is the normal vector to $\Gamma_{N_s}$ and $\|\cdot\|_2$ denotes the usual Euclidean norm. The linear form $\ell(v;\boldsymbol{\mu})$ is also transformed in the same manner.
We then replace all ``original'' $x^{\rm o}_1$ and $x^{\rm o}_2$ in the effective elastic tensor matrix $[\mathbf{S}_s^{a,\rm aff}(\boldsymbol{\mu})]$, effective load/output vectors $[\mathbf{S}^{f,\rm aff}_s(\boldsymbol{\mu})]$ and $[\mathbf{S}^{\ell,\rm aff}_s(\boldsymbol{\mu})]$ by \refeq{eqn:local_mapping} to obtain a $\mathbf{x}^{\rm o}$-free effective elastic tensor matrix and effective load/output vectors, respectively.\footnote{Here we note that, the Young's modulus $E$ in the isotropic and axisymmetric cases (or $E_1$, $E_2$ and $E_3$ in the orthotropic case only} in certain conditions) can be a polynomial function of the spatial coordinates $\mathbf{x}^{\rm o}$ as well, and we still be able to obtain our affine forms \refeq{eqn:affine}.
We next expand the bilinear form $a(w,v;\boldsymbol{\mu})$ by treating each entry of the effective elastic tensor matrix for each subdomain separately, namely
\begin{eqnarray}\label{eqn:bilinear_exp}
a(w,v;\boldsymbol{\mu}) &=& S_{1,11}^{a,\rm aff}(\boldsymbol{\mu})\int_{\Omega_1}\frac{\partial w_1}{\partial x_1}\frac{\partial v_1}{\partial x_1} + S_{1,12}^{a,\rm aff}(\boldsymbol{\mu})\int_{\Omega_1}\frac{\partial w_1}{\partial x_1}\frac{\partial v_1}{\partial x_2} + \ldots \\
&& + S_{L_{\rm reg},55}^{a,\rm aff}(\boldsymbol{\mu})\int_{\Omega_{L_{\rm reg}}}w_1w_1.
\end{eqnarray}
Note that here for simplicity, we consider the case where there is no spatial coordinates in $[\mathbf{S}^{\ell,\rm aff}_s(\boldsymbol{\mu})]$. In general (especially for axisymmetric case), some or most of the integrals may take the form of $\int_{\Omega_s}(x_1)^m(x_2)^n\dfrac{\partial w_i}{\partial x_j}\dfrac{\partial v_k}{\partial x_l}$, where $m,n \in \mathbb{R}$.
Taking into account the symmetry of the bilinear form and the effective elastic tensor matrix, there will be at most $Q^a = 7L_{\rm reg}$ terms in the expansion. However, in practice, most of the terms can be collapsed by noticing that not only there will be a lot of zero entries in $[\mathbf{S}_s^{a,\rm aff}(\boldsymbol{\mu})]$, $s = 1,\ldots,L_{\rm reg}$, but also there will be a lot of duplicated or ``linearly dependent'' entries, for example, $S_{1,11}^{a,\rm aff}(\boldsymbol{\mu}) = [{\rm Const}]S_{2,11}^{a,\rm aff}(\boldsymbol{\mu})$. We can then apply a symbolic manipulation technique \cite{rozza08:ARCME} to identify, eliminate all zero terms in \refeq{eqn:bilinear_exp} and collapse all ``linear dependent'' terms to end up with a minimal $Q^a$ expansion. The same procedure is also applied for the linear forms $f(\cdot;\boldsymbol{\mu})$ and $\ell(\cdot;\boldsymbol{\mu})$.
Hence the abstract formulation of the linear elasticity problem in the reference domain $\Omega$ reads as follow: given $\boldsymbol{\mu} \in \mbox{\boldmath$\mathcal{D}$}$, find
\begin{equation*}
s(\boldsymbol{\mu}) = \ell(u(\boldsymbol{\mu});\boldsymbol{\mu}),
\end{equation*}
where $u(\boldsymbol{\mu}) \in X$ satisfies
\begin{equation*}
a(u(\boldsymbol{\mu}),v;\boldsymbol{\mu}) = f(v;\boldsymbol{\mu}), \quad \forall v \in X,
\end{equation*}
where all the bilinear and linear forms are in affine forms,
\begin{eqnarray}\label{eqn:affine}
a(w,v;\boldsymbol{\mu}) &=& \sum_{q=1}^{Q^a}\Theta_q^a(\boldsymbol{\mu}) a_q(w,v), \nonumber \\
f(v;\boldsymbol{\mu}) &=& \sum_{q=1}^{Q^f}\Theta_q^f(\boldsymbol{\mu}) f_q(v), \nonumber \\
\ell(v;\boldsymbol{\mu}) &=& \sum_{q=1}^{Q^{\ell}}\Theta_q^{\ell}(\boldsymbol{\mu}) \ell_q(v), \quad \forall w,v, \in X.
\end{eqnarray}
Here $\Theta_q^a(\boldsymbol{\mu})$, $a_q(w,v)$, $q = 1,\ldots,Q^a$, $f_q(v)$; $\Theta_q^f(\boldsymbol{\mu})$, $f_q(v)$, $q = 1,\ldots,Q^f$, and $\Theta_q^{\ell}(\boldsymbol{\mu})$, $\ell_q(v)$, $q = 1,\ldots,Q^{\ell}$ are parameter-dependent coefficient and parameter-independent bilinear and linear forms, respectively.
We close this section by defining several useful terms. We first define our inner product and energy norm as
\begin{equation}\label{eqn:inner_prod}
(w,v)_X = a(w,v;\overline{\boldsymbol{\mu}})
\end{equation}
and $\|w\|_X = (w,w)^{1/2}$, $\forall w,v \in X$, respectively, where $\overline{\boldsymbol{\mu}} \in \mbox{\boldmath$\mathcal{D}$}$ is an arbitrary parameter. Certain other inner norms and associated norms are also possible \cite{rozza08:ARCME}. We then define our coercivity and continuity constants as
\begin{equation}\label{eqn:inf}
\alpha(\boldsymbol{\mu}) = \inf_{w\in X}\frac{a(w,v;\boldsymbol{\mu})}{\|w\|_X^2},
\end{equation}
\begin{equation}\label{eqn:sup}
\gamma(\boldsymbol{\mu}) = \sup_{w\in X}\frac{a(w,v;\boldsymbol{\mu})}{\|w\|_X^2},
\end{equation}
respectively. We assume that $a(\cdot,\cdot;\boldsymbol{\mu})$ is symmetric, $a(w,v;\boldsymbol{\mu}) = a(v,w;\boldsymbol{\mu})$, $\forall w,v \in X$, coercive, $\alpha(\boldsymbol{\mu}) > \alpha_0 > 0$, and continuous, $\gamma(\boldsymbol{\mu}) < \gamma_0 < \infty$; and also our $f(\cdot;\boldsymbol{\mu})$ and $\ell(\cdot;\boldsymbol{\mu})$ are bounded functionals. It follows that problem which is well-defined and has a unique solution. Those conditions are automatically satisfied given the nature of our considered problems \cite{sneddon1999mathematical, sneddon2000mathematical}.
\subsection{Truth approximation}
From now on, we shall restrict our attention to the ``compliance'' case ($f(\cdot;\boldsymbol{\mu}) = \ell(\cdot;\boldsymbol{\mu})$). Extension to the non-compliance case will be discuss in the Section~5.
We now apply the finite element method and we provide a matrix formulation \cite{CMCS-CONF-2009-002}: given $\boldsymbol{\mu} \in \mbox{\boldmath$\mathcal{D}$}$, we evaluate
\begin{equation}\label{eqn:FE_out}
s(\boldsymbol{\mu}) = [\mathbf{F}^\mathcal{N}(\boldsymbol{\mu})]^T[\mathbf{u}^\mathcal{N}(\boldsymbol{\mu})],
\end{equation}
where $[\mathbf{u}^\mathcal{N}(\boldsymbol{\mu})]$ represents a finite element solution $u^{\mathcal{N}}(\boldsymbol{\mu}) \in X^{\mathcal{N}} \in X$ of size $\mathcal{N}$ which satisfies
\begin{equation}\label{eqn:FE_stiff}
[\mathbf{K}^\mathcal{N}(\boldsymbol{\mu})][\mathbf{u}^\mathcal{N}(\boldsymbol{\mu})] = [\mathbf{F}^\mathcal{N}(\boldsymbol{\mu})];
\end{equation}
here $[\mathbf{K}^\mathcal{N}(\boldsymbol{\mu})]$, and $[\mathbf{F}^\mathcal{N}(\boldsymbol{\mu})]$ and the (discrete forms) stiffness matrix and load vector of $a(\cdot,\cdot;\boldsymbol{\mu})$, and $f(\cdot;\boldsymbol{\mu})$, respectively. Note that the stiffness matrix $[\mathbf{K}^\mathcal{N}(\boldsymbol{\mu})]$ is symmetric positive definite (SPD). By invoking the affine forms \refeq{eqn:affine}, we can express $[\mathbf{K}^\mathcal{N}(\boldsymbol{\mu})]$, and $[\mathbf{F}^\mathcal{N}(\boldsymbol{\mu})]$ as
\begin{eqnarray}\label{eqn:affine_FE}
[\mathbf{K}^\mathcal{N}(\boldsymbol{\mu})] &=& \sum_{q = 1}^{Q^a}\Theta_q^a(\boldsymbol{\mu})[\mathbf{K}^\mathcal{N}_q], \nonumber \\
\left[\mathbf{F}^\mathcal{N}(\boldsymbol{\mu})\right] &=& \sum_{q = 1}^{Q^f}\Theta_q^f(\boldsymbol{\mu})[\mathbf{F}^\mathcal{N}_q],
\end{eqnarray}
where $[\mathbf{K}^\mathcal{N}_q]$, $[\mathbf{F}^\mathcal{N}_q]$ and are the discrete forms of the parameter-independent bilinear and linear forms $a_q(\cdot,\cdot)$ and $f_q(\cdot)$, respectively. We also denote (the SPD matrix) $[\mathbf{Y}^\mathcal{N}]$ as the discrete form of our inner product \refeq{eqn:inner_prod}. We also assume that the size of of our FE approximation, $\mathcal{N}$ is large enough such that our FE solution is an accurate approximation of the exact solution.
\section{Reduced Basis Method}
In this Section we shall restrict our attention by recalling the RB method for the ``compliant'' output. We shall first define the RB spaces and the Galerkin projection. We then describe an Offline-Online computational strategy, which allows us to obtain $\mathcal{N}$-independent calculation of the RB output approximation \cite{hesthaven2015certified,NgocCuong2005}.
\subsection{RB Spaces and Greedy algorithm}
To define the RB approximation we first introduce a (nested) Lagrangian parameter sample for $1 \leq N \leq N_{\max}$,
\begin{equation*}
S_N = \{\boldsymbol{\mu}_1,\boldsymbol{\mu}_2,\ldots,\boldsymbol{\mu}_N\},
\end{equation*}
and associated hierarchical reduced basis spaces $(X_N^\mathcal{N} =) W^\mathcal{N}_N$, $1 \leq N \leq N_{\max}$,
\begin{equation*}
W^\mathcal{N}_N = {\rm span}\{u^{\mathcal{N}}(\boldsymbol{\mu}_n),1 \leq n \leq N\} ,
\end{equation*}
where $\boldsymbol{\mu}_n \in \mbox{\boldmath$\mathcal{D}$}$ are determined by the means of a Greedy sampling algorithm \cite{rozza08:ARCME, quarteroni2015reduced}; this is an interarive procedure where at each step a new basis function is added in order to improve the precision of the basis set.
The key point of this methodology is the availability of an estimate of the error induced by replacing the full space $X^{\mathcal{N}}$ with the reduced order one $W^\mathcal{N}_N$ in the variational formulation. More specifically we assume that for all $\boldsymbol{\mu} \in \mbox{\boldmath$\mathcal{D}$}$ there exist an estimator $\eta(\boldsymbol{\mu})$ such that
\begin{equation*}
|| u^{\mathcal{N}}(\boldsymbol{\mu}) - u^\mathcal{N}_{{\rm RB},N}(\boldsymbol{\mu})|| \leq \eta(\boldsymbol{\mu}) ,
\end{equation*}
where $u^{\mathcal{N}}(\boldsymbol{\mu}) \in X^{\mathcal{N}} \in X$ represents the finite element solution, $u^\mathcal{N}_{{\rm RB},N}(\boldsymbol{\mu})\in X_N^\mathcal{N} \subset X^\mathcal{N}$ the reduced basis one and we can choose either the induced or the energy norm.
During this iterative basis selection process and if at the j-th step a j-dimensional reduced basis space $W^\mathcal{N}_j$ is given, the next basis function is the one that maximizes the estimated model order reduction error given the j-dimensional space $W^\mathcal{N}_j$ over $\mbox{\boldmath$\mathcal{D}$}$. So at the $n+1$ iteration we select $$\boldsymbol{\mu}_{n+1} = arg \max_{\boldsymbol{\mu} \in \mbox{\boldmath$\mathcal{D}$}} \eta(\boldsymbol{\mu})$$ and compute $u^{\mathcal{N}}(\boldsymbol{\mu}_{n+1})$ to enrich the reduced space. This is repeated until the maximal estimated error is below a required error tolerance. With this choice the greedy algorithm always selects the next parameter sample point as the one for which the model error is the maximum as estimated by $\eta(\boldsymbol{\mu})$ and this yields a basis that aims to be optimal in the maximum norm over $\mbox{\boldmath$\mathcal{D}$}$.
Furthermore we can rewrite the reduced space as
\begin{equation*}
W^\mathcal{N}_N = {\rm span}\{\zeta^{\mathcal{N}}_n,1 \leq n \leq N\},
\end{equation*}
where the basis functions $\left\{\zeta^{\mathcal{N}}\right\}$ are computed from the snapshots $u^{\mathcal{N}}(\boldsymbol{\mu})$ by a Gram-Schmidt orthonormalization process such that $[\boldsymbol{\zeta}^{\mathcal{N}}_m]^T[\mathbf{Y}^\mathcal{N}][\boldsymbol{\zeta}^{\mathcal{N}}_n] = \delta_{mn}$, where $\delta_{mn}$ is the Kronecker-delta symbol. We then define our orthonormalized-snapshot matrix $[\mathbf{Z}_N] \equiv [\mathbf{Z}^\mathcal{N}_N] = [[\boldsymbol{\zeta}^{\mathcal{N}}_1]|\cdots|[\boldsymbol{\zeta}^{\mathcal{N}}_n]]$ of dimension $\mathcal{N} \times N$.
\subsection{Galerkin Projection}
We then apply a Galerkin projection on our ``truth'' problem \cite{almroth78:_autom,noor81:_recen,noor82,noor80:_reduc,rozza08:ARCME}: given $\boldsymbol{\mu} \in \mbox{\boldmath$\mathcal{D}$}$, we could evaluate the RB output as
\begin{equation*}
s_N(\boldsymbol{\mu}) = [\mathbf{F}^\mathcal{N}(\boldsymbol{\mu})]^T[\mathbf{u}^\mathcal{N}_{{\rm RB},N}(\boldsymbol{\mu})],
\end{equation*}
where
\begin{equation}\label{eqn:RB_sol}
[\mathbf{u}^\mathcal{N}_{{\rm RB},N}(\boldsymbol{\mu})] = [\mathbf{Z}_N][\mathbf{u}_N(\boldsymbol{\mu})]
\end{equation}
represents the RB solution $\mathbf{u}^\mathcal{N}_{{\rm RB},N}(\boldsymbol{\mu}) \in X_N^\mathcal{N} \subset X^\mathcal{N}$ of size $\mathcal{N}$. Here $[\mathbf{u}_N(\boldsymbol{\mu})]$ is the RB coefficient vector of dimension $N$ satisfies the RB ``stiffness'' equations
\begin{equation}\label{eqn:RB_semifull}
[\mathbf{K}_N(\boldsymbol{\mu})][\mathbf{u}_N(\boldsymbol{\mu})] = [\mathbf{F}_N(\boldsymbol{\mu})],
\end{equation}
where
\begin{eqnarray}\label{eqn:RB_comp1}
[\mathbf{K}_N(\boldsymbol{\mu})] &=& [\mathbf{Z}_N]^T[\mathbf{K}^\mathcal{N}(\boldsymbol{\mu})][\mathbf{Z}_N], \nonumber \\
\left[\mathbf{F}_N(\boldsymbol{\mu})\right] &=& [\mathbf{Z}_N]^T[\mathbf{F}^\mathcal{N}(\boldsymbol{\mu})].
\end{eqnarray}
Note that the system \refeq{eqn:RB_semifull} is of small size: it is just a set of $N$ linear algebraic equations, in this way we can now evaluate our output as
\begin{equation}\label{eqn:RB_outsemifull}
s_N(\boldsymbol{\mu}) = [\mathbf{F}_N(\boldsymbol{\mu})]^T[\mathbf{u}_N(\boldsymbol{\mu})].
\end{equation}
It can be shown \cite{patera07:book} that the condition number of the RB ``stiffness'' matrix $[\mathbf{Z}_N]^T[\mathbf{K}^\mathcal{N}(\boldsymbol{\mu})][\mathbf{Z}_N]$ is bounded by $\gamma_0(\boldsymbol{\mu})/\alpha_0(\boldsymbol{\mu})$, and independent of both $N$ and $\mathcal{N}$.
\subsection{Offline-Online Procedure}
Although the system \refeq{eqn:RB_semifull} is of small size, the computational cost for assembling the RB ``stiffness'' matrix (and the RB ``output'' vector $[\mathbf{F}^\mathcal{N}(\boldsymbol{\mu})]^T[\mathbf{Z}_N]$) is still involves $\mathcal{N}$ and costly, $O(N\mathcal{N}^2 + N^2\mathcal{N})$ (and $O(N\mathcal{N})$, respectively). However, we can use our affine forms \refeq{eqn:affine} to construct very efficient Offline-Online procedures, as we shall discuss below.
We first insert our affine forms \refeq{eqn:affine_FE} into the expansion \refeq{eqn:RB_semifull} and \refeq{eqn:RB_outsemifull}, by using \refeq{eqn:RB_comp1} we obtain
\begin{equation*}
\sum_{q = 1}^{Q^a}\Theta_q^a(\boldsymbol{\mu})[\mathbf{K}_{qN}][\mathbf{u}_N(\boldsymbol{\mu})]
= \sum_{q = 1}^{Q^f}\Theta_q^f(\boldsymbol{\mu})[\mathbf{F}_{qN}]
\end{equation*}
and
\begin{equation*}
s_N(\boldsymbol{\mu}) = \sum_{q = 1}^{Q^f}\Theta_q^f(\boldsymbol{\mu})[\mathbf{F}_{qN}][\mathbf{u}_N(\boldsymbol{\mu})],
\end{equation*}
respectively. Here
\begin{eqnarray*}
[\mathbf{K}_{qN}] &=& [\mathbf{Z}_N]^T[\mathbf{K}^\mathcal{N}_q][\mathbf{Z}_N], \quad 1 \leq q \leq Q^a \\
\left[\mathbf{F}_{qN}\right] &=& [\mathbf{Z}_N]^T[\mathbf{F}^\mathcal{N}_q], \quad 1 \leq q \leq Q^f,
\end{eqnarray*}
are parameter independent quantities that can be computed just once and than stored for all the subsequent $\boldsymbol{\mu}$-dependent queries.
We then observe that all the ``expensive'' matrices $[\mathbf{K}_{qN}]$, $1 \leq q \leq Q^a$, $1 \leq N \leq N_{\max}$ and vectors $[\mathbf{F}_{qN}]$, $1 \leq q \leq Q^f$, $1 \leq N \leq N_{\max}$, are now separated and parameter-independent, hence those can be \emph{pre-computed} in an Offline-Online procedure.
In the Offline stage, we first compute the $[\mathbf{u}^\mathcal{N}(\boldsymbol{\mu}^n)]$, $1 \leq n \leq N_{\max}$, form the matrix $[\mathbf{Z}_{N_{\max}}]$ and then form and store $[\mathbf{F}_{N_{\max}}]$ and $[\mathbf{K}_{qN_{\max}}]$. The Offline operation count depends on $N_{\max}$, $Q^a$ and $\mathcal{N}$ but requires only $O(Q^aN_{\max}^2 + Q^fN_{\max} + Q^\ell N_{\max})$ permanent storage.
In the Online stage, for a given $\boldsymbol{\mu}$ and $N$ ($1 \leq N \leq N_{\max}$), we retrieve the pre-computed $[\mathbf{K}_{qN}]$ and $[\mathbf{F}_{N}]$ (subarrays of $[\mathbf{K}_{qN_{\max}}]$, $[\mathbf{F}_{N_{\max}}]$), form $[\mathbf{K}_N(\boldsymbol{\mu})]$, solve the resulting $N \times N$ system \refeq{eqn:RB_semifull} to obtain $\{\mathbf{u}_N(\boldsymbol{\mu})\}$, and finally evaluate the output $s_N(\boldsymbol{\mu})$ from \refeq{eqn:RB_outsemifull}. The Online operation count is thus $O(N^3)$ and independent of $\mathcal{N}$. The implication of the latter is two-fold: first, we will achieve very fast response in the many-query and real-time contexts, as $N$ is typically very small, $N \ll \mathcal{N}$; and second, we can choose $\mathcal{N}$ arbitrary large -- to obtain as accurate FE predictions as we wish -- without adversely affecting the Online (marginal) cost.
\section{\emph{A posteriori} error estimation}
In this Section we recall the \emph{a posteriori} error estimator for our RB approximation. We shall discuss in details the computation procedures for the two ingredients of the error estimator: the dual norm of the residual and the coercivity lower bound. We first present the Offline-Online strategy for the computation of the dual norm of the residual; we then briefly discuss the Successive Constraint Method \cite{huynh07:cras} in order to compute the coercivity lower bound.
\subsection{Definitions}
We first introduce the error $e^\mathcal{N}(\boldsymbol{\mu}) \equiv u^\mathcal{N}(\boldsymbol{\mu}) - u^\mathcal{N}_{{\rm RB},N}(\boldsymbol{\mu}) \in X^\mathcal{N}$ and the
residual $r^\mathcal{N}(v;\boldsymbol{\mu}) \in (X^\mathcal{N})'$ (the dual space to $X^\mathcal{N})$, $\forall v \in X^\mathcal{N}$,
\begin{equation}\label{eqn:residual}
r^\mathcal{N}(v;\boldsymbol{\mu}) = f(v) - a(u^\mathcal{N}(\boldsymbol{\mu}),v;\boldsymbol{\mu}),
\end{equation}
which can be given in the discrete form as
\begin{equation}\label{eqn:FE_residual}
[\mathbf{r}^\mathcal{N}(\boldsymbol{\mu})] = [\mathbf{F}^\mathcal{N}(\boldsymbol{\mu})] - [\mathbf{K}^\mathcal{N}(\boldsymbol{\mu})][\mathbf{u}^\mathcal{N}_{{\rm RB},N}(\boldsymbol{\mu})].
\end{equation}
We then introduce the Riesz representation of $r^\mathcal{N}(v;\boldsymbol{\mu})$: $\hat{e}(\boldsymbol{\mu}) \in X^\mathcal{N}$ defined by $(\hat{e}(\boldsymbol{\mu}),v)_{X^\mathcal{N}} = r^\mathcal{N}(v;\boldsymbol{\mu})$, $\forall v \in X^\mathcal{N}$. In vector form, $\hat{e}(\boldsymbol{\mu})$ can be expressed as
\begin{equation}\label{eqn:rres}
[\mathbf{Y}^\mathcal{N}][\mbox{\boldmath$\hat{e}$}(\boldsymbol{\mu})] = [\mathbf{r}^{\mathcal{N}}(\boldsymbol{\mu})].
\end{equation}
We also require a lower bound to the coercivity constant
\begin{equation}\label{eqn:inf_FE}
\alpha^\mathcal{N}(\boldsymbol{\mu}) = \inf_{w \in X^{\mathcal{N}}}\frac{a(w,w;\boldsymbol{\mu})}{\|w\|^2_{X^{\mathcal{N}}}},
\end{equation}
such that $0< \alpha_{\rm LB}^\mathcal{N}(\boldsymbol{\mu}) \leq \alpha^\mathcal{N}(\boldsymbol{\mu})$, $\forall \boldsymbol{\mu} \in \mbox{\boldmath$\mathcal{D}$}$.
We may now define our error estimator for our output as
\begin{equation}
\Delta_N^s(\boldsymbol{\mu}) \equiv \frac{\|\hat{e}(\boldsymbol{\mu})\|^2_{X^\mathcal{N}}}{\alpha^\mathcal{N}_{\rm LB}},
\end{equation}
where $\|\hat{e}(\boldsymbol{\mu})\|_{X^\mathcal{N}}$ is the dual norm of the residual.
We can also equip the error estimator with an effectivity defined by
\begin{equation}
\eta_N^s(\boldsymbol{\mu}) \equiv \frac{\Delta_N^s(\boldsymbol{\mu})}{|s^\mathcal{N}(\boldsymbol{\mu})-s_N(\boldsymbol{\mu})|}.
\end{equation}
We can readily demonstrate \cite{rozza08:ARCME, patera07:book} that
\begin{equation*}
1 \leq \eta_N^s(\boldsymbol{\mu}) \leq \frac{\gamma_0(\boldsymbol{\mu})}{\alpha^\mathcal{N}_{\rm LB}(\boldsymbol{\mu})}, \quad \forall \boldsymbol{\mu} \in \mbox{\boldmath$\mathcal{D}$};
\end{equation*}
so that the error estimator is both \emph{rigorous} and \emph{sharp}. Note that here we can only claim the \emph{sharp} property for this current ``compliant'' case.
We shall next provide procedures for the computation of the two ingredients of our error estimator: we shall first discuss the Offline-Online strategy to compute the dual norm of the residual $\|\hat{e}(\boldsymbol{\mu})\|_{X^\mathcal{N}}$, and then provide the construction for the lower bound of the coercivity constant $\alpha^\mathcal{N}(\boldsymbol{\mu})$.
\subsection{Dual norm of the residual}
In discrete form, the dual norm of the residual $\varepsilon(\boldsymbol{\mu}) = \|\hat{e}(\boldsymbol{\mu})\|_{X^\mathcal{N}}$ is given by
\begin{equation}\label{eqn:dnres}
\varepsilon^2(\boldsymbol{\mu}) = [\mbox{\boldmath$\hat{e}$}(\boldsymbol{\mu})]^T[\mathbf{Y}^\mathcal{N}][\mbox{\boldmath$\hat{e}$}(\boldsymbol{\mu})].
\end{equation}
We next invoke \refeq{eqn:FE_residual}, \refeq{eqn:rres} and \refeq{eqn:dnres} to arrive at
\begin{eqnarray}\label{eqn:dnres_der1}
\varepsilon^2(\boldsymbol{\mu}) &=& \bigg([\mathbf{F}^\mathcal{N}(\boldsymbol{\mu})] - [\mathbf{K}^\mathcal{N}(\boldsymbol{\mu})][\mathbf{u}^\mathcal{N}_{{\rm RB},N}(\boldsymbol{\mu})])\bigg)^T[\mathbf{Y}^\mathcal{N}]^{-1}\bigg([\mathbf{F}^\mathcal{N}(\boldsymbol{\mu})] - [\mathbf{K}^\mathcal{N}(\boldsymbol{\mu})][\mathbf{u}^\mathcal{N}_{{\rm RB},N}(\boldsymbol{\mu})]\bigg) \nonumber \\
&=& [\mathbf{F}^\mathcal{N}(\boldsymbol{\mu})]^T[\mathbf{Y}^\mathcal{N}]^{-1}[\mathbf{F}^\mathcal{N}(\boldsymbol{\mu})] - 2[\mathbf{F}^\mathcal{N}(\boldsymbol{\mu})]^T[\mathbf{Y}^\mathcal{N}]^{-1}[\mathbf{K}^\mathcal{N}(\boldsymbol{\mu})] \nonumber \\
&& + [\mathbf{K}^\mathcal{N}(\boldsymbol{\mu})]^T[\mathbf{Y}^\mathcal{N}]^{-1}[\mathbf{K}^\mathcal{N}(\boldsymbol{\mu})].
\end{eqnarray}
We next defines the ``pseudo''-solutions $[\mathbf{P}^f_{q}] = [\mathbf{Y}^\mathcal{N}]^{-1}[\mathbf{F}_{q}^\mathcal{N}]$, $1 \leq q \leq Q^f$ and $[\mathbf{P}^a_{qN}] = [\mathbf{Y}^\mathcal{N}]^{-1}[\mathbf{K}_q^\mathcal{N}][\mathbf{Z}_N]$, $1 \leq q \leq Q^a$, then apply the affine form \refeq{eqn:affine_FE} and \refeq{eqn:RB_sol} into \refeq{eqn:dnres_der1} to obtain
\begin{eqnarray}\label{eqn:dnres_der2}
\varepsilon^2(\boldsymbol{\mu}) &=& \sum_{q=1}^{Q^f}\sum_{q'=1}^{Q^f}\Theta_q^f(\boldsymbol{\mu})\Theta_{q'}^f(\boldsymbol{\mu})\bigg([\mathbf{P}^f_{q}]^T[\mathbf{Y}^\mathcal{N}][\mathbf{P}^f_{q'}]\bigg) \\ && -2\sum_{q=1}^{Q^a}\sum_{q'=1}^{Q^f}\Theta_q^a(\boldsymbol{\mu})\Theta_{q'}^f(\boldsymbol{\mu})\bigg([\mathbf{P}^f_{q}]^T[\mathbf{Y}^\mathcal{N}][\mathbf{P}^a_{q'N}]\bigg)[\mathbf{u}_N^{RB}(\boldsymbol{\mu})] \nonumber \\
&&+\sum_{q=1}^{Q^a}\sum_{q'=1}^{Q^a}\Theta_q^a(\boldsymbol{\mu})\Theta_{q'}^a(\boldsymbol{\mu})[\mathbf{u}_N^{RB}(\boldsymbol{\mu})]^T\bigg([\mathbf{P}^a_{qN}]^T[\mathbf{Y}^\mathcal{N}][\mathbf{P}^a_{q'N}]\bigg)[\mathbf{u}_N^{RB}(\boldsymbol{\mu})] \nonumber.
\end{eqnarray}
It is observed that all the terms in bracket in \refeq{eqn:dnres_der2} are all parameter-independent, hence they can be \emph{pre-computed} in the Offline stage. The Offline-Online strategy is now clear.
In the Offline stage we form the parameter-independent quantities. We first compute the ``pseudo''-solutions $[\mathbf{P}^f_{q}] = [\mathbf{Y}^\mathcal{N}]^{-1}[\mathbf{F}_{q}^\mathcal{N}]$, $1 \leq q \leq Q^f$ and $[\mathbf{P}^a_{qN}] = [\mathbf{Y}^\mathcal{N}]^{-1}[\mathbf{K}_q^\mathcal{N}][\mathbf{Z}_N]$, $1 \leq q \leq Q^a$, $1 \leq N \leq N_{\max}$; and form/store $[\mathbf{P}^f_{q}]^T[\mathbf{Y}^\mathcal{N}][\mathbf{P}^f_{q'}]$, $1 \leq q, q' \leq Q^f$, $[\mathbf{P}^f_{q}]^T[\mathbf{Y}^\mathcal{N}][\mathbf{P}^a_{q'N}]$, $1 \leq q \leq Q^f$, $1 \leq q \leq Q^a$, $1 \leq N \leq N_{\max}$,\\ $[\mathbf{P}^a_{qN}][\mathbf{Y}^\mathcal{N}][\mathbf{P}^a_{q'N}]$, $1 \leq q, q' \leq Q^a$, $1 \leq N \leq N_{\max}$. The Offline operation count depends on $N_{\max}$, $Q^a$, $Q^f$, and $\mathcal{N}$.
In the Online stage, for a given $\boldsymbol{\mu}$ and $N$ ($1 \leq N \leq N_{\max}$), we retrieve the pre-computed quantities $[\mathbf{P}^f_{q}]^T[\mathbf{Y}^\mathcal{N}][\mathbf{P}^f_{q'}]$, $1 \leq q, q' \leq Q^f$, $[\mathbf{P}^f_{q}]^T[\mathbf{Y}^\mathcal{N}][\mathbf{P}^a_{q'N}]$, $1 \leq q \leq Q^f$, $1 \leq q \leq Q^a$, and $[\mathbf{P}^a_{qN}]^T[\mathbf{Y}^\mathcal{N}][\mathbf{P}^a_{q'N}]$, $1 \leq q, q' \leq Q^a$, and then evaluate the sum \refeq{eqn:dnres_der2}. The Online operation count is dominated by $O(((Q^a)^2+(Q^f)^2)N^2)$ and independent of $\mathcal{N}$.
\subsection{Lower bound of the coercivity constant}
We now briefly address some elements for the computation of the lower bound in the coercive case. In order to derive the discrete form of the coercivity constant $\refeq{eqn:inf_FE}$ we introduce the discrete eigenvalue problem: given $\boldsymbol{\mu} \in \mbox{\boldmath$\mathcal{D}$}$, find the minimum set $([\boldsymbol{\chi}_{\min}(\boldsymbol{\mu})],\lambda_{\min}(\boldsymbol{\mu}))$ such that
\begin{eqnarray}\label{eqn:inf_truth}
[\mathbf{K}^\mathcal{N}(\boldsymbol{\mu})][\boldsymbol{\chi}(\boldsymbol{\mu})] &=& \lambda_{\min}[\mathbf{Y}^\mathcal{N}][\boldsymbol{\chi}(\boldsymbol{\mu})], \nonumber \\
\left[\boldsymbol{\chi}(\boldsymbol{\mu})\right]^T[\mathbf{Y}^\mathcal{N}][\boldsymbol{\chi}(\boldsymbol{\mu})] &=& 1.
