text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
|---|---|---|
The 26th annual "Stamp Out Hunger" food drive, sponsored by the National Association of Letter Carriers, is set for Saturday, May 12, and is expected to once again have a huge impact in the High Country.
Considered the largest one-day event of its kind in the nation, more than 10,000 communities and cities across the country will participate in the food collection effort, in conjunction with Feeding America, the nation's largest domestic hunger relief organization.
Always held on the second Saturday of May each year, it's possibly the easiest way ever to help our neighbors who are dealing with food insecurity. And, yes, all food collected in the High Country area will stay in the community.
"This is a huge thing for us and the families we serve," said Elizabeth Young, executive director of the Hunger and Health Coalition in Boone, where the majority of local donations are delivered.
The timing could not be more perfect, Young said, as with the end of the school year at hand, many children who depend on schools for breakfast and lunch might possibly be left without adequate nourishment during the summer months.
With this year's goal to raise at least 9,000 pounds of food through the local effort, Young pleads with the community to participate in the food drive, and thanks everyone in advance for their generosity.
Last year's area collection netted approximately 7,900 pounds of food, Young said. "Because of our caring community, we were able to feed 675 families or about 1,982 people, through the provision of substantial food boxes," Young said.
"We are also very grateful to our letter carriers for hosting this food drive and making it all possible," said Young.
For more information, call the Hunger and Health Coalition at 828-262-1628.
On Saturday morning, May 12, simply hang a bag of nonperishable food on your mailbox or take your donation to your local post office.
Recommended items include the following: cereal, pasta, pasta/ spaghetti sauce, rice, canned fruits and vegetables, canned meals (such as soups, chili and pasta), 100% juice, peanut butter, macaroni & cheese, canned protein (tuna, chicken and turkey), beans (canned or dry).
Please do not attempt to donate: frozen food, homemade food or home-canned items, individual baby food products, items that have expired or are in glass containers, anything that is opened, damaged, out of code or does not have the official ingredients included.
Because all food collected during this food drive is given directly to non-profit charity food agencies in the community the food was collected, the donations are tax-deductible.
A spokesperson for the US Post Office in Boone said that local letter carriers and postal employees are happy to help with the food drive, and will work with volunteers to collect the food and deliver it "by the truck loads" to the food pantry at the Hunger and Health Coalition in Boone.
In 2017, the food drive collected 75 million pounds, the third-highest total in the event's history. Combined with the previous year's record of 80.1 million pounds, that makes a two-year total of 155 million pounds—the highest back-to-back total in the drive's history.
The Stamp Out Hunger Food Drive is held annually on the second Saturday in May in 10,000 cities and towns in all 50 states, the District of Columbia, Puerto Rico, the Virgin Islands and Guam.
The nation's 212,000 letter carriers collect the food donations that are left by the mail boxes and distribute them to local food agencies.. | {
"redpajama_set_name": "RedPajamaC4"
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@(repository: gitbucket.core.service.RepositoryService.RepositoryInfo, large: Boolean)(implicit context: gitbucket.core.controller.Context)
@import gitbucket.core.service.RepositoryService
@import context._
@import gitbucket.core.view.helpers._
@if(repository.repository.isPrivate){
<i class="@{if(large){"mega-"}}octicon octicon-lock"></i>
} else {
@if(repository.repository.originUserName.isDefined){
<i class="@{if(large){"mega-"}}octicon octicon-repo-forked"></i>
} else {
<i class="@{if(large){"mega-"}}octicon octicon-repo"></i>
}
}
| {
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} | 7,375 |
Das Unternehmen F.X. Gutter war ein deutscher Landtechnik- und Traktorenhersteller in Weißenhorn, Landkreis Neu-Ulm.
Geschichte
Die Firma Gutter wurde im Jahr 1864 gegründet und war in den Anfangsjahren als Reparaturwerkstatt für Landmaschinen tätig. Nach einem Umzug auf das Gelände einer ehemaligen Mühle (1898) begann das Unternehmen in Eigenregie landwirtschaftliche Geräte, wie zum Beispiel Pflüge, Dreschmaschinen und Schrotmühlen zu produzieren. Im Jahr 1929 beschlossen die Nachkommen des Unternehmensgründers die Aufteilung der Firma. Während Ludwig Gutter auf dem Gelände der alten Mühle verblieb und sich auf die Produktion von Schrotmühlen konzentrierte, baute Franz Xaver Gutter eine eigene Produktionsstätte unter dem Firmennamen "F.X. Gutter Dreschmaschinen-Fabrik" auf.
Traktorenproduktion
1936 erfolgte der Einstieg in den Traktorenbau. Die Schlepper stellten zu ihrer Zeit moderne Konstruktionen dar und wurden in Kleinserie produziert. Das Unternehmen stellte nur wenige Bauteile selbst her und verwendete fast ausschließlich Bauteile von Zulieferern, wie zum Beispiel Deutz, MWM, Prometheus, Bosch, Hurth und Haugg. Die Traktorenproduktion endete im Jahr 1958. Insgesamt hat die Firma F.X. Gutter zwischen 800 und 1000 Schlepper produziert.
Baureihen (Auswahl)
Einzelnachweise
Traktorenhersteller | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,664 |
Petrovouni (, meaning "rocky mountain", before 1927: Βασταβέτσι - Vastavetsi) is a village in the municipal unit of Tzoumerka (Ioannina, Epirus), Greece. It is situated on a mountainside of the Athamanika mountains, above the left bank of the river Arachthos, at 1,030 m elevation. It is 3 km north of Chouliarades, 9 km west of Syrrako and 17 km southeast of Ioannina. Its population is 79 people (2011 census).
Population
See also
List of settlements in the Ioannina regional unit
External links
Petrovouni at the GTP Travel Pages
References
Populated places in Ioannina (regional unit) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,664 |
package com.rubberduck.materialjumpstart.activities;
import android.annotation.TargetApi;
import android.content.Intent;
import android.content.res.Configuration;
import android.content.res.TypedArray;
import android.graphics.Bitmap;
import android.graphics.Color;
import android.graphics.drawable.BitmapDrawable;
import android.os.Build;
import android.os.Bundle;
import android.support.v7.app.ActionBarActivity;
import android.support.v7.graphics.Palette;
import android.support.v7.widget.LinearLayoutManager;
import android.support.v7.widget.Toolbar;
import android.util.Log;
import android.util.TypedValue;
import android.view.LayoutInflater;
import android.view.Menu;
import android.view.MenuItem;
import android.view.View;
import android.view.Window;
import android.view.WindowManager;
import android.widget.ImageView;
import android.widget.TextView;
import com.github.ksoichiro.android.observablescrollview.ObservableRecyclerView;
import com.github.ksoichiro.android.observablescrollview.ObservableScrollViewCallbacks;
import com.github.ksoichiro.android.observablescrollview.ScrollState;
import com.github.ksoichiro.android.observablescrollview.ScrollUtils;
import com.nineoldandroids.view.ViewHelper;
import com.rubberduck.materialjumpstart.R;
import com.rubberduck.materialjumpstart.adapters.DetailsRVAdapter;
import com.rubberduck.materialjumpstart.model.Dummy;
public class DetailsActivity extends ActionBarActivity implements ObservableScrollViewCallbacks {
private static final String TAG = "OfferDetailsActivity";
private static final float MAX_TEXT_SCALE_DELTA = 0.3f;
private static final boolean TOOLBAR_IS_STICKY = true;
private View toolbar;
private ImageView imageViewCover;
private View overlayView;
private View recyclerViewBackground;
private ObservableRecyclerView recyclerView;
private TextView textViewTitle;
private int actionBarSize;
private int flexibleSpaceImageHeight;
private int toolbarColor;
private Dummy currentDummy;
Palette palette;
Palette.Swatch vibrantSwatch, darkVibrantSwatch;
@Override
protected void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.activity_details);
Intent intent = getIntent();
currentDummy = (Dummy) intent.getParcelableExtra("dummyObj");
setSupportActionBar((Toolbar) findViewById(R.id.toolbar));
getSupportActionBar().setDisplayHomeAsUpEnabled(true);
flexibleSpaceImageHeight = getResources().getDimensionPixelSize(R.dimen.flexible_space_image_height);
actionBarSize = getActionBarSize();
Log.d(TAG, "Uri = " + currentDummy.getImageUri());
imageViewCover = (ImageView) findViewById(R.id.iv_cover);
imageViewCover.setImageResource((getImageResource(currentDummy.getImageUri())));
overlayView = findViewById(R.id.overlay);
toolbar = findViewById(R.id.toolbar);
toolbar.setBackgroundColor(Color.TRANSPARENT);
recyclerView = (ObservableRecyclerView) findViewById(R.id.rv_details);
recyclerView.setScrollViewCallbacks(this);
recyclerView.setLayoutManager(new LinearLayoutManager(this));
recyclerView.setHasFixedSize(false);
final View headerView = LayoutInflater.from(this).inflate(R.layout.recycler_header, null);
headerView.post(new Runnable() {
@Override
public void run() {
headerView.getLayoutParams().height = flexibleSpaceImageHeight;
}
});
DetailsRVAdapter adapter = new DetailsRVAdapter(this, currentDummy, headerView);
recyclerView.setAdapter(adapter);
Bitmap bitmap = ((BitmapDrawable)((ImageView)imageViewCover).getDrawable()).getBitmap();
if (bitmap != null) {
palette = Palette.generate(bitmap);
vibrantSwatch = palette.getVibrantSwatch();
darkVibrantSwatch = palette.getDarkVibrantSwatch();
}
//Set overlay color (for transition) and statusbar color
if (vibrantSwatch != null && darkVibrantSwatch != null) {
toolbarColor = getLighterShade(darkVibrantSwatch.getRgb());//vibrantSwatch.getRgb();
overlayView.setBackgroundColor(getLighterShade(darkVibrantSwatch.getRgb()));//(vibrantSwatch.getRgb());
setStatusBarColor(darkVibrantSwatch.getRgb());
}
textViewTitle = (TextView) findViewById(R.id.title);
textViewTitle.setText(currentDummy.getHeaderText());
setTitle(null);
// recyclerViewBackground makes RecyclerView's background except header view.
recyclerViewBackground = findViewById(R.id.list_background);
final View contentView = getWindow().getDecorView().findViewById(android.R.id.content);
contentView.post(new Runnable() {
@Override
public void run() {
// recylcerViewBackground's should fill its parent vertically
// but the height of the content view is 0 on 'onCreate'.
// So we should get it with post().
recyclerViewBackground.getLayoutParams().height = contentView.getHeight();
}
});
//since you cannot programatically add a headerview to a recyclerview we added an empty view as the header
// in the adapter and then are shifting the views OnCreateView to compensate
final float scale = 1 + MAX_TEXT_SCALE_DELTA;
recyclerViewBackground.post(new Runnable() {
@Override
public void run() {
ViewHelper.setTranslationY(recyclerViewBackground, flexibleSpaceImageHeight);
}
});
ViewHelper.setTranslationY(overlayView, flexibleSpaceImageHeight);
textViewTitle.post(new Runnable() {
@Override
public void run() {
ViewHelper.setTranslationY(textViewTitle, (int) (flexibleSpaceImageHeight - textViewTitle.getHeight() * scale));
ViewHelper.setPivotX(textViewTitle, 0);
ViewHelper.setPivotY(textViewTitle, 0);
ViewHelper.setScaleX(textViewTitle, scale);
ViewHelper.setScaleY(textViewTitle, scale);
}
});
}
@Override
public boolean onCreateOptionsMenu(Menu menu) {
// Inflate the menu; this adds items to the action bar if it is present.
getMenuInflater().inflate(R.menu.menu_offer_details, menu);
return true;
}
@Override
public boolean onOptionsItemSelected(MenuItem item) {
int id = item.getItemId();
if (id == R.id.action_settings) {
return true;
}
return super.onOptionsItemSelected(item);
}
@Override
public void onScrollChanged(int scrollY, boolean firstScroll, boolean dragging) {
// Translate overlay and image
float flexibleRange = flexibleSpaceImageHeight - actionBarSize;
int minOverlayTransitionY = actionBarSize - overlayView.getHeight();
ViewHelper.setTranslationY(overlayView, ScrollUtils.getFloat(-scrollY, minOverlayTransitionY, 0));
ViewHelper.setTranslationY(imageViewCover, ScrollUtils.getFloat(-scrollY / 2, minOverlayTransitionY, 0));
// Translate list background
ViewHelper.setTranslationY(recyclerViewBackground, Math.max(0, -scrollY + flexibleSpaceImageHeight));
// Change alpha of overlay
ViewHelper.setAlpha(overlayView, ScrollUtils.getFloat((float) scrollY / flexibleRange, 0, 1));
// Scale title text
float scale = 1 + ScrollUtils.getFloat((flexibleRange - scrollY) / flexibleRange, 0, MAX_TEXT_SCALE_DELTA);
setPivotXToTitle();
ViewHelper.setPivotY(textViewTitle, 0);
ViewHelper.setScaleX(textViewTitle, scale);
ViewHelper.setScaleY(textViewTitle, scale);
//Add left padding as user scrolls up
int leftPadding = convertPxToDp(
( (1 + MAX_TEXT_SCALE_DELTA - scale) * (100*(1 + MAX_TEXT_SCALE_DELTA)) ));
textViewTitle.setPadding(leftPadding, 0, textViewTitle.getPaddingRight(), 0);
// Translate title text
int maxTitleTranslationY = (int) (flexibleSpaceImageHeight - textViewTitle.getHeight() * scale);
int titleTranslationY = maxTitleTranslationY - scrollY;
if (TOOLBAR_IS_STICKY) {
titleTranslationY = Math.max(0, titleTranslationY);
}
ViewHelper.setTranslationY(textViewTitle, titleTranslationY);
if (TOOLBAR_IS_STICKY) {
// Change alpha of toolbar background
if (-scrollY + flexibleSpaceImageHeight <= actionBarSize) {
toolbar.setBackgroundColor(ScrollUtils.getColorWithAlpha(1, toolbarColor));
} else {
toolbar.setBackgroundColor(ScrollUtils.getColorWithAlpha(0, toolbarColor));
}
} else {
// Translate Toolbar
if (scrollY < flexibleSpaceImageHeight) {
ViewHelper.setTranslationY(toolbar, 0);
} else {
ViewHelper.setTranslationY(toolbar, -scrollY);
}
}
}
@Override
public void onDownMotionEvent() {
}
@Override
public void onUpOrCancelMotionEvent(ScrollState scrollState) {
}
@TargetApi(Build.VERSION_CODES.JELLY_BEAN_MR1)
private void setPivotXToTitle() {
Configuration config = getResources().getConfiguration();
if (Build.VERSION_CODES.JELLY_BEAN_MR1 <= Build.VERSION.SDK_INT
&& config.getLayoutDirection() == View.LAYOUT_DIRECTION_RTL) {
ViewHelper.setPivotX(textViewTitle, findViewById(android.R.id.content).getWidth());
} else {
ViewHelper.setPivotX(textViewTitle, 0);
}
}
private int convertPxToDp (float pixels) {
float scale = getResources().getDisplayMetrics().density;
return (int) (pixels * scale + 0.5f);
}
protected int getActionBarSize() {
TypedValue typedValue = new TypedValue();
int[] textSizeAttr = new int[]{R.attr.actionBarSize};
int indexOfAttrTextSize = 0;
TypedArray a = obtainStyledAttributes(typedValue.data, textSizeAttr);
int actionBarSize = a.getDimensionPixelSize(indexOfAttrTextSize, -1);
a.recycle();
return actionBarSize;
}
@TargetApi(Build.VERSION_CODES.LOLLIPOP)
private void setStatusBarColor(int color) {
if (Build.VERSION.SDK_INT >= 21) {
Window window = getWindow();
window.addFlags(WindowManager.LayoutParams.FLAG_DRAWS_SYSTEM_BAR_BACKGROUNDS);
window.clearFlags(WindowManager.LayoutParams.FLAG_TRANSLUCENT_STATUS);
window.setStatusBarColor(color);
}
}
private int getLighterShade(int rgb) {
float correctionFactor = 0.1f;
float red = (255 - Color.red(rgb)) * correctionFactor + Color.red(rgb);
float green = (255 - Color.green(rgb)) * correctionFactor + Color.green(rgb);
float blue = (255 - Color.blue(rgb)) * correctionFactor + Color.blue(rgb);
return Color.argb(Color.alpha(rgb), (int)red, (int)green, (int)blue);
}
private int getImageResource(String uri) {
int imageResource = getResources().
getIdentifier(uri, null, getPackageName());
return imageResource;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,509 |
Q: Swing vertical resizing one pane to expand other shrink (modified BorderLayout) Basically i would like to have BorderLayout that will expand north section vertically and shrink center section on resize. I think it's easier to explain with pictures.
So here is situation I have now:
But when dialog is resized i would like to have buttons expand vertically and to shrink down center pane (one with tabs). Currently when dialog is resized i get this (the rest of the buttons are "hidden":
My current layout is set like this:
contentPane - BorderLayout
panelTopButtons - FlowLayout (set in north section of conentPane)
panelContent - BorderLayout (set in center section of contentPane)
So how can i achieve when dialog is resized that buttons in north section take more vertical space (and thus all are visible), and content of panelContent is shrink to fit rest of dialog contentPane space?
A: Put the buttons in the CENTER and the tabbed pane in the PAGE_END of the BorderLayout.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,853 |
Q: how to get the center point of a scrollview subview relative to scrollview superview? I have a scollView(the blue one showing in the image below) containing 20 subviews(the white view).
the superView of scrollView is called pView, for example, the the center point of white view1 relative to pView is (10,10), and the center point of view6 is (496,10), after I scroll this the view as following:
the center point of view 6 is (10,10), but I don't know how to get it.
maybe I should use code like
CGPoint centerPoint = [view6 convertPoint:view6.center toView:pView];
but I didn't get the right point
A: view6.center is in the coordinate system of the superview of view6 (which is the UIScrollView). Try
CGPoint centerPoint = [scrollView convertPoint:view6.center toView:pView];
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,533 |
Ducati je italská firma zabývající se výrobou motocyklů. Založili ji v roce 1926 tři bratři Adriano, Marcello a Bruno Ducati v Bologni v Itálii a původně vyráběli elektronky, kondenzátory a různé součástky pro radiopřijímače. Motocykly se zde začaly vyrábět až po druhé světové válce. Motocykly Ducati jsou známé především desmodromickým rozvodem a svými úspěchy v Mistrovství světa superbiků. V současné době firmu vlastní společnost Audi.
Současná modelová řada Ducati
Panigale
Streetfighter
Multistrada
Monster
Hypermotard
Diavel
Nejznámější modely Ducati
Supersport
1098
999
916
851
Supersport
Reference
Externí odkazy
Italští výrobci motocyklů | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 265 |
<?xml version="1.0"?>
<project xmlns="http://maven.apache.org/POM/4.0.0" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://maven.apache.org/POM/4.0.0 http://maven.apache.org/maven-v4_0_0.xsd">
<parent>
<groupId>com.twelvemonkeys</groupId>
<artifactId>twelvemonkeys</artifactId>
<version>3.10.0-SNAPSHOT</version>
</parent>
<modelVersion>4.0.0</modelVersion>
<groupId>com.twelvemonkeys.servlet</groupId>
<artifactId>servlet</artifactId>
<name>TwelveMonkeys :: Servlet</name>
<dependencies>
<dependency>
<groupId>javax.servlet</groupId>
<artifactId>servlet-api</artifactId>
<version>2.4</version>
<scope>provided</scope>
</dependency>
<dependency>
<groupId>junit</groupId>
<artifactId>junit</artifactId>
<version>4.13.2</version>
<scope>test</scope>
</dependency>
<dependency>
<groupId>org.mockito</groupId>
<artifactId>mockito-core</artifactId>
<version>4.1.0</version>
<scope>test</scope>
</dependency>
</dependencies>
<build>
<plugins>
<plugin>
<artifactId>maven-source-plugin</artifactId>
</plugin>
<plugin>
<artifactId>maven-resources-plugin</artifactId>
<configuration>
<encoding>UTF-8</encoding>
</configuration>
</plugin>
<plugin>
<groupId>org.apache.maven.plugins</groupId>
<artifactId>maven-jar-plugin</artifactId>
<version>2.4</version>
<configuration>
<archive>
<manifestEntries>
<Implementation-Title>${project.name}</Implementation-Title>
<Implementation-Vendor>TwelveMonkeys</Implementation-Vendor>
<Implementation-Version>${project.version}</Implementation-Version>
<Implementation-URL>http://github.com/haraldk/TwelveMonkeys</Implementation-URL>
</manifestEntries>
</archive>
</configuration>
</plugin>
<plugin>
<groupId>org.apache.maven.plugins</groupId>
<artifactId>maven-shade-plugin</artifactId>
<version>3.2.2</version>
<executions>
<execution>
<id>jakarta</id>
<phase>package</phase>
<goals>
<goal>shade</goal>
</goals>
<configuration>
<shadedArtifactAttached>true</shadedArtifactAttached>
<shadedClassifierName>jakarta</shadedClassifierName>
<createDependencyReducedPom>false</createDependencyReducedPom>
<artifactSet>
<includes>
<include>${project.groupId}:${project.artifactId}</include>
</includes>
</artifactSet>
<relocations>
<relocation>
<pattern>javax.servlet</pattern>
<shadedPattern>jakarta.servlet</shadedPattern>
</relocation>
</relocations>
</configuration>
</execution>
</executions>
</plugin>
</plugins>
</build>
</project>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,607 |
import os
from os import path as op
import numpy as np
from .. import pick_types, pick_info
from ..io.pick import _has_kit_refs
from ..io import read_info, _loc_to_coil_trans, _loc_to_eeg_loc
from ..io.meas_info import Info
from ..io.constants import FIFF
from .forward import Forward, write_forward_solution, _merge_meg_eeg_fwds
from ._compute_forward import _compute_forwards
from ..transforms import (_ensure_trans, transform_surface_to, apply_trans,
_get_mri_head_t, _print_coord_trans,
_coord_frame_name, Transform)
from ..utils import logger, verbose
from ..source_space import _ensure_src, _filter_source_spaces
from ..surface import _normalize_vectors
from ..bem import read_bem_solution, _bem_find_surface, ConductorModel
from ..externals.six import string_types
_accuracy_dict = dict(normal=FIFF.FWD_COIL_ACCURACY_NORMAL,
accurate=FIFF.FWD_COIL_ACCURACY_ACCURATE)
@verbose
def _read_coil_defs(fname=None, elekta_defs=False, verbose=None):
"""Read a coil definition file.
Parameters
----------
fname : str
The name of the file from which coil definitions are read.
elekta_defs : bool
If true, use Elekta's coil definitions for numerical integration
(from Abramowitz and Stegun section 25.4.62).
verbose : bool, str, int, or None
If not None, override default verbose level (see mne.verbose).
Defaults to raw.verbose.
Returns
-------
res : list of dict
The coils. It is a dictionary with valid keys:
'cosmag' | 'coil_class' | 'coord_frame' | 'rmag' | 'type' |
'chname' | 'accuracy'.
cosmag contains the direction of the coils and rmag contains the
position vector.
"""
if fname is None:
if not elekta_defs:
fname = op.join(op.split(__file__)[0], '..', 'data',
'coil_def.dat')
else:
fname = op.join(op.split(__file__)[0], '..', 'data',
'coil_def_Elekta.dat')
big_val = 0.5
with open(fname, 'r') as fid:
lines = fid.readlines()
res = dict(coils=list())
lines = lines[::-1]
while len(lines) > 0:
line = lines.pop()
if line[0] != '#':
vals = np.fromstring(line, sep=' ')
assert len(vals) in (6, 7) # newer numpy can truncate comment
start = line.find('"')
end = len(line.strip()) - 1
assert line.strip()[end] == '"'
desc = line[start:end]
npts = int(vals[3])
coil = dict(coil_type=vals[1], coil_class=vals[0], desc=desc,
accuracy=vals[2], size=vals[4], base=vals[5])
# get parameters of each component
rmag = list()
cosmag = list()
w = list()
for p in range(npts):
# get next non-comment line
line = lines.pop()
while(line[0] == '#'):
line = lines.pop()
vals = np.fromstring(line, sep=' ')
assert len(vals) == 7
# Read and verify data for each integration point
w.append(vals[0])
rmag.append(vals[[1, 2, 3]])
cosmag.append(vals[[4, 5, 6]])
w = np.array(w)
rmag = np.array(rmag)
cosmag = np.array(cosmag)
size = np.sqrt(np.sum(cosmag ** 2, axis=1))
if np.any(np.sqrt(np.sum(rmag ** 2, axis=1)) > big_val):
raise RuntimeError('Unreasonable integration point')
if np.any(size <= 0):
raise RuntimeError('Unreasonable normal')
cosmag /= size[:, np.newaxis]
coil.update(dict(w=w, cosmag=cosmag, rmag=rmag))
res['coils'].append(coil)
logger.info('%d coil definitions read', len(res['coils']))
return res
def _create_meg_coil(coilset, ch, acc, t):
"""Create a coil definition using templates, transform if necessary"""
# Also change the coordinate frame if so desired
if t is None:
t = Transform('meg', 'meg', np.eye(4)) # identity, no change
if ch['kind'] not in [FIFF.FIFFV_MEG_CH, FIFF.FIFFV_REF_MEG_CH]:
raise RuntimeError('%s is not a MEG channel' % ch['ch_name'])
# Simple linear search from the coil definitions
for coil in coilset['coils']:
if coil['coil_type'] == (ch['coil_type'] & 0xFFFF) and \
coil['accuracy'] == acc:
break
else:
raise RuntimeError('Desired coil definition not found '
'(type = %d acc = %d)' % (ch['coil_type'], acc))
# Apply a coordinate transformation if so desired
coil_trans = np.dot(t['trans'], _loc_to_coil_trans(ch['loc']))
# Create the result
res = dict(chname=ch['ch_name'], coil_class=coil['coil_class'],
accuracy=coil['accuracy'], base=coil['base'], size=coil['size'],
type=ch['coil_type'], w=coil['w'], desc=coil['desc'],
coord_frame=t['to'], rmag=apply_trans(coil_trans, coil['rmag']),
cosmag=apply_trans(coil_trans, coil['cosmag'], False))
res.update(ex=coil_trans[:3, 0], ey=coil_trans[:3, 1],
ez=coil_trans[:3, 2], r0=coil_trans[:3, 3])
return res
def _create_eeg_el(ch, t=None):
"""Create an electrode definition, transform coords if necessary"""
if ch['kind'] != FIFF.FIFFV_EEG_CH:
raise RuntimeError('%s is not an EEG channel. Cannot create an '
'electrode definition.' % ch['ch_name'])
if t is None:
t = Transform('head', 'head', np.eye(4)) # identity, no change
if t.from_str != 'head':
raise RuntimeError('Inappropriate coordinate transformation')
r0ex = _loc_to_eeg_loc(ch['loc'])
if r0ex.shape[1] == 1: # no reference
w = np.array([1.])
else: # has reference
w = np.array([1., -1.])
# Optional coordinate transformation
r0ex = apply_trans(t['trans'], r0ex.T)
# The electrode location
cosmag = r0ex.copy()
_normalize_vectors(cosmag)
res = dict(chname=ch['ch_name'], coil_class=FIFF.FWD_COILC_EEG, w=w,
accuracy=_accuracy_dict['normal'], type=ch['coil_type'],
coord_frame=t['to'], rmag=r0ex, cosmag=cosmag)
return res
def _create_meg_coils(chs, acc=None, t=None, coilset=None):
"""Create a set of MEG or EEG coils in the head coordinate frame"""
acc = _accuracy_dict[acc] if isinstance(acc, string_types) else acc
coilset = _read_coil_defs(verbose=False) if coilset is None else coilset
coils = [_create_meg_coil(coilset, ch, acc, t) for ch in chs]
return coils
def _create_eeg_els(chs):
"""Create a set of MEG or EEG coils in the head coordinate frame"""
return [_create_eeg_el(ch) for ch in chs]
@verbose
def _setup_bem(bem, bem_extra, neeg, mri_head_t, verbose=None):
"""Set up a BEM for forward computation"""
logger.info('')
if isinstance(bem, string_types):
logger.info('Setting up the BEM model using %s...\n' % bem_extra)
bem = read_bem_solution(bem)
if not isinstance(bem, ConductorModel):
raise TypeError('bem must be a string or ConductorModel')
if bem['is_sphere']:
logger.info('Using the sphere model.\n')
if len(bem['layers']) == 0:
raise RuntimeError('Spherical model has zero layers')
if bem['coord_frame'] != FIFF.FIFFV_COORD_HEAD:
raise RuntimeError('Spherical model is not in head coordinates')
else:
if neeg > 0 and len(bem['surfs']) == 1:
raise RuntimeError('Cannot use a homogeneous model in EEG '
'calculations')
logger.info('Employing the head->MRI coordinate transform with the '
'BEM model.')
# fwd_bem_set_head_mri_t: Set the coordinate transformation
bem['head_mri_t'] = _ensure_trans(mri_head_t, 'head', 'mri')
logger.info('BEM model %s is now set up' % op.split(bem_extra)[1])
logger.info('')
return bem
@verbose
def _prep_meg_channels(info, accurate=True, exclude=(), ignore_ref=False,
elekta_defs=False, verbose=None):
"""Prepare MEG coil definitions for forward calculation
Parameters
----------
info : instance of Info
The measurement information dictionary
accurate : bool
If true (default) then use `accurate` coil definitions (more
integration points)
exclude : list of str | str
List of channels to exclude. If 'bads', exclude channels in
info['bads']
ignore_ref : bool
If true, ignore compensation coils
verbose : bool, str, int, or None
If not None, override default verbose level (see mne.verbose).
Defaults to raw.verbose.
Returns
-------
megcoils : list of dict
Information for each prepped MEG coil
compcoils : list of dict
Information for each prepped MEG coil
megnames : list of str
Name of each prepped MEG coil
meginfo : Info
Information subselected for just the set of MEG coils
"""
accuracy = 'accurate' if accurate else 'normal'
info_extra = 'info'
meg_info = None
megnames, megcoils, compcoils = [], [], []
# Find MEG channels
picks = pick_types(info, meg=True, eeg=False, ref_meg=False,
exclude=exclude)
# Make sure MEG coils exist
nmeg = len(picks)
if nmeg <= 0:
raise RuntimeError('Could not find any MEG channels')
# Get channel info and names for MEG channels
megchs = pick_info(info, picks)['chs']
megnames = [info['ch_names'][p] for p in picks]
logger.info('Read %3d MEG channels from %s'
% (len(picks), info_extra))
# Get MEG compensation channels
if not ignore_ref:
picks = pick_types(info, meg=False, ref_meg=True, exclude=exclude)
ncomp = len(picks)
if (ncomp > 0):
compchs = pick_info(info, picks)['chs']
logger.info('Read %3d MEG compensation channels from %s'
% (ncomp, info_extra))
# We need to check to make sure these are NOT KIT refs
if _has_kit_refs(info, picks):
err = ('Cannot create forward solution with KIT reference '
'channels. Consider using "ignore_ref=True" in '
'calculation')
raise NotImplementedError(err)
else:
ncomp = 0
_print_coord_trans(info['dev_head_t'])
# Make info structure to allow making compensator later
ncomp_data = len(info['comps'])
ref_meg = True if not ignore_ref else False
picks = pick_types(info, meg=True, ref_meg=ref_meg, exclude=exclude)
meg_info = pick_info(info, picks) if nmeg > 0 else None
# Create coil descriptions with transformation to head or MRI frame
templates = _read_coil_defs(elekta_defs=elekta_defs)
megcoils = _create_meg_coils(megchs, accuracy, info['dev_head_t'],
templates)
if ncomp > 0:
logger.info('%d compensation data sets in %s' % (ncomp_data,
info_extra))
compcoils = _create_meg_coils(compchs, 'normal', info['dev_head_t'],
templates)
logger.info('Head coordinate MEG coil definitions created.')
return megcoils, compcoils, megnames, meg_info
@verbose
def _prep_eeg_channels(info, exclude=(), verbose=None):
"""Prepare EEG electrode definitions for forward calculation
Parameters
----------
info : instance of Info
The measurement information dictionary
exclude : list of str | str
List of channels to exclude. If 'bads', exclude channels in
info['bads']
verbose : bool, str, int, or None
If not None, override default verbose level (see mne.verbose).
Defaults to raw.verbose.
Returns
-------
eegels : list of dict
Information for each prepped EEG electrode
eegnames : list of str
Name of each prepped EEG electrode
"""
eegnames, eegels = [], []
info_extra = 'info'
# Find EEG electrodes
picks = pick_types(info, meg=False, eeg=True, ref_meg=False,
exclude=exclude)
# Make sure EEG electrodes exist
neeg = len(picks)
if neeg <= 0:
raise RuntimeError('Could not find any EEG channels')
# Get channel info and names for EEG channels
eegchs = pick_info(info, picks)['chs']
eegnames = [info['ch_names'][p] for p in picks]
logger.info('Read %3d EEG channels from %s' % (len(picks), info_extra))
# Create EEG electrode descriptions
eegels = _create_eeg_els(eegchs)
logger.info('Head coordinate coil definitions created.')
return eegels, eegnames
@verbose
def _prepare_for_forward(src, mri_head_t, info, bem, mindist, n_jobs,
bem_extra='', trans='', info_extra='',
meg=True, eeg=True, ignore_ref=False, fname=None,
overwrite=False, verbose=None):
"""Helper to prepare for forward computation"""
# Read the source locations
logger.info('')
# let's make a copy in case we modify something
src = _ensure_src(src).copy()
nsource = sum(s['nuse'] for s in src)
if nsource == 0:
raise RuntimeError('No sources are active in these source spaces. '
'"do_all" option should be used.')
logger.info('Read %d source spaces a total of %d active source locations'
% (len(src), nsource))
# Delete some keys to clean up the source space:
for key in ['working_dir', 'command_line']:
if key in src.info:
del src.info[key]
# Read the MRI -> head coordinate transformation
logger.info('')
_print_coord_trans(mri_head_t)
# make a new dict with the relevant information
arg_list = [info_extra, trans, src, bem_extra, fname, meg, eeg,
mindist, overwrite, n_jobs, verbose]
cmd = 'make_forward_solution(%s)' % (', '.join([str(a) for a in arg_list]))
mri_id = dict(machid=np.zeros(2, np.int32), version=0, secs=0, usecs=0)
info = Info(nchan=info['nchan'], chs=info['chs'], comps=info['comps'],
ch_names=info['ch_names'], dev_head_t=info['dev_head_t'],
mri_file=trans, mri_id=mri_id, meas_file=info_extra,
meas_id=None, working_dir=os.getcwd(),
command_line=cmd, bads=info['bads'], mri_head_t=mri_head_t)
logger.info('')
megcoils, compcoils, megnames, meg_info = [], [], [], []
eegels, eegnames = [], []
if meg and len(pick_types(info, ref_meg=False, exclude=[])) > 0:
megcoils, compcoils, megnames, meg_info = \
_prep_meg_channels(info, ignore_ref=ignore_ref)
if eeg and len(pick_types(info, meg=False, eeg=True, ref_meg=False,
exclude=[])) > 0:
eegels, eegnames = _prep_eeg_channels(info)
# Check that some channels were found
if len(megcoils + eegels) == 0:
raise RuntimeError('No MEG or EEG channels found.')
# pick out final info
info = pick_info(info, pick_types(info, meg=meg, eeg=eeg, ref_meg=False,
exclude=[]))
# Transform the source spaces into the appropriate coordinates
# (will either be HEAD or MRI)
for s in src:
transform_surface_to(s, 'head', mri_head_t)
logger.info('Source spaces are now in %s coordinates.'
% _coord_frame_name(s['coord_frame']))
# Prepare the BEM model
bem = _setup_bem(bem, bem_extra, len(eegnames), mri_head_t)
# Circumvent numerical problems by excluding points too close to the skull
if not bem['is_sphere']:
inner_skull = _bem_find_surface(bem, 'inner_skull')
_filter_source_spaces(inner_skull, mindist, mri_head_t, src, n_jobs)
logger.info('')
rr = np.concatenate([s['rr'][s['vertno']] for s in src])
# deal with free orientations:
source_nn = np.tile(np.eye(3), (len(rr), 1))
update_kwargs = dict(nchan=len(info['ch_names']), nsource=len(rr),
info=info, src=src, source_nn=source_nn,
source_rr=rr, surf_ori=False, mri_head_t=mri_head_t)
return megcoils, meg_info, compcoils, megnames, eegels, eegnames, rr, \
info, update_kwargs, bem
@verbose
def make_forward_solution(info, trans, src, bem, fname=None, meg=True,
eeg=True, mindist=0.0, ignore_ref=False,
overwrite=False, n_jobs=1, verbose=None):
"""Calculate a forward solution for a subject
Parameters
----------
info : instance of mne.io.meas_info.Info | str
If str, then it should be a filename to a Raw, Epochs, or Evoked
file with measurement information. If dict, should be an info
dict (such as one from Raw, Epochs, or Evoked).
trans : dict | str | None
Either a transformation filename (usually made using mne_analyze)
or an info dict (usually opened using read_trans()).
If string, an ending of `.fif` or `.fif.gz` will be assumed to
be in FIF format, any other ending will be assumed to be a text
file with a 4x4 transformation matrix (like the `--trans` MNE-C
option). Can be None to use the identity transform.
src : str | instance of SourceSpaces
If string, should be a source space filename. Can also be an
instance of loaded or generated SourceSpaces.
bem : dict | str
Filename of the BEM (e.g., "sample-5120-5120-5120-bem-sol.fif") to
use, or a loaded sphere model (dict).
fname : str | None
Destination forward solution filename. If None, the solution
will not be saved.
meg : bool
If True (Default), include MEG computations.
eeg : bool
If True (Default), include EEG computations.
mindist : float
Minimum distance of sources from inner skull surface (in mm).
ignore_ref : bool
If True, do not include reference channels in compensation. This
option should be True for KIT files, since forward computation
with reference channels is not currently supported.
overwrite : bool
If True, the destination file (if it exists) will be overwritten.
If False (default), an error will be raised if the file exists.
n_jobs : int
Number of jobs to run in parallel.
verbose : bool, str, int, or None
If not None, override default verbose level (see mne.verbose).
Returns
-------
fwd : instance of Forward
The forward solution.
See Also
--------
do_forward_solution
Notes
-----
Some of the forward solution calculation options from the C code
(e.g., `--grad`, `--fixed`) are not implemented here. For those,
consider using the C command line tools or the Python wrapper
`do_forward_solution`.
"""
# Currently not (sup)ported:
# 1. --grad option (gradients of the field, not used much)
# 2. --fixed option (can be computed post-hoc)
# 3. --mricoord option (probably not necessary)
# read the transformation from MRI to HEAD coordinates
# (could also be HEAD to MRI)
mri_head_t, trans = _get_mri_head_t(trans)
bem_extra = 'dict' if isinstance(bem, dict) else bem
if fname is not None and op.isfile(fname) and not overwrite:
raise IOError('file "%s" exists, consider using overwrite=True'
% fname)
if not isinstance(info, (dict, string_types)):
raise TypeError('info should be a dict or string')
if isinstance(info, string_types):
info_extra = op.split(info)[1]
info = read_info(info, verbose=False)
else:
info_extra = 'info dict'
# Report the setup
logger.info('Source space : %s' % src)
logger.info('MRI -> head transform source : %s' % trans)
logger.info('Measurement data : %s' % info_extra)
if isinstance(bem, dict) and bem['is_sphere']:
logger.info('Sphere model : origin at %s mm'
% (bem['r0'],))
logger.info('Standard field computations')
else:
logger.info('BEM model : %s' % bem_extra)
logger.info('Accurate field computations')
logger.info('Do computations in %s coordinates',
_coord_frame_name(FIFF.FIFFV_COORD_HEAD))
logger.info('Free source orientations')
logger.info('Destination for the solution : %s' % fname)
megcoils, meg_info, compcoils, megnames, eegels, eegnames, rr, info, \
update_kwargs, bem = _prepare_for_forward(
src, mri_head_t, info, bem, mindist, n_jobs, bem_extra, trans,
info_extra, meg, eeg, ignore_ref, fname, overwrite)
del (src, mri_head_t, trans, info_extra, bem_extra, mindist,
meg, eeg, ignore_ref)
# Time to do the heavy lifting: MEG first, then EEG
coil_types = ['meg', 'eeg']
coils = [megcoils, eegels]
ccoils = [compcoils, None]
infos = [meg_info, None]
megfwd, eegfwd = _compute_forwards(rr, bem, coils, ccoils,
infos, coil_types, n_jobs)
# merge forwards
fwd = _merge_meg_eeg_fwds(_to_forward_dict(megfwd, megnames),
_to_forward_dict(eegfwd, eegnames),
verbose=False)
logger.info('')
# Don't transform the source spaces back into MRI coordinates (which is
# done in the C code) because mne-python assumes forward solution source
# spaces are in head coords.
fwd.update(**update_kwargs)
if fname is not None:
logger.info('writing %s...', fname)
write_forward_solution(fname, fwd, overwrite, verbose=False)
logger.info('Finished.')
return fwd
def _to_forward_dict(fwd, names, fwd_grad=None,
coord_frame=FIFF.FIFFV_COORD_HEAD,
source_ori=FIFF.FIFFV_MNE_FREE_ORI):
"""Convert forward solution matrices to dicts"""
assert names is not None
if len(fwd) == 0:
return None
sol = dict(data=fwd.T, nrow=fwd.shape[1], ncol=fwd.shape[0],
row_names=names, col_names=[])
fwd = Forward(sol=sol, source_ori=source_ori, nsource=sol['ncol'],
coord_frame=coord_frame, sol_grad=None,
nchan=sol['nrow'], _orig_source_ori=source_ori,
_orig_sol=sol['data'].copy(), _orig_sol_grad=None)
if fwd_grad is not None:
sol_grad = dict(data=fwd_grad.T, nrow=fwd_grad.shape[1],
ncol=fwd_grad.shape[0], row_names=names,
col_names=[])
fwd.update(dict(sol_grad=sol_grad),
_orig_sol_grad=sol_grad['data'].copy())
return fwd
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,044 |
\section{Effect of App Characteristics}
\par All the studied promotions caused an increase in the sales of the promoted apps, but was the extend of this increase correlated with any of the characteristics of the app\footnote{The author would like to thank the anonymous CIST reviewer that suggested this research idea}? For example, when offered for free, do apps that are normally more expensive benefit more or less from apps that are cheaper? How about the previous popularity of the promoted apps? Do apps that are already popular and established benefit more or less from a promotion? Insights towards this direction can help practitioners understand what type of apps can benefit the most from each type of promotions, and design their marketing strategy accordingly.
\par In order to examine this question, we define a new variable, called {\it boost}, as the difference between the average number of reviews an app received during the 7 days of the promotion and the 7 days before the promotion started.
\par We focus on four characteristics of an app: popularity before the start of the promotion, normal price, age (in days), and size (in Mega Bytes). Popularity before the start of the promotion will help us see if there are any `rich-get-richer' effects, where apps that were already popular benefit disproportionally from price reductions and promotions. The regular price of the app can capture if, amongst apps that are offered for the same price (free or otherwise), the apps that are more expensive normally will attract larger interest. For example, more users that held back their purchases for more expensive apps can now have the opportunity to get it at a discount. Furthermore, the regular price can also be correlated with the quality of the app. The age of the app can capture if an app is established and had more time to build their user base and word of mouth, and the size in MegaBytes of the app can signal the level of sophistication that app has, e.g., state of the art graphics or many levels for a game.
\subsection{Econometric Model}
\par Since we don't always have the date that the first version of an app appeared in the AppStore, we approximate the age of an app by calculating the number of days between the start of the promotion the app was featured in, and the date of the {\it first review} it received. Then we calculate the number of reviews that an app received during its lifetime up to the start of the promotion. We then divide this number by the app's age to get the average daily number of reviews for that app up to the start of the promotion. We use this metric as an indicator for the popularity of the app before the start of the promotion. The size and price are as observed at the time of scraping, which was January 2016. Even though, there may be some differences between the observed size and price and that of the offered version, we believe that they still provide a reasonable approximation to the characteristics of the offered version.
\par In order to find correlations between the boost and the above characteristics, we estimate the following model
\begin{equation}\label{model_4}
\mathrm{boost}= \alpha_0 + \beta_1\mathrm{POP} + \beta_2\mathrm{age}+\beta_3\mathrm{size} +\beta_4\mathrm{price}+e,
\end{equation}
where POP is the app's previous popularity, as calculated above, and age, size and price the characteristics of the app as discussed above. The error term is $e$. We estimate Model \ref{model_4} four times, once for each promotion and our results are summarized in Table \ref{regressions_characteristics}.
\begin{table*}[h!]
\begin{center}
\vspace{0.5cm}
\begin{tabular}{l c c c c c}
\toprule
{\it \small Promotion} &Intercept &POP & age & size & price\\[6pt]
\midrule
\multirow{2}{*}{Apple Weekly} & $557.7493^{***}$ &$10.5821$ &$-0.2063$ &0.0765 &-30.8789\\[6pt]
&(135.980) &(15.860) &(0.174) &(0.177 ) &(32.711) \\[6pt]
\midrule
\multirow{2}{*}{Starbucks Digital} & $134.03$ & $11.18$ &$0.01^{***}$ & -0.08 &-6.00 \\[6pt
&(97.27) &(6.18) &(0.09) &(1.4) & (15.01) \\[6pt]
\midrule
\multirow{2}{*}{\small Starbucks In-Store} & $84.3671$ & $14.2763^{**}$ &$-0.0105$ &-0.04 &-11.99\\[6pt]
&(58.85) &(4.33) &(0.07) &(0.30) &(17.17)\\[6pt]
\midrule
\multirow{2}{*}{Apple Amazing} & $-1.13$ & $7.30^{***}$ &$1.29^{*}$ & $0.002$ &5.43 \\[6pt
&(14.93) &(1.29) &(0.01) &(0.01) &(3.36) \\[6pt
\bottomrule
\end{tabular}
\end{center}
Values in parentheses are standard errors.\\
\small{$^{*}:p<0.05$, $^{**}: p<0.01$, $^{***}:p<0.001$}
\caption{Effect of app characteristics in the success of promotions}
\label{regressions_characteristics}
\end{table*}
\subsection{Results}
We discuss here the results of the estimations of Model \ref{model_4}.
\paragraph{Regular Price} Unlike promotions on platforms like Groupon, none of the promotions we study in this paper display the normal price of the featured app\footnote{We noticed only one case where Apple displayed the normal price, of \$9.99, for an app featured in the promotion.}. Hence we expect users to not be sensitive to that information but, instead, only on the price they are called to pay. Indeed we see in Table \ref{regressions_characteristics} that the coefficient of the variable price is not significantly different from zero for any of the four studied promotions. This means that no matter the regular price, the featured apps in the studied promotions benefit, on average, the same as the other apps featured in the same promotion.
\par Practitioners should be aware that even if they are cutting down their margins more than other apps featured in the same promotion, as long as the final discounted price is the same, they will not receive any advantage during the promotion.
\paragraph{Previous Popularity} Table \ref{regressions_characteristics} shows that the previous popularity of an app doesn't matter when the promotion is low barrier (i.e., digital and free) but it does when it's high barrier (i.e., offline redemption or not free). In low barrier promotions users are equally likely to acquire an app that they are aware of as they are to acquire an app they haven't heard of, exactly because the promote is low barrier; they have little reason not to go through with it. But with high barrier promotions, users are more likely to do the effort required to redeem the coupon or pay the (reduced) price if they are aware of the app already.
\par Hence, we see that high barrier promotions induce a `rich-get-richer' effect, where the increase an app observes during the promotion is proportional to their previous popularity. In contrast, apps participating in low barrier promotions receive, on average, the same increase no matter their previous popularity.
\paragraph{Age} The age of an app can be an indicator of how established it is and how much time it had to build its user base and word of mouth. We see in Table \ref{regressions_characteristics} that the coefficient for the age variable is not significantly different from zero for all of the studied promotions except the Apple Amazing one, the only promotion that offers partial discount. This could indicate the users are more willing to pay even a reduced price for apps that have been in the market longer, and hence, perhaps, had the time to release multiple versions fixing any bugs and improving their product. In contrast, in free promotions the app's success is not dependent on it's age.
\paragraph{Size} Finally, we use the size of the offered app as an indicator for the level of sophistication of the source code as well as characteristics like graphics, number of gameplay levels (if the app is a game) and more. We see in Table \ref{regressions_characteristics}, that the coefficient for the size variable is not significantly different from zero for any of the studied promotions. This means that when promoted, apps of all sizes benefit, on average, the same from the promotion.
\par Our insights from this section can help practitioners better design their own promotions according to their apps. For examplee, apps that are newer and not yet very succesfull should avoid being featured in high barrier promotions alongside with more successful apps, since the more successful apps will disproportionally benefit from the promotion.
\section{Discussion and Insights}
\par We studied four promotions offered on Apple's iOS AppStore, that varied in scale of exposure, level of price discount and ease of redemption. We aim to understand better the effects that these promotions have on the sales as well as the ratings of the featured apps. We found positive effects on the sales for all promotions and mixed effects on the ratings. Notably, the only promotion that was not full price discounted was that only one that had a negative effect on the ratings of its featured apps whereas the weekly promotion run by Apple was the one with the largest positive effect.
\par We also explored the correlation between app characteristics and how succesfull their promotion campaign is. We found that when a promotion is low barrier, i.e., digital and free, users are equally likely to get an app no matter the app's previous popularity, regular price and age. In contrast, promotions with higher barriers, such as non-zero price or non-trivial redemption process, induce a rich-get-effect where users are more likely to get a promoted app if it's already established and successful.
\par Finally, our study also explored the effect of the promotions on the sales of their competitors. We found that full price discounts decrease the sales of their competitors, perhaps because users have no incentive to search for further alternatives. In contrast, the only partially discounted promotion in our study, Apple Amazing, caused an increase in the sales of it's competition. This suggests that the promotion raised awareness for the featured apps, but the non-free price tag incentivized users to hold back on their purchase until they explore further alternatives.
\par These insights can help practitioners better design their promotion campaigns. Our work displays the benefits in sales and ratings of being featured in carefully selected and far-reaching promotions, but also the potential risks of not full price discounts. We also show that apps are newer and not yet very popular should avoid being featured in high barrier promotions alongside with more successful apps, since then the more succesfull apps will disproportionally benefit from the promotion. Finally, we exhibit the positive and negative externalities that various types of promotions can have, which make developers more aware of the full extend of the effect that their promotions can have on their competitors, as well as the other way around too.
\section{Effect of Promotions on Ratings}
\par Having studied the effect of the promotion on sales, we turn our attention in the effect on ratings. A promotion is usually designed to attract users that would otherwise not buy a product with the intention to turn them into long-term paying customers. Does this increase in sales need always come with the risk of lower ratings, as observed in \cite{byers2012groupon}, or can practitioners mitigate undesired effects by designing the promotion carefully? An extended literature has shown the importance of positive ratings in the economics success of products and services \citep{}, hence any effects that the promotions will have on the ratings of featured products and services, can continue affecting their success long after the promotions are over.
\par Even though our work is very similar in spirit with the work of \cite{byers2012groupon}, our setting has important differences. Perhaps the most important is that the promoted apps are selected by Apple or Starbucks, two companies with their own complicated objectives and incentives, that may not always align entirely with the objectives and incentives of the app developers. Unlike settings such as Groupon, a developer cannot just add their app in the Apple Weekly promotion, it needs to be selected (or at least agreed by) Apple. This induces a selection procedure that makes our predictions for the effects on the ratings harder. Take for example the Apple Weekly promotion. As can be seen in Figure~\ref{rev_volume}, Apple Weekly is the promotion that causes the largest increase in sales. This large influx of new users can be risky since some of them may not be in the target group of the promoted app. On the other hand, the promotion is on the front page of the AppStore and that means that the users that are exposed to the banner are customers that are actively browsing for apps to download. The fact that Apple is endorsing the promoted apps can also induce some positive `social influence'-type bias.
\par An exploratory analysis of the immediate effects of the promotions on the ratings of the promoted apps, shows that the apps featured in the Apple Weekly promotion experience, on average, an increase of 0.4 stars, from 3.86 in the week before the promotion starts to 4.27 for the week after the promotion started. In the same time period, apps featured in the Starbucks Digital promotion doesn't seem to have been affected, with their ratings increasing only by 0.01 star (from 4.21 to 4.22). The ratings for apps featured in the Starbucks In-Store promotion increased by 0.14 star (from 4.33 to 4.47) and for apps featured in the Apple Amazing promotion decreased by 0.34 star (from 4.3 to 3.96).
\par Figure \ref{per_rank_evolution} shows a more detailed view on the effect that the promotions had on the ratings. On the left hand side, is the evolution of the daily average rating for the two month period offset at the start of the promotion (all promotions run for 7 days) alongside the average rating for each 30 day period. On the right hand side, is the actual distributions of the ratings for each 30 day period (before and after the start of the promotion). Figure \ref{per_rank_evolution} provides further support for all the descriptive statistics we mentioned above: ratings become substantially more positive for the Apple Weekly apps, remain stable or go slightly up for the Starbucks promotions and there is a sudden decrease for the Apple Amazing apps, but one that seems to be partially corrected after the end of the promotion.
\begin{figure*}
\begin{center}
\begin{subfigure}[t]{\textwidth}
\centering
\begin{subfigure}[t]{0.35\textwidth}
\centering
\includegraphics[width=\textwidth]{per_day_around_offer_day_apple_weekly.pdf}
\end{subfigure}
\quad
\begin{subfigure}[t]{0.35\textwidth}
\includegraphics[width=\textwidth]{bfr_aft_dist_apple.pdf}
\end{subfigure}
\caption{Apple Weekly}
\label{apple_rating_evol}
\end{subfigure}
\quad
\begin{subfigure}[t]{\textwidth}
\centering
\begin{subfigure}[t]{0.35\textwidth}
\centering
\includegraphics[width=\textwidth]{per_day_around_offer_day_starb_dig.pdf}
\end{subfigure}
\quad
\begin{subfigure}[t]{0.35\textwidth}
\includegraphics[width=\textwidth]{bfr_aft_dist_starb_dig.pdf}
\end{subfigure}
\caption{Starbucks Digital}
\label{starb_dig}
\end{subfigure}
\quad
\begin{subfigure}[t]{\textwidth}
\centering
\begin{subfigure}[t]{0.35\textwidth}
\centering
\includegraphics[width=\textwidth]{per_day_around_offer_day_starb_instore.pdf}
\end{subfigure}
\quad
\begin{subfigure}[t]{0.35\textwidth}
\includegraphics[width=\textwidth]{bfr_aft_dist_starb_instore.pdf}
\end{subfigure}
\caption{Starbucks In-Store}
\label{starb_in_store}
\end{subfigure}
\quad
\begin{subfigure}[t]{\textwidth}
\centering
\begin{subfigure}[t]{0.35\textwidth}
\centering
\includegraphics[width=\textwidth]{per_day_around_offer_smaller.pdf}
\end{subfigure}
\quad
\begin{subfigure}[t]{0.35\textwidth}
\centering
\includegraphics[width=\textwidth]{bfr_aft_dist_smaller.pdf}
\end{subfigure}
\caption{Apple Amazing}
\label{apple_am}
\end{subfigure}
\end{center}
\textit{On the Left:} Evolution of daily average rating for a two month period centered around the offer date (dotted vertical line) for the four studied promotions.\\ \textit{On the right:} the distribution of ratings for the same time period.
\caption{Daily Average Star Rating, Before and After the Promotions}
\label{per_rank_evolution}
\end{figure*}
\subsection{The Econometric Model}
We estimate the following model in order to provide statistical support for the descriptive results shown in Figure \ref{per_rank_evolution}.
\begin{equation}\label{model_2}
\mathrm{rating}= \alpha_0 + \beta_1\mathrm{post} +\beta_2\mathrm{treat} + \beta_3\mathrm{post}\cdot\mathrm{treat}+e
\end{equation}
The post variable is a binary indicator if the datapoint is before the start date of the promotion and the treat variable is a binary indicator if the datapoint is from an app that was part of a promotion. The interaction variable post$\cdot$treat is the effect that is unique to the promoted apps after the promotion started. Hence, the coefficient of that variable estimates the effect on the promoted apps that cannot be explained by the control dataset, i.e., the causal effect of the promotion. The error term is $e$.
\par As we did with the regressions for the effect on sales, we restrict our attention on a period of two weeks before and two weeks after the start date of the promotions. This makes the time window long enough to capture the immediate effects of the promotions while the promotions are in progress as well as one week after they are over. For this time frame, our dataset is comprised of 88,172 ratings for the 297 treatment apps and 81,257 ratings for 2635 control apps.
\par We estimate Model \ref{model_volume} four times, once for each promotion and summarize our results in Table \ref{regressions_1}
\begin{table*}[h!]
\begin{center}
\vspace{0.5cm}
\begin{tabular}{l c c c c c}
\toprule
{\it \small Promotion} &Intercept &post &treat & post$\cdot$treat\\[6pt
\midrule
\multirow{2}{*}{Apple Weekly} & $4.0104^{***}$ &$-0.01084$ &$-0.1458^{**}$ & $0.4167^{***}$ \\[6pt]
&(0.017) &(0.024) &(0.046) &(0.050) \\[6pt]
\midrule
\multirow{2}{*}{\small Starbucks Digital} & $4.1758^{***}$ & $-0.0029$ &$0.052$ & -0.001\\[6pt]
&(0.021) &(0.03) &(0.05) &(0.05) &\\[6pt]
\midrule
\multirow{2}{*}{Starbucks In-Store} & $4.4737^{***}$ & $0.0307$ &$-0.1369^{***}$ & $0.11^{***}$ \\[6pt
&(0.014) &(0.018) &(0.037) &(0.04) &\\[6pt]
\midrule
\multirow{2}{*}{Apple Amazing} & $4.2258^{***}$ & $-0.1043^{***}$ &$0.0769$ & $-0.2320^{***}$ \\[6pt
&(0.022) &(0.029) &(0.045) &(0.054) & \\[6pt
\bottomrule
\end{tabular}
\end{center}
Values in parentheses are standard errors.\\
\small{$^{*}:p<0.05$, $^{**}: p<0.01$, $^{***}:p<0.001$}
\caption{Effect of the promotion on the short term ratings of the featured apps.}
\label{regressions_1}
\end{table*}
\subsection{Results}
We discuss here the results of our statistical analysis and how they tie up with our descriptive results.
\par The coefficient for post$\cdot$treat when Model \ref{model_2} is run on the Apple Weekly dataset is positive and highly significant. This confirms that our descriptive findings, that the Apple Weekly promotion has a positive effect on the ratings of the featured apps, is statistically significant. Figure \ref{apple_rating_evol} also shows that the positive effect in the ratings seems to be long term, one that remains even after the promotion is over. A look at the distributions of the ratings shows a substantial increase of 5-star ratings and decrease of 1, 2 and 3-star ratings, further confirming the shift towards more positive ratings.
\par The estimations of the coefficient for the post$\cdot$treat variable for the Starbucks promotions, shown in the second and third row of Table \ref{regressions_1}, provide statistical confirmation for our descriptive results about the effect of these promotions had on the ratings of the featured apps. For the digital version of the promotion the coefficient is not significantly different from zero but for the in-store version, the coefficient is positive and significant.
A look at the evolution of the daily average rating for the Starbucks Digital apps, in Figure \ref{starb_dig}, shows that that ratings didn't shift very much but seem to have been stabilized, perhaps because of the increased volume of reviews. The changes in the distributions of the ratings show a slight decline in 1 and 5-star ratings and a slight increase in 4-star ratings. For Starbucks In-Store, Figure \ref{starb_in_store} ,the average ratings seem to increase slightly and stabilize. Again, this stabilization could be because of the increased volume of reviews. Moreover, a comparison of the distributions of the ratings before and after the start of the promotion, show an increase of 4-star and decrease of 1-star ratings.
\par The in-store promotion involves a customer noticing the printed coupon in a physical location, picking it up and then redeeming it on their phone. In contrast, the digital version of the promotion offers the users the ability to claim their free app with only two taps from the within the Starbucks iOS app. Hence, we ask, what types of customers are more likely to go through each process? Customers are more likely to go through with the offline redemption procedure if they are positively predisposed towards the promoted app, e.g., they are aware of the app from their friends or from some other source. Hence, this small barrier seems to be effective in filtering out users that wouldn't be in the target audience of the group and wouldn't enjoy it.
\par Even though the Starbucks Digital and Apple Weekly promotion are both digital and free, only Apple's promotion causes an increase in the ratings. This could be due to few factors. First, it could be that the selection procedure of Apple is such that it discovers high quality apps and presents them to their users. From a private conversation the author had with a person in charge of the Starbucks promotions, the selections process for the Starbucks apps seems to be essentially the same as the one for the Apple promotions, with Apple needing to confirm the selections made by Starbucks before they go live on either of the two Starbucks promotions. Hence, another factor is the incentives and goals of each company. Starbucks seems to be offering the promotion as an additional service to their customers and our data show that they tend to choose apps that are higher rated than the Apple Weekly apps. This could perhaps be because Starbucks wants to offer their customers already popular and risk-free apps, whereas Apple could have incentives to choose more niche apps, in order to help their customers discover even more great apps than what they already know. Finally, users that are exposed to the Apple Weekly promotion are users that are already browsing the AppStore hence they are already actively looking for apps to purchase. This seamless and well timed integration of the promotion in the user experience could be why we see the different effects in the ratings of their featured apps. Moreover, the apps featured in the Apple Weekly promotion are promoted and endorsed by Apple itself, hence an iOS user could be predisposed positively towards them.
\par The coefficient of the post$\cdot$treat variable for the Apple Amazing promotion, is highly significant and the only, out of the four studied promotions, that is negative. This is in agreement with our descriptive results, shown in Figure \ref{apple_am}, that show a sudden decrease in the ratings immediately following the start of the Apple Amazing promotion. Figure \ref{apple_am} also shows that this decrease seems to be partially corrected as time goes on after the end of the promotion. A look at the distributions of the ratings before and after the promotion shows a decrease in 5-star and increase in 1-star ratings hence further confirming the shift towards lower ratings.
\par This decrease must have something to do with the fact that the Apple Amazing promotion is the only one not offering a full discount. Even though the discount is around 75\%, with the average normal price for the Apple Amazing promotions being around \$4, the \$0.99 price tag is still higher than the median (\$0) and only slightly lower than the mean (\$1.2) app price in the App Store. Note that users that are exposed to the Apple Amazing promotion don't get information about the app's original price. Hence, from a customer's point of view they are offered an app that in comparison to the average app in the store, is only slightly cheaper. Hence, the promotion and the endorsement by Apple may have incentivized some users to purchase the app even though they would not have buy otherwise. The price-tag may have then caused these users to be dissatisfied and submitting a low rating.
\par Note that the Apple Amazing promotion is the one most similar to promotions on platforms like Groupon, because it's the only one where the apps are not given out for free. It's interesting that it's the only promotion we find to have a negative effect on the ratings. We think this is a finding that can contribute in further understanding of the `Groupon phenomeon' \citep{byers2012groupon}, and one that practitioners should take into account when designing their promotions.
\section{A Tale of Four Promotions}
\section{Effect of Promotions on Sales}\label{sec_volume}
\par One of the first metrics that can help evaluate the success of a promotion is the increase in sales it caused for the participating apps. Hence we investigate here both the immediate as well as the longer term increase that the four studied promotions caused for their featured apps.
\par Generally, we would expect low barrier promotions, i.e., free and easy redemption process, to outperform higher barrier promotions such as promotions with less trivial redemption procedures or not fully discounted. Hence, we expect the Apple Weekly and Starbucks Digital promotions to cause a larger increase than the Starbucks In-Store and Apple Weekly. Furthermore, since the Apple Weekly promotion features a banner in the front page of the App Store driving awareness, we expect the Apple Weekly promotion to outperform the Starbucks Digital one. Amongst the two high barrier promotions, we expect that the less trivial redemption procedure will outperform partially discounted promotion.
\par An exploratory analysis of our dataset provides support for all the predictions above. The average Apple Weekly app goes from 1.45 reviews per day, in the 7 days before the promotion starts, to 53.6 reviews per day in the 7 days of the promotion ,an increase of almost 3600\%. For the same time periods, the Starbucks Digital apps go from 1.4 to 20 reviews per day (1800\% increase), the Starbucks In-Store apps go from 1.75 to 12 reviews per day (1025\%) and the Apple Amazing apps go from 2.6 to 5.75 reviews per day (124\% increase). Table~\ref{average_increase} displays the short term (one week after and one week before the start of the promotion) and long term (one month after and one month before the start of the promotion) increase in number of daily reviews. We show both the absolute as well as percentage increase. Note that all promotions studied in this paper run for 7 days.
\par Figure~\ref{rev_volume} displays the evolution of the average daily number of reviews per app for 60 day interval offset at the beginning of the promotions, and it provides further support for our exploratory findings.
\begin{figure}[h!]
\centering
\begin{subfigure}[t]{.6\textwidth}
\centering
\includegraphics[width=\textwidth]{review_volume.pdf}
\caption{Daily Average Review Volume, Before and After the Promotions}
\label{rev_volume}
\end{subfigure}
\quad
\begin{subfigure}[t]{.56\textwidth}
\centering
\begin{tabular}[b]{p{3.4cm} c c }
\toprule
&\small{7-day Effect} & \small{30-day Effect} \\[6pt]
\midrule
Apple Amazing &+52.15 (+3596\%) & +18.05 (+1612\%) \\[6pt]
Starbucks Digital &+18.6 (+1349.3\%) & +8.3 (+561.6\%) \\[6pt]
Starbucks~In-Store &+10.2 (+585.7\%) & +5.6 (+279.4\%) \\[6pt]
Apple Amazing &+3.2 (+124.1\%) & +1.34 (+61.8\%) \\[6pt]
\bottomrule
\end{tabular}
\caption{Increase in average daily volume of reviews per app. Values in parentheses represent how does the increase compare to the pre-promotion average}
\label{average_increase}
\end{subfigure}
\caption{The effect of the promotions on the sales of the apps}
\label{review_volume}
\end{figure}
\subsection{The Econometric Model}
In order to provide statistical rigor to our descriptive findings, we estimate the following model.
\begin{equation}\label{model_volume}
\mathrm{sales}= \alpha_0 + \beta_1\mathrm{post} +\beta_2\mathrm{treat} + \beta_3\mathrm{post}\cdot\mathrm{treat}+e.
\end{equation}
The post variable is an indicator if the datapoint is after the promotion's starting date. The treat variable is an indicator if the datapoint is from an app that was part of a promotion (i.e., treatment) or not (i.e., control). The interaction variable post$\cdot$treat is the one that estimates the effect of the promotion, i.e., the amount of variation in the data that cannot be explained by the control set of reviews. The error term is $e$.
\par For the purposes of the model estimations, we restrict our data only on the period of two weeks before and two weeks after the start of the promotion. This time frame captures the effect of the promotion, while the promotion is still active as well as one week after it's over. For this time period, our dataset is comprised of 88,172 ratings for 297 treatment apps and 81,257 ratings for 2635 control apps.
\par We estimate Model \ref{model_volume} four times, once for each promotion. In doing so, we estimate effect of each promotion on the sales of the featured apps. Table \ref{regressions_volume} summarizes our estimation results.
\begin{table*}[h!]
\begin{center}
\vspace{0.5cm}
\begin{tabular}{l c c c c c}
\toprule
{\it \small Promotion} &Intercept &post & treat & post$\cdot$ treat\\[6pt
\midrule
\multirow{2}{*}{Apple Weekly} & $5.6977^{***}$ &$-0.7120$ &$-2.3412$ & $35.6709^{***}$ \\[6pt]
&(0.924) &(1.295) &(2.208) &(2.671) \\[6pt]
\midrule
\multirow{2}{*}{Starbucks Digital} & $4.8775^{***}$ & $0.1978$ &$0.5511$ & $12.6284^{***}$\\[6pt]
&(0.613) &(0.852) &(1.589) &(1.897)\\[6pt]
\midrule
\multirow{2}{*}{Starbucks In-Store} & $5.6487$ & $1.7257$ &$-1.5137$ & $8.2381^{**}$ \\[6pt
&(0.849 &(1.170) &(2.053) &(2.540) &\\[6pt]
\midrule
\multirow{2}{*}{Apple Amazing} & $5.7345^{***}$ & $-0.1381$ &$-0.5999$ & $2.5726^{**}$ \\[6pt
&(0.295) &(0.404) &(0.620) &(0.813) \\[6pt
\bottomrule
\end{tabular}
\end{center}
Values in parentheses are standard errors.\\
\small{$^{*}:p<0.05$, $^{**}: p<0.01$, $^{***}:p<0.001$}
\caption{Effect of promotions on sales}
\label{regressions_volume}
\end{table*}
\subsection{Results}
\par We discuss here the results from the estimation of Model \ref{model_volume} and how they relate to our descriptive results.
\par The fourth column of Table \ref{regressions_volume} shows the coefficients for the post$\cdot$treat variable, which estimates the effect of the promotions, and it confirms our descriptive findings, shown in Figure \ref{rev_volume}.
\paragraph{Apple Weekly} As we expected, Apple Weekly has the largest coefficient for the post$\cdot$treat variable confirming that it's the promotion with the largest positive impact on the featured apps' sales. This can be explained not only by the low barrier nature of the promotion but also by the large awareness that the banner in the front page of the App Store brings. In addition to these factors, the app is promoted and appears to be endorsed by Apple itself, hence providing users with further confidence. As seen from Figure \ref{rev_volume} the large effects last around 2 weeks, after which a gradual descent `back to normal' begins. The promotion is only one week in duration, and hence we believe that the extra days of increased volume of sales is due to a combination of word-of-mouth as well as higher ranks in the top selling charts, which came as a result of the promotion.
\paragraph{Starbucks promotions} As can be seen in the second and third row of Table \ref{regressions_volume}, the coefficients of post$\cdot$treat for both of the Starbucks promotions are positive and highly significant, hence confirming the positive increase in sales we observed in Figure \ref{rev_volume}. Moreover, the coefficient is larger for the Starbucks Digital version compared to the Starbucks In-Store, confirming that low barrier promotions attract larger increases than higher barrier ones. Figure \ref{rev_volume} also shows that the increase in the digital promotion is much more sudden than the in the in-store promotion. This can be because customers redeeming a coupon for the in-store promotion can pick up the coupon from the physical locations but can wait up to three months before going through the process. In contrast, customers can redeem the digital promotion, right from their phones, only within the week of the promotion.
\par Similar to the Apple Weekly promotion, we expect that the Starbucks promotions caused an increase in word-of-mouth for the featured apps and helped them climb up the ranks of the top charts, which can explain why there are still some abnormally large sales (compared to before the promotion) even after the 7 days of the promotion were over.
\paragraph{Apple Amazing} Even though the Apple Amazing promotion is the only one not offering a full price discount, as can be seen in the fourth row of Table \ref{regressions_volume}, it still observes a positive and significant increase in sales. Confirming what we observed in Figure \ref{rev_volume}, this increase is the smallest amongst the studied promotions. Smaller increase in sales means smaller climb in the top-charts ranks which can also explain why this promotion seems to be the one returning to their normal pre-promotion sales faster from the other ones.
\par The results from this section show that low barrier promotions, i.e., digital and free, have the largest and most immediate increase in their sales whereas higher barrier promotions, i.e., partial discount or offline redemption procedure, have smaller and less sudden, but still substantial, increases.
\section{Future Directions}
\par We believe there are many interesting questions left to be studied in future work. First, most of the other major app store platforms have their own variation of a `Free App of the Week/Day' promotion. Investigating if similar results hold for the other app stores can increase our understanding on the subject and provide further insights for practitioners to better design their promotions on various platform.
\par Furthermore, it's interesting to see what is the effect of promotions that are not accompanied by any price discount, such as the `Editor's Choice' list. These promotions signal that a high profile entity with knowledge on the matter (usually the App Store itself) is endorsing an app for its high quality. What is the effect of such signals on the ratings and sales of the featured apps?
\par Moreover, promotions done by the developer's themselves can also be of interest. The four promotions studied in this paper all involve a selection process by a third party, but at any given day there are apps that are being sold in reduced prices, from their developers. These promotions are closer in spirit to the ones offered on platforms like Groupon since they are self-selected. Insights on the effect that these type of promotions have on the sales and ratings of the apps can help our understanding of the `Groupon effect' \cite{byers2012groupon}. It can also help developers explore better promotion strategies.
\par Finally, it will be of interest to see if effects similar to the ones discovered in this study arise when other types of digital goods, such as songs and movies, are offered in promotion.
\subsection{Data}
\par Our dataset is from the period of January 2013 to December 2015. Within this time frame, 59 and 71 apps were offered in the digital and in-store versions of the Starbucks promotion respectively, and 93 apps were offered in Apple's `App of the Week' promotion (roughly one per week). The Apple Amazing promotion is an aggregation of four promotions, that featured between them 68 unique apps.
\par Our first dataset consists of the app name, app id and the start date of the promotion for each of these 291 promoted apps. The app id is a unique string assigned by Apple to each app. In general, all apps were offered for one week each. For each promoted app, we also collect details about their characteristics such as category, price, size and number and price of in-app purchases. We use these characteristics to understand how the promoted apps may differ from other apps in the store.
\par On the iOS App Store a user can submit two types of reviews: A star rating on a scale from 1 to 5 or a star rating and accompanied by text. All ratings contribute to a displayed average star rating but only ratings accompanied by text are displayed in the app's review page alongside with their submitted date. Between them, the 291 (59 and 71 from the Starbucks promotions and 93 and 68 from Apple's) studied promoted apps had, as of January 2016, more than 665 thousand text reviews. For each of these text reviews we collected the accompanying text, the date the review was submitted, and their star rating. We don't have data on the ratings that were submitted without accompanying text review.
\par We also use two control datasets. One consists of 56 thousand uniformly sampled apps from the set of all available apps in the AppStore. For these control apps, we collected detailed information about their characteristics such as price, size, number of in-app purchases (if any), price of in-app purchases (if any) and more. We use this control dataset to get insights about how promoted apps differ from an average app from the App Store with respect to the aforementioned characteristics. From the 56 thousand uniformly sampled apps, we further uniformly sample 5 thousand apps for which we collect all text reviews. This is our second and main control dataset which we use to control for temporal and other trends when examining the effect of the promotions on the sales and ratings of the promoted (and competing to the promoted) apps.
\par Since we can't observe the number of sales, we use the number of text reviews to make indirectly detect relative changes in the sales of an app. This is an approached used also in \cite{} for data from the Google Play store. In both Google's and Apple's store, only users that have downloaded an app can submit a review, so the number of reviews an app receives is always a lower bound on it's sales.
\par Furthermore, we use the relative changes in the star average from text reviews to approximate relative changes in the overall star average.
\paragraph{Terminology}
Throughout the paper we will refer to any apps that were offered as part of the digital version of the Starbucks `iOS Pick of the week' promotion as \textit{Starbucks Digital apps} and to the promotion itself as \textit{Starbucks Digital}. Similarly, we will use the terms \textit{Starbucks In-Store apps} and \textit{Starbucks In-Store promotion}, \textit{Apple Weekly apps} and \textit{Apple Weekly promotion} and \textit{Apple Amazing apps} and \textit{Apple Amazing promotion}. There should be no confusion with apps \textit{made} by Apple or Starbucks, which are not part of any promotion discussed in this study. Finally, we will usually refer to a rating accompanied by text as \textit{text review} and simply as \textit{rating} otherwise.
\subsection{Initial Exploration}
We start by using our datasets to get insights on the general characteristics of the promoted apps and how they may differ from an average app in the App Store.
\par Table \ref{general} shows some general characteristics of the four sets of offered apps as well as of a control set of uniformly sampled apps. We see that the offered apps have a higher average price than the control and, amongst the offered apps, the ones that were offered on a non full price discount are the ones with the highest regular price. We also see that 75\% of the apps promoted in the Apple Weekly promotion offer in-app purchases, compared to 38\%-46\% for the other three promotions and only 12\% for the control. Furthermore, Apple Weekly apps offer the highest number of in-app purchases (5 compared to 2.47-2.73 for the other three promotions and 0.42 for the control) at the highest prices. This hints that the developers of the Apple Weekly apps prepare for a large influx of new users who will get the app for free and use in-app purchases to capitalize on them. In contrast, the two Starbucks and the Apple Amazing promotions have a smaller effect on the revenue of their developers since users that don't have redemption codes still pay full price for the app. Finally, the app size can provide an idea about how sophisticated the app is, for example in terms of state of the art graphics. Here too, we see that the Apple promotions feature apps that are generally much larger in size than the average app as well as the apps featured in the Starbucks promotions. This could hint toward Apple selecting and promoting state-of-the-art apps and games that, for example, take full advantage of the powerful graphics of the latest hardware releases.
\begin{table*}
\centering
\begin{tabular}{p{5cm} p{2cm} p{2cm} p{2cm} p{2.1cm} c }
\toprule
& \textbf{Apple Weekly} & \textbf{Starbucks In Store} & \textbf{Starbucks Digital} & \textbf{Apple 99\cent} &\textbf{Control}\\
\midrule
Number of apps & 93 & 71 & 59 & 68 & 56k\\[6pt]
Mean Price & 2.57 & 2.78 &2.73 &3.57 & 1.21 \\[6pt]
Median Price & 1.99 & 2.99 &1.99 &2.99 & 0 \\[6pt]
Offers In-App Purch. &75\% &43\% &46\% & 38\% & 12\% \\[6pt]
{\small Avg.~\#~of~In-App~Purchases} & 5 & 2.47 & 2.66 & 2.73 & 0.42 \\[6pt]
In-App Mean Price & 3.85 & 0.98 & 2.8 &2.07 & 0.62 \\[6pt]
App Size (MB) & 209 & 82.3 &43.7 &270.5 & 31.17 \\[6pt]
\bottomrule
\end{tabular}
\caption{Average General Characteristics}
\label{general}
\end{table*}
\section{Introduction}
\par Mobile apps have become an economy with a market size of \$25 Billion in 2013\footnote{Wall Street Journal (http://on.wsj.com/1Haufei)} and with a projected market size of \$77 Billion by 2017\footnote{entrepreneur.com (http://goo.gl/GdOa3P)}. As of mid 2016, The Google Play store for Android apps and iOS App Store for iOS apps, owned and operated by Google Inc. and Apple Inc. respectively, are the two largest mobile app stores, each with more than 1.5 Million mobile apps available to download\footnote{statista.com (http://goo.gl/dm1aw2)}. As of July 2013 the two mobile app stores had seen more than 50 Billion app downloads\footnote{statista.com (http://goo.gl/knUMyp)} \footnote{statista.com (http://goo.gl/dcqFks)} each. Apple announced100 Billion downloads in June 2015. With such volumes, it's impossible for users to be aware even of a small fraction of all the available apps: the average iOS and Android smartphone user used less than 30 apps per month at the end of 2013\footnote{nielsen.com (http://goo.gl/QkSPCw)}.
\par This incentivizes the owners of the stores to segment their content into easily navigable segments such as `Editor's Choice', `Essentials' or `New to the Store' to help their users find their most preferred apps. Similarly, creators of apps try to induce word of mouth and publicity for their apps, for example, via free `light' versions of their (not-free) app or by lowering the app's price for a limited time.
\par But promotions could come with their own caveats. For example, a price reduction could attracts users that are not in the target group of an app and hence lead to suboptimal user experiences which, in turn, can lead to lower online reviews. These lower online reviews, can significantly impact the future performance of any product and service. On the other hand, developers usually strive to be featured in their stores, perhaps without having a complete understanding of the benefits but also risks that such promotions can have. Our paper is contributing towards a better understanding of the effects that such promotions can have on the sales and ratings of featured apps, as well their competition.
\par We focus on four such promotion campaigns in the iOS App Store. Two of them are versions of a larger promotion offered by the coffee chain Starbucks, called `iOS Pick of the Week'. With both promotions, Starbucks offers a one-time redemption code that customers can use to get a featured app for free. The difference between the two promotions is that one distributes the redemption codes via printed coupons in physical stores (we will refer to this distribution method as `In-Store') and the other from within the Starbucks iOS app (we will refer to this distribution method as `Digital'). The sets of apps featured in these two versions of the promotion are non overlapping. The other promotions we study were offered by Apple itself.
The first, `App of the Week', is a recurring weekly free offering of an app that is generally not free otherwise.
The other ones, are four one-time promotions that run for one week each the third week of December 2014, the third week of July 2015, the fourth week of May 2015 and the third week of December 2015. Three of the promotions were called `Amazing Apps \& Games for 99\cent each' and the fourth was called `Handpicked Apps \& Games for \$0.99'. The general theme was the same in all four promotions: Apple chose 30 apps (usually 15 of them were games) and offered them for \$0.99 each for a week. Since the last four promotions are so similar in spirit and execution we bundle them together and refer to them simple as the Apple Amazing promotion.
\par One of the key differences between the two promotions offered by Apple and the two promotions offered by Starbucks is that the two promotions offered by Apple feature large banners on the front page of the App Store, and the featured apps were sold for the reduced price for all users (even those who were unaware of the promotion before their visit). In contrast, Starbucks' promotions were valid only for customers that had acquired a redemption code either in store or though the iOS app. All other users would have to pay the full price to get the app
\par Even though these promotions were offered both inside and outside of the United States, in this study we focus only on the US App Store and US Starbucks stores.
\subsection{Main Insights}
\par We focus on two main characteristics of the promotions: steepness of price discount and ease of redemption. We utilize the similarities and differences between the four promotions in our study to understand the effect each of these two promotion characteristics had on the sales and ratings of the promoted apps. We find that digital promotions, which are easy to redeem, accompanied by a full price discount are the promotions that cause the largest increase in sales, while having neutral or even positive effects on the ratings. In-store coupons, which are slightly less trivial to redeem, cause a smaller increase in sales but also have no effect on the ratings. Non-full price discounted promotions cause the smallest (but still positive) increase in sales but can also have a negative effect on the ratings.