\end{eqnarray}
We can then recover
\begin{equation}
\alpha^\mathcal{N}(\boldsymbol{\mu}) = \sqrt{\lambda_{\min}(\boldsymbol{\mu})}.
\end{equation}
However, the eigenproblem $\refeq{eqn:inf_truth}$ is of size $\mathcal{N}$, so using direct solution as an ingredient for our error estimator is very expensive. Hence, we will construct an inexpensive yet of good quality lower bound $\alpha_{\rm LB}^\mathcal{N}(\boldsymbol{\mu})$ and use this lower bound instead of the truth (direct) expensive coercivity constant $\alpha^\mathcal{N}(\boldsymbol{\mu})$ in our error estimator.
For our current target problems, our bilinear form is coercive and symmetric. We shall construct our coercivity lower bound by the Successive Constraint Method (SCM) \cite{huynh07:cras}. It is noted that the SCM method can be readily extended to non-symmetric as well as non-coercive bilinear forms \cite{huynh07:cras,rozza08:ARCME,patera07:book,huynh08:infsupLB}.
We first introduce an alternative (albeit not very computation-friendly) discrete form for our coercivity constant as
\begin{eqnarray}\label{eqn:inf_Y}
{\rm minimum} && \sum_{q = 1}^{Q^a}\Theta_q^a(\boldsymbol{\mu})y_q, \\
{\rm subject \ to} && y_q = \frac{[\mathbf{w}_q]^T[\mathbf{K}^\mathcal{N}_q][\mathbf{w}_q]}{[\mathbf{w}_q]^T[\mathbf{Y}^\mathcal{N}][\mathbf{w}_q]}, \quad 1 \leq q \leq Q^a, \nonumber
\end{eqnarray}
where $[\mathbf{w}_q]$ is the discrete vector of any arbitrary $w_y \in X^\mathcal{N}$.
We shall now ``relax'' the constraint in \refeq{eqn:inf_Y} by defining the ``continuity constraint box'' associated with $y_{q,\min}$ and $y_{q,\max}$, $1 \leq q \leq Q^a$ obtained from the minimum set $([\mathbf{y}_-(\boldsymbol{\mu})],y_{q,\min})$ and maximum set $([\mathbf{y}_+(\boldsymbol{\mu})],y_{q,\max})$ solutions of the eigenproblems
\begin{eqnarray*}
[\mathbf{K}^\mathcal{N}_q][\mathbf{y}_-(\boldsymbol{\mu})] &=& y_{q,\min}[\mathbf{Y}^\mathcal{N}][\mathbf{y}_-(\boldsymbol{\mu})], \\
\left[\mathbf{y}_-(\boldsymbol{\mu})\right][\mathbf{Y}^\mathcal{N}][\mathbf{y}_-(\boldsymbol{\mu})] &=& 1,
\end{eqnarray*}
and
\begin{eqnarray*}
[\mathbf{K}^\mathcal{N}_q][\mathbf{y}_+(\boldsymbol{\mu})] &=& y_{q,\max}[\mathbf{Y}^\mathcal{N}][\mathbf{y}_+(\boldsymbol{\mu})], \\
\left[\mathbf{y}_+(\boldsymbol{\mu})\right][\mathbf{Y}^\mathcal{N}][\mathbf{y}_+(\boldsymbol{\mu})] &=& 1,
\end{eqnarray*}
respectively, for $1 \leq q \leq Q^a$. We next define a ``coercivity constraint'' sample
\begin{equation*}
C_J = \{\boldsymbol{\mu}^{\rm SCM}_1 \in \mbox{\boldmath$\mathcal{D}$}, \ldots, \boldsymbol{\mu}^{\rm SCM}_J \in \mbox{\boldmath$\mathcal{D}$}\},
\end{equation*}
and denote $C_J^{M,\boldsymbol{\mu}}$ the set of $M$ $(1 \leq M \leq J)$ points in $C_J$ closest (in the usual Euclidean norm) to a given $\boldsymbol{\mu} \in \mbox{\boldmath$\mathcal{D}$}$. The construction of the set $C_J$ is done by means of a Greedy procedure \cite{huynh07:cras,rozza08:ARCME,patera07:book}. The Greedy selection of $C_J$ can be called the ``Offline stage'', which involves the solutions of $J$ eigenproblems \refeq{eqn:inf_truth} to obtain $\alpha^\mathcal{N}(\boldsymbol{\mu})$, $\forall \boldsymbol{\mu} \in C_J$.
We may now define our lower bound $\alpha^\mathcal{N}_{\rm LB}(\boldsymbol{\mu})$ as the solution of
\begin{eqnarray}\label{eqn:inf_LB}
{\rm minimum} && \sum_{q = 1}^{Q^a}\Theta_q^a(\boldsymbol{\mu})y_q, \\
{\rm subject \ to} && y_{q,\min} \leq y_q \leq y_{q,\max}, \quad 1 \leq q \leq Q^a, \nonumber \\
&& \sum_{q = 1}^{Q^a}\Theta_q^a(\boldsymbol{\mu}')y_q \geq \alpha^\mathcal{N}(\boldsymbol{\mu}'), \quad \forall \boldsymbol{\mu}' \in C_J^{M,\boldsymbol{\mu}}. \nonumber
\end{eqnarray}
We then ``restrict'' the constraint in \refeq{eqn:inf_Y} and define our upper bound $\alpha^\mathcal{N}_{\rm UB}(\boldsymbol{\mu})$ as the solution of
\begin{eqnarray}\label{eqn:inf_UB}
{\rm mininum} && \sum_{q = 1}^{Q^a}\Theta_q^a(\boldsymbol{\mu})y_{q,*}(\boldsymbol{\mu}'), \\
{\rm subject \ to} && y_{q,*}(\boldsymbol{\mu}') = [\boldsymbol{\chi}(\boldsymbol{\mu}')]^T[\mathbf{K}^\mathcal{N}_q][\boldsymbol{\chi}(\boldsymbol{\mu}')], \quad 1 \leq q \leq Q^a, \quad \forall \boldsymbol{\mu}' \in C_J^{M,\boldsymbol{\mu}}, \nonumber
\end{eqnarray}
where $[\boldsymbol{\chi}(\boldsymbol{\mu})]$ is defined by \refeq{eqn:inf_truth}. It can be shown \cite{huynh07:cras,rozza08:ARCME,patera07:book} that the feasible region of \refeq{eqn:inf_UB} is a subset of that of \refeq{eqn:inf_Y}, which in turn, is a subset of that of \refeq{eqn:inf_LB}: hence $\alpha^\mathcal{N}_{\rm LB}(\boldsymbol{\mu}) \leq \alpha^\mathcal{N}(\boldsymbol{\mu}) \leq \alpha^\mathcal{N}_{\rm UB}(\boldsymbol{\mu})$.
We note that the lower bound \refeq{eqn:inf_LB} is a linear optimization problem (or Linear Program (LP)) which contains $Q^a$ design variables and $2Q^a + M$ inequality constraints. Given a value of the parameter $\boldsymbol{\mu}$, the Online evaluation $\boldsymbol{\mu} \rightarrow \alpha^\mathcal{N}_{\rm LB}(\boldsymbol{\mu})$ is thus as follows: we find the subset $C_J^{M,\boldsymbol{\mu}}$ of $C_J$ for a given $M$, we then calculate $\alpha^\mathcal{N}_{\rm LB}(\boldsymbol{\mu})$ by solving the LP \refeq{eqn:inf_LB}. The crucial point here is that the online evaluation $\boldsymbol{\mu} \rightarrow \alpha^\mathcal{N}_{\rm LB}(\boldsymbol{\mu})$ is totally independent of $\mathcal{N}$. The upper bound \refeq{eqn:inf_LB}, however, can be obtained as the solution of just a simple enumeration problem; the online evaluation of $\alpha^\mathcal{N}_{\rm UB}(\boldsymbol{\mu})$ is also independent of $\mathcal{N}$. In general, the upper bound $\alpha^\mathcal{N}_{\rm UB}(\boldsymbol{\mu})$ is not used in the calculation of the error estimator, however, it is used in the Greedy construction of the set $C_J$ \cite{huynh07:cras}. In practice, when the set $C_J$ does not guarantee to produce a positive $\alpha^\mathcal{N}_{\rm LB}(\boldsymbol{\mu})$, the upper bound $\alpha^\mathcal{N}_{\rm UB}(\boldsymbol{\mu})$ can be used as a substitution for $\alpha^\mathcal{N}_{\rm UB}(\boldsymbol{\mu})$ since it approximates the ``truth'' $\alpha^\mathcal{N}(\boldsymbol{\mu})$ in a very way; however we will lose the rigorous property of the error estimators.
\section{Extension of the RB method to non-compliant output}
We shall briefly provide the extension of our RB methodology for the ``non-compliant'' case in this Section. We first present a suitable primal-dual formulation for the ``non-compliant'' output; we then briefly provide the extension to the RB methodology, including the RB approximation and its \emph{a posteriori} error estimation.
\subsection{Adjoint Problem}
We shall briefly discuss the extension of our methodology to the non-compliant problems. We still require that both $f$ and $\ell$ are bounded functionals, but now $(f(\cdot;\boldsymbol{\mu}) \neq \ell(\cdot;\boldsymbol{\mu}))$. We still use the previous abstract statement in Section~2. We begin with the definition of the dual problem associated to $\ell$: find $\psi(\boldsymbol{\mu}) \in X$ (our ``adjoint'' or ``dual'' field) such that
\begin{equation*}
a(v,\psi(\boldsymbol{\mu});\boldsymbol{\mu}) = -\ell(\boldsymbol{\mu}), \quad \forall v \in X.
\end{equation*}
\subsection{Truth approximation}
We now again apply the finite element method to the dual formulation: given $\boldsymbol{\mu} \in \mbox{\boldmath$\mathcal{D}$}$, we evaluate
\begin{equation*}
s(\boldsymbol{\mu}) = [\mathbf{L}^\mathcal{N}(\boldsymbol{\mu})]^T[\mathbf{u}^\mathcal{N}(\boldsymbol{\mu})],
\end{equation*}
where $[\mathbf{u}^\mathcal{N}(\boldsymbol{\mu})]$ is the finite element solution of size $\mathcal{N}$ satisfying \refeq{eqn:FE_stiff}. The discrete form of the dual solution $\psi^\mathcal{N}(\boldsymbol{\mu}) \in X^\mathcal{N}$ is given
\begin{equation*}
[\mathbf{K}^\mathcal{N}(\boldsymbol{\mu})][\boldsymbol{\psi}^\mathcal{N}(\boldsymbol{\mu})] = -[\mathbf{L}^\mathcal{N}(\boldsymbol{\mu})];
\end{equation*}
here $[\mathbf{L}^\mathcal{N}(\boldsymbol{\mu})]$ is the discrete load vector of $\ell(\cdot;\boldsymbol{\mu})$. We also invoke the affine forms \refeq{eqn:affine} to express $[\mathbf{L}^\mathcal{N}(\boldsymbol{\mu})]$ as
\begin{eqnarray}\label{eqn:affine_FE_out}
[\mathbf{L}^\mathcal{N}(\boldsymbol{\mu})] &=& \sum_{q = 1}^{Q^\ell}\Theta_q^\ell(\boldsymbol{\mu})[\mathbf{L}^\mathcal{N}_q],
\end{eqnarray}
where all the $[\mathbf{L}^\mathcal{N}_q]$ are the discrete forms of the parameter-independent linear forms $\ell_q(\cdot)$, $1 \leq q \leq Q^\ell$.
\subsection{Reduced Basis Approximation}
We now define our RB spaces: we shall need to define two Lagrangian parameter samples set, $S_{N^{\rm pr}} = \{\boldsymbol{\mu}_1,\boldsymbol{\mu}_2,\ldots,\boldsymbol{\mu}_{N^{\rm pr}}\}$ and $S_{N^{\rm du}}= \{\boldsymbol{\mu}_1,\boldsymbol{\mu}_2,\ldots,\boldsymbol{\mu}_{N^{\rm du}}\}$ corresponding to the set of our primal and dual parameter samples set, respectively. We also associate the primal and dual reduced basis spaces $(X_{N^{\rm pr}}^\mathcal{N} =) W^\mathcal{N}_{N^{\rm pr}}$, $1 \leq N \leq N^{\rm pr}_{\max}$ and $(X_{N^{\rm du}}^\mathcal{N} =) W^\mathcal{N}_{N^{\rm du}}$, $1 \leq N \leq N^{\rm du}_{\max}$ to our $S_{N^{\rm pr}}$ and $S_{N^{\rm du}}$ set, respectively, which are constructed from the primal $u^{\mathcal{N}}(\boldsymbol{\mu})$ and dual $\psi^{\mathcal{N}}(\boldsymbol{\mu})$ snapshots by a Gram-Schmidt process as in Section~3. Finally, we denote our primal and dual orthonormalized-snapshot as $[\mathbf{Z}^{\rm pr}_{N^{\rm pr}}]$ and $[\mathbf{Z}^{\rm du}_{N^{\rm du}}]$ basis matrices, respectively.
\subsection{Galerkin Projection}
We first denote the RB primal approximation to the primal ``truth'' approximation $u^\mathcal{N}(\boldsymbol{\mu})$ as $u_{{\rm RB},N}^\mathcal{N}(\boldsymbol{\mu})$ and the RB dual approximation to the primal ``truth'' dual approximation $\psi^\mathcal{N}(\boldsymbol{\mu})$ as $\psi_{{\rm RB},N}^\mathcal{N}(\boldsymbol{\mu})$: their discrete forms are given by $[\mathbf{u}_{RB,N^{\rm pr}}^\mathcal{N}(\boldsymbol{\mu})] = [\mathbf{Z}^{\rm pr}_{N^{\rm pr}}][\mathbf{u}_{N^{\rm pr}}(\boldsymbol{\mu})]$ and $[\boldsymbol{\psi}_{RB,N^{\rm du}}^\mathcal{N}(\boldsymbol{\mu})] = [\mathbf{Z}^{\rm du}_{N^{\rm du}}][\boldsymbol{\psi}_{N^{\rm du}}(\boldsymbol{\mu})]$, respectively.
We then apply a Galerkin projection (note that in this case, a Galerkin-Petrov projection is also possible \cite{rozza08:ARCME, patera07:book,benner2015}). given a $\boldsymbol{\mu} \in \mbox{\boldmath$\mathcal{D}$}$, we evaluate the RB output
\begin{equation*}
s_{N^{\rm pr},N^{\rm du}}(\boldsymbol{\mu}) = [\mathbf{L}^\mathcal{N}(\boldsymbol{\mu})]^T[\mathbf{u}^\mathcal{N}_{{\rm RB},N^{\rm pr}}(\boldsymbol{\mu})] - [\mathbf{r}^\mathcal{N}_{\rm pr}(\boldsymbol{\mu})]^T[\boldsymbol{\psi}^\mathcal{N}_{{\rm RB},N^{\rm du}}(\boldsymbol{\mu})],
\end{equation*}
recall that $[\mathbf{r}^\mathcal{N}_{\rm pr}(\boldsymbol{\mu})]$ is the discrete form of the RB primal residual defined in \refeq{eqn:residual}. The RB coefficient primal and dual are given by
\begin{eqnarray}\label{eqn:semifull_du}
\sum_{q=1}^{Q^a}\Theta_q^a(\boldsymbol{\mu})[\mathbf{K}_{qN^{\rm pr}N^{\rm pr}}][\mathbf{u}_{N^{\rm pr}}(\boldsymbol{\mu})] &=& \sum_{q=1}^{Q^f}\Theta_q^f(\boldsymbol{\mu})[\mathbf{F}_{qN^{\rm pr}}], \nonumber \\
\sum_{q=1}^{Q^a}\Theta_q^a(\boldsymbol{\mu})[\mathbf{K}_{qN^{\rm du}N^{\rm du}}[\boldsymbol{\psi}_{N^{\rm du}}(\boldsymbol{\mu})] &=& -\sum_{q=1}^{Q^\ell}\Theta_q^\ell(\boldsymbol{\mu})[\mathbf{L}_{qN^{\rm du}}].
\end{eqnarray}
Note that the two systems \refeq{eqn:semifull_du} are also of small size: their sizes are of $N^{\rm pr}$ and $N^{\rm du}$, respectively. We can now evaluate our output as
\begin{eqnarray}\label{eqn:RB_outsemifull_du}
s_{N^{\rm pr},N^{\rm du}}(\boldsymbol{\mu}) &=& \sum_{q=1}^{Q^\ell}\Theta_q^\ell(\boldsymbol{\mu})[\mathbf{L}_{qN^{\rm pr}}][\mathbf{u}_{N^{\rm pr}}(\boldsymbol{\mu})] - \sum_{q=1}^{Q^f}\Theta_q^f(\boldsymbol{\mu})[\mathbf{F}_{qN^{\rm du}}][\boldsymbol{\psi}_{N^{\rm du}}(\boldsymbol{\mu})] \nonumber\\
&&+\sum_{q=1}^{Q^a}\Theta_q^a(\boldsymbol{\mu})[\boldsymbol{\psi}_{N^{\rm du}}(\boldsymbol{\mu})]^T[\mathbf{K}_{qN^{\rm du}N^{\rm pr}}][\mathbf{u}_{N^{\rm pr}}(\boldsymbol{\mu})].
\end{eqnarray}
All the quantities in \refeq{eqn:semifull_du} and \refeq{eqn:RB_outsemifull_du} are given by
\begin{eqnarray*}
[\mathbf{K}_{qN^{\rm pr}N^{\rm pr}}] &=& [\mathbf{Z}^{\rm pr}_{N^{\rm pr}}]^T[\mathbf{K}_q][\mathbf{Z}^{\rm pr}_{N^{\rm pr}}], \quad 1 \leq q \leq Q^a, \ 1 \leq N^{\rm pr} \leq N^{\rm pr}_{\max},\\
\left[\mathbf{K}_{qN^{\rm du}N^{\rm du}}\right] &=& [\mathbf{Z}^{\rm du}_{N^{\rm du}}]^T[\mathbf{K}_q][\mathbf{Z}^{\rm du}_{N^{\rm du}}], \quad 1 \leq q \leq Q^a, \ 1 \leq N^{\rm du} \leq N^{\rm du}_{\max}, \\
\left[\mathbf{K}_{qN^{\rm du}N^{\rm pr}}\right] &=& [\mathbf{Z}^{\rm du}_{N^{\rm du}}]^T[\mathbf{K}_q][\mathbf{Z}^{\rm pr}_{N^{\rm pr}}], \quad 1 \leq q \leq Q^a, \ 1 \leq N^{\rm pr} \leq N^{\rm pr}_{\max}, \ 1 \leq N^{\rm du} \leq N^{\rm du}_{\max} \\
\left[\mathbf{F}_{qN^{\rm pr}}\right] &=& [\mathbf{Z}^{\rm pr}_{N^{\rm pr}}]^T[\mathbf{F}_q], \quad 1 \leq q \leq Q^f, \ 1 \leq N^{\rm pr} \leq N^{\rm pr}_{\max}, \\
\left[\mathbf{F}_{qN^{\rm du}}\right] &=& [\mathbf{Z}^{\rm du}_{N^{\rm du}}]^T[\mathbf{F}_q], \quad 1 \leq q \leq Q^f, \ 1 \leq N^{\rm du} \leq N^{\rm du}_{\max}, \\
\left[\mathbf{L}_{qN^{\rm pr}}\right] &=& [\mathbf{Z}^{\rm pr}_{N^{\rm pr}}]^T[\mathbf{L}_q], \quad 1 \leq q \leq Q^\ell, 1 \leq N^{\rm pr} \leq N^{\rm pr}_{\max}, \\
\left[\mathbf{L}_{qN^{\rm du}}\right] &=& [\mathbf{Z}^{\rm du}_{N^{\rm du}}]^T[\mathbf{L}_q], \quad 1 \leq q \leq Q^\ell, 1 \leq N^{\rm du} \leq N^{\rm du}_{\max}.
\end{eqnarray*}
The computation of the output $s_{N^{\rm pr},N^{\rm du}}(\boldsymbol{\mu})$ clearly admits an Offline-Online computational strategy similar to the one we discuss previously in Section~3.
\subsection{\emph{A posteriori} error estimation}
We now introduce the dual residual $r_{\rm du}^\mathcal{N}(v;\boldsymbol{\mu})$,
\begin{equation*}
r_{\rm du}^\mathcal{N}(v;\boldsymbol{\mu}) = -\ell(v) - a(v,\psi_{N^{\rm du}}^\mathcal{N}(\boldsymbol{\mu});\boldsymbol{\mu}), \quad \forall v \in X^\mathcal{N}.
\end{equation*}
and its Riesz representation of $r_{\rm du}^\mathcal{N}(v;\boldsymbol{\mu})$: $\hat{e}^{\rm du}(\boldsymbol{\mu}) \in X^\mathcal{N}$ defined by $(\hat{e}^{\rm du}(\boldsymbol{\mu}),v)_{X^\mathcal{N}} = r^\mathcal{N}_{\rm du}(v;\boldsymbol{\mu})$, $\forall v \in X^\mathcal{N}$.
We may now define our error estimator for our output as
\begin{equation}
\Delta_{N^{\rm pr}N^{\rm du}}^s(\boldsymbol{\mu}) \equiv \frac{\|\hat{e}^{\rm pr}(\boldsymbol{\mu})\|_{X^\mathcal{N}}}{(\alpha^\mathcal{N}_{\rm LB})^{1/2}}\frac{\|\hat{e}^{\rm du}(\boldsymbol{\mu})\|_{X^\mathcal{N}}}{(\alpha^\mathcal{N}_{\rm LB})^{1/2}},
\end{equation}
where $\hat{e}^{\rm pr}(\boldsymbol{\mu})$ is the Riesz representation of the primal residual. We then define the effectivity associated with our error bound
\begin{equation}
\eta_{N^{\rm pr}N^{\rm du}}^s(\boldsymbol{\mu}) \equiv \frac{\Delta_{N^{\rm pr}N^{\rm du}}^s(\boldsymbol{\mu})}{|s^\mathcal{N}(\boldsymbol{\mu})-s_{N^{\rm pr}N^{\rm du}}(\boldsymbol{\mu})|}.
\end{equation}
We can readily demonstrate \cite{rozza08:ARCME, patera07:book, grepl04:_reduc_basis_approx_time_depen} that
\begin{equation*}
1 \leq \eta_{N^{\rm pr}N^{\rm du}}^s(\boldsymbol{\mu}), \quad \forall \boldsymbol{\mu} \in \mbox{\boldmath$\mathcal{D}$};
\end{equation*}
note that the error estimator is still \emph{rigorous}, however it is less \emph{sharp} than that in the ``compliant'' case since in this case we could not provide an upper bound to $\eta_{N^{\rm pr}N^{\rm du}}^s(\boldsymbol{\mu})$.
The computation of the dual norm of the primal/dual residual also follows an Offline-Online computation strategy: the dual norm of the primal residual is in fact, the same as in Section~4.2; the same procedure can be applied to compute the dual norm of the dual residual.
\section{Numerical results}
In this sections we shall consider several ``model problems'' to demonstrate the feasibility of our methodology. We note that in all cases, these model problems are presented in non-dimensional form unless stated otherwise. In all problems below, displacement is, in fact, in non-dimensional form $u = {\tilde{u}\tilde{E}}/{\tilde{\sigma}_0}$, where $\tilde{u}$, $\tilde{E}$, $\tilde{\sigma}_0$ are the dimensional displacement, Young's modulus and load strength, respectively, while $E$ and $\sigma_0$ are our non-dimensional Young's modulus and load strength and usually are around unity.
We shall not provide any details for $\Theta_q^a(\boldsymbol{\mu})$, $\Theta_q^f(\boldsymbol{\mu})$ and $\Theta_q^\ell(\boldsymbol{\mu})$ and their associated bilinear and linear forms $a_q(\cdot,\cdot)$, $f_q(\cdot)$ and $\ell_q(\cdot)$ for any of the below examples as they are usually quite complex, due to the complicated structure of the effective elastic tensor and our symbolic manipulation technique. We refer the users to \cite{huynh07:ijnme, patera07:book, veroy03:_phd_thesis, milani08:RB_LE}, in which all the above terms are provided in details for some simple model problems.
In the below, the timing $t_{\rm FE}$ for an evaluation of the FE solution $\boldsymbol{\mu} \rightarrow s^\mathcal{N}(\boldsymbol{\mu})$ is the computation time taken by solving \refeq{eqn:FE_stiff} and evaluating \refeq{eqn:FE_out} by using \refeq{eqn:affine_FE} and \refeq{eqn:affine_FE_out}, in which all the stiffness matrix components, $[\mathbf{K}_q]$, $1\leq q\leq Q^a$, load and output vector components, $[\mathbf{F}_q]$, $1\leq q\leq Q^f$ and $[\mathbf{L}_q]$, $1\leq q\leq Q^\ell$, respectively, are pre-computed and pre-stored. We do not include the computation time of forming those components (or alternatively, calculate the stiffness matrix, load and output vector directly) in $t_{\rm FE}$.
Finally, for the sake of simplicity, we shall denote the number of basis $N$ defined as $N = N^{\rm pr} = N^{\rm du}$ in all of our model problems in this Section.
\subsection{The arc-cantilever beam}
We consider a thick arc cantilever beam correspond to the domain $\Omega^{\rm o}(\boldsymbol{\mu})$ representing the shape of a quarter of an annulus as shown in Figure~\ref{fig:ex1_model}. We apply (clamped) homogeneous Dirichlet conditions on $\Gamma^{\rm o}_D$ and non-homogeneous Neumann boundary conditions corresponding to a unit tension on $\Gamma^{\rm o}_N$. The width of the cantilever beam is $2d$, and the material is isotropic with $(E,\nu) = (1,0.3)$ under plane stress assumption. Our output of interest is the integral of the tangential displacement ($u_2$) over $\Gamma^{\rm o}_N$, which can be interpreted as the average tangential displacement on $\Gamma^{\rm o}_N$\footnote{The average tangential displacement on $\Gamma^{\rm o}_N$ is not exactly $s(\boldsymbol{\mu})$ but rather $s(\boldsymbol{\mu})/l_{\Gamma^{\rm o}_N}$, where $l_{\Gamma^{\rm o}_N}$ is the length of ${\Gamma^{\rm o}_N}$. It is obviously that the two descriptions of the two outputs, ''integral of'' and ``average of'', are pretty much equivalent to each other.}. Note that our output of interest is ``non-compliant''.
\begin{figure} [htbp]
\centering
\includegraphics[scale=0.5]{CMAME_cantilever}
\caption{The arc-cantilever beam}
\label{fig:ex1_model}
\end{figure}
The parameter is the half-width of the cantilever beam $\boldsymbol{\mu} = [\mu_1] \equiv [d]$. The parameter domain is chosen as $\mbox{\boldmath$\mathcal{D}$} = [0.3, 0.9]$, which can model a moderately thick beam to a very thick beam. We then choose $\boldsymbol{\mu}_{\rm ref} = 0.3$ and apply the domain decomposition and obtain $L_{\rm reg} = 9$ subdomains as shown in Figure~\ref{fig:ex1_mesh}, in which three subdomains are the general ``curvy triangles'', generated by our computer automatic procedure \cite{rozza08:ARCME}. Note that geometric transformations are relatively complicated, due to the appearances of the ``curvy triangles'' and all subdomains transformations are classified as the ``general transformation case'' \cite{patera07:book, huynh07:_phd_thesis}. We then recover our affine forms with $Q^a = 54$, $Q^f = 1$ and $Q^l = 1$.
We next consider a FE approximation where the mesh contains $n_{\rm node} = 2747$ nodes and $n_{\rm elem} = 5322$ $P_1$ elements, which corresponds to $\mathcal{N} = 5426$ degrees of freedoms\footnote{Note that $\mathcal{N} \neq 2n_{\rm node}$ since Dirichlet boundary nodes are eliminated from the FE system.} as shown in Figure~\ref{fig:ex1_mesh}. To verify our FE approximation, we compare our FE results with the approximated solution for thick arc cantilever beam by Roark \cite{roark01:roark_formula} for a $100$ uniformly distributed test points in $\mbox{\boldmath$\mathcal{D}$}$: the maximum difference between our results and Roark's one is just $2.9\%$.
\begin{figure} [htbp]
\centering
\includegraphics[scale=0.3]{CMAME_cantilever_mesh}
\caption{The arc-cantilever beam problem: Domain composition and FE mesh}
\label{fig:ex1_mesh}
\end{figure}
We then apply our RB approximation. We present in Table~\ref{tab:ex1_tab} our convergence results: the RB error bounds and effectivities as a function of $N (=N^{\rm pr} = N^{\rm du})$. The error bound reported, $\mathcal{E}_N = \Delta^s_N(\boldsymbol{\mu})/|s_N(\boldsymbol{\mu})|$ is the maximum of the relative error bound over a random test sample $\Xi_{\rm test}$ of size $n_{\rm test} = 100$. We denote by $\overline{\eta}_N^s$ the average of the effectivity $\eta_N^s(\boldsymbol{\mu})$ over $\Xi_{\rm test}$. We observe that average effectivity is of order $O(20-90)$, not very \emph{sharp}, but this is expected due to the fact that the output is ``non-compliant''.
\begin{table}
\centering
\begin{tabular}{|c||c|c|}
\hline
$N$ & $\mathcal{E}_N$ & $\overline{\eta}_N^s$ \\
\hline
2 & 3.57\texttt{E}+00 & 86.37 \\
4 & 3.70\texttt{E}-03 & 18.82 \\
6 & 4.07\texttt{E}-05 & 35.72 \\
8 & 6.55\texttt{E}-07 & 41.58 \\
10 & 1.99\texttt{E}-08 & 40.99 \\
\hline
\end{tabular}
\caption{The arc-cantilever beam: RB convergence}
\label{tab:ex1_tab}
\end{table}
As regards computational times, a RB online evaluation $\boldsymbol{\mu} \rightarrow (s_N(\boldsymbol{\mu}),\Delta_N^s(\boldsymbol{\mu}))$ requires just $t_{\rm RB} = 115$(ms) for $N = 10$; while the FE solution $\boldsymbol{\mu} \rightarrow s^\mathcal{N}(\boldsymbol{\mu})$ requires $t_{\rm FE} = 9$(s): thus our RB online evaluation is just $1.28\%$ of the FEM computational cost.
\subsection{The center crack problem}
We next consider a fracture model corresponds to a center crack in a plate under tension at both sides as shown in Figure~\ref{fig:ex2_modelfull}.
\begin{figure} [htbp]
\centering
\includegraphics[scale=0.5]{CMAME_crack_model}
\caption{The center crack problem}
\label{fig:ex2_modelfull}
\end{figure}
Due to the symmetry of the geometry and loading, we only consider one quarter of the physical domain, as shown in Figure~\ref{fig:ex2_modelfull}, note that the crack corresponds to the boundary segment $\Gamma^{\rm o}_C$. The crack (in our ``quarter'' model) is of size $d$, and the plate is of height $h$ (and of fixed width $w = 1$). We consider plane strain isotropic material with $(E,\nu) = (1,0.3)$. We consider (symmetric about the $x^{\rm o}_1$ direction and $x^{\rm o}_2$ direction) Dirichlet boundary conditions on the left and bottom boundaries $\Gamma^{\rm o}_L$ and $\Gamma^{\rm o}_B$, respectively; and non-homogeneous Neumann boundary conditions (tension) on the top boundary $\Gamma^{\rm o}_T$. Our ultimate output of interest is the stress intensity factor (SIF) for the crack, which will be derived from an intermediate (compliant) energy output by application of the virtual crack extension approach \cite{parks77:a_stiff_sif}. The SIF plays an important role in the field of fracture mechanics, for examples, if we have to estimate the propagation path of cracks in structures \cite{hutchingson79:fracture}. We further note that analytical result for SIF of a center-crack in a plate under tension is only available for the infinite plate \cite{murakami01:SIFhandbook}, which can be compared with our solutions for small crack length $d$ and large plate height $h$ values.
\begin{figure} [htbp]
\centering
\includegraphics[scale=0.5]{CMAME_crack}
\caption{The center crack problem}
\label{fig:ex2_model}
\end{figure}
Our parameters are the crack length and the plate height $\boldsymbol{\mu} = [\mu_1,\mu_2] \equiv [d, h]$, and the parameter domain is given by $\mbox{\boldmath$\mathcal{D}$} = [0.3,0.7] \times [0.5,2.0]$. We then choose $\boldsymbol{\mu}_{\rm ref} = [0.5,1.0]$ and apply a domain decomposition: the final setting contains $L_{\rm reg} = 3$ subdomains, which in turn gives us $Q^a = 10$ and $Q^f = 1$. Note that our ``compliant'' output $s(\boldsymbol{\mu})$ is just an intermediate result for the calculation of the SIF. In particular, the virtual crack extension method (VCE) \cite{parks77:a_stiff_sif} allows us to extract the ``Mode-I'' SIF though the energy $s(\boldsymbol{\mu})$ though the Energy Release Rate (ERR), $G(\boldsymbol{\mu})$, defined by
\begin{equation*}
G(\boldsymbol{\mu}) = -\bigg(\frac{\partial s(\boldsymbol{\mu})}{\partial \mu_1}\bigg).
\end{equation*}
In practice, the ERR is approximated by a finite-difference (FD) approach for a suitable small value $\delta\mu_1$ as
\begin{equation*}
\widehat{G}(\boldsymbol{\mu}) = -\bigg(\frac{s(\boldsymbol{\mu}+\delta\mu_1)-s(\boldsymbol{\mu})}{\delta\mu_1}\bigg),
\end{equation*}
which then give the SIF approximation $\widehat{\rm SIF}(\boldsymbol{\mu}) = \sqrt{\widehat{G}(\boldsymbol{\mu})/(1-\nu^2)}$.
We then consider a FE approximation with a mesh contains $n_{\rm node} = 3257$ nodes and $n_{\rm elem} = 6276$ $P_1$ elements, which corresponds to $\mathcal{N} = 6422$ degrees of freedoms; the mesh is refined around the crack tip in order to give a good approximation for the (singular) solution near this region as shown in Figure~\ref{fig:ex2_mesh}.
\begin{figure} [htbp]
\centering
\includegraphics[scale=0.3]{CMAME_crack_mesh}
\caption{The center crack problem: Domain composition and FE mesh}
\label{fig:ex2_mesh}
\end{figure}
We present in Table~\ref{tab:ex2_tab} the convergence results for the ``compliant'' output $s(\boldsymbol{\mu})$: the RB error bounds and effectivities as a function of $N$. The error bound reported, $\mathcal{E}_N = \Delta^s_N(\boldsymbol{\mu})/|s_N(\boldsymbol{\mu})|$ is the maximum of the relative error bound over a random test sample $\Xi_{\rm test}$ of size $n_{\rm test} = 200$. We denote by $\overline{\eta}_N^s$ the average of the effectivity $\eta_N^s(\boldsymbol{\mu})$ over $\Xi_{\rm test}$. We observe that the effectivity average is very sharp, and of order $O(10)$.
\begin{table}
\centering
\begin{tabular}{|c||c|c|}
\hline
$N$ & $\mathcal{E}_N$ & $\overline{\eta}_N^s$ \\
\hline
5 & 2.73\texttt{E}-02 & 6.16 \\
10 & 9.48\texttt{E}-04 & 8.47 \\
20 & 5.71\texttt{E}-06 & 7.39 \\
30 & 5.59\texttt{E}-08 & 7.01 \\
40 & 8.91\texttt{E}-10 & 7.54 \\
50 & 6.26\texttt{E}-11 & 8.32 \\
\hline
\end{tabular}
\caption{The center crack problem: RB convergence}
\label{tab:ex2_tab}
\end{table}
We next define the ERR RB approximation $\widehat{G}_N(\boldsymbol{\mu})$ to our ``truth'' (FE) $\widehat{G}^\mathcal{N}_{\rm FE}(\boldsymbol{\mu})$ and its associated ERR RB error $\Delta^{\widehat{G}}_N(\boldsymbol{\mu})$ by
\begin{eqnarray}\label{eqn:RB_SIF_err}
\widehat{G}_N(\boldsymbol{\mu}) &=& \frac{s_N(\boldsymbol{\mu}) - \Delta_N^s(\boldsymbol{\mu}+\delta\mu_1)}{\delta\mu_1}, \nonumber \\
\Delta^{\widehat{G}}_N(\boldsymbol{\mu}) &=& \frac{\Delta_N^s(\boldsymbol{\mu} + \delta\mu_1) + \Delta_N^s(\boldsymbol{\mu})}{\delta\mu_1}.
\end{eqnarray}
It can be readily proven \cite{rozza08:ARCME} that our SIF RB error is a rigorous bound for the ERR RB prediction $\widehat{G}_N(\boldsymbol{\mu})$: $|\widehat{G}_N(\boldsymbol{\mu}) - \widehat{G}^\mathcal{N}_{\rm FE}(\boldsymbol{\mu})| \leq \Delta^{\widehat{G}}_N(\boldsymbol{\mu})$. It is note that the choice of $\delta\mu_1$ is not arbitrary: $\delta\mu_1$ needed to be small enough to provide a good FD approximation, while still provide a good ERR RB error bound \refeq{eqn:RB_SIF_err}. Here we choose $\delta\mu_1 = 1\texttt{E}-03$.
We then can define the SIF RB approximation $\widehat{\rm SIF}_N(\boldsymbol{\mu})$ to our ``truth'' (FE) $\widehat{\rm SIF}^\mathcal{N}_{\rm FE}(\boldsymbol{\mu})$ and its associated SIF RB error estimation $\Delta^{\widehat{\rm SIF}}_N(\boldsymbol{\mu})$ as
\begin{eqnarray*}
\widehat{\rm SIF}_N(\boldsymbol{\mu}) &=& \frac{1}{2\sqrt{1-\nu^2}}\bigg\{\sqrt{\widehat{G}_N(\boldsymbol{\mu}) + \Delta^{\widehat{G}}_N(\boldsymbol{\mu})}+\sqrt{\widehat{G}_N(\boldsymbol{\mu}) - \Delta^{\widehat{G}}_N(\boldsymbol{\mu})}\bigg\}, \\
\Delta^{\widehat{\rm SIF}}_N(\boldsymbol{\mu}) &=& \frac{1}{2\sqrt{1-\nu^2}}\bigg\{\sqrt{\widehat{G}_N(\boldsymbol{\mu}) + \Delta^{\widehat{G}}_N(\boldsymbol{\mu})}-\sqrt{\widehat{G}_N(\boldsymbol{\mu}) - \Delta^{\widehat{G}}_N(\boldsymbol{\mu})}\bigg\}.
\end{eqnarray*}
It is readily proven in \cite{huynh07:ijnme} that $|\widehat{\rm SIF}_N(\boldsymbol{\mu}) - \widehat{\rm SIF}^\mathcal{N}_{\rm FE}(\boldsymbol{\mu})| \leq \Delta^{\widehat{\rm SIF}}_N(\boldsymbol{\mu})$.
We plot the SIF RB results $\widehat{\rm SIF}(\boldsymbol{\mu})$ with error bars correspond to $\Delta^{\widehat{\rm SIF}}_N(\boldsymbol{\mu})$, and the analytical results $\widehat{\rm SIF}(\boldsymbol{\mu})$ \cite{murakami01:SIFhandbook}
in Figure~\ref{fig:ex2_SIF15} for the case $\mu_1 \in [0.3,0.7]$, $\mu_2 = 2.0$ for $N = 15$. It is observed that the RB error is large since the small number of basis $N = 15$ does not compromise the small $\delta\mu_1 = 1\texttt{E}-03$ value.
\begin{figure} [htbp]
\centering
\includegraphics[scale=0.4]{CMAME_crack_SIF15}
\caption{The center crack problem: SIF solution for $N=15$}
\label{fig:ex2_SIF15}
\end{figure}
We next plot, in Figure~\ref{fig:ex2_SIF30}, SIF RB results and error for the same $\boldsymbol{\mu}$ range as in Figure~\ref{fig:ex2_SIF15}, but for $N = 30$. It is observed now that the SIF RB error is significantly improved -- thanks to the better RB approximation that compensates the small value $\delta\mu_1$. We also want to point out that, in both Figure~\ref{fig:ex2_SIF15} and Figure~\ref{fig:ex2_SIF30}, it is clearly shown that our RB SIF error is not a \emph{rigorous} bound for the \emph{exact} SIF values $\widehat{\rm SIF}(\boldsymbol{\mu})$ but rather is a \emph{rigorous} bound for the ``truth'' (FE) approximation $\widehat{\rm SIF}^\mathcal{N}_{\rm FE}(\boldsymbol{\mu})$. It is shown, however, that FE SIF approximation (which is considered in Figure~\ref{fig:ex2_SIF30} thanks to the negligible RB error) are of good quality compared with the exact SIF. The VCE in this case works quite well, however it is not suitable for complicate crack settings. In such cases, other SIF calculation methods and appropriate RB approximations might be preferable \cite{huynh07:ijnme, huynh07:_phd_thesis}.
\begin{figure} [htbp]
\centering
\includegraphics[scale=0.4]{CMAME_crack_SIF30}
\caption{The center crack problem: SIF solution for $N=30$}
\label{fig:ex2_SIF30}
\end{figure}
As regards computational times, a RB online evaluation $\boldsymbol{\mu} \rightarrow (\widehat{\rm SIF}_N(\boldsymbol{\mu}),\Delta_N^{\widehat{\rm SIF}}(\boldsymbol{\mu})$ requires just $t_{\rm RB} (= 25 \times) = 50$(ms) for $N = 40$; while the FE solution $\boldsymbol{\mu} \rightarrow \widehat{\rm SIF}^\mathcal{N}_{\rm FE}(\boldsymbol{\mu})$ requires $t_{\rm FE} (= 7 \times 2) = 14$(s): thus our RB online evaluation takes only $0.36\%$ of the FEM computational cost.
\subsection{The composite unit cell problem}
We consider a unit cell contains an ellipse region as shown in Figure~\ref{fig:ex3_model}. We apply (clamped) Dirichlet boundary conditions on the bottom of the cell $\Gamma^{\rm o}_B$ and (unit tension) non-homogeneous Neumann boundary conditions on $\Gamma^{\rm o}_T$. We denote the two semimajor axis and semiminor axis of the ellipse region as $d_1$ and $d_2$, respectively. We assume plane stress isotropic materials: the material properties of the matrix (outside of the region) is given by $(E_m,\nu_m) = (1,0.3)$, and the material properties of the ellipse region is given by $(E_f,\nu_f) = (E_f,0.3)$. Our output of interest is the integral of normal displacement ($u_1$) over $\Gamma^{\rm o}_T$. We note our output of interest is thus ``compliant''.
\begin{figure} [htbp]
\centering
\includegraphics[scale=0.5]{CMAME_platehole}
\caption{The composite unit cell problem}
\label{fig:ex3_model}
\end{figure}
We consider $P=3$ parameters $\boldsymbol{\mu} = [\mu_1,\mu_2,\mu_3] \equiv [d_1,d_2,E_f]$. The parameter domain is chosen as $\mbox{\boldmath$\mathcal{D}$} = [0.8, 1.2] \times [0.8,1.2] \times [0.2,5]$. Note that the third parameter (the Young modulus of the ellipse region) can represent the ellipse region from an ``inclusion'' (with softer Young's modulus $E_f<E_m (= 1)$) to a ``fiber'' (with stiffer Young's modulus $E_f>E_m (= 1)$).
We then choose $\boldsymbol{\mu}_{\rm ref} = [1.0,1.0,1.0]$ and apply the domain decomposition \cite{rozza08:ARCME} and obtain $L_{\rm reg} = 34$ subdomains, in which $16$ subdomains are the general ``curvy triangles'' ($8$ inward ``curvy triangles'' and $8$ outward curvy ``triangles'') as shown in Figure~\ref{fig:ex3_mesh}. However, despite the large number of ``curvy triangles'' in the domain decomposition, it is observed that almost all transformations are congruent, hence we expected a small number of $Q^a$ than (says), that of the ``arc-cantilever beam'' example, in which all the subdomains transformations are different. Indeed, we recover our affine forms with $Q^a = 30$ and $Q^f = 1$, note that $Q^a$ is relatively small for such a complex domain decomposition thanks to our efficient symbolic manipulation ``collapsing'' technique and those congruent ``curvy triangles''.
We next consider a FE approximation where the mesh contains $n_{\rm node} = 3906$ nodes and $n_{\rm elem} = 7650$ $P_1$ elements, which corresponds to $\mathcal{N} = 7730$ degrees of freedoms. The mesh is refined around the interface of the matrix and the inclusion/fiber.
\begin{figure} [htbp]
\centering
\includegraphics[scale=0.3]{CMAME_platehole_mesh}
\caption{The composite unite cell problem: Domain composition and FE mesh}
\label{fig:ex3_mesh}
\end{figure}
We then apply the RB approximation. We present in Table~\ref{tab:ex3_tab} our convergence results: the RB error bounds and effectivities as a function of $N$. The error bound reported, $\mathcal{E}_N = \Delta^s_N(\boldsymbol{\mu})/|s_N(\boldsymbol{\mu})|$ is the maximum of the relative error bound over a random test sample $\Xi_{\rm test}$ of size $n_{\rm test} = 200$. We denote by $\overline{\eta}_N^s$ the average of the effectivity $\eta_N^s(\boldsymbol{\mu})$ over $\Xi_{\rm test}$. We observe that our effectivity average is of order $O(10)$.
\begin{table}
\centering
\begin{tabular}{|c||c|c|}
\hline
$N$ & $\mathcal{E}_N$ & $\overline{\eta}_N^s$ \\
\hline
5 & 9.38\texttt{E}-03 & 8.86 \\
10 & 2.54\texttt{E}-04 & 7.18 \\
15 & 1.37\texttt{E}-05 & 5.11 \\
20 & 3.91\texttt{E}-06 & 9.74 \\
25 & 9.09\texttt{E}-07 & 6.05 \\
30 & 2.73\texttt{E}-07 & 10.64 \\
35 & 9.00\texttt{E}-08 & 10.17 \\
40 & 2.66\texttt{E}-08 & 10.35 \\
\hline
\end{tabular}
\caption{The composite unit cell problem: RB convergence}
\label{tab:ex3_tab}
\end{table}
As regards computational times, a RB online evaluation $\boldsymbol{\mu} \rightarrow (s_N(\boldsymbol{\mu}),\Delta_N^s(\boldsymbol{\mu}))$ requires just $t_{\rm RB} = 66$(ms) for $N = 30$; while the FE solution $\boldsymbol{\mu} \rightarrow s^\mathcal{N}(\boldsymbol{\mu})$ requires approximately $t_{\rm FE} = 8$(s): thus our RB online evaluation is just $0.83\%$ of the FEM computational cost.
\subsection{The multi-material plate problem}
We consider a unit cell divided into $9$ square subdomains of equal size as shown in Figure~\ref{fig:ex4_model}. We apply (clamped) Dirichlet boundary conditions on the bottom of the cell $\Gamma^{\rm o}_B$ and (unit tension) non-homogeneous Neumann boundary conditions on $\Gamma^{\rm o}_T$. We consider orthotropic plane stress materials: the Young's modulus properties for all $9$ subdomains are given in Figure, the Poisson's ratio is chosen as $\nu_{12,i} =0.3$, $i = 1,\ldots,9$ and $\nu_{21,i}$ is determined by \refeq{eqn:ortho_plane_stress}. The shear modulus is chosen as a function of the two Young's moduli as in \refeq{eqn:ortho_shear_modulus} for all $9$ subdomains. All material axes are aligned with the coordinate system (and loading). Our output of interest is the integral of normal displacement ($u_1$) over $\Gamma^{\rm o}_T$, which represents the average normal displacement on $\Gamma^{\rm o}_T$. We note our output of interest is thus ``compliant''.
\begin{figure} [htbp]
\centering
\includegraphics[scale=0.5]{CMAME_mulmat}
\caption{The multi-material problem}
\label{fig:ex4_model}
\end{figure}
We consider $P=6$ parameters $\boldsymbol{\mu} = [\mu_1,\ldots,\mu_6]$, correspond to the six Young's moduli values as shown in Figure~\ref{fig:ex4_model} (the two Young's moduli for each subdomain are shown in those brackets). The parameter domain is chosen as $\mbox{\boldmath$\mathcal{D}$} = [0.5, 2.0]^6$.
We then apply the domain decomposition \cite{rozza08:ARCME} and obtain $L_{\rm reg} = 18$ subdomains. Despite the large $L_{\rm reg}$ number of domains, there is no geometric transformation in this case. We recover our affine forms with $Q^a = 12$, $Q^f = 1$, note that all $Q^a$ are contributed from all the Young's moduli since there is no geometric transformation involved. Moreover, it is observed that the bilinear form can be, in fact, classified as a ``parametrically coercive'' one \cite{patera07:book}.
We next consider a FE approximation where the mesh contains $n_{\rm node} = 4098$ nodes and $n_{\rm elem} = 8032$ $P_1$ elements, which corresponds to $\mathcal{N} = 8112$ degrees of freedoms. The mesh is refined around all the interfaces between different subdomains as shown in Figure~\ref{fig:ex4_mesh}.
\begin{figure} [htbp]
\centering
\includegraphics[scale=0.3]{CMAME_mulmat_mesh}
\caption{The multi-material problem: Domain composition and FE mesh}
\label{fig:ex4_mesh}
\end{figure}
We then apply the RB approximation. We present in Table~\ref{tab:ex4_tab} our convergence results: the RB error bounds and effectivities as a function of $N$. The error bound reported, $\mathcal{E}_N = \Delta^s_N(\boldsymbol{\mu})/|s_N(\boldsymbol{\mu})|$ is the maximum of the relative error bound over a random test sample $\Xi_{\rm test}$ of size $n_{\rm test} = 200$. We denote by $\overline{\eta}_N^s$ the average of the effectivity $\eta_N^s(\boldsymbol{\mu})$ over $\Xi_{\rm test}$. We observe that our effectivity average is of order $O(10)$.
\begin{table}
\centering
\begin{tabular}{|c||c|c|}
\hline
$N$ & $\mathcal{E}_N$ & $\overline{\eta}_N^s$ \\
\hline
5 & 1.01\texttt{E}-02 & 8.11 \\
10 & 1.45\texttt{E}-03 & 11.16 \\
20 & 3.30\texttt{E}-04 & 11.47 \\
30 & 1.12\texttt{E}-04 & 12.59 \\
40 & 2.34\texttt{E}-05 & 11.33 \\
50 & 9.85\texttt{E}-06 & 12.90 \\
\hline
\end{tabular}
\caption{The multi-material problem: RB convergence}
\label{tab:ex4_tab}
\end{table}
As regards computational times, a RB online evaluation $\boldsymbol{\mu} \rightarrow (s_N(\boldsymbol{\mu}),\Delta_N^s(\boldsymbol{\mu}))$ requires just $t_{\rm RB} = 33$(ms) for $N = 40$; while the FE solution $\boldsymbol{\mu} \rightarrow s^\mathcal{N}(\boldsymbol{\mu})$ requires $t_{\rm FE} = 8.1$(s): thus the RB online evaluation is just $0.41\%$ of the FEM computational cost.
\subsection{The woven composite beam problem}
We consider a composite cantilever beam as shown in Figure~\ref{fig:ex5_model}. The beam is divided into two regions, each with a square hole in the center of (equal) size $2w$. We apply (clamped) Dirichlet boundary conditions on the left side of the beam $\Gamma^{\rm o}_L$, (symmetric about the $x^{\rm o}_1$ direction) Dirichlet boundary conditions on the right side of the beam $\Gamma^{\rm o}_R$, and (unit tension) non-homogeneous Neumann boundary conditions on the top side $\Gamma^{\rm o}_T$. We consider the same orthotropic plane stress materials for both regions: $(E_1, E_2) = (1, E_2)$, $\nu_{12} = 0.3$, $\nu_{21}$ is determined by \refeq{eqn:ortho_plane_stress} and the shear modulus $G_{12}$ is given by \refeq{eqn:ortho_shear_modulus}. The material axes of both regions are not aligned with the coordinate system and loading: the angles of the the material axes and the coordinate system of the first and second region are $\theta$ and $-\theta$, respectively. The setting represents a ``woven'' composite material across the beam horizontally. Our output of interest is the integral of the normal displacement ($u_1$) over the boundary $\Gamma^{\rm o}_O$. We note our output of interest is thus ``non-compliant''.
\begin{figure} [htbp]
\centering
\includegraphics[scale=0.5]{CMAME_composite}
\caption{The woven composite beam problem}
\label{fig:ex5_model}
\end{figure}
We consider $P=3$ parameters $\boldsymbol{\mu} = [\mu_1,\mu_2,\mu_3] \equiv [w, E_2, \theta]$. The parameter domain is chosen as $\mbox{\boldmath$\mathcal{D}$} = [1/6, 1/12] \times [1/2, 2] \times [-\pi/4, \pi/4]$.
We then apply the domain decomposition \cite{rozza08:ARCME} and obtain $L_{\rm reg} = 32$ subdomains, note that all subdomains transformations are just simply translations due to the ``added control points'' along the external (and interface) boundaries strategy \cite{rozza08:ARCME}. We recover the affine forms with $Q^a = 19$, $Q^f = 2$, and $Q^\ell = 1$.
We next consider a FE approximation where the mesh contains $n_{\rm node} = 3569$ nodes and $n_{\rm elem} = 6607$ $P_1$ elements, which corresponds to $\mathcal{N} = 6865$ degrees of freedoms. The mesh is refined around the holes, the interfaces between the two regions, and the clamped boundary as shown in Figure~\ref{fig:ex5_mesh}.
\begin{figure} [htbp]
\centering
\includegraphics[scale=0.5]{CMAME_composite_mesh}
\caption{The woven composite beam problem: Domain composition and FE mesh}
\label{fig:ex5_mesh}
\end{figure}
We then apply the RB approximation. We present in Table~\ref{tab:ex5_tab} our convergence results: the RB error bounds and effectivities as a function of $N$. The error bound reported, $\mathcal{E}_N = \Delta^s_N(\boldsymbol{\mu})/|s_N(\boldsymbol{\mu})|$ is the maximum of the relative error bound over a random test sample $\Xi_{\rm test}$ of size $n_{\rm test} = 200$. We denote by $\overline{\eta}_N^s$ the average of the effectivity $\eta_N^s(\boldsymbol{\mu})$ over $\Xi_{\rm test}$. We observe that our effectivity average is of order $O(5-25)$.
\begin{table}
\centering
\begin{tabular}{|c||c|c|}
\hline
$N$ & $\mathcal{E}_N$ & $\overline{\eta}_N^s$ \\
\hline
4 & 4.64\texttt{E}-02 & 22.66 \\
8 & 1.47\texttt{E}-03 & 7.39 \\
12 & 2.35\texttt{E}-04 & 9.44 \\
16 & 6.69\texttt{E}-05 & 14.29 \\
20 & 1.31\texttt{E}-05 & 11.41 \\
\hline
\end{tabular}
\caption{The woven composite beam problem: RB convergence}
\label{tab:ex5_tab}
\end{table}
As regards computational times, a RB online evaluation $\boldsymbol{\mu} \rightarrow (s_N(\boldsymbol{\mu}),\Delta_N^s(\boldsymbol{\mu}))$ requires just $t_{\rm RB} = 40$(ms) for $N = 20$; while the FE solution $\boldsymbol{\mu} \rightarrow s^\mathcal{N}(\boldsymbol{\mu})$ requires $t_{\rm FE} = 7.5$(s): thus our RB online evaluation is just $0.53\%$ of the FEM computational cost.
\subsection{The closed vessel problem}
We consider a closed vessel under tension at both ends as shown in Figure~\ref{fig:ex6_modelfull}.
\begin{figure} [htbp]
\centering
\includegraphics[scale=0.3]{CMAME_pipefull}
\caption{The closed vessel problem}
\label{fig:ex6_modelfull}
\end{figure}
The vessel is axial symmetric about the $x^{\rm o}_2$ axis, and symmetric about the $x^{\rm o}_1$ axis, hence we only consider a representation ``slice'' by our axisymmetric formulation as shown in Figure~\ref{fig:ex6_model}. The vessel is consists of two layered, the outer layer is of fixed width $w^{\rm out} = 1$, while the inner layer is of width $w^{\rm in} = w$. The material properties of the inner layer and outer layer are given by $(E^{\rm in},\nu) = (E^{\rm in},0.3)$ and $(E^{\rm out},\nu) = (1,0.3)$, respectively. We apply (symmetric about the $x^{\rm o}_2$ direction) Dirichlet boundary conditions on the bottom boundary of the model $\Gamma^{\rm o}_B$, (symmetric about the $x^{\rm o}_1$ direction) Dirichlet boundary conditions on the left boundary of the model $\Gamma^{\rm o}_L$ and (unit tension) non-homogeneous Neumann boundary conditions on the top boudanry $\Gamma^{\rm o}_T$. Our output of interest is the integral of the axial displacement ($u_r$) over the right boundary $\Gamma^{\rm o}_R$. We note our output of interest is thus ``non-compliant''.
\begin{figure} [htbp]
\centering
\includegraphics[scale=0.5]{CMAME_pipe}
\caption{The closed vessel problem}
\label{fig:ex6_model}
\end{figure}
We consider $P=2$ parameters $\boldsymbol{\mu} = [\mu_1,\mu_2] \equiv [w, E^{\rm in}]$. The parameter domain is chosen as $\mbox{\boldmath$\mathcal{D}$} = [0.1, 1.9] \times [0.1, 10]$.
We then apply the domain decomposition \cite{rozza08:ARCME} and obtain $L_{\rm reg} = 12$ subdomains as shown in Figure~\ref{fig:ex6_mesh}. We recover our affine forms with $Q^a = 47$, $Q^f = 1$, and $Q^\ell = 1$. Despite the small number of parameter (and seemingly simple transformations), $Q^a$ is large in this case. A major contribution to $Q^a$ come from the expansion of the terms $x^{\rm o}_1$ in the effective elastic tensor $[\mathbf{S}]$, which appeared due to the geometric transformation of the inner layer.
We next consider a FE approximation where the mesh contains $n_{\rm node} = 3737$ nodes and $n_{\rm elem} = 7285$ $P_1$ elements, which corresponds to $\mathcal{N} = 7423$ degrees of freedoms. The mesh is refined around the interfaces between the two layers.
\begin{figure} [htbp]
\centering
\includegraphics[scale=0.3]{CMAME_pipe_mesh}
\caption{The closed vessel problem: Domain composition and FE mesh}
\label{fig:ex6_mesh}
\end{figure}
We then apply the RB approximation. We present in Table~\ref{tab:ex6_tab} convergence results: the RB error bounds and effectivities as a function of $N$. The error bound reported, $\mathcal{E}_N = \Delta^s_N(\boldsymbol{\mu})/|s_N(\boldsymbol{\mu})|$ is the maximum of the relative error bound over a random test sample $\Xi_{\rm test}$ of size $n_{\rm test} = 200$. We denote by $\overline{\eta}_N^s$ the average of the effectivity $\eta_N^s(\boldsymbol{\mu})$ over $\Xi_{\rm test}$. We observe that our effectivity average is of order $O(50-120)$, which is quite large, however it is not surprising since our output is ``non-compliant''.
\begin{table}
\centering
\begin{tabular}{|c||c|c|}
\hline
$N$ & $\mathcal{E}_N$ & $\overline{\eta}_N^s$ \\
\hline
10 & 7.12\texttt{E}-02 & 56.01 \\
20 & 1.20\texttt{E}-03 & 111.28 \\
30 & 3.96\texttt{E}-05 & 49.62 \\
40 & 2.55\texttt{E}-06 & 59.96 \\
50 & 5.70\texttt{E}-07 & 113.86 \\
60 & 5.90\texttt{E}-08 & 111.23 \\
70 & 6.95\texttt{E}-09 & 77.12 \\
\hline
\end{tabular}
\caption{The closed vessel problem: RB convergence}
\label{tab:ex6_tab}
\end{table}
As regards computational times, a RB online evaluation $\boldsymbol{\mu} \rightarrow (s_N(\boldsymbol{\mu}),\Delta_N^s(\boldsymbol{\mu}))$ requires just $t_{\rm RB} = 167$(ms) for $N = 40$; while the FE solution $\boldsymbol{\mu} \rightarrow s^\mathcal{N}(\boldsymbol{\mu})$ requires $t_{\rm FE} = 8.2$(s): thus our RB online evaluation is just $2.04\%$ of the FEM computational cost.
\subsection{The Von {K{\'a}rm{\'a}n} plate problem}
We consider now a different problem that can be derived from the classical elasticity equations \cite{ciarlet1988three,ciarlet1997mathematical}. It turns out to be nonlinear and brings with it a lot of technical difficulties. Let us consider an elastic, bidimensional and rectangular plate $\Omega = [0,l] \times [0,1]$ in its undeformed state, subjected to a $\mu$-parametrized external load acting on its edge, then the \emph{Airy stress potential} and the deformation from its flat state, respectively $\phi$ and $u$ are defined by the Von {K{\'a}rm{\'a}n} equations
\begin{equation}
\label{karm}
\begin{cases}
\Delta^2u + \mu u_{xx} = \left[\phi, u\right] + f \ , \quad &\text{in}\ \Omega \\
\Delta^2\phi = -\left[u, u\right] \ , \quad &\text{in}\ \Omega
\end{cases}
\end{equation}
where $$\Delta^2 := \Delta\Delta = \left(\frac{\partial\,^2}{\partial\,x^2} + \frac{\partial\,^2}{\partial\,y^2}\right)^2 \ ,$$ is the biharmonic operator and
$$[u,\phi] := \frac{\partial\,^2u}{\partial\,x^2}\frac{\partial\,^2\phi}{\partial\,y^2} -2\frac{\partial\,^2u}{\partial x\partial y}\frac{\partial\,^2\phi}{\partial x\partial y} + \frac{\partial\,^2u}{\partial\,y^2}\frac{\partial\,^2\phi}{\partial\,x^2} \ ,$$
is the \emph{bracket of Monge-Amp\'ere}. So we have a system of two nonlinear and parametrized equations of the fourth order with $\mu$ the parameter that measures the compression along the sides of the plate.
\begin{figure} [htbp]
\centering
\input{CMAME_plate.pdf_tex}
\caption{A rectangular bidimensional elastic plate compressed on its edges}
\label{piastra}
\end{figure}
From the mathematical point of view, we will suppose the plate is simply supported, i.e. that holds boundary conditions $$u = \Delta u = 0, \qquad \phi = \Delta \phi = 0, \qquad \text{on} \ \partial\Omega .$$
In this model problem we are interested in the study of stability and uniqueness of the solution for a given parameter. In fact due to the nonlinearity of the bracket we obtain the so called \emph{buckling phenomena} \cite{PAMM:PAMM201310213}, that is the main feature studied in bifurcations theory.
What we seek is the critical value of $\mu$ for which the stable (initial configuration) solution become unstable while there are two new stable and symmetric solutions.
To detect this value we need a very complex algorithm that mixes a \emph{continuation method}, a nonlinear solver and finally a full-order method to find the buckled state. At the end for every $\mu \in \mbox{\boldmath$\mathcal{D}$}_{train}$ (a fine discretization of the parameter domain $\mbox{\boldmath$\mathcal{D}$}$) we have a loop due to the nonlinearity, for which at each iteration we have to solve the Finite Element method applied to the weak formulation of the problem.
Here we consider $P=1$ parameter $\mu$ and its domain is suitably chosen\footnote{It is possible to show that the bifurcation point is related to the eigenvalue of the linearized model \cite{berger}, so we are able to set in a proper way the range of the parameter domain.} as $\mbox{\boldmath$\mathcal{D}$} = [30, 70]$.
Also in this case we can simply recover the affine forms with $Q^a = 3$. For the rectangular plate test case with $l=2$ we applied the Finite Element method, with $n_{node} = 441$ nodes and $n_{elem} = 800$ $P_2$ elements, which corresponds to $\mathcal{N} = 6724$ degrees of freedom. We stress on the fact that the linear system obtained by the Galerkin projection has to be solved at each step of the nonlinear solver, here we chose the classic \emph{Newton method} \cite{QuateroniValli97}.
Moreover, for a given parameter, we have to solve a FE system until Newton method converges just to obtain one of the possible solutions of our model; keeping in mind that we do not know a priori where is the bifurcation point and we have to investigate the whole parameter domain. It is clear that despite the simple geometry and the quite coarse mesh, the reduction strategies are fundamental in this kind of applications.
For example, in order to plot a \emph{bifurcation diagram} like the one in Figure~\ref{fig:bifdia}, the full order code running on a standart computer takes approximately one hour.
\begin{figure} [htbp]
\centering
\includegraphics[scale=0.6]{Figure_1_bisbisbis}
\caption{Bifurcation diagram for a square plate and different initial guess for Newton method, on y-axis is represented the infinite norm of the solution}
\label{fig:bifdia}
\end{figure}
Once selected a specific parameter, $\lambda = 70$, we can see in Figure~\ref{fig:bifurc1} the two solutions that belong to the different branches of the plot reported in Figure~\ref{fig:bifdia}.
\begin{figure} [htbp]
\centering
\includegraphics[scale=0.4]{1cella_l1.png} \quad \includegraphics[scale=0.4]{2cella_l1.png}
\caption{Contour plot of the two solutions belonging to the green and red branches of the bifurcation diagram for $\lambda = 70$, respectively}
\label{fig:bifurc1}
\end{figure}
We then applied RB approximation and present in Table~\ref{tab:ex7_tab} a convergence results: the error between the truth approximation and the reduced one as a function of $N$. The error reported, $\mathcal{E}_N = \max_{\boldsymbol{\mu} \in \mbox{\boldmath$\mathcal{D}$}} ||\mathbf{u}^\mathcal{N}(\boldsymbol{\mu}) - \mathbf{u}^\mathcal{N}_{{\rm RB},N}(\boldsymbol{\mu})||_{X}$ is the maximum of the approximation error over a uniformly chosen test sample.
\begin{table}
\centering
\begin{tabular}{|c||c|}
\hline
$N$ & $\mathcal{E}_N$ \\
\hline
1 & 6.61\texttt{E}+00 \\
2 & 6.90\texttt{E}-01 \\
3 & 7.81\texttt{E}-02 \\
4 & 2.53\texttt{E}-02 \\
5 & 1.88\texttt{E}-02 \\
6 & 1.24\texttt{E}-02 \\
7 & 9.02\texttt{E}-03 \\
8 & 8.46\texttt{E}-03 \\
\hline
\end{tabular}
\caption{The Von {K{\'a}rm{\'a}n} plate problem : RB convergence}
\label{tab:ex7_tab}
\end{table}
As we can see in Figure~\ref{fig:1cell1} e obtain very good results with a low number of snapshots due to the strong properties of the underlying biharmonic operator.
\begin{figure} [htbp]
\centering
\includegraphics[scale=0.4]{lam65pfo.png} \quad \includegraphics[scale=0.4]{lam65pro.png}
\caption{Comparison between the full order solution (left) and reduced order one (right) for $\lambda = 65$ }
\label{fig:1cell1}
\end{figure}
A suitable extension for the a posteriori error estimate of the solution can be obtained by applying Brezzi-Rappaz-Raviart (BRR) theory on the numerical approximation of nonlinear problems \cite{Brezzi1980,Brezzi1981,Brezzi1982,Grepl2007,canuto2009}. However, the adaptation of BRR theory to RB methods in bifurcating problems is not straightforward, and we leave it for further future investigation \cite{pichirozza}.