\par Furthermore, we explore the effect of various app characteristics in the success of the app's promotion. We find that in digital and full-price discounted promotions, all featured apps receive on average the same increase in sales. But in promotions with higher barriers (such as less trivial redemption procedure or non-zero price) apps that were more popular before the promotion are the ones that will receive the largest increases in sales. This indicates that when the user is offered an app that is free and (literally) a few taps away, the specific characteristics of the app will not matter much on their decision to redeem the offer or not. But when the barriers are higher, users are more likely to do the effort (or pay the price) if it's a more established app that they are familiar with.
\par Finally, we are also interested in understanding the effect that these promotions had on the apps competing with the promoted apps. We find that full price discounted promotions to have a significant negative effect on the sales of their competition but not full price discounted promotions have a positive effect on the sales of their competition. We interpret this result as a sign that users are intrigued and interested when they are exposed to an app through a promotion. If the app is not free, the user will not commit immediately but instead will explore the alternatives. When the app is free, the user will have little incentive to search for alternatives.
\par All these insights can directly help practitioners better design their marketing campaigns.
\begin{figure}[h!]
\centering
\includegraphics[width=0.45\textwidth]{promotion_types_summary.pdf}
\caption{A summary of the characteristics of the promotions we study}
\label{summary}
\end{figure}
\subsection{Related Work}
There has been extensive work on the correlation between ratings and revenue, amongst others for the case of books \citep{chevalier2006effect}, movies \citep{chintagunta2010effects,liu2006word, basuroy2003critical,reinstein2005influence,duan2008dynamics,dellarocas2005using}, games \citep{zhu2010impact} as well as new products in general \citep{cui2012effect}. \cite{luca2011reviews} and \cite{anderson2012learning} apply a regression discontinuity technique on Yelp and find that a half star (on a 5 star scale) increase on a restaurant's ratings leads to a 5-9\% higher revenue and higher sell out rates for restaurants, respectively. \cite{engstrom2014demand} apply the same technique on Google Play and find that a half star increase on ratings (on a 5 star scale) leads to 3\% more downloads for the app. Furthermore, a 10 percentile increase on the displayed number of downloads can further increase the downloads by 20\%.
\par Another line of work tries to understand the profitability of discounts and promotions. For example, \cite{edelman2011groupon} argue that deep discounts work only if they reach customers with substantially lower valuations for the product than the regular customers. Especially relevant to our work, \cite{spriensma2012impact} estimates that being featured on Apple's App Store or Google Play can help an app gain +15 and +42 spots, respectively, on the store's top-seller rankings and \cite{carare2012impact} estimates that customers on Apple's App Store are willing to pay \$4.50 more on apps that are top-ranked than the same unranked ones. \cite{spriensma2012impact} also finds that putting apps on sale can be profitable especially if the price is cut in half or on the price points of \$0.99 or \$1.99. Hence, there are strong economic incentives for the developers to both strive to be featured on their respective stores and offer frequent discounts. \cite{ajorlou2014dynamic} even provide theoretical evidence for the optimality of a pricing strategy that drops the price to zero infinitely often in an environment where word of mouth effects are in play. Finally, \cite{adamopoulos2014social} study the effectiveness of a large scale marketing campaign carried out on Twitter and find significant returns on participating brands in terms of fan base and social media followers.
\par Our work is related to both of the aforementioned lines of research since it estimates the immediate effect of various promotions on the sales of mobile apps but also tries to understand the effect these promotions have on the ratings of the apps. The effect on ratings is important because the ratings can continue affecting the sales of the app long after the promotion is over. Our work is especially related to the work of \cite{byers2012groupon} who found that establishments that offer Groupon promotions find their Yelp rating decrease. But our work also differs from theirs in some crucial ways. First, we are studying promotions and reviews for {\it digital} and not physical goods (or services). Hence, some of the caveats that could apply for physical establishments don't apply here, e.g., all purchasers of a digital good get the exact same product no matter the price they pay, whereas owners of restaurants could discriminate against customers not paying full price. But perhaps the most important distinction between our work and the work of \cite{byers2012groupon} is that the four promotions studied in this paper are selected and offered by two third-party companies that have their own complicated incentives. Hence the selection process is not as endogenous as in platforms like Groupon.
\par By comparing four different promotions and the different effects they had on the sales and ratings of the featured promoted apps, our work, amongst other contributions, can lead to a better understanding of how to design effective marketing campaigns.
\section{Effect of the Promotions on the Competition}
\par Many online stores today use machine learning algorithms that, based on user's behavior on the site try to recommend items that the user might be interested in. For example, Amazon has a section called `Customers Who Bought This Item Also Bought' on the product pages of their item. Such recommendation algorithms can help users explore and discover new products of interest in an otherwise very large search space of options (e.g., Amazon and eBay have millions of products each for sale, Google's and Apple's App Store have more than 1.5 million apps each).
\par Apple's recommendation engine in the AppStore works mainly by displaying a `Customers also bough' tab on the product page of many apps. Even though sometimes these apps can be complimentary (such as Facebook and Messenger) most of the times they seem to be competing. For example, a casual game will have recommendations for other casual games and a productivity app will have recommendations for other apps with similar functions. These similar apps, compete for the user's attention, wallet as well as phone storage space (the 16GB model is the best selling amongst the iPhones).
\par For every promoted app we gathered the apps appearing in its `Customers also bought' tab'. We then collected the entire review history of every such app. We aim to study the effect of the promotions on these competing apps' sales. A negative effect on sales would suggest that the promotion causes customers to select the promoted app amongst the alternatives, possibly because of it's discounted price or to avoid the cost of searching further. A positive effect would suggest that the promoted app increases awareness for itself as well as apps similar to it and customers use the promoted app as a {\it starting point} to search further for the best alternative.
\subsection{Econometric Model}
\par In order to statistically test these findings, we estimate the following model on the volume of reviews for the apps similar to the four set of featured apps.
\begin{equation}\label{model_1}
\mathrm{sales}= \alpha_0 + \beta_1\mathrm{post} +\beta_2\mathrm{treat} + \beta_3\mathrm{post}\cdot\mathrm{treat}+e
\end{equation}
The post variable indicates if the datapoint is after the promotion, and treat is a binary variable indicating if the review is for an app in the treatment group or not. The post$\cdot$treat interaction term is the one that captures the variation that is not explained by the control dataset, hence it's the variable that estimates the causal effect of the promotion. Note that, unlike previous sections, the treatment group here is not the apps featured in the promotions but the apps that appear in the `Customers also bought' section of the the promoted apps.
\par Table \ref{regressions_comp} summarizes the results of the estimations of Model \ref{model_1} for the four studied promotions.
\begin{table*}[h!]
\begin{center}
\vspace{0.5cm}
\begin{tabular}{l c c c c c}
\toprule
{\it \small Promotion} &Intercept &post & treat & $\mathrm{post}\cdot\mathrm{treat}$\\[6pt
\midrule
\multirow{2}{*}{Apple Weekly} & 5.70 &-0.71 &$62.65^{***}$ & $-23.66^{**}$ \\[6pt]
&(3.19) &(4.47) &(5.02) &(6.85) \\[6pt]
\midrule
\multirow{2}{*}{\small Starbucks Digital} & $5.65^{***}$ & $1.73$ &$15.22^{***}$ & $-6.97^{*}$\\[6pt]
&(1.34) &(1.88) &(2.27) &(3.16) \\[6pt]
\midrule
\multirow{2}{*}{Starbucks In-Store} & $5.65^{***}$ & $1.73$ &$18.73^{***}$ & $-8.25^{**}$ \\[6pt
&(1.15) &(1.59) &(1.98) &(2.77) \\[6pt]
\midrule
\multirow{2}{*}{Apple Amazing} & $5.73^{***}$ & $-0.14$ &$11.523^{***}$ & $3.43^{**}$ \\[6pt
&(0.62) &(0.85) &(0.94) &(1.28) & \\[6pt
\bottomrule
\end{tabular}
\end{center}
Values in parentheses are standard errors.\\
\small{$^{*}:p<0.05$, $^{**}: p<0.01$, $^{***}:p<0.001$}
\caption{The effect of the promotions on the volume of reviews of their competitors}
\label{regressions_comp}
\end{table*}
\par The post$\cdot$treat coefficients are the ones estimating the extend of the effect that cannot be explained by the control dataset.
We see that the coefficients are negative and significant for Apple Weekly and Starbucks In-Store, negative and barely not significant ($p$=0.08) for Starbucks Digital and positive and significant for Apple Amazing.
\subsection{Results}
\par Since low barrier promotions make it very easy and costless for the user to acquire the promoted app, we expect that customers exposed to such suers will not explore for further alternatives and hence competing sales will decrease. Indeed, we find that apps competing with Apple Weekly promotions see their average daily number of reviews fall by XX in the 7 days of the promotion compared to the 7 days before. Similarly, we see a decrease of YY for apps competing with Starbucks Digital apps. Furthermore, we find that even apps competing with the Starbucks In-Store apps see a decrease of ZZ. This is even though Starbucks In-Store has a slightly non-trivial redemption procedure. Finally, and perhaps most interestingly, we find an {\it increase} of ZZ for the sales of the apps competing with Apple Amazing apps.
\par The negative and significant coefficients for the post$\cdot$treat variable for all full price discounted promotions, shown in Table \ref{regressions_comp}, confirm that such promotions have a negative effect on the sales of their competition. In contrast, the only partially discounted promotion causes an increase in the sales of its competition.
\par This suggests that, especially in electronic markets where such recommendation algorithms are in place, increasing awareness for a product may increase awareness and sales for it's competition. That is, unless the promotion is such that offers no incentives for users to explore for further alternatives, such as a full price discount.
\par We expect similar positive and negative externalities to exists in other types of markets as well, and in other types of marketing campaigns. Essentially, if we look at the recommendation graph produced by the centralized platform as a graph on which the user is walking a random walk, any increase in the probability of a specific point will have some effects on that point's neighbors.
\par This suggests that practitioners designing marketing campaigns for their products should be aware of the positive externalities they can have on their competition. In fact, phenomena similar to this has already been observed in the literature, \cite{lewis2014samsung}
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Światła września (hiszp. Las luces de septiembre) to powieść dla młodzieży hiszpańskiego pisarza Carlosa Ruiza Zafóna będąca trzecią i ostatnią częścią cyklu Trylogia mgły (hiszp. La Trilogía de la Niebla). Książka miała światową premierę w Hiszpanii w 1995 roku, natomiast w Polsce ukazała się nakładem wydawnictwa Muza w listopadzie 2011 roku.
Fabuła
Lazarus Jann jest projektantem i wykonawcą zabawek żyjącym w dużej willi w Cravenmoore, w której realizuje swoje projekty. Miał smutne dzieciństwo, gdyż mieszkał z matką z demencją, która zamykała go w ciemnym pokoju pozbawionym zabawek. Żyli w jednej z biednych dzielnic Paryża. Lazarus marzył o życiu. jakie wiódł bogaty przedsiębiorca, Daniel Hoffman, który miał fabrykę w sąsiedztwie i dawał dzieciom zabawki. Pewnego dnia matka zamknęła go w ciemnym pokoju i zapomniała o dziecku. Działo się to w chwili, gdy płonęła fabryka zabawek, a pożar niszczył całą okolicę. W ciemnym pokoju chłopcu ukazał się producent zabawek. Obiecał uczynić go swoim następcą w zamian za jego cień i wyrzeknięcie się miłości. Lazarus zgodził się na przedstawioną ofertę. Fabryka zabawek spłonęła i nie było żadnych wieści na temat producenta zabawek, którego uznano za zmarłego. Kilka lat później młody Lazarus pracuje jako przedsiębiorca i jest szczęśliwy. Zaręcza się z piękną dziewczyną i żeni się z nią. Po ślubie zaczynają nękać go problemy. Jego cień materializuje się i jako zabawka ściga go chcąc go zabić.
Wiele lat później, w roku 1936, do jego posiadłości przybywa nowa gospodyni, Simone Sauvelle, która po śmierci męża została praktycznie bez środków do życia. Kobieta przyjeżdża z dziećmi - córką Irene i synem Dorianem. W willi pracuje i mieszka również Hannah, młoda dziewczyna ze wsi, przez którą Irene poznaje Ismaela (kuzyna Hannah), który zostaje jej przyjacielem.
Pewnego dnia Ismael i Irene płyną łodzią do latarni morskiej, gdzie kiedyś w czasie burzliwej nocy zmarła młoda i piękna kobieta. Znajdują tam pamiętnik należący do Almy Maltisse. Kobieta opisała w nim, jak została zmuszona do ucieczki przez cień, który nawiedza ją i chce zabić. Alma płynie łodzią by spotkać się z mężem, który aby ją ocalić i odciągnąć od niej prześladujący ją cień, schronił się w latarni morskiej. Irene i Ismael czytają znaleziony pamiętnik i zbliżają się do siebie. Na wyspie ma miejsce ich pierwszy pocałunek, który zwiastuje zmianę relacji między nimi - z przyjaciół stają się parą.
Kilka dni później Hannah budzi hałas dochodzący z pokoju, w którym nikt nie mieszka. Za oknem szaleje burza, więc dziewczyna postanawia sprawdzić co się dzieje w pomieszczeniu. Zastaje umeblowany i sprawiający wrażenie zamieszkanego pokój. Zauważa otwarte okno i podchodzi, by je zamknąć. Słyszy przerażające dźwięki i ucieka z rezydencji do lasu. Mimo ucieczki nadal towarzyszą jej przerażające dźwięki. Następnego dnia zostaje znaleziona martwa.
Po śmierci Hannah Ismael i Irene zaczynają analizować fakty i obserwować Lazarusa. Pewnego dnia znajdują piwnicę pełną dziwnych zabawek, w tym metalowego anioła, który jednej nocy ścigał ich aż do jaskini, w której schronili się przed nim i cudem uniknęli śmierci. gdy woda z przypływu zaczęła wypełniać grotę. W ten sposób odkrywają istnienie cienia, ponieważ podczas pościgu anioł rozbija się, a cień opuszcza uszkodzone części metalu i śmieje się z nich.
Simone i Lazarus zostają przyjaciółmi. Mężczyzna przerażony zachowaniem cienia, zamyką ją na noc w piwnicy, by nic jej się nie stało. Simone nie rozumie jego zachowania i jest przerażona. Wreszcie Lazarus opowiada swoją historię Simone. Kobieta dowiaduje się, że prześladujący go cień teraz ściga ją i jej rodzinę, bo zobaczył, że Lazarus ich lubi. Po rozmowie projektant zamyka się w pokoju, gdzie trzyma i konserwuje ciało swojej zmarłej żony, Almy. Walczy z cieniem, aż jego dom i fabryka zostają spalone. Ostatecznie Lazarus wygrywa z Danielem Hoffmanem, ponieważ został bogatym i potężnym producentem zabawek oraz poznał smak miłości. Mimo wysokiej ceny, jaką musiał za to zapłacić, nie był w stanie o niej zapomnieć. Na końcu umiera.
Bohaterowie
Lazarus Jann: mąż Alexandry Almy Maltisse, zaproponował pracę i zakwaterowanie rodzinie Sauvelle;
Irene: piętnastoletnia córka Simone Sauvelle i siostra Doriana
Ismael: kuzyn Hannah i chłopak Irene
Hannah: kuzynka Ismaela i przyjaciółka Irene, pracuje dla Lazarusa
Simone: matka Irene i Doriana, zakochuje się w Lazarusie
Wydania
Książka została wydana w listopadzie 2011 roku przez wydawnictwo Muza w wersjach z twardą i miękką okładką. Ponadto w grudniu 2011 roku powieść ukazała się w formie audiobooka, który czyta Piotr Fronczewski.
Przypisy
Hiszpańskie powieści
Powieści z 1995 | {
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\section*{References}
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\title{Spatial Parrondo games \\ and an interacting particle system}
\author{Sung Chan Choi\thanks{Department of Mathematics, University of Utah, 155 S. 1400 E., Salt Lake City, UT 84112, USA. e-mail: choi@math.utah.edu}}
\date{}
\begin{document}
\maketitle
\begin{abstract}
Parrondo games with spatial dependence were introduced by Toral (2001) and have been studied extensively. In Toral's model $N$ players are arranged in a circle. The players play either game $A$ or game $B$. In game $A$, a randomly chosen player wins or loses one unit according to the toss of a fair coin. In game $B$, which depends on parameters $p_0,p_1,p_2\in[0,1]$, a randomly chosen player, player $x$ say, wins or loses one unit according to the toss of a $p_m$-coin, where $m\in\{0,1,2\}$ is the number of nearest neighbors of player $x$ who won their most recent game. In this paper, we replace game $A$ by a spatially dependent game, which we call game $A'$, introduced by Xie et al.~(2011). In game $A'$, two nearest neighbors are chosen at random, and one pays one unit to the other based on the toss of a fair coin. Game $A'$ is fair, so we say that the Parrondo effect occurs if game $B$ is losing or fair and the game $C'$, determined by a random or periodic sequence of games $A'$ and $B$, is winning. Here we give sufficient conditions for convergence as $N\to\infty$ of the mean profit per game played from game $C'$. This requires ergodicity of an associated interacting particle system (not necessarily a spin system), for which sufficient conditions are found using the basic inequality.
\end{abstract}
\section{Introduction}\label{sec-intro}
Toral (2001) introduced what he called \textit{cooperative} Parrondo games with spatial dependence. (We prefer the term \textit{spatially dependent} Parrondo games so as to avoid conflict with the field of cooperative game theory.) The games depend on an integer parameter $N\ge3$, the number of players, and four probability parameters, $p_0,p_1,p_2,p_3$. (This is a slight generalization of the model described in the abstract.) The players are arranged in a circle and labeled from 1 to $N$, so that players 1 and $N$ are adjacent. At each turn, a player is chosen at random to play. Suppose player $x$ is chosen. In game $A$, he tosses a fair coin. In game $B$, he tosses a $p_m$-coin (i.e., a coin whose probability of heads is $p_m$), where $m\in\{0,1,2,3\}$ depends on the winning or losing status of his two nearest neighbors. A player's status as winner (1) or loser (0) is decided by the result of his most recent game. Specifically,
$$
m=\begin{cases}0&\text{if $x-1$ and $x+1$ are both losers,}\\
1&\text{if $x-1$ is a loser and $x+1$ is a winner,}\\
2&\text{if $x-1$ is a winner and $x+1$ is a loser,}\\
3&\text{if $x-1$ and $x+1$ are both winners,}\end{cases}
$$
where $N+1:=1$ and $0:=N$ because of the circular arrangement of players.
Player $x$ wins one unit with heads and loses one unit with tails.
Game $A$ is fair, so we say that the Parrondo effect occurs if game $B$ is losing or fair and the game $C$, determined by a random or periodic sequence of games $A$ and $B$, is winning.
These games have been studied by Mihailovi\'c and Rajkovi\'c (2003) and Ethier and Lee (2012a,b, 2013a,b), and a strong law of large numbers and a central limit theorem were obtained. In particular, the asymptotic cumulative profits per turn exist and are the means in the SLLN. Further, it seems clear that these means converges as $N\to\infty$. This has been proved under certain conditions (see Ethier and Lee, 2013a).
Notice that Toral's (2001) game $A$ is not spatially dependent (the rules of the game do not depend on the spatial structure of the players). Xie et al.~(2011) proposed a modification of game $A$ that \textit{is} spatially dependent as well as being a fair game. To distinguish, we call that game $A'$. As before, the games depend on an integer parameter $N\ge3$, the number of players, and four probability parameters, $p_0,p_1,p_2,p_3$. The players are arranged in a circle and labeled from $1$ to $N$, so that players $1$ and $N$ are adjacent. At each turn, a player is chosen at random to play. Suppose player $x$ is chosen. In game $A'$, he chooses one of his two nearest neighbors at random and competes with that neighbor by tossing a fair coin. The result is a transfer of one unit from one of the players to the other, hence the wealth of the set of $N$ players is unchanged. Game $B$ is as before. Player $x$ wins one unit with heads and loses one unit with tails.
Game $A'$ is fair, so we say that the Parrondo effect occurs if game $B$ is losing or fair and the game $C'$, determined by a random or periodic sequence of games $A'$ and $B$, is winning.
These games were studied by Xie et al.~(2011), Li et al.~(2014), and Ethier and Lee (2015). Only the case of random sequences was treated, and convergence of the means has not yet been addressed. Our aim in this paper, together with Choi (2020), is to fill in these gaps in the literature.
The Markov chain formalized by Mihailovi\'c and Rajkovi\'c (2003) keeps track of the status (loser or winner, 0 or 1) of each of the $N\ge3$ players of game $B$. Its state space is the product space
$$
\{\eta=(\eta(1),\eta(2),\ldots,\eta(N)): \eta(x)\in\{0,1\}{\rm\ for\ }x=1,\ldots,N\}=\{0,1\}^N
$$
with $2^N$ states. Let $m_x(\eta):=2\eta(x-1)+\eta(x+1)\in\{0,1,2,3\}$. Of course $\eta(0):=\eta(N)$ and $\eta(N+1):=\eta(1)$ because of the circular arrangement of players. Also, let $\eta_x$ be the element of $\{0,1\}^N$ equal to $\eta$ except at the $x$th coordinate. For example, $\eta_1:=(1-\eta(1),\eta(2),\eta(3),\ldots,\eta(N))$.
The one-step transition matrix $\bm P_B$ for this Markov chain depends not only on $N$ but on four parameters, $p_0,p_1,p_2,p_3\in[0,1]$. It has the form
\begin{equation*}
P_B(\eta,\eta_x):=\begin{cases}N^{-1}p_{m_x(\eta)}&\text{if $\eta(x)=0$,}\\N^{-1}q_{m_x(\eta)}&\text{if $\eta(x)=1$,}\end{cases}\qquad x=1,\ldots,N,\;\eta\in \{0,1\}^N,
\end{equation*}
and
\begin{equation*}
P_B(\eta,\eta):=N^{-1}\bigg(\sum_{x:\eta(x)=0}q_{m_x(\eta)}+\sum_{x:\eta(x)=1}p_{m_x(\eta)}\bigg),\qquad \eta\in\{0,1\}^N,
\end{equation*}
where $q_m:=1-p_m$ for $m=0,1,2,3$ and empty sums are 0. The Markov chain is irreducible and aperiodic if $0<p_m<1$ for $m=0,1,2,3$. Under slightly weaker assumptions (see Ethier and Lee 2013a), the Markov chain is ergodic, which suffices. For example, if $p_0$ is arbitrary and $0<p_m<1$ for $m=1,2,3$, or if $0<p_m<1$ for $m=0,1,2$ and $p_3$ is arbitrary, then ergodicity holds.
A Markov chain in the same state space corresponds to game $A'$. As previously mentioned, if game $A'$ is played, the profit to the set of $N$ players is 0, since game $A'$ simply redistributes capital among the players. The transition probabilities require some new notation. Starting from state $\eta$, let $\eta^{x,x\pm1,\pm1}\in\{0,1\}^N$ represent the players' status after player $x$ plays player $x\pm1$ and wins (1) or loses ($-1$). Of course player 0 is player $N$ and player $N+1$ is player 1. For example, $\eta^{1,2,-1}=(0,1,\eta(3),\ldots,\eta(N))$ (player 1 competes against player 2 and loses, leaving player 1 a loser and player 2 a winner, regardless of their previous status). The transition probabilities have the form
\begin{align*}
&P_{A'}(\eta,\zeta)\\
&:=\frac{1}{4N}\sum_{x=1}^N[\delta(\eta^{x,x-1,-1},\zeta)+\delta(\eta^{x,x-1,1},\zeta)+\delta(\eta^{x,x+1,-1},\zeta)+\delta(\eta^{x,x+1,1},\zeta)],
\end{align*}
where $\delta(\eta,\zeta)=1$ if $\eta=\zeta$ and $=0$ otherwise.
Next, we want to regard the players, originally labeled from 1 to $N$, as labeled from $l_N$ to $r_N$, where
\begin{equation}\label{relabel}
l_N: = \begin{cases}
-(N-1)/2 &\text{if $N$ is odd,}\\
-N/2 &\text{if $N$ is even,}
\end{cases} \quad
r_N: = \begin{cases}
(N-1)/2 &\text{if $N$ is odd,}\\
N/2-1 &\text{if $N$ is even.}
\end{cases}
\end{equation}
Then we can speed up time, playing $N$ games per unit of time, and our process is described in the limit as $N\to\infty$ by an interacting particle system in the state space $\Sigma:=\{0,1\}^{\bf Z}$. The details of this limit operation are postponed to Section \ref{sec-convergence}. Initially, our concern is with the ergodicity of the limiting interacting particle system.
The generator $\Omega_{A'}$ of the interacting particle system corresponding to game $A'$ can be described as follows. For $\eta\in\Sigma:=\{0,1\}^{\bf Z}$ and $x\in{\bf Z}$, define $\eta_x$ and $_x\eta_{x+1}$ in $\Sigma$ by
\begin{equation*}
\eta_x(y):=\begin{cases}1-\eta(x)&\text{if $y=x$,}\\ \eta(y)&\text{otherwise,}\end{cases}\quad\text{and}\quad
_x\eta_{x+1}(y):=\begin{cases}\eta(x+1)&\text{if $y=x$,}\\ \eta(x)&\text{if $y=x+1$,}\\ \eta(y)&\text{otherwise.}\end{cases}
\end{equation*}
Then
\begin{equation*}
(\Omega_{A'} f)(\eta):=\sum_x c'(x,\eta)[f(\eta_x)-f(\eta)]+\frac12\sum_x[f(_x\eta_{x+1})-f(\eta)]
\end{equation*}
for $f\in C(\Sigma)$ depending on only finitely many coordinates, where
\begin{equation}\label{c'-def}
c'(x,\eta):=\frac{1}{2}[\bm{1}_{\{\eta(x)=\eta(x+1)\}}+\bm{1}_{\{\eta(x)=\eta(x-1)\}}].
\end{equation}
Note that the generator $\Omega_{A'}$ is the sum of a spin system generator and an exclusion process generator.
The generator $\Omega_B$ of the interacting particle system corresponding to game $B$ can be described as follows. For each $x\in{\bf Z}$, define $\eta_x$ as above. Then, given parameters $p_0,p_1,p_2,p_3\in[0,1]$,
\begin{equation*}
(\Omega_{B} f)(\eta):=\sum_x c(x,\eta)[f(\eta_x)-f(\eta)]
\end{equation*}
for $f\in C(\Sigma)$ depending on only finitely many coordinates, where
\begin{equation}\label{c-def}
c(x,\eta):=\begin{cases}p_{m_{x}(\eta)}&\text{if $\eta(x)=0$,}\\ q_{m_{x}(\eta)}&\text{if $\eta(x)=1,$}\end{cases}
\end{equation}
$q_m:=1-p_m$ for $m=0,1,2,3$, and $m_x(\eta):=2\eta(x-1)+\eta(x+1)\in\{0,1,2,3\}$.
Finally, the generator $\Omega_{C'}$ of the interacting particle system corresponding to game $C':=\gamma A'+(1-\gamma)B$ (denoting the random mixture of games $A'$ and $B$, i.e., the game in which a $\gamma$-coin determines whether game $A'$ or game $B$ is played) is
$$
\Omega_{C'}:=\gamma\Omega_{A'}+(1-\gamma)\Omega_B.
$$
The generator $\Omega_{C'}$, just like $\Omega_{A'}$, is the sum of a spin system generator and an exclusion process generator.
\section{Ergodicity}\label{sec-basic}
First, $\Omega_B$ is a spin system generator, so the basic inequality yields a sufficient condition for ergodicity (Liggett, 1985, Eq.~(III.0.6)):
\begin{equation}\label{erg-basic-B}
\sup_{x \in {\bf Z}}\sum_{u:u\neq x}\sup_{\eta \in \Sigma} |c(x,\eta)-c(x,\eta_u)| < \inf_{x \in {\bf Z},\; \eta \in \Sigma}[c(x,\eta)+c(x,\eta_{x})],
\end{equation}
where $c(x,\eta)$ is as in \eqref{c-def}.
Next, $\Omega_{C'}$, given by
\begin{align}\label{Omega-C'}
(\Omega_{C'} f)(\eta)&:=\gamma(\Omega_{A'}f)(\eta)+(1-\gamma)(\Omega_Bf)(\eta)\nonumber\\
&\;=\gamma\sum_x c'(x,\eta)[f(\eta_x)-f(\eta)]+\frac{\gamma}{2}\sum_x[f(_x\eta_{x+1})-f(\eta)]\nonumber\\
&\qquad\qquad+(1-\gamma)\sum_xc(x,\eta)[f(\eta_x)-f(\eta)],
\end{align}
is not a spin system generator, so \eqref{erg-basic-B} does not apply. Instead, we apply a more general form of the basic inequality. It assumes that the generator has the form
\begin{equation}\label{Omega-general}
(\Omega f)(\eta)=\sum_{T\subset{\bf Z}\text{ finite}}\int_{\{0,1\}^T}c_T(\eta,d\zeta)[f(\eta^\zeta)-f(\eta)],
\end{equation}
where $c_T(\eta,d\zeta)$ is a finite positive measure on $\{0,1\}^T$, and
$$
\eta^{\zeta}(x):=\begin{cases}\zeta(x)&\text{if $x \in T$}\\ \eta(x)&\text{if $x \notin T$}\end{cases}
$$
for $\zeta$ $\in \{0,1\}^T$. The interpretation is that $\eta$ is the current configuration, $c_T(\eta,\{0,1\}^T)$ is the rate at which a transition occurs involving coordinates in $T$, and $c_T(\eta,d\zeta)/c_T(\eta,\{0,1\}^T)$ is the distribution of the restriction to $T$ of the new configuration after a transition.
We conclude from \eqref{Omega-C'} and \eqref{Omega-general} that
\begin{align*}
c_{\{x\}}(\eta,G)&=\delta_{1-\eta(x)}(G)[\gamma c'(x,\eta)+(1-\gamma) c(x,\eta)],\\
c_{\{x,x+1\}}(\eta,H)&=\delta_{(\eta(x+1),\eta(x))}(H)\frac{\gamma}{2}\bm1_{\{\eta(x)\ne\eta(x+1)\}},
\end{align*}
where $\delta_u$ is the unit mass concentrated at $u$. Here $G\subset\{0,1\}$ and $H\subset\{(0,0),(0,1),(1,0),(1,1)\}$.
The sufficient condition for ergodicity is that $M<\varepsilon$ (Liggett, 1985, Theorem I.4.1), where $M$ and $\varepsilon$ are constants, defined below, that remain to be evaluated.
For $u\in {\bf Z}$ and finite $T\subset {\bf Z}$, let $c_T(u):=\sup \{\|c_T(\eta,d\zeta)-c_T(\eta',d\zeta)\|_{\text{TV}}: \eta,\eta'\in\Sigma,\eta(y)=\eta'(y)\,\forall\,y\neq u\}$, where $\|\cdot\|_{\text{TV}}$ denotes the total variation norm of a measure on $\{0,1\}^T$. Then
\begin{align} \label{M-for-C'}
M &:=\sup_{x \in {\bf Z}}\sum_{T \ni x}\sum_{u: u \neq x}c_T(u)\nonumber\\
&=\sup_{x \in {\bf Z}}\sum_{T \ni x}\sum_{u: u \neq x}\sup_{\eta,\eta'\in\Sigma,\eta(y)=\eta'(y)\,\forall\,y\neq u}\|c_T(\eta,d\zeta)-c_T(\eta',d\zeta)\|_{\text{TV}}\nonumber\\
&=\sup_{x \in {\bf Z}}\bigg[\sum_{u:u\neq x}\sup_{\eta \in \Sigma}\|c_{\{x\}}(\eta,d\zeta)-c_{\{x\}}(\eta_u,d\zeta)\|_{\text{TV}}\nonumber\\
&\qquad\;\;{}+\sum_{v: v\neq x}\sup_{\eta \in \Sigma}\|c_{\{x,x+1\}}(\eta,d\zeta)-c_{\{x,x+1\}}(\eta_v,d\zeta)\|_{\text{TV}}\nonumber\\
&\qquad\;\;{}+\sum_{w: w\neq x}\sup_{\eta \in \Sigma}\|c_{\{x-1,x\}}(\eta,d\zeta)-c_{\{x-1,x\}}(\eta_w,d\zeta)\|_{\text{TV}}\bigg]\nonumber\\
&=\sup_{x \in {\bf Z}}\bigg[\sum_{u:u\neq x}\sup_{\eta \in \Sigma} \sup_{G \subset \{0,1\}}|c_{\{x\}}(\eta,G)-c_{\{x\}}(\eta_u,G)|\nonumber\\
&\qquad\;\;{}+\sum_{v: v\neq x}\sup_{\eta \in \Sigma}\sup_{H \subset \{(0,0),(0,1),(1,0),(1,1)\}}|c_{\{x,x+1\}}(\eta,H)-c_{\{x,x+1\}}(\eta_v,H)|\nonumber\\
&\qquad\;\;{}+\sum_{w: w\neq x}\sup_{\eta \in \Sigma}\sup_{H \subset \{(0,0),(0,1),(1,0),(1,1)\}}|c_{\{x-1,x\}}(\eta,H)-c_{\{x-1,x\}}(\eta_w,H)|\bigg]\nonumber\\
&=\sup_{x \in {\bf Z}}\bigg[\sum_{u:u\neq x}\sup_{\eta \in \Sigma} |\gamma c'(x,\eta)+(1-\gamma) c(x,\eta)-[\gamma c'(x,\eta_{u})+(1-\gamma) c(x,\eta_{u})]|\nonumber\\
&\qquad\;\;{}+\sup_{\eta \in \Sigma}\sup_{H \subset \{(0,0),(0,1),(1,0),(1,1)\}}|c_{\{x,x+1\}}(\eta,H)-c_{\{x,x+1\}}(\eta_{x+1},H)|\nonumber\\
&\qquad\;\;{}+\sup_{\eta \in \Sigma}\sup_{H \subset \{(0,0),(0,1),(1,0),(1,1)\}}|c_{\{x-1,x\}}(\eta,H)-c_{\{x-1,x\}}(\eta_{x-1},H)|\bigg]\nonumber\\
&=\sup_{x \in {\bf Z}}\sum_{u=x\pm1}\sup_{\eta \in \Sigma} |\gamma c'(x,\eta)+(1-\gamma) c(x,\eta)-[\gamma c'(x,\eta_{u})+(1-\gamma) c(x,\eta_{u})]|\nonumber\\
&\qquad\qquad\qquad{}+2\cdot\frac{\gamma}{2}\nonumber\\
&=\sup_{x \in {\bf Z}}\bigg[\sup_{\eta \in \Sigma} |\gamma [c'(x,\eta)-c'(x,\eta_{x+1})]+(1-\gamma)[c(x,\eta)- c(x,\eta_{x+1})]|\nonumber\\
&\qquad\qquad{}+\sup_{\eta \in \Sigma} |\gamma [c'(x,\eta)-c'(x,\eta_{x-1})]+(1-\gamma)[c(x,\eta)- c(x,\eta_{x-1})]|\bigg]+\gamma\nonumber\\
&=\max\bigg[ \bigg|\frac{\gamma}{2}+(1-\gamma)(p_0-p_1)\bigg|,\bigg|\frac{\gamma}{2}+(1-\gamma)(p_2-p_3)\bigg|\bigg]\nonumber\\
&\qquad{} + \max\bigg[\bigg|\frac{\gamma}{2}+(1-\gamma)(p_0-p_2)\bigg|,\bigg|\frac{\gamma}{2}+(1-\gamma)(p_1-p_3)\bigg|\bigg]+\gamma,
\end{align}
where the last step requires clarification. Notice first that
$$
c'(x,\eta)=\frac{1}{2}[ \bm{1}_{\{\eta(x)=\eta(x+1)\}}+ \bm{1}_{\{\eta(x)=\eta(x-1)\}}],
$$
\begin{align*}
c'(x,\eta_{x+1})&=\frac{1}{2}[ \bm{1}_{\{\eta_{x+1}(x)=\eta_{x+1}(x+1)\}}+ \bm{1}_{\{\eta_{x+1}(x)=\eta_{x+1}(x-1)\}}]\\
&=\frac{1}{2}[ \bm{1}_{\{\eta(x)=1-\eta(x+1)\}}+ \bm{1}_{\{\eta(x)=\eta(x-1)\}}],
\end{align*}
\begin{align*}
c'(x,\eta_{x-1})&=\frac{1}{2}[ \bm{1}_{\{\eta_{x-1}(x)=\eta_{x-1}(x+1)\}}+ \bm{1}_{\{\eta_{x-1}(x)=\eta_{x-1}(x-1)\}}]\\
&=\frac{1}{2}[ \bm{1}_{\{\eta(x)=\eta(x+1)\}}+ \bm{1}_{\{\eta(x)=1-\eta(x-1)\}}],
\end{align*}
$$
c(x,\eta)=\begin{cases}p_{m_{x}(\eta)}&\text{if $\eta(x)=0$,}\\ q_{m_{x}(\eta)}&\text{if $\eta(x)=1$,}\end{cases}
$$
$$
c(x,\eta_{x+1})=\begin{cases}p_{m_{x}(\eta_{x+1})}&\text{if $\eta(x)=0$,}\\ q_{m_{x}(\eta_{x+1})}&\text{if $\eta(x)=1$,}\end{cases}\quad
c(x,\eta_{x-1})=\begin{cases}p_{m_{x}(\eta_{x-1})}&\text{if $\eta(x)=0$,}\\ q_{m_{x}(\eta_{x-1})}&\text{if $\eta(x)=1$,}\end{cases}
$$
where $q_m:=1-p_m$ for $m=0,1,2,3$ and $m_{x}(\eta):=2\eta(x-1)+\eta(x+1) \in \{0,1,2,3\}$. The last line of \eqref{M-for-C'} is by direct calculation.\bigskip
Next,
\begin{align} \label{eps-for-C'}
\varepsilon &:=\inf_{u \in {\bf Z}}\;\inf_{\eta=\eta'\text{ off }u,\;\eta(u)\neq\eta'(u)}
\sum_{T \ni u} [c_T(\eta,\{\zeta\in \{0,1\}^{T}: \zeta(u)=\eta'(u)\})\nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad{}+c_T(\eta',\{\zeta\in\{0,1\}^{T}:\zeta(u)=\eta(u)\})]\nonumber\\
&=\inf_{u \in {\bf Z}} \inf_{\eta\in \Sigma}\big[c_{\{u\}}(\eta,\{\eta_u(u)\})+c_{\{u\}}(\eta_u,\{\eta(u)\})\nonumber\\
&\qquad\qquad\quad{}+c_{\{u,u+1\}}(\eta,\{\zeta\in\{0,1\}^{\{u,u+1\}}:\zeta(u)=\eta_{u}(u)\})\nonumber\\
&\qquad\qquad\quad{}+c_{\{u,u+1\}}(\eta_{u},\{\zeta\in\{0,1\}^{\{u,u+1\}}:\zeta(u)=\eta(u)\})\nonumber\\
&\qquad\qquad\quad{}+c_{\{u-1,u\}}(\eta,\{\zeta\in\{0,1\}^{\{u-1,u\}}:\zeta(u)=\eta_{u}(u)\})\nonumber\\
&\qquad\qquad\quad{}+c_{\{u-1,u\}}(\eta_{u},\{\zeta\in\{0,1\}^{\{u-1,u\}}:\zeta(u)=\eta(u)\})\big]\nonumber\\
&=\inf_{u \in {\bf Z}} \inf_{\eta \in \Sigma}\big[c_{\{u\}}(\eta, \{1-\eta (u)\})+c_{\{u\}}(\eta_{u},\{1-\eta_{u}(u)\})\nonumber\\
&\qquad\qquad\quad{}+c_{\{u,u+1\}}(\eta,\{\zeta\in\{0,1\}^{\{u,u+1\}}:\zeta(u)=1-\eta(u)\})\nonumber\\
&\qquad\qquad\quad{}+c_{\{u,u+1\}}(\eta_{u},\{\zeta\in\{0,1\}^{\{u,u+1\}}:\zeta(u)=1-\eta_u(u)\})\nonumber\\
&\qquad\qquad\quad{}+c_{\{u-1,u\}}(\eta,\{\zeta\in\{0,1\}^{\{u-1,u\}}:\zeta(u)=1-\eta(u)\})\nonumber\\
&\qquad\qquad\quad{}+c_{\{u-1,u\}}(\eta_{u},\{\zeta\in\{0,1\}^{\{u-1,u\}}:\zeta(u)=1-\eta_u(u)\})\big]\nonumber\\
&=\inf_{u \in {\bf Z}} \inf_{\eta \in \Sigma}\big[\gamma[c'(u,\eta)+c'(u,\eta_{u})]+(1-\gamma)[c(u,\eta)+c(u,\eta_{u})]\big]+2\cdot\frac{\gamma}{2}\nonumber\\
&=1+\gamma,
\end{align}
where the last line of \eqref{eps-for-C'} follows from
\begin{align*}
c'(u,\eta)+c'(u,\eta_{u})&=\frac{1}{2}[ \bm{1}_{\{\eta(u)=\eta(u+1)\}}+ \bm{1}_{\{\eta(u)=\eta(u-1)\}}]\\
&\quad{}+\frac{1}{2}[ \bm{1}_{\{\eta_{u}(u)=\eta_{u}(u+1)\}}+ \bm{1}_{\{\eta_{u}(u)=\eta_{u}(u-1)\}}]\\
&=\frac{1}{2}[ \bm{1}_{\{\eta(u)=\eta(u+1)\}}+ \bm{1}_{\{\eta(u)=\eta(u-1)\}}]\\
&\quad{}+\frac{1}{2}[ \bm{1}_{\{\eta(u)=1-\eta(u+1)\}}+ \bm{1}_{\{\eta(u)=1-\eta(u-1)\}}]\\
&=1
\end{align*}
and $c(u,\eta)+c(u,\eta_{u})=1$. We have proved the following theorem.
\begin{theorem}
The interacting particle system in $\Sigma:=\{0,1\}^{\bf Z}$ with generator $\Omega_{C'}:=\gamma\Omega_{A'}+(1-\gamma)\Omega_B$, where $0<\gamma<1$, is ergodic if
\begin{align}\label{erg-ineq}
&\max\bigg[ \bigg|\frac{\gamma}{2}+(1-\gamma)(p_0-p_1)\bigg|,\bigg|\frac{\gamma}{2}+(1-\gamma)(p_2-p_3)\bigg|\bigg]\nonumber\\
&\qquad + \max\bigg[\bigg|\frac{\gamma}{2}+(1-\gamma)(p_0-p_2)\bigg|,\bigg|\frac{\gamma}{2}+(1-\gamma)(p_1-p_3)\bigg|\bigg]<1.
\end{align}
\end{theorem}
The volume of the subset of the parameter space $[0,1]^4$ for which \eqref{erg-ineq} holds with $\gamma=1/2$ is 5/6. If we assume that $p_1=p_2$, then the volume of the subset of the parameter space $[0,1]^3$ for which \eqref{erg-ineq} holds with $\gamma=1/2$ is 3/4. In fact, the volume is $3/4$ if and only if $\gamma \ge 1/3$.
\section{Convergence of means}\label{sec-convergence}
We would like to prove that $\lim_{N\rightarrow \infty}\mu^N_{(\gamma,1-\gamma)'}$ and $\lim_{N\rightarrow \infty}\mu^N_{[r,s]'}$ exist under certain conditions, where $\mu^N_{(\gamma,1-\gamma)'}$ denotes the mean profit per turn at equilibrium to the $N$ players playing the $(\gamma,1-\gamma)$ random mixture of games $A'$ and $B$ (the Parrondo games of Xie et al., 2011), and $\mu^N_{[r,s]'}$ denotes the mean profit per turn at equilibrium to the $N$ players playing games $A'$ and $B$ in the nonrandom periodic pattern $A', A', \dots, A'$ ($r$ times), $B, B, \dots, B$ ($s$ times), $A', A', \dots, A'$ ($r$ times), $B, B, \dots, B$ ($s$ times), and so on. The first result is relatively straightforward, while the second requires more work. The key step for the second result is to prove that the sequence of discrete generators converges to the generator of an interacting particle system.
We want to show that our sequence of discrete-time Markov chains, suitably rescaled, converges in distribution to an interacting particle system on ${\bf Z}$. The limiting process is characterized in terms of its generator. First, we need to define generators corresponding to game $A'$, game $B$, and game $C'$. The state space is
\begin{align*}
\Sigma&:=\{0,1\}^{{\bf Z}}\\
&\phantom{:}=\{\eta=(\ldots,\eta(-2),\eta(-1),\eta(0),\eta(1),\eta(2),\ldots):\eta(x)\in\{0,1\}\text{ for all }x\in{\bf Z}\}.
\end{align*}
In Section~\ref{sec-intro} we defined $\eta^{x,x\pm1,\pm1}$ for $\eta\in\{0,1\}^N$, and that definition is easily extended to $\eta\in\Sigma$. For $\eta\in\Sigma$ and $x\in{\bf Z}$ define $\eta^{x,-1}$ and $\eta^{x,1}$ to be the elements of $\Sigma$ given by
$$
\eta^{x,-1}(y):=\begin{cases}1&\text{if $y=x-1$,}\\ 0&\text{if $y=x$,}\\ \eta(y)&\text{otherwise,}\end{cases}\qquad
\eta^{x,1}(y):=\begin{cases}0&\text{if $y=x$,}\\ 1&\text{if $y=x+1$,}\\ \eta(y)&\text{otherwise.}\end{cases}
$$
For example, $\eta^{0,1}:=(\ldots,\eta(-2),\eta(-1), 0, 1, \eta(2),\eta(3), \ldots)$. And let $\eta_x$ be the element of $\Sigma$ equal to $\eta$ except at the $x$th coordinate. Then the generators are
\begin{align*}
&(\Omega_{A'}f)(\eta)\\
&:=\sum_{x\in{\bf Z}}\bigg[\frac14 f(\eta^{x,x-1,-1})+\frac14 f(\eta^{x,x-1,1})+\frac14 f(\eta^{x,x+1,-1})+\frac14 f(\eta^{x,x+1,1})-f(\eta)\bigg]\\
&=\frac12\sum_{x\in{\bf Z}}[f(\eta^{x-1,x,1})-f(\eta)]+\frac12\sum_{x\in{\bf Z}}[f(\eta^{x,x+1,-1})-f(\eta)]\\
&=\frac12\bigg[\sum_{x:(\eta(x-1),\eta(x))=(0,0)}[f(\eta_{x-1})-f(\eta)]+\sum_{x:(\eta(x-1),\eta(x))=(0,1)}[f(_{x-1}\eta_x)-f(\eta)]\\
&\quad{}+\sum_{x:(\eta(x-1),\eta(x))=(1,1)}[f(\eta_x)-f(\eta)]+\sum_{x:(\eta(x),\eta(x+1))=(0,0)}[f(\eta_{x+1})-f(\eta)]\\
&\quad{}+\sum_{x:(\eta(x),\eta(x+1))=(1,0)}[f(_x\eta_{x+1})-f(\eta)]+\sum_{x:(\eta(x),\eta(x+1))=(1,1)}[f(\eta_x)-f(\eta)]\bigg]\\
&=\frac12\bigg[\sum_{x:(\eta(x),\eta(x+1))=(0,0)}[f(\eta_x)-f(\eta)]+\sum_{x:(\eta(x),\eta(x+1))=(0,1)}[f(_x\eta_{x+1})-f(\eta)]\\
&\quad{}+\sum_{x:(\eta(x),\eta(x+1))=(1,1)}[f(\eta_{x+1})-f(\eta)]+\sum_{x:(\eta(x),\eta(x+1))=(0,0)}[f(\eta_{x+1})-f(\eta)]\\
&\quad{}+\sum_{x:(\eta(x),\eta(x+1))=(1,0)}[f(_x\eta_{x+1})-f(\eta)]+\sum_{x:(\eta(x),\eta(x+1))=(1,1)}[f(\eta_x)-f(\eta)]\bigg]\\
&=\frac12\bigg[\sum_{x:\eta(x)=\eta(x+1)}[f(\eta_x)-f(\eta)]+\sum_{x:\eta(x)=\eta(x+1)}[f(\eta_{x+1})-f(\eta)]\\
&\qquad\quad{}+\sum_{x\in{\bf Z}}[f(_x\eta_{x+1})-f(\eta)]\bigg]\\
&=\frac12\bigg[\sum_{x:\eta(x)=\eta(x+1)}[f(\eta_x)-f(\eta)]+\sum_{x:\eta(x-1)=\eta(x)}[f(\eta_x)-f(\eta)]\\
&\qquad\quad{}+\sum_{x\in{\bf Z}}[f(_x\eta_{x+1})-f(\eta)]\bigg]\\
&=\sum_{x\in{\bf Z}} c'(x,\eta)[f(\eta_x)-f(\eta)]+\frac12\sum_{x\in{\bf Z}}[f(_x\eta_{x+1})-f(\eta)],
\end{align*}
where $c'(x,\eta)$ is as in \eqref{c'-def},
\begin{equation*}
(\Omega_{B}f)(\eta):=\sum_{x \in {\bf Z}}c(x,\eta)\big[f(\eta_x)-f(\eta)\big],
\end{equation*}
where $c(x,\eta)$ is as in \eqref{c-def}, and
\begin{equation*}
(\Omega_{C'}f)(\eta):=\big[\gamma\Omega_{A'}f+(1-\gamma)\Omega_{B}f\big](\eta)
\end{equation*}
for functions $f\in C(\Sigma)$ depending on only finitely many coordinates.
Next, it is necessary to show that this interacting particle system is the limit in distribution of the $N$-player model as $N\to\infty$. Furthermore, we need to adjust the state space by relabeling the players. Specifically, we let
$$
\Sigma_N :=\{\eta =(\eta(l_{N}), \ldots, \eta(r_N)):\eta(x) \in \{0, 1\}\text{ for }x=l_N,\ldots, r_N \},
$$
where $l_N$ and $r_N$ are as in \eqref{relabel}. It should be noted that players $l_N$ and $r_N$ are nearest neighbors. We denote the Markov chain in $\Sigma_N$ by $\{X_k^{\cdot,N},\; k=0,1,2,\ldots\}$, where $A'$, $B$, or $C'$ appears in place of the dot.
First, let us analyze game $A'$.
The one-step transition matrix $\bm P_{A'}$ of the Markov chain in the state space $\Sigma_N$ has the form
\begin{align*}
P_{A'}(\eta, \xi)&:=\frac{1}{4N}\sum_{l_N\le x\le r_N}[\delta(\eta^{x,x-1,-1},\xi)+\delta(\eta^{x,x-1,1},\xi)\\
&\qquad\qquad\qquad\qquad{}+\delta(\eta^{x,x+1,-1},\xi)+\delta(\eta^{x,x+1,1},\xi)]\\
&\;=\frac{1}{2N}\sum_{l_N\le x\le r_N}\big[\delta(\eta^{x,-1},\xi)+\delta(\eta^{x,1},\xi)\big] ,
\end{align*}
where $\delta(\eta, \xi)$ is the Kronecker delta, which is $1$ if $\eta = \xi$ and is $0$ otherwise; the sum over $x$ ranges over $\{ l_{N}, \dots, r_{N} \}$, and $l_N-1:=r_N$ and $r_N+1:=l_N$. Next, we have the one-step transition matrix $\bm P_B$ of the form
$$
P_{B}(\xi, \zeta):=\frac{1}{N} \sum_y [1-c(y,\xi)]\delta(\xi, \zeta)+\frac{1}{N} \sum_y c(y,\xi)\delta(\xi_{y}, \zeta),
$$
where the sum over $y$ also ranges over $\{ l_{N}, \dots, r_{N} \}$; $c(y,\xi)$ is as in \eqref{c-def}, except that $l_N-1:=r_N$ and $r_N+1:=l_N$.
We speed up time in the $N$-player model so that $N$ one-step transitions occur per unit of time. Then the discrete generator corresponding to game $A'$ is
\begin{align*}
(\Omega^N_{A'}f)(\eta)&=N\text{E}\big[f(X_1^{A',N})-f(\eta)\mid X_0^{A',N}=\eta\big]\\
&=N\sum_{\xi\in\Sigma_N}P_{A'}(\eta, \xi)\big[f(\xi)-f(\eta)\big]\\
&=N\sum_{\xi \in \Sigma_N}\frac{1}{2N}\sum_{l_N\leq x \leq r_N}\big[\delta(\eta^{x,-1}, \xi)+\delta(\eta^{x,1}, \xi) \big]\big[f(\xi)-f(\eta)\big]\\
&=\frac12 \sum_{l_N\leq x \leq r_N}\big[f(\eta^{x,-1})-f(\eta)+f(\eta^{x,1})-f(\eta)\big]\\
&=\sum_{l_N\leq x \leq r_N}\bigg[\frac12f(\eta^{x,-1})+\frac12f(\eta^{x,1})-f(\eta)\bigg].
\end{align*}
The discrete generator corresponding to game $B$ is
\begin{align*}
(\Omega^N_{B}f)(\eta)&=N\text{E}\big[f(X_1^{B,N})-f(\eta)\mid X_0^{B,N}=\eta\big]\\
&=N\sum_{\xi\in\Sigma_N}P_B(\eta, \xi)\big[f(\xi)-f(\eta)\big]\\
&=\sum_{l_N\leq x \leq r_N: \eta(x)=0}p_{m_x(\eta)}\big[f(\eta_x)-f(\eta)\big]\\
&\qquad\quad+\sum_{l_N\leq x \leq r_N: \eta(x)=1}q_{m_x(\eta)}\big[f(\eta_x)-f(\eta)\big].
\end{align*}
Hence the discrete generator corresponding to game $C'$ is
\begin{align*}
(\Omega^N_{C'}f)(\eta)&=N\text{E}\big[f(X_1^{C',N})-f(\eta)\mid X_0^{C',N}=\eta\big]\\
&=\big[\gamma\Omega_{A'}^N f+(1-\gamma)\Omega_B^N f\big](\eta)\\
&=\gamma\sum_{l_N\leq x \leq r_N}\bigg[\frac12f(\eta^{x,-1})+\frac12f(\eta^{x,1})-f(\eta)\bigg]\\
&\qquad+(1-\gamma)\bigg[\sum_{l_N\leq x \leq r_N: \eta(x)=0}p_{m_x(\eta)}\big[f(\eta_x)-f(\eta)\big]\\
&\qquad\qquad\qquad\qquad+\sum_{l_N\leq x \leq r_N: \eta(x)=1}q_{m_x(\eta)}\big[f(\eta_x)-f(\eta)\big]\bigg].
\end{align*}
We define $\psi_N: B(\Sigma)\mapsto B(\Sigma_N)$ by
\begin{equation*}
(\psi_N f)(\eta(l_N),\ldots,\eta(r_N)):=f(\ldots,1,1,\eta(l_N),\ldots,\eta(r_N),1,1,\ldots).
\end{equation*}
\begin{lemma} \label{Omega-psi}
If $f\in C(\Sigma)$ depends on $\eta$ only through the $2K+1$ components $\eta(x)$ for $-K \leq x \leq K$, then
\begin{equation*}
(\Omega^N_{A'}\psi_Nf)(\eta)=\psi_N(\Omega_{A'}f)(\eta),
\end{equation*}
\begin{equation*}
(\Omega^N_{B}\psi_Nf)(\eta)=\psi_N(\Omega_{B}f)(\eta),
\end{equation*}
and
\begin{equation*}
(\Omega^N_{C'}\psi_Nf)(\eta)=\psi_N(\Omega_{C'}f)(\eta)
\end{equation*}
for all $\eta \in \Sigma_N$ and $N\geq 2K+4$.
\end{lemma}
\begin{proof}
The proof is straightforward.
\end{proof}
Lemma~\ref{Omega-psi} implies that the process $\{X_{\lfloor N t \rfloor}^{C',N}\}$ converges in distribution to the interacting particle system $\{X_t\}$ by Theorem 1.6.5 and 4.2.6 of Ethier and Kurtz (1986). More importantly, it implies that, if the interacting particle system has a unique stationary distribution, then the unique stationary distribution of the $N$-player Markov chain converges to it in the topology of weak convergence, essentially by Proposition I.2.14 of Liggett (1985). Let us assume that the interacting particle system with generator $\Omega_{C'}$ has a unique stationary distribution $\pi$, and let us denote the unique stationary distribution of the $N$-player Markov chain for the $(\gamma,1-\gamma)$ random mixture of games $A'$ and $B$ by $\pi^N$. Let us denote their $-1,1$ two-dimensional marginals by $\pi_{-1,1}^N$ and $\pi_{-1,1}$. Then we have
\begin{align}\label{mean-limit}
\mu_{(\gamma,1-\gamma)'}^N&=(1-\gamma)[\pi^N_{-1,1}(0,0)(2p_0-1)+\pi^N_{-1,1}(0,1)(2p_1-1)\nonumber\\
&\qquad{}+\pi^N_{-1,1}(1,0)(2p_2-1)+\pi^N_{-1,1}(1,1)(2p_3-1)]\nonumber\\
&\to(1-\gamma)[\pi_{-1,1}(0,0)(2p_0-1)+\pi_{-1,1}(0,1)(2p_1-1)\nonumber\\
&\qquad{}+\pi_{-1,1}(1,0)(2p_2-1)+\pi_{-1,1}(1,1)(2p_3-1)]\nonumber\\
&=:\mu_{(\gamma,1-\gamma)'},
\end{align}
where the first equality is based on the following idea: Suppose player 0 is chosen to play. If he plays game $A'$ (probability $\gamma$) he wins or loses 1 with probability 1/2 each, so his mean profit is 0; if he plays game $B$ (probability $1-\gamma$), he tosses a $p_m$ coins with $m$ determined by the status of his nearest neighbors, hence his expected profit is $2p_m-1$. We conclude that $\mu_{(\gamma,1-\gamma)'}^N$, the mean profit per turn at equilibrium to the $N$ players playing the $(\gamma,1-\gamma)$ random mixture of games $A'$ and $B$, converges as $N\to\infty$ to a limit that can be expressed in terms of an interacting particle system. We have proved the following.
\begin{theorem}\label{conv-N-gamma 1-gamma}
Fix $\gamma\in(0,1)$. Assume that the interacting particle system on ${\bf Z}$ with generator $\Omega_{C'}:=\gamma\Omega_{A'}+(1-\gamma)\Omega_B$ is ergodic with unique stationary distribution $\pi$. Then $\lim_{N\to\infty}\mu_{(\gamma,1-\gamma)'}^N=\mu_{(\gamma,1-\gamma)'}$, where $\mu_{(\gamma,1-\gamma)'}$ is as in \eqref{mean-limit}.
\end{theorem}
The discrete generator for the nonrandom periodic pattern $(A')^r B^s$ has the form, for $f\in B(\Sigma_N)$,
$$
(\Omega^N_{[r, s]'}f)(\eta^0)=\frac{N}{r+s}\sum_{\eta^{r+s}}[f(\eta^{r+s})-f(\eta^0)](\bm{P}^r_{A'}\bm{P}^s_{B})(\eta^0, \eta^{r+s}).
$$
We begin by evaluating
\begin{align}\label{P_A^rP_B^s}
&\!\!\!\!\!(\bm{P}^{r}_{A'}\bm{P}^{s}_{B})(\eta^0, \eta^{r+s})\nonumber\\
&=\sum_{\eta^1,\eta^2,\ldots, \eta^{r+s-1} } P_{A'}(\eta^0, \eta^1)P_{A'}(\eta^1, \eta^2)\cdots P_{A'}(\eta^{r-1}, \eta^r)\nonumber\\
&\quad\cdot P_B(\eta^r, \eta^{r+1})P_B(\eta^{r+1}, \eta^{r+2})\cdots P_B(\eta^{r+s-1}, \eta^{r+s})\nonumber\\
&=\sum_{\eta^1,\eta^2,\ldots, \eta^{r+s-1} }\prod_{i=1}^r P_{A'}(\eta^{i-1},\eta^i)\prod_{i=r+1}^{r+s}P_B(\eta^{i-1},\eta^i)\nonumber\\
&=\sum_{\eta^1,\eta^2,\ldots,\eta^{r+s-1}} \prod_{i=1}^{r}\bigg[\frac{1}{2N}\sum_{x_{i}}[\delta((\eta^{i-1})^{x_{i},-1}, \eta^i)+\delta((\eta^{i-1})^{x_{i},1}, \eta^i)]\bigg]\nonumber\\
&\quad\cdot \prod_{i=r+1}^{r+s}\bigg[\frac{1}{N}\sum_{x_{i}}(1-c(x_{i},\eta^{i-1}))\delta(\eta^{i-1}, \eta^{i})+\frac{1}{N}\sum_{x_{i}}c(x_{i},\eta^{i-1})\delta((\eta^{i-1})_{x_{i}}, \eta^{i})\bigg]\nonumber\\
&=\frac{1}{2^rN^{r+s}}\sum_{A\subset \{1,\ldots, r\}}\sum_{B\subset \{r+1,\ldots, r+s\}}\sum_{\eta^1,\eta^2,\ldots, \eta^{r+s-1} }
\prod_{i\in A^c}\bigg[\sum_{x_{i}}\delta((\eta^{i-1})^{x_{i},-1}, \eta^i)\bigg]\nonumber\\
&\quad\cdot \prod_{i\in A}\bigg[\sum_{x_{i}}\delta((\eta^{i-1})^{x_{i},1}, \eta^i)\bigg] \prod_{i\in B^c}\bigg[\sum_{x_{i}}(1-c(x_{i},\eta^{i-1}))\delta(\eta^{i-1}, \eta^{i})\bigg]\nonumber\\
&\quad\cdot \prod_{i\in B}\bigg[\sum_{x_{i}}c(x_{i},\eta^{i-1})\delta((\eta^{i-1})_{x_{i}}, \eta^{i})\bigg]\nonumber\\
&=\frac{1}{2^rN^{r+s}}\sum_{A\subset \{1,\ldots, r\}}\sum_{B\subset \{r+1,\ldots, r+s\}}\sum_{\eta^1,\eta^2,\ldots, \eta^{r+s-1}}\sum_{x_{i}: i\in A^c}\sum_{x_{i}: i\in A }\nonumber\\
&\quad\cdot \prod_{j\in A^c}\big[\delta((\eta^{j-1})^{x_{j},-1}, \eta^j)\big] \prod_{j\in A}\big[\delta((\eta^{j-1})^{x_{j},1}, \eta^j)\big]\sum_{x_{i}: i\in B^c}\sum_{x_{i}: i\in B}\nonumber\\
&\quad\cdot \prod_{j\in B^c}\big[(1-c(x_{j},\eta^{j-1}))\delta(\eta^{j-1}, \eta^{j})\big] \prod_{j\in B}\big[c(x_{j},\eta^{j-1})\delta((\eta^{j-1})_{x_{j}}, \eta^{j})\big]\nonumber\\
&=\frac{1}{2^rN^{r+s}}\sum_{A\subset \{1,\ldots, r\}}\sum_{B\subset \{r+1,\ldots, r+s\}}\sum_{x_{i}: i\in \{1,2,\ldots, r+s\}}\nonumber\\
&\quad\cdot\prod_{j\in B^c}\big[1-c(x_{j},((\cdots((\cdots(((\eta^{0})^{x_{1},a_1})^{x_{2},a_2})\cdots)^{x_{p},a_p})\cdots)^{x_{r},a_r})_{\{x_{l}: l \in B,l<j\}})\big]\nonumber\\
&\quad\cdot\prod_{j\in B}c(x_{j},((\cdots((\cdots(((\eta^{0})^{x_{1},a_1})^{x_{2},a_2})\cdots)^{x_{p},a_p})\cdots)^{x_{r},a_r})_{\{x_{l}: l \in B,l<j\}})\nonumber\\
&\qquad\times\delta(((\cdots((\cdots(((\eta^{0})^{x_{1},a_1})^{x_{2},a_2})\cdots)^{x_{p},a_p})\cdots)^{x_{r},a_r})_{\{x_{l}: l \in B\}}, \eta^{r+s}),
\end{align}
where $A^c:=\{1,2,\ldots, r\}-A$ and $B^c:=\{r+1,r+2,\ldots, r+s\}-B$; also $p \in \{1,2,\ldots, r\}$ and
$$
a_p = \begin{cases} -1 &\text{if $p \in A^c$}, \\ \phantom{-}1 & \text{if $p \in A$}. \end{cases}
$$
Here, for example, $\eta_{\{x_l:l\in B\}}$ denotes $\eta$ with the spins flipped at each site $x_l$ with $l\in B$. These site labels need not be distinct, so if there are multiple flips at a single site, only the parity of the number of flips is relevant.
Next, assume that $f\in B(\Sigma)$ depends only on $\eta(-(K-2)),\ldots,\eta(K-2)$ for some integer $K\ge2$, and put $f_N:=\psi_N f\in B(\Sigma_N)$. Then the discrete generator for the pattern $(A')^r B^s$, acting on $f_N$, reduces to
\begin{align}\label{Omega_[r,s]}
&(\Omega^N_{[r, s]'}f_N)(\eta^0)\nonumber\\
&=\frac{N}{r+s}\sum_{\eta^{r+s}}[f_N(\eta^{r+s})-f_N(\eta^0)](\bm{P}^{r}_{A'}\bm{P}^{s}_{B})(\eta^0, \eta^{r+s})\nonumber\\
&=\frac{1}{r+s}\ \frac{1}{2^rN^{r+s-1}}\sum_{A\subset \{1,\ldots, r\}}\sum_{B\subset \{r+1,\ldots, r+s\}}\sum_{x_{i}: i\in \{1,2,\ldots, r+s\}}\nonumber\\
&\quad\cdot \prod_{j\in B^c}\big[1-c(x_{j},((\cdots((\cdots(((\eta^{0})^{x_{1},a_1})^{x_{2},a_2})\cdots)^{x_{p},a_p})\cdots)^{x_{r},a_r})_{\{x_{l}: l \in B,l<j\}})\big]\nonumber\\
&\quad\cdot \prod_{j\in B}c(x_{j},((\cdots((\cdots(((\eta^{0})^{x_{1},a_1})^{x_{2},a_2})\cdots)^{x_{p},a_p})\cdots)^{x_{r},a_r})_{\{x_{l}: l \in B,l<j\}})\nonumber\\
&\quad\cdot [f_N(((\cdots((\cdots(((\eta^{0})^{x_{1},a_1})^{x_{2},a_2})\cdots)^{x_{p},a_p})\cdots)^{x_{r},a_r})_{\{x_{l}: l \in B\}})-f_N( \eta^{0})]
\end{align}
We replace $\sum_{x_{i}: i\in \{1,2,\ldots, r+s\}}$ in \eqref{Omega_[r,s]} by
\begin{align}\label{decomposition}
&\bigg(\sum_{i\in A^c}+\sum_{i\in A}\bigg)\sum_{|x_i|\leq K}\sum_{|x_m|>K: m\in \{1,2,\ldots,r+s\},m \neq i}\nonumber\\
&\quad{} + \sum_{i\in B}\sum_{|x_i|\leq K}\sum_{|x_m|>K: m\in \{1,2,\ldots,r+s\},m \neq i}
\end{align}
since each sum $\sum_{x_{i}}$ can be written as $\sum_{|x_i|\leq K}+\sum_{|x_i|> K}$ resulting in $2^{r+s}$ multiple sums in which each of those multiple sums with two or more sums of the form $\sum_{|x_i|\leq K}$ contributes at most $O(N^{-1})$ and those without the form $\sum_{|x_i|\leq K},$ where $i \in \{1,2,\ldots,r\}\cup B$ are $0$. So it is enough to analyze the cases in which only one of the $|x_i|$'s is less than or equal to $K$.
We consider first the first term in \eqref{decomposition}. It contributes
\begin{align*}
&\frac{1}{r+s}\;\frac{1}{2^rN^{r+s-1}}\sum_{A\subset \{1,\ldots, r\}}\sum_{B\subset \{r+1,\ldots, r+s\}}\bigg[\sum_{i \in A^c}\sum_{|x_{i}|\leq K}[f_N((\eta^0)^{x_i,-1})- f_N(\eta^0)]\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\sum_{i \in A}\sum_{|x_{i}|\leq K}[f_N((\eta^0)^{x_i,1})-f_N(\eta^0)]\bigg]\\
&\quad\cdot\sum_{|x_m|>K: m\in \{1,2,\ldots,r+s\},m \neq i}\prod_{j\in B^c}\big[1\\
&\qquad\qquad\quad{}-c(x_{j},((\cdots((\cdots(((\eta^{0})^{x_{1},a_1})^{x_{2},a_2})\cdots)^{x_{p},a_p})\cdots)^{x_{r},a_r})_{\{x_{l}: l \in B,l<j\}})\big]\\
&\quad\cdot\prod_{j\in B}c(x_{j},((\cdots((\cdots(((\eta^{0})^{x_{1},a_1})^{x_{2},a_2})\cdots)^{x_{p},a_p})\cdots)^{x_{r},a_r})_{\{x_{l}: l \in B,l<j\}})\\
&=\frac{1}{r+s}\,\frac{1}{2^rN^{r+s-1}}\!\!\!\sum_{A\subset\{1,\ldots,r\}}\sum_{B\subset\{r+1,\ldots,r+s\}}\bigg[\sum_{i\in A^c}\sum_{|x_{i}|\leq K}[f_N((\eta^0)^{x_i,-1})-f_N(\eta^0)]\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;{}+\sum_{i \in A}\sum_{|x_{i}|\leq K}[f_N((\eta^0)^{x_i,1})-f_N(\eta^0)]\bigg]\\
&\quad\cdot\sum_{|x_m|>K: m\in \{1,2,\ldots,r+s\},m \neq i}\prod_{j\in B^c}\big[1\\
&\qquad\qquad\qquad\qquad\qquad{}-c(x_{j},(\cdots((\cdots(((\eta^{0})^{x_{1},a_1})^{x_{2},a_2})\cdots)^{x_{p},a_p})\cdots)^{x_{r},a_r})\big]\\
&\quad\cdot\prod_{j\in B}c(x_{j},(\cdots((\cdots(((\eta^{0})^{x_{1},a_1})^{x_{2},a_2})\cdots)^{x_{p},a_p})\cdots)^{x_{r},a_r})\\
&=\frac{1}{r+s}\,\frac{1}{2^rN^{r+s-1}}\big[N-(2K+1)\big]^{r+s-1}\\
&\qquad\qquad\cdot{}\sum_{A\subset \{1,\ldots, r\}}\bigg[|A^c|\sum_{|x|\leq K}[f_N((\eta^0)^{x,-1})- f_N(\eta^0)]\\
&\qquad\qquad\qquad\qquad\qquad{}+|A|\sum_{|x|\leq K}[f_N((\eta^0)^{x,1})-f_N(\eta^0)]\bigg]\\
&=\frac{1}{r+s}\ \frac{1}{2^rN^{r+s-1}}\big[N-(2K+1)\big]^{r+s-1}\sum_{A\subset \{1,\ldots, r\}}\bigg[|A^c|\sum_{|x|\leq K}f_N((\eta^0)^{x,-1})\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad{}+|A|\sum_{|x|\leq K}f_N((\eta^0)^{x,1})-r\sum_{|x|\leq K}f_N(\eta^0)\bigg]\\
&=\frac{1}{r+s}\ \frac{1}{N^{r+s-1}}\big[N-(2K+1)\big]^{r+s-1} \bigg[\frac{r}{2}\sum_{|x|\leq K} f_N((\eta^0)^{x,-1})\\
&\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\frac{r}{2}\sum_{|x|\leq K} f_N((\eta^0)^{x,1})-r\sum_{|x|\leq K} f_N(\eta^0)\bigg]\\
&=\frac{r}{r+s}\sum_{|x|\leq K}\bigg[\frac{1}{2} f_N((\eta^0)^{x,-1})+\frac{1}{2} f_N((\eta^0)^{x,1})-f_N(\eta^0)\bigg]+O(N^{-1})\\
&=\frac{r}{r+s}\psi_N (\Omega_{A'}f)(\eta^0)+O(N^{-1}),
\end{align*}
where, in the second equality,
\begin{align*}
&\sum_{|x_m|>K: m\in \{1,2,\ldots,r+s\},m \neq i}\sum_{B\subset \{r+1,\ldots, r+s\}}\prod_{j\in B^c}\big[1\\
&\qquad\qquad\qquad\qquad\qquad{}-c(x_{j},(\cdots((\cdots(((\eta^{0})^{x_{1},a_1})^{x_{2},a_2})\cdots)^{x_{p},a_p})\cdots)^{x_{r},a_r})\big]\\
&\qquad\qquad\qquad\qquad\cdot \prod_{j\in B}c(x_{j},(\cdots((\cdots(((\eta^{0})^{x_{1},a_1})^{x_{2},a_2})\cdots)^{x_{p},a_p})\cdots)^{x_{r},a_r})\\
&=\sum_{|x_m|>K: m\in \{1,2,\ldots,r+s\},m \neq i} \prod_{j=r+1}^{r+s}\big[1\nonumber\\
&\qquad\qquad\qquad\qquad{}-c(x_{j},(\cdots((\cdots(((\eta^{0})^{x_{1},a_1})^{x_{2},a_2})\cdots)^{x_{p},a_p})\cdots)^{x_{r},a_r})\\
&\qquad\qquad\qquad\qquad{}+c(x_{j},(\cdots((\cdots(((\eta^{0})^{x_{1},a_1})^{x_{2},a_2})\cdots)^{x_{p},a_p})\cdots)^{x_{r},a_r})\big]\\
&=[N-(2K+1)]^{r+s-1},
\end{align*}
and in the fourth equality,
\begin{align*}
&\frac{1}{2^r}\sum_{A\subset \{1,\ldots, r\}}|A^c|=\frac{1}{2^r}\sum_{A\subset \{1,\ldots, r\}}|A|=\frac{1}{2^r}\sum_{i=0}^{r}i\ {r\choose i}=\frac{r}{2}.
\end{align*}
Next, we consider the second term in \eqref{decomposition}. It contributes
\begin{align*}
&\!\!\!\!\!\frac{1}{r+s}\ \frac{1}{2^rN^{r+s-1}}\sum_{A\subset \{1,\ldots, r\}}\sum_{B\subset \{r+1,\ldots, r+s\}}\sum_{i \in B}\sum_{|x_{i}|\leq K}[f_N((\eta^0)_{x_i})- f_N(\eta^0)]\\
&\quad\cdot\sum_{|x_m|>K: m\in \{1,2,\ldots,r+s\},m \neq i}\prod_{j\in B^c}\big[1\\
&\qquad\qquad{}-c(x_{j},((\cdots((\cdots(((\eta^{0})^{x_{1},a_1})^{x_{2},a_2})\cdots)^{x_{p},a_p})\cdots)^{x_{r},a_r})_{\{x_{l}: l \in B,l<j\}})\big]\\
&\quad\cdot \prod_{j\in B}c(x_{j},((\cdots((\cdots(((\eta^{0})^{x_{1},a_1})^{x_{2},a_2})\cdots)^{x_{p},a_p})\cdots)^{x_{r},a_r})_{\{x_{l}: l \in B,l<j\}})\\
&=\frac{1}{r+s}\ \frac{1}{2^rN^{r+s-1}}\sum_{A\subset\{1,\ldots, r\}}\sum_{i=r+1}^{r+s}\ \sum_{|x_{i}|\leq K}c(x_i,\eta^0)[f_N((\eta^0)_{x_i})- f_N(\eta^0)]\\
&\quad\cdot\sum_{|x_m|>K: m\in \{1,2,\ldots,r+s\},m \neq i}\sum_{B\subset \{r+1,\ldots, r+s\}: i\in B}\prod_{j\in B^c}\big[1-c(x_{j},\eta^{0})\big]\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad{}\cdot\prod_{j\in B-\{i\}}c(x_{j},\eta^{0})+O(N^{-1})\\
&=\frac{1}{r+s}\ \frac{1}{2^rN^{r+s-1}}\big[N-(2K+1)\big]^{r+s-1}\sum_{A\subset \{1,\ldots, r\}} \sum_{i=r+1}^{r+s}\\
&\qquad\qquad\qquad\qquad\qquad\qquad{}\cdot\sum_{|x_{i}|\leq K} c(x_i,\eta^0)[f_N((\eta^0)_{x_i})- f_N(\eta^0)]+O(N^{-1})\\
&=\frac{1}{r+s}\ \frac{1}{2^rN^{r+s-1}}\big[N-(2K+1)\big]^{r+s-1}2^r \sum_{i=r+1}^{r+s}\\
&\qquad\qquad\qquad\qquad\qquad\qquad{}\cdot\sum_{|x_{i}|\leq K} c(x_i,\eta^0)[f_N((\eta^0)_{x_i})- f_N(\eta^0)]+O(N^{-1})\\
&=\frac{1}{r+s}\ \frac{1}{N^{r+s-1}}\big[N-(2K+1)\big]^{r+s-1} s \sum_{|x|\leq K} c(x,\eta^0)[f_N((\eta^0)_{x})- f_N(\eta^0)]\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad{}+O(N^{-1})\\
&=\frac{s}{r+s}\ \sum_{|x|\leq K} c(x,\eta^0)[f_N((\eta^0)_{x})- f_N(\eta^0)]+O(N^{-1})\\
&=\frac{s}{r+s}\psi_N(\Omega_{B}f)(\eta^0)+O(N^{-1}),
\end{align*}
where the first and second equalities require clarification.
In the first equality we used
$$
\sum_{B\subset \{r+1,\ldots, r+s\}}\sum_{i \in B}=\sum_{i=r+1}^{r+s}\sum_{B \subset \{r+1,\ldots, r+s\}:i\in B}
$$
and
$$
c(x_{j},((\cdots((\cdots(((\eta^{0})^{x_{1},a_1})^{x_{2},a_2})\cdots)^{x_{p},a_p})\cdots)^{x_{r},a_r})_{\{x_{l}: l \in B,l<j\}})=c(x_j,\eta^{0})
$$
with possible exceptions if
$$
\{x_j-1,x_j,x_j+1\}\cap\bigg[\bigcup_{p\in A^c}\{x_p,x_p-1\}\cup\bigcup_{p\in A}\{x_p,x_p+1\}\cup\bigcup_{p\in B}\{x_p\}\bigg]\ne\varnothing.
$$
That excludes at most $4r+3s$ of the $N$ possible values of $x_j$, hence involves an error of at most $O(N^{-1})$.
In the second equality,
\begin{align*}
&\sum_{|x_m|>K: m\in \{1,2,\ldots,r+s\},m \neq i}\sum_{B\subset \{r+1,\ldots, r+s\}:i\in B}\prod_{j\in B^c}\big[1-c(x_{j},\eta^{0})\big] \prod_{j\in B-\{i\}}c(x_{j},\eta^{0})\nonumber\\
&=\sum_{|x_m|>K: m\in \{1,2,\ldots,r+s\},m \neq i}\prod_{j\in\{r+1,\ldots,r+s\}-\{i\}}\big[1-c(x_{j},\eta^{0})+c(x_{j},\eta^{0})\big]\\
&=[N-(2K+1)]^{r+s-1}.
\end{align*}
Therefore, we conclude that
\begin{equation} \label{Omega^N}
(\Omega^N_{[r, s]'}\psi_N f)(\eta^0)=\psi_{N}\bigg(\frac{r}{r+s}\Omega_{A'}f+\frac{s}{r+s}\Omega_{B}f\bigg)(\eta^0)+O(N^{-1}),
\end{equation} as desired.
Since \eqref{Omega^N} holds, uniformly over $\Sigma_N,$ the unique stationary distribution $\pi^N$ of $\bm P_{A'}^r \bm P_{B}^s$ converges weakly to the unique stationary distribution $\pi^{r/(r+s)}$ of the interacting particle system with generator $\Omega_{C'},$ provided that ergodicity holds for the limiting interacting particle system. Here, $\Omega_{C'}= \gamma \Omega_{A'}+(1-\gamma) \Omega_B$ with $\gamma:=r/(r+s)$, specifically
\begin{align}\label{Omega_C'2}
(\Omega_{C'} f)(\eta)&=\gamma\sum_x c'(x,\eta)[f(\eta_x)-f(\eta)]\nonumber\\
&\quad+\gamma\sum_x [f(_x \eta_{x+1})-f(\eta)]+(1-\gamma)\sum_x c(x,\eta)[f(\eta_x)-f(\eta)].
\end{align}
Our aim is to prove the following theorem.
\begin{theorem}\label{periodic-limit}
Fix integers $r,s\ge1$ and put $\gamma:=r/(r+s)$. Assume that the interacting particle system on ${\bf Z}$ with generator $\Omega_{C'}$ as in \eqref{Omega_C'2} is ergodic with unique stationary distribution $\pi^\gamma$. Then $\lim_{N\to\infty}\mu_{[r,s]'}^N=\mu_{(\gamma,1-\gamma)'}$, where $\mu_{(\gamma,1-\gamma)'}$ is as in \eqref{mean-limit}.
\end{theorem}
\begin{proof}
The mean profit per turn to the ensemble of $N$ players playing the nonrandom periodic pattern $(A')^rB^s$ is
\begin{equation} \label{mu^N}
\mu_{[r,s]'}^N =\frac{1}{r+s}\sum_{v=0}^{s-1}\sum_{\eta \in \Sigma_{N}}(\bm\pi^N\bm P^r_{A'}\bm P^v_{B})(\eta)\frac{1}{N}\sum_z[2p_{m_z(\eta)}-1].