As regards computational times, a RB online evaluation $\boldsymbol{\mu} \rightarrow \mathbf{u}^\mathcal{N}_{{\rm RB},N}(\boldsymbol{\mu})$ requires just $t_{\rm RB} = 100$(ms) for $N = 8$; while the FE solution $\boldsymbol{\mu} \rightarrow \mathbf{u}^\mathcal{N}(\boldsymbol{\mu})$ requires $t_{\rm FE} = 8.17$(s): thus our RB online evaluation is just $1.22\%$ of the FEM computational cost.
\section{Conclusions}
We have provided some examples of applications of reduced basis
methods in linear elasticity problems depending also on many parameters
of different kind (geometrical, physical, engineering) using
different linear elasticity approximations, a 2D Cartesian setting or a 3D axisymmetric one, different material models (isotropic and orthotropic), as well as an overview on nonlinear problems. Reduced basis methods have confirmed a
very good computational performance with respect to a classical
finite element formulation, not very suitable to solve
parametrized problems in the real-time and many-query contexts.
We have extended and generalized previous work \cite{milani08:RB_LE} with the possibility to treat with more complex outputs by introducing a
dual problem \cite{rozza08:ARCME}. Another
very important aspect addressed in this work is the certification of the errors in the
reduced basis approximation by means of a posteriori error
estimators, see for example \cite{huynh07:cras}. This work looks also at more complex 3D parametrized applications (not only in the special axisymmetric case) as quite promising problem to be solved with the same certified methodology \cite{chinesta2014separated,zanon:_phd_thesis}.
\section*{Acknowledgement}
We are sincerely grateful to Prof. A.T. Patera (MIT) and Dr. C.N. Nguyen (MIT) for important
suggestions, remarks, insights, and codevelopers of the \texttt{rbMIT} and \texttt{RBniCS} (http://mathlab.sissa.it/rbnics) software libraries used for the numerical tests. We acknowledge the European Research Council consolidator grant H2020 ERC CoG 2015 AROMA-CFD GA 681447 (PI Prof. G. Rozza).
\section*{Appendix}
\addcontentsline{toc}{section}{Appendix}
\section{Stress-strain matrices}
In this section, we denote $E_i$, $i = 1,3$ as the Young's moduli, $\nu_{ij}$; $i,j = 1,2,3$ as the Poisson ratios; and $G_{12}$ as the shear modulus of the material.
\subsection{Isotropic cases}
For both of the following cases, $E = E_1 = E_2$, and $\nu = \nu_{12} = \nu_{21}$.
Isotropic plane stress:
\begin{equation*}
[\mathbf{E}] = \frac{E}{(1-\nu^2)}\left[
\begin{array}{ccc}
1 & \nu & 0 \\
\nu & 1 & 0 \\
0 & 0 & 2(1+\nu) \\
\end{array}
\right].
\end{equation*}
Isotropic plane strain:
\begin{equation*}
[\mathbf{E}] = \frac{E}{(1+\nu)(1-2\nu)}\left[
\begin{array}{ccc}
1 & \nu & 0 \\
\nu & 1 & 0 \\
0 & 0 & 2(1+\nu) \\
\end{array}
\right].
\end{equation*}
\subsection{Orthotropic cases}
Here we assume that the orthotropic material axes are aligned with the axes used for the analysis of the structure. If the structural axes are not aligned with the orthotropic material axes, orthotropic material rotation must be rotated by with respect to the structural axes. Assuming the angle between the orthogonal material axes and the structural axes is $\theta$, the stress-strain matrix is given by $[\mathbf{E}] = [\mbox{\boldmath$T$}(\theta)][\hat{\mathbf{E}}][\mbox{\boldmath$T$}(\theta)]^T$, where
\begin{equation*}
[\mbox{\boldmath$T$}(\theta)] = \left[
\begin{array}{ccc}
\cos^2\theta & \sin^2\theta & -2\sin\theta\cos\theta \\
\sin^2\theta & \cos^2\theta & 2\sin\theta\cos\theta \\
\sin\theta\cos\theta & -sin\theta\cos\theta & \cos^2\theta-\sin^2\theta\\
\end{array}
\right].
\end{equation*}
Orthotropic plane stress:
\begin{equation*}
[\hat{\mathbf{E}}] = \frac{1}{(1-\nu_{12}\nu_{21})}\left[
\begin{array}{ccc}
E_1 & \nu_{12}E_1 & 0 \\
\nu_{21}E_2 & E_2 & 0 \\
0 & 0 & (1-\nu_{12}\nu_{21})G_{12} \\
\end{array}
\right].
\end{equation*}
Note here that the condition
\begin{equation}\label{eqn:ortho_plane_stress}
\nu_{12}E_1 = \nu_{21}E_2
\end{equation}
must be required in order to yield a symmetric $[\mathbf{E}]$.
Orthotropic plane strain:
\begin{equation*}
[\hat{\mathbf{E}}] = \frac{1}{\Lambda}\left[
\begin{array}{ccc}
(1-\nu_{23}\nu_{32})E_1 & (\nu_{12}+\nu_{13}\nu_{32})E_1 & 0 \\
(\nu_{21}+\nu_{23}\nu_{31})E_2 &(1-\nu_{13}\nu_{31})E_2 & 0 \\
0 & 0 & \Lambda G_{12} \\
\end{array}
\right].
\end{equation*}
Here $\Lambda = (1-\nu_{13}\nu_{31})(1-\nu_{23}\nu_{32})-(\nu_{12}+\nu_{13}\nu_{32})(\nu_{21}+\nu_{23}\nu_{31})$. Furthermore, the following conditions,
$$\nu_{12}E_1 = \nu_{21}E_2, \quad \nu_{13}E_1 = \nu_{31}E_3, \quad \nu_{23}E_2 = \nu_{32}E_3,$$
must be satisfied, which leads to a symmetric $[\mathbf{E}]$.
An reasonable good approximation for the shear modulus $G_{12}$ in orthotropic case is given by \cite{carroll98:fem} as
\begin{equation}\label{eqn:ortho_shear_modulus}
\frac{1}{G_{12}} \approx \frac{(1+\nu_{21})}{E_1} + \frac{(1+\nu_{12})}{E_2}.
\end{equation}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,658
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These rules are enacted pursuant to Article 46 of the Spatial Planning Act (hereinafter referred to as the Act).
The central competent authority may delegate other agencies or organizations to establish or revise the National spatial plan specified in Subparagraph 1 of Paragraph 1 of Article 4 of the Act.
The competent authorities of municipalities or counties (cities) may delegate other agencies or organizations to establish or revise of municipality and county (city) spatial plans specified in Subparagraph 1 of Paragraph 2 of Article 4 of the Act.
The central competent authority shall announce a spatial planning white paper described in Article 5 of the Act biennially. The contents shall include the current land use status and tendencies, basic land use management policies and related matters.
A municipality or county (city) government applying to the central competent authority for reconsideration of a ratified spatial plan in accordance with Article 14 of the Act shall state the reasons and submit related documents.
The central competent authority shall present reconsideration applications described in the preceding paragraph to be reviewed by the spatial planning committee.
2. If a notice described in the preceding subparagraph cannot be delivered, it may be left at the local village office and a public announcement shall also be put up at the offices of the competent authority and the village.
The provisions in the preceding paragraph shall apply mutatis mutandis when a competent authority acts according to Article 2 of these rules and delegates an agency or organization to be responsible for spatial plan establishment or revision.
As set forth in Item 3 of Subparagraph 1, Item 3 of Subparagraph 2, Item 3 of Subparagraph 3 and Item 3 of Subparagraph 4 of Paragraph 1 of Article 20 of the Act, the other necessary functional sub-zone shall comply with the demarcation principles specified in the said article. At the same time, environmental resource conditions, the current land use status, local characteristics and development needs shall also be taken into consideration. The other necessary functional sub-zone shall be specified in the National spatial plan or municipality or county (city) spatial plans.
The definition of land for designated uses as described in Paragraph 1 of Article 22 of the Act shall be conducted in accordance with spatial plans of all levels and characteristics of the land.
The definition of enhancing environmental conservation at any time as described in Paragraph 2 of Article 22 of the Act shall be revised functional zone or sub-zones to stricter zoning or sub-zoning according to the demarcation of functional zones in spatial plans of all levels.
The definition of the buildings and facilities constructed before implementation of regional plans as described in Paragraph 1 of Article 32 of the Act shall be the indigenous people's land that is not in the urban plan area, and constructed before defining the land for designated uses.
If any restoration plan to be established according to Article 36 of the Act involves indigenous people's land, the demarcating agency shall invite the indigenous tribes in concern to participate in plan establishment, execution and management. Written notices shall be issued 14 days before the corresponding meeting is held.
The contents of appropriate placement plans and complementary measures according to Paragraph 2 of Article 37 of the Act established by the concerned central competent authority or municipality and county (city) government shall include placement objects, approaches and locations, financial plans, social assistance and other related matters.
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Land use that are special or of scale up to a certain threshold that complies with the principles of functional zone and sub-zone without a permit as stated in Subparagraph 1 of Paragraph 2 of Article 38 of the Act shall be land use with no permit application filed according to the procedure specified in Paragraph 1 of Article 24 of the Act.
These rules shall take effect on the day the Act is enforced.
The amended articles of the Enforcement Rules shall take effect on the day of promulgation.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,742
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Q: Pass v-model through a child component to another one I currently have a component called generic-input. It basically is a input, which can also take a filter function as a prop and apply it to the input. This alone is working.
I now want to create concrete sub-components, which simply pass a predefined filter function to generic-input. This is not working.
My current layout (working example, before sub-components):
GenericInput.vue
<template>
<input ref="input" :value="objectValue" @input="inputHandler(§event.target.value)" />
</template>
<script>
export default {
props: {
value: undefined,
toObjectType: {
type: Function,
default: (x) => (x),
},
filter: {
type: Function,
default: (x) => (x),
},
},
data() {
return {
objectValue: null,
handler: true,
}
},
watch: {
value: {
immediate: true,
handler(val) {
let newValue = val;
if (newValue) {
newValue = this.toObjectType(newValue);
newValue = this.filter(newValue);
if (newValue !== val) {
this.$emit("input", newValue);
}
}
this.objectValue = newValue;
},
},
},
methods: {
inputHandler(val) {
this.updateValue(this.toObjectType(val), val);
},
updateValue: function (val, strVal = null) {
const oldVal = this.objectValue;
val = this.filter(val);
if (val === oldVal) {
this.$refs.input.value = strVal && val === this.toObjectType(strVal) ? strVal : null;
return;
}
this.objectValue = val;
this.$emit("input", val);
},
},
}
</script>
Breakdown:
Need to use objectValue, otherwise I get a mutable error on value. But now I have to use a watcher on value. inputHandlerand updateValue simply update the value according to the filter.
Inspiration
Usage: <generic-input v-model="myModel" />
Again, this works perfectly.
Now, I want to create sub-components of generic-input such as number-input. Those sub-components have predefined (dynamic) toObjectType and filter methods.
Not working minimal example:
NumberInput.vue
<template>
<generic-input />
</template>
I know this is not working, since the parent, which has the numeric-input-tag does not react to changes (v-if) . This does not happen when using generic-input instead (everything works as expected then).
→ Model-Variable is not being updated.
I have experimented, how to use number-input then. All didn't seem to work.
Ideas:
*
*v-model on generic-input or number-input
*.sync
I never want to change the value of my model in a sub-component of generic-input. I only want to pass it along the sub-component to generic-input.
A: As @kissu mentioned in a comment, simply use $emit. As I always emit an input-event in GenericInput.vue on a change anyway, I can simply add a few lines to my sub-component files.
A minimal working example would be:
NumberInput.vue
<template>
<generic-input :value="value" @input="$emit('input', $event)" />
</template>
<script>
export default {
props: {
value: Number,
},
}
</script>
Along with the generic-input as written in the question, you can use the sub-component number-input (in my case) with:
<number-input v-model="model" />
Thank you for the help!
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,161
|
Métodos de simulação estocástica são procedimentos que envolvem a geração de números aleatórios (pseudo-aleatórios) com o objectivo de explorar o espaço de incerteza ou campo de possibilidades de um dado fenómeno físico ou qualquer outro tipo de variável de estudo cujo comportamento possa ser quantificado matematicamente. Os métodos de simulação fazem parte da ciência de processos estocásticos e utilizam o método de Monte-Carlo nos seus algoritmos.
Nota histórica
Com a introdução da teoria do caos e especialmente a noção de efeito borboleta, a qual refere que o bater de asas de uma borboleta num lado do mundo poderá dar origem a um tornado no lado oposto, a introdução de aleatoriedade no estudo de fenómenos físicos começou a deixar de ser posta de lado. A noção que pequenas diferenças nas condições iniciais de um sistema podem evoluir para estados completamente diferentes implicariam que o grau de confiança numa estimação de determinista seria menor do que o expectável. Por esse motivo a quantificação e exploração do espaço de incerteza num dado procedimento passou a ser necessário. O progresso neste campo passou a ser tão notório que o professor Richard Forsyth escreveu num prefácio:
Discussão
A noção de incerteza é importante nos processo de simulação sequencial em geoestatística a qual utilizam simulação estocástica para gerar realizações equipróvaveis na estimação de uma variável de estudo num contexto espacial.
São também utilizados métodos deste tipo no campo da química, especialmente em operações de reação-difusão.
Em 1969 foi proposto pelo meteorologista Edward Epstein a realização de várias simulações para o estado da atmosfera no campo da previsão numérica do tempo de maneira a obter uma média e variância. Na década de 1990 tornou-se rotineiro a utilização de simulação estocástica nesta área científica.
Ver também
Processos estocásticos
Simulação sequencial
Estatística
Processos estocásticos
Simulação
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 2,517
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{"url":"http:\/\/www.r-bloggers.com\/fast-track-publishing-using-knitr-table-mania-part-iv\/","text":"# Fast-track publishing using knitr: table mania (part IV)\n\nJanuary 15, 2014\nBy\n\n(This article was first published on G-Forge \u00bb R, and kindly contributed to R-bloggers)\n\nFast-track publishing using knitr is a short series on how I use knitr to speedup publishing in my research. While illustrations (previous post) are optional, tables are not, and this fourth article is therefore devoted to tables. Tables through knitr is probably one of the most powerful fast-track publishing tools, in this article I will show (1) how to quickly generate a descriptive table, (2) how to convert your regression model into a table, and (3) worth knowing about table design and anatomy.\n\n## Data set preparation\n\nTo make this post more concrete I will use the melanoma data set in the boot package. Below I factor the variables:\n\n?View Code RSPLUS\n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 library(boot) \u00a0 # Set time to years instead of days melanoma$time_years <- melanoma$time \/ 365.25 \u00a0 # Factor the basic variables that # we're interested in melanoma$status <- factor(melanoma$status, levels=c(2,1,3), labels=c(\"Alive\", # Reference \"Melanoma death\", \"Non-melanoma death\")) melanoma$sex <- factor(melanoma$sex, labels=c(\"Male\", \"Female\")) \u00a0 melanoma$ulcer <- factor(melanoma$ulcer, labels=c(\"Present\", \"Absent\"))\n\n## Descriptive tables\n\nGenerating descriptive tables containing simple means, medians, ranges, frequencies, etc. should be fast and efficient. Decide on what you want in your columns and then structure your data into sections; I try to use the following structure:\n\n\u2022 Basic stats: e.g. sex, age.\n\u2022 Article specific stats: e.g. hip function, degree of osteoarthritis, type of surgery.\n\u2022 Outcomes: e.g. number of re-operations, mobility, pain.\n\nAfter deciding on the variables I often use the getDescriptionStatsBy function from my Gmisc-package to get the statistics into columns. I\u2019ve found that you almost always have more than one column, thereby comparing different groups. In an RCT you want to compare the treatment groups, in a case-control study you want to compare the cases to the controls, and in an observational survival study you usually want to compare those that survived with those that died (as in this example). If you are uncertain what groups to compare in your Table 1, then just compare those with complete data to those with missing data.\n\nThe getDescriptionStatsBy function has several settings that you may want to use:\n\n\u2022 P-values: While some despise the use of p-values in tables, I believe they can be useful in some cases and my function can therefore fetch fisher.test or wilcox.test p-values depending on the variable type by simply specifying statistics=TRUE.\n\u2022 Total-column: Adding a total-column may sometimes be useful, e.g. if you have by alive\/dead it is of interest to quickly get a total-column, while if you present your data by RCT-group then a total-column makes little sense.\n\u2022 Percentages for categorical variables: depending on the setting you may want your percentages to sum up horizontally or vertically, e.g. in an alive\/dead setting it makes sense to sum up the columns horizontally using hrzl_prop=TRUE while an RCT is better to sum up vertically where you want to show how many cemented, uncemented, mixed hip replacements were in each treatment arm.\n\nAs the getDescriptionStatsBy has plenty of options, I usually use a wrapper function like this:\n\n?View Code RSPLUS\n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 # A function that takes the variable name, # applies it to the melanoma dataset # and then runs the results by the status variable getT1Stat <- function(varname, digits=0){ getDescriptionStatsBy(melanoma[, varname], melanoma$status, add_total_col=TRUE, show_all_values=TRUE, hrzl_prop=TRUE, statistics=FALSE, html=TRUE, digits=digits) } # Save everything in a list # This simplifies the row grouping table_data <- list() # Get the basic stats table_data[[\"Sex\"]] <- getT1Stat(\"sex\") table_data[[\"Age\"]] <- getT1Stat(\"age\") table_data[[\"Ulceration\"]] <- getT1Stat(\"ulcer\") table_data[[\"Thicknessa\"]] <- getT1Stat(\"thickness\", 1) There is of course a myriad of alternatives for generating descriptive data. My function is trying to resemble the format for Table 1 in major medical journals, such as NEJM and Lancet. You can easily tailor it to your needs, for instance if you want to use median instead of mean for continuous variables, you provide it a different continuous function: ?View Code RSPLUS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 # A function that takes the variable name, # applies it to the melanoma dataset # and then runs the results by the status variable getT1Stat <- function(varname, digits=0){ getDescriptionStatsBy(melanoma[, varname], melanoma$status, add_total_col=TRUE, show_all_values=TRUE, hrzl_prop=TRUE, statistics=FALSE, html=TRUE, digits=digits, continuous_fn=describeMedian) }\n\nApart from my function I\u2019ve recently discovered the power of the plyr-package that can help you generate most table\/plot data. I strongly recommend having a closer look at the ddply function \u2013 it will save you valuable time.\n\nAfter running the previous code I loop through the list to extract the variable matrix and the rgroup\/n.rgroup variables that I then input to my htmlTable function:\n\n?View Code RSPLUS\n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 # Now merge everything into a matrix # and create the rgroup & n.rgroup variabels rgroup <- c() n.rgroup <- c() output_data <- NULL for (varlabel in names(table_data)){ output_data <- rbind(output_data, table_data[[varlabel]]) rgroup <- c(rgroup, varlabel) n.rgroup <- c(n.rgroup, nrow(table_data[[varlabel]])) } \u00a0 \u00a0 # Add a column spanner for the death columns cgroup <- c(\"\", \"Death\") n.cgroup <- c(2, 2) colnames(output_data) <- gsub(\"[ ]*death\", \"\", colnames(output_data)) \u00a0 htmlTable(output_data, align=\"rrrr\", rgroup=rgroup, n.rgroup=n.rgroup, rgroupCSSseparator=\"\", cgroup = cgroup, n.cgroup = n.cgroup, rowlabel=\"\", caption=\"Basic stats\", tfoot=\"a Also known as Breslow thickness\", ctable=TRUE)\n\nGenerating this beauty (the table is an image as the CSS for the site messes up the layout):\n\n## Regression tables\n\nI recently did a post on my printCrudeAndAdjustedModel-function where I showed how to output your model into a table. My function allows you to get both unadjusted and adjusted estimates into a table, adds the references, and allows can automatically attach the descriptive statistics:\n\n?View Code RSPLUS\n 1 2 3 4 5 6 7 8 9 10 11 # Setup needed for the rms coxph wrapper ddist <- datadist(melanoma) options(datadist = \"ddist\") fit <- cph(Surv(melanoma$time, melanoma$status==\"Melanoma death\") ~ sex + age + thickness + ulcer, data=melanoma) \u00a0 printCrudeAndAdjustedModel(fit, desc_digits=0, caption=\"Crude and adjusted estimates\", desc_column=TRUE, add_references=TRUE, ctable=TRUE)\n\nGives this:\n\nNow there are alternatives to my function. The texreg is an interesting package that is worth exploring and hopefully stargazer will eventually have an html\/markdown option. A minor note concerning these later packages where outputs contain R2 and more; I have never seen models presented in medical literature in that way and if you need to adjust the output you loose the fast-track idea.\n\n## Table design and anatomy\n\nTables are generally good for comparing a few values, while plots are better when you want to show a trend consisting of multiple values. Although you should avoid using tables to show trends, you can still have large tables with lots of data. When presenting a lot of data, you need to think about the table navigation:\n\n\u2022 Order: always report variables in the same order, e.g. sex, age, ulceration\u2026 should be at a similar location in each table\n\u2022 Precision: avoid unnecessary decimals\n\u2022 Markup: use headers and spanners\n\nThe first one we have already touched upon. For the second one, I often rely on the sprintf function. While round may seem like a natural option you will often want to show all decimals that you find of interest. For instance:\n\n?View Code RSPLUS\n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 round(3.901, digits=2) # 3.9 round(3.901, digits=3) # 3.901 \u00a0 # The format function works better although # you need to remember the nsmall option: # \"the minimum number of digits to the right of the decimal point\" format(3.901, digits=2) # 3.9 format(3.901, digits=2, nsmall=2) # 3.90 format(3.901, digits=3, nsmall=2) # 3.90 format(3.901, digits=4, nsmall=2) # 3.901 \u00a0 sprintf(\"%.2f\", 3.901) # 3.90 sprintf(\"%.1f HR (95 %% CI %.1f to %.1f)\", exp(coef(fit)), exp(confint(fit)[,1]), exp(confint(fit)[,2])) # \"0.9 HR (95 % CI 0.6 to 1.4)\" # \"1.0 HR (95 % CI 1.0 to 1.0)\" # \"1.0 HR (95 % CI 0.7 to 1.4)\"\n\nAlso a p-value converter is nice to have; here is a simple function that I use:\n\n?View Code RSPLUS\n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 pvalue_formatter <- function(pvalues, sig.limit = 0.001){ sapply(pvalues, function(x, sig.limit){ if (x < sig.limit) return(sprintf(\"< %s\", format(sig.limit))) # < stands for < and is needed # for the markdown\/html to work # and the format is needed to avoid trailing zeros \u00a0 # High p-values you usually want two decimals # otherwise report only one if (x > 0.01) return(format(x, digits=2)) \u00a0 return(format(x, digits=1)) }, sig.limit=sig.limit) } \u00a0 pv <- c(.133213, .0611233, .004233, .00000123123) pvalue_formatter(pv) # \"0.13\" # \"0.061\" # \"0.004\" # \"< 0.001\"\n\nThere are standard tools that you can us to help your readers to navigate the tables. I use stubs and column spanners as much as I can. A stub is a row header at the same column level as the actual rows, the rows differ by a small indentation of two white-spaces. This is an efficient way of grouping variables without making the table wider, while at the same time adding some white space around the numbers that help navigating. Similarly to stubs you can have column spanners that group columns. In my htmlTable these are called rgroup and cgroup arguments. They need to have the n.rgroup\/n.cgroup in order to let the function know how many rows\/columns each group should contain, see below example:\n\n?View Code RSPLUS\n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 col <- sprintf(\"Cell %d:%%d\", 1:9) vars <- sapply(1:3, function(i) sprintf(col, i)) \u00a0 rownames(vars) <- sprintf(\"Row no. %d\", 1:nrow(vars)) colnames(vars) <- sprintf(\"Column\nno. %d\", 1:3) cgroup <- c(\"\", \"Column spanner\") n.cgroup <- c(1, 2) rgroup <- c(\"Stub I\", \"\", \"Stub II\") n.rgroup <- c(2, 3, nrow(vars) - 2 - 3) htmlTable(vars, rowlabel=\"Row label\", cgroup=cgroup, n.cgroup=n.cgroup, rgroup=rgroup, n.rgroup = n.rgroup, rgroupCSSstyle=\"\", rgroupCSSseparator=\"\", caption=\"Basic table anatomy\", tfoot=\" Put your explanations in the table footer\", ctable=TRUE)\n\nAn alternative to using stubs is using row headers. The difference is that headers are located in a separate column, thus making the table wider. A benefit is that you can have infinite levels row group headers. Below is an example with two header levels using the xtable function:\n\n?View Code RSPLUS\n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 library(xtable) alt_vars <- cbind( Rowgrp1 = c(\"Mjr group 1\", \"\", \"\", \"Mjr group 2\", \"\", \"\", \"\", \"\", \"\"), Rowgrp2 = c(\"Group 1\", \"\", \"\", \"Group 2\", \"\", \"Group 3\", \"\", \"\", \"\"), Rownames= rownames(vars), vars) colnames(alt_vars) <- gsub(\"\n\", \"\\n\", colnames(alt_vars)) # rownames(vars) <- NULL options(xtable.html.table.attributes = list(style=sprintf(\"style='%s'\", paste(\"border:0\", \"border-top: 1px solid grey\", \"border-bottom: 1px solid grey\", sep=\"; \")))) print(xtable(alt_vars, caption=\"An xtable example\"), type=\"html\", include.rownames = FALSE)\n\nI hope you found this useful. In the next post I\u2019ll have a summary with an example for those of you new to knitr.","date":"2014-12-26 20:01:25","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.43683576583862305, \"perplexity\": 4268.79453127491}, \"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-2014-52\/segments\/1419447549662.85\/warc\/CC-MAIN-20141224185909-00019-ip-10-231-17-201.ec2.internal.warc.gz\"}"}
| null | null |
{"url":"https:\/\/www.gmlscripts.com\/forums\/viewtopic.php?pid=2740","text":"# GMLscripts.com\n\nDiscuss and collaborate on GML scripts\n\nYou are not logged in.\n\n## #1 2011-03-25 03:14:01\n\nxot\nRegistered: 2007-08-18\nPosts: 1,201\n\n### Further horrors of \"Treat uninitialized variables as value 0\"\n\nLucb1e pointed out a bug in my md5 script related to this blight of a feature. For speed, the script uses some precomputed tables. The first time it is called it checks for the existence of the tables and creates them as global arrays if they do not yet exist. The problem is when \"Treat uninitialized variables as value 0\" is set, the variable_global_exists() function always returns true. Because of this, the first time the script is called it thinks the tables have been created and proceeds to use tables filled with zeros to arrive at the wrong result. The workaround is to first check for the existence of an initialization flag, and then confirm that it is set to a non-zero or string value.\n\nThere are a few scripts posted here that will fail similarly.\n\nAbusing forum power since 1986.\n\nOffline","date":"2019-10-22 22:09:14","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.5624951720237732, \"perplexity\": 1745.1129276836525}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570987824701.89\/warc\/CC-MAIN-20191022205851-20191022233351-00168.warc.gz\"}"}
| null | null |
\section{Introduction}\label{s:intro}
In this work we consider the nonnegative solutions $u(t,x)$ of the heat equation with a logarithmic nonlinearity, namely
\begin{equation}
\label{eq}
\partial _t u=\partial _{xx} u +\lambda\, u \ln \(u^2\), \quad t>0,\; x\in \R,
\end{equation}
where $\lambda >0$ is a given parameter. Our primary goal is to
investigate the large time behavior of the solutions, since this study
reveals mechanisms which seem interesting to us. Equation \eqref{eq} shares some similarities with {\it bistable} equations modelling an {\it Allee effect} in population dynamics, but does not seem to correspond clearly to any model proposed in e.g. biology or chemistry. On the other hand, \eqref{eq} has challenging aspects from the mathematical point of view. Our results
may be extended to the multidimensional case, leading to a more
technical setting. We have chosen to stick to the one-dimensional
case to simplify the presentation, thus highlighting the main mechanisms.
Associated with \eqref{eq} is the following energy
\begin{equation}
\label{def-energie}
\mathcal E [u](t):=\frac 12 \int _\R (\partial _x u)^{2}(t,x)dx+\int _\R \frac \lambda 2 u^{2}\(1-\ln \(u^2\) \)(t,x)dx.
\end{equation}
Formally, solutions to \eqref{eq} satisfy
$$
\frac{d\mathcal E [u]}{dt}=-\int _\R (\partial _t u)^{2}(t,x)dx \leq 0.
$$
Many features make \eqref{eq} interesting from a mathematical
point of view. First, the nonlinearity is not Lipschitzean, which
causes difficulties already at the level of the local Cauchy
problem. Also, the second term in the energy \eqref{def-energie} has
no definite sign, which makes a priori estimates a delicate issue. Next, \eqref{eq} supports the
existence of Gaussian solutions. Last, the Cauchy problem may exhibit superexponential growth or decay.
\subsection{The Cauchy problem}
Such a logarithmic nonlinearity has been introduced in Physics in the
context of wave mechanics and optics \cite{BiMy75,BiMy76}. From a mathematical
point of view, the Cauchy problem for logarithmic Schr\"odinger
equations and logarithmic wave equations have been studied in
\cite{CaHa80,CaGa-p}: in the case of the logarithmic Schr\"odinger
equation, it is shown that
a unique, global weak solution can be constructed in a subset of $H^1$
(in any space dimension), whichever the sign of $\lambda$. For the
three dimensional wave equation and a suitable sign for $\lambda$, a
similar result is available. Due to the lack of regularity of the
nonlinearity, solutions are constructed by compactness methods, and
uniqueness is a rather unexpected property: in the case of
Schr\"odinger equation, it is a consequence of an elegant estimate in complex
analysis noticed in \cite{CaHa80}, while for the three dimensional
wave equation, it follows from fine properties of the wave equation
and a general result concerning the trace (see \cite{CaHa80} or
\cite{Ha81}).
\smallbreak
In the context of the heat equation like \eqref{eq}, the presence of a
logarithmic nonlinearity has been considered in \cite{ChLuLi15}, in
the case of a bounded domain $\Omega$, with Dirichlet boundary conditions. They
construct global solutions in $H^1_0(\Omega)$, and exhibit some
classes of solution growing or decaying (at least) exponentially in
time, thanks to variational arguments (potential well method). On the other hand, it seems very
delicate, if possible, to
construct a solution to \eqref{eq} by compactness methods on the whole line
$\R$. Also, uniqueness is missing in the Cauchy theory developed in
\cite{ChLuLi15}. We will see that this issue can be overcome by
changing functional spaces in which the Cauchy problem is studied.
\begin{defi}[Notion of solution]\label{def:sol-log}
Let $u_0$ be continuous and bounded, with continuous and bounded
derivative, and bounded and piecewise continuous second
derivative. A (global) solution to \eqref{eq} starting from $u_0$ is a
function $u:[0,\infty)\times \R\to \R$ which is continuous and bounded on
$[0,T]\times \R$, for which $u_t$, $u_x$ and $u_{xx}$ exist and
are continuous on $(0,T]\times \R$, such that $u(t,x)$ solves \eqref{eq} on $(0,T]\times \R$ (for any $T>0$), and $u_{\mid t=0}=u_0$. In addition, we require that
$u(t,x)$ is uniformly bounded as $|x|\to \infty$ for $t\in [0,T]$.
\end{defi}
\begin{prop}[Global well-posedness for \eqref{eq}]\label{prop:cauchy}
Let $u_0\geqslant 0$ be as in Definition ~\ref{def:sol-log}. Then \eqref{eq} has a unique solution $u$ starting from $u_0$, in
the sense of Definition~\ref{def:sol-log}.
\end{prop}
\subsection{Superexponential growth vs. decay}
In
\cite{JiYiCa16}, the presence of the logarithmic nonlinearity is
motivated as some limiting case for a nonlinearity of the form
$\lambda u^{1+\varepsilon}$ in the limit $\varepsilon\to 0$ (in the growth regime $u\to \infty$), still in a bounded domain
$\Omega\subset \R^N$ with Dirichlet boundary conditions. The authors show in
particular that time periodic solutions are highly unstable, in the
sense that a small perturbation of the initial data can lead to double
exponential growth or double exponential decay in time, see \cite[Theorem 1.1]{JiYiCa16}. Some of our results are qualitatively similar (superexponential
growth or decay), with the Gaussian steady state \eqref{def-varphi}
acting as a separation comparable to the time periodic solutions in
\cite{JiYiCa16}. Nevertheless, let us mention two main
differences. First, our results are valid on the whole line $\R$, and
not (only) on a bounded domain. On the other hand, we provide in Section \ref{s:data} initial data leading to superexponential growth or decay but that cannot be handled by \cite[Theorem 1.1]{JiYiCa16}. Roughly speaking, as can be seen from the proof, initial data of \cite[Theorem 1.1]{JiYiCa16} are multiples of the separating time periodic solution --- which is comparable to the present Remark \ref{rem:dichotomy}--- whereas initial data in Section \ref{s:data} are allowed to \lq\lq cross'' the separating Gaussian steady state \eqref{def-varphi} (see also Corollary \ref{cor:comp-1}).