\end{equation}
The sum over $\eta$ in \eqref{mu^N} can be expressed, using \eqref{P_A^rP_B^s}, as
\begin{align*}
&\!\!\!\!\!\sum_{\eta^0,\eta}\pi^N(\eta^0)(\bm P^r_{A'}\bm P^v_{B})(\eta^0, \eta)\frac{1}{N}\sum_{z}[2p_{m_z(\eta)}-1]\\
&=\frac{1}{2^rN^{r+v}}\ \sum_{\eta^0}\pi^N(\eta^0)\sum_{A\subset \{1,\ldots, r\}}\sum_{B\subset \{r+1,\ldots, r+v\}}\sum_{x_{i}: i\in \{1,2,\ldots, r+v\}}\\
&\quad\cdot \prod_{j\in B^c}\big[1-c(x_{j},((\cdots((\cdots(((\eta^{0})^{x_{1},a_1})^{x_{2},a_2})\cdots)^{x_{p},a_p})\cdots)^{x_{r},a_r})_{\{x_{l}: l \in B,l<j\}})\big]\\
&\quad\cdot \prod_{j\in B}c(x_{j},((\cdots((\cdots(((\eta^{0})^{x_{1},a_1})^{x_{2},a_2})\cdots)^{x_{p},a_p})\cdots)^{x_{r},a_r})_{\{x_{l}: l \in B,l<j\}})\\
&\quad\cdot\frac{1}{N}\sum_z[2p_{m_z(((\cdots((\cdots(((\eta^{0})^{x_{1},a_1})^{x_{2},a_2})\cdots)^{x_{p},a_p})\cdots)^{x_{r},a_r})_{\{x_{l}: l \in B\}})}-1]\\
&=\frac{1}{2^rN^{r+v}}\ \sum_{\eta^0} \pi^N(\eta^0)\sum_{A\subset \{1,\ldots, r\}}\sum_{B\subset \{r+1,\ldots, r+v\}}\sum_{x_{i}: i\in \{1,2,\ldots, r+v\}}\\
&\quad\cdot \prod_{j\in B^c}\big[1-c(x_{j},\eta^0)\big] \prod_{j\in B}c(x_{j},\eta^0)\frac{1}{N}\sum_z[2p_{m_z(\eta^0)}-1]+O(N^{-1})\\
&=\frac{1}{2^rN^{r+v}}\sum_{\eta^0} \pi^N(\eta^0)\ (2^r) (1)[N-(2k+1)]^{r+v}\frac{1}{N}\sum_z[2p_{m_z(\eta^0)}-1]+O(N^{-1})\\
&=\frac{1}{N}\sum_{z=1}^{N}\sum_{\eta^0} \pi^N(\eta^0)[p_{m_z(\eta^0)}-q_{m_z(\eta^0)}]+O(N^{-1})\\
&=\sum_{k=0}^{1} \sum_{l=0}^{1}(\pi^N)_{-1,1}(k, l)[2p_{2k+l}-1]+O(N^{-1})\\
&=\sum_{k=0}^{1} \sum_{l=0}^{1}(\pi^{r/(r+s)})_{-1,1}(k, l)[2p_{2k+l}-1]+o(1).
\end{align*}
So we have, with $\gamma:=r/(r+s)$,
$$
\mu^{N}_{[r,s]'}\rightarrow (1-\gamma)\sum_{k=0}^{1} \sum_{l=0}^{1}(\pi^\gamma)_{-1,1}(k, l)[2p_{2k+l}-1]=\mu_{(\gamma,1-\gamma)'},
$$
as required.
\end{proof}
\newpage
\begin{newreferences}
\item Choi, S. C. (2020) Spatial Parrondo games with spatially dependent game $A$, \url{http://arxiv.org/abs/2101.01172}.
\item Ethier, S. N. and Kurtz, T. G. (1986) {\em Markov Processes: Characterization and Convergence}, John Wiley \& Sons, New York.
\item Ethier, S. N. and Lee, J. (2009) Limit theorems for Parrondo's paradox, \textit{Electron. J. Probab.} \textbf{14} 1827--1862.
\item Ethier, S. N. and Lee, J. (2012a) Parrondo games with spatial dependence, \textit{Fluct. Noise Lett.} \textbf{11} 1250004.
\item Ethier, S. N. and Lee, J. (2012b) Parrondo games with spatial dependence, II, \textit{Fluct. Noise Lett.} \textbf{11} 1250030.
\item Ethier, S. N. and Lee, J. (2013a) Parrondo games with spatial dependence and a related spin system, \textit{Markov Process. Relat. Fields} \textbf{19} 163--194.
\item Ethier, S. N. and Lee, J. (2013b) Parrondo games with spatial dependence and a related spin system, II, \textit{Markov Process. Relat. Fields} \textbf{19} 667--692.
\item Ethier, S. N. and Lee, J. (2015) Parrondo games with spatial dependence, III, \textit{Fluct. Noise Lett.} \textbf{14} 1550039.
\item Liggett, T. M. (1985) {\em Interacting Particle Systems}, Springer-Verlag, New York.
\item Mihailovi\'c, Z. and Rajkovi\'c, M. (2003) One dimensional asynchronous cooperative Parrondo's games, \textit{Fluct. Noise Lett.} \textbf{3} L389--L398.
\item Toral, R. (2001) Cooperative Parrondo games, \textit{Fluct. Noise Lett.} \textbf{1} L7--L12.
\item Xie, N.-G., Chen, Y., Ye, Y., Xu, G., Wang, L.-G., and Wang, C. (2011) Theoretical analysis and numerical simulation of Parrondo's paradox game in space, \textit{Chaos Solitons Fractals} \textbf{44} 401--414.
\end{newreferences}
\end{document} | {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,222 |
Adam and Christine meet. Become friends. And maybe, if the time was right, they could become something more... But a terrible and dark secret slowly unfolds that will not allow a fairytale ending to this beautiful relationship. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,226 |
Q: Is it possible to edit and recompile an iOS Binary? I have an application and posted to Cydia recently. It has been cracked by someone else and posted it in torrent sites. I have a binary checksum verification mechanism inside and they were able to create a new checksum file based on the changes they have made to the binary. They have edited two functions and decompiled it and posted it to torrents.
I saw that it's possible to see the actual implementation of functions and classes. But in order to edit the functions they have to find the address of that function and edit it via HEX EDITOR. I don't want to make it "unhackable", but I really want to find out how they hack.
How can I edit a function in an iOS binary and re-compile it? For example I have a following method in one of my classes.
- (id) getSomething {
return @"Something";
}
I want to edit the return value of this function. Is that possible?
A: Usually, you don't "re-compile" it. Just feed the file to IDA, look for strings, function calls or whatever you are looking for and then use a hex editor or similar to edit the file on assembly level. In most cases it's enough to simply change a conditional jump into an unconditional jump or a nop (no operation). If you want to change return values, you have to put a little more effort into it, but in my experience you either edit the char sequence right inside the binary file, if it's specified as a constant or initial value - or you just write a completely new function and "copy" the assembler code of it into the original file. You just have to make sure your new function does not take more space than the original - or everything's getting a lot more complex.
I hope that's what you were asking for, otherwise just tell us which app you are talking about and we can look deeper into it :)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,613 |
Q: How to receive data from user Django Rest Framework I want to receive data from users and pass the processed version. For example user passed JSON:
{"first number": 3,
"second number": 4},
and the web API should response addition of numbers like {"result": 7}. How to do it without writing to the database?
My serializer.py looks like:
class AdditionSerializer(serializers.Serializer):
first = serializers.CharField()
second = serializers.CharField()
views.py:
class AdditionView(APIView):
@action(detail=False)
def get_addition(self, request):
try:
serializer = AdditionSerializer(data=request.data)
if serializer.is_valid():
first = serializer.validated_data(['first'])
second = serializer.validated_data(['second'])
response = {'result': first+second}
return Response(response, status=status.HTTP_200_OK)
else:
return Response({"message":"error"})
except:
None
but it's doesn't work
A: In your case, a custom view endpoint / API will help to get a response.
Since you are using APIView, you can override the post method with addition logic and send back JSON response.
A: @action decorator is not for APIView you must use action with viewsets remove that and change method to get
class AdditionView(APIView):
def get(self, request, *args, **kwargs):
try:
serializer = AdditionSerializer(data=request.data)
if serializer.is_valid():
first = serializer.validated_data(['first'])
second = serializer.validated_data(['second'])
response = {'result': first+second}
return Response(response, status=status.HTTP_200_OK)
else:
return Response({"message":"error"})
except:
None
Another thing I want mention is change serializer to use IntergerField rather than CharField
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,081 |
<?php
namespace TreeHouse\Feeder\Transport\Matcher;
use TreeHouse\Feeder\Exception\UnexpectedTypeException;
class CallbackMatcher implements MatcherInterface
{
/**
* @var callable
*/
protected $callback;
/**
* Constructor.
*
* @param callable $callback
*
* @throws UnexpectedTypeException
*/
public function __construct($callback)
{
if (!is_callable($callback)) {
throw new UnexpectedTypeException($callback, 'callback');
}
$this->callback = $callback;
}
public function match(array $files)
{
return call_user_func($this->callback, $files);
}
public function __toString()
{
return 'callback';
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,459 |
Q: How can I get an address of a non-extern function (normal function) from another program using C? Recently I've been trying to get the address of a function in aboot (android bootloader) . It is called mmc_read. It is not an extern function but it's a function from a header.How can I get the adress of mmc_read from aboot (without modifying aboot) from aboot memory (stack)?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,856 |
Q: How to remove first 4 bytes from a byte array in Java? I have a byte array named byteArr[].I need to remove first 4 bytes from it.My code is shown below. Here I use the byte array to store the input string.I got some unwanted byte with the output i.e the first four bytes is not needed from the fifth onwards is correct. My program is to take id from respected rfid tag using an rfid machine.
public class Serverc {
final protected static char[] hexArray = "0123456789ABCDEF".toCharArray();
public static String bytesToHex(byte[] bytes) {
char[] hexChars = new char[bytes.length * 2];
for (int j = 0; j < bytes.length; j++) {
int v = bytes[j] & 0xFF;
hexChars[j * 2] = hexArray[v >>> 4];
hexChars[j * 2 + 1] = hexArray[v & 0x0F];
}
return new String(hexChars);
}
public static void connection() throws IOException {
ServerSocket ss = new ServerSocket(9888);//exce
ss.setSoTimeout(300000000);//exce
System.out.println("Waiting for client on port " + ss.getLocalPort() + "...");
while (true) {
Socket server = ss.accept();//exce
System.out.println("Just connected to " + server.getRemoteSocketAddress());
int available = 0;
DataInputStream in = new DataInputStream(server.getInputStream());//exce
int input = 0;
//BufferedReader br = new BufferedReader(in);
byte byteArr[] = new byte[28];
try {
//read till the end of stream
//while((input = in.available()) != -1)
while ((input = in.read(byteArr)) != -1) {
System.out.println("Size read is " + input);
System.out.println("Data is " + bytesToHex(byteArr));
}
//System.out.println("inside finally");
server.close();//exce
//System.out.println("outside finally");
} catch (SocketTimeoutException ex) {
System.out.println("Socket timed out!");
}
try {
Thread.sleep(1000);
} catch (InterruptedException e) {
e.printStackTrace();
}
}
}
public static void main(String args[]) throws IOException {
Serverc obj = new Serverc();
obj.connection();
}
}
Here is my console
Waiting for client on port 9888...
Just connected to /106.208.71.50:61532
Size read is 28
Data is 55000016910001DB00FB63ABEEAFC1EC888F10263410050711148F3500000000
Size read is 28
Data is 55000016910001DB00FB63ABEEAFC1EC888F10263410050711148F3500000000
Size read is 28
Data is 55000016910001DB00FB63ABEEAFC1EC888F10263410050711148F3500000000
Size read is 28
Data is 55000016910001DB00FB63ABEEAFC1EC888F10263410050711148F3500000000
Size read is 28
Data is 55000016910001DB00FB63ABEEAFC1EC888F10263410050711148F3500000000
Here I need to remove 55000016 from the output.
Advance thanks
A: You could use Arrays.copyOfRange method to filter the unwanted bytes and save the result to a new byte array.
byte[] filteredByteArray = Arrays.copyOfRange(byteArr, 4, byteArr.length);
A: If you want to skip the first four bytes change this,
for (int j = 0; j < bytes.length; j++) {
to something like
for (int j = 4; j < bytes.length; j++) {
Or you might use String.substring(int)
bytesToHex(byteArr).substring(8); // <-- skip the first 4 bytes
A: It is best for you to skip the 4 bytes. But if you want to remove them, you should use this solution (your array may become massive). This algorithm is best you can get in Java for your problem. O(1) space , O(n) time.
void removeFourBytes(byte[] a) {
for (int i = 4; i < a.length; i++) {
a[i-4]=a[i];
}
for (int i = a.length - 1; i > a.length - 5; i--) {
a[i] = 0;
}
}
Second best option would be System.arrayCopy() - O(n) space, O(n) time.
A: You have a larger problem than skipping the 4 bytes at the start of each of your packets. Your code does this:
byte byteArr[] = new byte[28];
try {
while ((input = in.read(byteArr)) != -1) {
System.out.println("Size read is " + input);
System.out.println("Data is " + bytesToHex(byteArr));
}
(commented-out lines removed)
Note that you cannot guarantee that all 28 bytes will be read on each call to in.read(), but that your bytesToHex() method is assuming that 28 valid bytes are present in the array. You need to collect data that you read into the array until you have all of your packet before processing it, probably using a DataInputStream with the method readFully(byte[]). Then you can skip the first 4 bytes as suggested in the other answers.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,006 |
Father, son write about rare bird
NORWICH, Vt. (AP) – The lead author of an article about finches nesting on a Peruvian glacier in a scholarly bird journal is affiliated with the Geosciences Department at the University of Massachusetts, Amherst.
The other's affiliated with the Marion W. Cross School.
The citation in the quarterly Wilson Journal of Ornithology omits the word "elementary" from the name of the Norwich school.
Nor does it mention that it was the curiosity of 12-year-old Spencer Hardy that led he and his father, Douglas, to recognize the significance of the white-winged diuca finch nesting on the ice of a glacier.
But to the ornithological world, the age of Spencer is irrelevant to the phenomenon he helped document on a high-altitude ice field, in a mountain range he's never visited, thousands of miles from his Vermont home: It's the first well documented case in the world of a species other than penguins successfully nesting on ice.
"It gives you some idea of the adaptations that birds have undergone to utilize all spaces, all the niches that are out there, from living in Death Valley to living at 18,000 feet in the Andes in Peru," said Wilson Journal Editor Clait Braun, of Tucson, Ariz., who published the Hardys' paper in September.
It wasn't until Braun was finalizing the paper with the Hardys and he asked for the academic pedigree of Spencer that he found out the boy's age. "I was pretty well stunned," said Braun.
"We don't get very many papers from people like that," he said. He says most authors are graduate students or even older birders.
The journal itself was founded more than a century ago by young bird enthusiasts, Braun said. But by young, Braun meant men in their late teens and early 20s.
Now, Spencer – who's been interested in birds since he stared at flocks from his high chair – is 14, a ninth grader at Hanover (N.H.) High School, where he studies traditional high school science and other subjects. The birding is done outside school.
He seems unfazed by the attention his birding skills are bringing him.
"It's a neat experience. I don't have anything to compare it against," Spencer said. "Hopefully, it will be one of many to come."
His fascination with birds evolved into real-life study. At 6 or 7, he was working with established birders on local bird counts and breeding surveys.
Since he couldn't drive, his parents had to take him where he needed to go, getting his parents involved, too.
His Dad, a University of Massachusetts research scientist, started visiting the Quelccaya Ice Cap in Peru in 2003 as part of his work to help get a long-term climate history through dust trapped over the centuries in the ice at about 18,000 feet above sea level.
It was Spencer's fascination with birds that led the elder Hardy to take pictures of every bird he saw in Peru.
After Douglas Hardy came home, Spencer would pore over the photos and use bird books to identify the species.
It was during that process that Douglas Hardy started noticing the nests on the ice. So the Hardys shared their findings with Norwich neighbor George Clark, a retired biologist from the University of Connecticut who has helped groom the young ornithologist.
Then they had to figure out which species made the nests.
"We got it down to two species, mainly from the size (of the nests) and what was around and abundant in the area," Spencer said.
A feather expert at the Smithsonian Institution made the link to the finches.
"The discovery of this is really kind of a new frontier," said Clark. "Physically, we think much of the globe has been covered… This is an area that people just haven't visited."
Two years ago, Douglas Hardy queried the Wilson Journal about a paper on the ice birds. He didn't make an effort to list Spencer as a co-author until after he spoke with professional birders, who persuaded him that given Spencer's level of input, he merited a co-author tag.
"My first step was to go to the editor and say, 'I want to be totally honest with you, he's my son, I can't see objectively about this,"' Douglas Hardy said.
But Braun said the science – and Spencer's contribution to it – was sound.
"It's just not the casual person who can pick this stuff up and turn it into scientific prose," Braun said. "Spencer deserves a lot of credit to get his Dad to get more information and then helping his Dad get it turned into a scientific paper."
When the paper was finalized there was some discussion about which of Spencer's schools should be listed. They chose the school he attended when the journal first accepted the paper, the Norwich elementary school.
Spencer's birding is done outside his school, but he says his future, without a doubt, includes the formal study of ornithology.
Meanwhile, he'd like to visit Peru with his father, to see the nests he's become an expert on.
To do that, he's going to have to raise his own money to pay the expenses, but that's part of the world of science that he's becoming a part of.
Braun said the Hardys' paper is a building block for additional research.
"There's more that can be learned. Before you can go further you have to have the basic building blocks," Braun said. "It presents the basic blocks for other scientists to work on."
AP-ES-11-17-08 0006EST | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,320 |
Back to Awen
ABOUT AWEN
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Share your memories of the Muni Arts Centre in Pontypridd
In partnership with Awen Cultural Trust, Rhondda Cynon Taf County Borough Council is inviting residents to share their stories, memories and pictures of Pontypridd's Muni Arts Centre from years gone by – in the first stage of public engagement for the planned £4.5m investment in the building.
During late 2019, the Council announced Awen Cultural Trust as their preferred tenant and operator of the Muni – and jointly announced an ambitious long-term plan to refurbish the Muni in partnership with architects Purcell. The aim is to secure the building's heritage and celebrate the stunning gothic architecture for which it is recognised and listed – while securing a sustainable future that meets the needs of the local community and establishes the Muni as a unique regional arts and music venue once again.
This autumn, an important public consultation process is taking place with the community and key stakeholders, to bring residents up-to-date with progress and to gain important feedback to help shape the plans. It will consist of two main stages – a call for memories (starting October 2020) followed by public and stakeholder consultation exercises (starting on November 23, 2020).
This stage of the work is being funded by the Architectural Heritage Fund, along with additional funding from Rhondda Cynon Taf Council.
Residents are now welcome to share their stories and memories of the Muni's past. A dedicated webpage has been set up on the Council's website containing full details, which can be accessed here.
We want to hear as many memories as possible from the community – from when the Muni was used as a Wesleyan Chapel to the activities that took place there, the concerts enjoyed in the building and the weddings it hosted.
We also want to see as many pictures as possible, which may give further insight into the Muni's historic features and significance, as well as the social value of the Muni to the community. All submissions made in the consultation will help form part of a wider heritage significance assessment of the Muni.
Councillor Rhys Lewis, Rhondda Cynon Taf Council's Cabinet Member for Stronger Communities, Well-being and Cultural Services, said: "The Muni Arts Centre is certainly a much-loved venue by the people of Pontypridd and beyond, and there was lots of excitement when the Council, Awen and Purcell's ambitious plans were first shared at the end of last year. Cabinet recommended that the Council should engage with the wider community at the appropriate time, which is what is being brought forward this autumn.
"The first part of this consultation is inviting local residents to share their recollections of the Muni and its vast history, and it will be fascinating to hear the memories and see the photos that are shared. It will give the Council, Awen and Purcell further information about the Muni's heritage, which is important in the wider aim of revitalising the building as a fit-for-purpose regional arts venue which best meets the needs of local people.
"I'd urge anyone who has been a part of the Muni's past, as well as those interested in its future, to take part in the wider consultation process."
Richard Hughes, Chief Executive of Awen Cultural Trust, added: "We look forward with working with local groups and individuals to celebrate and share the Muni's history, heritage and cultural identity, and to find out what makes this much-loved venue so unique and special. There will be many people with their own stories to tell about visiting, working or performing at the Muni, whether it is a recent connection or a tale passed down over generations. We are excited to hear these memories, personal experiences and anecdotes, so that we can weave them into the future plans for the venue."
Following the call for memories, the process will move towards its second stage on November 23 – a public and stakeholder consultation. This will include virtual workshops, a presentation and a questionnaire for local people to get involved. Further details about this will be communicated in due course.
Awen Cultural Trust
Stable Offices, Bryngarw House,
Brynmenyn, CF32 8UU
enquiries@awen-wales.com
© Awen Cultural Trust, All Rights Reserved | Privacy Information
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MU AND THE VANISHING WORLD
Directed by Jessica W. Leung, Paco Beltrán
Burmese, English, Karenni, Kayan
West Coast premiere
A bonus Q&A is included as part of this film with Jessica W. Leung & Paco Beltrán, filmmakers of MU AND THE VANISHING WORLD and Brian Myers, filmmaker of short film FAR FROM KAWTHOOLEI, recorded on 10/31/2020. You can also find this recording on Pacific Arts Movement's Facebook page.
This film is a special presentation available to international audiences.
Ten years in the making, MU AND THE VANISHING WORLD is the story of one remarkable woman's journey from living as a tourist spectacle in Thailand to leaving everything behind to become an independent single mother in Bowling Green, Kentucky. At the outset of the film, Mu, a young Kayan refugee mother who fled Myanmar as a child, lives with other women who wear traditional brass neck coils in a fabricated "Long Neck Women" village set up by the Thai government. Although she is one of 150,000 people living in nine refugee camps along the Thai-Myanmar border in 2008, Mu pushes back against the traditions of her elders and the dissatisfaction of being a novelty to tourists. When an opening in a UN resettlement plan offers a chance to start a new life in the US, her curiosity and determination to forge her own path prompts her to cut the cord from her tribe and leave everything and everyone behind.
Once Mu arrives in Kentucky, the film follows her struggles adapting to a new country and the new community she must build to survive. Despite the real challenges of refugee resettlement, MU AND THE VANISHING WORLD is ultimately a portrait of true resilience: to give her son a better future, to find out where her true identity lies, and ultimately, to reclaim authorship over her own destiny.
–Carmela Prudencio
Co-Presented by Ethnic Minorities of Burma Advocacy and Resource Center, Gazunmyo, Burin Journey: the journey to sustainability, Mimi & Friends
Far From Kawthoolei
Directed by Brian Myers
In City Heights, San Diego, a tightknit refugee community center instills the Karen language, culture and legacy of the people with the youth. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
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{"url":"http:\/\/mathhelpforum.com\/advanced-statistics\/116811-solved-statistics-differance-mean-problem.html","text":"# Math Help - [SOLVED] Statistics - Differance in mean problem\n\n1. ## [SOLVED] Statistics - Differance in mean problem\n\nHi, please give me a help to find the answer for the following statistic problem,\n\nAs a production manager suppose you want to check whether the 3 filling\nmachines have different mean filling times. If you have assign 15 similarly trained &\nexperienced workers, 5 per machine, at the 5% level of significance, is there a\ndifference in mean filling times?\nMachine 1 Machine 2 Machine 3\n25.40 23.40 20.00\n26.31 21.80 22.20\n24.10 23.50 19.75\n23.74 22.75 20.60\n25.10 21.60 20.40\n\nSee the attached picture for better understanding of the problem\n\n2. Originally Posted by dhammikai\nHi, please give me a help to find the answer for the following statistic problem,\n\nAs a production manager suppose you want to check whether the 3 filling\nmachines have different mean filling times. If you have assign 15 similarly trained &\nexperienced workers, 5 per machine, at the 5% level of significance, is there a\ndifference in mean filling times?\nMachine 1 Machine 2 Machine 3\n25.40 23.40 20.00\n26.31 21.80 22.20\n24.10 23.50 19.75\n23.74 22.75 20.60\n25.10 21.60 20.40\n\nSee the attached picture for better understanding of the problem\nYou can compare two machines at a time by setting $H_0:\\mu_1=\\mu_2$ for no difference between two machines, and $h_1:\\mu_1 \\neq \\mu_2$ for there is significant difference between the two, using two tailed-test. In all tests, if $-z_{0.05} < z > z_{0.05}$, reject $H_0$; otherwise accept or withhold your decision.\n\n3. You should run an F test via ANOVA on this to test all three means at the same time.\nI just found this\nANOVA Test\nI'm not sure if it's any good, but the idea is correct.\n\n4. Mean1 24.93 Variance1 1.0648\nMean2 22.41 Variance2 1.483\nMean3 20.59 Variance3 0.9205\nVariation Between 23.7486667 Variation Within 1.1561\n\nF-Statistic 20.5420523 P-Value 0.00063\nConclusion Very strong evidence against the null hypothesis\n\n5. ## Thanks\n\nHi, I gut a answer now thanks matheagle, novice for excellent explanations!\n\n6. I would not do a Bonferroni test, where you test pairs of means.\nhttp:\/\/www.itl.nist.gov\/div898\/handb...on4\/prc473.htm\nYou should test...\n\n$H_0: \\mu_i=\\mu_j$ for all i,j\n\nvs.\n\n$H_a: \\mu_i\\ne\\mu_j$ for some $i\\ne j$","date":"2014-09-01 20:41:06","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\": 7, \"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\": 0.3862469494342804, \"perplexity\": 2947.8400741327423}, \"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-2014-35\/segments\/1409535919886.18\/warc\/CC-MAIN-20140909051415-00257-ip-10-180-136-8.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/ibpsonline.in\/questions\/IBPS-PO\/English-Language\/Test-44\/770","text":"# IBPS PO :: English Language :: Test 44 IBPS Recruitment Latest Govt Jobs\n\n## Home IBPS PO \/ English Language Test 44 Questions and Answers\n\n1 . Read the passage carefully and answer the questions given below it. Certain words\/expressions are given in bold in the passage to help you to locate them while answering some of the questions.\n\nRadically changing monsoon patterns, reduction in the winter rice harvest and a quantum increase in respiratory diseases-all part of the environmental doomsday scenario, which is reportedly playing out in South Asia. According to a United Nations Environment Programme report, a deadly 3 km deep blanket of pollution comprising a fearsome cocktail of ash, acids, aerosols and other particles has enveloped this region. For India, already struggling to cope with a drought , the i mplications of this are devastating and further crop failure will amount to a life and death question for many Indians. The increase in premature deaths will have adverse social and economic consequences and a rise in $morbidities$ will place an unbearable burden on our crumbling health system and, there is no one to blame but ourselves. Both , official and corporate India has always been $allergic$ to any mention off clean technology. Mostmechanical two wheelers roll of the assembly line without proper pollution control system. Little effort is made for R and D on simple technologies, which could make a vital difference to people's lives and the environment.\n\nHowever, while there is no denying that South Asia must clean up its act, skeptics might question the timing of the haze report. The Kyoto meet on climate change is just two weeks away and the stage is set for the usual battle between the developing world and the West, particularly the US President Mr. Obama has adamantly refused to sign any protocol, which would mean a change in American consumption level. UN environment report will likely find a place in the US arsenal as it plants an accusing finger towards controls like India and China. Yet the US can hardly deny its own $dubious$ role in the matter of erasi ng tradi ng quotas. Ri chercountries can simply buy up excess credits from poorer countries and continue to pollute. Ratherthan try to get the better of developing countries, who undoubtedly have taken up environmental shortcuts in their bid to catch up with the West, the US should take a look at the environmental $profligacy$,which is going on within. From opening up virgin territories for oil exploration to relaxing the standards for drinking water.\nIf the rate of premature deaths increase it will\nexert added burden on our crumbling economy\nhave adverse social and economic consequences\nmake positive effect on our efforts to control population\nhave less job aspirants in the society\nNone of the above\n2 . Read the passage carefully and answer the questions given below it. Certain words\/expressions are given in bold in the passage to help you to locate them while answering some of the questions.\n\nRadically changing monsoon patterns, reduction in the winter rice harvest and a quantum increase in respiratory diseases-all part of the environmental doomsday scenario, which is reportedly playing out in South Asia. According to a United Nations Environment Programme report, a deadly 3 km deep blanket of pollution comprising a fearsome cocktail of ash, acids, aerosols and other particles has enveloped this region. For India, already struggling to cope with a drought , the i mplications of this are devastating and further crop failure will amount to a life and death question for many Indians. The increase in premature deaths will have adverse social and economic consequences and a rise in $morbidities$ will place an unbearable burden on our crumbling health system and, there is no one to blame but ourselves. Both , official and corporate India has always been $allergic$ to any mention off clean technology. Mostmechanical two wheelers roll of the assembly line without proper pollution control system. Little effort is made for R and D on simple technologies, which could make a vital difference to people's lives and the environment.\n\nHowever, while there is no denying that South Asia must clean up its act, skeptics might question the timing of the haze report. The Kyoto meet on climate change is just two weeks away and the stage is set for the usual battle between the developing world and the West, particularly the US President Mr. Obama has adamantly refused to sign any protocol, which would mean a change in American consumption level. UN environment report will likely find a place in the US arsenal as it plants an accusing finger towards controls like India and China. Yet the US can hardly deny its own $dubious$ role in the matter of erasi ng tradi ng quotas. Ri chercountries can simply buy up excess credits from poorer countries and continue to pollute. Ratherthan try to get the better of developing countries, who undoubtedly have taken up environmental shortcuts in their bid to catch up with the West, the US should take a look at the environmental $profligacy$,which is going on within. From opening up virgin territories for oil exploration to relaxing the standards for drinking water.\nAccording to the passage, India connot tolerate any further\ncrop failure\ndeterioration of care health system\nincrease in respiratory diseases\nproliferation of nuclear devices\nsocial and economic consequences\n3 . Read the passage carefully and answer the questions given below it. Certain words\/expressions are given in bold in the passage to help you to locate them while answering some of the questions.\n\nRadically changing monsoon patterns, reduction in the winter rice harvest and a quantum increase in respiratory diseases-all part of the environmental doomsday scenario, which is reportedly playing out in South Asia. According to a United Nations Environment Programme report, a deadly 3 km deep blanket of pollution comprising a fearsome cocktail of ash, acids, aerosols and other particles has enveloped this region. For India, already struggling to cope with a drought , the i mplications of this are devastating and further crop failure will amount to a life and death question for many Indians. The increase in premature deaths will have adverse social and economic consequences and a rise in $morbidities$ will place an unbearable burden on our crumbling health system and, there is no one to blame but ourselves. Both , official and corporate India has always been $allergic$ to any mention off clean technology. Mostmechanical two wheelers roll of the assembly line without proper pollution control system. Little effort is made for R and D on simple technologies, which could make a vital difference to people's lives and the environment.\n\nHowever, while there is no denying that South Asia must clean up its act, skeptics might question the timing of the haze report. The Kyoto meet on climate change is just two weeks away and the stage is set for the usual battle between the developing world and the West, particularly the US President Mr. Obama has adamantly refused to sign any protocol, which would mean a change in American consumption level. UN environment report will likely find a place in the US arsenal as it plants an accusing finger towards controls like India and China. Yet the US can hardly deny its own $dubious$ role in the matter of erasi ng tradi ng quotas. Ri chercountries can simply buy up excess credits from poorer countries and continue to pollute. Ratherthan try to get the better of developing countries, who undoubtedly have taken up environmental shortcuts in their bid to catch up with the West, the US should take a look at the environmental $profligacy$,which is going on within. From opening up virgin territories for oil exploration to relaxing the standards for drinking water.\nAccording to the passage, two wheeler industry is not adequately concerned about\nlife cover insurance of the vehicle owners\npollution control system in the vehicles\nrising cost of the two wheelers\nrising cost of the petrol in the country\n4 . Read the passage carefully and answer the questions given below it. Certain words\/expressions are given in bold in the passage to help you to locate them while answering some of the questions.\n\nRadically changing monsoon patterns, reduction in the winter rice harvest and a quantum increase in respiratory diseases-all part of the environmental doomsday scenario, which is reportedly playing out in South Asia. According to a United Nations Environment Programme report, a deadly 3 km deep blanket of pollution comprising a fearsome cocktail of ash, acids, aerosols and other particles has enveloped this region. For India, already struggling to cope with a drought , the i mplications of this are devastating and further crop failure will amount to a life and death question for many Indians. The increase in premature deaths will have adverse social and economic consequences and a rise in $morbidities$ will place an unbearable burden on our crumbling health system and, there is no one to blame but ourselves. Both , official and corporate India has always been $allergic$ to any mention off clean technology. Mostmechanical two wheelers roll of the assembly line without proper pollution control system. Little effort is made for R and D on simple technologies, which could make a vital difference to people's lives and the environment.\n\nHowever, while there is no denying that South Asia must clean up its act, skeptics might question the timing of the haze report. The Kyoto meet on climate change is just two weeks away and the stage is set for the usual battle between the developing world and the West, particularly the US President Mr. Obama has adamantly refused to sign any protocol, which would mean a change in American consumption level. UN environment report will likely find a place in the US arsenal as it plants an accusing finger towards controls like India and China. Yet the US can hardly deny its own $dubious$ role in the matter of erasi ng tradi ng quotas. Ri chercountries can simply buy up excess credits from poorer countries and continue to pollute. Ratherthan try to get the better of developing countries, who undoubtedly have taken up environmental shortcuts in their bid to catch up with the West, the US should take a look at the environmental $profligacy$,which is going on within. From opening up virgin territories for oil exploration to relaxing the standards for drinking water.\nWhat could be the reason behind of the haze report just before the Kyoto meet, as indicated in the passage?\nUnited Nations is working hand-inglove with US\nOrganizers of the forthcoming meet to teach a lesson to the US\nDrawing attentions of the world towards devastating effects of environment degradation\nUS wants to use it as a handle against developing countries of the forthcoming meet\nThe meet is a part of political agenda of the UN\n5 . Read the passage carefully and answer the questions given below it. Certain words\/expressions are given in bold in the passage to help you to locate them while answering some of the questions.\n\nRadically changing monsoon patterns, reduction in the winter rice harvest and a quantum increase in respiratory diseases-all part of the environmental doomsday scenario, which is reportedly playing out in South Asia. According to a United Nations Environment Programme report, a deadly 3 km deep blanket of pollution comprising a fearsome cocktail of ash, acids, aerosols and other particles has enveloped this region. For India, already struggling to cope with a drought , the i mplications of this are devastating and further crop failure will amount to a life and death question for many Indians. The increase in premature deaths will have adverse social and economic consequences and a rise in $morbidities$ will place an unbearable burden on our crumbling health system and, there is no one to blame but ourselves. Both , official and corporate India has always been $allergic$ to any mention off clean technology. Mostmechanical two wheelers roll of the assembly line without proper pollution control system. Little effort is made for R and D on simple technologies, which could make a vital difference to people's lives and the environment.\n\nHowever, while there is no denying that South Asia must clean up its act, skeptics might question the timing of the haze report. The Kyoto meet on climate change is just two weeks away and the stage is set for the usual battle between the developing world and the West, particularly the US President Mr. Obama has adamantly refused to sign any protocol, which would mean a change in American consumption level. UN environment report will likely find a place in the US arsenal as it plants an accusing finger towards controls like India and China. Yet the US can hardly deny its own $dubious$ role in the matter of erasi ng trading quotas. Ri chercountries can simply buy up excess credits from poorer countries and continue to pollute. Ratherthan try to get the better of developing countries, who undoubtedly have taken up environmental shortcuts in their bid to catch up with the West, the US should take a look at the environmental $profligacy$,which is going on within. From opening up virgin territories for oil exploration to relaxing the standards for drinking water.\n\nChoose the word which is most similar in meaning to the word printed in bold as used in the passage.\n$Profligacy$\nWastefulness\nConservation\nUpliftment\nCriticality\nDenouncement\n6 . Read the passage carefully and answer the questions given below it. Certain words\/expressions are given in bold in the passage to help you to locate them while answering some of the questions.\n\nRadically changing monsoon patterns, reduction in the winter rice harvest and a quantum increase in respiratory diseases-all part of the environmental doomsday scenario, which is reportedly playing out in South Asia. According to a United Nations Environment Programme report, a deadly 3 km deep blanket of pollution comprising a fearsome cocktail of ash, acids, aerosols and other particles has enveloped this region. For India, already struggling to cope with a drought , the i mplications of this are devastating and further crop failure will amount to a life and death question for many Indians. The increase in premature deaths will have adverse social and economic consequences and a rise in $morbidities$ will place an unbearable burden on our crumbling health system and, there is no one to blame but ourselves. Both , official and corporate India has always been $allergic$ to any mention off clean technology. Mostmechanical two wheelers roll of the assembly line without proper pollution control system. Little effort is made for R and D on simple technologies, which could make a vital difference to people's lives and the environment.\n\nHowever, while there is no denying that South Asia must clean up its act, skeptics might question the timing of the haze report. The Kyoto meet on climate change is just two weeks away and the stage is set for the usual battle between the developing world and the West, particularly the US President Mr. Obama has adamantly refused to sign any protocol, which would mean a change in American consumption level. UN environment report will likely find a place in the US arsenal as it plants an accusing finger towards controls like India and China. Yet the US can hardly deny its own $dubious$ role in the matter of erasi ng trading quotas. Ri chercountries can simply buy up excess credits from poorer countries and continue to pollute. Ratherthan try to get the better of developing countries, who undoubtedly have taken up environmental shortcuts in their bid to catch up with the West, the US should take a look at the environmental $profligacy$,which is going on within. From opening up virgin territories for oil exploration to relaxing the standards for drinking water.\n\nChoose the word which is most similar in meaning to the word printed in bold as used in the passage.\n$Allergic$\nLiking\nPassionate\nPossessive\nCrumbling\nRepugnant\n7 . Read the passage carefully and answer the questions given below it. Certain words\/expressions are given in bold in the passage to help you to locate them while answering some of the questions.\n\nRadically changing monsoon patterns, reduction in the winter rice harvest and a quantum increase in respiratory diseases-all part of the environmental doomsday scenario, which is reportedly playing out in South Asia. According to a United Nations Environment Programme report, a deadly 3 km deep blanket of pollution comprising a fearsome cocktail of ash, acids, aerosols and other particles has enveloped this region. For India, already struggling to cope with a drought , the i mplications of this are devastating and further crop failure will amount to a life and death question for many Indians. The increase in premature deaths will have adverse social and economic consequences and a rise in $morbidities$ will place an unbearable burden on our crumbling health system and, there is no one to blame but ourselves. Both , official and corporate India has always been $allergic$ to any mention off clean technology. Mostmechanical two wheelers roll of the assembly line without proper pollution control system. Little effort is made for R and D on simple technologies, which could make a vital difference to people's lives and the environment.\n\nHowever, while there is no denying that South Asia must clean up its act, skeptics might question the timing of the haze report. The Kyoto meet on climate change is just two weeks away and the stage is set for the usual battle between the developing world and the West, particularly the US President Mr. Obama has adamantly refused to sign any protocol, which would mean a change in American consumption level. UN environment report will likely find a place in the US arsenal as it plants an accusing finger towards controls like India and China. Yet the US can hardly deny its own $dubious$ role in the matter of erasi ng trading quotas. Ri chercountries can simply buy up excess credits from poorer countries and continue to pollute. Ratherthan try to get the better of developing countries, who undoubtedly have taken up environmental shortcuts in their bid to catch up with the West, the US should take a look at the environmental $profligacy$,which is going on within. From opening up virgin territories for oil exploration to relaxing the standards for drinking water.\n\nChoose the word which is most Opposite in meaning to the word printed in bold as used in the passage.\n$Morbidity$\nPowerfulness\nHealthiness\nSoftness\nAcuteness\nPurposeful\n8 . Read the passage carefully and answer the questions given below it. Certain words\/expressions are given in bold in the passage to help you to locate them while answering some of the questions.\n\nRadically changing monsoon patterns, reduction in the winter rice harvest and a quantum increase in respiratory diseases-all part of the environmental doomsday scenario, which is reportedly playing out in South Asia. According to a United Nations Environment Programme report, a deadly 3 km deep blanket of pollution comprising a fearsome cocktail of ash, acids, aerosols and other particles has enveloped this region. For India, already struggling to cope with a drought , the i mplications of this are devastating and further crop failure will amount to a life and death question for many Indians. The increase in premature deaths will have adverse social and economic consequences and a rise in $morbidities$ will place an unbearable burden on our crumbling health system and, there is no one to blame but ourselves. Both , official and corporate India has always been $allergic$ to any mention off clean technology. Mostmechanical two wheelers roll of the assembly line without proper pollution control system. Little effort is made for R and D on simple technologies, which could make a vital difference to people's lives and the environment.\n\nHowever, while there is no denying that South Asia must clean up its act, skeptics might question the timing of the haze report. The Kyoto meet on climate change is just two weeks away and the stage is set for the usual battle between the developing world and the West, particularly the US President Mr. Obama has adamantly refused to sign any protocol, which would mean a change in American consumption level. UN environment report will likely find a place in the US arsenal as it plants an accusing finger towards controls like India and China. Yet the US can hardly deny its own $dubious$ role in the matter of erasi ng trading quotas. Ri chercountries can simply buy up excess credits from poorer countries and continue to pollute. Ratherthan try to get the better of developing countries, who undoubtedly have taken up environmental shortcuts in their bid to catch up with the West, the US should take a look at the environmental $profligacy$,which is going on within. From opening up virgin territories for oil exploration to relaxing the standards for drinking water.\n\nChoose the word which is most Opposite in meaning to the word printed in bold as used in the passage.\n$Dubious$\nUnquestionable\nDissimilar\nIllegal\nAntisocial\nInnovative\n9 . Rearrange the following six sentences (A), (B), (C), (D), (E), (F) and (G) in the proper sequence to form a meaningful paragraph and then answer the questions given below.\n\n(A) The history of that system is, however, a warning than a stimulus to reorganize a similar scheme.\n\n(B) However, we can't resign ourselves merely because there are no data.\n\n(C) One of the commonest risk which agricultural life is exposed to in this country is famine or failure of crops.\n\n(D) There are, however, no reliable data on which such a scheme of insurance can be based.\n\n(E) A kind of Famine Insurance System was attempted by the British Government of India in the last century.\n\n(F) Still, the need for such a scheme to cover the losses due to famine, cattle plague, crop pests, etc. can't be undermined.\n\n(G) It is obviously because of failure of rain and the consequence is starvation.\nWhich of the following should be the FOURTH sentence after rearrangement?\nA\nB\nC\nD\nE\n10 . Rearrange the following six sentences (A), (B), (C), (D), (E), (F) and (G) in the proper sequence to form a meaningful paragraph and then answer the questions given below.\n\n(A) The history of that system is, however, a warning than a stimulus to reorganize a similar scheme.\n\n(B) However, we can't resign ourselves merely because there are no data.\n\n(C) One of the commonest risk which agricultural life is exposed to in this country is famine or failure of crops.\n\n(D) There are, however, no reliable data on which such a scheme of insurance can be based.\n\n(E) A kind of Famine Insurance System was attempted by the British Government of India in the last century.\n\n(F) Still, the need for such a scheme to cover the losses due to famine, cattle plague, crop pests, etc. can't be undermined.\n\n(G) It is obviously because of failure of rain and the consequence is starvation.\nWhich of the following should be the SIXTH sentence after rearrangement?\nE\nD\nB\nC\nA","date":"2019-06-26 02:18:28","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.17678044736385345, \"perplexity\": 3211.1136825554904}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-26\/segments\/1560628000044.37\/warc\/CC-MAIN-20190626013357-20190626035357-00078.warc.gz\"}"} | null | null |
Q: Class not registered when I try to run setup I created a setup by Advanced Installer software for my program that I've written by c#. I installed my program on VMWare Windows7.
When I try to run it this message is displayed:
A: Here is a topic about COM registration: "Class not registered (Exception from HRESULT: 0x80040154 (REGDB_E_CLASSNOTREG))"
If it is a simple COM/Class registration, you should add them to COM page in your Advanced Installer project.
If you don't know what to add them to COM page you can also use this method to register them.
A: I used an API function in my program that checks internet connection,I think this function does not work on VmWare
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5/2011 vol. 7
Safety and impact on cardiovascular events of long-term multifactorial treatment in patients with metabolic syndrome and abnormal liver function tests: a post hoc analysis of the randomised ATTEMPT study
Vassilios G. Athyros
Olga Giouleme
Emmanouel S. Ganotakis
Moses Elisaf
Konstantinos Tziomalos
Themistoklis Vassiliadis
Evangelos N. Liberopoulos
Eleni Theocharidou
Asterios Karagiannis
Dimitri P. Mikhailidis
Arch Med Sci 2011;7(5):796–805
Non alcoholic-fatty-liver-disease
multifactorial treatment
Introduction : Non-alcoholic fatty liver disease (NAFLD), a hepatic manifestation of metabolic syndrome (MetS), is common and accounts for 80% of cases of elevated liver function tests (LFTs). We assessed the long-term effects of multifactorial intervention on LFTs and their association with cardiovascular disease (CVD) events in patients with MetS without diabetes mellitus or CVD.
Material and methods : This prospective, randomized, open label study included 1,123 patients (aged 45-65 years). Patients received intensive lifestyle intervention and pharmacotherapy: atorvastatin in all patients (low density lipoprotein cholesterol [LDL-C] targets of < 100 mg/dl [group A] or < 130 mg/dl [group B]), inhibitors of the renin-angiotensin-aldosterone axis for hypertension, metformin for dysglycaemia and orlistat for obesity.
Results : Among participants, 326 had modestly elevated LFTs and ultrasonographic (US) evidence of NAFLD (165 patients in group A2 and 161 patients in group B2). The NAFLD resolved during the 42-month treatment period in 86% of patients in group A2 and in 74% of patients in group B2 (p < 0.001). In both groups nearly 90% of patients attained lipid goals. Mean LDL-C and TG levels were higher in group B2 than in group A2 (p < 0.001). There were no CVD events in group A2 whereas 5 non-fatal events occurred in group B2 (log-rank- p = 0.024). There were no major side-effects.
Conclusions : Attaining multiple treatment targets is safe and beneficial in primary prevention patients with MetS and NAFLD. Lipid levels and LFTs normalized, US findings associated with NAFLD resolved and no CVD events occurred in patients with LDL-C levels < 100 mg/dl (group A2). Resolution of NAFLD might have contributed to the prevention of CVD events.
Implications of the 2021 ESC cardiovascular risk classification among 283,000 European immigrants living in a low-risk region: a population-based analysis in Catalonia
Telomere-telomerase system status in patients with acute myocardial infarction with ST-segment elevation (STEMI) -relationship with oxidative stress | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,321 |
{"url":"https:\/\/socratic.org\/questions\/how-do-you-simplify-m-30-m-12","text":"# How do you simplify m^30*m^12?\n\nApr 11, 2018\n\n${m}^{42}$\n${m}^{30} \\times {m}^{12}$ = ${m}^{30 + 12}$ = ${m}^{42}$\n${a}^{b} \\times {a}^{c}$ = ${a}^{b + c}$","date":"2019-10-24 00:34:46","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 6, \"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\": 0.422084242105484, \"perplexity\": 984.7207924417958}, \"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\/1570987836368.96\/warc\/CC-MAIN-20191023225038-20191024012538-00333.warc.gz\"}"} | null | null |
{"url":"https:\/\/indico.fysik.su.se\/event\/5892\/?print=1","text":"Molecular Physics seminar\n\n# Time-resolved photoelectron spectroscopy of solar cell materials using high harmonic generation\n\n## by Ute Cappel (Uppsala University)\n\nEurope\/Stockholm\nFB55 ()\n\n### FB55\n\nDescription\nSolar cells have a great potential in replacing fossil fuels in electricity generation, if requirements of low production costs can be met. In the last years, much research has been focused on developing new solar cells made from organic or hybrid materials, which can be fabricated by cheap methods. The success of this development crucially depends on understanding the energetics of the interfaces of different materials in a solar cell as well as the transfer kinetics of excited electrons across these interfaces. X-ray based techniques such as photoelectron spectroscopy are powerful for obtaining such information. In this talk, I will present a new approach for studying the time-dependence of the electronic structure by direct pump-probe measurements of the excited electrons in materials for solar energy conversion. Specifically, I will show results from extreme ultraviolet (XUV) based time-resolved photoelectron spectroscopy in the HELIOS laboratory at Uppsala University [1,2]. In these experiments, the fundamental of a femtosecond Ti:sapphire laser was used to generate XUV pulses in a high harmonic generation process. These pulses with an energy of 39 eV were used to emit photoelectrons from a sample, which were then measured with an angular resolved time-of-flight spectrometer [3]. Another part of the laser was used as the pump pulse to electronically excite the sample. By varying the relative arrival times of the pump and the probe at the sample, the electronic structure could be measured as a function of pump-probe delay time. In this presentation, I will show the successful application of this technique to the polymer PCPDTBT [2]. A clear signal - directly detecting the energy of the excited electrons - could be observed in the photoelectron spectra of this sample, when the pump arrived at the sample before the probe. We were then able to follow how the electronic structure redistributes and relaxes after excitation to either the 1st excited state or to a higher excited state as a function of electron energy and time. References [1] Plogmaker, S.; Terschl\u00fcsen, J. A.; Krebs, N.; Svanqvist, M.; Forsberg, J.; Cappel, U. B.; Rubensson, J.-E.; Siegbahn, H.; S\u00f6derstr\u00f6m, J. Rev. Sci. Instrum. 2015, 86 (12), 123107. [2] Cappel, U. B.; Plogmaker, S.; Terschl\u00fcsen, J. A.; Leitner, T.; Johansson, E. M. J.; Edvinsson, T.; Sandell, A.; Karis, O.; Siegbahn, H.; Svensson, S.; M\u00e5rtensson, N.; Rensmo, H.; S\u00f6derstr\u00f6m, J. Phys. Chem. Chem. Phys. 2016, 18, 21921-21929. [3] Ovsyannikov, R.; Karlsson, P.; Lundqvist, M.; Lupulescu, C.; Eberhardt, W.; F\u00f6hlisch, A.; Svensson, S.; M\u00e5rtensson, N. J. Electron Spectros. Relat. Phenomena 2013, 191, 92\u2013103.","date":"2020-04-07 11:57:06","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\": 0.8059519529342651, \"perplexity\": 3265.763513097458}, \"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-2020-16\/segments\/1585371700247.99\/warc\/CC-MAIN-20200407085717-20200407120217-00283.warc.gz\"}"} | null | null |
{"url":"http:\/\/cocoabeachsportspub.com\/pakistan-currency-pgze\/3-night-all-inclusive-vacation-packages-dd4f6f","text":"Warning: include(\/var\/chroot\/home\/content\/11\/10135011\/html\/wp-content_copy\/plugins\/google-map-direction\/option.php) [function.include]: failed to open stream: No such file or directory in \/home\/content\/11\/10135011\/html\/wp-config.php on line 19\n\nWarning: include() [function.include]: Failed opening '\/var\/chroot\/home\/content\/11\/10135011\/html\/wp-content_copy\/plugins\/google-map-direction\/option.php' for inclusion (include_path='.:\/usr\/local\/php5_3\/lib\/php') in \/home\/content\/11\/10135011\/html\/wp-config.php on line 19\n\nWarning: include(\/var\/chroot\/home\/content\/11\/10135011\/html\/wp-content\/plugins\/gallery-plugin\/stats.php) [function.include]: failed to open stream: No such file or directory in \/home\/content\/11\/10135011\/html\/wp-config.php on line 23\n\nWarning: include() [function.include]: Failed opening '\/var\/chroot\/home\/content\/11\/10135011\/html\/wp-content\/plugins\/gallery-plugin\/stats.php' for inclusion (include_path='.:\/usr\/local\/php5_3\/lib\/php') in \/home\/content\/11\/10135011\/html\/wp-config.php on line 23\n\nWarning: include(\/var\/chroot\/home\/content\/11\/10135011\/html\/wp-content\/plugins\/gallery-plugin\/stats.php) [function.include]: failed to open stream: No such file or directory in \/home\/content\/11\/10135011\/html\/wp-settings.php on line 10\n\nWarning: include() [function.include]: Failed opening '\/var\/chroot\/home\/content\/11\/10135011\/html\/wp-content\/plugins\/gallery-plugin\/stats.php' for inclusion (include_path='.:\/usr\/local\/php5_3\/lib\/php') in \/home\/content\/11\/10135011\/html\/wp-settings.php on line 10\n3 night all inclusive vacation packages Black Springbok Rugby Players, Yamata No Orochi Death, Examples Of Sea Stacks, Mark Wright Sister, Yamata No Orochi Death, The Loud House Racing Hearts Dailymotion, English Sleeping Sickness, Antrum In Telugu, Exmouth Australia Weather, English Editing Service, What Type Of Shelter Did Pharaohs Live In, Where Does Cecily Tynan Shop, \" \/>\n\n## 3 night all inclusive vacation packages\n\nTwo different formulas will be needed. We plot the graphs for the different formulas on a common set of axes, making sure each formula is applied on its proper domain. We can imagine graphing each function and then limiting the graph to the indicated domain. Advanced College Writing provides a program in critical reading and interpretive writing. It has been studied by mathematicians and philosophers for over two centuries, both for its own interest and for the insights it offered into mathematics and related fields. gee definition: 1. an expression of surprise or enthusiasm: 2. informal for grand informal (= \\$1,000 or \u00a31,000) 3\u2026. The Electoral College website now has an easy-to-remember address. Studying in Germany requires the graduate degree or an advance technical college entrance qualification. But this is bigger than a yes or no question. A cell phone company uses the function below to determine the cost, $C$, in dollars for $g$ gigabytes of data transfer. Each module covers the topic \u201cfrom scratch\u201d and provides students with the relevant analytical skills. College Algebra. Assessment is by a three-hour written examination. No. The Electoral College is a process, not a place. \u200bThe poems by Greiffenberg and Gryphius can be found here. It is one of the best place for finding expanded names. Alternative spelling of g\u0113 Module 5: Function Basics. 45 Group \u2013 A Answer any one question 1. universities and colleges based on grades. Each module covers the topic \u201cfrom scratch\u201d and provides students with the relevant analytical skills. and Colleges work. A preparatory reading list and module descriptions can be found\u00a0here. Explain the roles of theatre practitioners. pretty useful word in college, i guess. This will mean anyone\u2026 Aspects of Blended Learning North Orissa University Page 4 SEMESTER-I EDUCATION (HONS) ... - Meaning, Nature and purpose of Education Passing the GED Tests can mean a better job, further training or a pathway to higher education. In 2020-21, those modules are: History, Modern Literature II, Thought, and Medieval & Early Modern Literature (in this order). Paper: GE1.1Chg Module: II Subject: Statistics F.M. Goose63095. Find S.D. (GE1 PG1) 4. Each of the component functions is from our library of toolkit functions, so we know their shapes. For $n$-values under 10, $C=5n$. Definition from Wiktionary, the free dictionary. 12. Find out what is the full meaning of GE on Abbreviations.com! General Education. You will explore how you see yourself through the lenses of personal 8 STUDY GUIDE FOR MODULE NO. Ge definition: Georgia | Meaning, pronunciation, translations and examples Supervisions will be arranged by Directors of Studies and, wherever possible, will be timetabled to follow the relevant lectures. Arrangements for the 2020-21 exams have yet to be confirmed. Now that we have sketched each piece individually, we combine them in the same coordinate plane. 12. Top reasons to study in Melbourne ; World-class research capability ; Case studies and student stories ; Beyond Melbourne . The purpose of this module is to help you deepen your understanding and appreciation for who you are as a person. There are questions on all six modules and candidates can answer any three questions. Yes, you can go to college with a GED. It is the distance from 0 on the number line. C\\left(g\\right)=\\begin{cases}\\begin{align}{25} \\hspace{2mm}&\\text{ if }\\hspace{2mm}{ 0 }<{ g }<{ 2 }\\\\ { 25+10 }\\left(g - 2\\right) \\hspace{2mm}&\\text{ if }\\hspace{2mm}{ g}\\ge{ 2 }\\end{align}\\end{cases}. Do not graph two functions over one interval because it would violate the criteria of a function. N.B. Say your parents never went to college, you're one of three children, your older sister is in her second year at school, and you're just now filling out college applications: You're a first-generation college student, even though your sister went to college before you. Students will develop analytical writing and research skills by critically reading literature from three genres, writing analytical and interpretative essays, and conducting and documenting academic research, culminating in at least one research essay. GALAkti\u010dni ve\u010der GEA College V sredo, 16.12., smo ob ... Preberi ve\u010d > 16. Sign in. S = Supplemental Wages.The supplemental flat tax rate may be used if the payments is NOT paid with regular wages. State different measures of skewness and kurtosis. What should I do when my college doesn't allow reptile in dorm? College GEC abbreviation meaning defined here. College Bowl (also known as General Electric (G.E.) How do you think about the answers? The Educational system of Germany is underlying to continuous changes and many reforms. So it is desirable to introduce uniform grading system. Lv 4. Some GE2 courses have a prerequisite. We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain \u201cboundaries.\u201d For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. nursing, education, program. 4 years ago. Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. 17. In most cases, students will be seen by more than one supervisor over the course of their studies for this paper. All International support letters are available by filling in the form below. If you're a school administrator, teacher, or a librarian purchasing for your school, please contact the Educational Materials Advisor assigned to your school or fill up our inquiry form. 5 2. f\\left(x\\right)=\\begin{cases}\\begin{align}{ x}^{3} \\hspace{2mm}&\\text{ if }\\hspace{2mm}{ x }<{-1 }\\\\ { -2 } \\hspace{2mm}&\\text{ if } \\hspace{2mm}{ -1 }<{ x }<{ 4 }\\\\ \\sqrt{x} \\hspace{2mm}&\\text{ if }\\hspace{2mm}{ x }>{ 4 }\\end{align}\\end{cases}. (GE1, GE4 PG1, PG2) 5. The grading system is considered to be better than the conventional marks system and hence it has been followed in the top institutions in India and abroad. 26.70 and Rs. Term . If the payment is made with Regular pay, the payment is taxed based on the employee\u2019s W-4, otherwise the payment is taxed at the supplemental flat tax \u2026 What Does Ge Mean. If Mean and Median of a certain set of observations be Rs. As mentioned above, in order to conduct this test, two types of smears \u2013 Thick (greater volume of blood would mean higher chances of detecting malarial parasites) and Thin (fewer blood cells present in the sample allow the identification of the type of Plasmodium species causing the infection in the patient), are prepared. questions, How the 2020 Dan odprtih vrat \u2013 Inkubator GEA College. Source(s): https:\/\/shrinks.im\/a9fDt. For example, in the toolkit functions, we introduced the absolute value function $f\\left(x\\right)=|x|$. 9 answers. Definition. Dogodki & Seminarji. 12. f\\left(x\\right)=\\begin{cases}\\begin{align}{ x }^{2} \\hspace{2mm}&\\text{ if }\\hspace{2mm}{ x }\\le{ 1 }\\\\ { 3 } \\hspace{2mm}&\\text{ if }\\hspace{2mm} { 1 }<{ x }\\le 2\\\\ { x } \\hspace{2mm}&\\text{ if }\\hspace{2mm}{ x }>{ 2 }\\end{align}\\end{cases}. It is typically meant to mean four- dimensional Euclidean space, generalizing the rules of three-dimensional Euclidean space. Translate millions of words and phrases for free on SpanishDict, the world's largest Spanish-English dictionary and translation website. A piecewise function is a function in which more than one formula is used to define the output. The GED Testing Service claims that over 60% of recent GED recipients are currently enrolled in college, and over 97% of all colleges and employers accept a GED. \u2026 'General Electric' is one option -- get in to view more @ The Web's largest and most authoritative acronyms and abbreviations resource. 0 10-July-2020 Study Guide in GE1: Understanding the Self Module No. Modern Literature I: the 18th and 19th\u00a0Centuries, Annette von Droste-H\u00fclshoff:\u00a0Die Judenbuche, Modern Literature II: the 20th Century and Film, History:\u00a0Imperial Germany 1871-1918 \u2013 Authoritarianism and Modernism, Jacob and Wilhelm Grimm: Deutsches W\u00f6rterbuch, Marx and Engels:\u00a0Das Kommunistiche Manifest. 17 answers. Byline: Jacqueline Reis The state Board of Elementary and Secondary Education will devote a special meeting Monday to special education and a recently issued report that shows low-income, Latino and black students with disabilities are less likely to be included in general education classes than their white and Asian special education peers. Each value corresponds to one equation in a piecewise formula. RESTART SLOVENIA V ponedeljek, 21. decembra 2020 ob 15.00 ur... Preberi ve\u010d > 03. 5 3. Skip to content; Skip to navigation; Main Navigation Menu. Before they come to Cambridge, they should concentrate on the texts set for those modules for which lectures are scheduled in the first term (Michaelmas). 2020 Leadership 4.0. GEA1: Introduction to the German Language 1 - Listening Comprehension and Writing, GEA2: Introduction to the German Language 2 - Translation from and into German, GEA3: Introduction to German 3 - German Culture, GEB2: Translation from German into English and Oral Examination B, GE2: German History and Thought since 1750, GE4: The making of German culture overview, GEB3: Translation into German and a test in German through audio-visual media, DU5: Introduction to the language and literature of the Low Countries, GE6: Modern German Culture (2): 1890 - the Present Day, GE8: German literature, thought, and history from 1700 to 1815 (including Goethe's works to 1832), GE9: German literature, thought and history from 1815 to 1914, GE10: German literature, thought and history, since 1910, GE12: History and Identity in Germany, 1750 to the Present, GE13: Aspects of German-speaking Europe since 1945, GE14: German Literature, Thought and History in the Medieval and Early Modern Periods, GE14: German Literature, Thought and History in the Medieval and Early Modern Periods overview, GE15: Modern German Cultures of Performance, GE15: Modern German Cultures of Performance overview, Year Abroad Activities in German-Speaking Countries, The Germanic Pathway in the MPhil in Linguistics, Faculty of Modern and Medieval Languages and Linguistics, Guidelines for Incoming Erasmus Students in HML, Guidelines for Incoming Erasmus Students in MML, Student complaints and Examination Reviews, How the University College Writing develops critical reading, thinking, and writing skills for the purpose of exploring ideas and issues relevant in a participatory society. 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European kids but its for all the educational system of Germany is underlying to changes...","date":"2021-05-06 12:28:09","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\": 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\": 3, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.34463247656822205, \"perplexity\": 4638.783158970388}, \"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-2021-21\/segments\/1620243988753.97\/warc\/CC-MAIN-20210506114045-20210506144045-00582.warc.gz\"}"} | null | null |
Q: Which UI toolkit are you using with your Ember.js apps? I just started using Ember.js recently and I love the functionality. I'm wondering which UI toolkit you might be using to tie into design side of your applications.
A: Personaly, I am using Twitter's bootstrap library, which is quite low level, but pretty clean.
A: For Bootstrap integration with Ember, take a look at this project I started two days ago:
https://github.com/ember-addons/bootstrap-for-ember
It really fun and easy to use and lightly integrate bootstrap and ember components altogether.
A: Twitter Bootstrap is my preffered choice when it comes to UI especially when prototyping something quickly, recently i have started to use EmberJS and have looked into this as well. So far i have found https://github.com/emberjs-addons/ember-bootstrap
I will update this as my search continues.
Hope this helps with your project!
A: Twitter bootstrap is a great UI frameworks no doubts but I feel it is too mainstream these days. Hence my personal preference is Metro UI CSS, it's sleek and great for developing mobile applications using HTML5
A: I am just starting with emberjs also. Actually I use JQMobile. But I have some issues with it. As I want have a Mobile look and feel, I will try more.
But even if have not use bootstrap with EmberJs I think it will be easier to use as it's only css.
With a UI toolkit that use JavaScript and is owns attributs(exemple : data-role="List"... with JQuery Mobile) you can have rendering issues. I think this is because that Metamorphose/Handlebars and JQuery Mobile both modify the DOM on the fly and it can be tricky to get all work right.
But I am not a EmberJs or JQ Mobile Guru :-)
Sorry for my english, it isn't my mother tongue.
Just one Question .. what is a OSS framework and do you have the links on GitUb
A: This maybe old but I've used this addon on over 5 projects so far with great success. The project is well maintained and flexible. The maintainer is active and takes pull requests efficiently.
http://kaliber5.github.io/ember-bootstrap/
Disclaimer: I am not officiated with this project beyond that of an end consumer.
A: You could have a look at Ember Paper if you like Google Material:
http://miguelcobain.github.io/ember-paper/
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\section{Introduction}
The successful exfoliation of black phosphorus (BP)\cite{li14,xia1,liu14,koenig14} multilayers has triggered tremendous interests in this material, which is also one of the thermodynamically more stable phases of phosphorus, at ambient temperature and pressure.
In comparison to other 2D materials, such as graphene, hexagonal boron nitride (hBN), and transition metal dichalcogenides, multilayers BP has a direct bandgap which spans from $0.3\,{\rm eV} \sim 1.5\,{\rm eV}$ \cite{rudenko14,castellanos14}, hence making it an excellent candidate for infrared optoelectronics \cite{low14cond,low14photo,buscema14}. Since each BP layer forms a puckered surface due to $sp^3$ hybridization, it also reveals highly anisotropic electrical mobility \cite{morita86,li14,liu14}, linear dichroism in optical absorption spectra\cite{morita86,qiao14,low14cond,xia1,YRK15}, anisotropic excitonic structure\cite{tran14,chaves15} and anisotropic plasmons \cite{low14plas, rodin14}. However, BP is not stable in ambient \cite{favron14, island15},
which might render its electrical properties less than pristine.
Recently, encapsulation of BP with hexagonal boron nitride (hBN)\cite{cao15}, all within a controlled inert atmosphere, has allowed for higher carrier mobility in these BP devices\cite{exp1,exp2,exp3,exp4,cao15,bpqhe}. Similar hBN encapsulation has also been applied to other 2D materials such as graphene\cite{wang13} and transition metal dichalcogenides\cite{cui14} to achieve record mobilities. Indeed, the high quality BP has made possible the first observation of prominent quantum magneto-oscillations in these devices\cite{exp1,exp2,exp3,exp4,cao15} and quantum Hall effect\cite{bpqhe}. Hence, theoretical studies of BP in the presence of a magnetic field has also begun receiving attention\cite{zhou14,pereira15}.
In this paper, we examine the electronic properties of BP multilayer thin film in perpendicular magnetic field, such as its Landau level spectrum, ac conductivity, screening and its collective electronic excitations. In particular, we emphasize the manifestation of anisotropy in these experimentally observable quantities. We begin with a discussion of the model Hamiltonian used to describe multilayer BP in Section II. This is followed by the study of its electronic subband structure in Section III, and its Landau level spectrum in Section IV. Various experimentally relevant quantities such as ac conductivity, collective excitations and screening in the presence of magnetic field will be discussed in Section V-VII respectively.
\section{Model Hamiltonian}
In multilayer BP, broken translational symmetry in the out-of-plane $z$ direction renders the direct energy gap at the $\Gamma$ point instead of the Z point in bulk. The low energy Hamiltonian description of BP near $\Gamma$ point can be expressed as ${\cal H}={\cal H}_z + {\cal H}_{xy}$, with its out-of-plane and in-plane dynamics taken separately. Here, ${\cal H}_{xy}$ is given by\cite{rodin14h,zhou14},
\begin{eqnarray}
{\cal H}_{xy} = \left(
\begin{array}{cc}
E_c + \eta_c k_x^2 + \nu_c k_y^2 & \gamma k_x \\
\gamma k_x & E_v - \eta_v k_x^2 - \nu_v k_y^2
\end{array} \right)
\label{hamil}
\end{eqnarray}
where $\eta_{c,v}$ and $\nu_{c,v}$ are the respective band parameters, while $\gamma$ describe the effective coupling between the conduction and valence bands. $E_c$ and $E_v$ denotes the energies of the bulk conduction and valence band edges. We discuss our choice of these band parameters below.
Cyclotron resonance experiments on bulk BP \cite{narita83} found an out-of-plane electron and hole effective masses considerably smaller than that of layered tansition metal dichalcogenides materials \cite{mattheiss73}. In this work, we adopt an average of experimental \cite{narita83} and theoretically \cite{narita83,low14cond} predicted out-of-plane masses i.e. $m_{cz}\approx 0.2\,m_0$ and $m_{vz}\approx 0.4\,m_0$, $m_0$ being the electron mass. The out-of-plane Hamiltonian is given by,
\begin{eqnarray}
{\cal H}_{z} = -\frac{\hbar^2}{2}\left(
\begin{array}{cc}
m_{cz}^{-1}\partial_z^2 & 0 \\
0 & -m_{vz}^{-1}\partial_z^2
\end{array} \right)+eV(z)
\label{hamilz}
\end{eqnarray}
where $V(z)$ describes the out-of-plane electrostatic potential, typically induced by a bottom metal gate\cite{exp1,exp2,exp3,exp4,cao15}. For a finite BP thickness with given $V(z)$, Eq.\,(\ref{hamilz}) can be diagonalized numerically, leading to electronic subband structure. We denote these subband eigen-energies as $\delta E_{c,v}^j$, and their eigen-functions as $\phi_{c,v}^j(z)$, where $j$ is the subband index.
Close to the $\Gamma$ point, the in-plane band parameters, i.e. $\eta_{c,v}$, $\nu_{c,v}$ and $\gamma$, are related to the in-plane effective masses via\cite{rodin14h},
\begin{eqnarray}
m_{(c,v)x}=\frac{\hbar^2}{\pm 2(\gamma^2/E_g + \eta_{(c,v)})} &\mbox{ , }& m_{(c,v)y}=\frac{\hbar^2}{2\nu_{(c,v)}}
\end{eqnarray}
The band parameters $\eta_{c,v}$, $\nu_{c,v}$ and $\gamma$ are chosen such that they yield the known effective masses in the bulk BP limit i.e. $m_{cx}=m_{vx}=0.08\,m_0$, $m_{cy}=0.7\,m_0$ and $m_{vy}=1.0\,m_0$\cite{morita86,narita83}, and $m_{cx}=m_{vx}\approx 0.15\,m_0$ for monolayer BP\cite{rodin14}. $E_g$ is the electronic bandgap of the BP multilayer. The energy gap for bulk BP is $0.3\,$eV\cite{morita86}. While monolayer BP has not been ascertain experimentally, \emph{ab initio} calculation based on the GW method suggests an energy gap of $\sim 1.5-2\,$eV \cite{tran14,rudenko14}. In sum, the band parameters used in this work are; $\eta_c=\eta_v\approx 19.1\,$eV\AA$^2$, $\nu_c\approx 5.45\,$eV\AA$^2$, $\nu_v \approx 3.81\,$eV\AA$^2 $ and $\gamma=2.84\,$eV\AA.
\begin{figure}[t]
\centering
\scalebox{0.6}[0.6]{\includegraphics*[viewport=190 270 580 480]
{electrostatics.pdf}}
\caption{Electronic subband structure of $10\,$nm BP film
for electron (a) and hole (b), showing the lowest four subbands, $\delta E_{c,v}^j$. The red dashed lines corresponds to the Fermi energy $E_f$. These calculations assume $T=300\,$K.}
\label{fig1}
\end{figure}
\section{Subband Structure}\label{Sec:Subband}
Typical experimental device structure\cite{exp1,exp2,exp3,exp4,cao15} consists of multilayer BP on an insulating dielectric film on a back substrate, which serves also as a back gate, as sketched in the inset of Fig.\,\ref{fig1}(a). The electron density in BP along $z$, $n(z)$, can be obtained from\cite{stern67},
\begin{eqnarray}
\nonumber
n(z)
=\frac{k_B T}{\pi\hbar^{2}}\sum_{j}g_{s}m_{cd}\times\\
\mbox{ln}\left[\mbox{exp}\left(\frac{E_{f}-E_{c}-\delta E_{c}^j}{kT}\right)+1\right]\left|\phi_{c}^j(z)\right|^{2}
\label{nden}
\end{eqnarray}
where $g_s$ is the spin degeneracy, $k_B$ is the Boltzmann constant, $E_f$ is the Fermi level, $T$ is the temperature, and $m_{cd}$ refers to the density-of-states mass given by $\sqrt{m_{cx}m_{cy}}$. Solving Eq.\,(\ref{hamilz}) and the Poisson equation self-consistently, one can then arrives at the numerical solution for the BP electrostatics, an approach well-known in the context of semiconductor inversion layer\cite{stern72}.
The electron and hole subbands, $\delta E_{c,v}^j$, and the Fermi energy $E_f$, are plotted as function of carrier density $n$ in Fig.\,\ref{fig1}(a)-(b) respectively. These calculation assumes $T=300\,$K. The results indicate that across a wide range of carrier densities, only the first subband is occupied, and onset of second subband occupation takes place only when $n\geq 7\times 10^{12}\,$cm$^{-2}$ and $n\geq 5\times 10^{12}\,$cm$^{-2}$ for electron and hole respectively. This is consistent with recent experimental observations\cite{exp1,exp2,exp3,exp4,cao15} of a 2D electron gas in the quantum limit. Certainly, the transition to multi-subband occupation depends also on the BP thickness.
\begin{figure}[t]
\centering
\scalebox{0.48}[0.48]{\includegraphics*[viewport=110 90 610 490]
{Landau-level.pdf}}
\caption{Landau level spectrum for the conduction (a) and valence (b) subbands, calculated for a $10\,$nm thick BP film. The underlying subband structure corresponds to the case with a carrier density of $1\times 10^{12}\,$cm$^{-2}$. In (c), we show the probability distribution $\psi(\xi=\sqrt{\alpha}x,\sqrt{\alpha}y)$ for the two lowest Landau levels in conduction band under different Landau gauges at $B=10\,$T, where $\alpha=\omega_c/2\hbar\eta_c$. In (d), the probability distribution of the $i=0,\,5,\,10,\,15$ eigenstates over the bare oscillator eigenstates $\vert n \rangle$ of lowest conduction subband is plotted.}
\label{fig2}
\end{figure}
\section{Landau level Spectrum}
When an uniform magnetic field $B\hat{z}$ is applied perpendicular to the plane, we have $\vec{p}=\hbar\vec{k}\rightarrow \vec{\Pi}=\vec{p}+e\vec{A}$ by Peierls substitution. With the choice of Landau gauge $\vec{A}=Bx\hat{y}$, we can write
\begin{eqnarray}
\Pi_x=p_x, \Pi_y=p_y+exB,
\end{eqnarray}
which obey the commutation relation $[\Pi_x,\Pi_y]=-i\hbar eB$. It is useful to define,
\begin{eqnarray}
a=\frac{1}{\sqrt{\hbar\omega_c}}[\sqrt{\eta_c}\Pi_x-i\sqrt{\nu_c}\Pi_y]
\end{eqnarray}
where $\omega_c=2eB\sqrt{\eta_c\nu_c}$. It is easy to verify that $[a,a^\dagger]=1$, and
\begin{eqnarray}
a^{\dagger}a=\frac{1}{\hbar\omega_c}[\eta_c \Pi_x^2+ \nu_c \Pi_y^2]-\frac{1}{2}.
\end{eqnarray}
Introducing the parameters $s=\eta_v/\eta_c$, $t=\nu_v/\nu_c$ and $r=s/t$, we can express the dispersing term in valence band as:
\begin{eqnarray}
\eta_v\Pi_x^2+\nu_v\Pi_y^2=s(\eta_c\Pi_x^2+\nu_c\Pi_y^2)+(1-r)\nu_v\Pi_y^2
\end{eqnarray}
which allows us to obtain the effective Hamiltonian under magnetic field $B$ for each pair of electron-hole $j$ subbands,
\begin{eqnarray}
\nonumber
{\cal H}^j=\hbar\omega_c\left(%
\begin{array}{cc}
\epsilon_c^j +a^{\dagger}a+\frac{1}{2}&\tilde{\gamma}(a^{\dagger}+a)\\
\tilde{\gamma}(a^{\dagger}+a) & \epsilon_v^j +s(a^{\dagger}a+\frac{1}{2})+\tilde{r}(a^{\dagger}-a)^2\\
\end{array}%
\right)\\
\label{Heff}
\end{eqnarray}
where $\tilde{\gamma}=\frac{\gamma}{2\sqrt{\hbar \omega_c\eta_c}} $, $\tilde{r}=-\frac{1}{4}t(1-r)$ and $\epsilon_{c,v}^j=(E'_{c,v} + \delta E_{c,v}^j)/\hbar\omega_c$. We note that, for the representative numerical results presented here, the band edges $E_{c,v}$ are adjusted so as to reproduce the estimated electronic bandgap of $\approx 0.5\,$eV\cite{tran14} for a $10\,$nm BP film, i.e. $(E'_{c}+\delta E_c^1)-(E'_{v}+\delta E_v^1)=0.5\,$eV.
Now the matrix Hamiltonian is dimensionless, and the eigenvalue problem can be solved numerically. The procedure is as follows. First, we assume the following ansatz for the eigenvector of Eq.\,(\ref{Heff}),
\begin{eqnarray}
\psi(x)=\left(%
\begin{array}{c}
U(x)\\
V(x)\\
\end{array}%
\right)=\left(%
\begin{array}{c}
\sum_{n=0}^{n_{max}} U_n\phi_n(\alpha (x-x_0))\\
\sum_{n=0}^{n_{max}} V_n\phi_n(\alpha (x-x_0))\\
\end{array}%
\right).
\label{eigenfunction}
\end{eqnarray}
where $n_{max}$ sets the truncation of the expansion and $\phi_n(\alpha (x-x_0))$ is the $n$'th eigenstate of harmonic oscillator centered at $x_0=\dfrac{p_y}{eB}$ with $\alpha=\dfrac{\omega_c}{2\hbar\eta_c}$. Note the wave function will include $e^{ip_yy/\hbar}$ factor, associated to the good quantum number $p_y$. We can determine the coefficients $U_n$ and $V_n$ from the energy eigen-problem ${\cal H}^j\psi=E\psi$. Explicitly, we have the following relations
\begin{widetext}
\begin{eqnarray}
\begin{array}{c}
(\Delta_1+n-\tilde{E})U_n+\tilde{\gamma}(\sqrt{n}V_{n-1}+\sqrt{n+1}V_{n+1})=0\\
\tilde{\gamma}(\sqrt{n}U_{n-1}+\sqrt{n+1}U_{n+1})+ (\Delta_2+\tilde{s}n-\tilde{E})V_n+\tilde{r}(\sqrt{n(n-1)}V_{n-2}+\sqrt{(n+2)(n+1)}V_{n+2})=0 \\
\end{array}%
\label{eigen}
\end{eqnarray}
\end{widetext}
where $\Delta_1=\epsilon_{c}^j+\frac{1}{2} $, $\Delta_2=\epsilon_{v}^j+\frac{s}{2}-\tilde{r}$, $\tilde{s}=s-2\tilde{r}$ and
$\tilde{E}=E/\hbar \omega_c$, $E$ being the energy of Landau level corresponding to the eigen-function (\ref{eigenfunction}).
Numerically, we need to introduce a truncation condition, which we set to $U_{n>n_{max}}=V_{n>n_{max}}=0$.
\textcolor{black}{It is not difficult to check that for most physical relevant cases, $\Delta_{1,2}\gg n,\tilde{\gamma}$, which justifies a perturbative consideration for Eq.\,(\ref{eigen}). For the electron (hole) spectrum, we can write,
$\tilde{E}^{e/h}=\tilde{E}_{0}^{e/h}+\delta \tilde{E}^{e/h}$ with $\tilde{E}_{0}^{e}=\Delta_1+n$, $\tilde{E}_{0}^{h}=\Delta_2+\tilde{s}n$, and the second order perturbations terms are:
\begin{eqnarray}
\delta \tilde{E}^{e/h}= \pm\tilde{\gamma}^2(c_0+c_1n+c_2n^2)
\label{E2perturbation}
\end{eqnarray}
where $c_0=\dfrac{1}{(\Delta_1-\Delta_2)}$, $c_1=\dfrac{2}{(\Delta_1-\Delta_2)}-\dfrac{1-\tilde{s}}{(\Delta_1-\Delta_2)^2}$ and $c_2=\dfrac{-2(1-\tilde{s})}{(\Delta_1-\Delta_2)^2}$.
From Eq.(\ref{E2perturbation}), we may understand the deviation of resonant frequency of the ac conductivity from conventional 2D electron gas case, to be discussed in Section V.}
\textcolor{black}{From numerical recipe described below Eq.(\ref{eigen})}, we obtain the Landau level spectrum, as shown in Fig.\,\ref{fig2}(a)-(b) for conduction and valence bands respectively. The dispersion is typical of 2D electron gas system, exhibiting linear dependence with $B$. However, there are differences due to the finite inter-band coupling $\gamma$ and anisotropy of BP, which will be discussed later. In the calculations, we set the doping to be $1\times 10^{12}\,$cm$^{-2}$. We note that crossing of each Landau level acquires additional carrier density of $\Delta n = eBg/h\sim \tfrac{1}{2}B\times 10^{11}\,$cm$^{-2}$, where $g=2$ is the spin degeneracy. Hence, at the assumed doping of $1\times 10^{12}\,$cm$^{-2}$ and $B=10\,$T, the filling factor is $2$. For carrier densities larger than $1\times 10^{13}\,$cm$^{-2}$, the filling factor can be as large as $20$. It might then be possible to observe multi-subband phenomena, especially for the hole case. For the experimental doping range, only the Landau levels of the lowest two subbands are physically relevant.
The anisotropy of the problem is encoded in the wavefunctions. In Fig.\,\ref{fig2}(c) we plot the wavefunction probability along the two in-plane spatial coordinates for the first two Landau levels, expressed in their dimensionless coordinate (i.e., $\xi=\sqrt{\alpha}x$, $\xi=\sqrt{\alpha}y$). Two respective gauges are used, i.e., for gauges $\vec{A}=Bx\hat{y}$ and $\vec{A}=-By\hat{x}$, we get $\psi(x)$ and $\psi(y)$ respectively.
Due to the anisotropy inherent in the model Hamiltonian, we can clearly discern the difference between probability distribution over these two gauges. We will show that this anisotropy in wave function can result in prominent anisotropy in various experimental quantities such as ac conductivities and magneto-plasmons. In Fig.\,\ref{fig2}(d), the probability distribution over the eigenstates $\vert n\rangle$ of the \textit{bare} harmonic oscillator which forms the upper diagonal terms in Eq.\,(\ref{Heff}) is plotted. With the increase of Landau level, the probability distribution over $\vert n\rangle$ becomes more broadened. This can be understood from perturbation point of view, i.e., the matrix elements quantifying the perturbation upon the \textit{bare} eigenstates increases with factors $\sqrt{n}$'s.
\textcolor{black}{Before concluding this section, it is interesting to compare Landau levels in BP with the other electron gas system e.g. conventional 2D electron gas (i.e. Schr\"odinger fermions) and graphene (i.e. Dirac fermions), summarized as follows:
\begin{eqnarray}
E_n=\begin{cases} \hbar\omega_c(n+\tfrac{1}{2}): \mbox{Schr\"odinger fermions}
\\ \mbox{sgn}(n)v_f\sqrt{2e\hbar B|n|}: \mbox{Dirac fermions}
\\ \hbar\omega_c(n+\tfrac{1}{2}+\tilde{\gamma}^2(c_0+c_1n+c_2n^2)): \mbox{BP} \end{cases}
\label{Landaulevelcompare}
\end{eqnarray}}
\textcolor{black}{The BP effective Hamiltonian, described by Eq.(\ref{hamil}), have features not embodied in Schr\"odinger fermion description in the finer energy scale proportional to $\gamma^2$, and have experimental consequences discussed in Section V.
We note also that compared to other gapped Dirac systems, such as gaped graphene or transition metal dichalcogenides, BP's gap is placed at the time-reversal invariant $\Gamma$ point instead of the inequivalent $K$ and $K'$ points of the BZ. Hence, its Landau level spectrum resembles more to that of Schr\"odinger fermions rather than that of massive Dirac fermions.\cite{Goerbig14}}
\section{ac magneto-conductivity}
In this section, we study the ac magneto-conductivity $\sigma_{\alpha \beta}$ of the $10\,$nm BP multilayer film. Having the energy spectrum and wave-functions of Landau levels at hand, as presented in the previous section, we can numerically calculate the $\sigma_{\alpha \beta}$'s directly. According to the Kubo formula\cite{Hbert}, we have:
\begin{eqnarray}
\sigma_{\alpha \beta}(\omega)&=&\frac{-ie^2\hbar }{2\pi l_B^2}\sum_{snjs'n'j'}\frac{f(E_{snj})-f(E_{s'n'j'})}{E_{snj}-E_{s'n'j'}}\nonumber\\
&\times& \frac{\left \langle \Phi_{snj} \left|\hat{v}_{\alpha} \right| \Phi_{s'n'j'} \right \rangle \left \langle \Phi_{s'n'j'} \left|\hat{v}_{\beta} \right| \Phi_{snj} \right \rangle }{\hbar\omega- E_{snj}+E_{s'n'j'}+i\Gamma}
\label{eq4}
\end{eqnarray}
where the velocities are defined by $v_i=\frac{\partial H}{\partial p_i}$. Explicitly, we have:
\begin{eqnarray}
\begin{array}{c}
v_x=\left(%
\begin{array}{cc}
\sqrt{\hbar\omega_c\eta_c}(a+a^{\dagger})&\gamma\\
\gamma& \eta_{v}\sqrt{\hbar\omega_c/\eta_c}(a+a^{\dagger})\\
\end{array}%
\right)\\
v_y=\left(%
\begin{array}{cc}
\sqrt{\hbar\omega_c\nu_c}(a^{\dagger}-a)/i& 0\\
0& \nu_v\sqrt{\hbar\omega_c/\nu_c}(a^{\dagger}-a)/i\\
\end{array}%
\right)\\
\end{array}
\label{velocity}
\end{eqnarray}
\begin{figure}[t]
\centering
\scalebox{0.6}[0.6]{\includegraphics*[viewport=155 110 600 500]
{conductivity.pdf}}
\caption{ac conductivities as function of frequency $\omega$. The anisotropy between $\sigma_{xx}$ and $\sigma_{yy}$, at $B=10\,$T is displayed in (a), for different $n\rightarrow n+1$ transitions. Zooming in, we can observe slight shift of the resonant frequency for $\sigma_{xx}$ and $\sigma_{xy}$, as shown in (b), calculated for $B=20\,$T. The damping constant is set to be $\Gamma=10^{-4}\,$eV and temperature $T=10\,$K. The inset of (b) shows how the resonant frequency shift depends on filling factor $\nu$ and $B$. Conductivity are in units of $\sigma_0=e^2/\hbar$.}
\label{fig3}
\end{figure}
Our results for ac conductivity as a function of frequency are presented in Fig.\,\ref{fig3}. The ac longitudinal magneto-conductivities for filling factor $\nu=$ 1 to 3 (electron doped) is shown in Fig.\,\ref{fig3}(a). The inset depicts the transitions between the nearest Landau levels
for cases with $\nu=1$ to $3$, i.e. the resonance for each case corresponds to a particular transition process $\vert n\rangle$ to $\vert n+1\rangle$, and occurs at the terahertz frequencies.
Prominently, $\sigma_{xx}$ is about $5\sim 10$ times larger than $\sigma_{yy}$.
We note the spatial anisotropy in the wavefunctions of the Landau levels, albeit small, as shown in Fig.\,\ref{fig2}c. The anisotropy in the magneto-optical conductivity tensor arises mainly from the anisotropy in the velocity operators, i.e. $v_x$ and $v_y$, which accounts for the anisotropic optical transitions dipole.
We should point out here that the resonant structure observed here in BP is more like conventional 2D electron gas rather than in graphene\cite{Giant}, which has multiple resonant structures. Contrary to the conventional case, we find here that the resonant frequency is slightly red-shifted with increasing doping (or $\nu$). This is shown in Fig.\,\ref{fig3}(b), where the longitudinal conductivity $\sigma_{xx}$ and Hall conductivity $\sigma_{xy}$ are displayed for $\nu=1-4$, calculated for $B=20\,$T.
This red-shift also increases with magnetic field as depicted in the inset.
We note that the frequency shift increase in \textit{almost uniform} steps each time the filling factor decreases by $1$, \textcolor{black}{
Interestingly, this red-shifting behavior can be understood from the perturbation expression in Eq.\,(\ref{E2perturbation}) rather straightforwardly, from which we arrived at,
\begin{equation}
\tilde{E}^e(n+1)-\tilde{E}^e(n)=1+\tilde{\gamma}^2[c_1+c_2(2n+1)]
\end{equation}
This expression accounts for linear dependence of the resonance frequency on $n$ quantitatively, and is a direct result of the interband coupling $\gamma$. In other words, this red-shift in ac conductivity can be used to determine $\gamma$ in BP. }
As depicted in the inset of Fig.\,\ref{fig3}(b), it can be seen that the linear relation holds when $\nu$ is small. We anticipate experimental progress in magnetic oscillations of BP to shed light on this issue\cite{exp1,exp2,exp3,exp4,cao15}. It is also worthy to mention that the longitudinal and
Hall conductivities calculated here can be directly measured through absorption and Faraday rotation experiments\cite{Giant} via terahertz spectroscopy.
\section{Collective excitations}
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{MP-CB.pdf}
\caption{Excitation spectrum of BP, as obtained by density plots of ${\rm Im}\Pi({\bf q},\omega)$. (a) and (b)
correspond to non-interacting polarization of BP, Eq.
(\ref{Eq:Pi0}), for $q\parallel y$ and $q\parallel x$ respectively. Plots (c) and (d) include electron-electron
interactions in the RPA. The strength of the electron-electron interaction is chosen to be $r_s\approx 3$, and $N_F=3$.}
\label{Fig:MP}
\end{figure}
In this section we study the excitation spectrum of BP in the presence of a quantizing magnetic field applied perpendicular to the sample, including the effect of electron-electron interaction. As we have seen in Sec. \ref{Sec:Subband}, a 2DEG is created in a multi-layer BP, with occupation of only the first subband, unless the doping exceeds $\sim 5\times10^{12}~{\rm cm}^{-2}$. Therefore in this section we concentrate on the particle-hole excitation spectrum of the 2DEG formed by the carriers of a 10nm thick BP multi-layer. In the absence of Coulomb interaction, the particle-hole excitation spectrum, that enclose the region of the $\omega-q$ plane in which it is possible to excite electron-hole pairs, can be calculated from ${\rm Im}\Pi^0({\bf q},\omega)\ne 0$, where $\Pi^0({\bf q},\omega)$ is the non-interacting polarization function. The bare polarizability of BP in the quantum Hall regime can be expressed in terms of the standard result for a 2DEG\cite{KH84}
\begin{equation}\label{Eq:Pi0}
\Pi^0({\bf q},\omega)=\sum_{m=1}^{N_c}{\sum}'\frac{{\cal
F}_{n,m}({\bf q})}{\omega-m\omega_c+i\Gamma}+(\omega^+\rightarrow
-\omega^-)
\end{equation}
where $\sum'=\sum_{n=\max(0,N_F-m)}^{N_F}$ and
$\omega^+\rightarrow -\omega^-$ indicates the replacement
$\omega+i\Gamma\rightarrow -\omega-i\Gamma$, and $N_F$ is the index of the last occupied LL. As we have seen before, the anisotropy of the BP band structure is encoded in the wave-function, whose overlaps leads to different form factors in the $x$- and $y$-direction
\begin{equation}\label{FF2DEG}
{\cal F}_{n,m}({\bf q})=\frac{e^{-\alpha\frac{q^2l_B^2}{2}}}{2\pi
l_B^2}\frac{n!}{(n+m)!}\left (\alpha\frac{q^2l_B^2}{2}\right)^m
\left[L_n^m\left(\alpha\frac{q^2l_B^2}{2}\right )\right]^2.
\end{equation}
where $l_B=\sqrt{\hbar c/eB}$ is the magnetic length, $\alpha=\omega_c/2\hbar\eta_{(c,v)}$ for the case when $q_x$ is a good quantum number, and $\alpha=\omega_c/2\hbar\nu_{(c,v)}$ when the gauge chosen leads to a good $q_y$ quantum number. We note that the finite interband coupling, $\gamma$, can renormalize\cite{zhou14} the band parameters ($\eta_{(c,v)}$and $\nu_{(c,v)}$) but we checked that the effect is small in this case. A density plot of ${\rm Im}\,\Pi^0$ is shown in Fig. \ref{Fig:MP}(a) and (b) for electron doping with
$N_F=3$. In the presence of a quantizing magnetic field, ${\rm
Im}\,\Pi^0({\bf q},\omega)$ is a sum of Lorentzian peaks centered at
$\omega=m\omega_c$, where $m$ is the difference between
the LL indices of the electron $n'$ and the hole $n$, $m\equiv n'-n\ge 1$.\cite{GV05,RGF10} The
excitation spectrum is chopped into horizontal lines, separated by a constant
energy $\omega_c$. The peculiarities of the BP spectrum of Fig. \ref{Fig:MP}(a)-(b), like the presence of a superstructure of $N_F+1$ brighter regions and the nodes of the first horizontal line at $\omega=\omega_c$, are due to the form of the LL wavefunctions and have been studied in detail in Refs. \onlinecite{RFG09,RGF10}. As in the $B=0$ case,\cite{low14plas} anisotropy of BP band structure leads to a wider spectrum in the $q_y$ direction as compared to $q_x$ direction. Whereas we only show results for electron doping in Fig. \ref{Fig:MP}, a similar spectrum is found for the hole doped case, with the difference that the separation between consecutive horizontal lines is narrower for the latter, due to larger effective mass of the valence band as compared to the conduction band.
We next consider the effect of Coulomb interaction within the Random Phase Approximation (RPA), which {\it dresses} the electron-hole polarization function as
\begin{equation}\label{Eq:PiRPA}
\Pi({\bf q},\omega)=\frac{\Pi^0({\bf q},\omega)}{\varepsilon({\bf q},\omega)}
\end{equation}
where the dielectric function is obtained as
\begin{equation}
\varepsilon({\bf q},\omega)=1-v({\bf q})\Pi^0({\bf q},\omega),
\end{equation}
in terms of the non-interacting polarization function $\Pi^0$ and the two-dimensional Coulomb potential
in momentum space
\begin{equation}
v({\bf q})=\frac{2\pi e^2}{\kappa |{\bf q}|}
\end{equation}
where $\kappa$ is the background dielectric constant. The strength of the Coulomb interaction is usually expressed in terms of the dimensionless parameter $r_s=2m_be^2/\kappa k_F$, where the band mass in the present case is $m_b=\sqrt{m_xm_y}$, and $k_F=\sqrt{2N_F+1}/l_B=\sqrt{\pi n_{el}}$ in terms of the carrier density $n_{el}$. Long-range electron-electron interaction leads to the appearance of collective plasmon modes in the spectrum. Their dispersion relation can be calculated from the zeros of the dielectric function. In Fig. \ref{Fig:MP}(c) and (d) we show how the non-interacting electron-hole spectrum [shown in Fig. \ref{Fig:MP}(a) an (b)] is modified due to interactions. The electron-hole horizontal lines of Fig. \ref{Fig:MP}(a) and (b) acquire a dispersion, leading to the so-called magneto-excitons or magneto-plasmons. As in the $B=0$ case,\cite{low14plas} Coulomb interaction leads to highly anisotropic magneto-plasmons, with higher dispersion for $q\parallel x$ than for $q\parallel y$ directions. This is due to the fact that the {\it effective} mass is smaller in the $x$ direction than in the $y$ direction. We also notice that the excitation spectrum of BP greatly differs from that of doped graphene. In fact, the characteristic linear dispersion relation in graphene leads to a {\it relativistic} quantization of the spectrum into a set of non-equidistant LLs.\cite{G11} As a consequence, long range Coulomb interaction in graphene leads to a set of highly dispersing {\it linear} magneto-plasmons,\cite{RFG09} that differs from the equidistant magnetoexcitons (separated by a well defined cyclotron frequency energy, $\omega_c$) in BP.
\section{Static screening}
\begin{figure}[t]
\centering{
\includegraphics[width=0.4\textwidth]{StaticPi0-CB.pdf}
}
\centering{
\includegraphics[width=0.4\textwidth]{StaticEpsilon-CB.pdf}}
\caption{Static screening. Static polarization function $-\Pi^0({\bf q},0)$ (top) and dielectric function $\varepsilon({\bf q})$ (bottom) for the conduction band with ${\bf q} \parallel y$ (solid lines) and for ${\bf q} \parallel x$ (dashed lines). We have used $N_F=3$. }
\label{Fig:StaticScreening}
\end{figure}
In this section we focus on the properties of $\Pi^0({\bf q})=\Pi^0({\bf q},\omega=0)$ and $\varepsilon({\bf q})=\varepsilon({\bf q},\omega=0)$
in the static limit, for which the polarization function is entirely real.
The polarizability of BP in a strong magnetic field is shown in Fig. \ref{Fig:StaticScreening}(a). One first observe that, as in a standard 2DEG,\cite{GV05} the static polarizability
tends to zero as $\Pi^0({\bf q}\rightarrow 0)\propto q^2$ for
$B\neq 0$. The reason for this is that the main contribution to
$\Pi^0({\bf q})$ comes from ${\bf q}=0$ excitations in the vicinity of
the Fermi energy $E_f$. This differs from the $B=0$ case, for which the Fermi level cuts the band and
there are ${\bf q}=0$ excitations. In the integer quantum Hall regime, however,
$E_f$ lies in the cyclotron gap between the highest occupied LL
$N_F$ and the lowest unoccupied one $N_F+1$. Since this energy gap must be
overcome by excitations with ${\bf q}=0$, then its spectral weight tends
to zero. Notice that $\Pi^0(q=0)$ coincides with
the density of states at the Fermi energy because the latter
vanishes for $B> 0$ when $E_f$ lies in the gap. One also notices that the wave-vector at which the polarizability starts to vanish ($2k_F$) is larger for ${\bf q} \parallel y$ that for ${\bf q} \parallel x$, due to the anisotropy of the BP Fermi surface. One further notices the oscillatory behavior of the static polarizability, below $2k_F$, due to the wave-function overlap between the electron and the hole, leading to $N_F+1$ maxima.\cite{GV05,RGF10}
It is interesting to compare the screening properties of BP and graphene, and we start by briefly discussing the $B=0$ case. For $q\rightarrow 0$ we have $\varepsilon(q)\approx 1+q_{TF}/q$, where $q_{TF}\equiv
2\pi e^2 \rho(E_f)/\kappa$ is the 2D Thomas-Fermi
wave-vector, in terms of the density of states $\rho(E_f)$. Since the density of states (per unit area) is approximately a constant equal to $g_sm_b/2\pi$ for BP, where $g_s=2$ accounts for the spin degeneracy, whereas for graphene $\rho(E_f)$
is energy dependent and given approximately by $gE_f/(2\pi v_F^2)$, where $v_F$ is the Fermi velocity and
$g=g_sg_v=4$ accounts for spin and valley degeneracy, one obtains a density independent $q_{TF}$ for
BP, whereas it scales as $k_F$ for graphene. Therefore, in the two cases and for $B=0$, the dielectric function diverges as $\varepsilon \sim
q_{TF}/q\rightarrow \infty$ when $q\to 0$.\cite{low14plas} However, it is important to notice that
the density dependence in the numerator of
$\varepsilon(q\rightarrow 0)$ in graphene implies the
absence of screening in undoped graphene (i.e. for $k_F=0$) at long
distances. For short wavelengths $q\gg 2k_F$,
$\varepsilon(q)\rightarrow 1$ in a BP, whereas for graphene,
$\varepsilon(q)\rightarrow 1+\pi r_s/2$. The extra
contribution $\pi r_s/2$ to the dielectric function of graphene
at $q\gg 2k_F$ is related to virtual
inter-band particle-hole excitations.\cite{RGF10} In summary, at short
wavelengths and for $B=0$, BP (like a standard 2DEG) does not screen at all ($\varepsilon \to 1$),
whereas graphene screens as a dielectric due to its filled valence band.
The above picture changes in the presence of a quantizing magnetic field. In
Fig. \ref{Fig:StaticScreening} we have plotted the static polarization and
dielectric functions for BP, respectively. At
long wavelengths, $\varepsilon(q)-1\propto q$. In the limit of $N_F\gg 1$, the known result for a 2DEG and for graphene,\cite{RGF10} $\varepsilon(q)-1\propto r_sN_F^{3/2}ql_B$ as $q\rightarrow 0$, applies also for BP. The difference between graphene and BP is encoded in the density dependence of $r_s$ in the two cases: since $r_s\sim N_F^{-1/2}$ in BP, $\varepsilon$ grows linearly with $N_F$ in this case. However, $r_s\equiv e^2/\kappa v_F$ is density-independent in graphene, leading to a dielectric function proportional to $N_F^{3/2}$. On the other hand, in both graphene and BP,
$\varepsilon(q,\omega=0)\rightarrow 1$ as $q\rightarrow 0$,
which implies that there is no screening at long distances. Finally, the behavior of the dielectric function in a magnetic field at $q\gg 2k_F$
in BP is similar to the $B=0$ limit, corresponding to the standard metallic like screening governed by intra-band processes.\cite{low14plas}
\section{Conclusion}
In conclusion, we have examined the electronic properties of 2D electron gas in black phosphorus multilayers due to the presence of a perpendicular magnetic field. We highlight in this work the in-plane anisotropy reflected in various experimental quantities such as its ac magneto-conductivity, screening, and collective excitations. We found that resonant structures in the ac conductivity exhibits a red-shift with increasing doping due to interband coupling, suggesting possible electric modulation of light absorption and Faraday rotation. Coulomb interaction also leads to highly anisotropic magneto-plasmons.
\acknowledgments
We thank M. O. Goerbig and J.-N. Fuchs for useful conversations. FG and RR acknowledge support from the Spanish Ministry of Economy (MINECO) through Grant No. FIS2011-23713, the European Research Council Advanced Grant (contract 290846), and the European Commission under the Graphene Flagship, contract CNECT-ICT-604391. R.R. acknowledges financial support from the Juan de la Cierva Program. YJ and TL acknowledge support from University of Minnesota start-up fund. Y.J. acknowledge support from the National Natural Science Foundation of China under Grants No. 11474255 during intial part of the project.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,610 |
\section{Introduction}
In this series of papers we address a central problem in the flourishing area of metric Diophantine approximation on manifolds and measures: an attempt to exhibit a possibly widest natural class of sets and measures for which most points are not very well approximable by ones with rational coordinates.
Fix $d\in\N$. The quality of rational approximations to a vector $\xx\in\R^d$ can be measured by its \emph{exponent of irrationality}, which is defined by the formula
\[
\omega(\xx) = \limsup_{\pp/q\in\Q^d} \frac{-\log\|\xx - \pp/q\|}{\log(q)},
\]
where the limsup is taken over any enumeration of $\Q^d$, and $\|\cdot\|$ is any norm on $\R^d$. Another interesting quantity is the \emph{exponent of multiplicative irrationality}, which is the number
\[
\omega_\times(\xx) = \limsup_{\pp/q\in\Q^d} \frac{-\log\prod_{i = 1}^d |x_i - p_i/q|}{\log(q)}\cdot
\]
It follows from a pigeonhole argument that $\omega(\xx) \geq 1 + 1/d$ and $\omega_\times(\xx) \geq d + 1$. A vector $\xx$ is said to be \emph{very well approximable} if $\omega(\xx) > 1 + 1/d$, and \emph{very well multiplicatively approximable} if $\omega_\times(\xx) > d + 1$. We will denote the set of very well (multiplicatively) approximable vectors by $\mathrm{VW(M)A}_d$. It is well-known that $\VWA_d$ and $\VWMA_d$ are both Lebesgue nullsets of full Hausdorff dimension, and that $\VWA_d \subset \VWMA_d$.
A measure $\mu$ on $\R^d$ is \emph{extremal} if $\mu(\VWA_d) = 0$, and \emph{strongly extremal} if $\mu(\VWMA_d) = 0$. Extremality was first defined by V.~G.~Sprind\v zuk, who conjectured that the Lebesgue measure of any nondegenerate manifold is extremal. This conjecture was proven by D.~Y.~Kleinbock and G.~A.~Margulis \cite{KleinbockMargulis}, and later strengthened by D.~Y.~Kleinbock, E.~Lindenstrauss, and B.~Weiss (hereafter abbreviated ``KLW'') in \cite{KLW}, who considered a class of measures which they called ``friendly'' and showed that these measures are strongly extremal. However, their definition is somewhat rigid and many interesting measures, in particular ones coming from dynamics, do not satisfy their condition. In this paper, we study a much larger class of measures, which we call \emph{weakly quasi-decaying}, such that every weakly quasi-decaying measure is strongly extremal (Corollary \ref{corollaryKLW}). This class includes a subclass of \emph{quasi-decaying} measures, which are the analogue of KLW's ``absolutely friendly'' measures.\Footnote{The terminology ``absolutely friendly'' was not used by KLW and first appeared in \cite{PollingtonVelani}; however, several theorems about absolute friendliness had already appeared in \cite{KLW} without using the terminology.}
In the current paper (Part I), we demonstrate the most basic properties of the quasi-decay condition, including the facts that every exact dimensional measure is quasi-decaying, and that every quasi-decaying measure is extremal, which we prove using an elementary argument. We also prove the result stated above that every weakly quasi-decaying measure is strongly extremal (in particular verifying a conjecture of KLW), as well as considering the approximation properties of quasi-decaying measures on the space of matrices and on affine subspaces of $\R^d$. In particular we generalize results of some recent papers regarding approximation of friendly measures in the matrix framework \cite{KMW, ABRdS} (cf. Theorems \ref{theoremKLW} and \ref{theoremexponents} below).
\tableofcontents
{\bf Notation.} For the reader's convenience we summarize a list of notations and conventions:
\begin{convention}
The symbols $\lesssim$, $\gtrsim$, and $\asymp$ will denote coarse asymptotics; a subscript of $\plus$ indicates that the asymptotic is additive, and a subscript of $\times$ indicates that it is multiplicative. For example, $A\lesssim_\times B$ means that there exists a constant $C > 0$ (the \emph{implied constant}) such that $A\leq C B$.
If $\mu$ and $\nu$ are measures, then $\nu\lesssim_\times\mu$ means that there exists a constant $C > 0$ such that $\nu\leq C\mu$.
\end{convention}
\begin{convention}
In this paper, all measures and sets are assumed to be Borel, and measures are assumed to be locally finite.
\end{convention}
\begin{convention}
The symbol $\triangleleft$ will be used to indicate the end of a nested proof.
\end{convention}
\begin{figure}[h!]
\begin{tabular}{ll}
\hline
$\omega(\xx)$ & The \emph{exponent of irrationality} of $\xx\in\R^d$ defined as\\
& \hspace{0.1 in} $\omega(\xx) \df \limsup \left\{ \frac{-\log\|\xx - \pp/q\|}{\log(q)} : \pp/q\in\Q^d \right\}$\\
$\omega_\times(\xx)$ & The \emph{exponent of multiplicative irrationality} of $\xx\in\R^d$ defined as\\
& \hspace{0.1 in} $\omega_\times(\xx) \df \limsup \left\{ \frac{-\log\prod_{i = 1}^d |x_i - p_i/q|}{\log(q)} : \pp/q\in\Q^d \right\}$\\
$\mathrm{VWA}_d$ & The set of very well approximable vectors in $\R^d$\\
$\mathrm{VWMA}_d$ & The set of very well multiplicatively approximable vectors in $\R^d$\\
$\thickvar S\rho$ & The closed $\rho$-thickening of $S\subset\R^d$~:~ $\thickvar S\rho \df \{x\in\R^d:\dist(x,S) \leq \rho\}$\\
$\thickopenvar S\rho$ & The open $\rho$-thickening of $S\subset\R^d$~:~ $\thickopenvar{S}{\rho} \df \{x\in\R^d:\dist(x,S) < \rho\}$\\
$A\wedge B$ & The minimum of $A$ and $B$\\
$A\vee B$ & The maximum of $A$ and $B$\\
$\scrH$ & The collection of affine hyperplanes in $\R^d$\\
$\Supp(\mu)$ & The topological support of a measure $\mu$\\
$B(\xx,\rho)$ & The closed ball centered at $\xx \in \R^d$ of radius $\rho>0$\\
$\dist(\yy,\LL)$ & $\dist(\yy,\LL) \df \inf \{ d(\yy,\xx) : \xx \in \LL \}$\\
$\|d_\LL\|_{\mu,B}$ & For a hyperplane $\LL \in \HH$ and a ball $B$ centered at $\Supp(\mu)$\\
& \hspace{0.1 in} $\|d_\LL\|_{\mu,B} \df \sup\{\dist(\yy,\LL) : \yy\in B\cap\Supp(\mu)\}$\\
$\MM_{M,N}$ & The set of $M \times N$ matrices with real entries\\
$\omega(\bfA)$ & The \emph{exponent of irrationality} of $\bfA\in\MM$ defined as\\
& \hspace{0.1 in} $\omega(\bfA) \df \limsup \left\{ \frac{-\log\|\bfA\qq - \pp\|}{\log\|\qq\|} : \qq\in\Z^N\butnot\{\0\} , \pp\in\Z^M \right\}$\\
$\omega_\times(\bfA)$ & The \emph{exponent of multiplicative irrationality} of $\bfA\in\MM$ defined as\\
& \hspace{0.1 in} $\omega_\times(\bfA) \df \limsup \left\{ \frac{-\log\prod_{i = 1}^M |(\bfA\qq - \pp)_i|}{\log \prod_{j = 1}^N |q_j|\vee 1} : \qq\in\Z^N\butnot\{\0\} , \pp\in\Z^M \right\}$\\
$\mathrm{VWA}_{{M,N}}$ & The set of very well approximable $M\times N$ matrices in $\MM$\\
$\mathrm{VWMA}_{{M,N}}$ & The set of very well multiplicatively approximable $M\times N$ matrices in $\MM$\\
$\|f\|_B$ & $\|f\|_B \df \sup \{ \|f(\xx)\| : \xx\in B \}$\\
$\|f\|_{\CC^\epsilon,B}$ & $\|f\|_{\CC^\epsilon} \df \sup \left\{ \frac{\|f(\yy) - f(\xx)\|}{\|\yy - \xx\|^\epsilon} : \xx,\yy\in B \right\}$ for $f:B\to\R$ a function of class $\CC^{\ell + \epsilon}$\\
$\Delta,\|f\|,\|f\|_{\CC^\epsilon}$ & $\Delta \df B(\0,1)$, $\|f\| \df \|f\|_\Delta$, $\|f\|_{\CC^\epsilon} \df \|f\|_{\CC^\epsilon,\Delta}$\\
$\Lambda_0$ & $\Lambda_0 \df \Z^{M + N} \subseteq \R^{M + N}$\\
$u_\bfA, g_\tt$ & See Section \ref{subsectioncorrespondence}\\
$\mfa$ & $\mfa \df \{\tt\in\R^{M + N} : \sum t_i = 0\}$\\
$\mfa_+, \mfa_+ ^*$ & See Section \ref{subsectioncorrespondence}\\
$\omega(\bfA;\SS,s)$ & See Section \ref{subsectioncorrespondence}\\
$\Delta(\Lambda)$ & Given a lattice $\Lambda \subseteq \R^{M + N}$, $\Delta(\Lambda) \df -\log\min\big\{\|\rr\| : \rr\in\Lambda\butnot\{\0\}\big\}$\\
\hline
\end{tabular}
\end{figure}
\vskip13pt
{\bf Acknowledgements.} The authors thank Barak Weiss for helpful comments. The first-named author was supported in part by a 2014-2015 Faculty Research Grant from the University of Wisconsin--La Crosse. The second-named author was supported in part by the Simons Foundation grant \#245708. The third-named author was supported in part by the EPSRC Programme Grant EP/J018260/1. The fourth-named author was supported in part by the NSF grant DMS-1361677. The authors thank an anonynmous referee for helpful comments.
\subsection{Four conditions which imply strong extremality}
\label{subsectionfourconditions}
We begin by recalling the definitions of friendly and absolutely friendly measures, in order to compare these definitions with our new definitions of quasi-decaying and weakly quasi-decaying measures. The definitions given below are easily seen to be equivalent to KLW's original definitions in \cite{KLW}.
\begin{definition}
\label{definitionabsolutelyfriendly}
Let $\mu$ be a measure on an open set $U\subset\R^d$, and let $\Supp(\mu)$ denote the topological support of $\mu$.
\begin{itemize}
\item $\mu$ is called \emph{absolutely decaying (resp. decaying)} if there exist $C_1,\alpha > 0$ such that for all $\xx\in\Supp(\mu)$, $0 < \rho \leq 1$, $\beta > 0$, and $\LL\in\scrH$, if $B = B(\xx,\rho) \subset U$ then
\begin{align}
\label{absolutelydecaying}
\mu\big(\thickopenvar\LL{\beta\rho}\cap B\big) &\leq C_1 \beta^\alpha\mu(B) & \text{(absolutely decaying)}
\end{align}
or
\begin{align}
\label{decaying}
\mu\big(\thickopenvar\LL{\beta \|d_\LL\|_{\mu,B}}\cap B\big) &\leq C_1 \beta^\alpha\mu(B) & \text{(decaying)},
\end{align}
respectively, where
\[
\|d_\LL\|_{\mu,B} := \sup\{\dist(\yy,\LL):\yy\in B\cap\Supp(\mu)\}.
\]
\item $\mu$ is called \emph{nonplanar} if $\mu(\LL) = 0$ for all $\LL\in\scrH$. Note that every absolutely decaying measure is nonplanar. Moreover, the decaying and nonplanarity conditions can be combined notationally by using closed thickenings rather than open ones: a measure $\mu$ is decaying and nonplanar if and only if there exist $C_1,\alpha > 0$ such that for all $\xx\in\Supp(\mu)$, $0 < \rho \leq 1$, $\beta > 0$, and $\LL\in\scrH$, if $B = B(\xx,\rho) \subset U$ then
\begin{align}
\label{decayingprime}
\mu\big(\thickvar\LL{\beta \|d_\LL\|_{\mu,B}}\cap B\big) &\leq C_1 \beta^\alpha\mu(B). & \text{(decaying and nonplanar)}
\end{align}
\item $\mu$ is called \emph{Federer} (or \emph{doubling}) if for some (equiv. for all) $K > 1$, there exists $C_2 > 0$ such that for all $\xx\in\Supp(\mu)$ and $0 < \rho\leq 1$, if $B(\xx,K\rho)\subset U$ then
\begin{equation}
\label{federer}
\mu\big(B(\xx,K\rho)\big) \leq C_2\mu\big(B(\xx,\rho)\big).
\end{equation}
\end{itemize}
If $\mu$ is Federer, decaying, and nonplanar, then $\mu$ is called \emph{friendly}; if $\mu$ is both absolutely decaying and Federer, then $\mu$ is called \emph{absolutely friendly}.\Footnote{As KLW put it, the word ``friendly'' is ``a somewhat fuzzy abbreviation of \emph{Federer, nonplanar, and decaying}''.} When the open set $U$ is not explicitly mentioned, we assume that it is all of $\R^d$; otherwise we say that $\mu$ is absolutely decaying, friendly, etc. ``relative to $U$''.
\end{definition}
The main relations between friendly and absolutely friendly measures are as follows:
\begin{itemize}
\item[(i)] every absolutely friendly measure is friendly;
\item[(ii)] the Lebesgue measure of a nondegenerate submanifold of $\R^d$ (see Definition \ref{definitionnondegenerate} for the definition) is friendly but not absolutely friendly;
\item[(iii)] \cite[Theorem 2.1]{KLW} more generally, the image of an absolutely friendly measure under a nondegenerate embedding is friendly.
\end{itemize}
The main result of \cite{KLW} states that every friendly measure is strongly extremal; together with (ii), this provides a proof of Sprind\v zuk's conjecture.
The distinction between friendly and absolutely friendly measures is a fundamental part of the theory; for example, (iii) would be false if we replaced the hypothesis ``absolutely friendly'' by ``friendly''. So any good generalization of friendliness should also respect the ``friendliness-type condition/absolute friendliness-type condition'' distinction. Thus we will define two versions of the quasi-decay condition, one to correspond with friendliness and the other to correspond with absolute friendliness. Since, in our experience, the ``absolute'' versions of these conditions are more fundamental than the ``non-absolute'' versions, we call our condition which corresponds to absolute friendliness the ``quasi-decay'' condition and we call our condition which corresponds to friendliness the ``weak quasi-decay'' condition.
\begin{definition}
\label{definitionQD}
Let $\mu$ be a measure on $\R^d$ and consider $\xx\in E\subset\R^d$. We will say that $\mu$ is \emph{quasi-decaying (resp. weakly quasi-decaying) at $\xx$ relative to $E$} if for all $\gamma > 0$, there exist $C_1,\alpha > 0$ such that for all $0 < \rho \leq 1$, $0 < \beta \leq \rho^\gamma$, and $\LL\in\scrH$, if $B = B(\xx,\rho)$ then
\begin{align}
\label{QDwithE}
\mu\left(\thickvar{\LL}{\beta\rho}\cap B\cap E\right) &\leq C_1 \beta^\alpha \mu(B) & \text{(quasi-decaying)}
\end{align}
or
\begin{align}
\label{weakQDwithE}
\mu\left(\thickvar{\LL}{\beta\|\dist_\LL\|_{\mu,B}}\cap B\cap E\right) &\leq C_1 \beta^\alpha \mu(B) & \text{(weakly quasi-decaying)},
\end{align}
respectively. We will say that $\mu$ is \emph{(weakly) quasi-decaying relative to $E$} if for $\mu$-a.e. $\xx\in E$, $\mu$ is (weakly) quasi-decaying at $\xx$ relative to $E$. Finally, we will say that $\mu$ is \emph{(weakly) quasi-decaying} if there exists a sequence $(E_n)_n$ such that $\mu\left(\R^d\butnot\bigcup_n E_n\right) = 0$ and for each $n$, $\mu$ is (weakly) quasi-decaying relative to $E_n$.
\end{definition}
Let us briefly discuss several aspects of Definition \ref{definitionQD} which differ from Definition \ref{definitionabsolutelyfriendly}
\begin{itemize}
\item The uniform dependence of the constants $C_1$ and $\alpha$ on the point $\xx$ has been dropped. Moreover, the condition is only required to hold for $\mu$-a.e. every $\xx$, rather than for all $\xx$ in the support of $\mu$. This makes the quasi-decay conditions closer to the ``non-uniform'' versions of friendliness considered in \cite[\66]{KLW}. By itself, this relaxation does not seem to give any natural new examples of measures satisfying the condition, until it is combined with the other relaxations considered below.
\item The left-hand sides of \eqref{QDwithE} and \eqref{weakQDwithE} include an intersection with a set $E$ which has large but not full measure with respect to $\mu$. This change is done for two reasons:
\begin{itemize}
\item It makes quasi-decay into a measure class invariant. Note that the relaxation of uniformity is not itself enough to make the condition a measure class invariant (see Theorem \ref{theoremmeasureclass} below).
\item Sequences $(E_n)_n$ with the property described above often show up naturally in our proofs (see e.g. Theorem \ref{theoremexactdim}).
\end{itemize}
\item The inequalities \eqref{QDwithE} and \eqref{weakQDwithE} are only required to hold for $0 < \beta \leq \rho^\gamma$, rather than for all $\beta > 0$. This is probably the most unexpected aspect of our definition. It means that as the balls $B = B(\xx,\rho)$ get smaller, the thicknesses of hyperplane-neighborhoods whose measures can be bounded in terms of $\mu(B)$ get smaller not only in an absolute sense, but also relative to the radius $\rho$.
\item The Federer (doubling) condition has been dropped. The reason for this is that there is an analogue of the Federer condition (Lemma \ref{lemmaquasifederer}) which is good enough for our purposes and which holds for every measure on every doubling metric space, and in particular for every measure on $\R^d$.
\item The nonplanarity condition has been incorporated directly into the definition of weak quasi-decay by using closed thickenings rather than open ones. This difference is mathematically insignificant, but it is a notational convenience.
\end{itemize}
It is obvious that the following implications hold:
\begin{center}
\begin{tabular}{|ccc|}
\hline
Absolutely friendly & \implies & Friendly\\
$\Downarrow$ & & $\Downarrow$\\
Quasi-decaying & \implies & Weakly quasi-decaying\\
\hline
\end{tabular}
\end{center}
(The strictness of these implications is shown by examples in \cite[Figure 1]{DFSU_GE2}.) Moreover, the appropriate analogues of the friendliness/absolute friendliness relations hold:
\begin{itemize}
\item[(i)] every quasi-decaying measure is weakly quasi-decaying;
\item[(ii)] the Lebesgue measure of a nondegenerate submanifold of $\R^d$ is weakly quasi-decaying but not quasi-decaying;
\item[(iii)] more generally, the image of a quasi-decaying measure under a nondegenerate embedding is weakly quasi-decaying; more precisely:
\end{itemize}
\begin{theorem}[Proven in Section \ref{sectionbasic}]
\label{theoremQDembedding}
For all $\ell\in\N$ and $\epsilon > 0$, the image of a quasi-decaying measure under an $\ell$-nondegenerate embedding of class $\CC^{\ell + \epsilon}$ is weakly quasi-decaying.
\end{theorem}
In relation to extremality, we shall prove that every weakly quasi-decaying measure is strongly extremal (Corollary \ref{corollaryKLW}), thus generalizing the main result of \cite{KLW} and in particular providing a third proof of Sprind\v zuk's conjecture. This implication also proves a conjecture of KLW \cite[\610.5]{KLW} that nonplanar and decaying measures are strongly extremal, i.e. that the Federer condition is unnecessary in their main theorem. Although the proof of this result uses essentially the full machinery of the existing proofs of Sprind\v zuk's conjecture \cite{KleinbockMargulis,KLW}, it is worth noting that the following result (which does not imply Sprind\v zuk's conjecture) can be proven using only elementary real analysis together with the Simplex Lemma:
\begin{theorem}[Proven in Section \ref{sectionQDE}]
\label{theoremQDE}
Every quasi-decaying measure is extremal.
\end{theorem}
The idea of proving the extremality of measures using the Simplex Lemma is due to A. D. Pollington and S. L. Velani \cite[Theorem 1]{PollingtonVelani}. Proving Theorem \ref{theoremQDE} was a key step in our construction of the definition of the quasi-decay condition, since it allowed us to see what the minimal hypotheses on the measure were such that the proof would work. It was only later that we realized the Sprind\v zuk conjecture machinery developed in \cite{KleinbockMargulis, KLW} would work for our measures as well.
\subsection{Ahlfors regularity vs. exact dimensionality}
One way of thinking about the difference between KLW's conditions and our conditions is by comparing this difference with the difference between the classes of \emph{Ahlfors regular} and \emph{exact dimensional} measures, both of which are well-studied in dynamics (for more details see \cite{DFSU_GE2}). We recall their definitions:
\begin{definition*}
A measure $\mu$ on $\R^d$ is called \emph{Ahlfors $\delta$-regular} if there exists $C > 0$ such that for every ball $B(\xx,\rho)$ with $\xx\in\Supp(\mu)$ and $0 < \rho \leq 1$.
\[
C^{-1} \rho^\delta \leq \mu\big(B(\xx,\rho)\big) \leq C \rho^\delta.
\]
The measure $\mu$ is called \emph{exact dimensional of dimension $\delta$} if for $\mu$-a.e. $\xx\in\R^d$,
\begin{equation}
\label{exactdim}
\lim_{\rho\to 0} \frac{\log\mu\big(B(\xx,\rho)\big)}{\log\rho} = \delta.
\end{equation}
\end{definition*}
Every Ahlfors $\delta$-regular measure is exact dimensional of dimension $\delta$. The Hausdorff and packing dimensions of an exact dimensional measure of dimension $\delta$ are both equal to $\delta$ \cite[Theorem 4.4]{Young2}; for an Ahlfors $\delta$-regular measure, the Hausdorff, packing, and upper and lower Minkowski (box-counting) dimensions of the topological support are also equal to $\delta$. There are many dynamical examples of Ahlfors regular measures; there are also many examples of exact dimensional measures which are not Ahlfors regular. In Part II, the latter class of examples will prove to be a fruitful source of quasi-decaying measures which are not friendly.
The philosophical relations between Ahlfors regularity and exact dimensionality with absolute friendliness and quasi-decay, respectively, are:
\begin{equation}
\label{philosophical}
\begin{split}
\text{Ahlfors regular and ``nonplanar''} \;\; &\Rightarrow \;\; \text{Absolutely friendly}\\
\text{Exact dimensional and ``nonplanar''} \;\; &\Rightarrow \;\; \text{Quasi-decaying}
\end{split}
\end{equation}
Here ``nonplanar'' does not refer to nonplanarity as defined in Definition \ref{definitionabsolutelyfriendly}, but is rather something less precise (and stronger). This less precise definition should rule out examples like the Lebesgue measures of nondegenerate manifolds, since these are not quasi-decaying. One example of a ``sufficient condition'' for this imprecise notion of ``nonplanarity'' is simply the inequality $\delta > d - 1$, where $\delta$ is the dimension of the measure in question. In particular, in this context the relations \eqref{philosophical} are made precise by the following theorems:
\begin{theorem*}[{\cite[Proposition 6.3]{KleinbockWeiss1}}; cf. \cite{PollingtonVelani,Urbanski3}]
If $\delta > d - 1$, then every Ahlfors $\delta$-regular measure on $\R^d$ is absolutely friendly.
\end{theorem*}
\begin{theorem}[Proven in Section \ref{sectionQDE}]
\label{theoremexactdim}
If $\delta > d - 1$, then every exact dimensional measure on $\R^d$ of dimension $\delta$ is quasi-decaying.
\end{theorem}
\subsection{Further comparison of KLW's conditions vs. our conditions}
There are three axes on which we can compare our conditions against KLW's: Diophantine properties of measures satisfying the condition, examples of measures satisfying the condition, and stability properties. We deal with the first of these in \6\ref{subsectionadditional} below, and the second will be discussed at length in Part II. It remains to consider stability properties. The following proposition describes the stability properties of quasi-decaying and weakly quasi-decaying measures:
\begin{theorem}[Proven in Section \ref{sectionbasic}]
\label{theoremstabilityQD}~
\begin{itemize}
\item[(i)] The (weak) quasi-decay property does not depend on which norm $\|\cdot\|$ on $\R^d$ is used in Definition \ref{definitionQD}.
\item[(ii)] The product of any two (weakly) quasi-decaying measures is (weakly) quasi-decaying.
\item[(iii)] For all $\epsilon > 0$, the image of a quasi-decaying measure under a $\CC^{1 + \epsilon}$ diffeomorphism is quasi-decaying.
\item[(iv)] If $(U_i)_i$ is an open cover of $\R^d$, then $\mu$ is (weakly) quasi-decaying if and only if for each $i$, $\mu\given U_i$ is (weakly) quasi-decaying.
\item[(v)] Any measure absolutely continuous with respect to a (weakly) quasi-decaying measure is (weakly) quasi-decaying.
\end{itemize}
\end{theorem}
The first two properties are also satisfied for friendliness and absolute friendliness (for (ii) see \cite[Theorem 2.4]{KLW}). Property (iii) is not true for either friendliness or the weakly quasi-decay condition, since the image of the Lebesgue measure of a nondegenerate manifold under a diffeomorphism may be the Lebesgue measure of an affine hyperplane, which does not satisfy any of the four conditions (if the hyperplane is rational it is not even extremal). It is true for the absolute friendliness condition under the additional hypothesis that the measure is compactly supported \cite[Proposition 3.2]{FMS}.
Property (iv) is not true for either friendliness or absolute friendliness, but this can be fixed either by making a more careful statement which involves conditions holding ``relative to'' certain open sets in the sense of Definition \ref{definitionabsolutelyfriendly}, or else by considering ``non-uniform'' versions of the conditions, as is done in \cite[\66]{KLW}. It was hypothesized in \cite[para. after Theorem 6.1]{KLW} that a weak version of property (v) holds for the non-uniform versions of friendliness and absolute friendliness, namely that these conditions are measure class invariants. However, we can now show that this statement is false; see Appendix \ref{appendix1}.
We remark that stability properties (iii) and (iv) imply that it makes sense to talk about quasi-decaying measures on abstract differentiable manifolds, by calling a measure quasi-decaying if it is quasi-decaying on every coordinate chart. The non-uniform version of absolute friendliness can be also considered on manifolds. It doesn't make sense to talk about weakly quasi-decaying or friendly measures on abstract differentiable manifolds due to the failure of property (iii) for these classes.
\subsection{Additional Diophantine properties of quasi-decaying measures}
\label{subsectionadditional}
In addition to being extremal, the Lebesgue measure of a nondegenerate manifold has many other nice Diophantine properties which can also be generalized to weakly quasi-decaying measures. These improvements fall into three categories:
\begin{itemize}
\item those dealing with strong extremality rather than just extremality;
\item those dealing with matrices rather than just vectors;
\item those dealing with measures supported on proper affine subspaces of $\R^d$ (or in the case of matrices, of the space $\EE$ defined below).
\end{itemize}
Let us review the theory of Diophantine approximation of matrices. Fix ${M,N}\in\N$, let $\MM \equiv \MM_{M,N}$\Footnote{Here and elsewhere $A \equiv B$ means ``$A$ is shorthand for $B$''.} denote the set of $M\times N$ matrices, and fix $\bfA\in\MM$. Rather than approximating $\bfA$ by rational matrices, classically one considers ``approximations'' to $\bfA$ to be integer vectors $\qq\in \Z^N\butnot\{\0\}$ whose image under $\bfA$ is close to an integer vector. Thus the \emph{exponent of irrationality} of $\bfA$ is defined as
\[
\omega(\bfA) = \limsup_{\substack{\qq\in\Z^N\butnot\{\0\} \\ \pp\in\Z^M}} \frac{-\log\|\bfA\qq - \pp\|}{\log\|\qq\|},
\]
where the two $\|\cdot\|$s denote any two norms on $\R^M$ and $\R^N$, and the \emph{exponent of multiplicative irrationality} is the number\Footnote{This definition agrees with the multiplicative approximation framework considered in \cite{KMW}, but not the one considered in \cite{KleinbockMargulis}; see comments after Proposition \ref{propositiondynamicalinterpretation} for more details.}
\[
\omega_\times(\bfA) = \limsup_{\substack{\qq\in\Z^N\butnot\{\0\} \\ \pp\in\Z^M}} \frac{-\log\prod_{i = 1}^M |(\bfA\qq - \pp)_i|}{\log \prod_{j = 1}^N |q_j|\vee 1}\cdot
\]
Note that $\omega_\times(\bfA) \geq (M/N)\omega(\bfA)$. The relationship between matrix approximation and simultaneous approximation (i.e. the approximation of vectors in $\R^d$ by rational vectors described at the beginning of this paper) is as follows: if $N = 1$ and $\xx = \bfA\ee_1$, then $\omega(\bfA) = \omega(\xx) - 1$ and $\omega_\times(\bfA) = \omega_\times(\xx) - M$. The matrix $\bfA$ is called \emph{very well approximable} if $\omega(\bfA) > N/M$, and \emph{very well multiplicatively approximable} if $\omega_\times(\bfA) > 1$. As in the case of vectors, we denote the set of very well (multiplicatively) approximable $M\times N$ matrices by $\mathrm{VW(M)A}_{{M,N}}$, and we call a measure $\mu$ on $\MM$ \emph{extremal} if $\mu(\VWA_{{M,N}}) = 0$ and \emph{strongly extremal} if $\mu(\VWMA_{{M,N}}) = 0$. Also as before, the sets $\VWA_{{M,N}}$ and $\VWMA_{{M,N}}$ are both Lebesgue nullsets of full Hausdorff dimension which satisfy $\VWA_{{M,N}} \subset \VWMA_{{M,N}}$.
It turns out (cf. \cite{KMW, BKM, ABRdS,ABRdS3}) that the natural vector space structure of $\MM$ is not appropriate for determining extremality and strong extremality. Instead, it is better to identify $\MM$ with its image under the \emph{Pl\"ucker embedding} $\psi \equiv \psi_{{M,N}}:\MM_{{M,N}} \to \EE \equiv \EE_{{M,N}}$, where $\EE\subset \bigwedge^N \R^{M + N}$ is the subspace spanned by all basis vectors (vectors of the form $\ee_I = \bigwedge_{i\in I} \ee_i$\, where the product is taken in increasing order) other than $\bigwedge_{j = 1}^N (\0\oplus\ee_j)$, and
\begin{equation}
\label{plucker}
\psi(\bfA) = \bigwedge_{j = 1}^N (\bfA\ee_j\oplus\ee_j) - \bigwedge_{j = 1}^N (\0\oplus\ee_j) \in \EE.
\end{equation}
Concretely, $\psi$ is the map which sends a matrix to the list of the determinants of its minors.
\begin{remark*}
Technically, the map $\psi$ defined by \eqref{plucker} is not the Pl\"ucker embedding, but is related to it as follows. Let $\GG \equiv \GG(N,M + N)$ denote the Grassmannian space consisting of all $N$-dimensional subspaces of $\R^{M + N}$, and let $\PP$ denote the projectivization of the vector space $\bigwedge^N \R^{M + N}$. Consider the coordinate charts $\iota_1:\MM\to \GG$ and $\iota_2:\EE\to\PP$ defined by the formulas $\iota_1(\bfA) = (\bfA \oplus I_N)(\R^N)$ and $\iota_2(\omega) = (\bigwedge_1^N (\0\oplus\ee_j) + \omega)\R$. Then
\[
\iota_2\circ \psi = \wbar\psi\circ \iota_1,
\]
where $\wbar\psi:\GG\to\PP$ is the true Pl\"ucker embedding. Nevertheless, we shall continue to call the map defined by \eqref{plucker} the Pl\"ucker embedding.
\end{remark*}
Given a measure $\mu$ on $\MM$, we can ask about its geometric properties (e.g. friendliness, quasi-decay) either with respect to the natural vector space structure on $\MM$ or with respect to the natural identification of $\MM$ with a submanifold of $\EE$ via the Pl\"ucker embedding. When $N = 1$ or $M = 1$, the Pl\"ucker embedding is a linear isomorphism, so the geometric properties of $\mu$ do not depend on which way we consider $\MM$. In general, these properties may depend on which way we consider $\MM$, but due to the nondegeneracy of the Pl\"ucker embedding, the following relations hold (cf. \cite[Theorem 2.1]{KLW} and Theorem \ref{theoremQDembedding}):
\begin{itemize}
\item If $\mu$ is absolutely friendly with respect to the vector space structure of $\MM$, then $\mu$ is friendly when considered as a measure on $\EE$.
\item If $\mu$ is quasi-decaying with respect to the vector space structure of $\MM$, then $\mu$ is weakly quasi-decaying when considered as a measure on $\EE$.
\end{itemize}
In such a scenario, the following theorem implies that $\mu$ is strongly extremal:
\begin{theorem}[Corollary of Theorem \ref{theoremexponents} below]
\label{theoremKLW}
Let $\mu$ be a measure on $\MM$ which is weakly quasi-decaying when considered as a measure on $\EE$. Then $\mu$ is strongly extremal.
\end{theorem}
The special case of Theorem \ref{theoremKLW} which occurs when $\mu$ is friendly instead of weakly quasi-decaying was proven in \cite[Theorem 2.1]{KMW}.
Combining with Theorem \ref{theoremQDembedding} yields:
\begin{corollary}
\label{corollaryKLW}
Let $\mu$ be a measure on $\MM$ which is quasi-decaying with respect to the vector space structure of $\MM$. Then $\mu$ is strongly extremal. If $M = 1$ or $N = 1$, $\mu$ need only be weakly quasi-decaying.
\end{corollary}
Note that Corollary \ref{corollaryKLW} provides an alternate proof of Theorem \ref{theoremQDE}.
Although a measure $\mu$ supported on an affine subspace of $\EE$ cannot be weakly quasi-decaying, if $\mu$ is weakly quasi-decaying with respect to the affine subspace, then we can get information about the exponent of irrationality:
\begin{theorem}[Proven in Section \ref{sectionKLW}]
\label{theoremexponents}
Let $\mu$ be a measure on $\MM$ which is supported on an affine subspace $\AA\subset\EE$ and which is weakly quasi-decaying when interpreted as a measure on $\AA$. Then for $\mu$-a.e. $\bfA\in\MM$
\begin{align}
\label{exponents}
\omega(\bfA) &= \inf\{\omega(\bfB) : \bfB\in\MM\cap \AA\}.
\end{align}
Moreover, $\mu$ is strongly extremal if and only if $\MM\cap\AA\nsubset \VWMA$.
\end{theorem}
Note that Theorem \ref{theoremKLW} follows from Theorem \ref{theoremexponents} by taking $\AA = \EE$, since it is well-known that in this case the right hand side of \eqref{exponents} is $M/N$, and that $\MM \nsubset \VWMA$. It appears to be difficult to prove a multiplicative analogue of \eqref{exponents}, due to difficulties with providing a dynamical interpretation for the exponent of multiplicative irrationality function; cf. Footnote \ref{footnotetechnical}.
Historical note: A special case of \eqref{exponents}, where the condition of being weakly quasi-decaying is replaced by an analogue of a friendliness condition, was proven independently by Aka, Breuillard, Rosenzweig, and de Saxc\'e \cite[Theorem 5.2.5]{ABRdS3} (see also their earlier announcement of this result in \cite[Theorem 4.3]{ABRdS}). Their paper also contains other interesting information about the function $\AA\mapsto \inf\{\omega(\bfB) : \bfB\in\MM\cap \AA\}$, such as its value when $\AA$ is rational.
\subsection{An overview of Part II}\Footnote{In this subsection we refer to the references cited for the definitions of terms used in the theorems.}
\label{subsectionpreview}
Theorem \ref{theoremexactdim} (exact dimensional measures of sufficiently large dimension are quasi-decaying) already provides large classes of examples of quasi-decaying measures which are not known to be friendly. For example, the following result was proven by F. Hofbauer:
\begin{theorem*}[{\cite[Theorem 1]{Hofbauer}}]
Let $T:[0,1]\to[0,1]$ be a piecewise monotonic transformation whose derivative has bounded $p$-variation for some $p > 0$. Let $\mu$ be a measure on $[0,1]$ which is ergodic and invariant with respect to $T$. Let $h(\mu)$ and $\chi(\mu)$ denote the entropy and Lyapunov exponent of $\mu$, respectively. If $\chi(\mu) > 0$, then $\mu$ is exact dimensional of dimension
\[
\delta_\mu := \frac{h(\mu)}{\chi(\mu)}\cdot
\]
\end{theorem*}
Note that if $h(\mu) > 0$, then Ruelle's inequality \cite[Theorem 7.1]{BarrioJimenez} implies that $\chi(\mu) > 0$, so the above result applies and gives $\delta_\mu > 0 = d - 1$, so $\mu$ is quasi-decaying, and in particular extremal.\Footnote{The inequality $\chi(\mu) < \infty$ follows from the hypothesis that $T'$ has bounded $p$-variation, which in particular implies that $T'$ is bounded.}
There are numerous other classes of measures coming from dynamics which are known to be exact dimensional. A notable example is the theorem of Barreira, Pesin, and Schmeling \cite{BPS} to the effect that any measure ergodic, invariant, and hyperbolic with respect to a diffeomorphism is exact dimensional. Theorem \ref{theoremexactdim} applies directly to those measures whose dimension is sufficiently large, but in Part II we will mostly be interested in the question of what happens for measures whose dimension is not large enough. (We will also be interested in measures which are not necessarily exact dimensional but which nevertheless can be proved to be quasi-decaying.) As mentioned above, the philosophy is that some sort of ``nonplanarity'' assumption should be able to substitute for the large-dimension hypothesis. For inspiration we can turn to the known dynamical examples of absolutely friendly measures \cite{KLW, StratmannUrbanski1, Urbanski}, which share the property that the nonplanarity hypothesis takes the form: the dynamical system in question cannot preserve a manifold of strictly lower dimension than the ambient space.
Our next examples of quasi-decaying measures are generalizations of the known examples of absolutely friendly measures. For example, the following theorem generalizes the main result of \cite{Urbanski}:
\begin{theorem}[{\cite[Theorem 1.14]{DFSU_GE2}}]
\label{theoremgibbs}
Fix $d\in\N$, and let $(u_a)_{a\in A}$ be a (possibly infinite) irreducible conformal iterated function system (CIFS) on $\R^d$. Let $\phi:A^\N\to \R$ be a summable locally H\"older continuous potential function, let $\mu_\phi$ be a Gibbs measure of $\phi$, and $\pi:A^\N\to\R^d$ be the coding map. Suppose that the Lyapunov exponent
\begin{equation}
\label{lyapunov}
\chi_{\mu_\phi} = \int \log(1/|u_{\omega_1}'(\pi\circ\sigma(\omega))|) \;\; \dee\mu_\phi(\omega)
\end{equation}
is finite. Then $\pi_*[\mu_\phi]$ is quasi-decaying.
\end{theorem}
This theorem generalizes \cite{Urbanski} in two different ways:
\begin{itemize}
\item The CIFS can be infinite, as long as the Lyapunov exponent is finite.
\item The open set condition is no longer needed.
\end{itemize}
Note that if $\phi$ is the ``conformal potential'' $\phi(\omega) = -\log |u_{\omega_1}'(\pi(\sigma(\omega)))|$, then the convergence of \eqref{lyapunov} for some $\alpha$ is equivalent to the strong regularity of the CIFS $(u_a)_{a\in E}$. Thus the following is a corollary of Theorem \ref{theoremgibbs}:
\begin{corollary}[Conformal measures of infinite iterated function systems]
Fix $d\in\N$, and let $(u_a)_{a\in E}$ be a strongly regular conformal iterated function system acting irreducibly on an open set $W\subset \R^d$. Let $\mu$ be the conformal measure of $(u_a)_{a\in E}$. Then $\mu$ is quasi-decaying.
\end{corollary}
Our next example extends the result of \cite{StratmannUrbanski1} from the setting of convex-cocompact groups to the setting of geometrically finite groups:
\begin{theorem}[{\cite[Theorems 1.9 and 1.17]{DFSU_GE2}}]
Let $G$ be a geometrically finite group of M\"obius transformations of $\R^d$ which does not preserve any generalized sphere. Then the Patterson--Sullivan measure $\mu$ of $G$ is both quasi-decaying and friendly. However, $\mu$ is absolutely friendly if and only if every cusp of $G$ has maximal rank.
\end{theorem}
An interesting aspect of this example is that we are able to prove the extremality of the Patterson--Sullivan measure using KLW's condition; for this particular example it was not necessary to introduce the quasi-decay condition. However, proving quasi-decay has the advantage of also proving that the measure is extremal with respect to matrix approximations as well; cf. \6\ref{subsectionadditional} above.
In subsequent papers, we plan to find sufficient conditions for quasi-decay for many other classes of measures as well, but at this stage we cannot give precise theorem statements.
On the other hand, it is also interesting to consider dynamical measures which are not extremal. Three of the authors have already considered this question in \cite{FSU1}, where the following was proven:
\begin{theorem}[{\cite[Theorem 4.5]{FSU1}}]
\label{theoremFSU}
There exists a measure $\mu$ invariant with respect to the Gauss map which gives full measure to the Liouville numbers. In particular, $\mu$ is not extremal.
\end{theorem}
By \cite[Theorem 2.1]{FSU1}, the measure $\mu$ in Theorem \ref{theoremFSU} must have infinite Lyapunov exponent. In Part II, we show that for certain dynamical systems (namely hyperbolic toral endomorphisms), the class of invariant measures which give full measure to the Liouville points is not only nonempty but topologically generic:
\begin{theorem}[{\cite[Theorem 1.13]{DFSU_GE2}}]
Let $T:X\to X$ be a hyperbolic toral endomorphism, where $X = \R^d/\Z^d$. Let $\M_T(X)$ be the space of $T$-invariant probability measures on $X$. Then the set of measures which give full measure to the Liouville points is comeager in $\M_T(X)$.
\end{theorem}
{\bf Outline of the paper.}
In Section \ref{sectionQDE} we give elementary arguments proving that every exact-dimensional measure of dimension $>d - 1$ is quasi-decaying (Theorem \ref{theoremexactdim}), and that every quasi-decaying measure is extremal (Theorem \ref{theoremQDE}). In Section \ref{sectionbasic} we demonstrate the basic properties of the quasi-decay condition described in Theorems \ref{theoremQDembedding} and \ref{theoremstabilityQD}. In Section \ref{sectionKLW} we prove Theorem \ref{theoremexponents}, describing the Diophantine properties of weakly quasi-decaying measures with respect to matrix approximation.
\draftnewpage
\section{Proof of Theorems \ref{theoremQDE} and \ref{theoremexactdim} ($\delta > d - 1 \;\Rightarrow\text{ Quasi-decaying }\Rightarrow\text{ Extremal}$)}
\label{sectionQDE}
\begin{definition}
\label{definitionUQD}
Given a measure $\mu$ on $\R^d$ and a set $E\subset\R^d$, we will say that $\mu$ is \emph{uniformly quasi-decaying (resp. uniformly weakly quasi-decaying)} relative to $E$ if for all $\gamma > 0$, there exist $C_1,\alpha > 0$ such that for all $\xx\in E$, $0 < \rho \leq 1$, $0 < \beta \leq \rho^\gamma$, and $\LL\in\scrH$, if $B = B(\xx,\rho)$ then \eqref{QDwithE} (resp. \eqref{weakQDwithE}) holds.
\end{definition}
\begin{lemma}
\label{lemmaunifQD}
A measure $\mu$ is (weakly) quasi-decaying if and only if there exists a sequence $(E_n)_n$ such that $\mu(\R^d\butnot\bigcup_n E_n) = 0$ and for each $n$, $\mu$ is uniformly (weakly) quasi-decaying relative to $E_n$.
\end{lemma}
\begin{proof}
It suffices to show that if $\mu$ is (weakly) quasi-decaying relative to $E$, then there exists a sequence $(E_n)_n$ such that $\mu(E\butnot\bigcup_n E_n) = 0$ and for each $n$, $\mu$ is uniformly (weakly) quasi-decaying relative $E_n$. Indeed, for each $m,k\in\N$ let $E_{m,k}$ be the set of all $\xx\in E$ such that \eqref{QDwithE} (resp. \eqref{weakQDwithE}) holds for all $0 < \rho\leq 1$, $0 < \beta \leq \rho^\gamma$, and $\LL\in\scrH$, with $\gamma = 1/m$, $\alpha = 1/k$, and $B = B(\xx,\rho)$. Then for all $m$, $\mu(E\butnot\bigcup_k E_{m,k}) = 0$, so there exists $k_m\in\N$ such that $\mu(E\butnot E_{m,k_m}) \leq 2^{-m}$. Letting
\[
E_n \df \bigcap_{m > n} E_{m,k_m}
\]
completes the proof.
\end{proof}
Actually, uniformly quasi-decaying measures show up naturally in the analysis of exact dimensional measures:
\begin{proof}[Proof of Theorem \ref{theoremexactdim}]
Let $\mu$ be an exact dimensional measure on $\R^d$ of dimension $\delta > d - 1$, and we will show that $\mu$ is quasi-decaying. By Egoroff's theorem, there exists a sequence $(E_n)_n$ such that $\mu(\R^d\butnot \bigcup E_n) = 0$ and for all $n\in\N$, the limit \eqref{exactdim} holds uniformly on $E_n$. Fix $n$, and we will show that $\mu$ is uniformly quasi-decaying relative to $E_n$. Indeed, fix $\gamma > 0$, $\xx\in E_n$, $0 < \rho \leq 1$, $0 < \beta \leq \rho^\gamma$, and $\LL\in\scrH$. Let $(\xx_i)_1^N$ be a maximal $\beta\rho$-separated\Footnote{Recall that a set $S$ is said to be \emph{$\rho$-separated} if for all distinct $x,y\in S$, we have $\dist(x,y)\geq \rho$.} subset of $\thickvar\LL{\beta\rho}\cap B(\xx,\rho)\cap E_n$, and let $\lambda$ denote Lebesgue measure on $\R^d$. Then
\[
N (\beta\rho)^d \asymp_\times \sum_{i = 1}^N \lambda\big(B(\xx_i,\beta\rho/2)\big)
\leq \lambda\left(\thickvar\LL{2\beta\rho}\cap B(\xx,2\rho)\right)
\asymp_\times \rho^{d - 1} (\beta\rho) = \beta\rho^d,
\]
so $N\lesssim_\times \beta^{-(d - 1)}$. On the other hand, for all $\epsilon > 0$ we have
\begin{align*}
\mu\big(\thickvar{\LL}{\beta\rho}\cap B(\xx,\rho)\cap E_n\big)
&\leq_\pt \sum_{i = 1}^N \mu\big(B(\xx_i,\beta\rho)\big)\\
&\lesssim_\times N (\beta\rho)^{\delta - \epsilon} \by{\eqref{exactdim}}\\
&\lesssim_\times \beta^{-(d - 1) + (\delta - \epsilon)}\rho^{s - \epsilon}\\
&\lesssim_\times \beta^{\delta - (d - 1) - \epsilon} \rho^{-2\epsilon} \mu\big(B(\xx,\rho)\big). \by{\eqref{exactdim}}
\end{align*}
Letting $\alpha = \delta - (d - 1) > 0$ and $\epsilon = \alpha/(1 + 2/\gamma) > 0$, since $\rho \geq \beta^{1/\gamma}$ we get
\[
\mu\big(\thickvar{\LL}{\beta\rho}\cap B(\xx,\rho)\cap E_n\big)
\lesssim_\times \beta^{\alpha/2} \mu\big(B(\xx,\rho)\big).
\qedhere\]
\end{proof}
Next, we prove Theorem \ref{theoremQDE}. By Lemma \ref{lemmaunifQD}, it suffices to demonstrate the following:
\begin{theorem}
Let $\mu$ be a measure which is uniformly quasi-decaying relative to a set $E\subset\R^d$. Then
\[
\mu(\VWA_d\cap E) = 0 .
\]
\end{theorem}
\begin{proof}
For each $\gamma > 0$ let
\[
W_\gamma \df \{\xx\in\R^d: \omega(\xx) > (1 + 1/d)(1 + \gamma)\},
\]
so that $\VWA_d = \bigcup_{\gamma > 0} W_\gamma$. Fix $\gamma > 0$, and we will show that $\mu(W_\gamma\cap E) = 0$. Since $\mu$ is uniformly quasi-decaying relative to $E$, there exist $C_1,\alpha > 0$ such that for all $\xx\in E$, $0 < \rho \leq 1$, $0 < \beta \leq \rho^\gamma$, and $\LL\in\scrH$, if $B = B(\xx,\rho)$ then \eqref{QDwithE} holds.
To proceed further we recall the \emph{simplex lemma}, which is proven by a volume argument:
\begin{lemma}[{\cite[Lemma 4]{KTV}}]
\label{lemmasimplex}
Fix $d\in\N$. There exists $\epsilon_d > 0$ such that for all $\yy\in\R^d$ and $0 < \rho\leq 1$, the set
\[
S_{\yy,\rho} \df \left\{\frac{\pp}{q}\in\Q^d\cap B(\yy,\rho): q\leq \epsilon_d \rho^{-d/(d + 1)}\right\}
\]
is contained in an affine hyperplane $\LL_{\yy,\rho} \subset\R^d$.
\end{lemma}
\noindent Let $\epsilon_d > 0$ be as in Lemma \ref{lemmasimplex}. Fix $H > 1$, and for each $n\in\N$ let
\begin{align*}
Q_n &\df \epsilon_d H^{dn},&
\rho_n &\df \frac12H^{-(d + 1)n}.
\end{align*}
For each $n\in\N$ and $\yy\in\R^d$ let
\begin{align*}
S_{n,\yy} &\df S_{\yy,2\rho_n} = \left\{\frac{\pp}{q}\in\Q^d\cap B(\yy,2\rho_n): q\leq Q_n\right\},&
\LL_{n,\yy} &\df \LL_{\yy,2\rho_n}.
\end{align*}
Fix $n\in\N$, and let $E_n\subset W_\gamma\cap E$ be a maximal $\rho_n$-separated set.
\begin{claim}
\label{claimSbound}
\[
W_\gamma\cap E \subset \limsup_{n\to\infty}\bigcup_{\yy\in E_n}\left[\thickvar{\LL_{n,\yy}}{\rho_n^{1 + \gamma}}\cap B(\yy,\rho_n)\right].
\]
\end{claim}
\begin{subproof}
Fix $\xx\in W_\gamma\cap E$ and $\pp/q\in\Q^d$, and let $n\in\N$ satisfy $Q_{n - 1} \leq q < Q_n$. Then
\[
q^{-(1 + 1/d)(1 + \gamma)} \leq Q_{n - 1}^{-(1 + 1/d)(1 + \gamma)} \asymp_\times \rho_n^{1 + \gamma}
\]
and so since $\omega(\xx) > (1 + 1/d)(1 + \gamma)$, there exist infinitely many $\pp/q$ such that
\begin{equation}
\label{goodpq}
\|\xx - \pp/q\| < \rho_n^{1 + \gamma} < \rho_n.
\end{equation}
Fix $\pp/q$ satisfying \eqref{goodpq}. Since $\xx\in W_\gamma\cap E$, there exists $\yy\in E_n$ such that $\xx\in B(\yy,\rho_n)$. Then $\pp/q\in S_{n,\yy} \subset \LL_{n,\yy}$ and thus by \eqref{goodpq}, we have $\xx\in \thickvar{\LL_{n,y}}{\rho_n^{1 + \gamma}}$.
\end{subproof}
Without loss of generality we may assume that $E$ is bounded, so that $\mu\big(\thickvar \borel1\big) < \infty$. Then for each $n\in\N$, we have
\begin{align*}
&\sum_{\yy\in E_n} \mu\left(\thickvar{\LL_{n,\yy}}{\rho_n^{1 + \gamma}}\cap B(\yy,\rho_n)\cap E\right) \noreason \\
&\leq_\pt C_1 \rho_n^{\gamma\alpha} \sum_{\yy\in E_n} \mu\big(B(\yy,\rho_n)\big) \note{uniform quasi-decay}\\
&\lesssim_\times \rho_n^{\gamma\alpha} \mu\big(\thickvar \borel1\big) \asymp_\times \rho_n^{\gamma\alpha}. \note{bounded multiplicity}
\end{align*}
So by the Borel-Cantelli lemma,
\[
\mu\left(\limsup_{n\to\infty}\bigcup_{\yy\in E_n}\left[\thickvar{\LL_{n,\yy}}{\rho_n^{1 + \gamma}}\cap B(\yy,\rho_n)\cap E\right]\right) = 0,
\]
and so $\mu(W_\gamma\cap E) = 0$ by Claim \ref{claimSbound}.
\end{proof}
\section{Basic properties of the quasi-decay condition}
\label{sectionbasic}
Before proving Theorems \ref{theoremQDembedding} and \ref{theoremstabilityQD}, we need some preliminaries. The first, as mentioned in the introduction, is a substitute for the doubling condition which holds for every measure on a doubling metric space.
\begin{definition}
A metric space $X$ is \emph{doubling} if there exists a constant $N_X$ such that every ball $B(x,\rho) \subset X$ can be covered by at most $N_X$ balls of radius $\rho/2$.
\end{definition}
For example, $\R^d$ is a doubling metric space.
\begin{lemma}
\label{lemmaquasifederer}
Let $X$ be a doubling metric space, and let $\mu$ be a measure on $X$. Then for all $\epsilon > 0$, there exists $\delta > 0$ such that for $\mu$-a.e. $x\in X$, there exists $C_2 > 0$ such that for all $0 < \rho \leq 1$,
\begin{equation}
\label{quasifederer}
\mu\big(B(x,\rho^{1 - \delta})\big) \leq C_2 \rho^{-\epsilon} \mu\big(B(x,\rho)\big).
\end{equation}
\end{lemma}
\begin{proof}
Fix $\epsilon > 0$, and let $\delta = \epsilon/(2\log_2(N_X)) > 0$, where $N_X$ is the doubling constant of $X$. For each $n\in\N$ let $\rho_n = 2^{-n}$ and let
\[
S_n \df \{x\in X: \mu\big(B(x,\rho_n^{1 - \delta})\big) > \rho_n^{-\epsilon}\mu\big(B(x,\rho_{n + 1})\big)\}.
\]
\begin{claim}
If $E\subset X$ is bounded then $\sum_{n\in\N}\mu(S_n\cap E) < \infty$.
\end{claim}
\begin{subproof}
Fix $n$, and let $E_n$ be a maximal $\rho_{n + 1}$-separated subset of $S_n\cap E$, so that $S_n\cap E\subset \thickvar{E_n}{\rho_{n + 1}}$. We have
\begin{align*}
\mu(S_n\cap E) &\leq \sum_{x\in E_n} \mu\big(B(x,\rho_{n + 1})\big)\\
&\leq \rho_n^\epsilon \sum_{x\in E_n} \mu\big(B(x,\rho_n^{1 - \delta})\big)\\
&= \rho_n^\epsilon \int \#(E_n\cap B(x,\rho_n^{1 - \delta})) \dee\mu(x)\\
&\leq \rho_n^\epsilon \mu\big(\thickvar E 1\big) \max_{x\in X} \#(E_n\cap B(x,\rho_n^{1 - \delta})).
\end{align*}
Fix $x\in X$. Repeatedly applying the doubling condition shows that $B(x,\rho_n^{1 - \delta})$ can be covered by at most $N_X^m$ balls of radius $\rho_{n + 1}/3$, where $m \df \lceil \log_2(6\rho_n^{-\delta})\rceil$. But each of these balls intersects $E_n$ at most once, since $E_n$ is $\rho_{n + 1}$-separated. So
\[
\max_{x\in X} \#(E_n\cap B(x,\rho_n^{1 - \delta})) \leq N_X^m \asymp_\times N_X^{\log_2(\rho_n^{-\delta})} = \rho_n^{-\epsilon/2}.
\]
Thus $\mu(S_n) \lesssim_\times \rho_n^{\epsilon/2}$.
\end{subproof}
So by the Borel--Cantelli lemma, for $\mu$-a.e. $x\in X$ we have $\#\{n\in\N:x\in S_n\} < \infty$. Fix such an $x$, and fix $0 < \rho \leq 1$. Let $n\in\N$ satisfy $\rho_{n + 1} \leq \rho < \rho_n$. If $\rho$ is small enough, then $x\notin S_n$, which implies
\[
\mu\big(B(x,\rho^{1 - \delta})\big) \leq \mu\big(B(x,\rho_n^{1 - \delta})\big) \leq \rho_n^{-\epsilon} \mu\big(B(x,\rho_{n + 1})\big) \leq 2^\epsilon \rho^{-\epsilon} \mu\big(B(x,\rho)\big),
\]
demonstrating \eqref{quasifederer}. Larger values of $\rho$ can be accomodated by changing the constant appropriately.
\end{proof}
Let us call a measure satisfying the conclusion of Lemma \ref{lemmaquasifederer} \emph{quasi-Federer}, so that Lemma \ref{lemmaquasifederer} says that any measure on a doubling metric space is quasi-Federer. For the purposes of this paper this is a somewhat silly definition, since every measure on $\R^d$ is quasi-Federer. However, the following refinements of the quasi-Federer notion distinguish nontrivial classes of measures on $\R^d$:
\begin{definition}
Let $X$ and $\mu$ be as in Lemma \ref{lemmaquasifederer}. Given $E\in X$, we will say that $\mu$ is \emph{uniformly quasi-Federer relative to $E$} if for all $\epsilon > 0$, there exist $C_2,\delta > 0$ such that for all $x\inE$ and $0 < \rho \leq 1$, \eqref{quasifederer} holds. (Note however that $E$ does not occur on the left hand side of \eqref{quasifederer}, in contrast to \eqref{QDwithE}.) Similarly, if $x\in X$, we will say that $\mu$ is \emph{quasi-Federer at $x$} if $\mu$ is uniformly quasi-Federer relative to $\{x\}$.
Note that Lemma \ref{lemmaquasifederer} implies that there exists a sequence of sets $(E_n)_n$ such that $\mu\big(X \butnot \bigcup E_n\big) = 0$ and for each $n$, $\mu$ is uniformly quasi-Federer relative to $E_n$. In particular, $\mu$ is quasi-Federer at $\mu$-a.e. $x\in X$.
\end{definition}
We need two more preliminary results. The following lemma is an immediate consequence of Definitions \ref{definitionQD} and \ref{definitionUQD}:
\begin{lemma}
\label{lemmaQD}
~
\begin{itemize}
\item[(i)] If $\mu$ is uniformly quasi-decaying (resp. uniformly weakly quasi-decaying) relative to $E\subset\R^d$, then for all $\gamma > 0$ there exists $\alpha = \alpha(\gamma,\mu) > 0$ such that for all $C > 0$, there exists $C_1 > 0$ such that for all $\xx \in E$, $0 < \rho \leq 1$, $\beta \leq C \rho^\gamma$, and $\LL\in\scrH$, if $B = B(\xx,\rho)$ then \eqref{QDwithE} (resp. \eqref{weakQDwithE}) holds.
\item[(ii)] If $\mu$ is quasi-decaying (resp. weakly quasi-decaying) at $\xx\in\R^d$ relative to $E\subset\R^d$, then for all $\gamma > 0$ there exists $\alpha = \alpha(\gamma,\mu,\xx) > 0$ such that for all $C > 0$, there exists $C_1 > 0$ such that for all $0 < \rho \leq 1$, $\beta \leq C \rho^\gamma$, and $\LL\in\scrH$, if $B = B(\xx,\rho)$ then \eqref{QDwithE} (resp. \eqref{weakQDwithE}) holds.
\end{itemize}
\end{lemma}
Our last preliminary result is a generalization of the Lebesgue differentiation theorem:
\begin{theorem}[{\cite[Theorem 9.1]{Simmons1}}]
\label{theoremLD}
Let $\mu$ and $\nu$ be measures on $\R^d$ such that $\nu\lessless\mu$. Then the function
\begin{equation}
\label{lebesguederivative}
f(x) \df \lim_{\rho\to 0}\frac{\nu\big(B(x,\rho)\big)}{\mu\big(B(x,\rho)\big)}
\end{equation}
is well-defined for $\mu$-almost every $\xx\in \R^d$. Moreover, $\nu = f\mu$, i.e. $f$ is a Radon-Nikodym derivative of $\nu$ with respect to $\mu$.
\end{theorem}
We are now ready to prove Theorem \ref{theoremstabilityQD}; it clearly follows from Lemma \ref{lemmaquasifederer} and Theorem \ref{theoremLD} together with the following:
\begin{proposition}
\label{propositionstabilityQD2}~
\begin{itemize}
\item[(i)] Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be two norms on $\R^d$. If $\mu$ is (weakly) quasi-decaying and quasi-Federer at $\xx\in\R^d$ relative to $E\subset\R^d$ with respect to the norm $\|\cdot\|_1$, then $\mu$ is also (weakly) quasi-decaying at $\xx$ relative to $E$ with respect to the norm $\|\cdot\|_2$.
\item[(ii)] For each $i = 1,2$, fix $d_i\in\N$, and let $\mu_i$ be a measure on $\R^{d_i}$ which is (weakly) quasi-decaying and quasi-Federer at a point $\xx_i\in\R^{d_i}$ relative to a set $E_i\subset\R^{d_i}$. Let $d = d_1 + d_2$. Then $\mu = \mu_1\times \mu_2$ is (weakly) quasi-decaying at $\xx = (\xx_1,\xx_2)\in \R^d$ relative to $E = E_1\times E_2 \subset \R^d$.