\smallbreak
Let us recall that, in his seminal work \cite{Fuj-66}, Fujita considered solutions
$u(t,x)$ to the nonlinear heat equation
\begin{equation}
\label{eq-fujita}
\partial _t u=\Delta u+u^{1+p},\quad t>0,\; x\in \R^{N},
\end{equation}
supplemented with a nonnegative and nontrivial initial data. For $p>0$
solutions of the underlying ordinary differential equation (ODE)
problem --- namely $\frac{dn}{dt}=n^{1+p}$, $n(0)=n_0>0$ --- blow up in
finite time. The dynamics of the partial differential equation
\eqref{eq-fujita} is more complex
and rich. Precisely, there is a critical exponent $p_F:=\frac 2 N$,
referred to as the {\it Fujita exponent}, such that: If $0<p\leq p_F$
then any solution blows up in finite time, like those of the ODE. On the
other hand, if $p>p_F$ there is a balance between diffusion and
reaction. Solutions with large initial data blow up in finite time
whereas solutions with small initial data are global in time and go
extinct as $t\to\infty$. Those facts are proved in \cite{Fuj-66},
except the critical case $p=p_F$ which is studied in \cite{Hay-73}
when $N=1,2$, in \cite{Kob-Sir-Tun-77} when $N\geq 3$, and in
\cite{Wei-81} via a direct and simpler approach.
Concerning equation \eqref{eq}, the underlying ODE problem
\begin{equation}
\label{ode}
\frac{dn}{dt}=2\lambda n \ln n, \quad n(0)=n_0>0,
\end{equation}
is globally solved as
\begin{equation}
\label{sol-ode}
n(t)=e^{(\ln n_0)e^{2\lambda t}}.
\end{equation}
As $t\to \infty$, $n(t)\to 0$ if $0<n_0<1$ (extinction) whereas
$n(t)\to \infty$ if $n_0>1$ (blow up in infinite time). Hence the
dynamics of the ODE \eqref{ode} already shares some similarities with
the Fujita supercritical regime $p>p_F$ for the PDE
\eqref{eq-fujita}. The dynamics of the PDE \eqref{eq} is much richer
than the mechanism of \eqref{ode}, and our main goal is to understand its long time behavior for initial data $u_0$ ``crossing'' the equilibrium 1.
Notice also that the composition of exponential functions in
\eqref{sol-ode} is a strong indication that possible extinction or
growth phenomena are strong, and can thus hardly be
captured numerically. In practice, the superexponential growth may
appear like a blow-up phenomenon, while the superexponential decay may
be understood like a finite time extinction.
When possible, a second goal is to estimate
these rates of convergence.
\medskip
To give a flavor of the results established in the sequel, recall that the authors in \cite{ChLuLi15} consider
\eqref{eq} (possibly in multidimension) on a bounded domain, with
Dirichlet boundary conditions. By variational arguments, they exhibit
classes of initial data whose evolution under \eqref{eq} leads to (at
least) exponential decay in $L^2$, and another class of initial data
whose evolution leads to unboundedness of the $L^2$ norm in large
time. As a consequence of our analysis on the whole line $\R$, we actually provide more precise information on those phenomena for the equation on a bounded domain, say $(\alpha,\beta)$.
\begin{prop}[Growth/decay rates in a bounded domain]\label{prop:Omega}
Let $\alpha<\beta$ and $\Omega=(\alpha,\beta)$. Consider the mixed
problem
\begin{equation}\label{eq:Omega}
\left\{
\begin{aligned}
{\partial}_t u &= {\partial}_{xx}u + u\ln (u^2),\quad t>0,\ x\in \Omega,\\
u_{\mid {\partial} \Omega}&=0,\quad t>0,\\
u_{\mid t=0}&=u_0.
\end{aligned}
\right.
\end{equation}
There are nonnegative initial data $u_0\in C^1_c(\Omega)$ such that
\eqref{eq:Omega} has a unique solution, whose $L^2$ and $L^\infty$ norms decay at
least like a double exponential in time,
\begin{equation*}
\exists C,\eta>0,\quad \|u(t)\|_{L^2(\Omega)}\leqslant
|\Omega|^{1/2}\|u(t)\|_{L^\infty(\Omega)}\leqslant C e^{-\eta e^{2t}}.
\end{equation*}
There are nonnegative initial data $u_0\in C^1_c(\Omega)$ such that
\eqref{eq:Omega} has a unique solution, whose $L^2$ and $L^\infty$ norms grow at
least like a double exponential in time,
\begin{equation*}
\exists C,\eta>0,\quad |\Omega|^{1/2}\|u(t)\|_{L^\infty(\Omega)}
\geqslant \|u(t)\|_{L^2(\Omega)}\geqslant C e^{\eta e^{2t}}.
\end{equation*}
\end{prop}
\subsection{Changing the sign of the nonlinearity}
Let us observe that, for $\lambda >0$, the problem
\begin{equation}
\label{eq-moins}
\partial _t u=\partial _{xx} u -2\lambda\, u \ln u, \quad t>0,\; x\in \R,
\end{equation}
is of different nature. Indeed for the underlying ODE,
$$
n'(t)=-2\lambda n(t)\ln n(t),
$$
the equilibrium 0 is (very) unstable,
while 1 is stable. Hence, by the comparison principle, solutions are
{\it a priori} bounded between 0 and $\max (1,\Vert u_0\Vert
_{L^\infty})$. Moreover, by comparison with Fisher-KPP equations, much
can be said on the long time behavior of the Cauchy problem. For
instance, consider a nontrivial compactly supported initial data
$0\leq u_0\leq 1$. For any $r>0$, we can construct
$$
g_r:[0,1]\to \R \text{ concave with } g_r>0 \text{ on } (0,1), \; g_r(0)=g_r(1), \; r=g'_r(0)>0>g'_r(1),
$$
which is referred as to a Fisher-KPP nonlinearity, and such that
$g_r(u)\leq -2\lambda u \ln u$. By the comparison principle, we deduce
that $u_r(t,x)\leq u(t,x)\leq 1$, where $u_r$ is the solution of
$$
\partial _t u_r=\partial _{xx} u_r +g_r(u_r),
$$
starting from $u_0$. But it is known \cite{Aro-Wei-78} that the
spreading speed of this Fisher-KPP equation, with compactly supported
data, is $c_r^*:=2\sqrt{g_r'(0)}=2\sqrt r$, meaning that
$$
\text{ if } c>c_r^* \text{ then } u_r(t,x)\to 0 \text{ uniformly in } \{\vert x\vert \geq ct\} \text{ as } t\to \infty,
$$
$$
\text{ if } c<c_r^* \text{ then } u_r(t,x)\to 1 \text{ uniformly in } \{\vert x\vert \leq ct\} \text{ as } t\to \infty.
$$
Since this is true for any $r>0$ we get that
$$
\text{ for any } c>0,\, u(t,x)\to 1 \text{ uniformly in } \{\vert x\vert \leq ct\} \text{ as } t\to \infty,
$$
that is convergence to 1 with a superlinear speed.
\medskip
The organization of the paper is as follows. In Section
\ref{s:steady}, we enquire on steady states, proving existence of a
unique (Gaussian) nontrivial one. The well-posedness of the Cauchy
problem is established in Section~\ref{s:cauchy}. The long time behavior
(superexponential growth, decay or convergence to the steady state) is
studied in Section~\ref{s:gauss} (Gaussian initial data and
consequences), and Section~\ref{s:data} (more general data and
consequences).
\section{Steady states}\label{s:steady}
It is readily checked that the only constant steady states of
\eqref{eq} are $u\equiv 0$ and $u\equiv 1$.
\begin{prop}[Steady state]\label{prop:steady} There is a unique (up to
translation) nonnegative nontrivial steady state $\varphi$
solving \eqref{eq} and satisfying $\varphi(\pm \infty)=0$. It is the
Gaussian given by
\begin{equation}
\label{def-varphi}
\varphi(x)=e^{\frac 1 2}e^{-\frac \lambda 2 x^2}.
\end{equation}
\end{prop}
\begin{proof} Let $u=u(x)\geq 0$ be a nontrivial solution to \eqref{eq}, that is
\begin{equation}
\label{eq-steady}
u''(x)+2\lambda u(x)\ln u(x)=0, \quad \forall x \in \R,
\end{equation}
with $u(\pm \infty)=0$. If $u(x_0)=0$ for some $x_0\in\R$ then $u\equiv 0$ from the strong maximum principle. Hence $u>0$. Next, we multiply the equation by $u'$, integrate and infer that there is $C\in \R$ such that
$$
\({u'}\)^2(x)+ \lambda u^2(x)(2\ln u(x)-1)=C, \quad \forall x\in \R.
$$
{}From the above identity and since $u(\pm\infty)=0$, we deduce that
$u'(\pm \infty)$ must exist in $\R$ and, thus, be equal to 0
(otherwise we cannot have $u(\pm \infty)=0$). Hence $C=0$ and
\begin{equation}
\label{eq-steady-int}
\({u'}\)^2(x)=\lambda u^2(x)(1-2\ln u(x)), \quad \forall x\in \R.
\end{equation}
If $x\mapsto 1-2\ln u(x)$ never vanishes then this identity implies
that $u'$ has a constant sign, which contradicts $u(\pm
\infty)=0$. Hence, there exists $x_0\in \R$ such that $2\ln
u(x_0)=1$, and thus $u'(x_0)=0$ (from \eqref{eq-steady-int}),
$u''(x_0)<0$ (from \eqref{eq-steady}). In the sequel, we work
on $[x_0,+\infty)$, the arguments being similar on
$(-\infty,x_0]$.
\smallbreak
Assume that there is $x_1> x_0$ such that
$u'(x_1)=0$. From \eqref{eq-steady-int}, $2\ln u(x_1)=1$, and there
must be a point $x^{*}\in(x_0,x_1)$ where $u$ reaches a minimum
strictly smaller than $e^{\frac 1 2}$, which contradicts
\eqref{eq-steady-int}. Hence $u'<0$ on $(x_0,+\infty)$. It therefore
follows from \eqref{eq-steady-int} that $-u'(x)=\sqrt \lambda
u(x)\sqrt{1-2\ln u(x)}$ for $x\geq x_0$. Separating variables we get
$$
-\sqrt \lambda (x-x_0)=\int _{u(x_0)}^{u(x)}\frac{du}{u\sqrt{1-2\ln u}}=-\sqrt{1-2\ln u(x)},
$$
since $u(x_0)=e^{\frac 12}$. We end up with $u(x)=e^{\frac
12}e^{-\frac \lambda 2 (x-x_0)^{2}}$, which completes the proof.
\end{proof}
\section{Cauchy problem}
\label{s:cauchy}
As emphasized in the introduction, the Cauchy problem associated to
\eqref{eq} is not trivial, for two reasons:
\begin{itemize}
\item Local well-posedness: the nonlinearity is not Lipschitzean.
\item Global well-posedness: the potential energy in
\eqref{def-energie} has no definite sign.
\end{itemize}
The first aspect implies that constructing a solution certainly requires
compactness arguments, and uniqueness is not granted. The second
aspect shows that to have a solution defined for all $t\geqslant 0$, it may
be helpful that the first step yields this property ``for free''. This
is the strategy adopted in \cite{ChLuLi15}, where, on a bounded domain
$\Omega$, with Dirichlet boundary conditions, the authors construct a
solution in $H^1_0(\Omega)$ by Galerkin approximation. However,
uniqueness is not established in this case.
\medskip
In this section, we prove Proposition~\ref{prop:cauchy},
by showing that it fits perfectly into the framework of the PhD thesis
of J.~C.~Meyer \cite{Me13}.
Instead of working in spaces where the energy \eqref{def-energie}
is well defined, we adopt the approach of \cite{Me13}, see also \cite{MeNe15}. Consider more generally
the Cauchy problem
\begin{equation}
\label{eq:meyer}
u_t=u_{xx}+f(u),\quad 0<t\leq T,\; x\in \R,\quad u_{\mid t=0}=u_0,
\end{equation}
so we can emphasize which are the suitable assumptions of the
nonlinearity $f$ described in \cite{Me13}. Notice that, in \cite{Me13,MeNe15}, the standard examples, motivated by models
from Chemistry, are of the form $f(u) = \pm(u^p)^+$, $0<p<1$, and
$f(u)=(u^p)^+ ((1-u)^q)^+$, $0<p,q<1$.
The generalization of Definition~\ref{def:sol-log}, as introduced in
\cite{Me13}, is the following.
\begin{defi}[Notion of solution]\label{def:sol}
Let $u_0$ be continuous and bounded, with continuous and bounded
derivative, and bounded and piecewise continuous second
derivative. A solution to \eqref{eq:meyer} is a
function $u:[0,T]\times \R\to \R$ which is continuous and bounded on
$[0,T]\times \R$, for which $u_t$, $u_x$ and $u_{xx}$ exist and
are continuous on $(0,T]\times \R$, such that $u(t,x)$ satisfies
\eqref{eq}. In addition, we require that
$u(t,x)$ is uniformly bounded as $|x|\to \infty$ for $t\in [0,T]$.
\end{defi}
\begin{nota}
Following \cite{Me13,MeNe15}, we denote by ${\rm BPC}^2(\R)$ the set
of such initial data.
\end{nota}
Two notions are crucial, and correspond exactly to the type of logarithmic
nonlinearity considered in the present paper.
\begin{defi}[H\"older continuity]
Let $\alpha\in (0,1)$. A function $f:\R\to \R$ is said to be $\alpha$-\emph{H\"older continuous} if for any closed bounded interval
$E\subset \R$, there exists a constant $k_E>0$ such that for all
$x,y\in E$,
\begin{equation*}
|f(x)-f(y)|\leqslant k_E|x-y|^\alpha.
\end{equation*}
\end{defi}
A notion weaker than the standard notion of Lipschitz continuity turns
out to be rather interesting, as we will see below.
\begin{defi}[Upper Lipschitz continuity]
A function $f:\R\to \R$ is said to be \emph{upper Lipschitz
continuous} if $f$ is continuous, and for any closed bounded
interval $E\subset \R$, there exists a constant $k_E>0$ such that
for all $x,y\in E$, with $y\geqslant x$,
\begin{equation*}
f(y)-f(x)\leqslant k_E (y-x).
\end{equation*}
\end{defi}
Essentially, this property suffices to have a comparison principle,
hence a uniqueness result for \eqref{eq:meyer}.
\begin{example}
In the case of \eqref{eq}, $f(u) = \lambda u \ln(u^2)$. First, $f$ is
$\alpha$-H\"older continuous for any $\alpha\in (0,1)$. Indeed, for $y>x>0$, we have
$$
\vert f(y)-f(x)\vert=2\lambda \left\vert (y-x)\ln y +x\ln\left(1+\frac{y-x}{x}\right)\right\vert\leq 2\lambda \vert y-x\vert (\vert \ln y\vert +1),
$$
so that $\frac{\vert f(y)-f(x)\vert}{\vert y-x\vert^{\alpha}}\leq 2\lambda \vert y-x\vert ^{1-\alpha} (\vert \ln y\vert +1)\leq 2\lambda \vert y\vert ^{1-\alpha}(\vert \ln y\vert +1)$, which remains bounded as $y\to 0$. On the other
hand, even though $f$ is not Lipschitz continuous, we check that for
$\lambda>0$ (the case of interest in the present paper), $f$ is
\emph{upper} Lipschitz continuous. Indeed, for $x,y\in E$ bounded,
with $y>x>0$, Taylor formula yields
\begin{align*}
f(y)-f(x)
& =(y-x) \int_0^1f'\(x+\theta(y-x)\)d\theta\\
&=2\lambda(y-x)\int_0^1(1+\ln)
\(x+\theta(y-x)\)d\theta\\
&\leqslant 2\lambda(y-x) 2\lambda \(1+\sup_{z\in E}\ln z\).
\end{align*}
The last factor remains bounded as $x\to 0$. It would not be if the infimum was
considered: $f$ is not Lipschitz continuous.
\end{example}
\begin{defi}[Sub- and super-solutions]
Let $\underline u, \overline u:[0,T]\times \R$ be continuous on
$[0,T]\times \R$ and such that $\underline u_t,\underline
u_x,\underline u_{xx}, \overline u_t,\overline u_x,\overline u_{xx}$
exist and are continuous on $(0,T]\times\R$. If
\begin{align*}
&\underline u_t-\underline u_{xx}-f\(\underline u\)\leqslant 0\leqslant
\overline u_t-\overline u_{xx}-f\(\overline u\),\quad 0<t\leq T, \; x\in \R,\\
& \underline u(0,x)\leqslant u_0(x)\leqslant \overline u(0,x),\quad \forall x\in \R,
\end{align*}
and $\underline u, \overline u$ are uniformly bounded as $|x|\to
\infty$ for $t\in [0,T]$, then $\underline u$ is called a
\emph{regular sub-solution}, and $\overline u$ is called a
\emph{regular super-solution} to \eqref{eq:meyer}.
\end{defi}
\begin{theo}[Comparison; Theorem~7.1 from
\cite{Me13,MeNe15}]\label{theo:comparaison}
Let $f$ be upper Lipschitz continuous. If $\underline u$ and
$\overline u$ and regular sub and super-solutions on $[0,T]\times\R$, respectively,
then
\begin{equation*}
\underline u(t,x)\leqslant \overline u(t,x),\quad \forall (t,x)\in
[0,T]\times\R.
\end{equation*}
\end{theo}
\begin{theo}[Uniqueness; Theorem~7.2 from
\cite{Me13,MeNe15}]\label{theo:unicite}
Let $f$ be upper Lipschitz continuous. Then, for any $T>0$, \eqref{eq:meyer} has at
most one solution in $[0,T]\times\R$.
\end{theo}
The following statement is a slight modification from the original,
where we add a uniqueness assumption to simplify the presentation.
\begin{theo}[Existence; Theorem~8.1 and Corollary~8.6 from
\cite{Me13,MeNe15}]\label{theo:existence}
Suppose that $f$ is $\alpha$-H\"older continuous for some $\alpha\in
(0,1)$, and let $u_0\in {\rm BPC}^2(\R)$. Suppose that uniqueness
holds for
\eqref{eq:meyer}. Then \eqref{eq:meyer} has a (unique) solution
$u:[0,T^*[\times \R$. In addition, either $T^*=\infty$, or
$\|u(t,\cdot)\|_{L^\infty(\R)}$ is unbounded as $t\to T^*$.
\end{theo}
\begin{proof}[Proof of Proposition~\ref{prop:cauchy}] As emphasized above, the nonlinearity in \eqref{eq} is
both $\alpha$-H\"older continuous (for any $\alpha\in (0,1)$) and
upper Lipschitz continuous. Therefore, Theorem~\ref{theo:unicite}
implies uniqueness, and Theorem~\ref{theo:existence} yields a
(unique) maximal solution $u\in C([0,T^*)\times\R)$.
We conclude
by showing that the
solution is global ($T^*=\infty$)
thanks to a suitable a priori estimate. The solution of the ODE \eqref{ode} starting from $\Vert u_0\Vert _{L^{\infty}}$, namely
\begin{equation*}
\overline u(t) = e^{\ln \Vert u_0\Vert
_{L^{\infty}}e^{2\lambda t}}.
\end{equation*}
is a super-solution, while the zero function is obviously a
sub-solution. Theorem~\ref{theo:comparaison} implies that
\begin{equation*}
0\leqslant u(t,x)\leqslant \overline u(t),\quad \forall t\in [0,T^*).
\end{equation*}
We conclude that $T^*=\infty$, and the result follows.
\end{proof}
\begin{cor}[Initial data comparable to 1]\label{cor:comp-1}
Let $u_0 \in {\rm BPC}^2(\R)$, $u_0\geqslant 0$, and $\varepsilon\in (0,1)$.
\begin{itemize}
\item If $u_0(x)\geqslant 1+\varepsilon$ for all $x\in \R$, then $u$ grows at
least like a double exponential in time:
\begin{equation*}
u(t,x) \geqslant e^{\ln(1+\varepsilon)e^{2\lambda t}},\quad \forall t\geqslant 0, \
\forall x\in \R.
\end{equation*}
\item If $u_0(x)\leqslant 1-\varepsilon$ for all $x\in \R$, then $u$ decays at
least like a double exponential in time:
\begin{equation*}
u(t,x) \leqslant e^{\ln(1-\varepsilon)e^{2\lambda t}},\quad \forall t\geqslant 0,\
\forall x\in \R.
\end{equation*}
\end{itemize}
\end{cor}
\begin{proof}
This corollary is a straightforward consequence of
Proposition~\ref{prop:cauchy}, the comparison
principle (Theorem~\ref{theo:comparaison}), and the ODE case
\eqref{ode}--\eqref{sol-ode}.
\end{proof}
\section{Large time behavior: Gaussian data}\label{s:gauss}
Families of Gaussian solutions for nonlinear (and nonlocal) equations can be found in
\cite{B14}, \cite{Alf-Car-14, Alf-Car-17}, in the context of
evolutionary genetics. In the case of a logarithmic nonlinearity, for
the Schr\"odinger equation, it
was observed in \cite{BiMy76} that the flow preserves the Gaussian
structures, and so the resolution of the partial differential equation
boils down to the resolution of ordinary differential equations; see
\cite{CaGa-p} for more details. It is not surprising that the same
holds in the case of \eqref{eq}, and we have indeed:
\begin{prop}[Gaussian solutions]\label{prop:gauss} Let $b_0>0$ and $a_0>0$ be given. The solution of \eqref{eq} starting from the Gaussian
\begin{equation}
\label{gauss-u0}
u_0(x)=b_0 e^{-\frac{a_0}{2}x^2},
\end{equation}
is the Gaussian given by
\begin{equation}
\label{gauss-u}
u(t,x)=b(t)e^{-\frac{a(t)}{2}x^2}:=e^{\psi (t)e^{2\lambda t}}e^{-\frac{a(t)}{2}x^2},
\end{equation}
where
\begin{equation}
\label{psi(t)}
\psi(t)=\ln b_0-\frac{a_0}2\frac{\ln \lambda -\ln (a_0+(\lambda-a_0)e^{-2\lambda t})}{\lambda -a_0},
\end{equation}
with the natural continuation $\psi(t)=\ln b_0 -\frac 1
2(1-e^{-2\lambda t})$ if $a_0=\lambda$, and
\begin{equation}
\label{a(t)}
a(t)=\lambda \frac{a_0 e^{2\lambda t}}{\lambda -a_0+a_0e^{2\lambda t}}.
\end{equation}
\end{prop}
\begin{proof}
We plug the ansatz \eqref{gauss-u} into equation \eqref{eq}, we identify the $x^0$ and the $x^2$ coefficients to obtain two ordinary differential equations. The first one is the logistic equation
$$
a'(t)=2a(t)(\lambda -a(t)),
$$
whose solution, starting from $a(0)=a_0$, is given by \eqref{a(t)}. The second one is
$$
b'(t)=2\lambda b(t)\ln b(t)-a(t)b(t).
$$
Denoting $\phi(t):=\ln b(t)$ the above is recast
$$
\phi'(t)=2\lambda \phi(t)-a(t),
$$
whose solution, starting from $\phi(0)=\ln b_0$, is
\begin{equation}
\label{phi(t)}
\phi(t)=\left(\ln b_0 -\int _0^t e^{-2\lambda s}a(s)ds\right)e^{2\lambda t}.
\end{equation}
Next, using \eqref{a(t)} we get
\begin{align*}
\int _0^t e^{-2\lambda s}a(s)ds&= \int _0 ^{t} \frac{\lambda a_0 e^{-2\lambda s}}{a_0+(\lambda -a_0)e^{-2\lambda s}}ds
\\
&=\left\{
\begin{aligned}
&\frac{a_0}{-2(\lambda -a_0)}\left(\ln\left(a_0+(\lambda
-a_0)e^{-2\lambda t}\right)-\ln \lambda\right)\quad \text{ if
} a_0\neq \lambda \\
& \frac 1 2 (1-e^{-2\lambda t}) \quad \text{ if } a_0=\lambda,
\end{aligned}
\right.
\end{align*}
which we plug into \eqref{phi(t)} to get \eqref{psi(t)}.
\end{proof}
Clearly, the sign of $\psi_\infty:=\lim _{t\to \infty}\psi(t)$
decides between (superexponential) decay and growth of the Cauchy
problem starting from a Gaussian data, the critical case $\psi
_\infty=0$ leading to convergence to the steady state.
\begin{cor}[Gaussian data: three scenarii]\label{cor:supersonic-gauss} Let $b_0>0$ and $a_0>0$ be given. Define
\begin{equation}
\label{psi-infini}
\psi_\infty:=\ln b_0 -\frac{a_0}{2}\frac{\ln \lambda - \ln a_0}{\lambda -a_0},
\end{equation}
with the natural continuation $\psi _\infty= \ln b_0-\frac 12$ if $a_0=\lambda$.
Denote by $u(t,x)$ the Gaussian solution of Proposition \ref{prop:gauss}.
\begin{itemize}
\item [(i)] If $\psi _\infty <0$, then there is superexponential
decay in the sense that
$$
\Vert u(t,\cdot)\Vert _{L^{\infty}}=\max _{x\in \R} u(t,x) \Eq t\infty
e^{\frac 12}e^{\psi_\infty e^{2\lambda t}}, \quad \Vert
u(t,\cdot)\Vert _{L^1}\Eq t \infty \sqrt{\frac{2\pi}{\lambda}}e^{\frac 1 2} e^{\psi_\infty e^{2\lambda t}}.
$$
\item [(ii)] If $\psi _\infty =0$, then there is convergence to the steady state of Proposition~\ref{prop:steady} in the sense that
$$
\Vert u(t,\cdot)-\varphi \Vert _{L^{\infty}} +\Vert u(t,\cdot)-\varphi \Vert _{L^{1}} \Tend t \infty 0.
$$
\item [(iii)] If $\psi _\infty >0$, then there is superexponential growth in the sense that, for all $R>0$,
$$
\min _{\vert x\vert \leq R} u(t,x) \Eq t\infty e^{\frac
12}e^{\psi_\infty e^{2\lambda t}}e^{-\frac{\lambda}{2}R^{2}}, \quad
\Vert u(t,\cdot)\Vert _{L^1}\Eq t \infty \sqrt{\frac{2\pi}{\lambda}} e^{\frac 12}e^{\psi_\infty e^{2\lambda t}}.
$$
\end{itemize}
\end{cor}
\begin{proof}
One just has to use the asymptotic expansion
$\psi(t)=\psi_\infty +\frac 1 2 e^{-2\lambda t}+\mathcal O(e^{-4\lambda t})$ as $t\to \infty$, and perform straightforward estimates.
\end{proof}
In view of the comparison principle (Theorem~\ref{theo:comparaison})
and of Corollary~\ref{cor:supersonic-gauss}, we infer:
\begin{cor}[Initial data comparable to a Gaussian]\label{cor:comp-gauss}
Let $u_0 \in {\rm BPC}^2(\R)$, $u_0\geqslant 0$, and $\varepsilon\in (0,1)$. Let
$a_0,b_0>0$, and denote again
\begin{equation*}
\psi_\infty=\ln b_0 -\frac{a_0}{2}\frac{\ln \lambda - \ln a_0}{\lambda -a_0},
\end{equation*}
with the natural continuation $\psi _\infty= \ln b_0-\frac 12$ if $a_0=\lambda$.
\begin{itemize}
\item If $\psi_\infty<0$ and $u_0(x)\leqslant b_0e^{-a_0x^2/2}$, then $u$
decays at least like a double exponential in time,
$$
\Vert u(t,\cdot)\Vert _{L^{\infty}}\leqslant 2 e^{\frac
12}e^{\psi_\infty e^{2\lambda t}}, \quad \Vert u(t,\cdot)\Vert
_{L^1}\leqslant \sqrt{\frac{4\pi}{\lambda}}e^{\frac 1 2} e^{\psi_\infty
e^{2\lambda t}}, \quad \text{ as } t\to\infty.
$$
\item If $\psi_\infty>0$ and $u_0(x)\geqslant b_0e^{-a_0x^2/2}$, then $u$
grows locally at least like a double exponential in time: for all $R>0$,
$$
\min _{\vert x\vert \leq R} u(t,x) \geqslant \frac{1}{2} e^{\frac
12}e^{\psi_\infty e^{2\lambda t}}e^{-\frac{\lambda}{2}R^{2}}, \quad
\Vert u(t,\cdot)\Vert _{L^1}\geqslant \sqrt{\frac{\pi}{\lambda}} e^{\frac
12}e^{\psi_\infty e^{2\lambda t}}, \quad \text{ as } t\to\infty.
$$
\end{itemize}
\end{cor}
\begin{rem}\label{rem:dichotomy}
Observe that, if
$\psi_\infty=\ln b_0 -\frac{a_0}{2}\frac{\ln \lambda - \ln
a_0}{\lambda -a_0}=0$, then initial data $(1-\varepsilon)b_0e^{-a_0x^2/2}$
($0<\varepsilon<1$), and $(1+\varepsilon)b_0e^{-a_0x^2/2}$ ($\varepsilon>0)$, fall into the regime
$\psi_\infty<0$ (decay), and $\psi_\infty>0$ (growth),
respectively. Typical examples are $(1-\varepsilon)\varphi(x)$,
$(1+\varepsilon)\varphi(x)$, where $\varphi(x)=e^{\frac12}e^{-\lambda x^2/2}$
is the steady state from Proposition~\ref{prop:steady}.
\end{rem}
\section{Large time behavior in the case of more general data}\label{s:data}
We have seen that the comparison with the constant initial datum equal
to one leads to a strong dichotomy (Corollary~\ref{cor:comp-1}). The
same is true by comparison with an initial Gaussian, leading to a
larger variety of initial data (Corollary~\ref{cor:comp-gauss} and
Remark~\ref{rem:dichotomy}).
Now, we enquire on initial data that can be compared neither to 1 nor
to a Gaussian. In this direction, we can prove superexponential decay
for small initial data. To do so, we need the following standard
estimate, which stems from Young inequality applied to the formula
\begin{equation*}
e^{t{\partial}_{xx}}v_0(x) = \frac{1}{\sqrt{4\pi t}}\int_{\R}e^{-(x-y)^2/(4t)}v_0(y)dy.
\end{equation*}
\begin{lem}
\label{lem:heat} For any initial data $v_0\in L^1(\R )\cap
L^\infty(\R)$, the solution of the Cauchy problem $\partial _t
v=\partial _{xx}v$, $v_{\mid t=0}=v_0$, satisfies
$$
\Vert v(t,\cdot)\Vert _{L^\infty}\leq V(t):=\min\left(\Vert v_0\Vert _{L^\infty},\frac{\Vert v_0\Vert _{L^1}}{\sqrt{4\pi t}}\right), \quad \text{ for any } t\geq 0.
$$
\end{lem}
\begin{theo}[Superexponential decay for small data]
\label{th:donnee-pte} For a nonnegative initial datum $u_0$ in ${\rm
BPC}^2(\R)$, define $m_\infty:=\Vert u_0\Vert _{L^\infty}$ and
$m_1:=\Vert u_0\Vert _{L^1}$. Assume
\begin{equation}
\label{donnee-pte}
\begin{aligned}
\psi_\infty^*:=&\ln m_\infty -\int _{\tau }^{+\infty} \lambda
e^{-2\lambda s}\ln s \, ds+e^{-2\lambda \tau }\ln \sqrt \tau<0,\\
&\quad \text{ where } \tau:=\left(\frac{m_1}{\sqrt{4\pi}\, m_\infty}\right)^{2}.
\end{aligned}
\end{equation}
Then the solution of \eqref{eq}, starting from $u_0$, is decaying
superexponentially in the sense that
\begin{equation}
\label{u-donnee-pte}
0\leq u(t,x)\leq \min\left(m_\infty,\frac{m_1}{\sqrt{4\pi t}}\right) e^{\psi(t)e^{2\lambda t}},
\end{equation}
where $\psi(t)\to \psi _\infty^*<0$ as $t\to \infty$.
\end{theo}
\begin{rem} The above criterion provides new initial data leading to
superexponential decay. For instance, assume that $u_0$ has tails heavier than
Gaussian (so that domination by a Gaussian cannot be used), that
\begin{equation}\label{etoile}
1<m_\infty<e^{\int _1^{+\infty} \lambda e^{-2\lambda s}\ln s \, ds},
\end{equation}
(so that domination by the ODE cannot be used), and that
$\tau=1$. Then \eqref{donnee-pte} holds true, hence \eqref{u-donnee-pte}. A typical example could be
\begin{equation*}
u_0(x)= m_\infty e^{-\alpha\(\sqrt{1+x^2}-1\)},
\end{equation*}
with \eqref{etoile} and $\alpha>0$ adjusted so that
$\tau=1$.