\item[(iii)] Fix $\epsilon > 0$. Let $\mu_1$ be a measure on an open set $U_1\subset\R^d$ which is uniformly quasi-decaying relative to a set $E_1\subset U_1$. Let $\psi:U_1\to U_2\subset\R^d$ be a $\CC^{1 + \epsilon}$ diffeomorphism. Then if $\mu_2 = \psi(\mu_1)$ is quasi-Federer at $\xx_2\in U_2$, then $\mu_2$ is also quasi-decaying at $\xx_2$ relative to $E_2 = \psi(E_1)$.
\item[(iv)] Let $\mu$ be a measure on $\R^d$, and let $U\subset\R^d$ be an open set. Then $\mu$ is (weakly) quasi-decaying at a point $\xx\in U$ relative to a set $E\subset\R^d$ if and only if $\mu\given U$ is (weakly) quasi-decaying at $\xx$ relative to $E$.
\item[(v)] Let $\mu$ be a measure on $\R^d$ which is (weakly) quasi-decaying at a point $\xx\in\R^d$ relative to a set $E\subset\R^d$, and let $\nu$ satisfy $\nu\lesssim_\times \mu$ on $E$. If the limit \eqref{lebesguederivative} exists and is positive, then $\nu$ is (weakly) quasi-decaying at $\xx$ relative to $E$.
\end{itemize}
\end{proposition}
\begin{proof}~
\begin{itemize}
\item[(i)]
Let $C > 0$ be the implied constant in the asymptotic $\|\cdot\|_1 \asymp_\times \|\cdot\|_2$, which holds because any two norms on $\R^d$ are equivalent. Fix $\gamma > 0$, and let $\alpha \df \alpha(\gamma,\mu,\xx) > 0$ be as in Lemma \ref{lemmaQD}. Fix $0 < \rho\leq 1$, $0 < \beta \leq \rho^\gamma$, and $\LL\in\scrH$. Note that
\[
\|d_\LL^{(2)}\|_{\mu,B_2(\xx,\rho)} \leq C\|d_\LL^{(1)}\|_{\mu,B_1(\xx,C\rho)},
\]
where $B_i(\xx,\rho)$ denotes the ball $B(\xx,\rho)$ taken with respect to the norm $\|\cdot\|_i$, and similarly for $d_\LL^{(i)}$. If $\mu$ is weakly quasi-decaying, then
\begin{align*}
&\mu\big(\NN_2(\LL,\beta\|d_\LL^{(2)}\|_{\mu,B_2(\xx,\rho)})\cap B_2(\xx,\rho)\cap E\big) \noreason\\
&\leq_\pt \mu\big(\NN_1(\LL,\beta C^2\|d_\LL^{(1)}\|_{\mu,B_1(\xx,C\rho)})\cap B_1(\xx,C\rho)\cap E\big) \noreason\\
&\lesssim_\times \beta^\alpha\mu\big(B_1(\xx,C\rho)\big) \by{Lemma \ref{lemmaQD}}\\
&\lesssim_\times \beta^{\alpha/2} \mu\big(B_2(\xx,\rho)\big)/ \since{$\mu$ is quasi-Federer at $\xx$}
\end{align*}
If $\mu$ is quasi-decaying, then a similar argument shows that
\[
\mu\big(\NN_2(\LL,\beta\rho)\cap B_2(\xx,\rho)\cap E\big) \lesssim_\times \beta^{\alpha/2} \mu\big(B_2(\xx,\rho)\big).
\]
\item[(ii)]
By part (i), we can use any norm on $\R^d = \R^{d_1}\times \R^{d_2}$ in the proof. It is convenient to use the max norm $\|\cdot\|_\infty$, so that $B(\xx,\rho) = B(\xx_1,\rho)\times B(\xx_2,\rho)$ for all $\rho > 0$. Fix $\gamma > 0$, let $\alpha_i = \alpha(\gamma,\mu_i,\xx_i)$ be as in Lemma \ref{lemmaQD}, and let $\alpha = \alpha_1\wedge\alpha_2 > 0$. Fix $0 < \rho \leq 1$, $0 < \beta \leq \rho^\gamma$, and $\LL\in\scrH$. Write $B_i = B(\xx_i,\rho)$ and $B = B_1\times B_2$. There exist $\zz = (\zz_1,\zz_2)\in\R^d\butnot\{\0\}$ and $c\in\R$ such that
\[
\LL = \left\{\yy\in\R^d: \zz\cdot\yy = c \right\}.
\]
Without loss of generality suppose $\|\zz\|_1 = 1$. For each $\yy_1\in\R^{d_1}$, let
\[
\LL_{\yy_1} = \{\yy_2\in\R^{d_2} : (\yy_1,\yy_2)\in\LL\} = \{\yy_2\in\R^{d_2} : \zz_1\cdot \yy_1 + \zz_2\cdot\yy_2 = c\}.
\]
Note that for all $\yy = (\yy_1,\yy_2)\in\R^d$,
\begin{align} \label{dLz}
d_\LL(\yy) &= |\zz\cdot\yy - c|\\ \label{dLyz}
d_{\LL_{\yy_1}}(\yy_2) &= |\zz_1\cdot\yy_1 + \zz_2\cdot\yy_2 - c||/\|\zz_2\|_1.
\end{align}
In particular
\begin{equation}
\label{comparison}
d_{\LL_{\yy_1}}(\yy_2) = d_\LL(\yy_1,\yy_2) / \|\zz_2\|_1.
\end{equation}
We divide into cases:\\
\item[(iia)] Quasi-decaying case. Since $\|\zz_1\|_1 + \|\zz_2\|_1 = \|\zz\|_1 = 1$, there exists $i = 1,2$ such that $\|\zz_i\|_1\geq 1/2$. Without loss of generality, suppose $\|\zz_2\|_1\geq 1/2$. Then
\begin{align*}
&\mu\big(\thickvar\LL{\beta\rho}\cap B\cap E\big)\\
&=_\pt \int_{B_1\cap E_1} \mu_2\big(\big\{\yy_2\in B_2\cap E_2 : d_\LL(\yy_1,\yy_2)\leq \beta\rho\big\}\big) \;\dee\mu_1(\yy_1)\noreason\\
&\leq_\pt \int_{B_1} \mu_2\big(\thickvar{\LL_{\yy_1}}{2\beta\rho} \cap B_2\cap E_2\big) \;\dee\mu_1(\yy_1) \by{\eqref{comparison}}\\
&\lesssim_\times \beta^\alpha \int_{B_1} \mu_2(B_2) \;\dee\mu_1(\yy_1) = \beta^\alpha \mu(B). \note{Lemma \ref{lemmaQD}}
\end{align*}
Thus $\mu$ is quasi-decaying at $\xx$ relative to $E$.
\item[(iib)] Weakly quasi-decaying case. Let $\sigma = \|d_\LL\|_{\mu,B}$. We can assume that
\[
\thickvar\LL{(1/3)\sigma}\cap B\cap\Supp(\mu) \neq \emptyset,
\]
as otherwise \eqref{weakQDwithE} holds trivially. Then there exist $\aa,\bb\in B\cap\Supp(\mu)$ such that $d_\LL(\aa) \leq (1/3)\sigma \leq (2/3)\sigma \leq d_\LL(\bb)$. So by \eqref{dLz},
\[
|\zz\cdot \bb - \zz\cdot\aa| \geq (1/3)\sigma.
\]
Without loss of generality, we may suppose that
\[
|\zz_2\cdot\bb_2 - \zz_2\cdot\aa_2| \geq (1/6)\sigma.
\]
Then for all $\yy_1\in \R^d$, by \eqref{dLyz} we have
\[
d_{\LL_{\yy_1}}(\aa) + d_{\LL_{\yy_1}}(\bb) \geq (1/6)\sigma/\|\zz_2\|_1
\]
and thus
\[
\|d_{\LL_{\yy_1}}\|_{\mu_2,B_2} \geq (1/12)\sigma/\|\zz_2\|_1.
\]
Applying \eqref{comparison} gives
\[
\{\yy_2\in\R^{d_2}: (\yy_1,\yy_2)\in\thickvar\LL{\beta\sigma}\}
\subset \thickvar{\LL_{\yy_1}}{12\beta\|d_{\LL_{\yy_1}}\|_{\mu_2,B_2}},
\]
so
\begin{align*}
&\mu\big(\thickvar\LL{\beta\sigma}\cap B\cap E\big)\\
&=_\pt \int_{B_1\cap E_1} \mu_2\big(\big\{\yy_2\in B_2\cap E_2 : d_\LL(\yy_1,\yy_2)\leq \beta\sigma\big\}\big) \;\dee\mu_1(\yy_1)\noreason\\
&\leq_\pt \int_{B_1} \mu_2\big(\thickvar{\LL_{\yy_1}}{12\beta\|d_{\LL_{\yy_1}}\|_{\mu_2,B_2}} \cap B_2\cap E_2\big) \;\dee\mu_1(\yy_1) \by{\eqref{comparison}}\\
&\lesssim_\times \beta^\alpha \int_{B_1} \mu_2(B_2) \;\dee\mu_1(\yy_1) = \beta^\alpha \mu(B). \note{Lemma \ref{lemmaQD}}
\end{align*}
\item[(iii)]
The proof of (iii) is similar to the proof of Proposition \ref{propositionQDembedding} below. More precisely, in that proof we can replace $\|d_\LL\|_{\mu_2,B_2}$ by $\|d_\LL\|_{U_2\cap B_2}$ without affecting the argument. Here $\|d_\LL\|_B \df \sup_B \dist(\cdot,\LL)$. Since $U_2$ is open, for $\rho$ sufficiently small we have $\|d_\LL\|_{U_2\cap B_2} = \|d_\LL\|_{B_2} \geq \rho$. Thus in this case, the proof of Proposition \ref{propositionQDembedding} actually proves \eqref{QDwithE} rather than just \eqref{weakQDwithE}.
\item[(iv)]
This is immediate upon changing the implied constant of \eqref{QDwithE} or \eqref{weakQDwithE} appropriately to handle $0 < \rho \leq 1$ for which $B(\xx,\rho)\nsubset U$.
\item[(v)]
If the limit \eqref{lebesguederivative} exists and is positive, then $\mu(B(\xx,\rho)) \asymp_\times \nu\big(B(\xx,\rho)\big)$ for all $0 < \rho \leq 1$. The claim follows immediately.
\qedhere
\end{itemize}
\end{proof}
We now prepare for the proof of Theorem \ref{theoremQDembedding}. The key idea, already implicitly contained in the proofs of \cite[Theorem 7.6]{KLW} and \cite[Theorem 4.6]{FKMS}, is to cover the neighborhood of the zero set of a smooth function by neighborhoods of hyperplanes. We bring this idea to the foreground by stating the following lemma, in which we use the notation
\begin{align*}
\|f\|_B &\df \sup_{\xx\in B} |f(\xx)|,&
\|f\|_{\CC^\epsilon,B} &\df \sup_{\xx,\yy\in B} \frac{|f(\yy) - f(\xx)|}{\|\yy - \xx\|^\epsilon},&
\Delta &\df B(\0,1) \subset \R^d,\\
\|f\| &\df \|f\|_\Delta,&
\|f\|_{\CC^\epsilon} &\df \|f\|_{\CC^\epsilon,\Delta}
\end{align*}
\begin{lemma}
\label{lemmaZPbeta}
Fix $\ell\in\N$ and $0 < \epsilon\leq 1$, and let $f:\Delta\to\R$ be a function of class $\CC^{\ell + \epsilon}$ such that
\begin{equation}
\label{fellbound}
\|f^{(\ell)}\|_{\CC^\epsilon} \leq \kappa_\ell \|f\|,
\end{equation}
where $\kappa_\ell > 0$ is a small constant depending on $\ell$ and $\epsilon$. Then for all $\beta > 0$ sufficiently small (depending on $\ell$ and $\epsilon$), the set
\[
\ZZ(f,\beta) \df \{\xx\in\Delta : |f(\xx)| \leq \beta\|f\|\}
\]
can be covered by collections $\CC_1,\ldots,\CC_\ell$, where for each $k = 1,\ldots,\ell$, the collection $\CC_k$ takes the form
\begin{equation}
\label{Ck}
\CC_k \df \{\thickvar{\LL_j}{\beta_k^{1 + \epsilon/2}}\cap B(\pp_j,\beta_k) : j\in J_k\},
\end{equation}
where $\beta_k \df \beta^{1/2^{2k - 1}}$, $(\pp_j)_{j\in J_k}$ is a $\beta_k$-separated sequence in $\Delta$, and $(\LL_j)_{j\in J_k}$ is a sequence of affine hyperplanes.
\end{lemma}
\begin{proof}
We proceed by induction on $\ell$. If $\ell = 0$, then we let $\kappa_0 = 1/2$, which implies that $\ZZ(f,\beta) = \emptyset$ for all $\beta < 1/2$ and so the lemma is trivial. So suppose that $\ell \geq 1$ and $\ZZ(f,\beta)\neq\emptyset$. Then
\[
\inf_\Delta |f| \leq \beta \|f\| \leq (1/2) \|f\|,
\] so by the mean value inequality, there exists $i = 1,\ldots,d$ such that $\|\del_i f\| \gtrsim_\times \|f\|$. Let $C_1 > 0$ denote the implied constant and let $\kappa_\ell \leq \kappa_{\ell - 1}/C_1$. Then
\begin{align*}
\|\del_i f^{(\ell - 1)}\|_{\CC^\epsilon}
&\leq \|f^{(\ell)}\|_{\CC^\epsilon}
\leq \kappa_\ell \|f\| \leq C_1\kappa_\ell \|\del_i f\| \leq \kappa_{\ell - 1} \|\del_i f\|,
\end{align*}
so by the induction hypothesis, $\ZZ(\del_i f,\beta^{1/4})$ can be covered by collections $\CC_2,\ldots,\CC_\ell$ of the form \eqref{Ck}. If $\ell = 1$, then by the base case of the induction we have $\ZZ(\del_i f, \beta^{\epsilon/4}) = \emptyset$ assuming $\beta$ is sufficiently small. Let
\[
\gamma = \begin{cases}
\epsilon & \ell = 1\\
1 & \ell \geq 2
\end{cases},
\]
so that either way, $\ZZ(\del_i f,\beta^{\gamma/4})$ can be covered by collections $\CC_2,\ldots,\CC_\ell$ of the form \eqref{Ck}. So to complete the proof, we need to cover $\ZZ(f,\beta)\butnot \ZZ(\del_i f,\beta^{\gamma/4})$ by a collection $\CC_1$ of the form $\eqref{Ck}_{k = 1}$. Let $(\pp_j)_{j\in J_1}$ be a maximal $\beta_1 = \beta^{1/2}$-separated sequence in $\Delta\butnot \ZZ(\del_i f,\beta^{\gamma/4})$, and let $J = J_1$. Fix $j\in J$, so that
\[
|\del_i f(\pp_j)| > \beta^{\gamma/4} \|\del_i f\| \asymp_\times \beta^{\gamma/4} \|f\|.
\]
Let $B_j \df B(\pp_j,\beta_1)$.
\begin{claim}
For all $\yy\in B_j\cap \Delta$,
\[
|f(\yy) - f(\pp_j) - f'(\pp_j)[\yy - \pp_j]| \lesssim_\times \beta_1^{1 + \gamma} \|f\|.
\]
\end{claim}
\begin{subproof}
Elementary calculus gives
\begin{align*}
|f(\yy) - f(\pp_j) - f'(\pp_j)[\yy - \pp_j]|
&\leq \|\yy - \pp_j\| \sup_{\zz\in B_j\cap \Delta} \|f'(\zz) - f'(\pp_j)\|.
\end{align*}
Since $\|\yy - \pp_j\| \leq \beta_1$, to complete the proof we need to show that
\begin{equation}
\label{ETSZP}
\|f'(\zz) - f'(\pp_j)\| \lesssim_\times \beta_1^\gamma \|f\| \all\zz\in B_j\cap \Delta.
\end{equation}
If $\ell = 1$, then \eqref{ETSZP} follows directly from \eqref{fellbound}. So suppose that $\ell \geq 2$, and write $f = P + R$, where $P$ is the Taylor polynomial of $f$ at $\0$ of order $\ell$. By \eqref{fellbound} and the mean value inequality we have
\begin{equation}
\label{Rbounds}
\|R\| \lesssim_\times \cdots \lesssim_\times \|R^{(\ell)}\| \leq \|R^{(\ell)}\|_{\CC^\epsilon} \lesssim_\times \kappa_\ell \|f\|,
\end{equation}
so by making $\kappa_\ell$ sufficiently small we can guarantee that $\|R\| \leq (1/2)\|f\|$ and thus $\|f\| \asymp_\times \|P\|$. But $\|P\|$ is asymptotic to the maximum of the coefficients of $P$, which implies that $\|P^{(2)}\| \lesssim_\times \|P\|$. On the other hand, $\|R^{(2)}\| \lesssim_\times \|f\|$ by \eqref{Rbounds}, so overall we have $\|f^{(2)}\| \lesssim_\times \|f\|$. Applying the mean value inequality yields \eqref{ETSZP}.
\end{subproof}
Thus if we let $\LL_j = \{\yy : f(\pp_j) + f'(\pp_j)[\yy - \pp_j] = 0\}$, then for all $\yy\in B_j\cap \Delta$ we have
\[
\dist(\yy,\LL_j) = \frac{|f(\pp_j) - f'(\pp_j)[\yy - \pp_j]|}{\|f'(\pp_j)\|} \lesssim_\times \frac{|f(\yy)| + \beta_1^{1 + \gamma} \|f\|}{\beta_1^{\gamma/2}\|f\|}\cdot
\]
So for $\yy\in \ZZ(f,\beta)\cap B_j\cap \Delta$, we have $|f(\yy)|\leq \beta\|f\| \leq \beta_1^{1 + \gamma} \|f\|$ and thus $\dist(\yy,\LL_j) \lesssim_\times \beta_1^{1 + \gamma/2}$. So if $\beta$ is small enough, then
\[
B_j\cap \Delta\cap \ZZ(f,\beta) \subset \thickvar{\LL_j}{\beta_1^{1 + \gamma/3}}.
\]
Taking the union over $j\in J = J_1$ gives
\[
\Delta\cap \ZZ(f,\beta)\butnot \ZZ(\del_i f,\beta^{\gamma/4}) \subset \bigcup_{j\in J_1} \thickvar{\LL_j}{\beta_1^{1 + \gamma/3}}\cap B_j = \bigcup(\CC_1).
\qedhere\]
\end{proof}
We are almost ready to prove Theorem \ref{theoremQDembedding}. First, we recall the definition of a nondegenerate embedding:
\begin{definition}
\label{definitionnondegenerate}
Let $U\subset\R^d$ be an open set, and let $\psi:U\to\R^D$ be a map of class $\CC^1$. Suppose that $\psi$ is a \emph{smooth embedding}, i.e. that $\psi$ is a homeomorphism onto its image and that for each $\xx\in U$, the linear transformation $\psi'(\xx)$ is injective. Given $\xx\in U$ and $\ell\in\N$, we say that $\psi$ is \emph{$\ell$-nondegenerate at $\xx$} if $\psi$ is of class $\CC^\ell$ in a neighborhood of $\xx$ and
\[
\R^D = \psi'(\xx)[\R^d] + \psi''(\xx)[\R^d\otimes\R^d] + \cdots + \psi^{(\ell)}(\xx)[(\R^d)^{\otimes \ell}].
\]
If $\psi$ is $\ell$-nondegenerate at every point of $U$ (resp. at almost every point of $U$), then we say that $\psi$ is $\ell$-nondegenerate (resp. $\ell$-weakly nondegenerate), or just nondegenerate (resp. weakly nondegenerate). The manifold $\psi(U)$ will also be called $\ell$-nondegenerate (resp. $\ell$-weakly nondegenerate).
\end{definition}
It is not hard to see that if $U$ is connected and $\psi$ is a real-analytic smooth embedding, then $\psi$ is weakly nondegenerate if and only if $\psi(U)$ is not contained in any affine hyperplane. Even in the setting of smooth maps, examples of connected smooth embeddings which are strongly degenerate (i.e. not weakly nondegenerate) but not contained in any affine hyperplane are somewhat pathological \cite{Wolsson}.
Theorem \ref{theoremQDembedding} now follows from Lemmas \ref{lemmaunifQD} and \ref{lemmaquasifederer} together with the following:
\begin{proposition}
\label{propositionQDembedding}
Let $\mu_1$ be a measure on an open set $U\subset\R^d$, let $E_1\subset U$, and suppose that $\mu_1$ is uniformly quasi-decaying relative to $E_1$. Fix $\ell\in\N$ and $\epsilon > 0$, and let $\psi:U\to\R^D$ be a smooth embedding which is $\ell$-nondegenerate at a point $\xx_1\in U$, and of class $\CC^{\ell + \epsilon}$ in a neighborhood of $\xx_1$. Then if $\mu_2 = \psi(\mu_1)$ is quasi-Federer at $\xx_2 = \psi(\xx_1)$, then $\mu_2$ is weakly quasi-decaying at $\xx_2$ relative to $E_2 = \psi(E_1)$.
\end{proposition}
\begin{proof}
Fix $\gamma > 0$, $0 < \rho\leq 1$, $0 < \beta \leq \rho^\gamma$, $\LL\in\scrH$, and $B_2 \df B(\xx_2,\rho)$. Since $\psi$ is a smooth embedding, for some constant $C_1 > 0$ we have $\psi(B_1)\supset \psi(U)\cap B_2$, where $B_1 \df B(\xx_1,C_1\rho)$. Let $\pi:\R^d\to\R$ be an affine map such that for all $\yy\in\R^d$, $\dist(\yy,\LL) = |\pi(\yy)|$. Then
\begin{align*}
\mu_2\big(\thickvar{\LL}{\beta\|\dist_\LL\|_{\mu_2,B_2}}\cap B_2\cap E_2\big)
&\leq \mu_1\big(\big\{\yy\in B_1\cap E_1 : |\pi\circ\psi(\yy)| \leq \beta \|\pi\circ\psi\|_{B_1}\big\}\big).
\end{align*}
Let $T(\zz) = \xx_1 + C_1\rho\zz$, so that $T(\Delta) = B_1$. Let $f = \pi\circ\psi\circ T$, so that
\[
\big\{\yy\in B_1\cap E_1 : |\pi\circ\psi(\yy)| \leq \beta \|\pi\circ\psi\|_{B_1}\big\}
= T\big(\big\{\zz\in\Delta : |f(\zz)| \leq \beta \|f\|\big\}\big)\cap E_1.
\]
Let $\bfP:\R^d\to\R^D$ be the Taylor approximation of $\psi$ at $\xx_1$ to order $\ell$, and let $P = \pi\circ\bfP\circ T$. Since $\psi$ is of class $\CC^{\ell + \epsilon}$ in a neighborhood of $\xx_1$, we have
\[
\|f - P\| \lesssim_\times \rho^{\ell + \epsilon}.
\]
On the other hand, since by hypothesis $\bfP(\R^d)$ is not contained in any affine hyperplane, a compactness argument shows that $\|\pi\circ\bfP\|\asymp_\times 1$, and thus
\[
\|P\| \gtrsim_\times \rho^\ell \|\pi\circ\bfP\| \asymp_\times \rho^\ell.
\]
Thus if $\rho$ is sufficiently small, then $\|f\| \asymp_\times \|P\|$. We also have
\[
\|f^{(\ell)}\|_{\CC^\epsilon}
= \rho^{\ell + \epsilon} \|(\pi\circ\psi)^{(\ell)}\|_{\CC^\epsilon,B_1} \leq \rho^{\ell + \epsilon} \|\psi^{(\ell)}\|_{\CC^\epsilon,B_1} \lesssim_\times \rho^{\ell + \epsilon},
\]
so if $\rho$ is sufficiently small then \eqref{fellbound} holds. Let the collections $\CC_1,\ldots,\CC_\ell$ be given by Lemma \ref{lemmaZPbeta}. For each $k = 1,\ldots,\ell$ let $\beta_k$, $(\pp_j)_{j\in J_k}$, and $(\LL_j)_{j\in J_k}$ be as in \eqref{Ck}. Then
\begin{equation}
\label{whereweleftoff}
\begin{split}
&\mu_2\big(\thickvar{\LL}{\beta\|\dist_\LL\|_{\mu_2,B_2}}\cap B_2\cap E_2\big)\\
&\leq \mu_1\left(\bigcup_{k = 1}^\ell \bigcup_{j\in J_k} T\big(\thickvar{\LL_j}{\beta_k^{1 + \epsilon/2}}\cap B(\pp_j,\beta_k)\big)\cap E_1\right)\\
&\leq \sum_{k = 1}^\ell \sum_{j\in J_k} \mu_1\big(\thickvar{T(\LL_j)}{C_1 \beta_k^{1 + \epsilon/2}\rho}\cap B(T(\pp_j),C_1\beta_k \rho)\cap E_1\big)\\
&= \sum_{k = 1}^\ell \sum_{j\in J_k} \mu_1\big(\thickvar{T(\LL_j)}{\beta_k^{\epsilon/2}\rho_k}\cap B(T(\pp_j),\rho_k/2)\cap E_1\big),
\end{split}
\end{equation}
where $\rho_k = 2C_1 \beta_k \rho$.
Fix $k = 1,\ldots,\ell$ and $j\in J_k$. We claim that for some $\alpha > 0$ depending only on $\gamma$,
\begin{equation}
\label{Txy}
\mu_1\big(\thickvar{T(\LL_j)}{\beta_k^{\epsilon/2}\rho_k}\cap B(T(\pp_j),\rho_k/2)\cap E_1\big) \lesssim_\times \beta^\alpha \mu_1\big(B(T(\pp_j),2\rho_k)\big).
\end{equation}
To avoid trivialities, assume that the set in the left hand side is nonempty, and let $\yy$ be a member of that set. Then $B(T(\pp_j),\rho_k/2)\subset B(\yy,\rho_k) \subset B(T(\pp_j),2\rho_k)$, so it is enough to show that
\[
\mu_1\big(\thickvar{T(\LL_j)}{\beta_k^{\epsilon/2}\rho_k}\cap B(\yy,\rho_k)\cap E_1\big) \lesssim_\times \beta^\alpha \mu_1\big(B(\yy,\rho_k)\big).
\]
Write $\delta_k \df 1/2^{2k - 1}$, so that $\beta_k \df \beta^{\delta_k}$. Then
\begin{align*}
\beta_k &\leq \rho^{\gamma\delta_k},&
\rho_k &\lesssim_\times \beta_k^{1 + 1/(\gamma\delta_k)},&
\beta_k^{\epsilon/2} &\lesssim_\times \rho_k^{\frac{\epsilon/2}{1 + 1/(\gamma\delta_k)}}.
\end{align*}
Let $\alpha_k = \alpha\big(\frac{\epsilon/2}{1 + 1/(\gamma\delta_k)},\mu_1\big)$ (cf. Lemma \ref{lemmaQD}). Then since $\yy\in E_1$, we have
\[
\mu_1\big(\thickvar{T(\LL_j)}{\beta_k^{\epsilon/2}\rho_k}\cap B(\yy,\rho_k)\cap E_1\big) \lesssim_\times \beta_k^{\alpha_k \epsilon/2} \mu_1\big(B(\yy,\rho_k)\big),
\]
and letting $\alpha \df \min_k \delta_k \alpha_k \epsilon/2 > 0$ completes the proof of \eqref{Txy}.
We finish the proof with the following calculation:
\begin{align*
&\mu_2\big(\thickvar{\LL}{\beta\|\dist_\LL\|_{\mu_2,B_2}}\cap B_2\cap E_2\big)\\
&\lesssim_\times \sum_{k = 1}^\ell \sum_{j\in J_k} \beta^\alpha \mu_1\big(B(T(\pp_j),2\rho_k)\big) \by{\eqref{whereweleftoff} and \eqref{Txy}}\\
&\asymp_\times \beta^\alpha \sum_{k = 1}^\ell \mu_1\big(B(\xx_1,2 C_1 \rho)\big) \note{bounded multiplicity}
\end{align*}
\begin{align*}
&\lesssim_\times \beta^{\alpha} \mu_2\big(B(\xx_2,2C_2 \rho)\big) \note{for some $C_2 > 0$}\\
&\lesssim_\times \beta^{\alpha/2} \mu_2\big(B(\xx_2,\rho)\big). \since{$\mu_2$ is quasi-Federer at $\xx_2$}
&\qedhere\end{align*}
\end{proof}
\draftnewpage
\section{More refined Diophantine properties of quasi-decaying measures}
\label{sectionKLW}
In this section, we fix ${M,N}\in\N$, and let $\MM = \MM_{M,N}$ denote the set of $M\times N$ matrices as in \6\ref{subsectionadditional}. We will usually identify $\MM$ with its image under the Pl\"ucker embedding $\psi:\MM\to \EE$ defined by \eqref{plucker}; however, we will sometimes distinguish between $\bfA\in\MM$ and $\psi(\bfA)\in\EE$ for clarity. Our main goal in this section is to prove Theorem \ref{theoremexponents}, using the techniques of \cite{KleinbockMargulis2, KLW, BKM, KMW}. We begin by introducing a uniform framework with which to talk about exponents of irrationality and their multiplicative versions. Our tool for doing this is the Dani--Kleinbock--Margulis correspondence principle between Diophantine approximation and the dynamics of homogeneous flows \cite{Dani3, KleinbockMargulis}.
\subsection{The correspondence principle}
\label{subsectioncorrespondence}
To start with, we introduce the notations
\begin{align*}
\Lambda_0 &\df \Z^{M + N}\\
u_\bfA &\df \left[\begin{array}{cc}
I_M & \bfA \\
& I_N
\end{array}\right] & (\bfA\in\MM)\\
g_\tt &\df \left[\begin{array}{ccc}
e^{t_1} &&\\
& \ddots &\\
&& e^{t_{M + N}}
\end{array}\right] & (\tt\in\mfa),
\end{align*}
where $\mfa \df \{\tt\in\R^{M + N} : \sum t_i = 0\}$. Next let
\begin{align*}
\mfa_+ &\df \left\{\tt\in \mfa :
\begin{array}{c}
\text{$t_i\leq 0$ for $i \leq M$}\\
\text{$t_i\geq 0$ for $i > M$}
\end{array}\right\}\\
\mfa_+^* &\df \left\{\tt\in \mfa_+ :
\begin{array}{c}
t_1 = \cdots = t_M\\
t_{M + 1} = \cdots = t_{M + N}
\end{array}
\right\}.
\end{align*}
Finally, given $\SS\subset\mfa_+$ and a function $s:\SS\to\Rplus$, we let
\[
\omega(\bfA;\SS,s) \df \limsup_{\SS\ni\tt\to\infty} \frac{1}{s(\tt)}\Delta\big(g_\tt u_\bfA \Lambda_0\big),
\]
where
\[
\Delta(\Lambda) \df -\log\min\big\{\|\rr\| : \rr\in\Lambda\butnot\{\0\}\big\}.
\]
We can now state the following special case of the Dani--Kleinbock--Margulis correspondence principle:
\begin{proposition}[Corollary of {\cite[Theorem 8.5]{KleinbockMargulis}}]
\label{propositiondynamicalinterpretation}
For all $\bfA\in\MM$,
\begin{align*}
\omega(\bfA) &= \xi\big(\omega(\bfA;\mfa_+^*,s_0)\big),
\end{align*}
where
\begin{align*}
s_0\left(-\frac tN,\cdots,-\frac tN,\frac tM,\cdots,\frac tM\right) &= t,&
\xi(c) &= \frac NM \frac{1 + M c}{1 - N c}\cdot
\end{align*}
\end{proposition}
It is harder to state a multiplicative version of Proposition \ref{propositiondynamicalinterpretation}. In this context it is worth mentioning \cite[Theorem 9.2]{KleinbockMargulis}, which at first sight appears to be such a multiplicative analogue. However, the version of multiplicative approximation considered in \cite{KleinbockMargulis} differs in several senses from the version of multiplicative matrix approximation considered in this paper:
\begin{itemize}
\item the results there are for lattices rather than matrices, and the concepts become trivial when restricted to the ``usual example'' of lattices in the form $u_\bfA \Lambda_0$, as these lattices all contain vectors which lie in a coordinate subspace and are therefore $\psi$-MA for every $\psi$ in the sense of \cite[\69.1]{KleinbockMargulis}.
\item the ``height'' of a vector in a lattice is considered to be the maximum of its coordinates, whereas in our setup the height of the vector $(\pp,\qq)$ is considered to be the number $\prod_{j = 1}^N |q_j|\vee 1$. This change (by itself) does not affect which matrices are considered to be VWMA, but it does affect the exponent of multiplicative irrationality for those matrices which are VWMA.
\end{itemize}
A multiplicative version of Proposition \ref{propositiondynamicalinterpretation} which is closer to our setup appeared in \cite[Proposition 3.1]{KMW}:
\begin{proposition}[Corollary of {\cite[Proposition 3.1]{KMW}}]
\label{propositiondynintmult}
A matrix $\bfA\in\MM$ is VWMA if and only if
\[
\omega(\bfA;\mfa_+,s) > 0,
\]
where $s:\mfa\to\Rplus$ is any norm.
\end{proposition}
This theorem does not contain any information relating the exponent of multiplicative irrationality function $\omega_\times$ with functions of the form $\bfA\mapsto\omega(\bfA;\SS,s)$. This appears to be difficult or impossible to do for technical reasons.\Footnote{The integer point $\rr_1 = ((0,0),1) \in \R^{2 + 1}$ should be counted as a good multiplicative approximation of the matrix $\bfA = (1/2,\epsilon)^T \in \MM_{2,1}$ (since $u_\bfA \rr_1 = ((1/2,\epsilon),1)$ has a small second coordinate), but the point $\rr_2 = ((1,0),0)$ should not (since $u_\bfA \rr_2 = ((1,0),0)$ is independent of $\bfA$). But any $\tt\in\mfa$ which shrinks $u_\bfA \rr_1$ to a small size also shrinks $u_\bfA \rr_2$ to a small size. In \cite{KMW} this problem was circumvented by finding another approximant which can be shrunk to small size using only $\tt\in\mfa_+$, but this approximant may not be of as good quality as $\rr_1$. In some sense the real problem might be that the function $\Delta$ appearing in the definitions of $\omega$ and $\omega_\times$ does not give enough information as to how far a lattice is into the cusp.\label{footnotetechnical}}
\subsection{Computing the exponent of irrationality of an affine subspace of $\EE$}
In view of Propositions \ref{propositiondynamicalinterpretation} and \ref{propositiondynintmult}, and after replacing $\SS$ by a discrete approximation, to prove Theorem \ref{theoremexponents} it suffices to demonstrate the following:
\begin{theorem}
\label{theoremexponents2}
Let $\mu$ be a measure on $\MM$ which is supported on an affine subspace $\AA\subset\EE$ and which is weakly quasi-decaying relative to $E \subset \MM\cap\AA$ when interpreted as a measure on $\AA$. Fix $\SS\subset\mfa$ and $s:\SS\to\Rplus$ such that for all $\tt\in\SS$, we have $s(\tt) \asymp_\times \|\tt\|$. Then for $\mu$-a.e. $\bfA\in\MM\cap\AA$,
\begin{equation}
\label{exponents2}
\omega(\bfA;\SS,s) = \inf\{\omega(\bfB;\SS,s) : \bfB\in\MM\cap \AA\}.
\end{equation}
\end{theorem}
We now begin the preliminaries to the proof of this theorem, which involve finding an alternate expression for the right hand side of \eqref{exponents2}.
\begin{notation*}
Let $\VV$ denote the collection of all rational subspaces of $\R^{M + N}$. Note that $(\VV,\subset)$ is a partially ordered set whose maximal chains are all of length $(M + N)$. We will call the elements of $\VV$ ``vertices'', to emphasize that we are thinking about $\VV$ as a combinatorial object, namely a partially ordered set under inclusion. For each $V\in\VV$, $\bfA\in\MM$, and $\tt\in\mfa$, let
\[
f_{\tt,V}(\bfA) \equiv f_\tt(\bfA,V) \df \Covol\big(g_\tt u_\bfA(\Lambda_0\cap V)\big),
\]
where $\Covol$ denotes the covolume of a discrete subgroup of $\R^{M + N}$ with respect to some fixed norm on $\R^{M + N}$, relative to the $\R$-linear span of that discrete subgroup.
\end{notation*}
We will think of the number $f_{\tt,V}(\bfA)$ as a sort of ``accuracy of approximation'' of the rational subspace $V\leq \R^{M + N}$, relative to the window $\tt$, in analogy to how the number $\|g_\tt u_\bfA \rr\|$ can be thought of as the ``accuracy of approximation'' of an integer vector $\rr\in \Z^{M+N}$. The important thing is that smaller values of $f_{\tt,V}(\bfA)$ mean that $\bfA$ is more well approximable and larger values mean that it is less well approximable. The connection between the values of $f_{\tt,V}(\bfA)$ for various $\tt,V$ and the approximability of $\bfA$ in the sense of the Dani--Kleinbock--Margulis correspondence principle will be made more clear in the proof of Lemma \ref{lemmaeasydirection} below.
\begin{lemma}
\label{lemmaFtV}
For each $V\in\VV$ and $\tt\in\mfa$, there exists an affine map $F_{\tt,V}:\EE\to \bigwedge^{\dim(V)} \R^{M + N}$ such that for all $\bfA\in \MM$,
\[
f_{\tt,V}(\bfA) = \|F_{\tt,V}(\psi_{{M,N}}(\bfA))\|,
\]
where $\|\cdot\|$ is the wedge power of the norm used to define covolume.
\end{lemma}
\begin{proof}
Let $\bb_1,\ldots,\bb_v$ be an integral basis of $V$. Then the parallelepiped $\sum_{i = 1}^v [0,1]\bb_i$ is a fundamental domain for $\Lambda_0\cap V$. It follows that for each $\bfA\in\MM$, the parallelepiped $\sum_{i = 1}^v [0,1]g_\tt u_\bfA \bb_i$ is a fundamental domain for $g_\tt u_\bfA(\Lambda_0\cap V)$. Thus the covolume of $g_\tt u_\bfA(\Lambda_0\cap V)$ is equal to the volume of the parallelepiped, i.e.
\[
f_{\tt,V}(\bfA) = \Vol\left(\sum_{i = 1}^v [0,1]g_\tt u_\bfA \bb_i\right) = \|g_\tt u_\bfA \bb_1\wedge\cdots\wedge g_\tt u_\bfA \bb_v\|
= \|g_\tt u_\bfA(\bb_1\wedge\cdots\wedge\bb_v)\|.
\]
So to complete the proof, it suffices to show that for all $\tau\in \bigwedge^v \R^{M + N}$, the map
\[
F_\tau(\psi_{{M,N}}(\bfA)) \df u_\bfA(\tau)\in \bigwedge^v \R^{M + N}
\]
can be extended affinely to all of $\EE$. But this follows from the following explicit formula for $F_\tau$:
\[
F_\tau(\psi_{{M,N}}(\bfA))
= \sum_{\substack{I,J\subset\{1,\ldots,M + N\} \\ \#(I) = \#(J) = k \\ I\cap \{1,\ldots,M\} \subset J}} \epsilon_{IJ} \tau_I \big[\psi_{{M,N}}(\bfA)\big]_{K(I,J)} \ee_J
\]
where we use the notations
\begin{align*}
\ee_I \df \bigwedge_{i\in I} &\ee_i, \;\;\;
\tau \df \sum_{\substack{I\subset \{1,\ldots,M + N\} \\ \#(I) = v}} \tau_I \ee_I,\;\;\;
\epsilon_{IJ}\in\{\pm 1\}, \;\;\; \big[\psi_{{M,N}}(\bfA)\big]_\smallemptyset \df 1,\\
K(I,J) &\df (\{1,\ldots,M\}\cap (J\butnot I))\cup (\{M + 1,\ldots,M + N\}\butnot (I\butnot J)).
\qedhere\end{align*}
\end{proof}
In the sequel we will extend $f_{\tt,V}$ to $\EE$ by letting $f_{\tt,V}(\sigma) = \|F_{\tt,V}(\sigma)\|$ for all $\sigma\in\EE$.
Given an affine subspace $\AA\subset\EE$, a set $\SS\subset\mfa$, and a function $s:\SS\to\Rplus$, let
\begin{equation}
\label{omegaLdef}
\omega(\AA;\SS,s) \df \limsup_{\SS\ni\tt\to\infty} \sup_{V\in\VV} \frac{-\log \|F_{\tt,V}\given \AA\|}{s(\tt) \dim(V)},
\end{equation}
where
\[
\|F\given\AA\| \df \|F(\zero_\AA)\| \vee \sup_{\substack{\sigma\in \AA \\ \|\sigma - \zero_\AA\| \leq 1}} \|F(\sigma) - F(\zero_\AA)\|.
\]
Here $\zero_\AA\in\AA$ is chosen so as to minimize $\|\zero_\AA\|$. We will show that $\omega(\AA;\SS,s)$ is equal to the right hand side of \eqref{exponents2}. One direction we can show now, and the other direction will follow from the proof of Theorem \ref{theoremexponents2}.
\begin{lemma}
\label{lemmaeasydirection}
With the above notation,
\[
\inf\big\{\omega(\bfA;\SS,s) : \bfA\in \MM\cap \AA\} \geq \omega(\AA;\SS,s).
\]
\end{lemma}
\begin{proof}
Fix $\bfA\in\MM\cap \AA$, $\tt\in\SS$, and $V\in\VV$. Then
\[
f_{\tt,V}(\bfA) \leq \|F_{\tt,V}\given \AA\|\cdot(1 + \|\psi_{{M,N}}(\bfA)\|).
\]
By Minkowski's theorem, there exists a vector $\vv\in \Lambda_0\cap V$ such that
\[
\|g_\tt u_\bfA \vv\| \leq 2f_{\tt,V}(\bfA)^{1/\dim(V)},
\]
so
\[
\Delta\big(g_\tt u_\bfA \Lambda_0\big) \geq \frac{-\log\big(2^{\dim(V)}(1 + \|\psi_{{M,N}}(\bfA)\|)\cdot \|F_{\tt,V}\given \AA\|\big)}{\dim(V)}\cdot
\]
Dividing by $s(\tt)$ and taking the limsup over $\SS\ni\tt\to\infty$ completes the proof.
\end{proof}
\subsection{Proof of Theorem \ref{theoremexponents2}}
By Lemma \ref{lemmaeasydirection}, to prove Theorem \ref{theoremexponents2} it suffices to show that for $\mu$-a.e. $\bfA\in\MM\cap \AA$, we have $\omega(\bfA;\SS,s) \leq \omega(\AA;\SS,s)$. We now state a lemma which will allow us to prove this:
\begin{lemma}
\label{lemmaexponents}
Let $\mu$ be a measure on $\MM$ which is supported on an affine subspace $\AA\subset\EE$ and which is uniformly weakly quasi-decaying and uniformly quasi-Federer relative to $E\subset \MM\cap\AA$ when interpreted as a measure on $\AA$. Let $X = \Supp(\mu) \subset \MM\cap\AA$. Fix $\gamma > 0$ and a ball $B_0 = B_X(\bfA_0,\rho_0)$. Consider $\tt\in\mfa$ and $0 < \kappa \leq 1$ such that
\begin{align}
\label{KLWassumption}
\sup_{2B_0} f_{\tt,V} \geq \kappa^{\dim(V)}
\end{align}
for all $V\in\VV$, and let
\[
W_{\kappa,\tt} \df \{\bfA\in X : \exists \vv\in\Lambda_0\butnot\{\0\} \;\; \|g_\tt u_\bfA \vv\| \leq e^{-\gamma\|\tt\|} \kappa\}.
\]
Then there exists $\epsilon > 0$ (depending on $\mu,E,\gamma$ but not $\kappa,\tt$) such that
\[
\mu(W_{\kappa,\tt}\cap B_0\cap E) \lesssim_\times e^{-\epsilon\|\tt\|}.
\]
\end{lemma}
In this lemma and its proof, we understand the metric on $X$ to be the one inherited from the vector space $\EE$, not the one inherited from the vector space $\MM$.
\begin{proof}[Proof of Theorem \ref{theoremexponents2} assuming Lemma \ref{lemmaexponents}]
Let $E\subset \MM\cap\AA$, $X = \Supp(\mu)$, $\gamma > 0$, and $B_0 \subset B_X(\bfA_0,\rho_0)$ be as in Lemma \ref{lemmaexponents}, with the additional constraint that $E\cap 2B_0\neq \emptyset$. Let $\tau = \omega(\AA;\SS,s) + \gamma$, and for each $\tt\in\SS$, let $\kappa_\tt = e^{-\tau s(\tt)}$. Fix $\bfA\in E\cap 2B_0$. Since $\mu$ is weakly quasi-decaying at $\bfA$ relative to $E$, it follows that $2B_0 = B_\AA(\bfA_0,2\rho_0)\cap \Supp(\mu)$ cannot be contained in an affine hyperplane of $\AA$, so the affine span of $2B_0$ is equal to $\AA$. Thus for all $V\in\VV$ we have
\[
\sup_{2B_0} f_{\tt,V} \asymp_\times \|F_{\tt,V}\given \AA\|
\]
and so by \eqref{omegaLdef}, if $\tt\in\SS$ is sufficiently large then
\[
\sup_{2B_0} f_{\tt,V} \geq e^{-\tau s(\tt) \dim(V)} = \kappa_\tt^{\dim(V)}.
\]
So by Lemma \ref{lemmaexponents}, if $\mu$ is uniformly weakly quasi-decaying relative to a set $E\subset X$ when interpreted as a measure on $\AA$, then
\[
\mu(W_{\kappa_\tt,\tt}\cap B_0\cap E) \lesssim_\times e^{-\epsilon\|\tt\|}.
\]
Thus the Borel--Cantelli lemma implies that for $\mu$-a.e. $\bfA\in B_0\cap E$ we have
\begin{equation}
\label{borelcantelli}
\#\{\tt\in\SS : \bfA\in W_{\kappa_\tt,\tt}\} < \infty.
\end{equation}
But if $\bfA$ satisfies \eqref{borelcantelli}, then
\begin{align*}
\omega(\bfA;\SS,s) &\leq \limsup_{\SS\ni\tt\to\infty} \frac{-\log(e^{-\gamma\|\tt\|} \kappa_\tt)}{s(\tt)}
= \tau + \gamma\limsup_{\SS\ni\tt\to\infty} \frac{\|\tt\|}{s(\tt)} \noreason\\
&= \omega(\AA;\SS,s) + (1 + C_1) \gamma. \note{for some $C_1 > 0$}
\end{align*}
Since $\gamma$ and $B_0$ were arbitrary, we have $\omega(\bfA;\SS,s)\leq \omega(\AA;\SS,s)$ for $\mu$-a.e. $\bfA\in E$. Combining with Lemmas \ref{lemmaunifQD}, \ref{lemmaquasifederer}, and \ref{lemmaeasydirection} completes the proof.
\end{proof}
Now we need to prove Lemma \ref{lemmaexponents}. The idea, following \cite{KleinbockMargulis,KLW}, is to construct a cover of the set $W_{\kappa,\tt}\cap B_0$ whose measure can be bounded using fact that $\mu$ is uniformly weakly quasi-decaying relative to $E$. To construct this cover, we will first construct a tree $\TT$ such that each node $e\in\TT$ corresponds to a ball in $B_e\subset \AA$. We will also associate to $e$ a \emph{flag}, i.e. a set $\FF_e \equiv \{V_0,\ldots,V_\ell\} \subset \VV$ such that $\{\0\} = V_0 \propersubset V_1 \propersubset \cdots \propersubset V_\ell = \R^{M + N}$. The purpose of the flag $\FF_e$ is to separate potential approximants to points in $B_e$ into $\ell$ different classes: an approximant $\rr\in\Lambda_0\butnot\{\0\}$ is in exactly one of the sets $V_1\butnot V_0,\ldots,V_\ell\butnot V_{\ell - 1}$. In order for this separation to be useful, the flag $\FF_e$ should satisfy the following conditions:
\begin{itemize}
\item[(1)] The quality-of-approximation ratios $f_\tt(B_e,V_{i + 1})/f_\tt(B_e,V_i)$ should not be too large (in terms of $\omega(\AA;\SS,s)$), where
\begin{equation}
\label{ftSV}
f_\tt(S,V) \df \sup_S f_{\tt,V}.
\end{equation}
\label{page21}
\item[(2)] For each vertex $V\in\VV$ such that $V_i\subsetneqq V\subsetneqq V_{i + 1}$ for some $i$, the quality of approximation $f_\tt(B_e,V)$ should be bounded from below in terms of $f_\tt(B_e,V_i)$ and $f_\tt(B_e,V_{i + 1})$.
\end{itemize}
The idea of the tree is to give us a picture of what happens as we ``zoom in'' towards a point $\bfA\in X$. On the large scale, we will be able to find a flag $\FF$ which satisfies (1) and (2) which depends only on the Diophantine properties of the affine space $\AA$. As we zoom in, all vertices become better approximations (because the supremum in \eqref{ftSV} is taken over a smaller collection). If this causes a vertex to become a counterexample to (2), then we simply add it to our flag and create a new node on the tree. On the other hand, the probability that a vertex will become a counterexample to (1) (assuming that when we added the vertex to the flag, it satisfied (1)) is small because of the quasi-decay condition. So if $\bfA$ is a typical point, then after we finish the zooming process the flag will still satisfy (1). Lemma \ref{lemmaexistsV} below shows that in this case, $\bfA$ cannot be in $W_{\kappa,\tt}$.
We will encode the Diophantine properties of the root flag $\FF_\smallemptyset$ by defining a function $\eta: \{0,\ldots,M + N\}\to (0,\infty)$ such that for each $j$, $\eta(j)$ represents the quality of the ``best expected approximation'' in dimension $j$. As we zoom in, we will add the vertex $V$ to our flag at the exact moment when the quality of approximation of $V$ becomes better than $\eta(\dim(V))$. For a typical point, this strategy should create a final flag which satisfies condition 1.
\begin{figure}[h!]
\begin{center}
\begin{tabular}{@{}ll|@{}}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,scale=0.6]
\clip(-1.0744866397712483,-1.0652581632465994) rectangle (12.52932974604722,6.807415187466945);
\draw[->] (1.2,2.0)-- (11.4,2.0);
\draw (1.6296536816006593,1.9962393331765842)-- (2.69,0.42);
\draw (4.09,-0.39)-- (2.69,0.42);
\draw (4.09,-0.39)-- (6.89,0.16);
\draw (6.89,0.16)-- (9.694690100098676,1.9776280691815895);
\draw[->] (1.62,1.2)-- (1.62,5.950803061223502);
\draw[thick, dotted] (8.29,2.55)-- (8.29,6.54);
\draw[thick, dotted] (5.49,1.11)-- (5.49,6.11);
\begin{scriptsize}
\draw[color=black] (11.4,1.62) node {dimension};
\draw[color=black] (0.6,5.4) node {height};
\draw[color=black] (0.6,5) node {(log scale)};
\draw[color=black] (1.0,2.0) node {$0$};
\draw [fill=black] (1.62,1.99) circle (1.5pt);
\draw [fill=black] (2.69,0.42) circle (1.5pt);
\draw [fill=black] (4.09,-0.39) circle (1.5pt);
\draw [fill=black] (6.89,0.16) circle (1.5pt);
\draw [fill=black] (9.69,2.0) circle (1.5pt);
\draw [fill=black] (5.49,1.11) circle (1.5pt);
\draw [fill=black] (8.29,2.54) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}
\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,scale=0.6]
\clip(-1.0744866397712483,-1.0652581632465994) rectangle (12.52932974604722,6.807415187466945);
\draw[->] (1.2,2.0)-- (11.4,2.0);
\draw (1.6296536816006593,1.9962393331765842)-- (2.69,0.42);
\draw (4.09,-0.39)-- (2.69,0.42);
\draw (4.09,-0.39)-- (6.89,0.16);
\draw (6.89,0.16)-- (9.694690100098676,1.9776280691815895);
\draw[->] (1.62,1.2)-- (1.62,5.950803061223502);
\draw[thick, dotted] (8.29,1.55)-- (8.29,5.54);
\begin{scriptsize}
\draw[color=black] (11.4,1.62) node {dimension};
\draw[color=black] (0.6,5.4) node {height};
\draw[color=black] (0.6,5) node {(log scale)};
\draw[color=black] (1.0,2.0) node {$0$};
\draw [fill=black] (1.62,1.99) circle (1.5pt);
\draw [fill=black] (2.69,0.42) circle (1.5pt);
\draw [fill=black] (4.09,-0.39) circle (1.5pt);
\draw [fill=black] (6.89,0.16) circle (1.5pt);
\draw [fill=black] (9.69,2.0) circle (1.5pt);
\draw [fill=black] (5.49,-0.11) circle (1.5pt);
\draw [fill=black] (8.29,1.55) circle (1.5pt);
\end{scriptsize}
\end{tikzpicture}
\end{tabular}
\caption{Two possible plots of the set\[\hspace{-1 in}\{(\dim(V),\log f_\tt(B(\bfA,\rho),V)) : \text{$V\in\VV$ is $\FF$-addable or satisfies $V\in\FF$}\},\]
along with a graph of the piecewise linear function $\log\eta$. The displayed points represent the minimum plot points over each vertical strip, and the vertical ellipses represent additional ungraphed plot points. The two plots are taken with the same value of $\tt$ and $\bfA$ but different values for $\rho$. As $\rho$ decreases, all plot points will move down, but the probability that any given plot point jumps down a significant amount is small (under the assumption that $\bfA$ is $\mu$-random). Once a plot point ``crosses'' the graph of $\log\eta$, then its corresponding vertex is added to the flag, at which point it is unlikely to move down further. This explains why in the typical case the final plot is essentially the same as the graph of $\log\eta$ on the integers.}
\label{figureflag}
\end{center}
\end{figure}
\begin{definition*}
If $\FF\subset\VV$ is a flag, then the number $\ell \equiv \ell(\FF) \df \#(\FF) - 1$
is called the \emph{length} of the flag. A vertex $V\in\VV\butnot\FF$ is \emph{$\FF$-addable} if $\FF\cup\{V\}$ is a flag. A flag $\FF$ is called \emph{maximal} if $\ell(\FF) = M + N$, or equivalently if there is no $\FF$-addable vertex.
\end{definition*}
\begin{definition*}
Given $\eta:\{0,\ldots,M + N\}\to (0,\infty)$ and a vertex $V\in\VV$, a set $S\subset\EE$ is said to be \emph{$(\eta,V)$-approximable} if
\[
f_\tt(S,V) \leq \eta(\dim(V)).
\]
If $S\subset\EE$ is fixed, then the collection of vertices $V\in\VV$ such that $S$ is $(\eta,V)$-approximable will be denoted $\WW(\eta,S)$, and its complement will be denoted $\BB(\eta,S)$.
\end{definition*}
\begin{lemma}[Cf. {\cite[Proposition 5.1]{KLW}}]
\label{lemmaexistsV}
Let $\FF\subset \WW(\eta,S)$ be a maximal flag, and fix $\bfA\in W_{\kappa,\tt}\cap S$. Then there exists $V\in\FF\butnot\{\0\}$ such that
\[
f_{\tt,V}(\bfA) \leq e^{-\gamma\|\tt\|} \kappa \eta(\dim(V) - 1).
\]
\end{lemma}
\begin{proof}
Since $\bfA\in W_{\kappa,\tt}$, we have $\|g_\tt u_\bfA \vv\| \leq e^{-\gamma\|\tt\|} \kappa$ for some $\vv\in\Lambda_0$. Write $\FF = \{V_0,\ldots,V_{M + N}\}$ with $\{\0\} = V_0 \propersubset V_1 \propersubset \cdots \propersubset V_{M + N} = \R^{M + N}$. Let $i$ be the largest element of $\{0,\ldots,M + N\}$ such that $\vv\notin V_i$. Then $V_{i + 1} = V_i + \R\vv$. An argument based on the geometric significance of $f_\tt$ shows that
\[
f_\tt(\bfA,V_{i + 1}) \leq \|g_\tt u_\bfA \vv\| f_\tt(\bfA,V_i).
\]
On the other hand, since $\FF\subset \WW(\eta,S)$ and $\bfA\in S$ we have
\[
f_\tt(\bfA,V_i) \leq \eta(\dim(V_i)) = \eta(\dim(V_{i + 1}) - 1)
\]
and by the definition of $\vv$,
\[
\|g_\tt u_\bfA \vv\| \leq e^{-\gamma\|\tt\|} \kappa.
\]
Combining these inequalities completes the proof.
\end{proof}
\begin{definition*}
Fix $\lambda\geq 2$, a flag $\FF\subset\VV$, and a function $\eta:\{0,\ldots,M + N\}\to (0,\infty)$. A ball $B = B_X(\bfA,\rho)$ is said to be \emph{$(\FF,\eta,\lambda)$-permissible} if $\FF\subset \WW(2\eta,2B)$, but every $\FF$-addable vertex is in $\BB(\eta,\lambda B)$.
\end{definition*}
\begin{definition*}
Fix a flag $\FF\subset\VV$ and a function $\eta:\{0,\ldots,M + N\}\to (0,\infty)$. We say that $\eta$ is \emph{$\FF$-concave} if for all $j\notin \{\dim(V):V\in\FF\}$,
\[
\eta(j) \geq 8\sqrt{\eta(j - 1)\eta(j + 1)}.
\]
\end{definition*}
The purpose of concavity is to ensure that if $B$ is $(\FF,\eta,\lambda)$-permissible, then the flag $\FF$ will satisfy condition (2) on p.\pageref{page21}. The factor of $8$ will be important in the proof of \eqref{rx3t} below.
\begin{remark}
\label{remarketa}
If $\eta$ is $\FF$-concave and $\FF = \{V_0,\ldots,V_\ell\}$ with $V_0\propersubset \cdots \propersubset V_\ell$, then for each $i = 0,\ldots,\ell - 1$, if $j = \dim(V_i)$ and $m = \dim(V_{i + 1}) - \dim(V_i)$, then for each $0\leq k \leq m$ we have
\[
\eta(j + k) \geq 8^{k(m - k)} \eta(j + m)^{k/m} \eta(j)^{(m - k)/m}.
\]
\end{remark}
\begin{notation*}
For each $i = 0,\ldots,M + N$ let
\[
C_i \df 4^{i(M + N - i)}.
\]
Note that $C_0 = C_{M + N} = 1$, and $C_i = 4\sqrt{C_{i - 1} C_{i + 1}}$ for all $0 < i < M + N$.
\end{notation*}
\begin{lemma}[Base case]
\label{lemmabasecase}
Fix a ball $B_0 = B_X(\bfA_0,\rho_0)$. Then there exist a flag $\FF_0\subset\VV$ and an $\FF_0$-concave function $\eta:\{0,\ldots,M + N\}\to (0,\infty)$ such that:
\begin{itemize}
\item[(i)] $B_0$ is $(\FF_0,\eta,2)$-permissible for every $V\in\FF_0$,
\item[(ii)] $\FF_0\subset \BB(\eta,2B_0)$,
\item[(iii)] $\eta(j) \leq C_j/2 \all j$, and
\item[(iv)] $\eta(j + 1)/\eta(j) \gtrsim_\times \kappa \all j$.
\end{itemize}
\end{lemma}
\begin{proof}
For each $V\in\VV$ let
\begin{align*}
f(V) &\df f_\tt(2B_0,V),&
g(V) &\df \frac{\log(f(V)/C_{\dim(V)})}{\dim(V)},
\end{align*}
with the convention that $g(\{\0\}) = -\infty$. Note that by \eqref{KLWassumption}, $g(V)\gtrsim_\plus \log(\kappa)$ for all $V\propersupset \{\0\}$. Let $V_0 = \{\0\}$, and recursively define $V_1,\ldots,V_\ell$ by letting $V_{i + 1}\propersupset V_i$ satisfy
\[
g(V_{i + 1}) = \min\{g(V): V\propersupset V_i\}.
\]
This process halts when $V_\ell = \R^{M + N}$. Note that
\[
g(V_0) \leq g(V_1) \leq \cdots \leq g(V_\ell) = 0.
\]
Let $\FF_0 = \{V_0,\ldots,V_\ell\}$, and for each $i = 0,\ldots,\ell$ let $j_i = \dim(V_i)$ and
\begin{equation}
\label{etadef}
\eta(j_i) = f(V_i)/2.
\end{equation}
Extend $\eta$ to a map $\eta:\{0,\ldots,M + N\}\to(0,\infty)$ which is minimal subject to being $\FF_0$-concave. Equivalently, this extension can be described by the requirement that for each $i = 0,\ldots,\ell - 1$, the function
\[
\theta(j) = \log(2\eta(j)/C_j)
\]
is linear on $\{j_i,\ldots,j_{i + 1}\}$.
For all $i$, since $g(V_i)\leq 0$, we have $\theta(j_i)= j_i g(V_i) \leq 0$. So by linearity, we have $\theta(j)\leq 0$ for all $j$, i.e. $\eta(j)\leq C_j/2$. This demonstrates (iii).
By \eqref{etadef}, we have $\FF_0\subset\BB(\eta,2B_0) \cap \WW(2\eta,2B_0)$. This demonstrates (ii) and the first part of (i). To demonstrate the second part of (i), suppose $V$ is an $\FF$-addable vertex, and write $V_i\propersubset V\propersubset V_{i + 1}$ for some $i = 0,\ldots,\ell - 1$. By the definitions of $V_i$ and $V_{i + 1}$, we have
\[
g(V_i) \leq g(V_{i + 1}) \leq g(V)
\]
and thus
\begin{align*}
\theta(j_i) &\leq j_i g(V),&
\theta(j_{i + 1}) &\leq j_{i + 1} g(V).
\end{align*}
Since $\theta$ is linear on $\{j_i,\ldots,j_{i + 1}\}$, this implies that
\[
\theta(j) \leq j g(V) = \log(f(V)/C_j).
\]
Rearranging gives $\eta(j) \leq f(V)/2$, so $V\in\BB(\eta,2B_0)$ and thus $B_0$ is $(\FF_0,\eta,2)$-permissible.
Finally, to demonstrate (iv), we note that since
\[
\frac{\theta(j_i)}{j_i} = g(V_i) \leq g(V_{i + 1}) = \frac{\theta(j_{i + 1})}{j_{i + 1}},
\]
we have
\[
\frac{\theta(j_{i + 1}) - \theta(j_i)}{j_{i + 1} - j_i} \geq \frac{\theta(j_{i + 1}) - j_i\frac{\theta(j_{i + 1})}{j_{i + 1}}}{j_{i + 1} - j_i} = \frac{\theta(j_{i + 1})}{j_{i + 1}} = g(V_{i + 1}) \gtrsim_\plus \log(\kappa).
\]
By the piecewise linearity of $\theta$, we have $\theta(j + 1) - \theta(j) \gtrsim_\plus \log(\kappa)$ for all $j$, and writing this inequality in terms of $\eta$ yields (iv).
\end{proof}
\begin{lemma}[Inductive step]
\label{lemmainductivestep}
Fix $\lambda \geq 2$, a non-maximal flag $\FF\subset\VV$, an $\FF$-concave function $\eta:\{0,\ldots,M + N\}\to (0,\infty)$, and an $(\FF,\eta,\lambda)$-permissible ball $B = B_X(\bfA_0,\rho_0) \subset \AA$. Then for each $\bfA\in B$, there exists an $\FF$-addable vertex $V_\bfA$ and an $(\FF\cup\{V_\bfA\},\eta,8\lambda)$-permissible ball $B_\bfA = B_X(\bfA,\rho_\bfA)$ such that
\begin{align} \label{Vpermissible}
V_\bfA &\in\BB(\eta, 8\lambda B_\bfA) \\ \label{containment}
2B_\bfA &\subset 2B \\ \label{rx3t}
8\lambda \rho_\bfA &\geq 2^{-(M + N)} e^{-2\|\tt\|_\infty}.
\end{align}
\end{lemma}
\begin{remark*}
The condition \eqref{rx3t} is the key ``new'' element of the proof of Theorem \ref{theoremexponents2} which has no analogue in \cite{KleinbockMargulis2, KLW, BKM, KMW}; it will allow us to prove the bound $\beta \leq \rho^\gamma$ for the hyperplane-neighborhoods whose $\mu$-measures we want to bound, thus allowing the weak quasi-decay condition to be used as a substitute for friendliness.
\end{remark*}
\begin{proof}
For each $\FF$-addable vertex $V$ let $\rho_{\bfA,V}$ be the smallest value $\rho\in 2^\Z$ such that
\begin{equation}
\label{rhoAVdef}
V\in \BB(\eta,B_X(\bfA,8\lambda \rho)),
\end{equation}
with $\rho_{\bfA,V} = 0$ if \eqref{rhoAVdef} holds for all $\rho\in 2^\Z$. Let
\[
\rho_\bfA \df \max\{\rho_{\bfA,V}:\text{$V\in\VV$ is $\FF$-addable}\},
\]
let $V_\bfA$ be an $\FF$-addable vertex such that $\rho_\bfA = \rho_{\bfA,V_\bfA}$, and let $B_\bfA = B_X(\bfA,\rho_\bfA)$. Let $\FF_\bfA = \FF\cup\{V_\bfA\}$. By construction, the set $\BB(\eta,8\lambda B_\bfA)$ contains every $\FF$-addable vertex. In particular, \eqref{Vpermissible} holds.
On the other hand, the $(\FF,\eta,\lambda)$-permissibility of $B$ and the minimality of $\rho_{\bfA,V_\bfA}$ together imply that
\[
f_\tt\big(\lambda B,V_\bfA\big) > \eta(\dim(V_\bfA)) \geq f_\tt\big(4\lambda B_\bfA,V_\bfA\big);
\]
it follows that
\[
B_X(\bfA_0,\lambda \rho_0) \nsubset B_X(\bfA,4\lambda \rho_\bfA).
\]
Since $\dist_X(\bfA_0,\bfA) \leq \rho_0$, this implies
\[
4\lambda \rho_\bfA < (\lambda + 1) \rho_0 < 2\lambda \rho_0,
\]
so $\rho_\bfA < \rho_0/2$ and thus \eqref{containment} holds. In particular $\FF\subset \WW(\eta,2B_\bfA)$. On the other hand, $V_\bfA\in \WW(\eta,2B_\bfA)$ since $\rho_\bfA = \rho_{\bfA,V_\bfA}$. Moreover, as noted in the previous paragraph the set $\BB(\eta,8\lambda B_\bfA)$ contains every $\FF$-addable vertex and in particular every $\FF_\bfA$-addable vertex. Thus $B_\bfA$ is $(\FF_\bfA,\eta,8\lambda)$-permissible.
To demonstrate \eqref{rx3t}, we will find an $\FF$-addable vertex $V$ such that $4\lambda \rho_{\bfA,V} \geq 2^{-(M + N)} e^{-2\|\tt\|_\infty}$. Write $\FF = \{V_0,\ldots,V_\ell\}$ with $\{\0\} = V_0 \propersubset V_1 \propersubset \cdots \propersubset V_\ell = \R^{M + N}$. Since $\FF$ is not maximal, we have $m \df \dim(V_{i + 1}) - \dim(V_i) \geq 2$ for some $i$. Let $W_i \df g_\tt u_\bfA(V_i)$, $W_{i + 1} \df g_\tt u_\bfA(V_{i + 1})$, and $\Lambda \df g_\tt u_\bfA \Lambda_0$. Applying Minkowski's theorem to the vector space $W_{i + 1}/W_i$ with the lattice $(\Lambda\cap W_{i + 1})/W_i$, we see that there exists a vector $\ww = g_\tt u_\bfA \vv \in \Lambda\cap W_{i + 1}\butnot W_i$ such that
\[
\dist(\ww,W_i) \leq 2\Covol\big((\Lambda\cap W_{i + 1})/W_i\big)^{1/m}.
\]
Let $V \df V_i + \R\vv$ and $W \df g_\tt u_\bfA(V) = W_i + \R\ww$. Note that $V$ is an $\FF$-addable vertex. We have
\[
\Covol(\Lambda\cap W) = \dist(\ww,W_i)\Covol(\Lambda\cap W_i)
\]
and
\[
\Covol\big((\Lambda\cap W_{i + 1})/W_i\big)
= \frac{\Covol(\Lambda\cap W_{i + 1})}{\Covol(\Lambda\cap W_i)}\cdot
\]
It follows that
\begin{align*}
f_{\tt,V}(\bfA) &= \Covol(\Lambda\cap W)\\
&\leq 2\Covol(\Lambda\cap W_{i + 1})^{1/m}\Covol(\Lambda\cap W_i)^{(m - 1)/m}\\
&= 2f_\tt(\bfA,V_{i + 1})^{1/m} f_\tt(\bfA,V_i)^{(m - 1)/m}.
\end{align*}
Let $j = \dim(V_i)$. Since $\bfA$ is $(\FF,\eta)$-permissible, we have
\begin{equation}
\label{xVbound}
f_{\tt,V}(\bfA) \leq 4\eta(j + m)^{1/m} \eta(j)^{(m - 1)/m} \leq \frac{1}{2}\eta(j + 1)
\end{equation}
where the last inequality follows from Remark \ref{remarketa}. Let $\epsilon = 2^{-(M + N)}$, and note that $(1 + \epsilon)^{M + N} \leq 2$. For all $\bfB\in B_X(\bfA,\epsilon e^{-2\|\tt\|_\infty})$,
\begin{align*}
f_\tt(\bfB,V) &\leq \|g_\tt u_\bfB (g_\tt u_\bfA)^{-1}\|^{\dim(V)} f_{\tt,V}(\bfA)\\
&= \big\|u\big(\diag(e^{t_1},\ldots,e^{t_M})(\bfB - \bfA)\diag(e^{-t_{M + 1}},\ldots,e^{-t_{M + N}})\big)\big\|^{j + 1} f_{\tt,V}(\bfA)\\
&\leq (1 + e^{2\|\tt\|_\infty}\|\bfB - \bfA\|)^{j + 1} \frac12\eta(j + 1)\\
&\leq (1 + \epsilon)^{M + N} \frac12\eta(j + 1) = \eta(j + 1) = \eta(\dim(V)).
\end{align*}
Thus by definition, $8\lambda \rho_{\bfA,V} \geq \epsilon e^{-2\|\tt\|_\infty}$.
\end{proof}
For each $i = 0,\ldots,M + N$ write $\lambda_i = 2\cdot 8^i$. Let $B_0,\FF_0,\eta$ be as in Lemma \ref{lemmabasecase}, and let
\[
\PP \df \{(B,\FF) : \text{$B$ is $(\FF,\eta,\lambda_{\ell(\FF)})$-permissible, $B\subset B_0$, and $\FF\supset\FF_0$}\},
\]
so that $(B_0,\FF_0)\in \PP$. We will now construct a tree in $\PP$ with $(B_0,\FF_0)$ as the root node.
{\bf Construction of children.} Fix $(B,\FF)\in\PP$, and let $\lambda = \lambda_{\ell(\FF)}$. Since $\FF\supset\FF_0$, $\eta$ is $\FF$-concave, so Lemma \ref{lemmainductivestep} applies. For each $\bfA\in B\cap E$ let $\rho_\bfA > 0$ be as in Lemma \ref{lemmainductivestep}, so that $\{B_\bfA = B_X(\bfA,\rho_\bfA) : \bfA\in B\cap E\}$ is a cover of $B\cap E$. By the $4r$-covering lemma (see e.g. \cite[Theorem 8.1]{MSU}), there exists a finite set $(\bfA_i)_{i = 1}^n$ such that the collection $\{B_i = B_{\bfA_i} : i = 1,\ldots,n\}$ still covers $B\cap E$, but the collection $\{(1/4)B_i : i = 1,\ldots,n\}$ is disjoint. For each $i$, let $\FF_i = \FF\cup\{V_{\bfA_i}\}$, so that $(B_i,\FF_i)\in \PP$. Let
\[
\CC(B,\FF) \df \{(B_i,\FF_i) : i = 1,\ldots,n\} \subset \PP,
\]
and note that
\begin{equation}
\label{children}
B\subset \bigcup\{B_i : (B_i,\FF_i)\in \CC(B,\FF)\}.
\end{equation}
{\bf Covering argument.}
Let $\ell_0 = \ell(\FF_0)$, let $\TT_{\ell_0} \df \{(B_0,\FF_0)\}$, and for each $i = \ell_0 + 1,\ldots,M + N$ let
\[
\TT_i ~\df \bigcup_{(B,\FF)\in \TT_{i - 1}} \CC(B,\FF).
\]
Fix $\bfA\in B_0\cap E$. By \eqref{children}, can recursively define a sequence $(B_i,\FF_i)_{i = \ell_0}^{M + N}$ such that for each $i = \ell_0,\ldots,M + N$, we have $(B_i,\FF_i)\in\TT_i$, $\bfA\in B_i$, and if $i > \ell_0$, then
\[
(B_i,\FF_i) \in \CC(B_{i - 1},\FF_{i - 1}).
\]
Write $\FF_i = \FF_{i - 1}\cup\{V_i\}$, so that by Lemma \ref{lemmainductivestep}, $V_i\in \BB(\eta,\lambda_i B_i)$. Also write $\FF_0 = \FF_{\ell_0} = \{V_0,\ldots,V_{\ell_0}\}$, so that by Lemma \ref{lemmabasecase}, $V_i \in \BB(\eta,2 B_0) \subset \BB(\eta,\lambda_{\ell_0} B_{\ell_0})$ for all $i = 0,\ldots,\ell_0$.
If $\bfA\in W_{\kappa,\tt}$, then by Lemma \ref{lemmaexistsV} there exists $i = 0,\ldots,M + N$ such that
\[
f_\tt(\bfA,V_i) \leq e^{-\gamma\|\tt\|} \kappa \eta(\dim(V_i) - 1).
\]
Combining with part (iv) of Lemma \ref{lemmabasecase} gives
\[
f_\tt(\bfA,V_i) \lesssim_\times e^{-\gamma\|\tt\|} \eta(\dim(V_i)).
\]
To summarize,
\[
W_{\kappa,\tt}\cap B_0 \cap E \subset \bigcup_{i = \ell_0}^{M + N} \bigcup_{(B,\FF)\in\TT_i} \bigcup_{\substack{V\in\FF \\ V\in \BB(\eta,\lambda_i B_i)}} (W_{\kappa,\tt}(V)\cap B),
\]
where
\[
W_{\kappa,\tt}(V) \df \{\bfA\in B_0 : f_{\tt,V}(\bfA) \leq C e^{-\gamma\|\tt\|} \eta(\dim(V))\}
\]
for some $C > 0$.
\begin{claim}
Fix a ball $B \subset X$ and $V\in\BB(\eta,\lambda_{M + N} B)$. Then
\[
\mu(W_{\kappa,\tt}(V)\cap B\cap E) \lesssim_\times e^{-\alpha\|\tt\|} \mu(\lambda_{M + N} B)
\]
for some $\alpha > 0$ depending only on $\mu,E,\gamma$.
\end{claim}
\begin{subproof}
Since $V\in\BB(\eta,\lambda_{M + N} B)$, there exists $\bfA\in\lambda_{M + N} B$ such that $f_{\tt,V}(\bfA) > \eta(\dim(V))$. Let $F_{\tt,V}:\EE \to \EE_V \df \bigwedge^{\dim(V)}\R^{M + N}$ be as in Lemma \ref{lemmaFtV}, and let $\pi:\EE_V\to\R$ be a linear map such that
\[
|\pi\circ F_{\tt,V}(\bfA)| \asymp_\times \|F_{\tt,V}(\bfA)\| \text{ and } \|\pi\| = 1.
\]
Let $\LL = (\pi\circ F_{\tt,V})^{-1}(0)\in\scrH(\AA)$. Then there exists $c > 0$ (depending on $B,V$) such that for all $\bfB\in\MM\cap\AA$,
\[
\dist(\bfB,\LL) = c |\pi\circ F_{\tt,V}(\bfB)|.
\]
Then
\[
\|d_\LL\|_{\mu,\lambda_{M + N}B} \geq \dist(\bfA,\LL) = c |\pi\circ F_{\tt,V}(\bfA)| \asymp_\times c f_{\tt,V}(\bfA) > c\eta(\dim(V)).
\]
So for all $\bfB\in W_{\kappa,\tt}(V)$, we have
\[
\dist(\bfB,\LL) \leq c f_{\tt,V}(\bfB) \lesssim_\times e^{-\gamma\|\tt\|} c\eta(\dim(V)) \lesssim_\times e^{-\gamma\|\tt\|} \|d_\LL\|_{\mu,\lambda_{M + N}B}.
\]
Letting $C$ denote the implied constant, we have
\[
W_{\kappa,\tt} \subset \thickvar\LL{C e^{-\gamma\|\tt\|} \|d_\LL\|_{\mu,\lambda_{M + N} B}}.
\]
Let $\beta \df C e^{-\gamma\|\tt\|}$, and let $\rho$ be the radius of $B$. By \eqref{rx3t},
\begin{equation}
\label{r3t}
8\lambda_{M + N}\rho \geq 2^{-(M + N)} e^{-2\|\tt\|_\infty}
\end{equation}
and thus $\beta \lesssim_\times \rho^{\gamma/2}$. Letting $\alpha \df \alpha(\gamma/2,\mu) > 0$ (cf. Lemma \ref{lemmaQD}), we have
\[
\mu\big(W_{\kappa,\tt}(V)\cap B\cap E\big)
\leq \mu\big(\thickvar\LL{\beta \|d_\LL\|_{\mu,\lambda_{M + N}B}}\cap \lambda_{M + N} B\cap E\big)
\lesssim_\times e^{-\alpha\|\tt\|} \mu(\lambda_{M + N} B).
\qedhere\]
\end{subproof}
So we get
\[
\mu(W_{\kappa,\tt}\cap B_0\cap E) \lesssim_\times e^{-\alpha\|\tt\|} \sum_{i = \ell_0}^{M + N} \sum_{(B,\FF)\in\TT_i} \mu(\lambda_{M + N} B).
\]
Let $\epsilon = \alpha/(M + N + 1) > 0$. To complete the proof of Lemma \ref{lemmaexponents}, it suffices to show that for all $i = \ell_0,\ldots,M + N - 1$, we have
\begin{equation}
\label{indhyp}
\sum_{(B,\FF)\in\TT_i} \mu(\lambda_{M + N} B) \lesssim_\times e^{i\epsilon\|\tt\|}.
\end{equation}
We prove \eqref{indhyp} by induction on $i$. When $i = \ell_0$, it holds trivially since $B_0$ is fixed. If it holds for $i$, then
\begin{align*}
\sum_{(B,\FF)\in\TT_{i + 1}} \mu\big(\lambda_{M + N} B\big)
&\lesssim_\times e^{\epsilon\|\tt\|}\sum_{(B,\FF)\in\TT_{i + 1}}\mu\big((1/4) B\big) \by{\eqref{quasifederer} and \eqref{r3t}}\\
&=_\pt e^{\epsilon\|\tt\|} \sum_{(B,\FF)\in \TT_i} \sum_{(B',\FF')\in \CC(B,\FF)} \mu\big((1/4) B'\big) \noreason\\
&\leq_\pt e^{\epsilon\|\tt\|}\sum_{(B,\FF)\in\TT_i} \mu\big(\lambda_{M + N} B\big) \note{disjointness}\\
&\lesssim_\times e^{(i + 1)\epsilon\|\tt\|}, \by{\eqref{indhyp}}
\end{align*}
i.e. \eqref{indhyp} holds for $i + 1$. This completes the proof of Lemma \ref{lemmaexponents} and thus of Theorems \ref{theoremexponents2}, \ref{theoremexponents}, and \ref{theoremKLW}.
\draftnewpage
| {
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} | 8,101 |
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Gravity is Jennifer Holm's second EP, telling stories about real-life experiences through memorable melodies and compelling lyrics.
185 backers pledged $12,541 to help bring this project to life.
Since I was 17 (I won't tell you how long ago that was), I have had the honor of singing for other artists, both live and in the studio. Singing on other people's projects is a great joy for me. I love it. But I'm way overdue in pursuing music for myself.
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/* $Id: ClpConstraintQuadratic.cpp 1665 2011-01-04 17:55:54Z lou $ */
// Copyright (C) 2007, International Business Machines
// Corporation and others. All Rights Reserved.
// This code is licensed under the terms of the Eclipse Public License (EPL).
#include "CoinPragma.hpp"
#include "CoinHelperFunctions.hpp"
#include "CoinIndexedVector.hpp"
#include "ClpSimplex.hpp"
#include "ClpConstraintQuadratic.hpp"
#include "CoinSort.hpp"
//#############################################################################
// Constructors / Destructor / Assignment
//#############################################################################
//-------------------------------------------------------------------
// Default Constructor
//-------------------------------------------------------------------
ClpConstraintQuadratic::ClpConstraintQuadratic ()
: ClpConstraint()
{
type_ = 0;
start_ = NULL;
column_ = NULL;
coefficient_ = NULL;
numberColumns_ = 0;
numberCoefficients_ = 0;
numberQuadraticColumns_ = 0;
}
//-------------------------------------------------------------------
// Useful Constructor
//-------------------------------------------------------------------
ClpConstraintQuadratic::ClpConstraintQuadratic (int row, int numberQuadraticColumns ,
int numberColumns, const CoinBigIndex * start,
const int * column, const double * coefficient)
: ClpConstraint()
{
type_ = 0;
rowNumber_ = row;
numberColumns_ = numberColumns;
numberQuadraticColumns_ = numberQuadraticColumns;
start_ = CoinCopyOfArray(start, numberQuadraticColumns + 1);
int numberElements = start_[numberQuadraticColumns_];
column_ = CoinCopyOfArray(column, numberElements);
coefficient_ = CoinCopyOfArray(coefficient, numberElements);
char * mark = new char [numberQuadraticColumns_];
memset(mark, 0, numberQuadraticColumns_);
int iColumn;
for (iColumn = 0; iColumn < numberQuadraticColumns_; iColumn++) {
CoinBigIndex j;
for (j = start_[iColumn]; j < start_[iColumn+1]; j++) {
int jColumn = column_[j];
if (jColumn >= 0) {
assert (jColumn < numberQuadraticColumns_);
mark[jColumn] = 1;
}
mark[iColumn] = 1;
}
}
numberCoefficients_ = 0;
for (iColumn = 0; iColumn < numberQuadraticColumns_; iColumn++) {
if (mark[iColumn])
numberCoefficients_++;
}
delete [] mark;
}
//-------------------------------------------------------------------
// Copy constructor
//-------------------------------------------------------------------
ClpConstraintQuadratic::ClpConstraintQuadratic (const ClpConstraintQuadratic & rhs)
: ClpConstraint(rhs)
{
numberColumns_ = rhs.numberColumns_;
numberCoefficients_ = rhs.numberCoefficients_;
numberQuadraticColumns_ = rhs.numberQuadraticColumns_;
start_ = CoinCopyOfArray(rhs.start_, numberQuadraticColumns_ + 1);
int numberElements = start_[numberQuadraticColumns_];
column_ = CoinCopyOfArray(rhs.column_, numberElements);
coefficient_ = CoinCopyOfArray(rhs.coefficient_, numberElements);
}
//-------------------------------------------------------------------
// Destructor
//-------------------------------------------------------------------
ClpConstraintQuadratic::~ClpConstraintQuadratic ()
{
delete [] start_;
delete [] column_;
delete [] coefficient_;
}
//----------------------------------------------------------------
// Assignment operator
//-------------------------------------------------------------------
ClpConstraintQuadratic &
ClpConstraintQuadratic::operator=(const ClpConstraintQuadratic& rhs)
{
if (this != &rhs) {
delete [] start_;
delete [] column_;
delete [] coefficient_;
numberColumns_ = rhs.numberColumns_;
numberCoefficients_ = rhs.numberCoefficients_;
numberQuadraticColumns_ = rhs.numberQuadraticColumns_;
start_ = CoinCopyOfArray(rhs.start_, numberQuadraticColumns_ + 1);
int numberElements = start_[numberQuadraticColumns_];
column_ = CoinCopyOfArray(rhs.column_, numberElements);
coefficient_ = CoinCopyOfArray(rhs.coefficient_, numberElements);
}
return *this;
}
//-------------------------------------------------------------------
// Clone
//-------------------------------------------------------------------
ClpConstraint * ClpConstraintQuadratic::clone() const
{
return new ClpConstraintQuadratic(*this);
}
// Returns gradient
int
ClpConstraintQuadratic::gradient(const ClpSimplex * model,
const double * solution,
double * gradient,
double & functionValue,
double & offset,
bool useScaling,
bool refresh) const
{
if (refresh || !lastGradient_) {
offset_ = 0.0;
functionValue_ = 0.0;
if (!lastGradient_)
lastGradient_ = new double[numberColumns_];
CoinZeroN(lastGradient_, numberColumns_);
bool scaling = (model && model->rowScale() && useScaling);
if (!scaling) {
int iColumn;
for (iColumn = 0; iColumn < numberQuadraticColumns_; iColumn++) {
double valueI = solution[iColumn];
CoinBigIndex j;
for (j = start_[iColumn]; j < start_[iColumn+1]; j++) {
int jColumn = column_[j];
if (jColumn >= 0) {
double valueJ = solution[jColumn];
double elementValue = coefficient_[j];
if (iColumn != jColumn) {
offset_ -= valueI * valueJ * elementValue;
double gradientI = valueJ * elementValue;
double gradientJ = valueI * elementValue;
lastGradient_[iColumn] += gradientI;
lastGradient_[jColumn] += gradientJ;
} else {
offset_ -= 0.5 * valueI * valueI * elementValue;
double gradientI = valueI * elementValue;
lastGradient_[iColumn] += gradientI;
}
} else {
// linear part
lastGradient_[iColumn] += coefficient_[j];
functionValue_ += valueI * coefficient_[j];
}
}
}
functionValue_ -= offset_;
} else {
abort();
// do scaling
const double * columnScale = model->columnScale();
for (int i = 0; i < numberCoefficients_; i++) {
int iColumn = column_[i];
double value = solution[iColumn]; // already scaled
double coefficient = coefficient_[i] * columnScale[iColumn];
functionValue_ += value * coefficient;
lastGradient_[iColumn] = coefficient;
}
}
}
functionValue = functionValue_;
offset = offset_;
CoinMemcpyN(lastGradient_, numberColumns_, gradient);
return 0;
}
// Resize constraint
void
ClpConstraintQuadratic::resize(int newNumberColumns)
{
if (numberColumns_ != newNumberColumns) {
abort();
#ifndef NDEBUG
int lastColumn = column_[numberCoefficients_-1];
#endif
assert (newNumberColumns > lastColumn);
delete [] lastGradient_;
lastGradient_ = NULL;
numberColumns_ = newNumberColumns;
}
}
// Delete columns in constraint
void
ClpConstraintQuadratic::deleteSome(int numberToDelete, const int * which)
{
if (numberToDelete) {
abort();
int i ;
char * deleted = new char[numberColumns_];
memset(deleted, 0, numberColumns_ * sizeof(char));
for (i = 0; i < numberToDelete; i++) {
int j = which[i];
if (j >= 0 && j < numberColumns_ && !deleted[j]) {
deleted[j] = 1;
}
}
int n = 0;
for (i = 0; i < numberCoefficients_; i++) {
int iColumn = column_[i];
if (!deleted[iColumn]) {
column_[n] = iColumn;
coefficient_[n++] = coefficient_[i];
}
}
numberCoefficients_ = n;
}
}
// Scale constraint
void
ClpConstraintQuadratic::reallyScale(const double * )
{
abort();
}
/* Given a zeroed array sets nonquadratic columns to 1.
Returns number of nonlinear columns
*/
int
ClpConstraintQuadratic::markNonlinear(char * which) const
{
int iColumn;
for (iColumn = 0; iColumn < numberQuadraticColumns_; iColumn++) {
CoinBigIndex j;
for (j = start_[iColumn]; j < start_[iColumn+1]; j++) {
int jColumn = column_[j];
if (jColumn >= 0) {
assert (jColumn < numberQuadraticColumns_);
which[jColumn] = 1;
which[iColumn] = 1;
}
}
}
int numberCoefficients = 0;
for (iColumn = 0; iColumn < numberQuadraticColumns_; iColumn++) {
if (which[iColumn])
numberCoefficients++;
}
return numberCoefficients;
}
/* Given a zeroed array sets possible nonzero coefficients to 1.
Returns number of nonzeros
*/
int
ClpConstraintQuadratic::markNonzero(char * which) const
{
int iColumn;
for (iColumn = 0; iColumn < numberQuadraticColumns_; iColumn++) {
CoinBigIndex j;
for (j = start_[iColumn]; j < start_[iColumn+1]; j++) {
int jColumn = column_[j];
if (jColumn >= 0) {
assert (jColumn < numberQuadraticColumns_);
which[jColumn] = 1;
}
which[iColumn] = 1;
}
}
int numberCoefficients = 0;
for (iColumn = 0; iColumn < numberQuadraticColumns_; iColumn++) {
if (which[iColumn])
numberCoefficients++;
}
return numberCoefficients;
}
// Number of coefficients
int
ClpConstraintQuadratic::numberCoefficients() const
{
return numberCoefficients_;
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,170 |
{"url":"https:\/\/planetmath.org\/CayleyTable","text":"# Cayley table\n\nA Cayley table for a group is essentially the \u201cmultiplication table\u201d of the group.11A caveat to novices in group theory: multiplication is usually used notationally to represent the group operation, but the operation needn\u2019t resemble multiplication in the reals. Hence, you should take \u201cmultiplication table\u201d with a grain or two of salt. The columns and rows of the table (or matrix) are labeled with the elements of the group, and the cells represent the result of applying the group operation to the row-th and column-th elements.\n\nFormally, let $G$ be our group, with operation $\\circ$ the group operation. Let $C$ be the Cayley table for the group, with $C(i,j)$ denoting the element at row $i$ and column $j$. Then\n\n $C(i,j)=e_{i}\\circ e_{j}$\n\nwhere $e_{i}$ is the $i$th element of the group, and $e_{j}$ is the $j$th element.\n\nNote that for an Abelian group, we have $e_{i}\\circ e_{j}=e_{j}\\circ e_{i}$, hence the Cayley table is a symmetric matrix.\n\nAll Cayley tables for isomorphic groups are isomorphic (that is, the same, invariant of the labeling and ordering of group elements).\n\n## 0.1 Examples.\n\n\u2022 The Cayley table for $\\mathbb{Z}_{4}$, the group of integers modulo 4 (under addition), would be\n\n $\\left(\\begin{array}[]{c|cccc}&[0]&[1]&[2]&[3]\\\\ \\hline\\;[0]&[0]&[1]&[2]&[3]\\\\ \\;[1]&[1]&[2]&[3]&[0]\\\\ \\;[2]&[2]&[3]&[0]&[1]\\\\ \\;[3]&[3]&[0]&[1]&[2]\\end{array}\\right)$\n\u2022 The Cayley table for $S_{3}$, the permutation group of order 3, is\n\n $\\left(\\begin{array}[]{c|cccccc}&(1)&(123)&(132)&(12)&(13)&(23)\\\\ \\hline(1)&(1)&(123)&(132)&(12)&(13)&(23)\\\\ (123)&(123)&(132)&(1)&(13)&(23)&(12)\\\\ (132)&(132)&(1)&(123)&(23)&(12)&(13)\\\\ (12)&(12)&(23)&(13)&(1)&(132)&(123)\\\\ (13)&(13)&(12)&(23)&(123)&(1)&(132)\\\\ (23)&(23)&(13)&(12)&(132)&(123)&(1)\\end{array}\\right)$\nTitle Cayley table CayleyTable 2013-03-22 13:06:44 2013-03-22 13:06:44 akrowne (2) akrowne (2) 11 akrowne (2) Definition msc 20A99 Cayley-table","date":"2018-11-20 11:45:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 16, \"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\": 0.8992266654968262, \"perplexity\": 425.9106646005197}, \"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-2018-47\/segments\/1542039746386.1\/warc\/CC-MAIN-20181120110505-20181120132505-00400.warc.gz\"}"} | null | null |
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We're your Denver moving company. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,484 |
\section{HISTORY OF THE PROBLEM}
The usefulness of regularly varying (RV) functions in economics seems to be discussed for the first time during modeling of the wealth in our society by Pareto distribution, called to the name of Vilfredo Pareto (1897). J. Karamata (1933) provides their definition and integral representation. Later on the Convergence to types theorem, proved by R. A. Fisher, L. H. C. Tippett (1928), and B.V. Gnedenko (1948) plays a key role for their future applications. It is well known that this class of distributions describes very well the domain of attraction of stable distribution (see Mandelbrot (1960) \cite{Mandelbrot1960}) and max-domain of attraction of Fr$\acute{e}$chet distribution (see M. Fr$\acute{e}$chet (1927)). Laurens de Haan (1970) and co-authors \cite{dH70,deHaanStadtmueller,deHaanFerreira} develop the main machinery for working with cumulative distribution functions(c.d.fs.) with such tail behaviour. Let us remind that the c.d.f. $F$ has regularly varying right tail with parameter $\alpha > 0$, if
$$\lim_{t\to \infty} \frac{1-F(xt)}{1-F(t)}= x^{-\alpha}, \quad \forall x > 0.$$
After their works the topic spread over the world very fast and many estimators of the index of regular variation are proposed, see e.g. Hill (1975) \cite{Hill}, Pickands (1975)\cite{Pickands} and Deckers-Einmahl-de Haan (1989) \cite{Dekkers1989}, t-Hill (Stehlik and co-authors (2010) \cite{Stehlik2010,Fabian,Stehlik2012}, and Pancheva and Jordanova (2012) \cite{JordanovaPancheva, JordanovaMilan2012}), among others.
Here we show the usefulness of functions of two central order statistics in estimating the parameter of regular variation. Under very general settings we show that the logarithm of the fraction of two specific central order statistics is an weakly consistent and asymptotically normal estimators of the logarithm of the corresponding theoretical quantiles. Then we use these functions and obtain our estimator for $\alpha$. Its main advantage is that it is very flexible and provides a useful accuracy given mid-range and small samples.
Pareto case, considered in Section 3 motivates our investigation. First we define a biased form of the estimator. Then using results about order statistics, which could be seen e.g. in Nevzorov (2001) \cite{Nevzorov} we obtain explicit formulae for its mean and variance. This allows us to define unbiased correction which is asymptotically efficient. Then we prove asymptotic normality and obtain large sample confidence intervals. Our simulation study depicts the advantages of the considered estimators over Hill, t-Hill, and Deckers-Einmahl-de Haan estimators. The paper finishes with some conclusive remarks.
Trough the paper we assume that $\mathbf{X}_1, \mathbf{X}_2, ..., \mathbf{X}_n$ are independent observations on a random variable(r.v.) $\mathbf{X}$, and denote by $\mathbf{X}_{(1, n)} \leq \mathbf{X}_{(2, n)} \leq ... \leq \mathbf{X}_{(n, n)}$ the corresponding increasing order statistics.
$$H_{n, m} = 1 + \frac{1}{2^m} + \frac{1}{3^m} + ... + \frac{1}{(n-1)^m} + \frac{1}{n^m}, \quad n = 1, 2, ..., $$
denotes the $n$-the Generalized harmonic number of power $m = 1, 2, ...$, and $H_n = H_{n, 1}$, $n = 1, 2, ...$ is for the well-known $n$-th harmonic number.
The main object of interest in this point are the statistics
$$ Q_{k,s} : = \frac{log\frac{\mathbf{X}_{(ks, (s+1)k-1)}}{\mathbf{X}_{(k, (s+1)k-1)}}}{H_{ks-1} - H_{k-1}}, \quad Q_{k,s}^* = \frac{log\frac{\mathbf{X}_{(ks, (s+1)k-1)}}{\mathbf{X}_{(k, (s+1)k-1)}}}{log(s)}, \quad s = 2, 3, ...$$
The estimator $Q_{k,3}^*$ it is obtained in Jordanova et al. \cite{jordanova2017measuring} via quantile matching procedure. About the last procedure see e.g. Sgouropoulos et al. (2015) \cite{sgouropoulos2015matching}.
Along the paper $\stackrel{d}{\to}$ means convergence in distribution.
\section{GENERAL RESULTS}
In 1933 - 1949 Smirnoff \cite{smirnov1949limit} shows that in case of central order statistics, and more precisely for $k, n$ and $p$ such that $\frac{k}{n} \to p \in (0, 1)$ and $\sqrt{n}\left(\frac{k}{n} - p\right) \to \mu \in (-\infty, \infty)$, the asymptotic distribution of $\frac{\sqrt{n}[X_{(k, n)} - F^\leftarrow(p)]}{\sqrt{\frac{p(1-p)}{f^2[F^\leftarrow(p)]}}}$ is a standard normal. Moreover it seems that he has a similar results about bivariate order statistics. It could be seen e.g. in Arnold et al. (1992) \cite{arnold1992first}, p. 226, Mosteller (1946) \cite{Mosteller1946} p.338, Nair \cite{Nair2013}, p.330, or Wilks \cite{Wilks1948} among others. The multivariate delta method is a very powerful technique for obtaining confidence intervals in such cases. In the next theorem we apply them and obtain the limiting distribution of the logarithmic differences of central order statistics.
{\bf Smirnoff's theorem.} Assume for $n \to \infty$, $\frac{k_1(n)}{n} \to p_1 \in (0, 1)$, $\frac{k_2(n)}{n} \to p_2 \in (0, 1)$, $f\left[F^\leftarrow(p_1)\right] \in (0, \infty)$, and $f\left[F^\leftarrow\left(p_2\right)\right] \in (0, \infty)$. Then
$$\sqrt{n}\left(
\begin{array}{c}
X_{(k_1, n)} - F^\leftarrow(p_1) \\
X_{(k_2, n)} - F^\leftarrow(p_2)\\
\end{array}
\right)
\stackrel{d}{\to} \left(
\begin{array}{c}
\theta_1 \\
\theta_2 \\
\end{array}
\right), \quad \left(
\begin{array}{c}
\theta_1 \\
\theta_2 \\
\end{array}
\right) \in N(0, V)
$$
where the covariance matrix
$$V = \left(
\begin{array}{cc}
\frac{p_1(1-p_1)}{f^2[F^\leftarrow(p_1)]} & \frac{p_1(1-p_2)}{f^2[F^\leftarrow(p_1)]f^2[F^\leftarrow(p_2)]} \\
\frac{p_1(1-p_2)}{f^2[F^\leftarrow(p_1)]f^2[F^\leftarrow(p_2)]} & \frac{p_2(1-p_2)}{f^2[F^\leftarrow(p_2)]} \\
\end{array}
\right).
$$
We apply this theorem together with the Multivariate delta method and obtain asymptotic normality of the estimators, discussed in this paper.
{\bf Theorem 1.} Consider a sample of $n = (s+1)k-1$, $s = 2, 3, ...$ independent observations on a r.v. $X$ with c.d.f. $F$ and p.d.f. $f = F'$. If there exists $0 < f\left[F^\leftarrow(\frac{1}{s+1})\right] < \infty$ and $0 < f\left[F^\leftarrow(\frac{s}{s+1})\right] < \infty$, then for $k \to \infty$
\begin{equation}\label{GeneralAN}
T_{k,s} := \sqrt{(s+1)k-1}\left[log\left(\frac{X_{(ks, (s+1)k-1)}}{X_{(k, (s+1)k-1)}}\right) - log\left(\frac{F^\leftarrow(\frac{s}{s+1})}{F^\leftarrow(\frac{1}{s+1})}\right)\right] \stackrel{d}{\to } N\left( 0; V\right).
\end{equation}
The variance $V$ in (\ref{GeneralAN}) is $V = \frac{1}{(s+1)^2} \left(\frac{s}{a_{F,s}^2} - \frac{2}{a_{F,s}b_{F,s}} + \frac{s}{b_{F,s}^2}\right)$, where
$a_{F,s} = F^\leftarrow(\frac{1}{s+1})f\left[F^\leftarrow(\frac{1}{s+1})\right] = \frac{1}{\{log[ F^\leftarrow(p)]\}'|_{p = \frac{1}{s+1}}}$, and $b_{F,s} = F^\leftarrow(\frac{s}{s+1})f\left[F^\leftarrow(\frac{s}{s+1})\right] =$ $ \frac{1}{\{log[ F^\leftarrow(p)]\}'|_{p = \frac{s}{s+1}}}.$
{\bf Proof:} We will apply the Theorem of Smirnoff for $p_1 = \frac{1}{s+1}$ and $p_2 = \frac{s}{s+1}$ and Multivariate delta method.
By assumptions the conditions $f\left[F^\leftarrow\left(\frac{1}{s+1}\right)\right] \in (0, \infty)$, $f\left[F^\leftarrow\left(\frac{s}{s+1}\right)\right] \in (0, \infty)$ are satisfied. And for $k \to \infty$ we have $\frac{sk}{(s+1)k-1} \to \frac{s}{s+1}$, $\frac{s}{(s+1)k-1} \to \frac{1}{s+1}$, $\sqrt{(s+1)k-1}\left(\frac{sk}{(s+1)k-1} - \frac{s}{s+1}\right) \to 0$ and $\sqrt{(s+1)k-1}\left(\frac{k}{(s+1)k-1} - \frac{s}{s+1}\right) \to 0$, therefore the Smirnoff's theorem on the joint asymptotic normality of the order statistics, says that
$$\sqrt{(s+1)k-1}\left(\begin{array}{c}
X_{(k, (s+1)k-1)} - F^\leftarrow(\frac{1}{s+1}) \\
X_{(ks, (s+1)k-1)} - F^\leftarrow(\frac{i}{s+1}) \\
\end{array}
\right) \stackrel{d}{\to} N\left[\left(\begin{array}{c}
0\\
0 \\
\end{array}\right); D\right], \quad k \to \infty,$$
where the asymptotic covariance matrix of this bivariate distribution is
$$D = \frac{1}{(s+1)^2}\left(\begin{array}{cc}
\frac{s}{f^2\left[F^\leftarrow(\frac{1}{s+1})\right]} & \frac{1}{f\left[F^\leftarrow(\frac{1}{s+1})\right]f\left[F^\leftarrow(\frac{s}{s+1})\right]} \\
\frac{1}{f\left[F^\leftarrow(\frac{1}{s+1})\right]f\left[F^\leftarrow(\frac{s}{s+1})\right]} & \frac{s}{f^2\left[F^\leftarrow(\frac{s}{s+1})\right]} \\
\end{array}\right)$$
and the asymptotic correlation between these two order statistics is $\frac{1}{s}$.
Consider the function $g(x, y) = log\left(\frac{y}{x}\right)$. For $x > 0$ and $y > 0$ it is continuously differentiable.
The Jacobian of the transformation is
$$J : = \left[\frac{\partial g(x, y)}{\partial x}, \frac{\partial g(x, y)}{\partial y}\right] = \left(-\frac{1}{x}, \frac{1}{y}\right).$$
The asymptotic mean is
$$ \lim_{k \to \infty} E log\left(\frac{X_{(ks, (s+1)k-1)}}{X_{(k, (s+1)k-1)}}\right) = g\left[F^\leftarrow(\frac{1}{s+1}), F^\leftarrow(\frac{s}{s+1})\right] = log\left(\frac{F^\leftarrow(\frac{s}{s+1})}{F^\leftarrow(\frac{1}{s+1})}\right).$$
Now we apply the Multivariate Delta method, which could be seen e.g. in Sobel (1982) \cite{MultivariateDeltaMethod}, and obtain that the asymptotic variance of $T_{k,s}$ is
\begin{eqnarray*}
V : &=& J \times D \times J' = \left(-\frac{1}{x}, \frac{1}{y}\right)\left|_{x = F^\leftarrow(\frac{1}{s+1}), y = F^\leftarrow(\frac{s}{s+1})}\right. \\
&\times& \frac{1}{(s+1)^2}\left(\begin{array}{cc}
\frac{s}{f^2\left[F^\leftarrow(\frac{1}{s+1})\right]} & \frac{1}{f\left[F^\leftarrow(\frac{1}{s+1})\right]f\left[F^\leftarrow(\frac{s}{s+1})\right]} \\
\frac{1}{f\left[F^\leftarrow(\frac{1}{s+1})\right]f\left[F^\leftarrow(\frac{s}{s+1})\right]} & \frac{s}{f^2\left[F^\leftarrow(\frac{s}{s+1})\right]} \\
\end{array}\right) \left(
\begin{array}{c}
-\frac{1}{x} \\
\frac{1}{y} \\
\end{array}
\right)\left|_{x = F^\leftarrow(\frac{1}{s+1}), y = F^\leftarrow(\frac{s}{s+1})}\right. \\
&=& \frac{1}{(s+1)^2} \left[\left(-\frac{1}{x}, \frac{1}{y}\right)\left(\begin{array}{cc}
\frac{s}{f^2(x)} & \frac{1}{f(x)f(y)} \\
\frac{1}{f(x)f(y)} & \frac{s}{f^2(y)} \\
\end{array}\right) \left(
\begin{array}{c}
-\frac{1}{x} \\
\frac{1}{y} \\
\end{array}
\right)\right]\left|_{x = F^\leftarrow(\frac{1}{s+1}), y = F^\leftarrow(\frac{s}{s+1})}\right. \\
&=& \frac{1}{(s+1)^2} \left[\frac{s}{x^2f^2(x)} - \frac{2}{xyf(x)f(y)} + \frac{s}{y^2f^2(y)}\right] \left|_{x = F^\leftarrow(\frac{1}{s+1}), y = F^\leftarrow(\frac{s}{s+1})}\right. =\frac{1}{(s+1)^2} \left(\frac{s}{a_{F,s}^2} - \frac{2}{a_{F,s}b_{F,s}} + \frac{s}{b_{F,s}^2}\right).
\end{eqnarray*}
\hfill Q.A.D.
Slutsky's theorem about continuous functions together with the definition of convergence in probability, application of quantile transform, and Smirnoff's theorem about a.s. convergence of empirical quantiles to corresponding theoretical one, lead us to the following result. Without lost of generality we consider only a.s. positive r.vs, however the result could be easily transformed for $P(X < \mu) = 1$ or $P(X > \mu) = 1$, $\mu > 0$.
{\bf Theorem 2.} Assume $P(X > 0) = 1$. If $f\left[F^\leftarrow\left(\frac{1}{s+1}\right)\right] \in (0, \infty)$, $f\left[F^\leftarrow\left(\frac{s}{s+1}\right)\right] \in (0, \infty)$, then for $s = 2, 3, ...$
\begin{equation}\label{logfraction}
log\left(\frac{X_{(ks, (s+1)k-1)}}{X_{(k, (s+1)k-1)}}\right) \stackrel{P}{\to } log\left[\frac{F^\leftarrow(\frac{s}{s+1})}{F^\leftarrow(\frac{1}{s+1})}\right], \quad k \to \infty.
\end{equation}
\section{PARETO CASE}
In this section we assume that $\mathbf{X}_1, \mathbf{X}_2, ..., \mathbf{X}_n$ are independent observations on a r.v. $\mathbf{X}$ with Pareto c.d.f.
\begin{equation}\label{Pareto}
F_{\mathbf{X}}(x) = \left\{
\begin{array}{ccc}
0 & , & x \leq \delta \\
1 - \left(\frac{\delta}{x}\right)^{\alpha} & , & x > \delta
\end{array}
\right., \quad \alpha > 0, \quad \delta > 0.
\end{equation}
Briefly we will denote this by $\mathbf{X} \in Par(\alpha, \delta)$. Different generalizations of this distributions could be seen in Arnold (2015) \cite{Arnold2015}. The number $-\alpha$ is called "index of regular variation of the tail of c.d.f.". It determines the tail behaviour of the c.d.f. See e.g. de Haan and Ferreira \cite{deHaanFerreira}, Resnick \cite{Resnick87}, or Jordanova \cite{Jordanova2019}.
Denote by $\mathbf{X} \in Exp(\lambda)$, $\lambda > 0$ the fact that the r.v. $\mathbf{X}$ has c.d.f.
\begin{equation}\label{Exponential}
F_{\mathbf{X}}(x) = \left\{
\begin{array}{ccc}
0 & , & x \leq 0 \\
1 - e^{-\lambda x} & , & x > 0
\end{array}
\right..
\end{equation}
The results in the following theorem allow us later on, in Corollaries 1 and 2, to obtain unbiased, consistent, and asymptotically efficient estimators of the parameter $\alpha$.
{\bf Theorem 3.} Assume $\mathbf{X}_{(1, n)} \leq \mathbf{X}_{(2, n)} \leq ... \leq \mathbf{X}_{(n, n)}$ are order statistics of independent observations on a r.v. $\mathbf{X} \in Par(\alpha, \delta)$, $\alpha > 0$, $\delta > 0$, and $1 \leq i < j \leq n$ are integer.
\begin{description}
\item[i)] Denote by $\rho$ a Beta distributed with parameters $n - j + 1$, and $j - i$. Then
$$log\left(\frac{X_{(j, n)}}{X_{(i, n)}}\right) \stackrel{d}{=} - \frac{1}{\alpha} \log(\rho) \stackrel{d}{=} E_{(j - i, n-i)} \stackrel{d}{=} \frac{1}{\alpha} E_{(j - i, n-i)}^*,$$
where $E_{(j - i, n - i)}$ is the $j - i$-th order statistics in a sample of $n - i$ independent observations on i.i.d. Exponential r.vs. with parameter $\alpha$, and $E_{(j - i, n - i)}^*$ is the $j - i$ - th order statistic of a sample of $n-i$ independent observations on exponentially distributed r.v. with parameter $1$. Its probability density function is
$$f_{log\frac{\mathbf{X}_{(j, n)}}{\mathbf{X}_{(i, n)}}}(x) = \frac{\alpha (n-i)!}{(j-i-1)!(n-j)!}(1-e^{-\alpha x})^{j-i-1}e^{-\alpha x(n-j + 1
)}, \quad x > 0.$$
\item[ii)] $\mathbb{E} \left[log\frac{\mathbf{X}_{(j, n)}}{\mathbf{X}_{(i, n)}}\right] = \frac{1}{\alpha}\left(H_{n-i} - H_{n-j}\right),$ and
$\mathbb{D} \left[log\frac{\mathbf{X}_{(j, n)}}{\mathbf{X}_{(i, n)}}\right] = \frac{1}{\alpha^2}\left(H_{n-i, 2} - H_{n-j, 2}\right).$
\end{description}
{\bf Proof:} Let us fix $1 \leq i < j \leq n$, integers.
Because of $g(x) = e^x$ is a strictly increasing function, it is well known that the probability quantile transform, entails
$$ \left(\frac{\mathbf{X}_{(1, n)}}{\delta}, \frac{\mathbf{X}_{(2, n)}}{\delta}, ..., \frac{\mathbf{X}_{(n, n)}}{\delta
}\right) \,\,{\mathop{=}\limits_{}^{d}} \,\,(e^{\mathbf{E}_{(1, n)}}, e^{\mathbf{E}_{(2, n)}}, ... e^{\mathbf{E}_{(n, n)}}),$$
where $\mathbf{E}_{(1, n)} \leq \mathbf{E}_{(2, n)} \leq ... \leq \mathbf{E}_{(n, n)}$ are order statistics of independent identically distributed (i.i.d.) r.vs. with $\mathbf{E}_1 \in Exp(\alpha)$. Then, because of the multiplicative property of the exponential distribution
$$\left(\frac{\mathbf{X}_{(1, n)}}{\delta}, \frac{\mathbf{X}_{(2, n)}}{\delta}, ..., \frac{\mathbf{X}_{(n, n)}}{\delta
}\right) \,\, {\mathop{=}\limits_{}^{d}}\,\, \left(e^{\frac{1}{\alpha}\mathbf{E}_{(1, n)}^*}, e^{\frac{1}{\alpha}\mathbf{E}_{(2, n)}^*}, ... e^{\frac{1}{\alpha}\mathbf{E}_{(n, n)}^*}\right),$$
where $\mathbf{E}_{(1, n)}^* \leq \mathbf{E}_{(2, n)}^* \leq ... \leq \mathbf{E}_{(n, n)}^*$ are order statistics of i.i.d. r.vs. with $\mathbf{E}_1^* \in Exp(1)$. See e.g. de Haan and Ferreira \cite{deHaanFerreira}. Denote the logarithm with basis $e$ by log. Because of $e > 1$, $h(x) = log(x)$, is an increasing function, thus
$$log\left(\frac{\mathbf{X}_{(j, n)}}{\mathbf{X}_{(i, n)}}\right) \,\, {\mathop{=}\limits_{}^{d}}\,\, \frac{1}{\alpha}\left(\mathbf{E}_{(j, n)}^* - \mathbf{E}_{(i, n)}^*\right) \,\, {\mathop{=}\limits_{}^{d}}\,\, \frac{1}{\alpha}\mathbf{E}_{(j-i, n-i)}^*.$$
The last equality could be seen e.g. in de Haan and Ferreira \cite{deHaanFerreira} or Arnold et al. (1992) \cite{arnold1992first}.
{\bf i)} Follows by the equality $P_{\frac{E_{(j - i, n-i)}^*}{\alpha}}(x) = \alpha P_{E_{(j - i, n-i)}^*}(\alpha \, x)$, the well known relation $- \log(\rho) \stackrel{d}{=} E_{(j - i, n-i)}^*$ and the formula for probability density function (p.d.f.) of order statistics of a sample of i.i.d. r.vs. See e.g. p. 7 Nevzorov \cite{Nevzorov}.
{\bf ii)}
The mean, and the variance of the last order statistics are very well investigated. See e.g. Nevzorov \cite{Nevzorov}, p.23. Using his results and the main properties of the expectation and the variance we obtain:
\begin{eqnarray*}
\mathbb{E}\left[log\left(\frac{\mathbf{X}_{(j, n)}}{\mathbf{X}_{(i, n)}}\right)\right] &=& \mathbb{E}\left(\frac{1}{\alpha} E_{(j - i, n-i)}^*\right) = \frac{1}{\alpha} \mathbb{E}\left( E_{(j - i, n-i)}^*\right) = \frac{1}{\alpha}\left(H_{n-i} - H_{n-j}\right)\\
\mathbb{D}\left[log\left(\frac{\mathbf{X}_{(j, n)}}{\mathbf{X}_{(i, n)}}\right)\right] &=& \mathbb{D}\left(\frac{1}{\alpha} E_{(j - i, n-i)}^*\right) = \frac{1}{\alpha^2} \mathbb{D}\left( E_{(j - i, n-i)}^*\right) = \frac{1}{\alpha^2}\left(H_{n-i, 2} - H_{n-j, 2}\right).
\end{eqnarray*}
\hfill Q.A.D.
In the next corollary is useful when working with finite samples. We obtain that for any $s = 2, 3, ...$, and for fixed $k = 1, 2, ...$ the estimators $Q_{k,s}$ are unbiased for $\frac{1}{\alpha}$. The accuracy of these estimators in that case is explicitly calculated.
However these estimators are applicable also for large enough samples, because for $k \to \infty$ they are weakly consistent and asymptotically efficient.
{\bf Corollary 1.} Assume $n = k(s+1) - 1$, $\mathbf{X}_{(1, n)} \leq \mathbf{X}_{(2, n)} \leq ... \leq \mathbf{X}_{(n, n)}$ are order statistics of independent observations on a r.v. $\mathbf{X} \in Par(\alpha, \delta)$, $\alpha > 0$, $\delta > 0$. Then, for all $s = 2, 3, ...$, and $k \in \mathbb{N}$,
\begin{description}
\item[i)] Denote by $\rho$ a Beta distributed with parameters $k$, and $(s-1)k$. Then
$$Q_{k,s} \stackrel{d}{=} \frac{-\log(\rho)}{\alpha(H_{ks-1} - H_{k-1})} \stackrel{d}{=} \frac{E_{((s-1)k, ks-1)}}{H_{ks-1} - H_{k-1}} \stackrel{d}{=} \frac{E_{((s-1)k, ks-1)}^*}{\alpha(H_{ks-1} - H_{k-1})} ,$$
where $E_{((s-1)k, ks - 1)}$ is the $(s-1)k$-th order statistics in a sample of $ks - 1$ independent observations on i.i.d. Exponential r.vs. with parameter $\alpha$. $E_{((s-1)k, ks-1)}^*$ is the $(s-1)s$ - th order statistic of a sample of $ks-1$ independent observations on exponentially distributed r.v. with parameter $1$. Its probability density function is
$$f_{Q_{k,s}}(x) = \frac{\alpha (H_{ks-1} - H_{k-1}) (ks-1)!}{[(s-1)k-1]!(k-1)!}(1-e^{-\alpha (H_{ks-1} - H_{k-1}) x})^{(s-1)k-1}e^{-k \alpha (H_{ks-1} - H_{k-1}) x}, \quad x > 0.$$
\item[ii)] $\mathbb{E} Q_{k,s} = \frac{1}{\alpha},$ and
$\mathbb{D} Q_{k,s} = \frac{H_{ks-1, 2} - H_{k-1, 2}}{\alpha^2(H_{ks-1} - H_{k-1})^2}.$
\item[iii)] For all $\varepsilon > 0$,
$$P\left[\left|Q_{k,s} - \frac{1}{\alpha}\right| > \varepsilon \right] \leq \frac{H_{ks-1, 2} - H_{k-1, 2}}{\alpha^2\varepsilon^2(H_{ks-1} - H_{k-1})^2}.$$
\item[iv)] The estimator $Q_{k,s}$ is asymptotically efficient. For
$k \to \infty$,
$$\mathbb{D} Q_{k,s} \sim \frac{H_{ks-1, 2} - H_{k-1, 2}}{\alpha^2} \left[log\left(\frac{ks-1}{k}\right)\right]^{-2}, \quad \lim_{k \to \infty} \mathbb{D} Q_{k,s} = 0.$$
\item[v)] The estimator $Q_{k,s}$ is weekly consistent. More precisely, for all $\varepsilon > 0$,
$\lim_{k\to \infty} P\left[\left|Q_{k,s} - \frac{1}{\alpha}\right| > \varepsilon \right] = 0.$
\end{description}
{\bf Proof:} i) and ii) follow by Theorem 1, definition of $Q_{k,s}$ and the relations
$$f_{Q_{k,s}}(x) = (H_{ks-1} - H_{k-1}) f_{log\frac{\mathbf{X}_{(ks, k(s+1)-1)}}{\mathbf{X}_{(k, k(s+1)-1)}}}[x(H_{ks-1} - H_{k-1})], \quad \mathbb{E} Q_{k,s} = \frac{\mathbb{E}log\frac{\mathbf{X}_{(ks, k(s+1)-1)}}{\mathbf{X}_{(k, k(s+1)-1)}}}{H_{ks-1} - H_{k-1}}, \quad \mathbb{D} Q_{k,s} = \frac{\mathbb{D}log\frac{\mathbf{X}_{(ks, k(s+1)-1)}}{\mathbf{X}_{(k, k(s+1)-1)}}}{(H_{ks-1} - H_{k-1})^2}$$
iii) is corollary of ii) and Chebyshev's inequality.
iv) It is well known that $\lim_{n \to \infty} [H_n - log(n)] = \gamma,$ where $\gamma = -\Gamma'(1) = \psi(1) \approx 0,5772$ is the Euler$-$Mascheroni constant, $\Gamma(\alpha) = \int_0^\alpha x^{\alpha-1} e^{-x} dx$, and $\psi$ is the Digamma function. By ii) for any fixed $s = 2, 3, ...$, we have
\begin{eqnarray}
\lim_{k \to \infty} \mathbb{D} Q_{k,s} &=& \frac{1}{\alpha^2} \lim_{k \to \infty} \frac{H_{ks-1, 2} - H_{k-1, 2}}{(H_{ks-1} - H_{k-1})^2} = \frac{1}{\alpha^2} \lim_{k \to \infty} \frac{H_{ks-1, 2} - H_{k-1, 2}}{\left\{H_{ks-1} - log(ks-1) - [H_{k-1} - log(k)] + log\left(\frac{ks-1}{k}\right)\right\}^2} \\
&=& \frac{1}{\alpha^2} \lim_{k \to \infty} \frac{H_{ks-1, 2} - H_{k-1, 2}}{\left[log\left(\frac{ks-1}{k}\right)\right]^2} = \frac{1}{\alpha^2} \frac{\lim_{k \to \infty} H_{ks-1, 2} - \lim_{k \to \infty} H_{k-1, 2}}{[log(s)]^2} = 0
\end{eqnarray}
In the last equality we have used the well known solution of the Basel problem, and more precisely the limit $\lim_{n \to \infty} H_{n,2} = \frac{\pi^2}{6}.$
v) is a consequence of ii), iii) and iv). \hfill Q.A.D.
In the previous proof we have seen that for any fixed $s = 2, 3, ...$, $\lim_{k \to \infty} (H_{ks-1} - H_{k-1}) = log(s)$. Therefore, although
$Q_{k,s}^*$ are biased, they are asymptotically unbiased, asymptotically normal, weakly consistent and asymptotically efficient estimators for $\frac{1}{\alpha}$. The next conclusions follow by the relation $Q_{k,s}^* = \frac{Q_{k,s} (H_{ks-1} - H_{k-1})}{log(s)}$, and the main properties of the mean and the variance.
{\bf Corollary 2.} Assume $n = k(s+1) - 1$, $\mathbf{X}_{(1, n)} \leq \mathbf{X}_{(2, n)} \leq ... \leq \mathbf{X}_{(n, n)}$ are order statistics of independent observations on a r.v. $\mathbf{X} \in Par(\alpha, \delta)$, $\alpha > 0$, $\delta > 0$.
\begin{description}
\item[i)] Denote by $\rho$ a Beta distributed with parameters $k$, and $(s-1)k$. Then, for all $k \in \mathbb{N}$,
$$Q_{k,s}^* \stackrel{d}{=} - \frac{\log(\rho)}{\alpha log(s)} \stackrel{d}{=} \frac{E_{((s-1)k, ks-1)}}{log(s)} \stackrel{d}{=} \frac{E_{((s-1)k, ks-1)}^*}{\alpha log(s)} ,$$
where $E_{((s-1)k, ks - 1)}$ is the $(s-1)k$-th order statistics in a sample of $ks - 1$ independent observations on i.i.d. Exponential r.vs. with parameter $\alpha$. $E_{((s-1)k, ks-1)}^*$ is the $(s-1)s$ - th order statistic of a sample of $ks-1$ independent observations on exponentially distributed r.v. with parameter $1$. Its probability density function is
$$f_{Q_{k,s}^*}(x) = \frac{\alpha\,\, log(s) (ks-1)!}{[(s-1)k-1]!(k-1)!}(1-s^{-\alpha x})^{(s-1)k-1}s^{-\alpha k x}, \quad x > 0.$$
\item[ii)] For all $k \in \mathbb{N}$, $\mathbb{E} Q_{k,s}^* = \frac{H_{ks-1} - H_{k-1}}{\alpha \,\,log(s)},$ and
$\mathbb{D} Q_{k,s}^* = \frac{H_{ks-1, 2} - H_{k-1, 2}}{\alpha^2[log(s)]^2}.$
\item[iii)] For all $\varepsilon > 0$, and $k \in \mathbb{N}$,
$$P\left[\left|Q_{k,s}^* - \frac{1}{\alpha}\right| > \varepsilon \right] \leq \frac{H_{ks-1, 2} - H_{k-1, 2}}{\alpha^2\varepsilon^2[log(s)]^2}.$$
\item[iv)]$Q_{k,s}^*$ estimator is asymptotically unbiased and asymptotically efficient. More precisely
$$\lim_{k\to \infty} \mathbb{E}Q_{k,s}^* = \frac{1}{\alpha}, \quad \lim_{k\to \infty} \mathbb{D} Q_{k,s}^* = 0.$$
\item[v)] $Q_{k,s}^*$ estimator is weekly consistent. For all $\varepsilon > 0$,
$\lim_{k\to \infty} P\left[\left|Q_{k,s}^* - \frac{1}{\alpha}\right| > \varepsilon \right] = 0.$
\end{description}
Applications of the previous results require knowledge about confidence intervals. Therefore, in the the next theorem, we obtain asymptotic normality of these estimators which allows us later on to construct large sample confidence intervals.
{\bf Theorem 4.} If $\mathbf{X} \in Par(\alpha, \delta)$, $\alpha > 0$, $\delta > 0$, then for all $s = 2, 3, ...$, and $k \to \infty$,
\begin{equation}\label{ANi}
\sqrt{k(s+1) - 1}\left[log\left(\frac{X_{(ks, k(s+1) - 1)}}{X_{(k, k(s+1) - 1)}}\right)-\frac{1}{\alpha}log(s)\right] \stackrel{d}{\to} \eta_1, \quad \eta_1 \in N\left(0, \frac{s^2-1}{s\alpha^2}\right),
\end{equation}
\begin{equation}\label{ANii}
\sqrt{k(s+1) - 1}(H_{ks-1} - H_{k-1})\left[\alpha Q_{k,s} - \frac{log(s)}{H_{ks-1} - H_{k-1}}\right] \stackrel{d}{\to} \eta_2, \quad \eta_2 \in N\left(0, \frac{s^2-1}{s}\right),
\end{equation}
\begin{equation}\label{ANiii}\sqrt{k(s+1) - 1}\left[\alpha Q_{k,s}^* - 1\right] \stackrel{d}{\to} \eta_3, \quad \eta_3 \in N\left(0, \frac{s^2-1}{s[log(s)]^2}\right).
\end{equation}
{\bf Proof:} In this case $F^\leftarrow(p) = \frac{\delta}{\sqrt[\alpha]{1 - p}}$, and $f(x) = \frac{\alpha \delta^\alpha}{x^{\alpha + 1}}$. Therefore, $F^\leftarrow\left(\frac{1}{s+1}\right) = \delta \sqrt[\alpha]{\frac{s+1}{s}}$, $F^\leftarrow\left(\frac{s}{s+1}\right) = \delta \sqrt[\alpha]{s+1}$,
$$f\left[F^\leftarrow\left(\frac{1}{s+1}\right)\right] = \frac{\alpha s^{1/\alpha + 1}}{\delta(s + 1)^{1/\alpha + 1}} \in (0, \infty),\quad f\left[F^\leftarrow\left(\frac{s}{s+1}\right)\right] = \frac{\alpha}{\delta(s + 1)^{1/\alpha + 1}}\in (0, \infty),$$
For $k \to \infty$ we have $\frac{s}{(s+1)k-1} \to \frac{1}{s+1}$, $\frac{sk}{(s+1)k-1} \to \frac{s}{s+1}$, $\sqrt{(s+1)k-1}\left(\frac{k}{(s+1)k-1} - \frac{s}{s+1}\right) \to 0$, and $\sqrt{(s+1)k-1}$ $\left(\frac{sk}{(s+1)k-1} - \frac{s}{s+1}\right) \to 0$ therefore we can apply Smirnoff's theorem about the joint asymptotic normality of the order statistics and Theorem 1. In order to determine $a_{F,s}$ and $b_{F,s}$ let us note that $log[F^\leftarrow(p)]' = \frac{1}{\alpha(1-p)}$. Therefore
$$a_{F,s} = \frac{1}{\{log[ F^\leftarrow(p)]\}'|_{p = \frac{1}{s+1}}} = \frac{\alpha s}{s+1}, \quad b_{F,s} = \frac{1}{\{log[ F^\leftarrow(p)]\}'|_{p = \frac{s}{s+1}}}= \frac{\alpha}{s+1}$$
The equalities
$$V = \frac{1}{(s+1)^2} \left(\frac{s}{a_{F,s}^2} - \frac{2}{a_{F,s}b_{F,s}} + \frac{s}{b_{F,s}^2}\right) = \frac{1}{\alpha^2}\left(\frac{1}{s} -\frac{2}{s} + s\right) = \frac{s^2-1}{\alpha^2 s}, \quad log\left(\frac{F^\leftarrow(\frac{s}{s+1})}{F^\leftarrow(\frac{1}{s+1})}\right)
= \frac{1}{\alpha}log(s).$$
lead us to (\ref{ANi}). When we multiply the numerator in (\ref{ANi}) by $\alpha[H_{ks-1} - H_{k-1}]$, and the denominator by $H_{ks-1} - H_{k-1}$, and use that $D\eta_2 = D(\alpha\eta_1) = \alpha^2 D\eta_1$ we obtain (\ref{ANii}). If we multiply both sides of (\ref{ANi}) by $\frac{\alpha}{log(s)}$, and use that $D\eta_3 = D\left(\frac{\alpha}{log(s)}\eta_1\right) = \frac{\alpha^2}{[log(s)]^2} D\eta_1$ we obtain (\ref{ANiii}).
\hfill Q.A.D.
Now we are ready to compute the corresponding confidence intervals. Let us chose $\alpha_0 \in (0, 1)$ and denote by $z_{1-\frac{\alpha_0}{2}}$, $1-\frac{\alpha_0}{2}$ quantile of the standard normal distribution. Using (\ref{ANiii}), and the definition of $Q_{k,s}^*$ we obtain
$$P\left[-z_{1-\frac{\alpha_0}{2}} \leq log(s)\sqrt{\frac{s[k(s+1) - 1]}{s^2-1}}\left(\alpha Q_{k,s}^* - 1\right)
\leq z_{1-\frac{\alpha_0}{2}}\right] \to 1-\alpha_0, \quad k \to \infty.$$
$$P\left[\frac{1}{Q_{k,s}^*}-\frac{z_{1-\frac{\alpha_0}{2}}}{Q_{k,s}^* log(s)} \sqrt{\frac{s^2-1}{s[k(s+1) - 1]}}\leq \alpha \leq \frac{1}{Q_{k,s}^*} +\frac{z_{1-\frac{\alpha_0}{2}}}{Q_{k,s}^* log(s)}\sqrt{\frac{s^2-1}{s[k(s+1) - 1]}}\right] \to 1-\alpha_0, \quad k \to \infty.$$
Therefore for any fixed $s = 2, 3, ...$, the corresponding asymptotic confidence intervals for $\alpha$ when $k \to \infty$ are:
\begin{equation}\label{ciPareto}
\left[\frac{log(s)}{log\frac{\mathbf{X}_{(ks, (s+1)k-1)}}{\mathbf{X}_{(k, (s+1)k-1)}}} -\frac{z_{1-\frac{\alpha_0}{2}}}{log\frac{\mathbf{X}_{(ks, (s+1)k-1)}}{\mathbf{X}_{(k, (s+1)k-1)}}} \sqrt{\frac{s^2-1}{s[k(s+1) - 1]}}; \frac{log(s)}{log\frac{\mathbf{X}_{(ks, (s+1)k-1)}}{\mathbf{X}_{(k, (s+1)k-1)}}} +\frac{z_{1-\frac{\alpha_0}{2}}}{log\frac{\mathbf{X}_{(ks, (s+1)k-1)}}{\mathbf{X}_{(k, (s+1)k-1)}}}\sqrt{\frac{s^2-1
}{s[k(s+1) - 1]}}\right].
\end{equation}
{\it Simulation study}
Let us now depict the rate of convergence of $1/Q_{k,s}^*$ for different values of $\alpha = 0.3, 0.5, 1, 1.5$ and $s = 2, 3, 4, 5$. Figures \ref{fig:Pareto0}-\ref{fig:Pareto1} represent the dependence of $1/Q_{k,s}^*$ and the corresponding confidence intervals, on $k$. They are plotted via software R \cite{R}. The real values of $\alpha$ are plotted via straight dense line. In order to visualise the values of the estimators $\frac{1}{Q_{k,s}^*}$ for any of the lines we have simulated 100 samples of $500(s+1)-1$ realizations of Pareto distributed r.v. with c.d.f. (\ref{Pareto}), correspondingly for $\alpha = 0.3, 0.5, 1, 1.5$. Separately for any fixed $k = 1, 2, ...$ and $s$ the values of $\frac{1}{Q_{k,s}^*}$ are averaged over these 100 samples and presented correspondingly by dense $(s = 2)$, dashed $(s = 3)$, dash-dot $(s = 4)$, and dotted $(s = 5)$ lines. Then, for any fixed $k$, and $s$ we have computed and plotted also $0.95$-confidence intervals (red lines) for $\alpha$, calculated by formula (\ref{ciPareto}) using the averaged values of $1/Q_{k,s}^*$ instead of separate estimators $1/Q_{k,s}^*$. We observe that when $\alpha$ decreases and $s$ increases, the accuracy of the estimators improves. However, because of the sample size $k(s+1)-1$ increases with $s$ we can not chose too big $s$ for small samples.
\begin{figure}
\begin{minipage}[t]{0.5\linewidth}
\includegraphics[scale=.56]{Pareto03}\vspace{-0.3cm}
\end{minipage}
\begin{minipage}[t]{0.49\linewidth}
\includegraphics[scale=.56]{Pareto05}\vspace{-0.3cm}
\end{minipage}
\caption{Pareto case: Dependence of $1/Q_{k,s}^*$ and the corresponding confidence intervals(\ref{ciPareto}) on $k$, for different values of $\alpha$ and $s = 2$(solid lines), $s = 3$(dashed lines), $s = 4$(dash-dot lines), $s = 5$(dotted lines).\label{fig:Pareto0}}
\end{figure}
\begin{figure}
\begin{minipage}[t]{0.5\linewidth}
\includegraphics[scale=.56]{Pareto1}\vspace{-0.3cm}
\end{minipage}
\begin{minipage}[t]{0.49\linewidth}
\includegraphics[scale=.56]{Pareto15}\vspace{-0.3cm}
\end{minipage}
\caption{Pareto case: Dependence of $1/Q_{k,s}^*$ and the corresponding confidence intervals(\ref{ciPareto}) on $k$, for different values of $\alpha$ and $s = 2$(solid lines), $s = 3$(dashed lines), $s = 4$(dash-dot lines), $s = 5$(dotted lines).\label{fig:Pareto1}}
\end{figure}
If we compare these results with those about the well known Hill \cite{Hill}, t-Hill\cite{Stehlik2010,JordanovaPancheva}, Deckers-Einmahl-de Haan \cite{Dekkers1989,EinmahlGuillou}, or Pickands \cite{Pickands} estimators, described very well in Embrechts et al. \cite{embrechts2013modelling}, we observe that in this case $1/Q_{k,s}^*$ estimator have better properties, especially given a small sample.
\section{CONCLUSIONS}
The paper points out good properties of couples of central order statistics for obtaining consistent and asymptotically normal estimators of the parameter of regular variations of the tail of the c.d.f. of the observed r.v. We consider more thoroughly Pareto case, where we transform the logarithm of the fraction of the order statistics in such a way that to obtain at least asymptotically unbiased and asymptotically efficient estimator. However our results about the general case show that an analogous approach could be applied in many other cases of distributions with regularly varying tails of the c.d.f. For example: Fr\'{e}chet, Pareto, Log-logistic, Hill-horror among others. The biggest advantage of the proposed estimators is that they can be very useful for working with relatively small samples.
\section{ACKNOWLEDGMENTS}
The authors are grateful to the bilateral projects Bulgaria - Austria, 2016-2019, Feasible statistical modelling for extremes in ecology and finance, BNSF, Contract number 01/8, 23/08/2017.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,484 |
require 'spec_helper'
describe MediaMagick do
describe ApplicationHelper do
describe 'attachmentUploader' do
let(:album) { Album.new }
before do
album.stub(id: '12345678')
end
def uploader_container_html(model, relation, type, data_attributes, &block)
id = "#{model}-#{relation.to_s}-#{type.to_s}"
classes = "attachmentUploader"
content_tag(:div, nil, id: id, class: classes, data: data_attributes) do
class_eval(&block) if block_given?
end
end
def loader_container_html(model, relation, data_attributes, &block)
id = "#{model}-#{relation.to_s}-loadedAttachments"
classes = "#{relation.to_s} loadedAttachments"
content_tag(:div, nil, id: id, class: classes, data: data_attributes) do
class_eval(&block) if block_given?
end
end
context 'without block' do
before do
# album.stub(id: '12345678')
helper.stub(:render)
end
it 'should render /uploader partial' do
helper.should_receive(:render).with('/uploader')
helper.attachment_uploader(album, :photos, :image)
end
it 'should create a div with data attributes' do
data_attributes = { id: 12345678, model: 'Album', relation: 'photos', partial: '/uploader' }
html = uploader_container_html('Album', 'photos', 'image', data_attributes)
helper.attachment_uploader(album, :photos, :image).should eq(html)
end
context 'using partials' do
it 'creates a div with data-partial attribute' do
html = uploader_container_html('Album', 'photos', 'image', { id: 12345678, model: 'Album', relation: 'photos', partial: 'albums/photo'})
helper.attachment_uploader(album, :photos, :image, partial: 'albums/photo').should eq(html)
end
it 'creates a div with data-loader_partial attribute' do
attrs = { id: 12345678, model: 'Album', relation: 'photos', partial: '/uploader', loader_partial: 'custom_loader'}
html = uploader_container_html('Album', 'photos', 'image', attrs)
helper.attachment_uploader(album, :photos, :image, loader_partial: 'custom_loader').should eq(html)
end
it 'includes partial option on data attributes' do
helper.should_receive(:render).with('albums/photo')
helper.attachment_uploader(album, :photos, :image, partial: 'albums/photo')
end
end
# context 'partial for images' do
# xit 'should create a div with data-partial attributes' do
# helper.attachment_container(album, :photos, as: 'file').should eq(uploader_container_html('album', 'photos', { id: 12345678, model: 'Album', relation: 'photos', partial: '/file'}))
# end
# xit 'should include partial option on data attributes' do
# helper.should_receive(:render).with('/upload', model: album, relations: :photos, newAttachments: {}, loadedAttachments: {}, partial: '/file')
# helper.attachment_container(album, :photos, as: 'file')
# end
# end
context 'embbeded models' do
let(:track) { album.tracks.new }
before do
track.stub(id: '87654321')
end
it 'should create a div with data-embedded-in-id and data-embedded-in-model attributes' do
html = uploader_container_html('Track', 'files', 'file', { id: 87654321, model: 'Track', embedded_in_id: 12345678, embedded_in_model: 'Album', relation: 'files', partial: '/uploader'})
helper.attachment_uploader(track, :files, :file, embedded_in: album).should eq(html)
end
xit 'should render /upload partial' do
helper.should_receive(:render).with('/upload', model: track, relations: :files, newAttachments: {}, loadedAttachments: {}, partial: '/uploader')
helper.attachment_container(track, :files, embedded_in: album)
end
end
context 'customizing newAttachments element' do
xit 'should create a div with data attributes' do
helper.attachment_container(album, :photos, newAttachments: { class: 'thumbnails' }).should eq(uploader_container_html('album', 'photos', { id: 12345678, model: 'Album', relation: 'photos', partial: '/image' }))
end
xit 'should render /upload partial with newAttachments attributes' do
helper.should_receive(:render).with('/upload', model: album, relations: :photos, newAttachments: { class: 'thumbnails' }, loadedAttachments: {}, partial: '/image')
helper.attachment_container(album, :photos, newAttachments: { class: 'thumbnails' })
end
end
context 'customizing loadedAttachments element' do
xit 'should create a div with data attributes' do
helper.attachment_container(album, :photos, loadedAttachments: { class: 'span3' }).should eq(uploader_container_html('album', 'photos', { id: 12345678, model: 'Album', relation: 'photos', partial: '/image' }))
end
xit 'should render /upload partial with loadedAttachments attributes' do
helper.should_receive(:render).with('/upload', model: album, relations: :photos, newAttachments: {}, loadedAttachments: { class: 'span3' }, partial: '/image')
helper.attachment_container(album, :photos, loadedAttachments: { class: 'span3' })
end
end
end
context 'with block' do
xit 'should create a div with data attributes and content inside' do
expected = uploader_container_html('album', 'photos', { id: 12345678, model: 'Album', relation: 'photos', partial: '/image' }) { 'template here' }
helper.attachment_container(album, :photos) { 'template here' }.should eq(expected)
end
end
context 'when model is inside a module' do
let(:model) { Store::Product.new }
before do
helper.stub(:render)
end
describe 'uploader' do
it 'converts string :: to - from model name to create id attribute' do
data_attributes = { id: model.id.to_s, model: 'Store::Product', relation: 'images', partial: '/uploader' }
html = uploader_container_html('Store-Product', 'images', 'image', data_attributes)
helper.attachment_uploader(model, :images, :image).should eq(html)
end
end
describe 'loader' do
it 'converts string :: to - from model name to create id attribute' do
data_attributes = { id: model.id.to_s, model: 'Store::Product', relation: 'images', partial: '/loader' }
html = loader_container_html('Store-Product', :images, data_attributes)
helper.attachment_loader(model, :images).should eq(html)
end
end
end
end
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,055 |
I'm doing the 10×10 spring remix hosted by Un-Fancy and StyleBee. I'm using 10 pieces of clothing to create 10 different outfits, using self-imposed limitations to spark some creativity. All the details – and the clothes (all thrifted)– here.
Day 10 – the finale!
I had planned to go with my Delft-blue Chico's tunic top today to round out using each of my blouses two times, but the navy J. Crew skirt with which I had planned to pair it is out of commission (it needs its lining tacked up). Plus that top feels really summery/casual, more of a Friday or summer slow day top than a mid-week top.
It didn't hurt that my remix has stretched into a third week (due to not working a few days last week); wearing it three times in two weeks might've been a little much.
I'll tell you a secret, though – I topped it off (or will if it gets chilly in the office) with something outside my 10×10 capsule. Egads!
Close up on that pattern – I guess it's more of an orange-y red than a pink!
As Caroline of Un-Fancy likes to remind us, this challenge isn't about sticking to the rules for rules' sake, but learning about our wardrobes and what works/doesn't work for us. The fact that my grey blazer didn't feel spring-y enough is definitely useful learning – a piece of information I'll save for evaluating the blazer's place in my wardrobe, since I had planned to use it as a spring/fall workhorse.
Styling: I went with my simple grey belt to keep within a palette of muted colors (yes I know my jacket is pink and blue, but they're not bright pink and blue, or even particularly saturated).
I notched another belt loop using a tack to tighten my belt up a bit – great trick for belts that fit your hips but that you'd occasionally like to use for your waist. Just make sure to make the hole starting from the finished side – the side that the world sees – so that any rough edges are out of sight.
Finally, I rolled the blazer cuffs to the bracelet length I love, both to show off my cuff bracelet and to make it feel slightly less formal.
I like its cream/off-white color with the blazer's muted pink/blue even though I didn't really need a second layer today – our highs are in the 80s all week. That's spring in Atlanta for ya!
I'll be back tomorrow with what I learned from this experience. In the meantime, scroll down to comment and tell me what you think of this outfit, whether you're a rules person or a rule-breaker/bender, and whether you'd like to see me do 2 or 3 outfit posts like this per week. Now that I have a consistent set up (not glamorous but it works!) and a routine going, I'd be happy to do more of these if you all enjoy them!
I like that dress on you, it looks particularly pretty with the pinkish jacket. That outfit could take you many places! I am probably a rule-bender as regards my clothing, primarily because I have trouble remembering what all the rules are. If the outfit looks pleasing to me in the mirror I figure I am OK-just don't alert the fashion police!
Thanks Jill! I think your take on what makes an outfit a winner is spot on – I think what shines through most is when you feel good in your clothes, whether they break "rules" or not! | {
"redpajama_set_name": "RedPajamaC4"
} | 7,670 |
\subsection{Supplementary materials in Optica Publishing Group journals}
\begin{backmatter}
\bmsection{Disclosures} The authors declare no conflicts of interest.
\bmsection{Data availability} Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
\end{backmatter}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,816 |
{"url":"https:\/\/en.academic.ru\/dic.nsf\/enwiki\/128787","text":"# Orthonormal basis\n\n\ufeff\nOrthonormal basis\n\nIn mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal.[1][2][3] For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for Rn arises in this fashion.\n\nFor a general inner product space V, an orthonormal basis can be used to define normalized orthogonal coordinates on V. Under these coordinates, the inner product becomes dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a finite-dimensional inner product space to the study of Rn under dot product. Every finite-dimensional inner product space has an orthonormal basis, which may be obtained from an arbitrary basis using the Gram\u2013Schmidt process.\n\nIn functional analysis, the concept of an orthonormal basis can be generalized to arbitrary (infinite-dimensional) inner product spaces (or pre-Hilbert spaces).[4] Given a pre-Hilbert space H, an orthonormal basis for H is an orthonormal set of vectors with the property that every vector in H can be written as an infinite linear combination of the vectors in the basis. In this case, the orthonormal basis is sometimes called a Hilbert basis for H. Note that an orthonormal basis in this sense is not generally a Hamel basis, since infinite linear combinations are required. Specifically, the linear span of the basis must be dense in H, but it may not be the entire space.\n\n## Examples\n\n\u2022 The set of vectors {e1\u00a0=\u00a0(1,\u00a00,\u00a00), e2\u00a0=\u00a0(0,\u00a01,\u00a00), e3\u00a0=\u00a0(0,\u00a00,\u00a01)} (the standard basis) forms an orthonormal basis of R3.\nProof: A straightforward computation shows that the inner products of these vectors equals zero, <e1, e2> = <e1, e3> = <e2, e3> = 0 and that each of their magnitudes equals one, ||e1|| = ||e2|| = ||e3|| = 1. This means {e1, e2, e3} is an orthonormal set. All vectors (xyz) in R3 can be expressed as a sum of the basis vectors scaled\n$(x,y,z) = xe_1 + ye_2 + ze_3, \\,$\nso {e1,e2,e3} spans R3 and hence must be a basis. It may also be shown that the standard basis rotated about an axis through the origin or reflected in a plane through the origin forms an orthonormal basis of R3.\n\u2022 The set {fn\u00a0: nZ} with fn(x) = exp(2\u03c0inx) forms an orthonormal basis of the complex space L2([0,1]). This is fundamental to the study of Fourier series.\n\u2022 The set {eb\u00a0: bB} with eb(c)\u00a0=\u00a01 if b\u00a0=\u00a0c and 0 otherwise forms an orthonormal basis of \u00a02(B).\n\u2022 Eigenfunctions of a Sturm\u2013Liouville eigenproblem.\n\u2022 An orthogonal matrix is a matrix whose column vectors form an orthonormal set.\n\n## Basic formula\n\nIf B is an orthogonal basis of H, then every element x of H may be written as\n\n$x=\\sum_{b\\in B}{\\langle x,b\\rangle\\over\\lVert b\\rVert^2} b.$\n\nWhen B is orthonormal, we have instead\n\n$x=\\sum_{b\\in B}\\langle x,b\\rangle b$\n\nand the norm of x can be given by\n\n$\\|x\\|^2=\\sum_{b\\in B}|\\langle x,b\\rangle |^2.$\n\nEven if B is uncountable, only countably many terms in this sum will be non-zero, and the expression is therefore well-defined. This sum is also called the Fourier expansion of x, and the formula is usually known as Parseval's identity. See also Generalized Fourier series.\n\nIf B is an orthonormal basis of H, then H is isomorphic to \u00a02(B) in the following sense: there exists a bijective linear map \u03a6\u00a0: H -> \u00a02(B) such that\n\n$\\langle\\Phi(x),\\Phi(y)\\rangle=\\langle x,y\\rangle$\n\nfor all x and y in H.\n\n## Incomplete orthogonal sets\n\nGiven a Hilbert space H and a set S of mutually orthogonal vectors in H, we can take the smallest closed linear subspace V of H containing S. Then S will be an orthogonal basis of V; which may of course be smaller than H itself, being an incomplete orthogonal set, or be H, when it is a complete orthogonal set.\n\n## Existence\n\nUsing Zorn's lemma and the Gram\u2013Schmidt process (or more simply well-ordering and transfinite recursion), one can show that every Hilbert space admits a basis and thus an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same cardinality (this can be proven in a manner akin to that of the proof of the usual dimension theorem for vector spaces, with separate cases depending on whether the larger basis candidate is countable or not). A Hilbert space is separable if and only if it admits a countable orthonormal basis.\n\n## As a homogeneous space\n\nThe set of orthonormal bases for a space is a principal homogeneous space for the orthogonal group O(n), and is called the Stiefel manifold $V_n(\\mathbf{R}^n)$ of orthonormal n-frames.\n\nIn other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point: given an orthogonal space, there is no natural choice of orthonormal basis, but once one is given one, there is a one-to-one correspondence between bases and the orthogonal group. Concretely, a linear map is determined by where it sends a given basis: just as an invertible map can take any basis to any other basis, an orthogonal map can take any orthogonal basis to any other orthogonal basis.\n\nThe other Stiefel manifolds $V_k(\\mathbf{R}^n)$ for k < n of incomplete orthonormal bases (orthonormal k-frames) are still homogeneous spaces for the orthogonal group, but not principal homogeneous spaces: any k-frame can be taken to any other k-frame by an orthogonal map, but this map is not uniquely determined.\n\n## References\n\n1. ^ Lay, David C. (2006). Linear Algebra and Its Applications (3rd ed.). Addison\u2013Wesley. ISBN\u00a00-321-28713-4.\n2. ^ Strang, Gilbert (2006). Linear Algebra and Its Applications (4th ed.). Brooks Cole. ISBN\u00a00-03-010567-6.\n3. ^ Axler, Sheldon (2002). Linear Algebra Done Right (2nd ed.). Springer. ISBN\u00a00-387-98258-2.\n4. ^ Rudin, Walter (1987). Real & Complex Analysis. McGraw-Hill. ISBN\u00a00-07-054234-1.\n\nWikimedia Foundation. 2010.\n\n### Look at other dictionaries:\n\n\u2022 Basis (linear algebra) \u2014 Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference. In linear algebra, a basis is a set of linearly independent vectors that, in a linear\u2026 \u2026 \u00a0 Wikipedia\n\n\u2022 Orthonormal function system \u2014 An orthonormal function system (ONS) is an orthonormal basis in a vector space of functions. See basis (linear algebra), Fourier analysis, square integrable, Hilbert space for more. Categories: Mathematics stubsLinear algebraFunctional analysis \u2026 \u00a0 Wikipedia\n\n\u2022 orthonormal \u2014 adjective Date: 1932 1. of real valued functions orthogonal with the integral of the square of each function over a specified interval equal to one 2. being or composed of orthogonal elements of unit length < orthonormal basis of a vector space > \u2026 \u00a0 New Collegiate Dictionary\n\n\u2022 orthonormal \u2014 \u02cc adjective Etymology: orth + normal 1. of real valued functions : orthogonal with the integral of the square of each function over a specified interval equal to one 2. : being or composed of orthogonal elements of unit length orthonormal basis\u2026 \u2026 \u00a0 Useful english dictionary\n\n\u2022 Orthonormal \u2014 Als orthonormal (genauer: zueinander orthonormal) werden in der Mathematik Vektoren bezeichnet, die zueinander orthogonal sind und alle die Norm (anschaulich: L\u00e4nge) eins besitzen. Eine Basis eines Vektorraums aus orthonormalen Vektoren bildet\u2026 \u2026 \u00a0 Deutsch Wikipedia\n\n\u2022 Orthonormal frame \u2014 In Riemannian geometry and relativity theory, an orthonormal frame is a tool for studying the structure of a differentiable manifold equipped with a metric. If M is a manifold equipped with a metric g, then an orthonormal frame at a point P of M\u2026 \u2026 \u00a0 Wikipedia\n\n\u2022 Basis function \u2014 In mathematics, particularly numerical analysis, a basis function is an element of the basis for a function space. The term is a degeneration of the term basis vector for a more general vector space; that is, each function in the function space\u2026 \u2026 \u00a0 Wikipedia\n\n\u2022 Standard basis \u2014 In mathematics, the standard basis (also called natural basis or canonical basis) of the n dimensional Euclidean space Rn is the basis obtained by taking the n basis vectors:{ e i : 1leq ileq n}where e i is the vector with a 1 in the ith\u2026 \u2026 \u00a0 Wikipedia\n\n\u2022 Change of basis \u2014 In linear algebra, change of basis refers to the conversion of vectors and linear transformations between matrix representations which have different bases. Contents 1 Expression of a basis 2 Change of basis for vectors 2.1 Tensor proof \u2026 \u00a0 Wikipedia\n\n\u2022 Schauder basis \u2014 In mathematics, a Schauder basis or countable basis is similar to the usual (Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This \u2026 \u00a0 Wikipedia\n\nWe are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.","date":"2020-04-02 21:52:39","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 7, \"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\": 0.9400208592414856, \"perplexity\": 462.11403216733044}, \"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-2020-16\/segments\/1585370508367.57\/warc\/CC-MAIN-20200402204908-20200402234908-00077.warc.gz\"}"} | null | null |
This is the code that creates the shapefile.
Now, I'm trying to combine the geometries based on the value of group_var, so California should stand alone while Oregon and Washington get lumped together into a single geometry. Unfortunately, st_combine doesn't take a grouping variable, and although the aggregate function in sf looks promising, code like this throws an error that group_var cannot be found.
Furthermore, aggregate requires a function as a third argument, presumably because aggregate in the stats package applies the function to the data, but in this case, there isn't any data to apply a function to. I'm just trying to combine/aggregate the shapefiles.
How to analyze and combine overlaping elevation maps into one?
Finding variable names for SF1 Census files?