\end{rem}
\begin{proof} Following \cite{Gar-Qui-10} or \cite{Alf-fujita}, we
look for a supersolution to \eqref{eq} in the form $g(t)v(t,x)$,
where $g(t)>0$ is to be determined (with $g(0)=1$), and $v(t,x)$ is
the solution of the heat equation $\partial _t v=\partial _{xx}v$
with $u_0$ as initial datum. A straightforward computation shows
that to construct a supersolution,
it is enough to have
$$
\frac{g'(t)}{g(t)}-2\lambda \ln g(t)\geq 2\lambda \ln v(t,x).
$$
By Lemma \ref{lem:heat}, it is therefore enough to have $g(t)=e^{\phi(t)}$ where
$$
\phi'(t)-2\lambda \phi (t)=2\lambda \ln V(t), \quad \phi(0)=0,
$$
that is
$$
\phi(t)=e^{2\lambda t}\int _0 ^{t}2\lambda e^{-2\lambda s}\ln V(s)ds.
$$
Observe that $V(t)=m_\infty$ when $t\leq \tau$ while
$V(t)=\frac{m_1}{\sqrt{4\pi t}}$ when $t\geq \tau$. Cutting the above
integral and performing straightforward computations, we end up with
$g(t)=e^{\psi(t)e^{2\lambda t}}$, where
$$
\psi(t):=(\ln m_\infty) (1-e^{-2\lambda \tau})+\ln \frac{m_1}{\sqrt{4\pi}}(e^{-2\lambda \tau}-e^{-2\lambda t})-\int _\tau ^{t}\lambda e^{-2\lambda s}\ln s \, ds,
$$
which tends to $\psi _\infty^*$ as $t\to \infty$. It therefore follows
from the comparison principle (Theorem~\ref{theo:comparaison}) that
$u(t,x)\leq g(t)v(t,x)$, which
yields \eqref{u-donnee-pte}.
\end{proof}
In the context of bounded solutions, typically for Lipschitz {\it
ignition} or {\it bistable} nonlinearities, some threshold results
between extinction and convergence to an equilibrium, say 1, are known
to exist \cite{Zla-06}, \cite{Du-Mat-10}, \cite{Mat-Pol-16}. For
equation \eqref{eq}, we can prove a threshold result between decay and
growth. We first need to construct compactly supported sub-solutions.
\begin{lem}[High plateaux as sub-solutions]\label{lem:plateaux} Let $L>0$ and $0<\varepsilon<L$ be given. Let $\Theta \in C^\infty([-L,L])\cap {\rm BPC}^2(\R)$ be
such that
\begin{align*}
&\Theta(x)=0\text{ for }|x|\geqslant L,\\
& \Theta >0\text{ on }(-L,L),\\
& \Theta(\pm L)=\Theta '(\pm L)=\Theta ''(\pm L)=0,\\
& \gamma:=\Theta'''(-L)=-\Theta ''' (L)>0,\\
& \Theta \equiv 1 \text{ on }[-L+\varepsilon,L-\varepsilon].
\end{align*}
Then there is $K_0>1$
such that, for any $K\geq K_0$, the function $\Theta _K:=K\Theta$
satisfies
\begin{equation}\label{sub-plateau}
\Theta _K'' +2\lambda \Theta _K \ln \Theta _K \geq 0,\text{ on }\R.
\end{equation}
Hence, $\Theta _K$ is a sub-solution to \eqref{eq}.
\end{lem}
\begin{proof}
By our assumption $\Theta (x)\sim \frac 1 6 \gamma (x+L)^3$,
$\Theta''(x)\sim \gamma (x+L)$ for $ 0<x+L\ll 1$, where we
thus have
$$
\Theta _K''(x) +2\lambda \Theta _K(x) \ln \Theta _K(x) \geq K(\Theta''(x)+2\lambda \Theta (x)\ln \Theta (x))\sim K\gamma (x+L).
$$
As result, there is $\delta >0$ such that \eqref{sub-plateau} holds
on $(-L,-L+\delta)$ and, by symmetry, on $(L-\delta,L)$.
Next, denoting $\theta ^*:=\min _{-L+\delta \leq x\leq L -\delta}\Theta (x)>0$, we have, for $x\in [-L+\delta,L-\delta]$,
\begin{align*}
\Theta _K''(x) +2\lambda \Theta _K(x) \ln \Theta _K(x)&= K(\Theta''(x)+2\lambda \Theta (x)\ln \Theta (x)+2\lambda \Theta (x) \ln K)\\
&\geq K(-\Vert \Theta''+2\lambda \Theta \ln \Theta \Vert _{L^{\infty}}+2\lambda \theta ^{*}\ln K)
\end{align*}
which is nonnegative if $K>1$ is large enough.
\end{proof}
\begin{theo}[Threshold phenomena for compactly supported
data]\label{th:threshold}
Let $0<\varepsilon<L<L'$ be given. Select $K\geq
K_0>1$, where $K_0$ is given by Lemma~\ref{lem:plateaux}.
Let $u_0\in {\rm BPC}^2(\R)$ be such that $u_0>0$ on $(-L',L')$ and $u_0\equiv
0$ on $(-\infty,-L']\cup[L',+\infty)$. For $M>0$, we denote by
$u_M(t,x)$ the solution of \eqref{eq} starting from $Mu_0$.
\begin{itemize}
\item [(i)] There is $M_{\rm decay}>0$ such that, for any $0<M<M_{\rm
decay}$, the solution $u_M(t,x)$ is decaying superexponentially in time.
\item [(ii)] There is $M_{\rm growth}>0$ such that, for any $M>M_{\rm
growth}$, the solution $u_M(t,x)$ grows locally superexponentially
in time, in the sense that
\begin{equation}
\label{local-blowup}
u_M(t,x)\geq Ke^{\left(\ln \frac{K+1}{K}\right)e^{2\lambda t}}, \quad
\forall x\in [-L+\varepsilon,L-\varepsilon].
\end{equation}
\end{itemize}
\end{theo}
\begin{proof} The first point is a consequence of
Corollary~\ref{cor:comp-1}, that is comparison with the ODE, provided we choose $M_{\rm
decay}=1/\|u_0\|_{L^\infty}$. We now prove $(ii)$. Since $\min
_{-L\leq x\leq L} u_0(x)>0$, there is $M_{\rm growth}>0$ such that, for
all $M\geq M_{\rm growth}$, $Mu_0\geq (K+1) \Theta$. Next we take $n(t)$
as the solution of the underlying ODE starting from
$n_0:=\frac{K+1}{K}$, see \eqref{ode} and \eqref{sol-ode}, that is
$$
n(t)=e^{\left(\ln \frac{K+1}{K}\right)e^{2\lambda t}}.
$$
Now we define
$$
w(t,x):=\Theta _K(x)n(t)=K\Theta (x) n(t).
$$
We have $w(0,\cdot)=(K+1)\Theta \leq Mu_0$ and
\begin{align*}
&\partial_t w(t,x)-\partial _{xx}w(t,x)-2\lambda w(t,x)\ln w(t,x)\\
&= \Theta _K (x)n'(t)-\Theta _K ''(x)n(t)-2\lambda \Theta _K (x) n(t)\ln n(t) -2\lambda \Theta _K (x) n(t)\ln \Theta _K (x)\\
&= n(t)(-\Theta _K ''(x)-2\lambda \Theta _K(x)\ln \Theta _k(x))\\
&\leq 0,
\end{align*}
by Lemma \ref{lem:plateaux}. It therefore follows from the comparison principle that $u_M\geq w$. In particular, since $\Theta \equiv 1$ on $[-L+\varepsilon,L+\varepsilon]$, we get \eqref{local-blowup}. \end{proof}
Proposition~\ref{prop:Omega}, concerned with a bounded domain, is a straightforward consequence of the
above result. One just has to use a translation in space $c\neq 0$ if necessary so that $c+[-L',L']\subset
(\alpha,\beta)$, and note that the quantities involved in
Theorem~\ref{th:threshold} control the $L^2$ norm on $\Omega$.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 4,181
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Widmo masowe – widmo powstałe poprzez rozdzielenie strumienia jonów według stosunku ich masy (atomowej lub cząsteczkowej) m do ładunku elektrycznego z.
Na podstawie widm masowych ustalono procentowe zawartości poszczególnych izotopów w pierwiastkach, a także precyzyjne wyznaczono masy jąder atomowych.
Przypisy
Spektrometria mas
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 567
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Q: "Winter Bash 2022" or "Winter Summer Bash 2022"? The countdown for this year's Winter Bash (2022) is started.
This time on the countdown page, the image is displayed as "Winter Summer Bash 2022".
So shall we call it "Winter Bash 2022" or "Winter Summer Bash 2022"?
Also instead of the tag winter-bash-2022, shall we use winter-summer-bash-2022?
Screenshot from the countdown page:
A:
Also instead of the tag winter-bash-2022, shall we use winter-summer-bash-2022?
There isn't any need for one or the other. Make one a synonym of the other.
I suggest winter-summer-bash-2022 be the main one if that is the name of the event. It is also more inclusive to people in the Southern Hemisphere.
A: Since it is an annual event around the globe, why not call it Annual Bash, Stack Bash, or simply Bash, and forsake the seasonal label altogether?
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,798
|
Prevue Hendryx Signature Series Large Bird Cage - Colorful Options!
The Prevue Hendryx Signature Series Large Bird Cage is ideal for Conures, Senegals and similarly-sized parrots.
Provide sizable birds with space they need by selecting the wrought iron Signature Series Large Bird Cage from Prevue Hendryx, one of the oldest manufacturers of quality pet cages and accessories.
Sleek and stylish - features include top and bottom pull-out drawers, four stainless steel cups, bottom pull-out grille, rounded seed guards, top playpen, two wooden perches, and a durable cage stand featuring casters for optimal movement from room to room.
This wonderful enclosure is available in five great colors.
The cage also includes a heavy-duty push button lock for total pet safety, meaning birds can't get out and predator pets can't get in. Depending on the model, your birds will enjoy either hoop or flat bar ladders.
Optimal species for this cage include African Grey, Amazon, Aracari, Caique, Cockatiel, Conure, Eclectus, Pionus and Rosella. Provide your birds with space they want with this stylish option!
The cage is capable of accommodating up to three birds, and does not include a storage shelf or a divider.
Shipping cost: Free standard shipping within mainland USA.
Allow 2 busines days for processing and 1 week for delivery.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,769
|
Q: User Interface functionality modelling languages? I am looking for a UI functionality modelling language (UML-alike "thing", but for user interfaces) that is already accepted and maybe has its design patterns and handles the problem better than state or activity diagram.
This question came to mind as a result of a discovery that UML and its diagrams fails at describing complicated UI functionality with event-driven flow of execution (ie. javascript/jQuery big projects)
EDIT:
I've been thinking of using BPMN but It was not created for this purpose.
A: Maybe user interface prototypes or storyboards can be helpful ...
they are not part of a "modelling language" but very well proved techniques for designing GUI ...
A: One thing that comes to mind is Jesse James Garrett's Visual Vocabulary for Information Architecture.
A: I don't think there are any standards out there for that specific purpose (I was thinking of the same thing the other day). SysML is close, I think, although it is definitely overkill.
Mainly, my thought was that if a UML profile or metamodel is defined with core UI components and events ("text-field", "single-click", etc.), different UI implementations (HTML, Swing, AJAX) could be generated using transformations on an instance model's XMI. Barring that, at least there would be a more clear and formal way of describing the functionality of a given UI.
A: You can use traditional modeling notations for UI modeling, but this soon ends up with messy and useless models.
You should think to domain-specific models like WebML (soon to become a OMG standard under the name of IFML). In this case you also get a free modeling tool called WebRatio that provides quick prototyping and integration with BPMN specifications.
[Disclaimer: I'm with Politecnico di Milano and WebRatio, and among the inventors of WebML/IFML]
A: I just stumbled onto your question and it's an issue I take seriously. Here's my answer to that question, and I've used it in various forms for over 20 years.
Basically, here are the criteria I seek in such a descriptive language:
*
*The language should not be watered-down and incapable of things like access to data or basic flow-of-control primitives like IF, FOR, and subroutine calls. I accomplish this by building the language on top of the underlying standard language, by means of macros and function definitions. Thus, it requires no parser or interpreter, has direct access to application data, and has the control-flow primitives of the underlying language (but only some of them).
The reason for including flow-control primitives in the descriptive language is for their descriptive utility. The IF(test)-ELSE-END construct is a way of saying that one of two sets of controls is to be displayed depending on the value of (test). The FOR - END construct is a way of saying that a collection of controls to be displayed in a multiplicity, such as a linear array of controls. These can be nested to get a 2-D matrix of controls, if desired. A subroutine (with parameters) can display a set of controls, and can then be invoked in multiple places to replicate that set multiple times. Without such primitives in a DSL, structures like that are difficult to specify.
*The language should not require the person specifying the UI to have to deal with issues that only exist for implementation's sake, such as input event handling, creation, deletion, and naming of controls, and movement of data between controls and application data. So, for example, each edit, button, or other control is a single line of code. Code to handle events such as a button-click is written directly adjacent to the line of code specifying the button (not in a separate function or closure). Binding of controls to application data is handled "underneath" and is not the concern of the UI programmer. In order to make all this work, it is based on a control structure called Differential Execution that I stumbled upon in 1986. It is based on incremental re-execution of a program in such a way that it can incrementally update its output. In this case the output is a collection of controls on the surface of a window. Those controls are automatically created, deleted, moved, or otherwise modified in response to changes in the application state, without the UI programmer having to think in terms of state changes.
I have only used this in desktop UIs, and I have done almost no web development. I'm pretty sure the same principles could be applied to web UIs, and that it hasn't yet been done.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,805
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My parents built my childhood home, the house my father still lives in today, in the early 1970's, for just over $30,000. My grandfather convinced my parents that a fourth bedroom wasn't worth the extra money, a decision that turned out to be ill-informed when I made my surprise appearance a few years after my two sisters. He redeemed himself when I was a toddler, by paying for the addition of an in-ground pool in the backyard. That pool came to define our summers. Days, weeks, and months were spent playing sharks and minnows and agonizing over the 15 minute wait to get back in the pool after each meal, swim lessons were held there for all the neighborhood kids, and countless bbqs were thrown together on a whim, with my mom firmly at the helm. For a city girl, she thrived in her yard and by the pool---both of which required a staggering amount of work, as my sisters and I are finding out years later. She weeded and edged, power-washed, and for her pièce de résistance, she mowed the lawn in her bathing suit and bare feet, as evidenced by the color of her feet all season long. She never had a good explanation for her mowing uniform, beyond It's hot out! What do you want from me?, but told us years later that it was the only time she had to herself when we were little. It wasn't all work and no play, though. As we swam away our days, my mom entertained neighbors and friends with gin wedges and an endless supply of potato salad, melon, and veggie platters, making it seem as though they just appeared out of the ether. Her open door policy was known throughout the neighborhood and beyond---what would start as a small gathering inevitably became, in her words, a cast of thousands.
My mom was famous for her bright red geraniums, transplanted from large hanging plants and placed in pots around the pool. The years she tried something different---begonias, dahlias, petunias even --- were busts, and she always went back to her beloved geraniums. She surveyed those flowers daily, methodically---and, if you knew my mom, without remorse---getting rid of dead blooms with a flick of her wrist. There is an area of the yard, behind one corner of the fence surrounding the pool, that courtesy of my cousin became known as the "Geranium Graveyard," where the dead blooms went to spend their final days. It is only fitting that we plan to place some of my mom's ashes there, forever memorializing the spot. In the first few days following my mom's death, those geraniums came up in conversation several times. Family and friends wanted to make sure that my sisters and I would still plant them; no one could imagine the backyard without those pops of red.
We all chipped in to open the pool this year---my sisters and I, along with our significant others and my dad, with the help of neighbors who have themselves swam in the pool since childhood. My sisters took charge of the geraniums, and the good news is that all but two of the plants are surviving their first summer without my mom's care. The unfortunate ones are victims of my dad's valiant effort to water them using chlorinated pool water.
There have been barbeques and gatherings already this summer, and the youngest generation now whiles away their endless summer days in the pool, just as we did a lifetime ago. To celebrate the 4th, we invited friends and family over for what felt like just another Brady barbeque, but with me in charge instead of my mom. I grocery shopped, I straightened the house, and I made burgers, salads and snacks, all the while cursing my father and husband, who were relaxing and playing golf, respectively. How did my mom do it all those years?---this was the question I asked repeatedly throughout the day. But deep down, I already knew the answer. She did it because it was more important to bring family and friends together than to lounge by the pool; she did it because it was always a few good laughs; she did it because she didn't know how not to do it. It's a burden and a blessing, this legacy of ours, but I don't have time to worry about that. I'm busy planning our next party.
On (New) Marriage and the Ever-Elusive "Home"
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{
"redpajama_set_name": "RedPajamaC4"
}
| 4,087
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Q: TypeError using isin I try to run the following code.
path="path"
df=pd.read_csv(path, header=0, delimiter=',',dtype=str)
df=df.loc[~df["x"].isin(df["y"]),:]
It yields the following error
TypeError: '>' not supported between instances of 'str' and 'float'
After a bit of investigation I found out that the dtype is in fact an object. Could this be causing the error ?
Is there a way to fix this without having to itterate over 7 columns 3 million rows to fix the datatype ?
The data looks like:
x y z
DE120 UK354 4506
UK354 AT235 9783
FE560 DK645 4652
IT456 NL978 7831
I want to exclude all obsevations where x is also in y. (z is just 1 of the 5 other columns that is otherwise completely irrelevant to this story, just want you to know it's there)
x y z
DE120 UK354 4506
FE560 DK645 4652
IT456 NL978 7831
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,181
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{"url":"https:\/\/vru.vibrationresearch.com\/lesson\/aliasing\/","text":"# Aliasing\n\nMarch 29, 2018\n\nDifferent sine waves can have identical sample values for a fixed sample rate. This means, in general, the frequency value of an FFT result is not unique. This is illustrated with a sample rate of 2,000Hz in Figure 2.11. The digital samples of the two sine waves with different frequencies are identical. For the FFT computation, the higher frequency looks like the lower frequency, thus the origin of the name \u201caliasing.\u201d\n\nFigure 2.11. Aliasing of digitally-sampled sine waves at a sample rate of 2,000Hz.\n\n### Low-pass Filtering\n\nTo avoid the ambiguity in the FFT, the lowest frequency values are assumed. For a given sample rate, these frequencies lie between 0 and SR\/2 = fq. Therefore, all frequencies above the Nyquist frequency need to be removed.\n\nNormally, the frequencies are removed by passing the signal through an analog low-pass filter before the signal is converted to digital samples by an A\/D convertor. A typical low-pass filter begins attenuating the signal at about 80% of the nominal cut-off frequency, so most systems provide good data for frequencies up to f\u00a0= 0.4 SR.\n\nNote that the aliasing occurs as a mirror image about the Nyquist frequency (and odd number multiples). The frequency fa\u00a0=\u00a0fq\u00a0+\u00a0df\u00a0 looks the same as the frequency\u00a0fb\u00a0=\u00a0fq\u00a0\u2013\u00a0df\u00a0 (with 0 <\u00a0df\u00a0<\u00a0f\u00a0q\u00a0) for a sample rate SR = 2\u00a0fq.\u00a0 As does\u00a0 fc\u00a0= 3\u00a0fq\u00a0\u2013\u00a0df, fd\u00a0= 3\u00a0fq\u00a0+\u00a0df, and so on for all odd-number multiples of fq.","date":"2020-06-06 19:56:13","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8233619928359985, \"perplexity\": 760.7821692436769}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-24\/segments\/1590348519531.94\/warc\/CC-MAIN-20200606190934-20200606220934-00034.warc.gz\"}"}
| null | null |
Edna McCarter
By Mark Darnell | November 22, 2022
Edna Maurell Davis McCarter was born on November 15, 1929, in Clay County, Tennessee, to the late Parlie Davis and Matilda Susan Browning Davis. She departed this life on November 13, 2022, just two days before her ninety-third birthday, making her earthly stay Ninety-two years, eleven months, and twenty-nine days.
Edna married James Carson McCarter on July 21, 1945, and they were blessed with ten children and fifty-three years together before his passing on December 12, 1998. Along with her parents and dear husband, Edna was also preceded in death by her Son, Jimmie Dudney McCarter; Daughters, Pamela Kay McCarter, Mary Ruth Spivey and LaDora Jean Carter; Grandsons, Howard Christopher McCarter, Eric Glen LaFevers and Donnie Ray Marsh; Brothers, Odell Davis, Mizell Davis, Shelba Davis, and her twin, Leslie Coell Davis; Sisters, Estelle Russell, Lorene Shoulders, Cornell Clark, Elise Deckard and Beatrice Ritter; and Sons-in-Law, Ralph Cassetty and Jerry Morgan.
Edna was a member of Drapers Cross Roads Church of Christ. She worked as a sewing machine operator for over fifty-three years, beginning her career at Imperial Reading and finally retiring from Formfit Rogers. After retiring from the garment industry, Edna became a caregiver, assisting many families in the care of their loved ones. In her spare time, Edna loved sewing, quilting, and embroidery. She enjoyed growing flowers as well.
Edna Davis McCarter is survived by; Sons, Howard McCarter and wife, Carol, and Mark McCarter and wife, Betty; Daughters, Patricia Cassetty, Sharon Morgan, Shelta Shrum and husband, Gary, and Renita Marsh; Sons-in-law, Jimmy Spivey and Tommy Carter; Daughter-in-law, Janice McCarter; Grandchildren, Kelly McCarter, Blake Spivey, Corey Spivey, Seth Marsh, Jamie McCarter, Joshua McCarter, Matthew Morgan, Zackary Marsh, Natasha Roberts, Stephanie Lamb, Laura Kendall, and Cecilia Cash; Grandsons by Heart, David Roberts and Alex Coimbre. Eighteen Great-Grandchildren, Two Great-Great-Grandchildren, and Numerous Nieces and Nephews also survive.
Funeral services for Ms. Edna Davis McCarter were conducted on Tuesday, November 15, 2022, from The Chapel of Alexander Funeral Home with Jack Honeycutt officiating. Interment followed in The Leonard Cemetery. Pallbearers were her grandsons. Alexander Funeral Home, Directors, in charge of arrangements. 615-666-2189 or www.alexanderfh.com
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\section{Introduction}
In the hamiltonian formulation of General Relativity
coupled to matter fields, the Euler-Lagrange equations
split into 6 dynamical equations plus the equations for
the matter fields and 4 constraints which express the
invariance of the theory under local changes of coordinates.
Therefore, when quantizing this system in a gauge
invariant manner, i.e. \`a la Dirac, the ``wave function''
must be annihilated by these four constraints that are
now operator valued.
This clear structure dictated by the reparametrization invariance
turns out to lead to very complicated problems concerning the
interpretation of this ``wave function''.
By interpretation, one designates the problem of extracting
(probabilistic) predictions concerning the evolution
of matter and gravity configurations.
The origin of the difficulties can be blamed on the absence
of time and the accompanying unitary evolution on which
quantum mechanics is based\cite{isham}\cite{hartle}\cite{Hal}\cite{kiefer2}\cite{vil}.
In order to describe more precisely some of these difficulties,
from now on we shall pursue the discussion in mini-superspace.
This drastic restriction to homogeneous and isotropic
three-geometries offers the double advantage of removing the
U.V. problem that plagues the local theory while keeping
the problem concerning the interpretation of the wave function.
In this restricted configuration space,
gravity is described by the scale factor $a$, matter by homogeneous
fields that we shall denoted collectively by $\phi$ and the wave
$\Psi(a, \phi)$ is constrained to satisfy a single global
Wheeler De Witt (WDW) equation.
The questions
are the following:
How to proceed to read from $\Psi(a, \phi)$ predictions concerning
quantum events such as,
for instance, transitions rates among the $\phi$
fields ?
and: Are those transitions described by a unitary evolution ?
Both questions have received attention and
many different schemes
have been proposed\cite{isham}.
Let us mention here only the scheme based on the
hypothesis that $\vert \Psi(a, \phi) \vert^2$ does posses
a probabilistic interpretation\cite{hartle} (at least in terms of conditional
probabilities\cite{isham})
and the scheme based on the conserved current\cite{vil}
$\Psi^*i\!\!\lr{\partial_{a}}\!\! \Psi$.
The answers to the second question are
partially related with this choice
and range from, ``Yes, the evolution is unitary\cite{vent}\cite{kim}'',
to ``no, unitarity is violated\cite{kiefer2}'',
and includes the middle attitude: ``unitarity is only approximatively conserved
\cite{vil}''.
The peculiar aspect of these widely distributed answers is that they
arise from the same starting point:
the WDW equation and its solutions.
The disagreements build up with the choice of the
treatment
required to extract information from $\Psi(a, \phi)$.
In the present article, we clarify the mathematical
aspects of these treatments by analysing matter
interactions from the solutions of the WDW equation.
More precisely, by studying transitions, we identify the coefficient
${\cal{C}}_n(a)$ that weights the n-th state at $a$
and that replaces the amplitude $c_n(t)$ to be in the n-th
matter state at time $t$
in conventional quantum mechanics.
The unambiguous identification is based on two criteria:
1) In the absence of transitions, ${\cal{C}}_n(a)$ must be constant.
2) When one simplifies the equation governing their dependence in $a$
by treating gravity in the background field
approximation (BFA), the resulting equation must be the
Schroedinger equation.
This mathematical procedure of extracting
the ${\cal{C}}_n(a)$
is based on \cite{wdwgf}\cite{wdwpt}\cite{time}
and does not require an {\it a priori} interpretation of $\Psi ( a, \phi )$.
On the contrary, our program is first to determine
the properties of the coefficients ${\cal{C}}_n(a)$
and only then to examine the question of its interpretation
in the light of these
properties.
They
reveal the existence of three regimes
which are delineated by the values of the
parameters describing matter transitions in quantum cosmology.
In the case of weak interactions occurring close to
equilibrium in a macroscopic\cite{vil}\cite{BV},
the coefficients ${\cal{C}}_n(a)$ are equal to the
amplitudes $c_n(t)$, solutions of the corresponding
Schroedinger equation evaluated in the geometry described by
$a(t)$.
In the second regime, the
departure from equilibrium and/or the
importance of the
interactions lead to ${\cal{C}}_n(a)$ that no longer coincides
with $c_n(t)$. However when gravity is still correctly described by
WKB waves, the ${\cal{C}}_n(a)$ still satisfy
the ``unitary'' equation $\sum_n \vert {\cal{C}}_n(a) \vert^2 = 1$,
up to negligeable corrections.
In the third regime, the interactions are so violent that
the propagation of $a$ is affected by the quantum transition acts.
In that case, $\sum_n \vert {\cal{C}}_n(a) \vert^2 \neq 1$.
This ``violation'' is a direct manifestation
of the modification of the propagation of $a$ by the
transitions themselves. It is
also kinematically related to the conservation
of the current carried by $\Psi ( a, \phi )$.
Under these extreme conditions,
there is no possibility of defining a background.
Neither, therefore, should there be any possibility
of interpreting $\Psi (a, \phi )$ using the conventional rules
of quantum mechanics. This does not mean that no predictions
can be made, it simply means that the conventional
analysis cannot be performed {\it{in situ.}}
By
propagating $\Psi (a, \phi )$
outside this regime, one can then perfectly determine its
physical outcome.
In conclusion, in this article, we shall show
that when one requires that the conventional description
of matter transitions is recovered from
quantum cosmology,
the probabilistic interpretation of $\Psi (a, \phi )$ must be based
on its current and not on its norm.
We shall also show that the interpretation of ${\cal{C}}_n(a)$ as the amplitude
of probability to find the n-th matter state at $a$
is valid as long as the propagation of gravity
is not significantly affected by the interactions.
Both aspects have been put forward by Vilenkin in \cite{vil}.
However, to the knowledge of the author, they have never been
made as explicit as in the present paper. Furthermore,
in contradistinction to \cite{vil},
our small parameter is the coupling constant among the quantum systems
and not their energy.
This allows to reach more general conclusions.
\section{The identification, the evolution and the meaning
of the coefficients ${\cal{C}}_n(a)$}
As said in the Introduction, we shall use perturbation theory applied
to matter interactions as a guide to identify the coefficients
${\cal{C}}_n(a)$. Before accomplishing this program,
we briefly present the kinematical properties at work in Quantum Cosmology,
in the absence of these interactions, see \cite{wdwgf} for more details.
\vskip.5 truecm
{\bf{The coefficients in absence of interactions}}
\vskip.2 truecm
\noindent
For simplicity, the matter system is chosen in such a
way that the free hamiltonian $H^{0}_m(a)$ does not
lead to transitions. Thus, the general solution of the Schroedinger equation
is
\be
\chi(t, \phi) = \sum_n c_n e^{-i\!\int^t dt' E_n(t')} \scal{\phi}{{n}}
\label{onee}
\ee
where the eigenstates of the free hamiltonian $H^{0}_m(a)$
satisfy
\be
H^{0}_m(a(t)) \ket{n} = E_n(a(t)) \ket{n}
\label{one}
\ee
Their time dependence arises only through the
equation governing the background
propagation $a = a(t)$.
The coefficients $c_n$ are interpreted as the amplitudes
of probability to find the matter system in the n-th state.
Therefore, by convention, they are
normalized so that $\sum_n
\vert c_n \vert^2 =1$. Furthermore, in the present case, they are constant.
In this respect, notice that one must consider either interactions
with the external world, or self interactions, or non adiabatic
transitions\cite{wdwpc} in order to
give physical substance to
the probabilistic interpretation of
$c_n$.
Indeed, in the absence of interactions,
no interference among the $c_n$ will show up.
We shall now determine to what
extend these
properties are
recovered in quantum cosmology from the solutions of the
Wheeler De Witt equation.
In minisuperspace, when matter is characterized by an
energy $E_n(a)$, the gravitational part of the action
satisfies the Hamilton-Jacobi constraint
\ba
H_G(a) + E_n(a)=
{ -G^2 ( \partial_a S_G (a) )^2
+ \kappa a^2 + \Lambda a^4 \over 2 Ga } +
E_n(a) =0
\label{M12}
\ea
where $G$ is Newton's constant, $\kappa$
is equal to $\pm 1$ or $0$ for respectively open,
closed and flat three surfaces and $\Lambda $ is the
cosmological constant. The
solution of this equation
is simply $S_n (a)
= \int^a da' p_n(a')$ where the momentum of $a$
driven by the $E_n(a)$
\be
p_n(a) = \mp G^{-1} \sqrt{
\kappa a^2 + \Lambda a^4 + 2 Ga E_n(a) }
\label{M8}
\ee
where $\mp $ correspond respectively to expanding and
contracting universes.
Upon quantizing gravity, the Hamilton-Jacobi constraint
becomes the operator valued equation (the WDW equation)
\be
\left[
H_G(a) + H^0_m(a , \phi) \right] \Psi ( a, \phi ) =0
\label{wdw}
\ee
When the quantum matter states are characterized
by constants of motion,
$\Psi ( a, \phi )$ can be decomposed as
\be
\Psi ( a, \phi ) = \sum_{n} {\cal{C}}_{n}
\Psi ( a; n ) \scal{\phi}{{n}}
\label{M13}
\ee
where the n-th gravitational wave,
entangled\footnote{
This is the main reason for which we have chosen to work with
matter states characterized
by constants of motion. It allows an unambiguous
decomposition of the total wave in terms of products. This is
to be compared with the difficulties
to perform this decomposition in the general case wherein
no clear principle seems to exist,
see e.g. \cite{kiefer2} after eq. (2.35) ``We {\it{choose}} $D$ in such a way that
the equations become simple''.
Very important also is the fact that our decomposition
keeps the {\it linearity} of the WDW equation when used
in a perturbative treatment.} to the n-th matter state, is a solution of
\be
\left[
G^2
\partial^2 _a
+ \kappa a^2 + \Lambda a^4 + 2 Ga E_n (a) \right]
\Psi ( a; n ) =0
\label{M14}
\ee
This equation is second order in $\partial _a $ and has
therefore two independent solutions.
This has to be the case since classically we can work
either with expanding or contracting universes.
Indeed when using the WKB approximation
\be
\Psi ( a; n ) = { e^{ i\!\int^{a}
p_n(a') da' } \over \sqrt{2 \vert p_n(a) \vert }}
\label{M144}
\ee
one verifies that solutions with positive (negative)
wronskian
\be
W_n = \Psi^*( a; n)\
i\!\!\lr{\partial_{a}} \Psi ( a;n )
\label{M1444}
\ee
correspond to expanding (contracting)
universes.
At this stage, several remarks should be made.
Firstly, the decomposition of the general
solution of the WDW equation is performed by using the
same set of quantum numbers $n$ that the one
used in eq. (\ref{one}) wherein gravity was treated in the background
field approximation (BFA). The enlargement of the dynamics
to $a$ is compensated by the WDW constraint.
Secondly, it is now through the sign of the
Wronskian rather than at the classical level
that one now chooses expanding or contracting universes.
Thirdly, with our definition of
$\Psi ( a;n )$, the coefficients ${\cal{C}}_{n}$ are
constant.
Having recall these kinematical properties, we can
now formulate precisely our mathematical claim\cite{wdwgf}:
in order for ${\cal{C}}_{n} $
to satisfy the
Schroedinger equation
when gravity is treated in the BFA,
all Wronskians $W_n$
must be equal to the same constant.
This can be already guessed by considering the
{conserved} current\cite{vil} carried by $\Psi( a, \phi)$
\ba
J = \int d\phi \ \Psi^*( a, \phi)
i\!\!\lr{\partial_{a}} \Psi( a, \phi)
= \sum_{n} \vert {\cal{C}}_{n} \vert^2 \ W_n
\label{curr}
\ea
To work with both $J=1$ and unit Wronskians suggests
an identification
of ${\cal{C}}_{n}$ with $c_n$.