Drawing border using R in shapefile based on dummy variable? | {
"redpajama_set_name": "RedPajamaC4"
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using MoonSharp.Interpreter.Debugging;
using MoonSharp.Interpreter.Execution;
using MoonSharp.Interpreter.Execution.VM;
namespace MoonSharp.Interpreter.Tree.Statements
{
internal class WhileStatement : Statement
{
private readonly Statement m_Block;
private readonly Expression m_Condition;
private readonly SourceRef m_End;
private readonly RuntimeScopeBlock m_StackFrame;
private readonly SourceRef m_Start;
public WhileStatement(ScriptLoadingContext lcontext)
: base(lcontext)
{
var whileTk = CheckTokenType(lcontext, TokenType.While);
m_Condition = Expression.Expr(lcontext);
m_Start = whileTk.GetSourceRefUpTo(lcontext.Lexer.Current);
//m_Start = BuildSourceRef(context.Start, exp.Stop);
//m_End = BuildSourceRef(context.Stop, context.END());
lcontext.Scope.PushBlock();
CheckTokenType(lcontext, TokenType.Do);
m_Block = new CompositeStatement(lcontext);
m_End = CheckTokenType(lcontext, TokenType.End).GetSourceRef();
m_StackFrame = lcontext.Scope.PopBlock();
lcontext.Source.Refs.Add(m_Start);
lcontext.Source.Refs.Add(m_End);
}
public override void Compile(ByteCode bc)
{
var L = new Loop
{
Scope = m_StackFrame
};
bc.LoopTracker.Loops.Push(L);
bc.PushSourceRef(m_Start);
var start = bc.GetJumpPointForNextInstruction();
m_Condition.Compile(bc);
var jumpend = bc.Emit_Jump(OpCode.Jf, -1);
bc.Emit_Enter(m_StackFrame);
m_Block.Compile(bc);
bc.PopSourceRef();
bc.Emit_Debug("..end");
bc.PushSourceRef(m_End);
bc.Emit_Leave(m_StackFrame);
bc.Emit_Jump(OpCode.Jump, start);
bc.LoopTracker.Loops.Pop();
var exitpoint = bc.GetJumpPointForNextInstruction();
foreach (var i in L.BreakJumps)
i.NumVal = exitpoint;
jumpend.NumVal = exitpoint;
bc.PopSourceRef();
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 6,425 |
.class Landroid/app/LoaderManagerImpl;
.super Landroid/app/LoaderManager;
.source "LoaderManager.java"
# annotations
.annotation system Ldalvik/annotation/MemberClasses;
value = {
Landroid/app/LoaderManagerImpl$LoaderInfo;
}
.end annotation
# static fields
.field static DEBUG:Z = false
.field static final TAG:Ljava/lang/String; = "LoaderManager"
# instance fields
.field mActivity:Landroid/app/Activity;
.field mCreatingLoader:Z
.field final mInactiveLoaders:Landroid/util/SparseArray;
.annotation system Ldalvik/annotation/Signature;
value = {
"Landroid/util/SparseArray",
"<",
"Landroid/app/LoaderManagerImpl$LoaderInfo;",
">;"
}
.end annotation
.end field
.field final mLoaders:Landroid/util/SparseArray;
.annotation system Ldalvik/annotation/Signature;
value = {
"Landroid/util/SparseArray",
"<",
"Landroid/app/LoaderManagerImpl$LoaderInfo;",
">;"
}
.end annotation
.end field
.field mRetaining:Z
.field mRetainingStarted:Z
.field mStarted:Z
.field final mWho:Ljava/lang/String;
# direct methods
.method static constructor <clinit>()V
.locals 1
.prologue
.line 202
const/4 v0, 0x0
sput-boolean v0, Landroid/app/LoaderManagerImpl;->DEBUG:Z
return-void
.end method
.method constructor <init>(Ljava/lang/String;Landroid/app/Activity;Z)V
.locals 2
.param p1, "who" # Ljava/lang/String;
.param p2, "activity" # Landroid/app/Activity;
.param p3, "started" # Z
.prologue
const/4 v1, 0x0
.line 533
invoke-direct {p0}, Landroid/app/LoaderManager;-><init>()V
.line 207
new-instance v0, Landroid/util/SparseArray;
invoke-direct {v0, v1}, Landroid/util/SparseArray;-><init>(I)V
iput-object v0, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
.line 213
new-instance v0, Landroid/util/SparseArray;
invoke-direct {v0, v1}, Landroid/util/SparseArray;-><init>(I)V
iput-object v0, p0, Landroid/app/LoaderManagerImpl;->mInactiveLoaders:Landroid/util/SparseArray;
.line 534
iput-object p1, p0, Landroid/app/LoaderManagerImpl;->mWho:Ljava/lang/String;
.line 535
iput-object p2, p0, Landroid/app/LoaderManagerImpl;->mActivity:Landroid/app/Activity;
.line 536
iput-boolean p3, p0, Landroid/app/LoaderManagerImpl;->mStarted:Z
.line 537
return-void
.end method
.method private createAndInstallLoader(ILandroid/os/Bundle;Landroid/app/LoaderManager$LoaderCallbacks;)Landroid/app/LoaderManagerImpl$LoaderInfo;
.locals 3
.param p1, "id" # I
.param p2, "args" # Landroid/os/Bundle;
.annotation system Ldalvik/annotation/Signature;
value = {
"(I",
"Landroid/os/Bundle;",
"Landroid/app/LoaderManager$LoaderCallbacks",
"<",
"Ljava/lang/Object;",
">;)",
"Landroid/app/LoaderManagerImpl$LoaderInfo;"
}
.end annotation
.prologue
.local p3, "callback":Landroid/app/LoaderManager$LoaderCallbacks;, "Landroid/app/LoaderManager$LoaderCallbacks<Ljava/lang/Object;>;"
const/4 v2, 0x0
.line 554
const/4 v1, 0x1
:try_start_0
iput-boolean v1, p0, Landroid/app/LoaderManagerImpl;->mCreatingLoader:Z
.line 555
invoke-direct {p0, p1, p2, p3}, Landroid/app/LoaderManagerImpl;->createLoader(ILandroid/os/Bundle;Landroid/app/LoaderManager$LoaderCallbacks;)Landroid/app/LoaderManagerImpl$LoaderInfo;
move-result-object v0
.line 556
.local v0, "info":Landroid/app/LoaderManagerImpl$LoaderInfo;
invoke-virtual {p0, v0}, Landroid/app/LoaderManagerImpl;->installLoader(Landroid/app/LoaderManagerImpl$LoaderInfo;)V
:try_end_0
.catchall {:try_start_0 .. :try_end_0} :catchall_0
.line 559
iput-boolean v2, p0, Landroid/app/LoaderManagerImpl;->mCreatingLoader:Z
return-object v0
.end local v0 # "info":Landroid/app/LoaderManagerImpl$LoaderInfo;
:catchall_0
move-exception v1
iput-boolean v2, p0, Landroid/app/LoaderManagerImpl;->mCreatingLoader:Z
throw v1
.end method
.method private createLoader(ILandroid/os/Bundle;Landroid/app/LoaderManager$LoaderCallbacks;)Landroid/app/LoaderManagerImpl$LoaderInfo;
.locals 2
.param p1, "id" # I
.param p2, "args" # Landroid/os/Bundle;
.annotation system Ldalvik/annotation/Signature;
value = {
"(I",
"Landroid/os/Bundle;",
"Landroid/app/LoaderManager$LoaderCallbacks",
"<",
"Ljava/lang/Object;",
">;)",
"Landroid/app/LoaderManagerImpl$LoaderInfo;"
}
.end annotation
.prologue
.line 545
.local p3, "callback":Landroid/app/LoaderManager$LoaderCallbacks;, "Landroid/app/LoaderManager$LoaderCallbacks<Ljava/lang/Object;>;"
new-instance v0, Landroid/app/LoaderManagerImpl$LoaderInfo;
invoke-direct {v0, p0, p1, p2, p3}, Landroid/app/LoaderManagerImpl$LoaderInfo;-><init>(Landroid/app/LoaderManagerImpl;ILandroid/os/Bundle;Landroid/app/LoaderManager$LoaderCallbacks;)V
.line 546
.local v0, "info":Landroid/app/LoaderManagerImpl$LoaderInfo;
invoke-interface {p3, p1, p2}, Landroid/app/LoaderManager$LoaderCallbacks;->onCreateLoader(ILandroid/os/Bundle;)Landroid/content/Loader;
move-result-object v1
.line 547
.local v1, "loader":Landroid/content/Loader;, "Landroid/content/Loader<Ljava/lang/Object;>;"
iput-object v1, v0, Landroid/app/LoaderManagerImpl$LoaderInfo;->mLoader:Landroid/content/Loader;
.line 548
return-object v0
.end method
# virtual methods
.method public destroyLoader(I)V
.locals 5
.param p1, "id" # I
.prologue
.line 723
iget-boolean v2, p0, Landroid/app/LoaderManagerImpl;->mCreatingLoader:Z
if-eqz v2, :cond_0
.line 724
new-instance v2, Ljava/lang/IllegalStateException;
const-string v3, "Called while creating a loader"
invoke-direct {v2, v3}, Ljava/lang/IllegalStateException;-><init>(Ljava/lang/String;)V
throw v2
.line 727
:cond_0
sget-boolean v2, Landroid/app/LoaderManagerImpl;->DEBUG:Z
if-eqz v2, :cond_1
const-string v2, "LoaderManager"
new-instance v3, Ljava/lang/StringBuilder;
invoke-direct {v3}, Ljava/lang/StringBuilder;-><init>()V
const-string v4, "destroyLoader in "
invoke-virtual {v3, v4}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v3
invoke-virtual {v3, p0}, Ljava/lang/StringBuilder;->append(Ljava/lang/Object;)Ljava/lang/StringBuilder;
move-result-object v3
const-string v4, " of "
invoke-virtual {v3, v4}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v3
invoke-virtual {v3, p1}, Ljava/lang/StringBuilder;->append(I)Ljava/lang/StringBuilder;
move-result-object v3
invoke-virtual {v3}, Ljava/lang/StringBuilder;->toString()Ljava/lang/String;
move-result-object v3
invoke-static {v2, v3}, Landroid/util/Log;->v(Ljava/lang/String;Ljava/lang/String;)I
.line 728
:cond_1
iget-object v2, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v2, p1}, Landroid/util/SparseArray;->indexOfKey(I)I
move-result v0
.line 729
.local v0, "idx":I
if-ltz v0, :cond_2
.line 730
iget-object v2, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v2, v0}, Landroid/util/SparseArray;->valueAt(I)Ljava/lang/Object;
move-result-object v1
check-cast v1, Landroid/app/LoaderManagerImpl$LoaderInfo;
.line 731
.local v1, "info":Landroid/app/LoaderManagerImpl$LoaderInfo;
iget-object v2, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v2, v0}, Landroid/util/SparseArray;->removeAt(I)V
.line 732
invoke-virtual {v1}, Landroid/app/LoaderManagerImpl$LoaderInfo;->destroy()V
.line 734
.end local v1 # "info":Landroid/app/LoaderManagerImpl$LoaderInfo;
:cond_2
iget-object v2, p0, Landroid/app/LoaderManagerImpl;->mInactiveLoaders:Landroid/util/SparseArray;
invoke-virtual {v2, p1}, Landroid/util/SparseArray;->indexOfKey(I)I
move-result v0
.line 735
if-ltz v0, :cond_3
.line 736
iget-object v2, p0, Landroid/app/LoaderManagerImpl;->mInactiveLoaders:Landroid/util/SparseArray;
invoke-virtual {v2, v0}, Landroid/util/SparseArray;->valueAt(I)Ljava/lang/Object;
move-result-object v1
check-cast v1, Landroid/app/LoaderManagerImpl$LoaderInfo;
.line 737
.restart local v1 # "info":Landroid/app/LoaderManagerImpl$LoaderInfo;
iget-object v2, p0, Landroid/app/LoaderManagerImpl;->mInactiveLoaders:Landroid/util/SparseArray;
invoke-virtual {v2, v0}, Landroid/util/SparseArray;->removeAt(I)V
.line 738
invoke-virtual {v1}, Landroid/app/LoaderManagerImpl$LoaderInfo;->destroy()V
.line 740
.end local v1 # "info":Landroid/app/LoaderManagerImpl$LoaderInfo;
:cond_3
iget-object v2, p0, Landroid/app/LoaderManagerImpl;->mActivity:Landroid/app/Activity;
if-eqz v2, :cond_4
invoke-virtual {p0}, Landroid/app/LoaderManagerImpl;->hasRunningLoaders()Z
move-result v2
if-nez v2, :cond_4
.line 741
iget-object v2, p0, Landroid/app/LoaderManagerImpl;->mActivity:Landroid/app/Activity;
iget-object v2, v2, Landroid/app/Activity;->mFragments:Landroid/app/FragmentManagerImpl;
invoke-virtual {v2}, Landroid/app/FragmentManagerImpl;->startPendingDeferredFragments()V
.line 743
:cond_4
return-void
.end method
.method doDestroy()V
.locals 4
.prologue
.line 838
iget-boolean v1, p0, Landroid/app/LoaderManagerImpl;->mRetaining:Z
if-nez v1, :cond_2
.line 839
sget-boolean v1, Landroid/app/LoaderManagerImpl;->DEBUG:Z
if-eqz v1, :cond_0
const-string v1, "LoaderManager"
new-instance v2, Ljava/lang/StringBuilder;
invoke-direct {v2}, Ljava/lang/StringBuilder;-><init>()V
const-string v3, "Destroying Active in "
invoke-virtual {v2, v3}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v2
invoke-virtual {v2, p0}, Ljava/lang/StringBuilder;->append(Ljava/lang/Object;)Ljava/lang/StringBuilder;
move-result-object v2
invoke-virtual {v2}, Ljava/lang/StringBuilder;->toString()Ljava/lang/String;
move-result-object v2
invoke-static {v1, v2}, Landroid/util/Log;->v(Ljava/lang/String;Ljava/lang/String;)I
.line 840
:cond_0
iget-object v1, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v1}, Landroid/util/SparseArray;->size()I
move-result v1
add-int/lit8 v0, v1, -0x1
.local v0, "i":I
:goto_0
if-ltz v0, :cond_1
.line 841
iget-object v1, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v1, v0}, Landroid/util/SparseArray;->valueAt(I)Ljava/lang/Object;
move-result-object v1
check-cast v1, Landroid/app/LoaderManagerImpl$LoaderInfo;
invoke-virtual {v1}, Landroid/app/LoaderManagerImpl$LoaderInfo;->destroy()V
.line 840
add-int/lit8 v0, v0, -0x1
goto :goto_0
.line 843
:cond_1
iget-object v1, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v1}, Landroid/util/SparseArray;->clear()V
.line 846
.end local v0 # "i":I
:cond_2
sget-boolean v1, Landroid/app/LoaderManagerImpl;->DEBUG:Z
if-eqz v1, :cond_3
const-string v1, "LoaderManager"
new-instance v2, Ljava/lang/StringBuilder;
invoke-direct {v2}, Ljava/lang/StringBuilder;-><init>()V
const-string v3, "Destroying Inactive in "
invoke-virtual {v2, v3}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v2
invoke-virtual {v2, p0}, Ljava/lang/StringBuilder;->append(Ljava/lang/Object;)Ljava/lang/StringBuilder;
move-result-object v2
invoke-virtual {v2}, Ljava/lang/StringBuilder;->toString()Ljava/lang/String;
move-result-object v2
invoke-static {v1, v2}, Landroid/util/Log;->v(Ljava/lang/String;Ljava/lang/String;)I
.line 847
:cond_3
iget-object v1, p0, Landroid/app/LoaderManagerImpl;->mInactiveLoaders:Landroid/util/SparseArray;
invoke-virtual {v1}, Landroid/util/SparseArray;->size()I
move-result v1
add-int/lit8 v0, v1, -0x1
.restart local v0 # "i":I
:goto_1
if-ltz v0, :cond_4
.line 848
iget-object v1, p0, Landroid/app/LoaderManagerImpl;->mInactiveLoaders:Landroid/util/SparseArray;
invoke-virtual {v1, v0}, Landroid/util/SparseArray;->valueAt(I)Ljava/lang/Object;
move-result-object v1
check-cast v1, Landroid/app/LoaderManagerImpl$LoaderInfo;
invoke-virtual {v1}, Landroid/app/LoaderManagerImpl$LoaderInfo;->destroy()V
.line 847
add-int/lit8 v0, v0, -0x1
goto :goto_1
.line 850
:cond_4
iget-object v1, p0, Landroid/app/LoaderManagerImpl;->mInactiveLoaders:Landroid/util/SparseArray;
invoke-virtual {v1}, Landroid/util/SparseArray;->clear()V
.line 851
return-void
.end method
.method doReportNextStart()V
.locals 3
.prologue
.line 826
iget-object v1, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v1}, Landroid/util/SparseArray;->size()I
move-result v1
add-int/lit8 v0, v1, -0x1
.local v0, "i":I
:goto_0
if-ltz v0, :cond_0
.line 827
iget-object v1, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v1, v0}, Landroid/util/SparseArray;->valueAt(I)Ljava/lang/Object;
move-result-object v1
check-cast v1, Landroid/app/LoaderManagerImpl$LoaderInfo;
const/4 v2, 0x1
iput-boolean v2, v1, Landroid/app/LoaderManagerImpl$LoaderInfo;->mReportNextStart:Z
.line 826
add-int/lit8 v0, v0, -0x1
goto :goto_0
.line 829
:cond_0
return-void
.end method
.method doReportStart()V
.locals 2
.prologue
.line 832
iget-object v1, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v1}, Landroid/util/SparseArray;->size()I
move-result v1
add-int/lit8 v0, v1, -0x1
.local v0, "i":I
:goto_0
if-ltz v0, :cond_0
.line 833
iget-object v1, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v1, v0}, Landroid/util/SparseArray;->valueAt(I)Ljava/lang/Object;
move-result-object v1
check-cast v1, Landroid/app/LoaderManagerImpl$LoaderInfo;
invoke-virtual {v1}, Landroid/app/LoaderManagerImpl$LoaderInfo;->reportStart()V
.line 832
add-int/lit8 v0, v0, -0x1
goto :goto_0
.line 835
:cond_0
return-void
.end method
.method doRetain()V
.locals 5
.prologue
.line 799
sget-boolean v2, Landroid/app/LoaderManagerImpl;->DEBUG:Z
if-eqz v2, :cond_0
const-string v2, "LoaderManager"
new-instance v3, Ljava/lang/StringBuilder;
invoke-direct {v3}, Ljava/lang/StringBuilder;-><init>()V
const-string v4, "Retaining in "
invoke-virtual {v3, v4}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v3
invoke-virtual {v3, p0}, Ljava/lang/StringBuilder;->append(Ljava/lang/Object;)Ljava/lang/StringBuilder;
move-result-object v3
invoke-virtual {v3}, Ljava/lang/StringBuilder;->toString()Ljava/lang/String;
move-result-object v3
invoke-static {v2, v3}, Landroid/util/Log;->v(Ljava/lang/String;Ljava/lang/String;)I
.line 800
:cond_0
iget-boolean v2, p0, Landroid/app/LoaderManagerImpl;->mStarted:Z
if-nez v2, :cond_2
.line 801
new-instance v0, Ljava/lang/RuntimeException;
const-string v2, "here"
invoke-direct {v0, v2}, Ljava/lang/RuntimeException;-><init>(Ljava/lang/String;)V
.line 802
.local v0, "e":Ljava/lang/RuntimeException;
invoke-virtual {v0}, Ljava/lang/RuntimeException;->fillInStackTrace()Ljava/lang/Throwable;
.line 803
const-string v2, "LoaderManager"
new-instance v3, Ljava/lang/StringBuilder;
invoke-direct {v3}, Ljava/lang/StringBuilder;-><init>()V
const-string v4, "Called doRetain when not started: "
invoke-virtual {v3, v4}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v3
invoke-virtual {v3, p0}, Ljava/lang/StringBuilder;->append(Ljava/lang/Object;)Ljava/lang/StringBuilder;
move-result-object v3
invoke-virtual {v3}, Ljava/lang/StringBuilder;->toString()Ljava/lang/String;
move-result-object v3
invoke-static {v2, v3, v0}, Landroid/util/Log;->w(Ljava/lang/String;Ljava/lang/String;Ljava/lang/Throwable;)I
.line 812
.end local v0 # "e":Ljava/lang/RuntimeException;
:cond_1
return-void
.line 807
:cond_2
const/4 v2, 0x1
iput-boolean v2, p0, Landroid/app/LoaderManagerImpl;->mRetaining:Z
.line 808
const/4 v2, 0x0
iput-boolean v2, p0, Landroid/app/LoaderManagerImpl;->mStarted:Z
.line 809
iget-object v2, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v2}, Landroid/util/SparseArray;->size()I
move-result v2
add-int/lit8 v1, v2, -0x1
.local v1, "i":I
:goto_0
if-ltz v1, :cond_1
.line 810
iget-object v2, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v2, v1}, Landroid/util/SparseArray;->valueAt(I)Ljava/lang/Object;
move-result-object v2
check-cast v2, Landroid/app/LoaderManagerImpl$LoaderInfo;
invoke-virtual {v2}, Landroid/app/LoaderManagerImpl$LoaderInfo;->retain()V
.line 809
add-int/lit8 v1, v1, -0x1
goto :goto_0
.end method
.method doStart()V
.locals 5
.prologue
.line 766
sget-boolean v2, Landroid/app/LoaderManagerImpl;->DEBUG:Z
if-eqz v2, :cond_0
const-string v2, "LoaderManager"
new-instance v3, Ljava/lang/StringBuilder;
invoke-direct {v3}, Ljava/lang/StringBuilder;-><init>()V
const-string v4, "Starting in "
invoke-virtual {v3, v4}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v3
invoke-virtual {v3, p0}, Ljava/lang/StringBuilder;->append(Ljava/lang/Object;)Ljava/lang/StringBuilder;
move-result-object v3
invoke-virtual {v3}, Ljava/lang/StringBuilder;->toString()Ljava/lang/String;
move-result-object v3
invoke-static {v2, v3}, Landroid/util/Log;->v(Ljava/lang/String;Ljava/lang/String;)I
.line 767
:cond_0
iget-boolean v2, p0, Landroid/app/LoaderManagerImpl;->mStarted:Z
if-eqz v2, :cond_2
.line 768
new-instance v0, Ljava/lang/RuntimeException;
const-string v2, "here"
invoke-direct {v0, v2}, Ljava/lang/RuntimeException;-><init>(Ljava/lang/String;)V
.line 769
.local v0, "e":Ljava/lang/RuntimeException;
invoke-virtual {v0}, Ljava/lang/RuntimeException;->fillInStackTrace()Ljava/lang/Throwable;
.line 770
const-string v2, "LoaderManager"
new-instance v3, Ljava/lang/StringBuilder;
invoke-direct {v3}, Ljava/lang/StringBuilder;-><init>()V
const-string v4, "Called doStart when already started: "
invoke-virtual {v3, v4}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v3
invoke-virtual {v3, p0}, Ljava/lang/StringBuilder;->append(Ljava/lang/Object;)Ljava/lang/StringBuilder;
move-result-object v3
invoke-virtual {v3}, Ljava/lang/StringBuilder;->toString()Ljava/lang/String;
move-result-object v3
invoke-static {v2, v3, v0}, Landroid/util/Log;->w(Ljava/lang/String;Ljava/lang/String;Ljava/lang/Throwable;)I
.line 781
.end local v0 # "e":Ljava/lang/RuntimeException;
:cond_1
return-void
.line 774
:cond_2
const/4 v2, 0x1
iput-boolean v2, p0, Landroid/app/LoaderManagerImpl;->mStarted:Z
.line 778
iget-object v2, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v2}, Landroid/util/SparseArray;->size()I
move-result v2
add-int/lit8 v1, v2, -0x1
.local v1, "i":I
:goto_0
if-ltz v1, :cond_1
.line 779
iget-object v2, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v2, v1}, Landroid/util/SparseArray;->valueAt(I)Ljava/lang/Object;
move-result-object v2
check-cast v2, Landroid/app/LoaderManagerImpl$LoaderInfo;
invoke-virtual {v2}, Landroid/app/LoaderManagerImpl$LoaderInfo;->start()V
.line 778
add-int/lit8 v1, v1, -0x1
goto :goto_0
.end method
.method doStop()V
.locals 5
.prologue
.line 784
sget-boolean v2, Landroid/app/LoaderManagerImpl;->DEBUG:Z
if-eqz v2, :cond_0
const-string v2, "LoaderManager"
new-instance v3, Ljava/lang/StringBuilder;
invoke-direct {v3}, Ljava/lang/StringBuilder;-><init>()V
const-string v4, "Stopping in "
invoke-virtual {v3, v4}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v3
invoke-virtual {v3, p0}, Ljava/lang/StringBuilder;->append(Ljava/lang/Object;)Ljava/lang/StringBuilder;
move-result-object v3
invoke-virtual {v3}, Ljava/lang/StringBuilder;->toString()Ljava/lang/String;
move-result-object v3
invoke-static {v2, v3}, Landroid/util/Log;->v(Ljava/lang/String;Ljava/lang/String;)I
.line 785
:cond_0
iget-boolean v2, p0, Landroid/app/LoaderManagerImpl;->mStarted:Z
if-nez v2, :cond_1
.line 786
new-instance v0, Ljava/lang/RuntimeException;
const-string v2, "here"
invoke-direct {v0, v2}, Ljava/lang/RuntimeException;-><init>(Ljava/lang/String;)V
.line 787
.local v0, "e":Ljava/lang/RuntimeException;
invoke-virtual {v0}, Ljava/lang/RuntimeException;->fillInStackTrace()Ljava/lang/Throwable;
.line 788
const-string v2, "LoaderManager"
new-instance v3, Ljava/lang/StringBuilder;
invoke-direct {v3}, Ljava/lang/StringBuilder;-><init>()V
const-string v4, "Called doStop when not started: "
invoke-virtual {v3, v4}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v3
invoke-virtual {v3, p0}, Ljava/lang/StringBuilder;->append(Ljava/lang/Object;)Ljava/lang/StringBuilder;
move-result-object v3
invoke-virtual {v3}, Ljava/lang/StringBuilder;->toString()Ljava/lang/String;
move-result-object v3
invoke-static {v2, v3, v0}, Landroid/util/Log;->w(Ljava/lang/String;Ljava/lang/String;Ljava/lang/Throwable;)I
.line 796
.end local v0 # "e":Ljava/lang/RuntimeException;
:goto_0
return-void
.line 792
:cond_1
iget-object v2, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v2}, Landroid/util/SparseArray;->size()I
move-result v2
add-int/lit8 v1, v2, -0x1
.local v1, "i":I
:goto_1
if-ltz v1, :cond_2
.line 793
iget-object v2, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v2, v1}, Landroid/util/SparseArray;->valueAt(I)Ljava/lang/Object;
move-result-object v2
check-cast v2, Landroid/app/LoaderManagerImpl$LoaderInfo;
invoke-virtual {v2}, Landroid/app/LoaderManagerImpl$LoaderInfo;->stop()V
.line 792
add-int/lit8 v1, v1, -0x1
goto :goto_1
.line 795
:cond_2
const/4 v2, 0x0
iput-boolean v2, p0, Landroid/app/LoaderManagerImpl;->mStarted:Z
goto :goto_0
.end method
.method public dump(Ljava/lang/String;Ljava/io/FileDescriptor;Ljava/io/PrintWriter;[Ljava/lang/String;)V
.locals 5
.param p1, "prefix" # Ljava/lang/String;
.param p2, "fd" # Ljava/io/FileDescriptor;
.param p3, "writer" # Ljava/io/PrintWriter;
.param p4, "args" # [Ljava/lang/String;
.prologue
.line 866
iget-object v3, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v3}, Landroid/util/SparseArray;->size()I
move-result v3
if-lez v3, :cond_0
.line 867
invoke-virtual {p3, p1}, Ljava/io/PrintWriter;->print(Ljava/lang/String;)V
const-string v3, "Active Loaders:"
invoke-virtual {p3, v3}, Ljava/io/PrintWriter;->println(Ljava/lang/String;)V
.line 868
new-instance v3, Ljava/lang/StringBuilder;
invoke-direct {v3}, Ljava/lang/StringBuilder;-><init>()V
invoke-virtual {v3, p1}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v3
const-string v4, " "
invoke-virtual {v3, v4}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v3
invoke-virtual {v3}, Ljava/lang/StringBuilder;->toString()Ljava/lang/String;
move-result-object v1
.line 869
.local v1, "innerPrefix":Ljava/lang/String;
const/4 v0, 0x0
.local v0, "i":I
:goto_0
iget-object v3, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v3}, Landroid/util/SparseArray;->size()I
move-result v3
if-ge v0, v3, :cond_0
.line 870
iget-object v3, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v3, v0}, Landroid/util/SparseArray;->valueAt(I)Ljava/lang/Object;
move-result-object v2
check-cast v2, Landroid/app/LoaderManagerImpl$LoaderInfo;
.line 871
.local v2, "li":Landroid/app/LoaderManagerImpl$LoaderInfo;
invoke-virtual {p3, p1}, Ljava/io/PrintWriter;->print(Ljava/lang/String;)V
const-string v3, " #"
invoke-virtual {p3, v3}, Ljava/io/PrintWriter;->print(Ljava/lang/String;)V
iget-object v3, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v3, v0}, Landroid/util/SparseArray;->keyAt(I)I
move-result v3
invoke-virtual {p3, v3}, Ljava/io/PrintWriter;->print(I)V
.line 872
const-string v3, ": "
invoke-virtual {p3, v3}, Ljava/io/PrintWriter;->print(Ljava/lang/String;)V
invoke-virtual {v2}, Landroid/app/LoaderManagerImpl$LoaderInfo;->toString()Ljava/lang/String;
move-result-object v3
invoke-virtual {p3, v3}, Ljava/io/PrintWriter;->println(Ljava/lang/String;)V
.line 873
invoke-virtual {v2, v1, p2, p3, p4}, Landroid/app/LoaderManagerImpl$LoaderInfo;->dump(Ljava/lang/String;Ljava/io/FileDescriptor;Ljava/io/PrintWriter;[Ljava/lang/String;)V
.line 869
add-int/lit8 v0, v0, 0x1
goto :goto_0
.line 876
.end local v0 # "i":I
.end local v1 # "innerPrefix":Ljava/lang/String;
.end local v2 # "li":Landroid/app/LoaderManagerImpl$LoaderInfo;
:cond_0
iget-object v3, p0, Landroid/app/LoaderManagerImpl;->mInactiveLoaders:Landroid/util/SparseArray;
invoke-virtual {v3}, Landroid/util/SparseArray;->size()I
move-result v3
if-lez v3, :cond_1
.line 877
invoke-virtual {p3, p1}, Ljava/io/PrintWriter;->print(Ljava/lang/String;)V
const-string v3, "Inactive Loaders:"
invoke-virtual {p3, v3}, Ljava/io/PrintWriter;->println(Ljava/lang/String;)V
.line 878
new-instance v3, Ljava/lang/StringBuilder;
invoke-direct {v3}, Ljava/lang/StringBuilder;-><init>()V
invoke-virtual {v3, p1}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v3
const-string v4, " "
invoke-virtual {v3, v4}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v3
invoke-virtual {v3}, Ljava/lang/StringBuilder;->toString()Ljava/lang/String;
move-result-object v1
.line 879
.restart local v1 # "innerPrefix":Ljava/lang/String;
const/4 v0, 0x0
.restart local v0 # "i":I
:goto_1
iget-object v3, p0, Landroid/app/LoaderManagerImpl;->mInactiveLoaders:Landroid/util/SparseArray;
invoke-virtual {v3}, Landroid/util/SparseArray;->size()I
move-result v3
if-ge v0, v3, :cond_1
.line 880
iget-object v3, p0, Landroid/app/LoaderManagerImpl;->mInactiveLoaders:Landroid/util/SparseArray;
invoke-virtual {v3, v0}, Landroid/util/SparseArray;->valueAt(I)Ljava/lang/Object;
move-result-object v2
check-cast v2, Landroid/app/LoaderManagerImpl$LoaderInfo;
.line 881
.restart local v2 # "li":Landroid/app/LoaderManagerImpl$LoaderInfo;
invoke-virtual {p3, p1}, Ljava/io/PrintWriter;->print(Ljava/lang/String;)V
const-string v3, " #"
invoke-virtual {p3, v3}, Ljava/io/PrintWriter;->print(Ljava/lang/String;)V
iget-object v3, p0, Landroid/app/LoaderManagerImpl;->mInactiveLoaders:Landroid/util/SparseArray;
invoke-virtual {v3, v0}, Landroid/util/SparseArray;->keyAt(I)I
move-result v3
invoke-virtual {p3, v3}, Ljava/io/PrintWriter;->print(I)V
.line 882
const-string v3, ": "
invoke-virtual {p3, v3}, Ljava/io/PrintWriter;->print(Ljava/lang/String;)V
invoke-virtual {v2}, Landroid/app/LoaderManagerImpl$LoaderInfo;->toString()Ljava/lang/String;
move-result-object v3
invoke-virtual {p3, v3}, Ljava/io/PrintWriter;->println(Ljava/lang/String;)V
.line 883
invoke-virtual {v2, v1, p2, p3, p4}, Landroid/app/LoaderManagerImpl$LoaderInfo;->dump(Ljava/lang/String;Ljava/io/FileDescriptor;Ljava/io/PrintWriter;[Ljava/lang/String;)V
.line 879
add-int/lit8 v0, v0, 0x1
goto :goto_1
.line 886
.end local v0 # "i":I
.end local v1 # "innerPrefix":Ljava/lang/String;
.end local v2 # "li":Landroid/app/LoaderManagerImpl$LoaderInfo;
:cond_1
return-void
.end method
.method finishRetain()V
.locals 4
.prologue
.line 815
iget-boolean v1, p0, Landroid/app/LoaderManagerImpl;->mRetaining:Z
if-eqz v1, :cond_1
.line 816
sget-boolean v1, Landroid/app/LoaderManagerImpl;->DEBUG:Z
if-eqz v1, :cond_0
const-string v1, "LoaderManager"
new-instance v2, Ljava/lang/StringBuilder;
invoke-direct {v2}, Ljava/lang/StringBuilder;-><init>()V
const-string v3, "Finished Retaining in "
invoke-virtual {v2, v3}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v2
invoke-virtual {v2, p0}, Ljava/lang/StringBuilder;->append(Ljava/lang/Object;)Ljava/lang/StringBuilder;
move-result-object v2
invoke-virtual {v2}, Ljava/lang/StringBuilder;->toString()Ljava/lang/String;
move-result-object v2
invoke-static {v1, v2}, Landroid/util/Log;->v(Ljava/lang/String;Ljava/lang/String;)I
.line 818
:cond_0
const/4 v1, 0x0
iput-boolean v1, p0, Landroid/app/LoaderManagerImpl;->mRetaining:Z
.line 819
iget-object v1, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v1}, Landroid/util/SparseArray;->size()I
move-result v1
add-int/lit8 v0, v1, -0x1
.local v0, "i":I
:goto_0
if-ltz v0, :cond_1
.line 820
iget-object v1, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v1, v0}, Landroid/util/SparseArray;->valueAt(I)Ljava/lang/Object;
move-result-object v1
check-cast v1, Landroid/app/LoaderManagerImpl$LoaderInfo;
invoke-virtual {v1}, Landroid/app/LoaderManagerImpl$LoaderInfo;->finishRetain()V
.line 819
add-int/lit8 v0, v0, -0x1
goto :goto_0
.line 823
.end local v0 # "i":I
:cond_1
return-void
.end method
.method public getLoader(I)Landroid/content/Loader;
.locals 3
.param p1, "id" # I
.annotation system Ldalvik/annotation/Signature;
value = {
"<D:",
"Ljava/lang/Object;",
">(I)",
"Landroid/content/Loader",
"<TD;>;"
}
.end annotation
.prologue
.line 751
iget-boolean v1, p0, Landroid/app/LoaderManagerImpl;->mCreatingLoader:Z
if-eqz v1, :cond_0
.line 752
new-instance v1, Ljava/lang/IllegalStateException;
const-string v2, "Called while creating a loader"
invoke-direct {v1, v2}, Ljava/lang/IllegalStateException;-><init>(Ljava/lang/String;)V
throw v1
.line 755
:cond_0
iget-object v1, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v1, p1}, Landroid/util/SparseArray;->get(I)Ljava/lang/Object;
move-result-object v0
check-cast v0, Landroid/app/LoaderManagerImpl$LoaderInfo;
.line 756
.local v0, "loaderInfo":Landroid/app/LoaderManagerImpl$LoaderInfo;
if-eqz v0, :cond_2
.line 757
iget-object v1, v0, Landroid/app/LoaderManagerImpl$LoaderInfo;->mPendingLoader:Landroid/app/LoaderManagerImpl$LoaderInfo;
if-eqz v1, :cond_1
.line 758
iget-object v1, v0, Landroid/app/LoaderManagerImpl$LoaderInfo;->mPendingLoader:Landroid/app/LoaderManagerImpl$LoaderInfo;
iget-object v1, v1, Landroid/app/LoaderManagerImpl$LoaderInfo;->mLoader:Landroid/content/Loader;
.line 762
:goto_0
return-object v1
.line 760
:cond_1
iget-object v1, v0, Landroid/app/LoaderManagerImpl$LoaderInfo;->mLoader:Landroid/content/Loader;
goto :goto_0
.line 762
:cond_2
const/4 v1, 0x0
goto :goto_0
.end method
.method public hasRunningLoaders()Z
.locals 5
.prologue
.line 889
const/4 v3, 0x0
.line 890
.local v3, "loadersRunning":Z
iget-object v4, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v4}, Landroid/util/SparseArray;->size()I
move-result v0
.line 891
.local v0, "count":I
const/4 v1, 0x0
.local v1, "i":I
:goto_0
if-ge v1, v0, :cond_1
.line 892
iget-object v4, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v4, v1}, Landroid/util/SparseArray;->valueAt(I)Ljava/lang/Object;
move-result-object v2
check-cast v2, Landroid/app/LoaderManagerImpl$LoaderInfo;
.line 893
.local v2, "li":Landroid/app/LoaderManagerImpl$LoaderInfo;
iget-boolean v4, v2, Landroid/app/LoaderManagerImpl$LoaderInfo;->mStarted:Z
if-eqz v4, :cond_0
iget-boolean v4, v2, Landroid/app/LoaderManagerImpl$LoaderInfo;->mDeliveredData:Z
if-nez v4, :cond_0
const/4 v4, 0x1
:goto_1
or-int/2addr v3, v4
.line 891
add-int/lit8 v1, v1, 0x1
goto :goto_0
.line 893
:cond_0
const/4 v4, 0x0
goto :goto_1
.line 895
.end local v2 # "li":Landroid/app/LoaderManagerImpl$LoaderInfo;
:cond_1
return v3
.end method
.method public initLoader(ILandroid/os/Bundle;Landroid/app/LoaderManager$LoaderCallbacks;)Landroid/content/Loader;
.locals 4
.param p1, "id" # I
.param p2, "args" # Landroid/os/Bundle;
.annotation system Ldalvik/annotation/Signature;
value = {
"<D:",
"Ljava/lang/Object;",
">(I",
"Landroid/os/Bundle;",
"Landroid/app/LoaderManager$LoaderCallbacks",
"<TD;>;)",
"Landroid/content/Loader",
"<TD;>;"
}
.end annotation
.prologue
.line 599
.local p3, "callback":Landroid/app/LoaderManager$LoaderCallbacks;, "Landroid/app/LoaderManager$LoaderCallbacks<TD;>;"
iget-boolean v1, p0, Landroid/app/LoaderManagerImpl;->mCreatingLoader:Z
if-eqz v1, :cond_0
.line 600
new-instance v1, Ljava/lang/IllegalStateException;
const-string v2, "Called while creating a loader"
invoke-direct {v1, v2}, Ljava/lang/IllegalStateException;-><init>(Ljava/lang/String;)V
throw v1
.line 603
:cond_0
iget-object v1, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v1, p1}, Landroid/util/SparseArray;->get(I)Ljava/lang/Object;
move-result-object v0
check-cast v0, Landroid/app/LoaderManagerImpl$LoaderInfo;
.line 605
.local v0, "info":Landroid/app/LoaderManagerImpl$LoaderInfo;
sget-boolean v1, Landroid/app/LoaderManagerImpl;->DEBUG:Z
if-eqz v1, :cond_1
const-string v1, "LoaderManager"
new-instance v2, Ljava/lang/StringBuilder;
invoke-direct {v2}, Ljava/lang/StringBuilder;-><init>()V
const-string v3, "initLoader in "
invoke-virtual {v2, v3}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v2
invoke-virtual {v2, p0}, Ljava/lang/StringBuilder;->append(Ljava/lang/Object;)Ljava/lang/StringBuilder;
move-result-object v2
const-string v3, ": args="
invoke-virtual {v2, v3}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v2
invoke-virtual {v2, p2}, Ljava/lang/StringBuilder;->append(Ljava/lang/Object;)Ljava/lang/StringBuilder;
move-result-object v2
invoke-virtual {v2}, Ljava/lang/StringBuilder;->toString()Ljava/lang/String;
move-result-object v2
invoke-static {v1, v2}, Landroid/util/Log;->v(Ljava/lang/String;Ljava/lang/String;)I
.line 607
:cond_1
if-nez v0, :cond_4
.line 609
invoke-direct {p0, p1, p2, p3}, Landroid/app/LoaderManagerImpl;->createAndInstallLoader(ILandroid/os/Bundle;Landroid/app/LoaderManager$LoaderCallbacks;)Landroid/app/LoaderManagerImpl$LoaderInfo;
move-result-object v0
.line 610
sget-boolean v1, Landroid/app/LoaderManagerImpl;->DEBUG:Z
if-eqz v1, :cond_2
const-string v1, "LoaderManager"
new-instance v2, Ljava/lang/StringBuilder;
invoke-direct {v2}, Ljava/lang/StringBuilder;-><init>()V
const-string v3, " Created new loader "
invoke-virtual {v2, v3}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v2
invoke-virtual {v2, v0}, Ljava/lang/StringBuilder;->append(Ljava/lang/Object;)Ljava/lang/StringBuilder;
move-result-object v2
invoke-virtual {v2}, Ljava/lang/StringBuilder;->toString()Ljava/lang/String;
move-result-object v2
invoke-static {v1, v2}, Landroid/util/Log;->v(Ljava/lang/String;Ljava/lang/String;)I
.line 616
:cond_2
:goto_0
iget-boolean v1, v0, Landroid/app/LoaderManagerImpl$LoaderInfo;->mHaveData:Z
if-eqz v1, :cond_3
iget-boolean v1, p0, Landroid/app/LoaderManagerImpl;->mStarted:Z
if-eqz v1, :cond_3
.line 618
iget-object v1, v0, Landroid/app/LoaderManagerImpl$LoaderInfo;->mLoader:Landroid/content/Loader;
iget-object v2, v0, Landroid/app/LoaderManagerImpl$LoaderInfo;->mData:Ljava/lang/Object;
invoke-virtual {v0, v1, v2}, Landroid/app/LoaderManagerImpl$LoaderInfo;->callOnLoadFinished(Landroid/content/Loader;Ljava/lang/Object;)V
.line 621
:cond_3
iget-object v1, v0, Landroid/app/LoaderManagerImpl$LoaderInfo;->mLoader:Landroid/content/Loader;
return-object v1
.line 612
:cond_4
sget-boolean v1, Landroid/app/LoaderManagerImpl;->DEBUG:Z
if-eqz v1, :cond_5
const-string v1, "LoaderManager"
new-instance v2, Ljava/lang/StringBuilder;
invoke-direct {v2}, Ljava/lang/StringBuilder;-><init>()V
const-string v3, " Re-using existing loader "
invoke-virtual {v2, v3}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v2
invoke-virtual {v2, v0}, Ljava/lang/StringBuilder;->append(Ljava/lang/Object;)Ljava/lang/StringBuilder;
move-result-object v2
invoke-virtual {v2}, Ljava/lang/StringBuilder;->toString()Ljava/lang/String;
move-result-object v2
invoke-static {v1, v2}, Landroid/util/Log;->v(Ljava/lang/String;Ljava/lang/String;)I
.line 613
:cond_5
iput-object p3, v0, Landroid/app/LoaderManagerImpl$LoaderInfo;->mCallbacks:Landroid/app/LoaderManager$LoaderCallbacks;
goto :goto_0
.end method
.method installLoader(Landroid/app/LoaderManagerImpl$LoaderInfo;)V
.locals 2
.param p1, "info" # Landroid/app/LoaderManagerImpl$LoaderInfo;
.prologue
.line 564
iget-object v0, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
iget v1, p1, Landroid/app/LoaderManagerImpl$LoaderInfo;->mId:I
invoke-virtual {v0, v1, p1}, Landroid/util/SparseArray;->put(ILjava/lang/Object;)V
.line 565
iget-boolean v0, p0, Landroid/app/LoaderManagerImpl;->mStarted:Z
if-eqz v0, :cond_0
.line 569
invoke-virtual {p1}, Landroid/app/LoaderManagerImpl$LoaderInfo;->start()V
.line 571
:cond_0
return-void
.end method
.method public restartLoader(ILandroid/os/Bundle;Landroid/app/LoaderManager$LoaderCallbacks;)Landroid/content/Loader;
.locals 7
.param p1, "id" # I
.param p2, "args" # Landroid/os/Bundle;
.annotation system Ldalvik/annotation/Signature;
value = {
"<D:",
"Ljava/lang/Object;",
">(I",
"Landroid/os/Bundle;",
"Landroid/app/LoaderManager$LoaderCallbacks",
"<TD;>;)",
"Landroid/content/Loader",
"<TD;>;"
}
.end annotation
.prologue
.local p3, "callback":Landroid/app/LoaderManager$LoaderCallbacks;, "Landroid/app/LoaderManager$LoaderCallbacks<TD;>;"
const/4 v6, 0x0
.line 649
iget-boolean v3, p0, Landroid/app/LoaderManagerImpl;->mCreatingLoader:Z
if-eqz v3, :cond_0
.line 650
new-instance v3, Ljava/lang/IllegalStateException;
const-string v4, "Called while creating a loader"
invoke-direct {v3, v4}, Ljava/lang/IllegalStateException;-><init>(Ljava/lang/String;)V
throw v3
.line 653
:cond_0
iget-object v3, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v3, p1}, Landroid/util/SparseArray;->get(I)Ljava/lang/Object;
move-result-object v1
check-cast v1, Landroid/app/LoaderManagerImpl$LoaderInfo;
.line 654
.local v1, "info":Landroid/app/LoaderManagerImpl$LoaderInfo;
sget-boolean v3, Landroid/app/LoaderManagerImpl;->DEBUG:Z
if-eqz v3, :cond_1
const-string v3, "LoaderManager"
new-instance v4, Ljava/lang/StringBuilder;
invoke-direct {v4}, Ljava/lang/StringBuilder;-><init>()V
const-string/jumbo v5, "restartLoader in "
invoke-virtual {v4, v5}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v4
invoke-virtual {v4, p0}, Ljava/lang/StringBuilder;->append(Ljava/lang/Object;)Ljava/lang/StringBuilder;
move-result-object v4
const-string v5, ": args="
invoke-virtual {v4, v5}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v4
invoke-virtual {v4, p2}, Ljava/lang/StringBuilder;->append(Ljava/lang/Object;)Ljava/lang/StringBuilder;
move-result-object v4
invoke-virtual {v4}, Ljava/lang/StringBuilder;->toString()Ljava/lang/String;
move-result-object v4
invoke-static {v3, v4}, Landroid/util/Log;->v(Ljava/lang/String;Ljava/lang/String;)I
.line 655
:cond_1
if-eqz v1, :cond_3
.line 656
iget-object v3, p0, Landroid/app/LoaderManagerImpl;->mInactiveLoaders:Landroid/util/SparseArray;
invoke-virtual {v3, p1}, Landroid/util/SparseArray;->get(I)Ljava/lang/Object;
move-result-object v0
check-cast v0, Landroid/app/LoaderManagerImpl$LoaderInfo;
.line 657
.local v0, "inactive":Landroid/app/LoaderManagerImpl$LoaderInfo;
if-eqz v0, :cond_b
.line 658
iget-boolean v3, v1, Landroid/app/LoaderManagerImpl$LoaderInfo;->mHaveData:Z
if-eqz v3, :cond_4
.line 663
sget-boolean v3, Landroid/app/LoaderManagerImpl;->DEBUG:Z
if-eqz v3, :cond_2
const-string v3, "LoaderManager"
new-instance v4, Ljava/lang/StringBuilder;
invoke-direct {v4}, Ljava/lang/StringBuilder;-><init>()V
const-string v5, " Removing last inactive loader: "
invoke-virtual {v4, v5}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v4
invoke-virtual {v4, v1}, Ljava/lang/StringBuilder;->append(Ljava/lang/Object;)Ljava/lang/StringBuilder;
move-result-object v4
invoke-virtual {v4}, Ljava/lang/StringBuilder;->toString()Ljava/lang/String;
move-result-object v4
invoke-static {v3, v4}, Landroid/util/Log;->v(Ljava/lang/String;Ljava/lang/String;)I
.line 664
:cond_2
const/4 v3, 0x0
iput-boolean v3, v0, Landroid/app/LoaderManagerImpl$LoaderInfo;->mDeliveredData:Z
.line 665
invoke-virtual {v0}, Landroid/app/LoaderManagerImpl$LoaderInfo;->destroy()V
.line 666
iget-object v3, v1, Landroid/app/LoaderManagerImpl$LoaderInfo;->mLoader:Landroid/content/Loader;
invoke-virtual {v3}, Landroid/content/Loader;->abandon()V
.line 667
iget-object v3, p0, Landroid/app/LoaderManagerImpl;->mInactiveLoaders:Landroid/util/SparseArray;
invoke-virtual {v3, p1, v1}, Landroid/util/SparseArray;->put(ILjava/lang/Object;)V
.line 711
.end local v0 # "inactive":Landroid/app/LoaderManagerImpl$LoaderInfo;
:cond_3
:goto_0
invoke-direct {p0, p1, p2, p3}, Landroid/app/LoaderManagerImpl;->createAndInstallLoader(ILandroid/os/Bundle;Landroid/app/LoaderManager$LoaderCallbacks;)Landroid/app/LoaderManagerImpl$LoaderInfo;
move-result-object v1
.line 712
iget-object v3, v1, Landroid/app/LoaderManagerImpl$LoaderInfo;->mLoader:Landroid/content/Loader;
:goto_1
return-object v3
.line 671
.restart local v0 # "inactive":Landroid/app/LoaderManagerImpl$LoaderInfo;
:cond_4
iget-boolean v3, v1, Landroid/app/LoaderManagerImpl$LoaderInfo;->mStarted:Z
if-nez v3, :cond_6
.line 675
sget-boolean v3, Landroid/app/LoaderManagerImpl;->DEBUG:Z
if-eqz v3, :cond_5
const-string v3, "LoaderManager"
const-string v4, " Current loader is stopped; replacing"
invoke-static {v3, v4}, Landroid/util/Log;->v(Ljava/lang/String;Ljava/lang/String;)I
.line 676
:cond_5
iget-object v3, p0, Landroid/app/LoaderManagerImpl;->mLoaders:Landroid/util/SparseArray;
invoke-virtual {v3, p1, v6}, Landroid/util/SparseArray;->put(ILjava/lang/Object;)V
.line 677
invoke-virtual {v1}, Landroid/app/LoaderManagerImpl$LoaderInfo;->destroy()V
goto :goto_0
.line 682
:cond_6
sget-boolean v3, Landroid/app/LoaderManagerImpl;->DEBUG:Z
if-eqz v3, :cond_7
const-string v3, "LoaderManager"
const-string v4, " Current loader is running; attempting to cancel"
invoke-static {v3, v4}, Landroid/util/Log;->v(Ljava/lang/String;Ljava/lang/String;)I
.line 686
:cond_7
iget-object v3, v1, Landroid/app/LoaderManagerImpl$LoaderInfo;->mPendingLoader:Landroid/app/LoaderManagerImpl$LoaderInfo;
if-eqz v3, :cond_9
.line 687
sget-boolean v3, Landroid/app/LoaderManagerImpl;->DEBUG:Z
if-eqz v3, :cond_8
const-string v3, "LoaderManager"
new-instance v4, Ljava/lang/StringBuilder;
invoke-direct {v4}, Ljava/lang/StringBuilder;-><init>()V
const-string v5, " Removing pending loader: "
invoke-virtual {v4, v5}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v4
iget-object v5, v1, Landroid/app/LoaderManagerImpl$LoaderInfo;->mPendingLoader:Landroid/app/LoaderManagerImpl$LoaderInfo;
invoke-virtual {v4, v5}, Ljava/lang/StringBuilder;->append(Ljava/lang/Object;)Ljava/lang/StringBuilder;
move-result-object v4
invoke-virtual {v4}, Ljava/lang/StringBuilder;->toString()Ljava/lang/String;
move-result-object v4
invoke-static {v3, v4}, Landroid/util/Log;->v(Ljava/lang/String;Ljava/lang/String;)I
.line 688
:cond_8
iget-object v3, v1, Landroid/app/LoaderManagerImpl$LoaderInfo;->mPendingLoader:Landroid/app/LoaderManagerImpl$LoaderInfo;
invoke-virtual {v3}, Landroid/app/LoaderManagerImpl$LoaderInfo;->destroy()V
.line 689
iput-object v6, v1, Landroid/app/LoaderManagerImpl$LoaderInfo;->mPendingLoader:Landroid/app/LoaderManagerImpl$LoaderInfo;
.line 691
:cond_9
sget-boolean v3, Landroid/app/LoaderManagerImpl;->DEBUG:Z
if-eqz v3, :cond_a
const-string v3, "LoaderManager"
const-string v4, " Enqueuing as new pending loader"
invoke-static {v3, v4}, Landroid/util/Log;->v(Ljava/lang/String;Ljava/lang/String;)I
.line 692
:cond_a
invoke-direct {p0, p1, p2, p3}, Landroid/app/LoaderManagerImpl;->createLoader(ILandroid/os/Bundle;Landroid/app/LoaderManager$LoaderCallbacks;)Landroid/app/LoaderManagerImpl$LoaderInfo;
move-result-object v3
iput-object v3, v1, Landroid/app/LoaderManagerImpl$LoaderInfo;->mPendingLoader:Landroid/app/LoaderManagerImpl$LoaderInfo;
.line 695
iget-object v2, v1, Landroid/app/LoaderManagerImpl$LoaderInfo;->mPendingLoader:Landroid/app/LoaderManagerImpl$LoaderInfo;
.line 696
.local v2, "pending":Landroid/app/LoaderManagerImpl$LoaderInfo;
invoke-virtual {v1}, Landroid/app/LoaderManagerImpl$LoaderInfo;->cancel()V
.line 698
iget-object v3, v2, Landroid/app/LoaderManagerImpl$LoaderInfo;->mLoader:Landroid/content/Loader;
goto :goto_1
.line 705
.end local v2 # "pending":Landroid/app/LoaderManagerImpl$LoaderInfo;
:cond_b
sget-boolean v3, Landroid/app/LoaderManagerImpl;->DEBUG:Z
if-eqz v3, :cond_c
const-string v3, "LoaderManager"
new-instance v4, Ljava/lang/StringBuilder;
invoke-direct {v4}, Ljava/lang/StringBuilder;-><init>()V
const-string v5, " Making last loader inactive: "
invoke-virtual {v4, v5}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
move-result-object v4
invoke-virtual {v4, v1}, Ljava/lang/StringBuilder;->append(Ljava/lang/Object;)Ljava/lang/StringBuilder;
move-result-object v4
invoke-virtual {v4}, Ljava/lang/StringBuilder;->toString()Ljava/lang/String;
move-result-object v4
invoke-static {v3, v4}, Landroid/util/Log;->v(Ljava/lang/String;Ljava/lang/String;)I
.line 706
:cond_c
iget-object v3, v1, Landroid/app/LoaderManagerImpl$LoaderInfo;->mLoader:Landroid/content/Loader;
invoke-virtual {v3}, Landroid/content/Loader;->abandon()V
.line 707
iget-object v3, p0, Landroid/app/LoaderManagerImpl;->mInactiveLoaders:Landroid/util/SparseArray;
invoke-virtual {v3, p1, v1}, Landroid/util/SparseArray;->put(ILjava/lang/Object;)V
goto/16 :goto_0
.end method
.method public toString()Ljava/lang/String;
.locals 2
.prologue
.line 855
new-instance v0, Ljava/lang/StringBuilder;
const/16 v1, 0x80
invoke-direct {v0, v1}, Ljava/lang/StringBuilder;-><init>(I)V
.line 856
.local v0, "sb":Ljava/lang/StringBuilder;
const-string v1, "LoaderManager{"
invoke-virtual {v0, v1}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
.line 857
invoke-static {p0}, Ljava/lang/System;->identityHashCode(Ljava/lang/Object;)I
move-result v1
invoke-static {v1}, Ljava/lang/Integer;->toHexString(I)Ljava/lang/String;
move-result-object v1
invoke-virtual {v0, v1}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
.line 858
const-string v1, " in "
invoke-virtual {v0, v1}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
.line 859
iget-object v1, p0, Landroid/app/LoaderManagerImpl;->mActivity:Landroid/app/Activity;
invoke-static {v1, v0}, Landroid/util/DebugUtils;->buildShortClassTag(Ljava/lang/Object;Ljava/lang/StringBuilder;)V
.line 860
const-string/jumbo v1, "}}"
invoke-virtual {v0, v1}, Ljava/lang/StringBuilder;->append(Ljava/lang/String;)Ljava/lang/StringBuilder;
.line 861
invoke-virtual {v0}, Ljava/lang/StringBuilder;->toString()Ljava/lang/String;
move-result-object v1
return-object v1
.end method
.method updateActivity(Landroid/app/Activity;)V
.locals 0
.param p1, "activity" # Landroid/app/Activity;
.prologue
.line 540
iput-object p1, p0, Landroid/app/LoaderManagerImpl;->mActivity:Landroid/app/Activity;
.line 541
return-void
.end method
| {
"redpajama_set_name": "RedPajamaGithub"
} | 742 |
Protect your little one from the sun with this adorable baby boy's bucket hat by Dozer. This reversible swim hat provides UPF 50+ sun protection to guard against sun burn and keep your little beach buddy comfortable. The adjustable chin strap will keep the hat in place, making it a practical addition to your summer gear. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,835 |
Q: replace string after each second occurence of character Working with a javascript-variable, I want to replace each occurence of a word (in the example: 'OR' with the ||-sign), but only if it does NOT occur in a doublequoted string.
Is it possible using a regular expression?
An example of what I'm looking upon:
'one OR " two OR " OR three OR five OR " OR six "'.replace( regex , ???) is giving
'one || " two OR " || three || five || " OR six "'
I've tried with:
regex = new RegExp("([^\"]*)(OR?)([^\"]*\"[^\"]*\")" , "gi");
text = text.replace( regex , "$1||$3");
result:
'one || " two OR " OR three OR five || " OR six "'
which leaves me with some ORs to be replaced.
If I try to do the replacement one more time, to get rid of the remaining ORs, the replacement does (of course) not work:
'||ne || " two OR " OR three || five || " OR six "'
The letter 'o' in 'one' has been replaced with the ||-sign, which wasn't intended!
Is it possible to do with with an regExp?
A: OKAY you want regex? I will give you regex:
var str = 'one OR " two OR " OR three OR five OR " OR six "';
var res = str.replace(/OR/g, '||');
console.log(res.replace(/(.*?)(\".*?)\|\|(.*?\")/g, '$1$2OR$3'));
ENJOY MADI !
A: Rather than try to do this with a single regex, I would split up the work:
spl = text.split(/(".*?")/);
result = "";
spl.forEach(function (piece) {
if (~piece.indexOf('"')) {
result += piece;
}
else {
result += piece.replace("OR", "||");
}
});
http://jsfiddle.net/UkBZF/
A: Use this:
var str = 'one OR " two OR " OR three OR five OR " OR six "';
var res = str.replace(/or/gi,"||");
This will replace all or words case-insensitive.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,369 |
Toki may refer to:
People
The Toki clan, a Japanese samurai clan
, Japanese decathlete
, Japanese sumo wrestler
Palnatoki, a legendary Danish hero and chieftain
Toki (also spelled Toqui), the title of a selected leader of the Mapuche (indigenous Chilean people) during a time of war
Valmaine Toki, New Zealand barrister and solicitor
Places
Toki, Gifu, a city in Gifu prefecture, Japan
Toki, Subcarpathian Voivodeship, in south-east Poland
Toki, Masovian Voivodeship, in east-central Poland
Public institutions
TOKİ, Turkey's government-run public housing authority
Fictional characters
Kamen Rider Tōki, a fictional character in Kamen Rider Hibiki
Princess Toki, a character in Naruto
Toki, the second of the four brothers of Hokuto Shinken in Fist of the North Star
Toki, a character from Hayao Miyazaki's 1997 animated film Mononoke Hime
Toki Fujiwara, a character in the anime Code:Breaker
Toki Wartooth, the rhythm guitarist of the metal band Dethklok in Metalocalypse
Other
Toki (train), the name of a train service in Japan
Toki (video game), an arcade game, originally in Japan, featuring an eponymous enchanted ape as the main character
Toki, the Japanese name of the Crested ibis
Toki, shamans in the ancient Ryukyu Islands of Japan
Toki, a brand of Japanese whisky produced by Beam Suntory
Toki, a variety of Japanese apple
Toki, the word for speech/language, and for hello in Toki Pona, a language created in 2001
Toki, a company founded by Rob Monster creating servers to be distributed to Africa and Asia
Toki, a prehistoric Māori adze
Japanese-language surnames | {
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} | 4,030 |
Insurance is a great product; but it is often sold to the wrong people for the wrong reasons. Insurance makes most sense to buy when the odds of incurring the expense are low but the cost of incurring expense are catastrophic. A great example of this would be term life insurance for a single mother who is the sole bread winner for the family.
If something were to happen to her, the survivors would be financially devastated. In this case, the family needs some inexpensive term insurance on mom.
Remember, insurance is a business decision – not an emotional one. Chances are that your kids will live long and healthy lives. In fact, the odds of children under four dying are less than .25%.
And since we're talking about business, let's put this in dollars and cents. If heaven forbid your child dies prematurely, you probably would not incur tremendous financial losses because you don't depend on that child's income.
Now, if there is a genetic problem that runs in the family that exposes your children to higher mortality risk and/or final costs would devastate your finances, it might make sense to consider a small policy. It also could make sense to buy a policy if you think there is a high chance that your child will be uninsurable later on.
But unless you have specific information that makes you think your child is more susceptible to major illness or an early death, life insurance for children is a waste of money. Yes, there is a chance that this could go horribly wrong. But unless the financial costs would clean you out, invest the money instead for your child's college education. It's a far better use of the money.
Most stores push extended warranties on consumers for almost any purchase above $100. In most all cases, they are not needed and are a waste of your money.
Before you buy something, do your research to determine how long it should last and the main reasons why similar products fail. If the product is known to create problems even for people who use it carefully, that's one thing and the warranty might pay off. But if the product is fragile and can be destroyed easily by spilling something on it or by dropping it, the extended warranty may not help. Remember that most warranties exclude coverage for problems that are not a result of defective materials and workmanship under normal operation.
Purchasing electronics items with a credit card that will double the life of the manufacturer's warranty is a much better idea. That is a free way to get some peace of mind when purchasing a product as opposed to shelling out extra for a warranty that rarely pays off.
This is one of the most egregious insurance rip-offs there is. My father died in an accident when he was in his 40s. The financial cost to the family was the same whether his death was a result of an accident or an illness. What difference did it make to us from a financial standpoint? None. Unfortunately, dead is dead.
If you need to protect your family in case of your premature death, buy a low-cost term life insurance policy. Skip the accidental death and mortgage life insurance policies. They are designed to separate you from your money.
Unlike the policies described above, umbrella policies do have their place for some consumers. Just not everybody.
Umbrella policies take over where your basic coverage ends. So, for example, let's say you have a renter's insurance policy that covers you for up to $100,000 liability. If you think you could get sued for an amount greater than that $100,000 cap, you might consider buying an umbrella policy.
If you did that, you could extend your liability coverage by $1,000,000 or more. That umbrella policy could provide extended liability coverage for injuries sustained on your property or as a result of a car or boating accident. It also covers your dependent children in case they cause harm to others and may cover liabilities that occur on your rental property and personal injury lawsuits filed against you.
Typically, these policies have a high deductible and the premiums are very low – maybe a couple hundred dollars a year.
With all those benefits, who wouldn't want an umbrella policy? Well, you possibly.
First, consider increasing the coverage on the basic policies you already have in place. That could be enough. For example, it might be cheaper to bump up that liability on your auto insurance from $100,000 to $250,000 or $500,000 than to buy a separate umbrella policy. And your homeowners or renters policies might provide the same opportunity. First check there.
In addition, if you are retired on a fixed income and have very modest assets, you might not need the umbrella. That's because would-be plaintiffs probably won't come after you anyway.
However, even if your assets are sparse, you may want to consider this kind of policy if you are at high risk and you still earn wages. That's because people who win a law suit against you may be able to garnish your wages if they win a claim you can't pay.
Life insurance is meant to indemnify your family should they suffer a catastrophic financial loss at your death. That's what term insurance is good at. Whole life – like other types of permanent life insurance — however, is very bad at doing this. That's because whole life is sold as an investment vehicle as well as an insurance product. But it does a terrible job on both fronts.
First, the investment angle fails because the costs associated with whole life and variable life are extraordinarily high. As a result, relatively little of your premium goes to the investment. The rest gets eaten up in costs.
It also fails on the insurance side because of the cost. Whole life premiums are sometimes 10 times greater than similar term policies. As a result, many people who get stuck with whole life usually can't afford to have all the life insurance they need.
For example, Joe might be offered a $100,000 term insurance policy for $30 a month whereby he might have to put in $300 a month for a similar whole life policy. If Joe doesn't have $300 extra each month, the agent might talk him into putting his $30 into a whole life policy which provides 1/10th the coverage he really needs.
Insurance is an important part of your financial foundation. Use it to protect yourself and your family against the real risks you face but do so prudently. Before you make a decision about any insurance, speak to an objective financial planner and make sure you aren't just getting sold a bill of goods. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,281 |
ZENDAYA CLOSE UP//VIVIENNE WESTWOOD//RANKIN
Fashion, Film & Photography
November 12, 2015 by Diane Pernet
To see the full film and read the full interview with Zendaya go to www.hungertv.com –
This exclusive fashion film with Zendaya Hunger 9's cover star directed by Rankin, styled by Hunger's fashion director Kim Howells and features a selection of key looks from Vivienne Westwood's collection. The film's glitchy, slick aesthetic is matched by Zendaya's fierce attitude throughout.
The film features the song "Close Up", one of the lead tracks from Zendaya's upcoming sophomore album, due for release in early 2016. Produced by Timbaland it is a marked evolution from the artist's previous work, likely to catapult her into well-deserved superstardom.
A former Disney star, the 19-year-old actress, singer, producer and designer Zendaya has recently been named as one of Time magazine's 30 most influential teenagers alongside the likes of Malala Yousafzei, Malia Obama, Jaden Smith and Amandla Stenberg. It is Zendaya's dignified, outspoken and intelligent nature that has set her apart from her contemporaries and made her the perfect celebrity to front Hunger's Autumn/Winter issue, featuring on four limited edition covers of "Right Way Up"
Credits Styled by Kim Howells Fashion Director at Hunger. Directed by Rankin. Creative Director Vicky Lawton
Quotes from Zendaya's interview in Hunger 9 Copyright Hunger Magazine interviewed by Holly Fraser Hunger Editor
"Of course I've had a problem with people taking me seriously because of my age", Zendaya tells us in an exclusive interview. "People are always going do that because you're less experienced, you haven't lived as much. And in a sense it is true that there are certain things that you only learn with time. But I think that there are also a lot of things that young people know that older people don't, and older people will never understand again because they're in a different section of their lives. I think when you're young you're a lot more open-minded, and sometimes you're a lot more perceptive about what's going on in the world."
"Sometimes we think we know too much, but there are people who have done the research, and have used social media for good. So of course young people are not always going to be taken as seriously, but that's when you have to prove them wrong."
"It's important to raise your voice in things you feel passionate about, and things that you know about. Don't raise your voice just to raise your voice, if have nothing behind it and don't know what you're talking about. But if you take the time to learn about something, and educate yourself, then you have a voice, and you're allowed to use it."
Exaggerated shapes and nipple cleavage at Pressait PFW FW 22/23 – text by Leticia Dare
PLAN 8 SHOWROOMS
Digital Paris Fashion Week J W Anderson for LOEWE and his signature collecion – NVU with Canal Plus
Walter van Beirendonck has the slighted idea by Lily Templeton | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,876 |
Elisabet oder Elisabeth ist im Neuen Testament die Mutter Johannes' des Täufers. Sie wird in einigen christlichen Konfessionen als Heilige verehrt.
Etymologie
Der griechische Personenname Ἐλισάβετ 'Elisabet' stammt aus dem Hebräischen. Die Septuaginta gibt mit Ελισαβεθ 'Elisabeth' den nur in vorkommenden Namen wieder. Elischeba ist ein Nominalsatzname, bestehend aus Subjekt und Prädikat. Subjekt und zugleich theophores Element ist אֱל 'äl "Gott", Prädikat ist das Substantiv שֶׁבַע šæva' "Fülle". Das Hebräische, wie es in den Personennamen vorliegt, konserviert eine ältere Sprachstufe als das Bibelhebräische. Dies äußert sich darin, dass es sich bei dem -î-, welches an das erste Substantiv angehängt ist, wohl nicht um eine Endung der 1. Person Singular ("mein Gott") handelt, sondern um einen funktionslosen Bindevokal und möglicherweise Überrest einer alten Kasusendung. Der Name bedeutet daher "Gott ist Fülle". Vergleichen lassen sich die Personennamen Batseba (בַּת־שֶׁבַע bat šæva' "Tochter der Fülle") und Joscheba (יְהֹושֶׁבַע jəhôšæva' bzw. יְהֹושַׁבְעַת jəhôšav'at "JHWH ist Fülle"), die ebenfalls von Frauen getragen wurden.
Die Vulgata gibt den Namen im Alten Testament als Elisabe und im Neuen Testament als Elisabeth wieder, der Samaritanische Pentateuch als 'Ēlīšābā.
Biblischer Bericht
Nach dem Lukasevangelium stammte sie aus dem Geschlecht Aarons, nach dessen Stammmutter Elischeba sie genannt war. Sie war mit dem Priester Zacharias verheiratet. Die Ehe blieb lange kinderlos, da Elisabet unfruchtbar war, bis der Engel Gabriel dem Zacharias die Geburt eines Sohnes voraussagte, den er Johannes nennen sollte . Weiter erwähnt wird Elisabet bei der Heimsuchung Mariä und bei der Geburt des Johannes .
Gedenktage
armenisch: 9. April
koptisch: 10. Februar
römisch-katholisch: 23. September
orthodox: 5. September, 24. Juni
evangelisch: 5. November im Kalender der Lutherischen Kirche – Missouri-Synode
Siehe auch
Elisabeth
Literatur
Art. אֱלִישֶׁבַע, In: Gesenius. 18. Aufl. 2013, S. 65.
Martin Noth: Die israelitischen Personennamen im Rahmen der gemeinsemitischen Namengebung. Kohlhammer, Stuttgart 1928, S. 146.237.
Hans Rechenmacher: Althebräische Personennamen. Münster 2012, S. 117.199.
Weblinks
Einzelnachweise
Person im Neuen Testament
Jude (Altertum)
Heiliger (1. Jahrhundert)
Johannes der Täufer
Person des evangelischen Namenkalenders
Geboren im 1. Jahrhundert v. Chr.
Gestorben im 1. Jahrhundert v. Chr. oder 1. Jahrhundert
Frau
Lukasevangelium
Frauen in der Bibel
gl:Isabel#Relixión | {
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est le dix-huitième album studio de Yes, sorti le .
C'est le premier de deux albums du groupe avec six musiciens, avec la même formation, l'autre étant le live House of Yes: Live from House of Blues. Il s'est classé au UK Albums Chart et au Billboard 200.
Histoire
Sur cet album, Billy Sherwood délaisse les claviers pour se concentrer sur la guitare, puisque le claviériste russe Igor Khoroshev a officiellement rejoint le groupe, il est donc maintenant un membre à part entière de Yes. Le multiinstrumentiste canadien , qui a joué avec Aerosmith sur leur album (Pump, 1989) ainsi que les Cranberries (To the Faithful Departed, 1996), participe également à l'enregistrement de l'album. Steve Howe joue le koto sur la pièce It Will Be a Good Day (The River), pendant l'introduction, vers le milieu et tout à la fin de la chanson. Il en joue aussi sur une pièce, Moss Garden, de l'album de son fils Dylan Howe Subterraneans sortit en 2007, qui reprend en version plus jazzy des chansons de David Bowie de l'époque de sa trilogie berlinoise.
Homeworld (The Ladder), le premier titre de l'album et le plus long, se rapporte au jeu Homeworld créé par Sierra Entertainment. Jon Anderson a écrit les paroles en fonction de l'histoire du jeu, dont la bande originale contient cette chanson. Pour la première fois, le groupe fait appel à une section de cuivres les Marguerita Horns, pour la chanson Lightning Strikes, comme Genesis l'avait fait avec les Phenix Horns pour deux chansons, Paperlate et No Reply at All. Et le piccolo au début de cette chanson emprunte la mélodie d'ouverture d'une chanson des Kinks, Phenomenal Cat de l'album The Kinks Are the Village Green Preservation Society. À noter que tous les membres du groupe participent aux chœurs sur l'album, incluant le batteur Alan White ainsi que le claviériste Igor Khoroshev.
The Messenger est un hommage à Bob Marley.
La chanson la plus courte de l'album, Can I?, est une relecture de We Have Heaven, parue en 1971 sur l'album Fragile.
Titres
Musiciens
Jon Anderson : chant, bâton de pluie, percussions
Steve Howe : guitares acoustique et électrique, guitare pedal steel, mandoline, koto, chœurs
Billy Sherwood : guitares rythmiques, tambourins, shakers, chœurs
Chris Squire : basse, chœur
Igor Khoroshev : orgue Hammond B3, piano, synthétiseurs, mellotron, chœurs
Alan White : batterie, percussions, chœurs
Musiciens additionnels
Randy Raine-Reusch : Tambûr, guzheng, cymbales ching, Rhombe, didjeridoo, percussions
Rhys Fulber : boucles
The Marguerita Horns sur Lightning Strikes :
Tom Keenlyside : piccolo, saxophone ténor
Tom Colclough : saxophone alto
Derry Burns : trompette
Rod Murray : trombone
Neil Nicholson : tuba
Notes et références
Album de Yes
Album musical sorti en 1999
Album produit par Bruce Fairbairn
Album avec pochette par Roger Dean | {
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Alfio Piva Mesén (Goicoechea, 9 de enero de 1940) es un científico y ambientalista costarricense, médico veterinario de profesión y doctor en fisiología animal. Rector de la Universidad Nacional de Costa Rica y director ejecutivo del InBio. Electo Primer Vicepresidente de la República de Costa Rica en la administración de Laura Chinchilla Miranda (2010-2014).
Es hijo del italiano Alfio Piva Cugola, nativo de Ostiglia (Mantua), que emigró a Costa Rica en 1914
siendo muy niño y de Carmen Mesén. Desde el 7 de diciembre de 1966 está casado con Ginette Rodríguez Vargas y tienen tres hijos, Alfio, Alessandro y Giancarlo Piva Rodríguez. Este último falleció en un accidente de tránsito en enero de 2009.
La Universidad Nacional
Fue miembro de la comisión que sentó las bases de la Universidad Nacional, junto a figuras como Óscar Arias Sánchez, Francisco Pacheco Fernández, Rosemary Karpinsky Dodero, Roberto Murillo y el que finalmente fue su primer rector, el Presbítero Benjamín Núñez.
Con apenas 37 años, Alfio Piva sucede al Presbítero Benjamín Núñez como rector de la UNA. Originalmente se discutió que la sucesora fuese la escritora Carmen Naranjo, pero ella declinó la postulación al estimar que no tenía la fuerza política para resolver la crisis financiera de la UNA en ese momento. Los postulados como candidatos fueron Roberto De la Ossa, Director de la Escuela de Relaciones Internacionales y Alfio Piva, Director de la Escuela de Veterinaria. Las elecciones se llevan a cabo el 4 de noviembre de 1977.
Otros cargos docentes y administrativos
Alfio Piva también trabajó como académico en la Escuela de Agronomía de la Universidad de Costa Rica, donde también fue director fundador de la carrera de Zootecnia; enseñó en el Centro Agronómico Tropical de Investigación y Enseñanza (CATIE), presidió el Consejo Nacional para Investigaciones Científicas y Tecnológicas (CONICIT), y desde finales de los 80 trabajó en el (INBio), donde actualmente fungió como director ejecutivo.
En 1995 ganó el Premio Príncipe de Asturias de Investigación Científica y Técnica en conjunto con sus colegas del Instituto Nacional de Biodiversidad (INBio). La Universidad Nacional lo declara profesor emérito en agosto de 2009.
Primer Vicepresidente de la República de Costa Rica
Al postular su candidatura para la Presidencia de la República para el período (2010-2014), Laura Chinchilla lo presenta como candidato a Primer Vicepresidente de la República de Costa Rica, junto a Luis Liberman Ginsburg, Segundo Vicepresidente. La fórmula resulta elegida el 7 de febrero de 2010 y toman posesión de sus cargos el 8 de mayo de 2010, en sucesión del gobierno de Óscar Arias Sánchez.
Véase también
Política de Costa Rica
Anexo:Gabinete de Gobierno Chinchilla Miranda 2010-1014
Referencias
Enlaces externos
Políticos del Partido Liberación Nacional
Administración Chinchilla Miranda 2010-2014
Premio Príncipe de Asturias de Investigación Científica y Técnica
Científicos de Costa Rica
Vicepresidentes de Costa Rica
Ecologistas de Costa Rica
Rectores de la Universidad Nacional de Costa Rica | {
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\section{Introduction}
One of the goals of knot theory is to distinguish between different knots.
The most convenient and universal ways to do this is to calculate and compare
some polynomial knot invariants.
Powerful enough while still directly calculable are the HOMFLY-PT polynomials \cite{HOMFLY,PT},
which are (in a proper normalization) polynomials of two variables $q$ and $A$.
From the physical point of view, they are observables (Wilson loop averages)
in $3d$ Chern-Simons theory.
At the specialization $A=q^N$, these polynomials are observables (Wilson loop averages) in Chern-Simons theory with the gauge group $SU(N)$ or $SL(N)$ and $q:=\exp\Big({2\pi i\over \kappa +N}\Big)$, where $\kappa$ is the coupling constant \cite{Witt}. These polynomials also depend on the representation
$R$ of the gauge group, running along the Wilson loop.
The most difficult knots to distinguish are the pairs of mutant knots.
These are families of knots which can be transformed into each other
using a special mutation transformation (see section 2).
These knots have the same HOMFLY-PT polynomials in all symmetric
and even all rectangular representations $R$ \cite{mor,mor1} (in fact, these are the representations whose decomposition of the tensor square does not contain non-trivial multiplicities \cite{NRV,Rama1}).
Thus, in order to distinguish these knots, one needs to study mixed representations.
The simplest of them is representation $R=[2,1]$, and it indeed allows one to distinguish between some mutant knots \cite{NRV,Rama1}.
However, as was explained by H. Morton \cite{mor2} (see also \cite{Rama1}),
there are mutants that possess even higher degree of symmetry.
These mutants are not distinguished (resolved) by the representation $R=[2,1]$
and one needs at least $R=[4,2]$ to this end.
The study of these mutant knot polynomials and the differences between them is
a challenging problem, interesting both from the point of view of knot theory and
of representation theory.
At the moment no pair of mutants is known, which is {\it not} resolved even by $R=[4,2]$.
The most efficient method to calculate the HOMFLY-PT polynomials is to apply the Reshetikhin-Turaev (RT) approach, first proposed in \cite{RT}-\cite{RT3} and based on the use of the ${\cal R}$-matrix
for the quantum group $U_q(SL(N))$.
Its topical form, which is sometimes called the modern RT, uses a specific $N$-independent basis for
${\cal R}$-matrices.
It was developed in a series of papers \cite{GKR1}-\cite{NRZ}, \cite{ModernRT1}-\cite{mut21},
applied to knot polynomials calculations for a variety of knots and links,
and proved to be technically much more powerful in most cases.
However, as we emphasize in this letter, in some problems the original RT formulation turns to be more straightforward and fast,
but the explicit polynomial in variables $q,A$ may not be possible.
This is because the modern RT technique requires knowledge of the Racah matrices, which are very hard to find in the case under consideration.
We provide more details on these approaches in section \ref{secRT}.
We used the RT approach to calculate the differences between the polynomials of mutant knots in representations $R=[3,1]$ and $R=[4,2]$ at some particular values of $N$.
For $[3,1]$, we have managed to do it up to $N=7$.
This allowed us to construct the general answers for any $N$ in this case,
and, hence, to evaluate the corresponding HOMFLY-PT polynomial.
For representation $R=[4,2]$, we managed to evaluate the differences only for $N=3,\ 4$.
We studied the properties of these differences (see section 4) and their differential expansions.
The differential expansion \cite{DGR,Diff,Diff1}, \cite{MorK}-\cite{DEnonrect}, which is a relatively new and powerful tool in knot theory, often allows one to guess the unavailable answers for knot polynomials,
study their various properties and generally gives many insights.
It is rather simple for the defect zero knots \cite{MorK} and becomes less trivial in other cases.
Unfortunately, the mutant knots usually have non-zero defects.
Our results obtained here demonstrate that the differential expansion of the mutant knots
exhibits quite interesting properties, we discuss them in section \ref{de}.
\section{Mutant knots }
\label{secmut}
Let us first discuss in detail what are the mutant knots. Mutant knots are families of knots which are related to each other by a special operation called mutation (see Fig.\ref{fmutation}). This operation means that one cuts a part of a knot
with two ingoing and two outgoing lines inside
3-sphere. Such a cut portion of the knot, technically referred to as two-tangle, is rotated by 180 deg and glued back.
The resulting knot is a mutant to the initial knot.
\begin{figure}{}
\centering
\begin{center}
\includegraphics[scale=0.25]{mutantprocedure.eps}
\end{center}
\caption{1. The mutation procedure\label{fmutation}}
\end{figure}
Obviously, for these knots to be distinct, both the part in the 3-sphere and outside it should be non-trivial. For this reason, the mutant pairs are knots with many crossings.
The simplest mutant pairs have at least 11 crossings, one of them is formed by the well-known Kinoshita-Terasaka and Conway knots.
However in the case of 11 crossings knots, mutants come only in pairs. We believe that there could be many mutation operations resulting in a family of mutant knots with more than 11 crossings. This is beyond the scope of the present letter.
There is a family of knots called pretzel knots drawn in Fig.2 which includes many new mutants from the mutation operation on any two-tangle.
These are the generalization of the torus knots, which we know a lot about.
The pretzel knots can be put on the genus $g$ surface.
However, unlike torus knots, one puts only two strands on each handle, see Fig.\ref{fpretzel}.
The pretzel knot is parameterized by the numbers of crossings on each handle.
\begin{figure}{}
\centering
\begin{center}
\includegraphics[scale=0.4]{Pretzel_knot_4.eps}
\end{center}
\caption{2. Pretzel knot $K(n_1,n_2,\ldots,n_g,n_{g+1})$ \label{fpretzel}}
\end{figure}
It is easy to see that interchanging numbers between handles provides exactly a mutation. Thus, starting from genus $4$, the mutant pairs begin to appear. For three and two handles, the mutation gives just the same knot, while for higher genera there are wider sets of mutants.
Note that, among these Pretzel mutant pairs, some get into the class of those possessing even a higher degree of symmetry and distinguishable by representation $R=[4,2]$ only (see Fig.\ref{fmorton} and \cite{Rama1}).
\begin{figure}{}
\centering
\begin{center}
\includegraphics[scale=0.2]{morton.eps}
\end{center}
\caption{3. Mutant knots with higher degree of symmetry from \cite{mor2}\label{fmorton}}
\end{figure}
\section{RT $\mathcal{R}$-matrix approach}
\label{secRT}
Reshetikhin-Turaev (RT) approach naturally arises
if the Wilson loop average \cite{Witt} is evaluated in the temporal gauge \cite{RT4}-\cite{RT5}.
Then each crossing of the knot diagram (knot projection on the two-dimensional plane)
is associated with an $\mathcal{R}$-matrix.
One can start with the universal quantum $\mathcal{R}$-matrix for $U_q(SL(N))$ and calculate the $\mathcal{R}$-matrix in concrete representations using the generators of the quantized universal enveloping algebra:
\begin{equation}
\mathcal{R} = \mathcal{P} q^{\sum_{i,j} a^{-1}_{i,j} h_i \otimes h_j} \overrightarrow{\prod_{\beta \in \Phi^{+}}} {\rm exp}_{q} \left ( (q - q^{-1}) E_{\beta} \otimes F_{\beta} \right),
\end{equation}
where $E_{\beta}$, $F_{\beta}$ and $h_{\beta}$ are the generators of the quantized Universal enveloping algebra:
\begin{equation}
\begin{array}{ll}
\ [h_i, E_j] = a_{ij} E_j, & [h_i, h_j] = 0,
\\ \
[h_i, F_j] = -a_{ij} F_j, & [E_i, F_j] = \delta_{ij} \frac{q^{h_i}-q^{-h_i}}{q-q^{-1}}.
\end{array}
\end{equation}
Convolution along the edges of $\mathcal{R}$-matrices at the crossings of the
knot diagram and calculating a weighted trace \cite{RT} provides the HOMFLY-PT polynomial.
Unlike the original definition through the skein relations \cite{HOMFLY},
this approach works equally well for any representation $R$.
However, if one uses the universal $\mathcal{R}$-matrix,
the calculation has to be done separately for each $N$.
A more advanced modern RT approach \cite{ModernRT1}-\cite{ModernRT2} uses the $\mathcal{R}$-matrix
in the basis of intertwining operators.
When the $\mathcal{R}$-matrix acts
on the tensor product of two representations $V_1$ and $V_2$,
one can consider its decomposition into the irreducible ones, $V_1\otimes V_2=\oplus_Q {\cal M}^Q_{V_1V_2}\cdot Q$.
Since the $\mathcal{R}$-matrix commutes with the co-product \cite{FRT},
its action on the irreducible component is just a number $\pm q^{C_2(Q)}$, \cite{Rev}
where $C_2(Q)$ is the eigenvalue of second Casimir operator in representation $Q$.
Let us now consider an $\mathcal{R}$-matrix in the basis of irreducible representations, or, better to say, in the space of intertwining operators, and realize the knot as a closed $n$-strand braid. Then, the answer for the HOMFLY-PT polynomial $\mathcal{H}^{\mathcal{K}}_V(A,q)$ of the knot ${\cal K}$ colored with representation $V$ is separated into two parts: the one dependent on the group and the one dependent on the knot \cite{ModernRT1}-\cite{ModernRT2}:
\begin{equation}\label{H}
\mathcal{H}^{\mathcal{K}}_V(A,q)=\sum\limits_{Q\in V^{\otimes n}} S^*_Q(A,q) B^{\mathcal{K}}_Q(q).
\end{equation}
From now on, we associate with the representation $Q$ the corresponding Young diagram.
Here $S^*_Q$ is the quantum dimension of the representation $Q$ of the quantum group $U_q(SL(N))$. The quantum dimension is equal to the Schur polynomial $S_{Q}$ at a special point \cite[sec.7.1.6]{qD}. It does not depend on the knot, and its dependence on $N$ is through the polynomial variable $A=q^N$.
On the other hand, $B^{\mathcal{K}}_Q$ is calculated for a particular knot as a trace of the product of $\mathcal{R}$-matrices and does not depend on $A$ or $N$. What is important, both these factors depend only on the Young diagram $Q$, and all the dependence on $A$ is hidden into the fixed polynomials of $A$, quantum dimensions. This means that this modern RT approach allows one to evaluate the HOMFLY-PT polynomial at all $N$ at once, unlike the standard RT approach that we use in this letter.
Nevertheless, there is a big hindrance in the modern RT approach. It requires, first, a braid representation of the knot which, in the case of mutant knots, requires many strands.
Second, though $\mathcal{R}$-matrices in the space of intertwining operators are simple,
one needs to rotate the basis moving from one to another crossing between different pairs of braids \cite{ModernRT1}-\cite{ModernRT2}. This rotation is provided by the Racah matrices, and evaluating them is very tedious, especially in the case of higher representations \cite{JL1,JL2} and higher number of strands \cite{ModernRT1}, which we are interested in here.
In the calculations reported in this letter,
we do not use the basis of irreducible representations for the $\mathcal{R}$-matrix
and study the knot polynomials
and their differences for particular values of $N$,
which in some cases can be extended to arbitrary $N$.
\section{Mutant knot polynomial differences}
\label{secmd}
In \cite{NRV,Rama1,mut21}, we used the modern RT approach to evaluate mutant differences for the 11 crossing mutant knots. We found that these differences are highly factorizable. They are equal to
\begin{equation}
\Delta H_{[2,1]}^{mutant}=A^\gamma\cdot f(A,q)\cdot {\rm Mt}_{_{[2,1]}}(q)
\end{equation}
where $\gamma$ is an integer, ${\rm Mt}_{_{[2,1]}}(q)$ is a function of only $q$, and
\begin{equation}
f(A,q):=\{q\}^{4}\cdot [3]^2 D_{3}^2D_{2}D_0D_{-2}D_{-3}^2,
\label{ffactor}
\end{equation}
where $[...]$ denotes the $q$-number, $\{q\}:=q-q^{-1}$, and the factors
\begin{equation}
D_k:=Aq^k-A^{-1}q^{-k}
\end{equation}
are called differentials. Note that the $H$ in eqn.(4) refer to the reduced (normalized) HOMFLY-PT polynomials in variance with the unreduced polynomial (Wilson average) ${\cal H}$ in (\ref{H}).
Now, using the approach described in the previous section and based on the $\mathcal{R}$-matrix at concrete $N$, we evaluated the differences between the polynomials of the mutant knots in representations $[3,1]$ and $[4,2]$, the answers are rather long and can be found in a detailed publication \cite{Mila2} and on a special internet resource \cite{knotebook}.
For representation $[3,1]$, we first managed to calculate the differences for several values $N$. This allowed us to construct the full answers for any $N$ and, hence, to obtain the $[3,1]$-colored HOMFLY-PT polynomial. These differences, however, do not factorize as completely as in the representation $[2,1]$ case. Still there is some structure of factorized differentials:
\begin{equation}
\Delta H_{[3,1]}^{mutant}=\{q\}^4\cdot [4]^2[2]D_{4}D_{3}D_{0}D_{-2}\cdot {\rm Mt}_{_{[3,1]}}(A,q)
\end{equation}
Whenever the differential $D_{-i}$ appears as a factor,
it means the difference vanishes for the $U_q(SL(i))$ group.
Thus we see that
the differences between the mutant knot polynomials disappear for representation $[2,1]$
for the groups $U_q(SL(2))$ and $U_q(SL(3))$,
and for representation $[3,1]$ for the group $U_q(SL(2))$. It is trivial for the group $U_q(SL(2))$, since the difference disappears for any symmetric representation as was explained in the Introduction. It is less trivial for $U_q(SL(3))$ (see \cite{mor}).
The differential $D_i$ with positive $i$
has the same implication for transposed representation $R$.
Representation $R=[2,1]$ does not change under this transposition
and representation $R=[3,1]$ turns into $R=[2,1,1]$.
Thus we see that the differences vanishes for $R=[2,1,1]$ for the group $U_q(SL(4))$.