However, at this moment, no physically relevant
conclusion\footnote{
It is nevertheless interesting to compare the present treatment based
on the current $J$ to the one in which it is the norm
that determines probabilities.
In that latter case, by definition, the probability to be in the
n-th state is given by $\vert \scal{n}{\Psi( a, \phi)}
\vert^2 / \scal{\Psi}{\Psi} = \vert {\cal{C}}_{n} \vert^2 p^{-1}_n(a)
/ \sum_{m} \vert {\cal{C}}_{m} \vert^2
/p^{-1}_m(a)$, where we have used the WKB form for the
waves $\Psi( a ;n )$. This ``probability'' depends on $a$
through the $a$-dependent norm of each $\Psi( a; n )$,
a feature that I find unattractive. Notice
however, that in the doubled limit of {\it well grouped} states
living in a {\it macroscopic} universe, this dependence in $a$ vanishes,
see latter in the text for
more precision concerning these limits.} can be
made concerning a probabilistic interpretation of the ${\cal{C}}_{n}$.
Indeed it is mandatory
to consider self interactions leading to transitions
in order for the ${\cal{C}}_{n}$
to vary
and interfere, see the remark made after eq. (\ref{one}).
\vskip.5truecm
{\bf{The $a$-dependence of the ${\cal{C}}_{n}$}}
\vskip.2truecm
\noindent
We return for a moment to quantum mechanics
and consider the time
dependent perturbation theory
for allowing
a comparison with the corresponding equations
derived in quantum cosmology.
Upon introducing an interacting
hamiltonian $H_{int}$ possessing non vanishing matrix
elements $\expect{n}{H_{int}}{m}$, the time dependence of the
coefficients
$c_n$ is given by
\ba
i \partial_t c_n(t) = \sum_m
\expect{n}{H_{int}}{m}\ c_m(t) \
e^{-i \int^t dt' [E_m(t') - E_n(t')]}
\label{schr}
\ea
Since this equation is linear in $c_n$ and first order in $i \partial_t$,
when the hamiltonian $H_{int}$ is hermitian, one obtains
$\Sigma_n \vert c_{n}(t) \vert^2 = 1$,
a necessary condition to keep the probabilistic interpretation of
$\vert c_n(t)\vert^2$.
In quantum cosmology, the fact of taking into account
the
hamiltonian $H_{int}$
is quite different from what we just did in usual
quantum mechanics wherein one has an external
time parameter at our disposal.
Indeed,
$H_{int}$ modifies the propagation of
both gravity and matter through the modified WDW equation
given by
\be
\left[
H_G(a) + H^0_m(a , \phi) + H_{int}(a , \phi)
\right] \Xi ( a, \phi ) =0
\label{wdw'}
\ee
By decomposing the interacting wave $\Xi ( a, \phi ) $
in terms of the free components $\Psi ( a; n)$
\be
\Xi ( a, \phi ) = \sum_{m} {\cal{C}}_{m}(a) \
\Psi ( a; m ) \scal{\phi}{{m}}
\label{M13'}
\ee
and by projecting eq. (\ref{wdw'}) into the bra $\bra{n}$,
we obtain\footnote{Notice that this development does not coincide
with the Born-Oppenheimer treatment. What would be
closer to that treatment, would consist in working with
states which diagonalise the total hamiltonian
$H^0_m(a , \phi) + H_{int}(a , \phi)$ at fixed $a$.
Together with S. Massar, we shall present this adiabatic
treatment applied to quantum cosmology in \cite{wdwpc}.}
\be
{\partial_a \Psi ( a; n ) \over \Psi ( a ; n )}
\partial_a {\cal{C}}_{n}(a) + {1 \over 2} \partial^2_a {\cal{C}}_{n}(a)
+ {a \over G} \sum_{m} {\cal{C}}_{m}(a) \ \expect{n}{H_{int}}{m}
{ \Psi ( a ; m )\over \Psi ( a ; n )} = 0
\label{wdwg}
\ee
Since this equation is
second order in $\partial_a$, some analysis is required
in order to reveal the properties of the evolution it encodes.
To this end,
we shall first simplify it by making use of three approximations.
In the next subsection, we shall evaluate the errors they induce
on the basis of the analysis of \cite{wdwgf}\cite{wdwpt}.
The first approximation consists in using
the WKB approximation for $\Psi ( a; n)$.
Its validity requires $\partial_a p_n(a) / p^2_n(a) <\!\!< 1$,
an inequality which
is satisfied when the second condition is met.
This second condition requires that the universe be
macroscopic \cite{vil}\cite{BV},
i.e. that the matter sources driving gravity must be macroscopic.
By denoting $M$ the rest mass of the atoms and
$\Delta m$ the energy change induced by the transitions
engendered by $H_{int}$, the macroscopic limit
guarantees that $\Delta m /E_n \simeq \Delta m/ N_M M <\!\!< 1$
since $N_M$, the total number of atoms, satisfies $N_M >\!\!>1$.
Thirdly, we
require that the dimensionless constant $g^2$ that characterizes
the transition rates be smaller or comparable
to unity ($g$ is related to $H_{int}$
by $\expect{n}{H_{int}}{m} \simeq g \Delta m$).
By applying these three approximations to eq. (\ref{wdwg}),
one can drop the second term and write the two
other terms as
\be
i \partial_a {\cal{C}}_{n}(a) = \sum_{m}
{a \over G}{\expect{n}{H_{int}}{m} \over \sqrt{p_n(a) p_m(a)}}\
{\cal{C}}_{m}(a) \ e^{i\!\int^a da'[p_m(a')- p_n(a') ] }\
{\sqrt{W_m \over W_n}}
\label{niceeq}
\ee
This equation
deserves a few comments.
Firstly, as eqs. (\ref{wdw'}, \ref{wdwg}),
it is linear in ${\cal{C}}_{n}(a)$.
We point out this fact since
many approximation schemes\cite{kiefer2}\cite{vent}\cite{kim}\cite{BV},
as the semi-classical treatment,
destroy the linearity of the WDW equation and therefore
the superposition principle. The loss
of linearity in these treatments results
from a quantum averaging performed at an earlier stage.
Secondly, in order for eq. (\ref{niceeq}) to coincide with
eq. (\ref{schr}) in the background
field approximation, the Wronskians must satisfy ${W_m / W_n} =1$
for all $m, n$. The validity of the further simplification which consists in treating
eq. (\ref{niceeq}) in the BFA, requires that the ${\cal{C}}_{n}(a) \neq 0$
be grouped together such that their spread around the mean energy
$E_{\bar n}$ satisfies $(E_n - E_{\bar n})/E_{\bar n} <\!\!< 1$\cite{vil}\cite{BV}.
Only then can one correctly describe the evolution in terms of a
single
time parameter\cite{banks} defined, from the propagation of
$a$, by
\be
t_{\bar n}(a) = \int^a_{a_0} da' {a' \over G p_{\bar n}(a')}
\label{ta}
\ee
and develop the $a$-dependent phase of eq. (\ref{niceeq})
to first order in $E_m - E_n$ around $E_{\bar n}$, see \cite{wdwpt}.
Using
eq. (\ref{M8}) and $t_{\bar n}(a) $,
one finds, for an expanding universe,
\be
-\int^a_{a_0} da'(p_m(a')- p_n(a')) =
\int_0^{t_{\bar n}(a)} dt' (E_m(t')- E_n(t') ) + O((E_m - E_n)/ E_{\bar n})
\label{ta2}
\ee
Thirdly, when ${W_m / W_n} =1$, eq. (\ref{niceeq}) leads
to ``unitary'' evolution in the sense that $\Sigma_n \vert
{\cal{C}}_{n}(a) \vert^2 = 1$. We emphasize that this does not imply
that, starting with $ {\cal{C}}_{n}$ that coincide with $c_n$
at $a_0$ ($t=0),
{\cal{C}}_{n}(a)$ will evolve like $c_n(t(a))$. Indeed, as
shown in eq. (\ref{ta2}), the
phases of eq. (\ref{niceeq}) equal the corresponding phases
of eq. (\ref{schr}) only when developed to first order in $\Delta E $
and evaluated for the mean energy $E_{\bar n}$.
Therefore,
the non-linear terms will induce increasing additional phase
shifts.
(This is not particular to quantum cosmology.
Indeed, whenever the quantum dynamics is enlarged to a variable formerly
treated classically, non-linear phase shifts appear,
see \cite{rec}.)
Then, after a certain lapse of $t(a)$, the interferences amongst
the ${\cal{C}}_{n}(a)$ will posses no relation to those amongst the $c_n(t(a))$.
Moreover, remote configurations evolve with their own time\cite{vil},
see eq. (\ref{ta}) for the dependence on $E_n$ in $t_n$.
Fourthly, eq. (\ref{niceeq}) might have been obtained by
``first quantizing'' the wave $\Xi(a, \phi)$. In that framework,
one
postulates that the fundamental equation
is
\be
-i \partial_a \Xi(a, \phi) = \sqrt{ H_0 (a, \phi) + H_{int}(a, \phi)} \ \Xi(a, \phi)
\label{firstq}
\ee
instead of eq. (\ref{wdw'}), see \cite{isham}.
Then, by developing the square root to first order in $ H_{int}$
one obtains eq. (\ref{niceeq}) exactly like eq. (\ref{schr}) is obtained\cite{wdwpt}.
The main weakness of this {\it ad hoc} approach is that there is no a priori
justification to eliminate half of the solutions of the Hamilton Jacobi
equation before quantization. Furthermore, whether or not it offers a good
approximation, the importance of the neglected terms cannot be estimated
without considering the solutions to eq. (\ref{wdw'}).
Finally, the interpretation
of the wave $\Xi ( a, \phi )$ as the conditional amplitude
of probability to find $\phi$ at $a$ is rejected by the present
analysis
of the
solutions of eq. (\ref{wdw'}).
Indeed, by adopting this interpretation,
one would obtain a non-linear equation
for the conditional amplitudes
since these are non linearly related to ${\cal{C}}_{n}(a)$
--recall the presence
of the normalisation factor $p_n^{-1/2}(a)$
stemming from current conservation, see the second footnote--.
Notice that this would not have been the case if one
would have worked with the solutions of eq. (\ref{firstq}).
Notice also that, in contrast to what is presented in
\cite{Hal}, our derivation of eqs. (\ref{wdwg}, \ref{niceeq})
which encode the correlations between matter and gravity
in no way requires that $\Xi(a, \phi)$ be peaked around certain
configurations. But it does require that gravity be modified
by the transitions, otherwise $a$ could not be used to parametrize
their amplitudes.
We now address the problem of the corrections to eq. (\ref{niceeq}).
\vskip .5 truecm
{\bf The $a$-dependence of $\Sigma_n \vert
{\cal{C}}_{n}(a) \vert^2$}
\vskip .2 truecm
\noindent
Our aim is to determine how the terms neglected in passing from
eq. (\ref{wdw'}) to eq. (\ref{niceeq}) affect $\Sigma_n \vert
{\cal{C}}_{n}(a) \vert^2$.
This is most easily achieved by using the {\it exact} conservation
law of the current carried by $\Xi ( a, \phi )$.
Indeed the current carried by $\Xi ( a, \phi )$ is
\ba
J &=& \int d\phi \left( \Xi^*( a, \phi)
i\!\!\lr{\partial_{a}} \Xi( a, \phi) \right)
\nonumber \\
&=& \sum_{n} \vert {\cal{C}}_{n} \vert^2 + \sum_{n} \left( {\cal{C}}_{n}^* (a)
i\!\!\lr{\partial_{a}} {\cal{C}}_{n} (a)\right)
\vert \Psi (a; n) \vert^2 = 1
\label{curr'}
\ea
We first notice that there is no
systematic departure from unity. Indeed, when
the interactions cease to act, due for instance to some
cooling in an expanding universe, the final value of $\Sigma_n \vert
{\cal{C}}_{n}(a) \vert^2 $ coincides with the initial one.
Before discussing the reasons that lead to the
local departure from unity,
we estimate the importance of this departure. To this end,
it is sufficient to use eq. (\ref{niceeq}) and the
WKB expression for the free waves $\Psi(a, \phi)$. One should also
further specify the characteristics of the solution under
examination. As an example, in the case of a matter dominated
universe composed of $N_M$ ``heavy'' atoms of mass $M$
having transitions between inner states characterized by
energy gaps $\Delta m$,
this correction is
\ba
\sum_{n} \left( {\cal{C}}_{n}^*
i\!\!\lr{\partial_{a}} {\cal{C}}_{n} \right)
\vert \Psi (a; n) \vert^2 \simeq
2 \sum_{n} \sum_{m} {\cal{C}}_{m } {\cal{C}}_{n}^* {a \expect{n}{H_{int}}{m}
\over G p_n(a) p_m(a)}
\simeq g {a N_M^{1/2 }\Delta m \over G p^2_n(a)}
\simeq g { \Delta m \over M } {1 \over N_M^{1/2 } }
\label{depart}
\ea
far from a turning point\cite{km} and at equilibrium.
(The incoherence of the transitions
close or at equilibrium brings the factor
$N_M^{1/2 }$ in the second approximation.)
The departure from unity is therefore small for three independent reasons: the
weakness of the transitions rates controlled by $g$, the
smallness of the $\Delta m / M$ which allows to neglect the recoil
of the atoms induced by the transitions and the macroscopic character
of the universe $ N_M >\!\!> 1$.
The mechanism that leads to the
departure from unity is clear: when the ${\cal{C}}_{n}$
depend on $a$, they carry some current and therefore their norm
should vary accordingly.
Indeed, when the ``potential term'' of
a second order differential equation varies,
the induced modifications of the norm and the phase of the solutions
are correlated since the Wronskian is conserved.
The physical origin of
these correlated modifications
comes from the fact that the WDW imposes
that the kinetic energy of gravity be
modified by the presence of $H_{int}$, see
eq. (\ref{wdw'}).
This in turn modifies the norm of the solution.
Therefore, $\Sigma_n \vert {\cal{C}}_{n}(a) \vert^2 \neq 1$
is a manifestation of the modification of the
propagation of gravity induced by the interactions
themselves.
To obtain explicitly the relation between the
changes in phase and in amplitude, it is instructive
to consider the simple case wherein the interactions
induce a diagonal level shift $E_n(a) \to E_n(a) + \delta_n(a)$.
Using eq. (\ref{M8}) in eq. (\ref{M144}), one immediately gets the desired
relation through the change of the $p_n(a)$.
Upon considering matter transitions, this modification of $p_n(a)$
has the following physical meaning.
The transition probability is given by an integral over $a$ whose
norm is $a da/p_n(a)=dt_n$, see \cite{wdwpt}. Therefore,
the modification of $p_n(a)$ is
{\it necessary} in that it guarantees
that the Golden Rule is obtained, i.e. that
the transition probability increases linearly with the proper time lapse,
eq. (\ref{ta}),
evaluated in the universe wherein the transition occurs.
In view of this, it is inviting to work with eigenstates of
the total matter hamiltonian $H_0 + H_{int}$ rather than to work
in perturbation theory with the free states. Indeed, when working with
these eigenstates,
one automatically includes
the backreaction of the (adiabatic part of the)
interaction hamiltonian in the definition of
the WKB momentum $p_n(a)$.
Since the induced modification of $p_n(a)$
will no longer be found, its contribution
to $\Sigma_n \vert {\cal{C}}_{n}(a) \vert^2 \neq 1$ will be also
absent.
This strongly suggests not to interpret
$\Sigma_n \vert {\cal{C}}_{n}(a) \vert^2 \neq 1$
as a violation of unitarity since the magnitude of the
departure from unity depends
on the perturbative scheme adopted.
Together with S. Massar, we shall return to these aspects
in \cite{wdwpc}.
\vskip .4 truecm
{\bf Conclusions and additional remarks}
\vskip .1 truecm
\noindent
In resume, our analysis of ${\cal{C}}_{n}(a)$ shows that there are three
regimes delineated by the values of the coupling constant $g$, the relative
transition energy $\Delta m/M$ and the macroscopic character of the universe,
here controlled by $N_M$.
In order to be in the first regime, the initial values ${\cal{C}}_{n}(a_0) \neq 0$
must be grouped together so that the ``mean'' time parameter, eq. (\ref{ta}),
correctly parametrizes the
evolution of the whole system.
One must also requires that the universe be macroscopic
and that the transitions be not too violent.
In that regime, ${\cal{C}}_{n}(a) = c_n(t_{\bar n}
(a))$. Therefore, ${\cal{C}}_{n}(a)$ {\it is} the amplitude of probability to
find the matter system in the n-th state at $a$.
In the second regime, the spread of ${\cal{C}}_{n}(a_0) \neq 0$
and/or the violence of the interactions and/or the ``smallness'' of the universe
leads to an evolution that cannot be obtained from a Schroedinger equation
based on a single time parameter. Nevertheless, when the
WKB approximation for describing the free
evolution of gravity is still valid, one still obtains a linear equation
for the propagation of the ${\cal{C}}_{n}(a)$
that guarantees that $\Sigma_n \vert {\cal{C}}_{n}(a) \vert^2 = 1$.
Moreover, upon considering a set of neighbouring ${\cal{C}}_{n}(a)$ for
a sufficiently small amount of time, the evolution of this restricted set
still obeys a Schroedinger equation.
Finally, as usually in quantum mechanics, remote configurations do not interfere.
Therefore, it is still perfectly correct to interpret
${\cal{C}}_{n}(a)$ as the amplitude of probability to
find the n-th state at $a$.
Notice however that the decoherence of the background time, that is the
fact that remote configurations evolve with their own time, would leave no
physical meaning to mean values of matter operators in which
non-interfering configurations would contribute. Only summations
over restricted sets of interfering configurations make physical sense.
In this we differ from \cite{hartle}\cite{Hal} since,
in these works, physical predictions are based on the peaks
of $\Xi(a, \phi)$ which include summation over all states.
In the third regime, the importance of the interactions leads to
backreaction effects on the propagation of gravity such that
$\Sigma_n \vert {\cal{C}}_{n}(a) \vert^2$ depends on $a$.
A part of this dependence comes from the (adiabatic\cite{wdwpc}) dressing energy
brought in by the interactions. It reflects the fact that
the relationship between proper time lapse and the momentum $p(a)$
has been modified by this dressing energy. Therefore the deviation from unity
it engenders should certainly not be interpreted as a violation of unitarity.
This deviation can be also understood from the necessity of applying a ``reduction
formula'' to the universe's wave function in order to obtain
transition amplitudes amongst its constituents, see \cite{wdwpt}.
Far more difficult to interpret are the consequences of the
corrections to the WKB approximation.
Indeed upon dealing with exact solutions for
the propagation of $a$, initially expanding (forward)
solutions contain, at other radii, backward waves
corresponding to contracting universes.
Moreover, the conservation of the Wronskian tells
us that the current of the forward part has increased.
The only way to confront this ``Klein paradox'' seems
to proceed to the so called third quantization\cite{isham}.
However, we want to emphasize that by adopting
this framework, one has not solve the question of extracting transition amplitudes
occurring {\it within} expanding universes.
Indeed, additional choices
must be specified in order to know how to extract from superpositions
of expanding and contracting universes predictions concerning
transitions in the expanding sector.
Without having further specified these choices and
having considered combinations of forward and backward solutions,
it is overhasty to conclude that unitarity will (not) be violated in Quantum
Cosmology.
\vskip 1. truecm
\vskip .5 truecm
{\bf Acknowlegdments}
\noindent
I am indebted to R. Brout for the clarifying discussions
that we had upon writing the review article\cite{time}
wherein similar problems are addressed.
I am also indebted to S. Massar for helpful remarks
provided during the last stage of the writing.
Finally, I wish to thank Y. Aharonov, Cl. Bouchiat, A. Casher, Fr. Englert,
J. Iliopoulos and T. Jacobson
for useful discussions.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,034
|
<?php
class Connect {
public $DEFAULT_URL = 'http://connect.reviewpro.com';
public $REVIEW_SUMMARIES_URL = '/v1/lodging/review/summaries';
public $CSQ_URL = '/v1/lodging/csq';
public $REVIEW_AVAILABLE_SRC_URL = '/v1/lodging/sources/available';
public $PUBLISHED_REVIEWS_SRC_URL = '/v1/lodging/review/published';
public $MANAGEMENT_RESPONSES_URL = '/v1/lodging/review/responses';
public $PIDS_FOR_ACCOUNT_URL = '/v1/account/lodgings';
public $DAILY_INDEX_URL = '/v1/lodging/index/daily';
public $LODGING_DIST_URL = '/v1/lodging/review/rating/distribution';
private $api_key;
private $api_sec;
function Connect($api_key, $api_sec) {
$this->api_key = $api_key;
$this->api_sec = $api_sec;
}
function __construct($api_key, $api_sec) {
$this->Connect($api_key, $api_sec);
}
function generate_signature() {
return hash('sha256', $this->api_key.$this->api_sec.time());
}
/*
* Creates the context for http requests
* @return stream_context with protocol, headers and content to upload
*/
private function generate_http_context($method, $data="") {
$context = stream_context_create(
array(
"http" => array(
"method" => $method,
"header" => "Content-Type: application/json",
"content" => $data
)
)
);
return $context;
}
/*
* Makes a request to the desired url with errors control (retries X times the same request)
* @return string http response with the content requestesd (if GET), reponse code (if POST)
*/
private function make_api_request($url, $method, $data, $max_error, $endpoint) {
echo "Request to: ".$url."\n";
$error_count = 0;
while ($error_count < $max_error) {
$context = $this->generate_http_context($method, $data);
$resp = file_get_contents($url, FALSE, $context);
if ($resp == TRUE) {
return $resp;
} else {
$error_count++;
echo sprintf("endpoint \"%s\" responded with error code \"%s\". Retry %d out of %d. Sleeping %d seconds...\n",
$endpoint, $http_response_header[0], $error_count, $max_error, pow($error_count, 2));
sleep(pow($error_count, 2));
}
}
return false;
}
/*
* makes a GET request to retrieve data
*/
private function make_get_request($url, $max_error, $endpoint) {
return $this->make_api_request($url, "GET", "", $max_error, $endpoint);
}
/*
* makes a POST request to upload data
*/
private function make_post_request($url, $data, $max_error, $endpoint) {
return $this->make_api_request($url, "POST", $data, $max_error, $endpoint);
}
/*
* Generic request method
* @param string $params string with needed params "fd=2013-01-01&td=2015-01-01&rt=OVERALL"
* @return string request response
*/
function fetch_api_data($pid, $params, $endpoint, $max_error=3) {
$url = "$this->DEFAULT_URL$endpoint?pid=$pid&$params&api_key=$this->api_key";
return $this->make_get_request($url, $max_error, $endpoint);
}
/*
* THOSE ARE PREDEFINED REQUESTS TO THE API
*/
// Lodging Rating Indexes
function fetch_daily_index_for_rating($pid, $fd, $td, $rt, $max_error=3) {
$endpoint = $this->DAILY_INDEX_URL;
$url = "$this->DEFAULT_URL$endpoint?pid=$pid&fd=$fd&td=$td&rt=$rt&api_key=$this->api_key";
return $this->make_get_request($url, $max_error, $endpoint);
}
// Lodging Rating Distribution
function fetch_daily_distribution($pid, $fd, $td, $rt, $max_error=3) {
$endpoint = $this->LODGING_DIST_URL;
$url = "$this->DEFAULT_URL$endpoint?pid=$pid&fd=$fd&td=$td&rt=$rt&api_key=$this->api_key";
return $this->make_get_request($url, $max_error, $endpoint);
}
// Sources
function fetch_available_sources($pid, $fd, $td, $max_error=3) {
$endpoint = $this->REVIEW_AVAILABLE_SRC_URL;
$url = "$this->DEFAULT_URL$endpoint?pid=$pid&fd=$fd&td=$td&api_key=$this->api_key";
return $this->make_get_request($url, $max_error, $endpoint);
}
// Reviews
function fetch_published_reviews($pid, $max_error=3) {
$endpoint = $this->PUBLISHED_REVIEWS_SRC_URL;
$url = "$this->DEFAULT_URL$endpoint?pid=$pid&api_key=$this->api_key";
return $this->make_get_request($url, $max_error, $endpoint);
}
// Reviews Summary
function fetch_review_summaries($pid, $fd, $td, $rw=10, $st=0, $max_error=3) {
$endpoint = $this->REVIEW_SUMMARIES_URL;
$url = "$this->DEFAULT_URL$endpoint?pid=$pid&fd=$fd&td=$td&rw=$rw&st=$st&api_key=$this->api_key";
return $this->make_get_request($url, $max_error, $endpoint);
}
// Account
function fetch_pids_for_account($max_error=3) {
$endpoint = $this->PIDS_FOR_ACCOUNT_URL;
$url = "$this->DEFAULT_URL$endpoint?api_key=$this->api_key";
return $this->make_get_request($url, $max_error, $endpoint);
}
// Management Responses
function fetch_management_responses($pid, $fd, $td, $max_error=3) {
$endpoint = $this->MANAGEMENT_RESPONSES_URL;
$url = "$this->DEFAULT_URL$endpoint?pid=$pid&fd=$fd&td=$td&api_key=$this->api_key";
return $this->make_get_request($url, $max_error, $endpoint);
}
/*
* Upload a CSQ Review, $data must be a well-formed JSON.
* @return string json response with response code.
* */
function pushCSQ($pid, $data, $max_error=3) {
$endpoint = $this->CSQ_URL;
$sig = $this->generate_signature();
$url = "$this->DEFAULT_URL$endpoint?pid=$pid&api_key=$this->api_key&sig=$sig";
return $this->make_post_request($url, $data, $max_error, $endpoint);
}
/*
* Transforms a JSON object (or string) in a human readable string.
*/
function pretty_print_json($json) {
try {
if (gettype($json) == "string") {
$json = json_decode($json);
}
return json_encode($json, JSON_PRETTY_PRINT);
} catch (Exception $e) {
echo "Content not parseable to json.";
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,284
|
© 2018 by TMS Medical Associates of New York, PLLC | TMS Upper East Side | Upper East Side TMS Therapy
TMS Upper East Side
Alan Manevitz, M.D.
Dr. Manevitz is recognized nationally for his clinical skills and excellence in helping patients who are not receiving full benefit of current treatments. He is often consulted by colleagues for illnesses difficult to diagnose and new treatment approaches for treatment resistant disease. He and his colleague, James Halper M.D., are recognized by patients and colleagues as one of the most internationally experienced clinical providers of TMS (Transcranial Magnetic Stimulation).
Dr. Manevitz has been named amongst the "Top Doctors in America" by Castle Connolly Medical Ltd., "Top Doctors: New York Metro Area", "Top Doctors on Long Island" New York Time's "Super Doctors", "New York Magazine's Best Doctors", Vital Sign's "Best Physician", and "Best Doctors of America".
Dr. James Halper is board-certified in both Internal Medicine and Psychiatry. He attended Columbia College and College of Physicians and Surgeons Columbia University, did his residency training in Medicine at Columbia Presbyterian and Stanford University Hospitals, and trained in Psychiatry at Payne Whitney Clinic (New York Hospital). He completed research fellowships at the National Institutes of Health, Rockefeller University, and Payne Whitney Clinic and is the recipient of numerous research awards and the author of many scientific and medical articles and book chapters.
Dr. Halper is well known for his innovative use of a variety of treatment modalities for patients with refractory psychiatric disorders. He is considered an expert in neurological functioning of the "brain."
James Halper, MD
Leading TMS Therapy Providers
We are the first clinic to provide TMS therapy treatments in the New York Metro area and one of the first 10 clinical pioneers in the United States. At this time we have performed close to 3000 treatments in about 100 patients.
Ready to learn more or schedule an appointment? We are pleased to be able to offer Neurostar TMS Therapy and Brainsway Deep TMS Therapy to our patients.We have extended daily office hours and will accommodate you for weekends and after hours.
Call Now 212-751-5072 to schedule consultation!
Ready to Learn More About TMS Therapy?
We are here to help. Contact us or Schedule a Consultation. Call 646-452-4536
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,161
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RT: How is it possible the US claims it doesn't know anything about this airstrike and that they're still gathering information?
MO: They're still gathering information? How come they did an airstrike without the information? I am sorry but we are not idiots. Our minds are not somewhere backwards. We know what is happening. We know that every flight that is being made costs more than $200,000 for a jet to go up in the air. And I am very sure that any state that gets a jet up in the air to bomb someone knows exactly what they are bombing. What they did is a… retaliatory move, a fast move just to breed blood in the streets of Syria just to say to the French and the US public: "Look, we are fighting ISIS." This is not how they fight ISIS. They fight ISIS by stopping the Turkish state from opening its borders where they infiltrate, by stopping the funding, by stopping the arming and by stopping the so-called moderate rebels who yesterday killed a 12-year-old boy. These are the moderates of the US. The US recognizes the support for this Nour al-Din al-Zenki brigade that killed the boy. And trust me, they know who they are because today they told them: "We are going to stop the funding if you continue doing this." They have been doing this for the past five years. They know everything that is going on. I surely appreciate what the Syrian government is doing by sending this message to the UN but this is not enough... This is definitely not enough. Syrians are suffering; they are dying by the hundreds. People need to wake up and see what their governments are doing and stop them. Because it is their tax payments that are causing the suffering over the world, especially in Syria and Iraq. I am very sure that they know who they are fighting and bombing, but they just pretend that they don't know.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,818
|
Walter Winterbottom (Oldham, 31 de marzo de 1913 — Guildford, 16 de febrero de 2002) apodado Walter el Magnífico, fue un entrenador y futbolista británico. En 1946 fue nombrado primer seleccionador en la historia de Inglaterra, cargo que ocupó durante 16 años hasta su dimisión en 1962. En todo ese tiempo dirigió al combinado nacional en cuatro Copas Mundiales y ganó 13 campeonatos de la British Home Championship.
Trayectoria como futbolista
Walter Winterbottom nació en Oldham, Lancashire en 1913. En su infancia sobresalió como un estudiante de matrícula, estudió magisterio en la Chester Diocesan Teachers Training College y se graduó como el primero de la promoción de 1933. Al tiempo que daba clases en una escuela local, jugaba al fútbol en equipos amateur como el Royton Amateurs o el Mossley FC. Después de hacer pruebas en el Oldham Athletic y en el Manchester City fue descubierto por Louis Rocca, el ojeador del Manchester United, quien le convenció en 1936 para firmar un contrato a tiempo parcial que le permitiría seguir siendo profesor.
La carrera profesional de Winterbottom fue muy corta. En su primera temporada jugó 21 partidos de liga y dos de la FA Cup, la mayoría de las veces en la posición de centrocampista derecho. Pero en su segundo año se vio afectado por una enfermedad reumática en la espalda que redujo sus apariciones a cuatro partidos. El estallido de la Segunda Guerra Mundial en 1939 supuso su retirada definitiva al enrolarse en el ejército británico. Durante el conflicto fue director de educación física en la Royal Air Force y después se marchó al Ministerio del Aire para trabajar en la formación de instructores.
En 1942 contrajo matrimonio con Ann Richards, con la que tuvo dos hijas (Janet y Brenda) y un varón (Alan).
Trayectoria como entrenador
En 1946 el secretario de la Asociación de Fútbol, Stanley Rous, convenció a la junta directiva para que Walter Winterbottom fuese el primer director técnico de la selección de fútbol de Inglaterra, ocupándose tanto del primer equipo como de las categorías amateur y juveniles. A día de hoy sigue siendo el seleccionador más joven en la historia del país (38 años), así como en el único sin experiencia previa.
Su debut tuvo lugar el 28 de septiembre de 1946 frente a la Isla de Irlanda en Windsor Park (Belfast), con victoria por 2:7.
Winterbottom asumió responsabilidades técnicas y tácticas pero no podía seleccionar a los futbolistas, pues esa labor correspondía a un comité especial. Al mismo tiempo, trabajó con la Asociación de Fútbol en implementar los primeros cursos de formación para futuros entrenadores. Una de sus primeras medidas fue crear un «Equipo B» que diese oportundiades a los jugadores más jóvenes.
En 1947 Inglaterra derrotó a en Lisboa por un contundente 10:0 y un año después venció en Turín a por 4:0. Estos resultados, y el hecho de contar con jugadores como Billy Wright y Stanley Matthews, otorgaron al país la vitola de favoritos en la Copa Mundial de Fútbol de 1950, la primera de su historia. No obstante, sufrieron una sorprendente derrota por 1:0 frente a y después cayeron contra por el mismo marcador. Aunque esa eliminación fue un duro golpe, el punto de inflexión para modernizar del fútbol inglés fue el partido contra del 25 de noviembre de 1953: el llamado «Equipo de oro» venció a los ingleses en Wembley por 6:3, la primera derrota en casa frente a un equipo de Europa continental.
Walter fue también el seleccionador de en los Juegos Olímpicos de Helsinki 1952. Con un equipo formado por aficionados debido a las limitaciones olímpicas de la época, los británicos perdieron en la primera fase contra .
En todo este tiempo, el seleccionador se implicó en la modernización del fútbol inglés, que entonces se estaba viendo superado por otros países de Europa y Sudamérica. Si bien no pudo eliminar el comité de selección, tuvo cierta libertad para configurar la plantilla inglesa en la Copa Mundial de 1958, al que Inglaterra llegó con el desastre aéreo de Múnich aún en mente. El seleccionador dejó fuera a Nat Lofthouse, no alineó en ningún partido al joven Bobby Charlton pese a ser convocado, y quedó eliminado de la fase de grupos por la . Sin hacer caso a las críticas de la prensa, se mantuvo al frente y logró la cuarta clasificación consecutiva para el Mundial de 1962 en Chile, donde cayeron en cuartos de final.
Winterbottom dejó el cargo a finales de 1962 y convenció a la Asociación de Fútbol de que el próximo entrenador debía tener libertad para convocar jugadores, eliminando así el comité especial de selección. Aunque se especuló en la prensa que sería el sustituto de Stanley Rous en la secretaría de la Asociación, perdió la votación y terminó dejando el organismo. En 1963 la federación inglesa eligió como nuevo entrenador a Alf Ramsey, procedente del Ipswich Town.
Durante los 16 años que estuvo al frente, Winterbottom dirigió a Inglaterra en 139 partidos con un récrod de 78 victorias, 33 empates y 28 derrotas. Disputó un total de cuatro Copas Mundiales de Fútbol en 1950, 1954, 1958 y 1962, aunque en ninguna de ellas superó los cuartos de final. A nivel británico el país tuvo más éxito y se proclamó campeón de la British Home Championship en 13 ocasiones (siete indiscutidas y seis compartiendo el primer lugar).
En su encuentro de despedida, disputado el 21 de noviembre de 1962, Inglaterra se impuso por 4:0 a Gales en el estadio de Wembley.
Resto de su vida
Tras abandonar la Asociación de Fútbol, Winterbottom siguió siendo una figura de autoridad en el ámbito deportivo como asesor del gobierno británico en esa materia, director del Consejo de Deportes, secretario del Sport and Recreation Alliance y presidente del grupo de estudios técnicos de la FIFA desde 1966 hasta 1978.
En 1963 fue nombrado oficial de la Orden del Imperio Británico, comendador en 1972 y caballero comendador en 1978. Es uno de los cuatro miembros del Manchester United que ha obtenido tal distinción, junto con Matt Busby, Bobby Charlton y Alex Ferguson.
Falleció el 16 de febrero de 2002 a los 88 años en el Royal Surrey Hospital de Guildford, víctima de un cáncer de colon. A su funeral asistieron numerosos representantes del deporte británico.
Clubes
Como futbolista
Como entrenador
Bibliografía
Morse, Graham: Sir Walter Winterbottom: The Father of Modern English Football. 2013, Editado por John Blake Publishing. ISBN 978-1782191384
Referencias
Enlaces externos
Walter Winterbottom en el sitio web de la Asociación de Fútbol (en inglés)
Futbolistas de Inglaterra
Futbolistas del Manchester United Football Club
Entrenadores de la selección de fútbol de Inglaterra
Entrenadores en la Copa Mundial de Fútbol de 1950
Entrenadores en la Copa Mundial de Fútbol de 1954
Entrenadores en la Copa Mundial de Fútbol de 1958
Entrenadores en la Copa Mundial de Fútbol de 1962
Caballeros comendadores de la Orden del Imperio Británico
Nacidos en Oldham
Fallecidos en Guildford
|
{
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| 7,923
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PressClub Global · Article.
Festive Joy. The BMW Lifestyle Collection.
There are surprises for adults and children alike – from Snow Racer sleds to USB sticks, from watches to back packs. The extensive BMW Lifestyle Collection includes gift ideas for both young and old, and it's available at selected BMW dealerships and on the Internet at www.bmw-shop.com.
Gesa Pruene
Tel: +49-89-382-94081
Fax: +49-89-382-20626
Attachments(1x, ~135.5 KB)
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Munich. There are surprises for adults and children alike – from Snow Racer sleds to USB sticks, from watches to back packs. The extensive BMW Lifestyle Collection includes gift ideas for both young and old, and it's available at selected BMW dealerships and on the Internet at www.bmw-shop.com.
Hot on the slopes: the BMW Snow Racer.
Every child who finds this speedster under their Christmas tree will hope that winter never comes to an end. The BMW Snow Racer,with its smooth, replaceable metal runners, is pure winter joy. The sporty sled isn't just quick off the mark and a stunner on the slopes – it also offers safe fun in the snow. Its precise steering mechanism with suspension soaks up small bumps as you go and the non-slip wheel, which has a limited steering lock, prevents overly risky manoeuvres. An ergonomic seat and broad runners where kids can place their feet ensure a stable sitting position. And the horn on the steering wheel means that you can warn Sunday strollers to get out of the way.
This combination of fun, safety, and high-quality materials (the BMW Snow Racer is made of cold- and UV-resistant plastic) has been safety-approved by TÜV Rheinland. The silver Bobby Bob in BMW Motorsport design is suitable for children from four to ten years old.
BMW Snow Racer
Max. load: 50 kg
A bag that's full of joy: the BMW Joy back pack.
Santa Claus would be well advised to upgrade to a BMW Joy back pack for Christmas. The specially ventilated, thickly padded back of the BMW Joy back pack makes it exceptionally comfortable to wear. Different sized compartments lined with fleece provide room for documents, drinks, computer equipment, and other travelling items. With premium black coloring, decorated with Joy lettering, and also coming with a bag tag and BMW badge, its outside is as much fun as its inside.
Time to celebrate: the BMW Classic ladies' watch.
The BMW Classic ladies' watch shimmers in its classical look. Its elegant black dial, with three white hands and date display, is enclosed in a 30 mm stainless steel casing that is water resistant up to 10 atm. The fastener and clasp are made of the same fine metal and, offset against a black leather strap, a classic look is emphasized. A discrete BMW logo on the dial, and the laser-engraved BMW logo on the back of the precision Swiss Quartz mechanism, complete this decorative piece. The watch is supplied in a high-quality leather case.
BMW Classic ladies' watch
Material: stainless steel, leather
Motorsport for the little ones: the BMW M3 GT2 in toy form.
The BMW M3 GT2 asconvertible version for children - this kid's racer is a perfect replica of the legendary BMW racing car. Even subtle details like the M logo on the doors and the sponsor's branding beneath the radiator grill have been reproduced. Little racing car drivers can sit comfortably on their BMW M3 GT2's adjustable seat.
The BMW M3 GT2 is available for racing drivers from three to five years old – in two versions. There's a motorized version with a maximum speed of 2.5 m/h, forward and reverse gears, and a 1.5-hour driving time before being recharged. With the pedal version, the diminutive drivers determine the speed and rest periods for themselves.
Motorized version
Pedal version
The key to data storage: the BMW USB stick
Would you like a BMW for Christmas? Wouldn't it be nice to find a little package containing a BMW key beneath your Christmas tree? The BMW USB stick looks like a genuine vehicle key but, in reality, it's an original data-storage device for when you're on the go.
The BMW USB stick boasts exceptional design, functionality, and unique details: a gentle click on the trunk-button slides out the well-protected USB plug, making the stick ready for storing up to 8GB of data. At the risk of mistaking it for your vehicle key, you should nevertheless take the BMW USB stick with you in your car – especially when it's loaded with your music along with other documents and photos. The stick fits perfectly into the BMW USB audio port or the USB port of other modern navigation systems. Simply plug it in, turn on your audio system, and enjoy your own personal music mix while you drive.
Festive Joy. The BMW Lifestyle Collection. DOC, EN, 135.5 KB
The following applies to consumption figures for vehicles with new type approval, September 2017 onward: The figures for fuel consumption, CO2 emissions and energy consumption are obtained in accordance with the specified measuring procedure (EC Regulation No. 715/2007), as issued and amended. The figures are for a basic-version vehicle in Germany. The bandwidths allow for differences in the choice of wheel and tire sizes and items of optional equipment and can be changed by the configuration.
Obtained on the basis of the new "Worldwide harmonized Light vehicles Test Procedure" (WLTP), the figures are converted back to the "New European Driving Cycle" (NEDC) for the sake of comparability. Values other than those stated here may be used for the purposes of taxation and for other vehicle-related duties relating to CO2 emissions.
More information about official fuel consumption figures and the official specific CO2 emissions of new passenger cars can be obtained from the "guideline on fuel consumption, CO2 emissions and current consumption of new passenger cars", available here: https://www.dat.de/co2/.
Specifications of the BMW 4 Series Convertible, valid from 07/2021.
Specifications of the BMW 4 Series Coupe, valid from 07/2021.
BMW Group and Daimler Mobility join forces with bp as a partner for Digital Charging Solutions Gm...
Specifications of the BMW X6 M, valid from 03/2021.
BMW M Automobiles
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Q: How can I redirect to the log in page when I press the log out button When I'm using the following code I get a blank page
<form action="login.php ">
<input type="submit" name="log out" value="Log Out">
A: first of all you have to use sessions if you use session then print following code
<?php
session_start();
unset($_SESSION["name"]);
header("Location:index.php");
?>
if you are not using session then simply use redirect in php
1.create logout.php page code
<?php
header("Location:index.php");
?>
2.after that
<form action=logout.php method=post>
<input type="submit" name="log out" value="Log Out">
</form>
A: In your code there was no closing tag for form (</form>).May be that is making issue..
Also always add all the attributes to your html elements like 'name'.You missed form name. this is a good practice.
Try this
<form name="formname" action="data.php" method="POST">
UserName <input type="text" name="username" placeholder="username"><br>
PassWord <input type="password" name="password"><br>
Remember Me <input type="checkbox" value="1" name="remember"><br>
<input type="submit" name="login" value="LogIn">
</form>
Hope this helps...
A: use
header("location:login.php");
that will automatically redirect you on that page
A: Use this
<button type="button" onclick="document.location='login_page.php'">Login Page</button>
It will redirect you when the button clicked
A: You have to use a session variable. And unset it. And do redirect to any page you want.
session_start();
unset($_SESSION["nome"]); // where $_SESSION["nome"] is your own variable. if you do not have one use only this as follow **session_unset();**
header("Location: homepage.php");
Edited: Always use Session variable. Because it gets set at server side and as an admin, you have more control over it.
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{
"redpajama_set_name": "RedPajamaStackExchange"
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<resources>
<string name="app_name">TwitterCover</string>
</resources>
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{
"redpajama_set_name": "RedPajamaGithub"
}
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\section{Introduction}
\label{sec:intro}
In a recent series of papers Fern\'{a}ndez and Garcia\cite{FG14b,FG14c} and
Amore et al\cite{AFG14b,AFG15} applied group theory to non-Hermitian coupled
oscillators with the purpose of determining the conditions under which the
well known parity-time (PT) symmetry\cite{B07} as well as its generalization
space-time (ST) symmetry\cite{KC08} are unbroken. One of the examples
discussed by Fern\'{a}ndez and Garcia\cite{FG14b} is a pair of harmonic
oscillators coupled by an imaginary quadratic interaction
H_{2D}=p_{x}^{2}+p_{y}^{2}+\omega _{x}^{2}x^{2}+\omega _{y}^{2}y^{2}+iaxy$.
They found that when the two oscillators are identical ($\omega
_{x}^{2}=\omega _{y}^{2}$) the eigenvalues $E_{mn}$ are real when $m=n$ and
pairs of complex conjugate numbers when $m\neq n$. On the other hand, when
\omega _{x}^{2}\neq \omega _{y}^{2}$ all the eigenvalues are real for
certain combinations of the oscillator parameters $\omega _{x}^{2}$, $\omega
_{y}^{2}$ and $a$. That is to say: in the latter case there is a sort of
phase transition from real to complex eigenvalues.
More recently, Beygi et al\cite{BKB15} discussed a family of $N$ coupled
oscillators that contain the example discussed by Fern\'{a}ndez and Garcia
when $N=2$. These authors assumed that each oscillator interacts only with
its nearest neighbors in a sort of linear chain; that is to say: the $i$-th
oscillator interacts only with the $(i-1)$-th and $(i+1)$-th ones. They
studied both quantum-mechanical as well as classical oscillators and arrived
at exactly the same conclusions drawn earlier by Fern\'{a}ndez and Garci
\cite{FG14b} for the former case. According to Beygi et al those
non-Hermitian coupled harmonic oscillators exhibit partial PT
symmetry which is a concept introduced somewhat earlier by
Yang\cite{Y14}. Both PT symmetry and partial PT symmetry are
particular cases of the ST symmetry introduced by Klainman and
Cederbaun\cite{KC08} and all of them are examples of the
antiunitary symmetry discussed by Wigner\cite{W60} long time
before. It is well known that if $U$ is a unitary transformation
in configuration space and $T$ is the time reversal operation then
$A_{U}=UT$ is an antiunitary operator\cite{W60}. Therefore, it
seems more reasonable to speak of antiunitary symmetry as the most
general concept. When $A_{U}$ leaves the non-Hermitian Hamiltonian
operator $H$ invariant ($A_{U}HA_{U}^{-1}=H$) we say that $H$
exhibits antiunitary symmetry. In the particular case that $U$ is
the parity operation $P$ then we are in the presence of PT
symmetry. According to Yang\cite{Y14} when the potential energy
function $V(x,y)$ of a
two-dimensional quantum-mechanical model satisfies $V(-x,y)^{*}=V(x,y)$ or
V(x,-y)^{*}=V(x,y)$ then the model satisfies partial PT symmetry (note that
V(-x,-y)^{*}=V(x,y)$ describes the usual PT symmetry). Obviously, it is a
particular case of the antiunitary symmetry generated by the unitary
operators $U_{x}:(x,y)\rightarrow (-x,y)$ or $U_{y}:(x,y)\rightarrow (x,-y)
, respectively. Note, for example, that the Hamiltonian $H_{2D}$ is
invariant with respect to the antiunitary operators $A_{x}=U_{x}T$ and
A_{y}=U_{y}T$. In fact, Klainman and Cederbaum\cite{KC08} chose
H=H_{0}+i\lambda xy$ as a simple prototypical example of ST symmetry. The
exactly-solvable $N$-dimensional harmonic oscillators chosen by Beygi et a
\cite{BKB15} that exhibit partial PT symmetry are also particular examples
of antiunitary symmetry.
By means of group theory and perturbation theory Amore et al\cite{AFG14b}
conjectured that of all the examples of antiunitary symmetry studied so far,
PT symmetry appears to be the less likely to be broken. Their conclusion was
based on the role of $U$ in the symmetry group $G_{0}$ for the unperturbed
Hermitian oscillator $H_{0}$.
In a later communication Fern\'{a}ndez and Garcia\cite{FG15} discussed three
PT-symmetric models with completely different spectra. They are of the form
H=H_{0}+igz$, where $H_{0}$ is an Hermitian operator with a central-field
potential $V(r)$. The first one is exactly solvable and exhibits a real
spectrum for all values of $g$. On the other side, the second one exhibits
complex eigenvalues for all values of $g$. Finally, with an intermediate
behavior, the third one shows the well known phase transition typical of
PT-symmetric quantum-mechanical models\cite{BW12}. The second model is most
interesting because the search carried out by Fern\'{a}ndez and Garcia\cite
{FG14b,FG14c} and Amore et al\cite{AFG14b,AFG15} failed to produce any
PT-symmetric Hamiltonian with completely broken PT symmetry. In this case
the phase transition takes place at the trivial Hermitian limit $g=0$. The
remarkable difference in the spectra of those non-Hermitian operators can be
traced back to the symmetry of $H_{0}$. The higher this symmetry the more
probable the appearance of complex eigenvalues\cite{FG15}.
One of the aims of this paper is to analyse the system of coupled
oscillators proposed by Beygi et al\cite{BKB15} from the point of view of
group theory and perturbation theory. In Sec.~\ref{sec:A_symmetry} we
develop the concept of antiunitary symmetry in a way that generalizes and
extends the original idea of Klainman and Cederbaum\cite{KC08} and improves
the point-group analysis carried out by Amore et al\cite{AFG14b}. We also
show that the combination of group theory and perturbation theory yields a
fairly good idea about the kind of spectrum that one expects of a given
non-Hermitian operator. In Sec.~\ref{sec:harm_osc} we study the $N
-dimensional coupled harmonic oscillators of Beygi et al\cite
{BKB15} from the point of view of point-group theory. In addition
to finding the point group for these oscillators we show that
their antiunitary symmetry is greater than the partial PT symmetry
discussed so far. Finally, in Sec.~\ref{sec:conclusions} we
summarize the main results of the paper and draw conclusions.
\section{Antiunitary symmetry}
\label{sec:A_symmetry}
Consider a Hamiltonian operator of the form
\begin{equation}
H(\lambda )=H_{0}+\lambda H^{\prime }, \label{eq:H_general}
\end{equation}
where $H_{0}$ is Hermitian and $H^{\prime }$ is real and linear. Suppose
that there is a group $G=\left\{ U_{1},U_{2},\ldots ,U_{n}\right\} $ of
unitary operators ($U_{i}^{\dagger }=U_{i}^{-1}$) that leave both $H_{0}$
and $H^{\prime }$ invariant:
\begin{equation}
U_{i}H_{0}U_{i}^{-1}=H_{0},\;U_{i}H^{\prime }U_{i}^{-1}=H^{\prime }.
\label{eq:G_for_H}
\end{equation}
This group of operators is commonly called the symmetry group for $H$\cite
{C90,T64}. Suppose that there is a set of unitary operators $S_{W}=$
\left\{ W_{1},W_{2},\ldots ,W_{m}\right\} $ such that
\begin{equation}
W_{i}H_{0}W_{i}^{-1}=H_{0},\;W_{i}H^{\prime }W_{i}^{-1}=-H^{\prime }.
\label{eq:S_W}
\end{equation}
Since $U_{i}W_{j}\in S_{W}$ and $W_{i}W_{j}\in G$ we conclude that
G_{0}=G\cup S_{W}$ is a group of unitary operators that leave $H_{0}$
invariant. Obviously, $G_{0}$ is at least a subgroup of the symmetry group
for $H_{0}$.
Let us now consider the set of antiunitary operators $S_{A}=$ $\left\{
A_{1},A_{2},\ldots ,A_{m}\right\} $, where $A_{i}=W_{i}T=TW_{i}$ and $T$ is
the time-reversal operator\cite{W60}. If $\lambda ^{*}=-\lambda $ then
A_{i}H(\lambda )A_{i}^{-1}=H(-\lambda ^{*})=H(\lambda )$. Besides, since
U_{i}A_{j}\in S_{A}$ and $A_{i}A_{j}\in G$ we conclude that $G_{A}=G\cup
S_{A}$ is a group of operators that leave $H(\lambda )$ invariant
\begin{equation}
KH(\lambda )K^{-1}=H(\lambda ),\;K\in G_{A},\;\lambda ^{*}=-\lambda .
\label{eq:G_ST}
\end{equation}
We call $G_{A}$ the antiunitary-symmetry group for $H$ as a generalization
of the concept introduced by Klainman and Cederbaum\cite{KC08}. In this
sense, PT symmetry is a particular case of antiunitary symmetry where $P\in
S_{W}$, $P:\mathbf{x}\rightarrow -\mathbf{x}$. Another particular case of
antiunitary symmetry is the partial PT symmetry introduced by Yang\cite{Y14}
which for two-dimensional models takes the form $W_{x}:(x,y)\rightarrow
(-x,y)$ or $W_{y}:(x,y)\rightarrow (x,-y)$. As indicated in the introduction
Klainman and Cederbaum\cite{KC08} already chose these coordinate
transformations to illustrate the concept of ST symmetry (they called them
P_{x}$ and $P_{y}$, respectively). This kind of symmetry was recently
extended by Beygi et al\cite{BKB15} to $N$-dimensional oscillators.
If we apply $W\in S_{W}$ to the eigenvalue equation
\begin{equation}
H(\lambda )\psi _{n}=E_{n}(\lambda )\psi _{n}, \label{eq:Schr_gen}
\end{equation}
we have
\begin{equation}
WH(\lambda )W^{-1}W\psi _{n}=H(-\lambda )W\psi _{n}=E_{n}(\lambda )W\psi
_{n}. \label{eq:SHpsi}
\end{equation}
It is clear that
\begin{equation}
E_{n}(\lambda )=E_{m}(-\lambda ), \label{eq:En(lam)=Em(-lam)}
\end{equation}
where $E_{m}(-\lambda )$ is an eigenvalue of $H(-\lambda )$. Since this
equation should be valid for all $\lambda $, then when $\lambda \rightarrow
0 $ we have $E_{n}(0)=E_{m}(0)$. If the eigenvalue $E_{n}(0)$ of $H_{0}$ is
nondegenerate, then $m=n$, $E_{n}(-\lambda )=E_{n}(\lambda )$ and
perturbation theory yields the formal power series
\begin{equation}
E_{n}(\lambda )=\sum_{j=0}^{\infty }E_{n}^{(j)}\lambda ^{2j}.
\label{eq:En_series_even}
\end{equation}
Klaiman et al\cite{KGM08} obtained a similar result by means of a lengthier
argument. If the radius of convergence of this series is finite, then we
conclude that $E_{n}(\lambda )$ is real for sufficiently small $|\lambda |$
when $\lambda =-\lambda ^{*}$. If, on the other hand, the eigenvalue
E_{n}(0)$ is degenerate, then the perturbation series for
$E_{n}(\lambda )$ may exhibit odd powers of $\lambda $ and one
expects complex eigenvalues under such conditions. This argument
is also expected to apply to the more general case of a divergent
perturbation series provided it is asymptotic to $E_{n}(\lambda )$
because we only have to consider sufficiently small values of
$|\lambda |$ to prove that the eigenvalue is complex. We
appreciate that the occurrence of real eigenvalues of a
non-Hermitian operator $H(ig)$, $g$ real, is strongly dependent on
the form of the spectrum of $H_{0}$. This approach based on
perturbation theory is not new\cite{AFG14b} but we outline it here
for completeness.
\section{Coupled harmonic oscillators}
\label{sec:harm_osc}
In this section we focus our attention on the set of adjacently coupled
harmonic oscillators studied by Beygi et al\cite{BKB15}
\begin{equation}
H(\lambda )=\frac{1}{2}\sum_{i=1}^{N}\left( p_{i}^{2}+\omega
_{i}^{2}x_{i}^{2}\right) +\lambda \sum_{j=1}^{N-1}x_{j}x_{j+1},
\label{eq:H_CHO}
\end{equation}
where $p_{j}=-i\partial /\partial x_{j}$ in the coordinate representation.
As indicated in the introduction, the particular case $N=2$ was studied
earlier by Fern\'{a}ndez and Garcia\cite{FG14b}. If we choose the oscillator
frequencies $\omega _{j}$ so that the spectrum of $H_{0}=H(0)$ is
nondegenerate then, arguing as in Sec.~\ref{sec:A_symmetry}, the spectrum of
$H(ig)$ is expected to be real for sufficiently small values of $g$. When
some of the frequencies are equal, degeneracy emerges as well as the chance
of complex eigenvalues. Consequently, one expects phase transitions; that is
to say surfaces in the $\omega $-plane that separate regions of real and
complex eigenvalues. Fern\'{a}ndez and Garcia\cite{FG14b} and more generally
and exhaustively Beygi et al\cite{BKB15} discussed some particular cases in
detail.
From the point of view of symmetry the case of equal frequencies is the most
interesting one; therefore in what follows we choose $\omega _{j}=1$ for all
$j$. We first obtain the unitary operators $U_{j}$ that leave $H^{\prime }$
invariant. They are given by the coordinate transformations
\begin{eqnarray}
U_{1} &:&(x_{1},x_{2},\ldots ,x_{N})\rightarrow (x_{1},x_{2},\ldots ,x_{N}),
\nonumber \\
U_{2} &:&(x_{1},x_{2},\ldots ,x_{N})\rightarrow (-x_{1},-x_{2},\ldots
,-x_{N}), \nonumber \\
U_{3} &:&(x_{1},x_{2},\ldots ,x_{N})\rightarrow (x_{N},x_{N-1},\ldots
,x_{1}), \nonumber \\
U_{4} &:&(x_{1},x_{2},\ldots ,x_{N})\rightarrow (-x_{N},-x_{N-1},\ldots
,-x_{1}). \label{eq:Uj_group}
\end{eqnarray}
The set $G_{4}=\left\{ U_{1},U_{2},U_{3},U_{4}\right\} $ is an
Abelian group isomorphic to $D_{2}$, $C_{2v}$ and
$C_{2h}$\cite{C90,T64}.
In addition to these four operators there are other four ones that change
the sign of $H^{\prime }$ leaving $H_{0}$ invariant:
\begin{eqnarray}
W_{1} &:&x_{j}\rightarrow (-1)^{j}x_{j}, \nonumber \\
W_{2} &:&x_{j}\rightarrow (-1)^{j+1}x_{j}, \nonumber \\
W_{3} &=&U_{3}W_{1}, \nonumber \\
W_{4} &=&U_{3}W_{2}. \label{eq:S_W_set}
\end{eqnarray}
We can thus construct four antiunitary operators $A_{j}=W_{j}T$, $j=1,2,3,4$
that leave $H(ig)$ invariant and describe its antiunitary symmetry. Beygi et
al\cite{BKB15} only considered $A_{1}$ and $A_{2}$ in their discussion of
partial PT symmetry. If $A_{1}$ and $A_{2}$ describe partial PT symmetry,
how do we call the antiunitary symmetries associated to $A_{3}$ and $A_{4}$:
reverse-order partial PT symmetry? Instead of adding more fancy names to new
antiunitary transformations of the Hamiltonian operator we suggest the
general term antiunitary symmetry.
The set of eight operators $G_{8}=\left\{
U_{1},U_{2},U_{3},U_{4},W_{1},W_{2},W_{3},W_{4}\right\} $ is also
a group, with a structure that depends on $N$. These operators
obviously leave $H_{0} $ invariant because $G_{8}$ is a subgroup
of the actual symmetry group for $H_{0}$ that we do not discuss
here. $G_{8}$ is isomorphic to $C_{4v}$ when $N$ is even and to
$D_{2h}$ when $N$ is odd\cite{C90,T64}.
Each eigenvalue $E_{k}^{(0)}$ of $H_{0}$ is $g_{k}$-fold degenerate, where
\begin{eqnarray}
E_{k}^{(0)} &=&k+\frac{N}{2},\;k=\sum_{j=1}^{N}n_{j},\;n_{j}=0,1,\ldots ,
\nonumber \\
g_{k} &=&\frac{(k+N-1)!}{k!(N-1)!}. \label{eq:E^0_k,g_k}
\end{eqnarray}
Such great degeneracy is the reason why so many eigenvalues of $H(ig)$ are
complex. For example, when $N=2$ all the eigenvalues with $n_{1}\neq n_{2}$
are complex\cite{FG14b,BKB15}. For any value of $N$ the lowest eigenvalue
E_{0}^{(0)}$ is nondegenerate and $E_{0}(ig)$ is real for all $g$\cite{BKB15}
in agreement with the perturbation analysis outlined in Sec.~\ref
{sec:A_symmetry}.
\section{Conclusions}
\label{sec:conclusions}
Instead of inventing new names for new antiunitary symmetries that may
appear in future investigations we suggest to resort to the quite general
concept of antiunitary symmetry. It is perfectly reasonable to keep PT
symmetry for historical reasons but it is far more convenient to avoid
particular names for all its possible variants such as partial PT symmetry,
reverse-order partial PT symmetry, etc. and simply use antiunitary symmetry.
To this end we have tried to formalize the concept of antiunitary symmetry
by means of group theory.
We have also tried to show that perturbation theory is useful for predicting
whether a given non-Hermitian Hamiltonian will exhibit antiunitary symmetry,
at least for some kind of examples. Although the approach is not completely
rigorous it nevertheless provides a fairly good idea of what to expect. As a
general rule, the greater the symmetry of $H_{0}$ the more probable the
occurrence of complex eigenvalues for all values of $g$. This point of view
was used successfully in earlier papers\cite{FG14b,FG14c,AFG14b,AFG15}.
As an illustrative example of the main theoretical ideas we have chosen the
coupled harmonic oscillators studied by Beygi et al\cite{BKB15} and
disclosed their point-group symmetry as well as their antiunitary symmetry.
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"redpajama_set_name": "RedPajamaArXiv"
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Q: Why Gibbs Sampling for mixture models? I am studying MCMC and in the book I'm reading there is this example on Gibbs algorithm for inferring the posterior of a gaussian mixture. I understand how the algorithm works and the fact that its convenience relies in the simplicity of sampling from the full conditionals for the parameters, however I was wondering whether and why this method should be preferable to others that were introduced in previous chapters and with which I am less familiar (like Variational methods, or others I don't know).
Furthermore, the book starts by explicitly writing the full joint $p(x,z,\mu,\Sigma, \pi)$ for the gmm (assuming semi-conjugate prior), am I right to say that even knowing this explicitly, sampling directly (using for instance rejection sampling) is inefficient because of the curse of dimensionality, while Gibbs suffer far less from this phenomenon? Is this the only reason why Gibbs is to be preferred?
Any help on understanding pros and cons of the various existing methods for inferring the posterior is well accepted.
A: Gibbs sampling is possibly the first MCMC algorithm implemented for mixture models (Gelber, Gelman and Goldhirsch, 1989), inspired from the data augmentation of Tanner and Wong (1987) and ultimately from the EM algorithm. It takes advantage of the latent variable structure in producing nice, low dimension, and natural conditionals that can be simulated quite efficiently, much more than a default MCMC algorithm like random walk Metropolis-Hastings.
However, here is a slide from my MCMC course, where I illustrate the potential pitfall of using plain Gibbs for a mixture of two Gaussians with unknown means, namely that it may get trapped in a local mode because the latent variables are practically if not theoretically stuck at a fixed value.
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San Bonoso y San Maximiano, mártires cristianos del siglo IV de Arjona (Jaén). Su fiesta se celebra el 21 de agosto.
Historia
Los santos gloriosos Bonoso y Maximiano, padecieron el martirio durante la décima persecución contra los cristianos, promovida por los emperadores Diocleciano y Maximiano. Eran naturales de Iliturgi, colonia romana muy cerca de la actual Mengíbar. En su juventud siguieron la carrera de las armas, e intervinieron con valentía para sofocar la rebelión del Prefecto de Itálica, Numeriano, que se había sublevado contra Roma. Por este hecho de armas fueron distinguidos e invitados por el presidente Daciano, a dar gracias a las divinidades romanas, ofreciéndoles incienso y sacrificios en el Alcázar de Arjona, donde estaba su tribunal para juzgar a los cristianos. Como estos jóvenes se negaran a obedecer las órdenes del presidente y
confesaran públicamente, que ellos "eran soldados de Cristo antes que de los emperadores", fueron puestos en prisión, cargados de grillos y cadenas, azotados y sometidos a tormentos de la tróclea; hasta que decapitados, entregaron sus vidas a Dios el 21 de agosto del año 308 de nuestra era, siendo Bonoso de 21 años y Maximiano de 18.
Apariciones
Las reliquias de Bonoso y Maximiano, junto con las de otros muchos mártires, que sufrieron tormentos en el mismo lugar, aparecieron el 14 de octubre de 1628 y años subsiguientes, entre los muros del Alcázar, con prodigios y señales celestes.
Patronazgo
El cardenal Moscoso y Sandoval, obispo de Jaén, los declaró patronos de Arjona, calificando sus reliquias, y edificó para su culto un suntuoso santuario en el siglo XVII.
Fiestasantos
Fiestasantos es la fiesta más importantes de las celebradas en Arjona a lo largo del año. Se realizan en honor de los patronos del municipio, San Bonoso y San Maximiano, los santos de Arjona. De ahí el curioso nombre de «Fiestasantos», con que se conocen estos días festivos. Son jornadas en que los diversos actos religiosos, deportivos, lúdicos y culturales se suceden casi ininterrumpidamente, del 11 al 24 de agosto. El día más importante es el 21, en que tiene lugar la procesión de las imágenes y las reliquias de los santos. A lo largo de esta semana, tienen lugar distintas tradiciones que se mantienen vivas desde sus orígenes legendarios tales como:
El repique de la Campanica del Turrón, cada mañana (de 11:30 a 12:00 horas) desde el inicio de las fiestas hasta que acaban, seguida del gran repique de las campanas de las demás Parroquias y disparo de cohetes.
Traslado de las Sagradas Reliquias (11 de agosto) a la iglesia de Santa María, donde están ubicados los santos.
Los Pesos, para recolectar donativos pesando a personas en una romana, quienes entregan el equivalente en trigo o su valor en dinero, y echar las Banderas, cubriendo con las banderas de los Santos, acompañado con el himno de los Santos, a los presentes en las calles de Arjona.
La Procesión de la Luminaria (19 de agosto): niños y niñas procesionan con faroles fabricados a partir de un melón, en recuerdo de las luces que en 1628 señalaron dónde se encontraban los restos de los mártires.
La Quema de Daciano (19 de agosto), delegado de Roma en Urgavo que martirizó a los Santos sometiéndolos a cruenta tortura y posterior decapitación. Por ello se quema a un muñeco vestido de romano en el lugar del martirio, lo que se conoce como el Cementerio de los Santos, junto a la Plaza de Santa María.
Día más importante del año en arjona (21 de agosto) los santos junto a sus reliquias pasan por las calles más importantes hasta subir por unas escaleras ("escalericas") para llegar a la iglesia de Santa Maria, que después de varias entradas y salidas se encierran los santos.
Último traslado de las Sagradas Reliquias a su lugar de origen (22 de agosto).
Fuegos artificiales en el Paseo Nuevo para cerrar las fiestas (24 de agosto).
Enlaces externos
FIESTAS EN HONOR A SAN BONOSO, SAN MAXIMIANO Y SUS SANTAS RELIQUIAS
Arjona
Santos de la provincia de Jaén
Mártires cristianos de la Antigua Roma del siglo IV
Santos cristianos de la Antigua Roma del siglo IV
Santos católicos de España del siglo IV
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