For the representation $[4,2]$, we were not able to construct the universal answer for all $N$. We, however, managed to calculate the answers for pretzel mutant knots in the case of $U_q(SL(3))$ and $U_q(SL(4))$ groups.
\section{Differential expansion}
\label{de}
The colored HOMFLY-PT polynomial possesses an additional structure called differential expansion (DE) \cite{DGR,Diff,Diff1}, \cite{MorK}-\cite{DEnonrect}, which is related with the representation theory \cite{Diff1,Diff2}.
The simplest example of DE appears already in the fundamental representation:
since for the abelian $U(1)$ Chern-Simons theory, i.e. for $A=q$, the reduced
polynomial in the topological framing is trivial, we have
\begin{equation}
H_{[1]}(A,q)= 1+D_1D_{-1}\cdot F_{[1]}(A,q)
\end{equation}
with a new, simpler, Laurent polynomial $F_{[1]}(A,q)$.
Continuing further and looking at other $N$, one comes to the general structure of expansion
of colored polynomials in products
of the knot-independent combinations $Z_R^Q$ of various differentials $D_k$:
\begin{equation}
H_{R}^K(A,q)= \sum_{Q\in M_R} Z_R^Q(A,q) \cdot F_{Q}^K(A,q)
\end{equation}
Important parameter for the differential expansion is the defect $\delta_K$ of a knot $K$.
It is defined by degree of the Alexander polynomial,
i.e. the specialization of the fundamental HOMFLY-PT polynomial at $A=1$:
\begin{equation}
H_{[1]}^K(A,q)\Big|_{A=1}=\sum_{j=-\delta_K-1}^{\delta_k+1}a_jq^{2j}
\end{equation}
The differential expansion is more involved in the case of non-vanishing defect, and this is exactly the case for mutant knots. A general theory of differential expansion in this case will be reported elsewhere, here we just discuss a concrete problem arising for the mutant pairs of knots.
Let us consider the difference of differential
expansions of the HOMFLY-PT polynomials in the
mutant pair. We denote this difference by $\Delta$. One can assume that contributing to it are only the pairs of non-diagonal composite representations, like $X_2:=([2],[1,1])\oplus ([1,1],[2])$ and $X_3:=([3],[2,1]) \oplus ([2,1],[3])$. We, however, allow also an additional adjustment of some DE coefficients, which remains unobservable in the leading order, but can show up for higher representations, this adjustment will be denoted by small $\delta$. In the first mixed representation, from \cite[eq.(106)]{Rama1} and \cite[eqs.(14)-(17)]{DEnonrect}, one gets an expression, depending on two unknown functions of the form
$$
\Delta H_{[2,1]} =
\frac{[3]}{[2]^2}\Big(\underbrace{D_0^2}_{{\footnotesize \times} 0}
+ \underbrace{[3]D_2D_{-2}}_{{\footnotesize \times}\{q\}^4[2]^2\cdot\delta F_{[1]}}\Big)
\oplus \underbrace{\{q\}^4[3]^2D_2D_{-2}}_{{\footnotesize \times}(\Delta F_{X_2} - \delta F_{[1]})}
=
$$
\begin{equation}
=\{q\}^4\cdot [3]^2D_3^2D_2D_0D_{-2}D_{-3}^2 \cdot {\rm Mt}_{_{[2,1]}}
\end{equation}
\noindent
$\delta F_1$ denotes a possible redistribution of the coefficients
between the different terms of the differential expansion for two mutants,
which does not affect the r.h.s., but can show up in the mutant difference
for higher representations.
It is natural to assume that it vanishes, but we keep this option open.
Similarly, for the next mixed representation
{\footnotesize
$$
\Delta H_{[3,1]} =
\frac{[4]}{[3]}\Big(\underbrace{D_1D_0}_0
+ \underbrace{\frac{[4]}{[2]}D_3D_{-2}}_{{\footnotesize \times}[2]^2\{q\}^4\cdot\delta F_{[1]}}\Big) \oplus
\underbrace{\{q\}^4[4]^2[2]D_3D_{-2}}_{{\footnotesize \times}(\Delta F_{X_2} - \delta F_{[1]})}
+
$$
$$
+ \frac{[4]}{[3]^2}\Big(\!\underbrace{D_3D_1^{\overline 2}D_0}_{{\footnotesize \times} 0}
+ \underbrace{[4][2]D_4D_3\overline{D_0}D_{-2}}_{{\footnotesize \times}\{q\}^4[3]^2\cdot\delta F_{[2]}}\Big)
\oplus
\underbrace{\{q\}^4 [4]^2[2]D_4D_3\overline{D_0}D_{-2}}_{
{\footnotesize \times}(\Delta F_{X_3} - \delta F_{[2]})}
$$
}
\begin{equation}
=\{q\}^4\cdot [4]^2[2]D_4D_3D_0D_{-2} \cdot {\rm Mt}_{_{[3,1]}}
\end{equation}
Now we have a problem:
everything in the last line is divisible by $D_4$,
but $\Delta F_{X_2}\neq 0$ is not.
There are at least two possible ways out.
One possibility is to allow $\delta F_{[1]}\neq 0$.
For example, take
$$
\Delta F_{X_2} = D_3^2D_0D_{-3}^2 \cdot {\rm Mt}_{_{[2,1]}} \\
\delta F_{[1]} = D_3D_2D_1D_{-3}^2 \cdot {\rm Mt}_{_{[2,1]}}
$$
\noindent
so that $\Delta F_{X_2}-\delta F_{[1]}=-[2]\{q\}^2D_3D_{-3}^2\cdot {\rm Mt}_{_{[2,1]}}$.
Then we get:
$$
\{q\}^4[4]^2[2]D_2D_{-2}\Big(\Delta F_{X_2}
-\underbrace{\Big(1-\frac{1}{[3]}\Big)}_{\frac{[4]}{[3][2]}} \delta F_{[1]}\Big)=
$$
\vspace{-0.3cm}
$$
= \{q\}^4[4]^2[2] D_2D_{-2} D_3D_{-3}^2
\underbrace{\left(D_3D_0 -\frac{[4]}{[3][2]}D_2D_1\right)}_{\frac{D_4D_{-1}}{[3]}}\cdot {\rm Mt}_{_{[2,1]}}
$$
\noindent
which is divisible by $D_{4}$.
However, now arises a new potential problem: \ we get $D_4D_{-1}$ rather than $D_4D_0$,
but overlined $D_0$ can actually be absent from the differential expansion for non-vanishing defect.
In any case, still many more things need to match...
Another possibility is to note that $F_{[3,1]}$,
which was not taken into account in above differences,
can also be different for the two mutants.
In the case of $H_{[2,1]}$, we had $\Delta F_{[2,1]}=0$,
because it also enters the expansion of rectangular $H_{[2,2]}$
which does {\it not} distinguish mutants, i.e. $\Delta H_{[2,2]}=0$.
However, $H_{[3,3]}$
contains contributions from {\it two}
non-rectangular structures $F_{[3,1]}$ and $F_{[3,2]}$,
thus $\Delta F_{[3,1]}$ and $\Delta F_{[3,2]}$ can be non-vanishing
and compensate each other in the vanishing rectangular $\Delta H_{[3,3]}$.
\section{Conclusion}
This letter is a brief summary of our results for
the HOMFLY-PT polynomials of the mutant knots.
These polynomials and especially the differences between them are of great interest
from many points of view.
We managed to construct the differences between these polynomials
in representation $R=[3,1]$ for all 11-crossing mutant knots.
These are much less structured than in the representation $[2,1]$ case,
nevertheless the representation dependence is not quite trivial.
We also studied the differential expansion of these differences, which is related to their representation properties.
In particular, we realize a subtle point in the differential expansion of mutants that requires further development in the case of knots with non-vanishing defect. It remains to be seen what happens for higher representations.
We also evaluated differences between the polynomials of mutant knots in representation $[4,2]$, but only for the $U_q(SL(3))$ and $U_q(SL(4))$ groups, which did not allow us to find the general answer, though allowed us to distinguish between mutants. To find the whole HOMFLY-PT invariant in these cases, one needs either some new approaches or
serious optimization of computer programs.
\section*{Acknowledgements}
Our work is supported in part by the grant of the
Foundation for the Advancement of Theoretical Physics ``BASIS" (L.B., A.M.'s, A.S.),
by President of Russian Federation grant MK-2038.2019 (L.B., An.M.),
by RFBR grants 19-01-00680 (A.Mir.), 19-02-00815 (A.Mor.), 20-01-00644 (An. Mor., A.S.)
18-31-20046-mol-a-ved (A.S.),
by joint grants 19-51-50008-YaF-a (L.B., A.Mir., An.Mor.), 19-51-53014-GFEN-a, 18-51-05015-Arm-a, 18-51-45010-IND-a (L.B., A.M.'s, A.S.), PR, VKS and SD acknowledge DST-RFBR grant (INT/RUS/RFBR/P-231) for support.
The work was also partly funded by RFBR and NSFB according
to the research project 19-51-18006 (A.Mir., A.Mor., An.Mor.). VKS would like to thank IISER, Pune (India) where part of
this work was done during his visit as visiting fellow. A.Mir., A.Mor. and PR also acknowledge
the hospitality of KITP and partial support by the National Science Foundation under Grant No. NSF PHY1748958.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,662 |
**The Accused**
Jeffrey Archer's latest play, _The Accused_ , is a tense courtroom drama with a difference. The audience will act as the jury, as if they were in the Central Criminal Court at the Old Bailey.
You will have to decide...
Did Dr Sherwood murder his wife?
Was Jennifer Mitchell his mistress?
Which of his alibis should you believe?
The choice will keep you on the edge of your seats, and at the end of the trial you will be invited to deliver your verdict of guilty or not guilty. Once you have made that decision, the play will continue - with one of two different endings, depending on your verdict. Only then will you finally discover the truth.
**Jeffrey Archer** is one of Britain's top-selling novelists. He is published in 63 countries and 32 languages, and has international sales passing one hundred and twenty million copies. He is a former Member of Parliament and Deputy Chairman of the Conservative Party, and was created a Life Peer in the Queen's Birthday Honours of 1992. He is also an amateur auctioneer, conducting some 60 charity auctions each year.
He has written ten novels, several of which have been serialised for television and radio, four sets of short stories, and three plays. His first - _Beyond Reasonable Doubt_ , starring Frank Finlay and Wendy Craig, ran at the Queen's Theatre, in London's West End, for over 600 performances. His second play, _Exclusive_ , which ran at the Strand Theatre, starred Paul Scofield, Eileen Atkins and Alec McCowen.
Jeffrey Archer is now working on his eleventh novel, _Serendipity_ , which will be published in May 2002.
_By the same author_
_Novels_
Not a Penny More Not a Penny Less
Shall We Tell the President?
Kane and Abel
The Prodigal Daughter
First Among Equals
A Matter of Honour
As the Crow Flies
Honour Among Thieves
The Fourth Estate
The Eleventh Commandment
_Short stories_
A Quiver Full of Arrows
A Twist in the Tale
Twelve Red Herrings
The Collected Short Stories
To Cut a Long Story Short
_Plays_
Beyond Reasonable Doubt
Exclusive
# **The Accused**
**_by_**
**Jeffrey Archer**
### **Contents**
Act One
Scene One
Scene Two
Act Two
Scene One
Act Three
Scene One
Scene Two
**The Accused**
_The Accused_ was presented by Lee Menzies at the Theatre Royal, Haymarket, London, on 5 December 2000, having received its world premiere performance at the Theatre Royal, Windsor, on 26 September 2000.
The cast was as follows (in order of appearance)
**Jury Baliff** | Edward de Souza
---|---
**Anthony Kersley QC** | Michael Feast
**Alison Ashton** | Janet Wantling
**Sir James Barrington QC** | Edward Petherbridge
**Andrew Jarvis** | Dominic Kemp
**Court Usher** | Neil France
**Mr Justice Cartwright** | Tony Britton
**Patrick Sherwood** | Jeffrey Archer
**Guard** | Richard Hodder
**Detective Chief Inspector Payne** | Douglas Fielding
**Albert Webster** | David Weston
**Masood Hussein** | Madhav Sharma
**Professor Alistair Forsyth** | David Collings
**Jennifer Mitchell** | Emma Davies
**Foreman of the Jury** | The voice of Ken
|
Livingstone
**Second Guard** | Gary laylor
All the action takes place in Court Number One at the Old Bailey.
Time: the present day.
_Director_ | Val May
---|---
_Designer_ | Simon Higlett
_Lighting Designer_ | Vince Herbert
_Sound Designer_ | Frank Bradley
_Company Stage Manager_ | Debbie Cronshaw
_Deputy Stage Manager_ | Jane Allen
_Assistant Stage Managers_ | Angharad Watson,
|
Rebecca Kilgariff,
|
Nathalie Hobday
### **_Act One_**
#### **Scene One**
_The Jury Room, Central Criminal Court, the Old Bailey. Morning_.
_When the audience take their seats, there is no curtain. The stage is dark, but at the front is a door marked JURY ROOM. On the wall is a clock showing 9.45 a.m. When the house lights go down, the door opens and the_ **Jury Bailiff** _steps through it and addresses the audience_.
**Jury Bailiff** Good morning Ladies and Gentlemen of the jury. Welcome to the Old Bailey. My name is Gilbert Pierce and I have been appointed your jury bailiff. You have been summoned this day to appear at the Central Criminal Court. The trial you have been selected for is the Crown versus Mr Patrick Sherwood. Mr Sherwood is charged with murder. In a few moments I will take you through to Court Number One. But as I suspect this is the first time you've been called on to give jury service, you may well have some questions, and as this is my one hundred and fourteenth murder, I may just have some of the answers.
Now the judge in this case is Mr Justice Cartwright who considers himself to be the natural successor to Judge Jeffreys and his views on the restoration of the death penalty are well documented. However, you will be relieved to learn that he's considerate with jurors, but uncompromising when it comes to barristers - especially those who try to take advantage of him. And while I'm on the subject of barristers, both the Crown and the Defence in this case are represented by two of the best - and they both know it.
Sir James Barrington leads for the Defence, while Mr Anthony Kersley, a former Member of Parliament, appears for the Crown. If you sense any friction between them, don't be surprised: they loathe each other. It's no secret that Sir James defeated Mr Kersley by one vote to become Chairman of the Bar Council, and was awarded the knighthood that goes with it. It's also common knowledge that they haven't exchanged a civil word since.
_The spotlight falls on_ **Kersley** _and his junior_ , **Ashton,** _who are moving towards the courtroom in a corridor of light_.
**Ashton** I see we're up against your old sparring partner.
**Kersley** Are you referring to Barrington, or His Lordship Mr Justice Cartwright?
**Ashton** I thought you and Sir James...
**Kersley** Frankly I've never cared for either of them, and I can assure you the feeling is mutual. You see, I didn't go to the right school, old boy.
_The attention moms to_ **Barrington** _and_ **Jarvis**.
**Jarvis** Have you appeared before Mr Justice Cartwright before?
**Barrington** Many times, and on this occasion we start with a home advantage.
**Jarvis** Why's that?
**Barrington** Because he disapproves of Kersley almost as much as I do, so all we have to do is massage the old boy's ego.
**Jarvis** But surely Kersley will do exactly the same thing?
**Barrington** Not a chance. He'll pick a quarrel with Cartwright at the first possible opportunity - he believes in the old adage that a good row with the judge makes up for a weak case.
**Jury Bailiff** _looks up from his clipboard_.
**Jury Bailiff** Once Mr Kersley has made his opening statement for the Crown, he will proceed to call his witnesses. At the moment there are five on the list, but I'm not expecting to get through all of them today. After the adjournment you may return home, when you will quickly discover that friends and relatives will want to discuss your views on the case and will be only too willing to offer theirs. But unlike you, they will not have heard all the evidence, so their opinions are, at best, worthless and, at worst, injurious. The safest bet is don't talk to anyone and don't allow anyone to talk to you
**Usher** Mr Pierce, I am instructed by His Lordship to ask if the jury are assembled and ready?
**Jury Bailiff** They are all in attendance, Mr Usher.
**Usher** Then will you please accompany the jury to the courtroom?
**Jury Bailiff** ( _bows, and the Usher returns the bow_. **Jury Bailiff** _turns back to face the audience_ ) Ladies and Gentlemen of the Jury, please follow me.
**Kersley** _and_ **Ashton** _walk through the jury door towards the courtroom in a corridor of light_.
**Ashton** Do you think in the twenty-first century anyone gives a damn which school you went to?
**Kersley** Mr Justice Cartwright doesn't belong in the twenty-first century - as you're about to find out - which may even work to our advantage.
**Ashton** What do you mean?
**Kersley** Simply that if the jury considers the judge is favouring one side, they quite often react against it.
**Ashton** I can't believe he'll make it that obvious.
**Kersley** Then I'll have to make sure he does.
**Barrington** _and_ **Jarvis** _follow in the corridor of light_.
**Barrington** Andrew, have you had a chance to speak to our client this morning?
**Jarvis** Yes, when I left him he was pacing up and down his cell and feeling rather sorry for himself. ( _Pauses_.) I have to admit I rather like him.
**Barrington** After thirty years in this game Andrew, I can tell you that murderers are a far nicer class of person - it's the barristers you have to watch out for.
**Jarvis** ( _laughs_ ) Are you referring to Mr Kersley, by any chance?
**Barrington** I never refer to Kersley if I can avoid it. However, don't lower your guard even for a moment, because when it comes to punching, no one has explained to Kersley where the belt is.
_As_ **Barrington** _enters the courtroom, the two QCs come face to face for the first time. They give each other a cursory nod. A few moments later there is a triple knock on the outside of the door_.
**Usher** Be upstanding in the court. All persons having anything to do before my Lords, the Queen's Justices, oyer and terminer, and general gaol delivery for the jurisdiction of the Central Criminal Court, draw near and give your attendance. God save the Queen.
**Mr Justice Cartwright** _enters and takes his place in the centre of the bench. All bow. The_ **Judge** _returns their bow_.
**Usher** Bring up the prisoner.
**Sherwood** _steps into the dock, and the Jury Bailiff walks across to stand in front of him_.
**Jury Bailiff** Is your name Patrick Hugh Sherwood?
**Sherwood** Yes.
**Jury Bailiff** Patrick Sherwood, you stand charged with murder. The particulars of the offence are that, on the twenty-first of March 1999 in the county of London, you administered a fatal dose of poison to your wife, Elizabeth Sherwood. How say you - Guilty or Not Guilty?
**Sherwood** Not guilty.
**Jury Bailiff** ( _walks to the front of the stage_ , _faces the audience and bows_ ) Members of the Jury, the defendant stands before you charged with one count - that of murder. To this count he has pleaded Not Guilty. By his plea he has cast himself upon his country, which country ye are. Your charge, therefore, is to hearken unto the evidence and say whether he be guilty or no. ( _He turns his attention away from the audience and returns to his place_.)
**Judge** Members of the Jury, you have sworn to try this case on the evidence. You must therefore give heed only to what takes place in this court and ignore anything you have read in the press. Mr Kersley, you may proceed with the prosecution.
**Kersley** May it please Your Lordship, Members of the Jury. The murder of Elizabeth Sherwood was a crime planned by a man with a brilliant and subtle mind, who set out to dupe his colleagues, so that when his wife died, no one would suspect him of being involved in such an evil enterprise.
But unfortunately for him, even the cleverest of murderers make mistakes which, like pieces in a jigsaw, end up revealing the true picture.
The Crown will produce five pieces of that jigsaw, which will show the lengths to which Mr Sherwood was willing to go in order to cover his tracks. Once those five pieces are in. place, I believe you will come to one conclusion: that Mr Patrick Sherwood is guilty of murder.
My Lord, we call our first witness, Chief Inspector Payne.
**Usher** Call Chief Inspector Payne.
**Guard** Chief Inspector Payne.
_The_ **Usher** _stands and announces_ **Chief Inspector Payne,** _as he will do for all ather witnesses from the lobby outside the double doors. He then shuts the door and returns to his seat. The_ **ChiefInspector** _enters and takes his place in the witness box. Whenever witnesses speak, they should face the jury - the audience_.
**Usher** Take the testament in your right hand and read from the card.
**Payne** I swear by Almighty God that the evidence I shall give shall be the truth, the whole truth and nothing but the truth.
**Kersley** Is your name Alan Payne and are you a Detective Chief Inspector with the Central Area major incident team?
**Payne** Yes, I am, sir.
**Kersley** Chief Inspector, can you tell the court how you became involved in this case.
**Payne** ( _Checks his notebook and faces the_ **Judge.)** May I refer to my notebook, My Lord?
**Judge** ( _nods_ ) You may, Chief Inspector.
**Payne** On the evening of March the twenty-first 1999 we received an emergency call at Wimbledon police station, from a Mr Albert Webster, the porter of a block of flats in the division. He informed us that he thought a burglary had taken place and, as there had been several in the area recently, we immediately went round to Arcadia Mansions to investigate.
**Kersley** And what did you find when you arrived at the Sherwoods' flat?
**Payne** I found Mrs Sherwood lying on the floor, covered in a blanket. She was sobbing and holding up her right arm, which appeared to be badly bruised. I assumed that she must have taken an intruder by surprise, but when I asked her what had happened she was incoherent. She kept pointing to a glass of wine on a side table. A few moments later an ambulance crew arrived and after they had examined her, told me that she must be taken into hospital immediately.
**Kersley** Did you follow the ambulance to St George's?
**Payne** No, my first priority was to take a statement from the porter before checking over the flat.
**Kersley** And once you had done that, were you still of the opinion that a burglary had taken place?
**Payne** No, I was puzzled. There was clearly evidence of a struggle, but I could find no sign of a forced entry. All the locks and windows were intact, with the exception of a kitchen window, which led on to the fire escape and was slightly open. No drawers appeared to have been left open, which is what one would expect if a burglary had taken place.
**Kersley** Did anything else arouse your suspicions?
**Payne** A rubber glove that had been dropped on the kitchen floor, and I was also curious about the glass of wine Mrs Sherwood had been pointing to, so I instructed forensics to have the glove and the wineglass sent to the lab for testing.
**Kersley** Did you then go to the hospital?
**Payne** Yes, as I hoped Mrs Sherwood might have recovered sufficiently to answer some questions.
**Kersley** And was she able to do so?
**Payne** No, she died a few minutes after I arrived at St George's.
**Kersley** And the cause of death, Chief Inspector? How was it entered on the death certificate?
**Payne** Congestive cardiac failure, but as Mrs Sherwood had suffered in the past from ( _Checks his notebook_.) atherosclerosis, her heart attack was not considered to be suspicious.
**Barrington** My Lord, I apologise for interrupting, I simply wish to seek your guidance.
**Judge** ( _suspiciously_ ) Yes, I'm listening, Sir James.
**Barrington** Is the Chief Inspector a qualified medical practitioner? Because if he isn't, perhaps we could dispense with his medical opinions and stick to the facts in this case.
**Judge** You were not seeking my guidance, Sir James, but simply wishing to make a point. No doubt you feel you have. Carry on, Mr Kersley.
**Kersley** Thank you, My Lord. Chief Inspector, you were telling the court, before we were interrupted by my learned friend, that you went to the hospital in the hope of asking Mrs Sherwood some questions. When you arrived, did you find Mrs Sherwood in her husband's arms?
**Judge (Barrington** _rises_ ) Mr Kersley, that was an outrageous leading question whose only purpose was to prejudice the jury.
**Kersley** I do apologise, My Lord. I must have been distracted by Sir James's unnecessary interruption. Chief Inspector, when you arrived at the hospital, was Mr Sherwood at his wife's bedside?
**Payne** No, sir, a member of the hospital staff was trying to locate him.
**Kersley** I see. So who signed the death certificate?
**Payne** Her local GP, a Dr Haslam.
**Kersley** And was he satisfied that Mrs Sherwood had suffered a heart attack?
**Payne** He had no reason to believe otherwise. He'd been treating her for a heart condition for some time.
**Barrington** My Lord, he is at it again.
**Judge** And so are you, Sir James. Carry on, Mr Kersley.
**Kersley** Did you also accept this judgement?
**Payne** I saw no reason to question their professional opinion.
**Kersley** So what caused you to change your mind and open a murder inquiry?
**Payne** Some weeks later I received a call from the dangerous drugs division of the Home Office. Following that call, I visited a chemist in Wellingborough to check their Controlled Drugs Register. It showed that a Mr Sherwood had been regularly having prescriptions of Potassium Chloride made up, that fell into the category of poisons under the 1994 Drug Trafficking Act.
**Kersley** Why should that make you consider a crime had been committed? After all, Mr Sherwood has the authority to write out such prescriptions.
**Payne** Yes, but why have them made up outside London when St George's Hospital has a large pharmacy of its own?
**Kersley** Why indeed, and did Mr Sherwood collect these prescriptions himself?
**Payne** No, over a period of three months he used a Ms Jennifer Mitchell, a junior staff nurse at St George's, to collect them on a Saturday in Wellingborough and then return the ampoules of Potassium to Mr Sherwood's office on the Monday morning.
**Kersley** Chief Inspector, would I be correct in saying that if Ms Mitchell had not volunteered a statement, you would never have considered charging the defendant?
**Payne** That is correct. Her evidence was to prove vital. She stated that over a period of twelve weeks, between January and March of 1999, Mr Sherwood had instructed her on six occasions to pick up ten-millilitre ampoules of Potassium Chloride. But he only handed her the prescriptions on a Friday evening, when he knew she would be visiting her parents in Wellingborough.
**Judge** Wellingborough keeps cropping up, Chief Inspector. Does it have some particular significance in this case?
**Kersley** Wellingborough has no relevance in itself, My Lord. All Mr Sherwood needed was for the prescriptions to be dispensed at a chemist outside London, so that none of his colleagues at St George's would be aware of what he was really up to.
**Judge** Ah, so it could well have been Milton Keynes or Henley?
**Kersley** Yes, My Lord, but only if Ms Mitchell had lived in Milton Keynes or Henley.
**Judge** Ah, yes, I see. Carry on, Mr Kersley.
**Kersley** Chief Inspector, were any of these prescriptions for Potassium Chloride made out for Mrs Sherwood?
**Payne** No, they were all prescribed for his private patients.
**Kersley** So what made you think that they might not have reached those patients?
**Payne** When the results of the lab test came back, they showed that there were traces of Potassium Chloride on the rubber glove found on the Sherwoods' kitchen floor.
**Kersley** On the glove, I see... and did the lab tests reveal anything else of significance?
**Payne** Yes, the glass of wine found on the table by Mrs Sherwood's side contained thirty millilitres of Temazepam, which is three times the recommended dosage.
**Kersley** And did you identify any fingerprints on that glass?
**Payne** Yes, Mrs Sherwood's.
**Kersley** But did you also check the fingerprints on the wine bottle?
**Payne** Yes, I did, and we could only find Mr Sherwood's.
**Kersley** Only Mr Sherwood's. But what about the rest of the flat?
**Payne** The only other fingerprints identifiable were those of the porter, Albert Webster.
**Kersley** So there was no reason to believe there had ever been an intruder?
**Payne** None that we could discover, sir.
**Kersley** Once you had gathered all your evidence, what did you do next?
**Payne** I obtained a warrant for the arrest of Patrick Sherwood, which I executed on June the ninth at St George's Hospital. I cautioned him, and then charged him with the murder of his wife, Elizabeth Sherwood.
**Kersley** Thank you, Chief Inspector. No more questions, My Lord.
**Judge** Sir James, do you wish to cross-examine the Chief Inspector?
**Barrington** I most certainly do, My Lord. Chief Inspector, I must begin by asking you if it is usual to open a full murder inquiry on information supplied by one young woman.
**Payne** No, but...
**Barrington** No buts, Chief Inspector, it was a simple enough question and the answer was no. My next question is equally simple. The bruises on Mrs Sherwood's arm. Surely the most likely explanation is that they resulted from a struggle with an intruder? You told us there had been several burglaries in the area recently.
**Payne** I could find no evidence of a burglary, sir.
**Barrington** Chief Inspector, is it common for burglars to leave their fingerprints all over the place, hoping you will find them?
**Payne** No, but.,.
**Barrington** I thought we'd agreed on no buts, Chief Inspector, Can I also confirm, that it was you who discovered the kitchen window open?
**Payne** Someone had unlatched it from the inside.
**Barrington** It hardly matters who unlatched it, Chief Inspector, only who might have used it as a means of entry, and I do hope you're not going to suggest that Mr Sherwood entered his own flat by the fire escape, when he could so easily have walked in through the front door.
**Payne** Unless he had a reason for not wanting to be seen walking in through the front door.
**Barrington** And on that flimsy supposition you decided to charge Mr Sherwood with murder?
**Payne** No, that decision was made by the Crown Prosecution Service after they had considered all the evidence.
**Barrington** I see. So let me finally ask you, Chief Inspector, when you charged Mr Sherwood, did he make any statement?
**Payne** Yes. ( _Checks his notebook_.) He said, 'This is ridiculous. I adored my wife; someone must have been feeding you with false information'.
**Barrington** Someone must have been feeding you with false information. Now, I wonder who that can have been? No further questions, My Lord. ( _He resumes his seat_.)
**Judge** Do you wish to re-examine, Mr Kersley?
**Kersley** ( _rises slightly_ ) No, thank you, My Lord.
**Judge** Thank you, Chief Inspector. You may leave the witness box. ( _The_ **Chief Inspector** _leaves the witness box and the courtroom_.) Perhaps you'd like to call your next witness, Mr Kersley.
**Kersley** Yes, My Lord. I call Mr Albert Webster.
**Usher** Call Mr Albert Webster.
**Guard** Mr Albert Webster.
**Webster** _is a man aged between forty-five and fifty. He is wearing a T-shirt and a well-worn suit. He enters the courtroom, baking lost, and the_ **Usher** _has to guide him to the witness box_.
**Usher** This way, sir. Please take the testament in you right hand and read from the card.
**Webster** I never bothered with the reading.
**Usher** Then repeat after me, I swear by Almighty God.
**Webster** I swears by Almigh'y God.
**Usher** That the evidence I shall give.
**Webster** That the evidence I shall give.
**Usher** Shall be the truth, the whole truth and nothing but the truth.
**Webster** Shall be the truth, the 'ole truth and nothin' but the truth. ( _He stares at the_ **Judge**.)
**Judge** Mr Webster, there's no need to address your remarks to me. It is the jury who will want to hear all your evidence.
**Kersley** Is your name Albert Philip Webster?
**Webster** You got it, mate.
**Kersley** And where do you live?
**Webster** Arcadia Mansions, Arcadia Road, Wimbledon.
**Kersley** And what is your occupation?
**Webster** I'm the resident porter - 'ave been for the past twelve years, 'aven't I, ever since I came out of the army.
**Kersley** Mr Webster, can you tell the court why the late evening of March twenty-first 1999 is etched on your memory?
**Webster** I dunno if it's etched on me memory, but I won't never forge' it.
**Kersley** So please tell the court what happened that night, Mr Webster.
**Webster** It must 'ave been around 'alf past ten, 'cos that's when I goes on me night round 'fore turnin' in. Always starts at the top of the building and works down to the bottom, where I live. It's only logic, innit? When I reached the landin' of the sixth that night, I 'eard noises comin' from the floor below.
**Kersley** Can you describe those noises?
**Webster** Yes, it was as if someone was 'avin a row, and then I 'eard a crash - like a chair turnin' over.
**Kersley** A row? Could it have been a quarrel between a man and his wife?
**Barrington** ( _rises_ ) My Lord, how can Mr Webster possibly know the answer to that question?
**Webster** Cos I 'eard voices.
**Judge** Voices? Can you be certain, Mr Webster, that you heard more than one voice coming from the flat?
**Webster** No, I can't be certain, but why would Mrs Sherwood want to shout at 'erself?
**Kersley** Why indeed. And did you hear anything she said?
**Webster** Yes, I'm pretty sure I 'eard 'er say ' 'ow did you get in?'.
**Kersley** 'How did you get in.' And did she sound surprised?
**Webster** Too bloody right she did.
**Judge** Moderate your language in my court, Mr Webster.
**Webster** Sorry, Guv.
**Judge** You address me as My Lord, and counsel as Sir.
**Webster** ( _looking directly at the_ **Judge** ) Right you are, sir. ( _He turns his attention back to_ **Kersley.)**
**Kersley** Had you been inside Mrs Sherwood's flat before?
**Webster** Oh, yes, when I gets to 'er floor, she'd often ask me in for a cup'a tea, I think she liked the company.
**Kersley** But wasn't Mr Sherwood around most evenings?
**Webster** No, not regular. In any case, I wouldn't 'ave gone in if he was around, not after that time he came back unexpected and told me to bugger off. ( **Judge** _and_ **Webster** _look at each other_.)
**Kersley** And how often did Mr Sherwood go out in the evenings?
**Webster** Quite a lot. What with his private patients and so on.
**Kersley** How did you know that he was going out to see one of his private patients?
**Webster** I used to watch 'im leavin' from my flat in the basement, didn't I.
**Kersley** Yes, I'm sure you did, but that doesn't explain how you knew that he was visiting a patient?
**Webster** Common sense, inn't ( _He touches his nose_.) Whenever he was visitin' a patient, he'd take his doctor's bag with him, wouldn't he.
**Kersley** And was he carrying his doctor's bag that night?
**Webster** No, he wasn't.
**Kersley** What time was it when he left the building?
**Webster** It must 'ave bin a few minutes after seven.
**Kersley** How can you be so sure?
**Webster** Cos I'd just come out of the lavatory, 'adn't I. ( **Kersley** _looks puzzled_.) I phones me ol' mum at six, tea at 'alf past, go to the toilet at seven, _Coronation Street_ at seven thirty - regular as clockwork. Old army training, innit.
**Kersley** And when did Mr Sherwood return that night?
**Webster** No idea, Guv, but it can't 'ave been before eleven could it, cos he wasn't there when they carted his wife off to 'ospital.
**Kersley** Quite. So when you heard the row, what did you do next?
**Webster** I ran down the stairs, fast as I could. I bangs on the door, but no one answers.
**Kersley** Was that when you called the police?
**Webster** No, Guv, I decided this 'ad to be one of them emergencies, where I'm expected to use me master key. In the Pioneer Corps it's what we used to call initiative.
**Webster** _holds up a bunch of keys, showing one in particular, and waits while everyone hangs on his words_.
**Kersley** But you told the judge that before you unlocked the door, you had heard more than one voice coming from the flat?
**Webster** Yes, I'm pretty ( _He hesitates_.) sure about that.
**Kersley** Could one of those voices have been Mr Sherwood's?
**Webster** Doesn't seem likely on account of the fact that he'd already gone out.
**Kersley** But could he have returned without you seeing him?
**Webster** Only if he used the fire escape.
**Kersley** Well, that would certainly explain Mrs Sherwood's words, 'How did you get in?' So, Mr Webster, when you unlocked the door, what did you find?
**Webster** Some furniture had been knocked over and Mrs Sherwood was lyin' on the floor moanin'.
**Kersley** And was she on her own?
**Webster** Yes, as far as I could tell.
**Judge** What do you mean, Mr Webster, by 'as far as I could tell'?
**Webster** Cos the door on the far side of the room slammed shut the minute I walked in, didn't it. ( **Judge** _makes a note_.)
**Kersley** Slammed shut the minute you walked in - as if someone had hurriedly pulled it closed?
**Webster** Yes, you got the idea.
**Kersley** So what did you do next? ( **Barrington** _nods_.)
**Webster** I dials 999 and tells 'em to send round an ambulance an' the police sharpish, an' then I gets a blanket and covers 'er up.
**Kersley** Did she give any reason why the furniture had been knocked over?
**Webster** No, she was just lying there, moanin' and rubbin' 'er arm what was bruised, so I offered 'er the glass of wine that was on the table 'opin it would 'elp, but she just pushed it away, and then she began cryin' even louder. So I wondered if someone 'ad put somethin' in the wine that had made her ill.
**Barrington** ( _rises_ ) My Lord...
**Judge** Yes, yes, Sir James. ( _Faces the audience_.) Members of the Jury, you should ignore that comment - it is nothing more than speculation. Carry on, Mr Kersley.
**Kersley** No more questions, My Lord. I think the jury has taken the point.
**Webster** _starts to leave the witness box_.
**Judge** Mr Webster, please remain in the box for a moment, as I have a feeling that Sir James might want to ask you a question or two.
**Barrington** You are quite right, My Lord. Mr Webster, may I begin by congratulating you on your remarkable memory.
**Webster** Thank you, Guv. I s'pose you can put it down to my army trainin'.
**Barrington** Quite so, but even I was puzzled, Mr Webster, as to how you could be so sure that when Mr Sherwood left the building, on the night in question, he was not carrying his doctor's bag. ( **Kersley** _smiles_.)
**Webster** To be honest, Guv, I wasn't sure at the time.
**Barrington** You weren't sure at the time, but you stated categorically...
**Webster** No, I wasn't categoric, not until I phoned for the ambulance.
**Barrington** Not until you phoned for the ambulance. I'm not altogether certain I'm following you, Mr Webster.
**Webster** Well, you see, that's when I first saw the doctor's bag. He'd left it on the table by the phone, so he couldn't 'ave taken it wiv 'im, could he?
**Barrington** I see. Mr Webster you told the court that you thought someone else might have been in the room when you first unlocked the door to the apartment.
**Webster** Yeah, I did.
**Barrington** And your immediate reaction when you saw the overturned furniture was that it must have been a burglar whom Mrs Sherwood had been shouting at?
**Webster** Yes, cos there 'ave been a lot of break-ins durin' the past year, aven't there?
**Barrington** Have there? Now, Mr Webster, remembering what a good memory you have, is it possible you can tell the court when you heard the sentence 'How did you get in?'. Was it before you unlocked the door, as you opened the door, or after you had entered the room?
**Webster** Before I unlocked the door.
**Barrington** So, some time before you stepped into the room?
**Webster** ( _hesitates_ ) Yes, I think so.
**Barrington** When you walked in, you told my learned friend that you saw the door on the far side of the room slam shut?
**Webster** Yeah, I did.
**Barrington** Could it have been the wind?
**Webster** I s'pose so.
**Barrington** Did you check to see if anyone was hiding in the kitchen?
**Webster** No. Why should I?
**Barrington** Because it leads to the kitchen window and fire escape.
**Kersley** My Lord, I am enthralled by Sir James's gift for storytelling and indeed I would go as far as to suggest that were he to submit this particular scenario to the BBC they might well consider it for _A Book at Bedtime_. But I'm bound to ask what it has to do with the case now being tried before Your Lordship?
**Barrington** It goes to the very heart of this case, My Lord, because the Prosecution are claiming that the defendant poisoned his wife, when there is no evidence to show that he was even in the building when she collapsed. It now seems there is a distinct possibility that someone else was and therefore the police could well have arrested the wrong person.
**Judge** Ingenious, Sir James. But I do feel the jury might require a little more proof of the existence of your phantom intruder.
**Barrington** You may even see them in the witness box, My Lord...
**Kersley** ( _leaps up_ ) Is Sir James suggesting that it was Mr Webster who murdered Mrs Sherwood?
**Webster** Are you accusin' me?
**Barrington** If Mr Kersley had allowed me to finish my sentence, I would have added 'before this trial is over'. No further questions, My Lord.
**Webster** I did hear her say ' 'ow did you get in?'.
**Judge** Do you wish to re-examine, Mr Kersley?
**Kersley** No, thank you, My Lord.
**Webster** I wasn't tellin' no porkies.
**Judge** You may leave the box now, Mr Webster.
**Webster** And I _did_ see the door slam. ( _Remains in the witness box_.)
**Judge** Mr Webster, will you please leave the courtroom. ( **Webster** _reluctantly leaves_.)
**Webster** On my mother's life, I swear I saw it slam shut! ( _He walks off, aided by_ **Guard.)**
**Guard** This way, sir.
**Webster** I only told 'em what you told me to say!
_The lights fade on all members of the court as_ **Webster** _exits. Blackout as sound of bells is heard_.
#### **Scene Two**
_Later that afternoon_.
_When the lights come up_ **Mr Hussein** _is in the witness box. He is between forty and fifty, formally dressed, and speaks with a pronounced Indian accent. The Koran should be wrapped in a cloth_.
**Usher** How will you take the oath, Mr Hussein.
**Hussein** On the Koran, sir.
**Usher** Take the Koran in your right hand and read from the card.
**Hussein** I swear by Almighty Allah that the evidence I shall give shall be the truth, the whole truth and nothing but the truth.
**Kersley** Your name is Masood Hussein, and you are the proprietor of Hussein the Chemist, 141 High Street, Wellingborough?
**Hussein** Yes, sir I am.
**Kersley** Mr Hussein, perhaps you could tell the court how you became involved in this case.
**Hussein** I read in one of the medical journals of the premature death of Mrs Elizabeth Sherwood. I wouldn't have given the matter a second thought, had it not been for an accompanying photograph of the mourners attending the funeral.
**Kersley** Why were the mourners of any significance, Mr Hussein?
**Hussein** I observed a lady standing a few paces behind Mr Sherwood whom I thought I recognised. I studied her face more closely with a magnifying glass and realised I had seen her before, but couldn't remember where.
**Kersley** And did you eventually remember?
**Hussein** Yes, it was some days later when I was making an entry in my register of poisons and came across the signature of a Ms J. Mitchell at the top of the page, and recalled that she had visited the pharmacy several times.
**Kersley** Can you be certain that it was the same Ms Mitchell whose photograph you saw?
**Hussein** Oh, yes, I checked back through the register and discovered that Ms Mitchell had called in to the pharmacy on six separate occasions during a period of three months, and always on a Saturday.
**Kersley** But that doesn't prove she was the woman in the photograph?
**Hussein** But Mr Sherwood's signature does.
**Kersley** Mr Sherwood's signature?
**Hussein** Yes, he had countersigned all the prescriptions.
**Kersley** Had he? And what were these prescriptions made out for?
**Hussein** Ten millilitres of Potassium Chloride.
**Kersley** And did you keep ai six prescriptions?
**Hussein** I most certainly did, Mr Kersley. I retain all prescriptions for controlled drugs for the required period of five years.
**Kersley** You say you served Ms Mitchell on no fewer than six occasions. Do you recall anything in particular about her?
**Hussein** She was a self-confident young woman and although she knew exactly what she wanted, she appeared tense, even a little nervous.
**Kersley** And is that what made you suspicious?
**Hussein** No, I was more puzzled than suspicious, because I couldn't work out why Mr Sherwood would want to have his prescriptions made up in my small pharmacy in Wellingborough, when St George's has a large pharmacy of its own. From my experience, doctors want drugs immediately, not some days later.
**Kersley** Mr Hussein, an ampoule of Potassium Chloride was found in Mr Sherwood's bag on the night of his wife's death. Could it have came from your shop?
**Judge** How can Mr Hussein know the answer to that question?
**Hussein** ( _turning to face the_ **Judge)** My Lord, if I could see the ampoule I would be able to tell if it had come from my shop.
**Judge** How is that possible, Mr Hussein?
**Hussein** Because every dangerous drug has a number displayed on its packaging which, when it is sold, has to be entered in my poisons register.
**Kersley** My Lord, both the poisons register and the ampoule of Potassium found in Mr Sherwood's bag are in the court's possession. They are numbers eleven and twenty-six on the court's list of exhibits. Perhaps Your Lordship would be kind enough to check the entry in the register, while I ask Mr Hussein to read out the number on the ampoule of Potassium. ( **Judge** _nods his agreement_.)
**Barrington** My Lord, I must object. This parlour game adds nothing to the evidence. After all, the ampoule of Potassium found in Mr Sherwood's bag was unopened so what possible bearing can it have on the case?
**Judge** A great deal Sir James, because if this is not one of the ampoules collected from Wellingborough, it will surely work in favour of your client, as it will show that there is no proof that the other five ever left the hospital. ( **Jury Bailiff** _hands the register up to the Bench, while the_ **Usher** _hands over the ampoule of Potassium to_ **Hussein**.) Please read out the number on your package, Mr Hussein.
**Hussein** 107293 AZ.
**Judge** ( _nods, and faces the audience_ ) They are identical.
**Kersley** So there can be no doubt that the ampoule of Potassium picked up by Ms Mitchell on March the nineteenth from Mr Hussein's shop, was the one found in the defendant's bag on the night of Mrs Sherwood's death, and therefore we can assume that the other five...
**Barrington** My Lord, that is outrageous. We can assume nothing...
**Kerlsey** Other than that my learned friend will always interrupt whenever he finds his client in any real trouble. No more questions, My Lord. ( _Resumes his seat_.)
**Judge** Do you wish to question this witness, Sir James?
**Barrington** I most certainly do, My Lord. Mr Hussein, do you keep a box of dusting powder and a brush under your shop counter?
**Hussein** ( _puzzled_ ) No, sir, I do not.
**Barrington** What about a fingerprint pad?
**Hussein** A fingerprint pad?
**Barrington** A pair of handcuffs, perhaps?
**Hussein** Why should I do that?
**Barrington** Because you seem to enjoy playing the amateur detective.
**Hussein** I'm not sure I know what you mean, Sir James.
**Barrington** Then allow me to explain. You come across a photograph of Mr Sherwood in a medical journal attending his wife's funeral With the help of a magnifying glass you spot a lady in the crowd whom you think you recognise. You discover her name in your drugs register and suddenly you're the Sherlock Holmes of Wellingborough.
**Hussein** ( _looks puzzled_ ) I don't know a Mr Holmes.
**Barrington** You don't?
**Hussein** I don't recall him ever coming into the shop.
**Barrington** When did you arrive in this country, Mr Hussein?
**Hussein** Just over two years ago.
**Barrington** Only two years ago, so you're probably still unfamiliar with our ways, not to mention our literature?
**Hussein** I have long been an admirer of the British, Sir James.
**Barrington** I feel sure you have, Mr Hussein, but that doesn't make you British. Did you qualify in this country?
**Hussein** No, sir, I did not.
**Barrington** Then why are you allowed to dispense dangerous drugs?
**Hussein** Because I have been a member of the Royal Pharmaceutical Society for the past twenty years.
**Barrington** On what grounds, may I ask?
**Hussein** Because in my own country I am a qualified doctor.
**Barrington** But you are not in your own country, Mr Hussein, so I must ask you, do your qualifications permit you to practise, as a doctor, in this country?
**Hussein** No, sir, but... that is purely...
**Barrington** No buts, Mr Hussein. They do not permit you to practise as a doctor in this country!
**Hussein** Sadly, not yet.
**Barrington** Or perhaps not so sadly, for the patients.
**Kersley** My Lord, is my learned friend going to be allowed continually to insult the witness in this manner?
**Judge** Mr Kersley, this is a charge of murder and, within the bounds of reason, I shall allow the Defence every latitude.
**Kersley** Is that latitude to be extended to Sir James asking questions to which he already knows the answer?
**Judge** Only a foolish lawyer asks questions to which he does not know the answer, Mr Kersley. Please carry on, Sir James.
**Barrington** Thank you, My Lord. Mr Hussein, have you ever visited St George's Hospital?
**Hussein** No, sir, though I have...
**Barrington** Have you ever come into contact with Mr Sherwood?
**Hussein** Yes, I attended a lecture he gave to the King's Fund.
**Barrington** Along with how many other people, may I ask?
**Hussein** There must have been over a hundred people present.
**Barrington** I doubt if even Mr Sherlock Holmes would have gleaned enough evidence from going to a lecture, attended by over a hundred people, to conclude that the lecturer should be charged with murdering his wife. If that were sufficient evidence, you'd be claiming we're old friends simply on the strength of this cross-examination.
**Hussein** I wouldn't dream of doing so, Sir James, especially as we have met before.
**Barrington** And when, pray, was that?
**Hussein** It was just over a year ago, when you were chairman of the Bar Council and addressed the Anglo-Indian Society - of which I have the honour of being Secretary.
**Barrington** And how many people were present on _that_ occasion.
**Hussein** Just over three hundred.
**Barrington** Just over three hundred.
**Hussein** Yes, but we did sit next to each other during dinner. At the time you left the Society in no doubt that you felt most Indians ran corner shops and you thought it amusing that you were still unable to tell one from another. However, I would not ask Sir Arthur Conan Doyle to consider that as evidence of anything.
**Barrington** I'm delighted that we've finally found something we can agree on, Mr Hussein, because convicting Mr Sherwood will depend on substantiated evidence, and not on the arm's-length opinion of an unqualified chemist. No further questions, My Lord.
**Judge** Do you wish to re-examine, Mr Kersley?
**Kersley** Yes, thank you My Lord. I do have one question for Dr Hussein... I beg your pardon, My Lord... Mr Hussein, may I ask you why you were so willing to assist the police in their enquiries, even agreeing to appear in this case as a Crown witness?
**Hussein** To do one's duty as a good citizen and respect the law is the British way, Mr Kersley. Or that's what Sir James assured us when he delivered his lecture to the Anglo-Indian Society last year.
**Kersley** No more questions, My Lord.
**Judge** Thank you Mr Hussein, you may leave the witness box. (Hussein _leaves the witness box and passes in front of_ **Kersley**.).
**Kersley** ( _aside to_ **Hussein)** Dr Hussein, you're wasted on the medical profession - you should have joined us at the Bar.
**Hussein** Oh, no, Mr Kersley, I'm far too honest to make a success of your chosen profession.
**Judge** You may call your next witness, Mr Kersley.
**Kersley** Thank you, My Lord. I call Professor Alistair Forsyth.
**Usher** Call Alistair Forsyth.
**Guard** Alistair Forsyth.
**Professor Forsyth** _is a Scot of around fifty-five to sixty, very formal and slightly pompom. He enters the courtroom and goes straight to the witness box. He takes the card in his right hand, as if he has been through the process many times before. He does not wait for the_ **Usher** _to instruct him_.
**Forsyth** I swear by Almighty God that the evidence I shall give shall be the truth, the whole truth and nothing but the truth.
**Kersley** Professor, I would like to establish with the jury the particular expertise you bring to this case. ( **Forsyth** _nods_.) You were educated at Edinburgh Academy, from where you won a scholarship to Cambridge to read medicine?
**Forsyth** Yes, Mr Kersley, that is correct.
**Kersley** At Cambridge you graduated with honours in Pharmacology and went on to do research for an MD?
**Forsyth** That is also correct.
**Kersley** On completing your MD, you took up a Fellowship at King's College, London, where you continued your research. May I enquire what your specialist subject was?
**Forsyth** Toxicology, the study of poisons.
**Kersley** You were offered the Chair of Toxicology at London University, and you have since written several books on the subject, which are acknowledged as the recognised text for any student reading for a medical degree?
**Forsyth** Only three of my works are set texts, Mr Kersley. The rest of them would be far beyond the understanding of the average undergraduate.
**Kersley** Quite so. You are a Fellow of the Royal Society and have recently been awarded a CBE for services to medicine?
**Judge** I do believe, Mr Kersley, that you have established beyond peradventure the credentials of your expert witness, so perhaps the time has come to get on with the case in hand.
**Kersley** I am delighted to learn, My Lord, that you feel Professor Forsyth's credentials need no further claims on my part, as I believe your endorsement can only give the jury added confidence in his opinions.
**Judge** ( _scowls_ ) Mr Kersley. Get on with it.
**Kersley** First let me ask you, Professor, how you became involved in this case?
**Forsyth** The Crown Prosecution Service invited me to make a report on Mrs Sherwood's medical history. I began by reading all Mrs Sherwood's files held at St George's.
**Kersley** And would I be correct in thinking that you sought a Home Office order to exhume the body?
**Forsyth** I would have done so, Mr Kersley, had Mr Sherwood not given instructions for the body to be cremated a few days after her death.
**Kersley** Really. Despite this setback, were you able to discover any new evidence?
**Forsyth** No, because Mr Sherwood had misled his colleagues into believing that his wife's previous heart attack explained her premature death. They also emphasised that as she was married to a surgeon who specialised in the subject, her aftercare treatment could hardly have been better.
**Kersley** Or worse, as the case may be.
**Judge** Mr Kersley, you will in future desist from making these _sotto voce_ remarks. ( _Turns to the audience_.) Members of the Jury, Mr Kersley's comment should be ignored.
**Kersley** But not forgotten, I suspect.
**Judge** Did you wish to say something, Mr Kerlsey?
**Kersley** My Lord, I was simply at pains to point out that...
**Judge** It is not your responsibility to point out anything, Mr Kersley, merely to ask questions, which may elicit answers that in turn might possibly assist the jury.
**Kersley** But... My Lord... if I am to discharge...
**Judge** No buts, Mr Kersley, as Sir James has so properly reminded us. From you, I only require questions. I expect the answers to come from the witnesses.
**Kersley** So be it, My Lord. Professor, would it be possible for an experienced doctor to poison a patient while at the same time fooling his colleagues?
**Forsyth** Yes, nowadays that would be easy enough for anyone with Mr Sherwood's experience. There are three known poisons - only one available on prescription - that would kill an intended victim without leaving any clue that a murder had taken place.
**Kersley** Well, I will deal only with the one poison that is available on prescription - Potassium Chloride. Professor, could you poison someone with Potassium Chloride and hope to get away with it?
**Forsyth** Oh, yes, it's the most satisfactory of all poisons for a would-be murderer. Once injected, the victim will suffer a cardiac arrest, showing absolutely no sign of being poisoned.
**Kersley** So what led you to suspect that this was not a death by natural causes?
**Forsyth** The discovery of a deposit of Potassium Chloride on the rubber glove found on the floor of the Sherwoods' kitchen.
**Kersley** And how much poison was discovered on the glove?
**Forsyth** One milligram, which is an amount consistent with checking that a hypodermic needle was working effectively.
**Kersley** Would you care to demonstrate to the jury exactly what you mean by that, Professor?
**Forsyth** Certainly. ( _Pulls on a rubber glove and demonstrates to the audience_.) Just before injecting a patient, you press the plunger thus, to ensure that the liquid is flowing. ( _He allows it to fly into the air, landing on his glove_.) As you observe, some droplets end up on the glove.
**Kersley** And this led you to believe that Mrs Sherwood had probably received an injection of Potassium Chloride just before her death?
**Forsyth** Yes, it did.
**Kersley** And could such an injection also have caused the bruising on her arm?
**Forsyth** Most certainly it could, especially if she had offered any resistance.
**Kersley** Professor, I should now like to ask you about the glass of wine found on the table by Mrs Sherwood's side. Have you been able to analyse its contents?
**Forsyth** Yes I have, and they revealed large deposits of Temazepam - a particularly strong sedative, available only on prescription. There was enough left in the glass to have knocked out a heavyweight boxer.
**Kersley** Which would, had she taken it, have made injecting her all the more easy.
**Barrington** My Lord, I was accused by my learned friend of a plot worthy of _A Book at Bedtime_. Following Mr Kersley's flight of fancy, once this trial is over can I assume he will be applying to become an investigative journalist with the _News of the World?_
**Judge** We will leave the jury to decide which one they consider the better qualified for that job, Sir James. Carry on, please, Mr Kersley.
**Kersley** Professor, can you confirm that six ampoules of Potassium Chloride were collected by Ms Mitchell from a chemist in Wellingborough?
**Forsyth** Yes, I can. I studied the poison register and checked all six entries against the prescriptions collected by Miss Mitchell and they all tallied.
**Kersley** And as a leading authority on the subject, would you now tell the court how many ampoules of Potassium Chloride it would take to cause a fatal heart attack?
**Forsyth** ( _hesitates_ ) Four ampoules would be certain to cause cardiac arrest, but a fifth would leave no hope of survival.
**Kersley** And how would the victim die, Professor?
**Forsyth** In great pain, before the heart finally gave out.
**Kersley** But surely the post-mortem would reveal strong traces of Potassium Chloride that would cause the examining doctor to become suspicious?
**Forsyth** Unfortunately not. A heart attack causes an unusual amount of Potassium to be released into the bloodstream, which would be regarded as quite normal by any doctor conducting a post-mortem.
**Kersley** And Mr Sherwood would have been aware of this?
**Forsyth** A first-year medical student would have been aware of it.
**Kersley** Professor, what would be your opinion of a doctor who took advantage of such specialised knowledge?
**Forsyth** It betrays the very principles of the Hippocratic Oath, 'Whatever house I enter, there will I go for the benefit of the sick, refraining from all wrong doing.' The meaning could not be clearer.
**Kersley** No more questions, My Lord.
**Judge** Sir James. Do you wish to cross-examine?
**Barrington** Thank you, My Lord. Dr Forsyth.
**Forsyth** Professor.
**Barrington** I do apologise, Professor. May I begin by congratulating you on such an illustrious career, detailed so laboriously by my learned friend. But do you consider, as a scientist, you are also qualified to pass moral judgements on a colleague without relying on a shred of evidence?
**Forsyth** The Hippocratic Oath is the very foundation of a doctor's code of practice.
**Barrington** And there is nothing in Mr Sherwood's equally distinguished career to suggest that he doesn't agree with you. So let us now consider the facts, Professor. You told the court that none of the doctors at St George's gave you any reason to believe that Mrs Sherwood had died in unusual circumstances.
**Forsyth** That is correct, but none of them was aware that Mr Sherwood had been collecting ampoules of Potassium Chloride from a chemist outside London.
**Barrington** I will come to that, Professor. Now, in your long report commissioned by the Crown Prosecution Service, you also confirm ( _Holds up the report_.) that Mrs Sherwood's GP had her on the correct programme of medication for the particular heart problem she was suffering from?
**Forsyth** Yes, but Dr Haslam, was not...
**Barrington** I wonder, Professor, if you would be kind enough to confine yourself to answering my questions and not making speeches. This courtroom is not an extension of your lecture theatre and I am not one of your undergraduates. So allow me to move on to the constituents of Potassium Chloride and, may I say, Professor, how much we all enjoyed your little demonstration with the hypodermic needle, which you claimed was consistent with the amount of the chemical found on the kitchen glove.
**Forsyth** ( _now angry_ ) It most certainly was.
**Barrington** But tell me, Professor, as an acknowledged expert on the subject, would it not also be consistent with the amount of Potassium found in this bottle of grapefruit juice, which is more likely to be located in a kitchen?
**Forsyth** Yes, but...
**Barrington** Search as I might, I couldn't find any reference to grapefruit juice in your hundred-and-thirty-nine-page report.
**Forsyth** My report was not concerned with... the contents...
**Barrington** Then perhaps it should have been. Professor, you told the court that the first thing you did when you were asked to look into this case was to study Mrs Sherwood's medical history.
**Forsyth** And I did so.
**Barrington** And so did I, Professor, and I discovered that Mrs Sherwood's father had died of a heart attack at the age of fifty-eight. Why didn't you consider this possible hereditary condition worthy of mention?
**Forsyth** Because I could find no connection between the death of a thirty-seven-year old woman and her father's demise at fifty-eight. Had you been one of my undergraduates Sir James, you would have learnt that research is one thing, being able to draw scientific conclusions from it is quite another.
**Barrington** Well, let us consider some of your scientific conclusions, shall we, Professor, and try to find out what you have learnt from them. On how many occasions have you testified for the crown in murder trials where poisoning was involved?
**Forsyth** A dozen - more, perhaps.
**Barrington** And was one of those cases 'The Crown versus Mr Roger Latham'?
**Forsyth** ( _embarrassed_ ) Yes, it was.
**Barrington** And were you called by the prosecuting counsel as an expert witness?
**Forsyth** Yes, I was.
**Barrington** And was it your evidence that influenced the jury to return a verdict of guilty?
**Kersley** ( _leaps up_ ) My Lord, is my learned friend questioning Professor Forsyth's integrity?
**Judge** Are you, Sir James?
**Barrington** Certainly not. But I would refer your Lordship to the Judge's summing up, and I quote, ( _He picks up a book and quotes from the trial_.) 'I find the evidence presented by Professor Forsyth as compelling, and feel it should weigh heavily with the jury when they come to consider their verdict.'
**Judge** In the judge's summing up. I see. Please continue, Sir James.
**Barrington** Did you tell the court on that occasion that after you had carried out extensive laboratory tests, you were in no doubt that the liquid Mr Roger Latham poured into his brother's coffee not only poisoned him, but was responsible for his premature death?
**Forsyth** That was my opinion at the time.
**Barrington** At the time. I see. And did another professor later prove that when the liquid was poured into hot coffee it was immediately neutralised and couldn't have poisoned a mouse?
**Forsyth** Yes, but there was no way of knowing that then. It was some years later...
**Barrington** I was not suggesting, even for one moment, Professor, that you were culpable, only human, like the rest of us and therefore capable of making mistakes. Remind me, what verdict did the jury reach on that occasion?
**Forsyth** Guilty.
**Barrington** And did Mr Latham die in gaol, having served fourteen years of his life sentence?
**Forsyth** ( _nods_ ) Yes, I believe he did.
**Barrington** And, two years after his death, did his family receive an unconditional pardon from the Home Secretary?
**Forsyth** ( _softly_ ) Yes, but the antitoxic properties of that particular substance were not discovered until...
**Barrington** Until it was too late, Professor. But fortunately it is not too late in this case. Professor, you have already confirmed that the amount of Potassium Chloride found in Mrs Sherwood's bloodstream was consistent with a heart attack.
**Forsyth** What I actually said was....
**Barrington** Yes or no, Professor? It either was consistent with a heart attack or it wasn't.
**Forsyth** ( _hesitates_ ) Yes, it was.
**Barrington** Could someone with Mrs Sherwood's medical history have suffered a heart attack, if she had been surprised by an intruder? Yes or no?
**Forsyth** Yes, it's possible.
**Barrington** And are you aware of any other doctors who have prescriptions made up outside of the hospital they work in? Yes or no?
**Forsyth** Yes.
**Barrington** And could the amount of Potassium Chloride found on the rubber glove have been concentrated grapefruit juice? Yes or no?
**Forsyth** Yes, I suppose it could.
**Barrington** And if Potassium Chloride is taken in small doses, isn't it harmless - and in certain cases even beneficial? Yes or no?
**Forsyth** Yes, but...
**Barrington** Shall we dispense with all these buts, Professor, and remove any doubt in the jury's mind once and for all? I wonder, My Lord, if I might be shown exhibit twenty-six, the ampoule of Potassium Chloride that was found in Mr Sherwood's bag, which Mr Hussein identified as coming from his shop.
**Judge** For what purpose, Sir James?
**Barrington** Like the distinguished professor, My Lord, I wish to conduct an experiment.
**Judge** I do hope you're not wasting the court's time, Sir James.
**Barrington** Heaven forbid, My Lord. I simply wish to prove my client's innocence.
_The_ **Judge** _nods to the_ **Usher** _who removes the ampoule from_ **Sherwood** ' _s bag and hands it over to_ **Barrington,** _who breaks the ampoule and drinks it slowly_.
**Barrington** A little dry for my taste, but as you can see, Professor, not fatal. So let us hope that once again it will not be your expert evidence who allows an innocent man to rot in gaol for the rest of his life.
**Kersley** My Lord, this goes far beyond the bounds of... legitimate cross-examination...
**Judge** You are quite right, Mr Kersley. Sir James, that comment was unforgivable, and I must insist that you withdraw it immediately.
**Barrington** ( _pauses_ ) My Lord, I am unable to do so.
**Judge** And why is that, Sir James?
**Barrington** I was the defence counsel in The Crown versus Mr Roger Latham and I will go to my grave aware that my feeble advocacy failed to save the life of an innocent man. I am determined that it will not happen a second time. No further questions, My Lord.
_The lights slowly dim and everyone's eyes remain on_ **Barrington**. _The curtain falls, but goes back up during the interval, to reveal the jury door back in place_.
CURTAIN
### **_Act Two_**
#### **Scene One**
_The following morning_.
_The stage remains dark, but the audience can still see the jury room door. On the wall, the clock is showing 9.45 a.m. When the house lights dim, the_ **Jury Bailiff** _steps through the door and addresses the audience_.
**Jury Bailiff** Good Morning, Ladies and Gentlemen of the Jury and thank you for reporting back in good time. The second day of the trial will begin with Mr Kersley's final witness, Ms Jennifer Mitchell, Ms Mitchell is the Crown's principal witness, so I would not be surprised if she was in the witness box for most of the day.
Please continue to be vigilant when it comes to discussing this case with anyone not on the jury. If there are no questions, we should make our way back to Court Number One. Ladies and Gentlemen of the Jury, please follow me.
_The_ **Jury Bailiff** _steps through the door, and when the lights come up, we are back in Court Number One, where we discover the barristers are talking among themselves while they await the entrance of the_ **Judge** _and jury_.
**Ashton** I've just bumped into Ms Mitchell as she was coming up the steps.
**Kersley** What's she wearing?
**Ashton** Smartly tailored blue suit, very conservative, and virtually no make-up.
**Kersley** Good, that's exactly the image I want fixed in the jury's mind.
**Ashton** And I reminded her to refer to her father, Councillor Mitchell, as often as possible.
**Kersley** And under no circumstances to mention the reason she left St George's at such short notice?
**Ashton** It was the last thing I emphasised.
**Kersley** Good. Then we can only hope that Sir James doesn't know the real reason.
_The attention moves across to_ **Barrington** and _his junior_ , **Jarvis**.
**Barrington** And so we finally come up against their star witness.
**Jarvis** And how do we feel about that?
**Barrington** If Ms Mitchell is as innocent as Mr Kersley would have us believe, we're in a lot of trouble.
**Jarvis** Surely not, after the roasting you gave Webster and Forsyth yesterday.
**Barrington** Yes, but try not to forget the doctor's bag. Kersley made me feel like a pupil who'd just arrived in chambers, so if under his guidance Ms Mitchell proves a little too convincing, I may have to take the odd risk.
**Jarvis** What do you have in mind?
**Barrington** I might even consider asking the occasional question to which I do not know the answer. So if I put out my hand, ( _He makes a gesture_.) make sure you pass me a blank sheet of paper.
**Usher** Be upstanding in the court. All persons having anything to do before my Lords, the Queen's Justices, draw near and give your attendance. God save the Queen.
_All rise as_ **Mr Justice Cartwright** _enters and resumes his place. All bow and he returns the bow_.
**Usher** Bring up the prisoner. ( **Sherwood** _enters the dock_.)
**Judge** ( _to the audience_ ) Ladies and Gentlemen of the Jury, good morning. Mr Kersley, you may call your next witness.
**Kersley** Thank you, My Lord. I call Jennifer Mitchell.
**Usher** Call Jennifer Mitchell. ( _A woman of thirty, attractive and dressed in smart suit, enters the witness box_.)
**Guard** Jennifer Mitchell.
**Usher** Take the testament in your right hand and read from the card.
**Mitchell** ( _quietly_ ) I swear by Almighty God that the evidence I shall give shall be the truth, the whole truth and nothing but the truth.
**Kersley** Is your name Jennifer Alice Mitchell and are you presently working as a senior staff nurse at Wellingborough Cottage Hospital?
**Mitchell** Yes, I did.
**Kersley** Did you previously work at St George's Hospital, Tooting?
**Mitchell** Yes, I did.
**Kersley** And was the consultant in charge of the cardiac unit the defendant, Mr Patrick Sherwood?
**Mitchell** ( _avoids looking at the dock_ ) Yes, he was.
**Judge** Can you please speak up, Ms Mitchell, The jury ( _He waves a hand, sweeping the audience_.) will need to hear every word you have to say. ( **Mitchell** _nods_.)
**Kersley** When you first began working for Mr Sherwood, what were your responsibilities?
**Mitchell** I was a junior staff nurse attached to the cardiac unit.
**Kersley** And did Mr Sherwood ever ask you to carry out any duties not directly related to your work on the cardiac unit?
**Mitchell** Yes, a few months after I'd started working at St George's, Mr Sherwood asked me to pick up a sedative for his wife from the hospital pharmacy, which I was happy to do.
**Kersley** Understandably. You would want to please your boss.
**Judge** Mr Kersley, that was both leading as well as an opinion. Do not further try my patience.
**Kersley** ( _facing the judge_ ) I will attempt very hard not to do so, My Lord. ( _Turning back to_ **Mitchell.)** Ms Mitchell, did Mr Sherwood ask you to collect any other prescriptions from the hospital pharmacy?
**Mitchell** Yes, but that would be quite normal practice for any of the nurses.
**Kersley** But then one Friday evening he asked you to pick up a prescription from outside London?
**Mitchell** Yes, that was just before I was leaving to spend the weekend with my parents in Wellingborough and he instructed me _not_ to have that prescription made up at the hospital pharmacy.
**Sherwood** I did no such thing - I've never asked you to pick up any of my prescriptions in Wellingborough.
**Mitchell** But you did tell me _not_ to have them made up at St George's.
**Sherwood** I most certainly did not...
**Judge** Mr Sherwood, you must not interrupt the witness while she is being questioned by counsel, now sit down ( **Sherwood** _reluctantly sits down)_. Carry on, Mr Kersley.
**Kersley** Thank you, My Lord. Did Mr Sherwood give any explanation as to why you shouldn't have the prescription made up at the hospital pharmacy?
**Mitchell** No, he just said that it was for a private patient.
**Kersley** Is that also normal practice?
**Mitchell** No, I had never been asked to do that before by any doctor.
**Kersley** Did you question Mr Sherwood about this?
**Mitchell** No one questioned Mr Sherwood about anything - not even Sister.
**Kersley** And was this prescription also for a sedative?
**Mitchell** No, it was for ten millilitres of Potassium Chloride.
**Kersley** Did he ever ask you to pick up any more ampoules of Potassium Chloride?
**Mitchell** Yes, he did. It must have been about two weeks later - also on a Friday evening, and he asked me to drop it back to his office on Monday morning.
**Kersley** And how many times did he ask you to have prescriptions for poison made up outside London?
**Mitchell** Half a dozen times during the next three months.
**Kersley** I apologise, Ms Mitchell, but I must now ask you an embarrassing question as I want the jury to understand fully why you became so willing to fall in with Mr Sherwood's plans. During the time leading up to Mr Sherwood asking you to collect these prescriptions from Wellingborough, did he maintain a professional relationship with you?
**Mitchell** To begin with he did, but then he started to give me presents, send me flowers and even invited me out to dinner.
**Kersley** And did you accept any of these invitations?
**Mitchell** No, not immediately, but then he became more and more insistent, and it's quite difficult for a junior nurse to go on refusing the senior consultant. So I finally gave in and agreed to have dinner with him at a restaurant in Fulham, where he assured me no one would recognise us.
**Kersley** So he chose the restaurant, but did you make the booking?
**Mitchell** No, he did. And he paid for the meal in cash - he explained that way it couldn't be traced back to him.
**Kersley** And did he invite you out again?
**Mitchell** Oh, yes. After that, Patrick regularly invited me out for a meal, or to the theatre. He kept telling me how lonely he was. And then one night on the way home, he said he was falling in love with me.
**Kersley** What was your reaction?
**Mitchell** I was very flattered at the time, and when we arrived back at my flat he asked if he could come up for coffee.
**Kersley** And did you agree to his request?
**Mitchell** No, I made some excuse about having to be up early for morning rounds. I didn't mind having dinner with him, but I wasn't willing to begin a relationship with a married man, especially one who was working at the same hospital.
**Kersley** Quite understandably. But that later changed?
**Mitchell** Yes, it was a few weeks later, at the staff Christmas party - he pulled me under the mistletoe and started kissing me.
**Kersley** But isn't that normal at a Christmas party, Ms Mitchell?
**Mitchell** Yes, but it isn't normal for a doctor to put a hand on your breast.
**Kersley** And how did you respond?
**Mitchell** I pushed him away and then left the party as quickly as possible, hoping that none of the other nurses had noticed.
**Kersley** And did you go straight home?
**Mitchell** I would have done so, but he followed me out. He apologised immediately and asked if he could walk me back to my flat.
**Kersley** Did you let him?
**Mitchell** Yes, I lived less than a mile away and he seemed very contrite.
**Kersley** And did he leave you once you arrived back at your flat, Ms Mitchell?
**Mitchell** No, he asked if he could come in for coffee.
**Kersley** And did you agree this time?
**Mitchell** Yes, I'm afraid I did. You see, on the way home he never stopped telling me how much he adored me and longed to be with me. And in any case, it had started to rain and I felt rather sorry for him.
**Kersley** So after he'd joined you in the flat, what happened next?
**Mitchell** I made us some coffee, but he hung around long after he'd drunk it. I pointed out that it had stopped raining, but he just wouldn't go.
**Kersley** And did he once again try to kiss you?
**Mitchell** Yes, he did.
**Kersley** Forgive me for being indelicate, Ms Mitchell, but how far did it go?
**Mitchell** ( _looks at the_ **Judge** ) Do I have to answer that question, My Lord?
**Judge** Yes, indeed you do, Ms Mitchell. You see the jury need to know exactly what took place that night.
**Mitchell** ( _hesitates_ ) We ended up making love.
**Sherwood** We've never made love and you know it.
**Judge** Mr Sherwood, I have already spoken to you about interrupting this witness.
**Sherwood** My Lord, do I have to sit in silence while this woman goes on telling lies about me?
**Judge** You must listen to all the witnesses without interrupting, Mr Sherwood. You will in time be given an opportunity to answer questions from the witness box under oath, if you choose so to do. But until then, you will remain silent. ( **Sherwood** _sits down_.) Carry on, Mr Kersley.
**Kersley** So, after Mr Sherwood had made love to you, did he eventually leave?
**Mitchell** Yes, it must have been about three in the **morning**.
**Kersley** And how did he treat you the following day?
**Mitchell** He was very courteous and professional during ward rounds, but that changed when he asked me to join him in his office.
**Kersley** Why, what happened, Ms Mitchell?
**Mitchell** As soon as I walked in, he pulled me towards him and started to undo my uniform. I warned him that anyone might come in.
**Kersley** What did he say to that?
**Mitchell** He didn't, he just smiled and locked the door.
**Kersley** And did he continue to try to undress you?
**Mitchell** Yes, we ended up making love on his couch.
**Kersley** And did this become a regular occurrence?
**Mitchell** Oh, yes. After that, Patrick would often turn up late at night, explaining that he told Elizabeth he was out visiting a patient. He made love with such a passion that I was convinced he no longer had a sexual relationship with his wife.
**Kersley** And how long did this affair go on for?
**Mitchell** About three months.
**Kersley** And he asked you to pick up the prescriptions from Wellingborough soon after you started sleeping with him?
**Mitchell** Within days. But by then Patrick knew only too well that I nearly always visited my parents at the weekend.
**Kersley** And were these prescriptions ever made out for Mrs Sherwood?
**Mitchell** No, they were all for private patients.
**Kersley** Can you recall the date of the first one?
**Mitchell** Yes. It was about three months before his wife died.
**Kersley** How can you be so sure?
**Mitchell** Because I made a diary entry that Patrick had given me a ten-pound note to cover the cost, and there was four pounds change, which I returned to him on the Monday morning.
**Kersley** But if you were so surprised to be asked to carry out Mr Sherwood's instructions, why didn't you report your misgivings to anyone in authority?
**Mitchell** Because by then I'd fallen in love with him.
**Kersley** And did he ever give any hint that he might have fallen in love with you?
**Mitchell** Oh, yes. Patrick regularly told me he was sick of his wife, and he couldn't wait to be rid of her so he could spend the rest of his life with me. He hated the way she constantly belittled him in front of the staff. He even talked about moving out of Arcadia Mansions, finding a larger apartment and starting a family. I didn't take it too seriously to begin with. But then one night he proposed.
**Kersley** He asked you to marry him?
**Mitchell** Yes, he did.
**Kersley** While his wife was still alive?
**Mitchell** Yes, it was after we'd made love - I'll never forget it, because he left a few minutes later.
**Kersley** So would that have been around three in the morning?
**Mitchell** No, just after ten.
**Kersley** But you told the court that he usually left around three in the morning, so that no one would realise you were having an affair.
**Mitchell** Yes, but that night he told me he had to visit a patient in Westminster and he would come back later, which puzzled me.
**Kersley** Why did it puzzle you, Ms Mitchell?
**Mitchell** Because he didn't have his doctor's bag with him.
**Kersley** And did he come back later?
**Mitchell** Yes, a few minutes after eleven.
**Kersley** And did you notice any change in him when he returned?
**Mitchell** Yes, he seemed very nervous. And when I asked if there was a problem, he told me that the patient had died.
**Kersley** He told you that the patient had died?
**Mitchell** Yes, I tried to calm him, but he just paced around the room mumbling to himself and then suddenly left without warning.
**Kersley** And did he ever raise the subject of marriage again?
**Mitchell** No. Once his wife had died, Patrick refused even to speak to me.
**Kersley** But this was the man who had regularly shared your bed and even asked you to be his wife.
**Mitchell** Yes. ( _Near to tears_.) Which is why it came as such a terrible shock when he dropped me without any warning. Of course, relationships can come to an end, but that doesn't mean you can't remain on good terms.
**Kersley** But you did attend Mrs Sherwood's funeral?
**Mitchell** Yes, but Patrick ignored me and didn't even invite me back to his flat to join the other guests.
**Kersley** Was there anything else you can remember about the funeral?
**Mitchell** Yes, I was surprised that Mrs Sherwood was cremated and it made me anxious for the first time.
**Kersley** What were you anxious about?
**Mitchell** Well, I began to wonder if Patrick was making sure that no one could ever check what had caused the bruising on her arm.
**Sherwood** _half rises, catches_ **Judge** 's _eye and sits back down_.
**Barrington** My Lord, this is a disgraceful slur on my client. There is absolutely no proof that...
**Judge** I agree, Sir James. The jury will disregard Ms Mitchell's last statement. Ms Mitchell, this court is not interested in your opinions, only in facts. And Mr Kersley, you are as much to blame. You must have known only too well where that question was leading.
**Kersley** I did, My Lord, but then I considered Mrs Sherwood's cremation to be a fact, and one the jury might even find relevant.
**Judge** Mr Kersley, you are bordering on impertinence. Stick to facts in future.
**Kersley** Ms Mitchell, is it a fact that while you continued working at St George's Mr Sherwood began to threaten you?
**Mitchell** Yes, he did. When I raised the subject of the prescriptions he flew into a rage, warning me that if I mentioned them to anyone, not only would I lose my job at St George's but he would personally make sure that no other hospital would ever employ me again.
**Kersley** So you decided to leave St George's.
**Mitchell** I didn't have a lot of choice, as he'd made it abundantly clear that I had no hope of promotion while he remained the head of department.
**Kersley** So what did you do next?
**Mitchell** I resigned and took a short holiday abroad. While I was away, a vacancy arose at Wellingborough Cottage Hospital for the position of senior staff nurse. I was delighted when they offered me the post and hoped this would be my chance to start a new life.
**Kersley** But that didn't prove possible, did it, Ms Mitchell, because Mr Sherwood got in touch with you again. When was that?
**Mitchell** It must have been about a month after I'd started my new job. He began phoning me at the hospital.
**Sherwood** ( _rises_ ) I never phoned you. I didn't even know you'd got another job.
**Judge** Mr Sherwood, this is the last time I shall warn you. If you interrupt again, I will have you taken below as long as Ms Mitchell remains in the witness box. Do I make myself clear?
**Sherwood** Yes My Lord, but...
**Barrington** My Lord, may I be allowed to have a word with the defendant?
**Judge** For what purpose, Sir James?
**Barrington** To leave Mr Sherwood in no doubt of the harm it will do his cause if he is unable to hear all the evidence offered by this particular witness.
**Judge** Do you have any objection, Mr Kersley?
**Kersley** If it means I can continue uninterrupted, My Lord, I would welcome it. However, it may be that these unseemly outbursts, repeated by Mr Sherwood at regular intervals despite Your Lordship's continual warnings, will only serve to shed light on the character of the accused and thus assist the jury when the time comes to consider their verdict.
**Barrington** My Lord, my learned friend can never resist making a speech and this one did indeed shed some light, if not on Mr Sherwood's character, then on the Crown's lack of any real evidence, which they try to cover up by resorting to personal abuse. However, I will have a word with my client My Lord, and warn him of the gravity of your words. ( **Barrington** _goes over to the dock_.) Very good, but that was one more interruption than we agreed on, so don't do it again. ( **Sherwood** _nods and sits back down_ , **Barrington** _returns to his place_.) I'm obliged, My Lord. I think Mr Sherwood now fully understands what is expected of him.
**Judge** Thank you, Sir James. Carry on, Mr Kersley.
**Kersley** You were telling us, Ms Mitchell, that Mr Sherwood telephoned you at the hospital - for what purpose?
**Mitchell** To warn me that the police had begun an investigation into his wife's death and might want to question me about our relationship.
**Kersley** Really? Did he say anything else?
**Mitchell** Yes, he begged me not to mention the fact that I'd been with him on the night his wife died, as he had already come up with a more convincing alibi.
**Kersley** Why would he need another alibi, if he had been with you?
**Mitchell** Because he didn't want the police to know that we'd been having an affair.
**Kersley** And how did you respond to this request?
**Mitchell** I told him to go to hell and rang off.
**Kersley** Did he call again that day?
**Mitchell** Yes, about an hour later, but I refused to speak to him.
**Kersley** How did you avoid it?
**Mitchell** I made some excuse to Sister about not feeling well, and went home early. But that night I was so distressed I just couldn't get to sleep, so in the morning I told my parents everything.
**Kersley** And it was your father, Councillor Mitchell, who left you in no doubt as to where your responsibility lay?
**Mitchell** Yes. He advised me to tell the police everything I knew, otherwise I could be an accessory to the crime and just as guilty as he was.
**Kersley** And when the police got in touch with you, you immediately volunteered a statement?
**Mitchell** Yes, I did.
**Kersley** And has Mr Sherwood tried to contact you again since you made that statement?
**Mitchell** He never stops phoning, sometimes twice a day, but I haven't spoken to him since that morning he rang the hospital. ( **Sherwood** _rises but then sits back down_.)
**Kersley** And despite all that you've been put through these last six months, you still have no regrets about taking your father's advice?
**Mitchell** None whatsoever. My only regret... ( _She hesitates_.)
**Kersley** Yes, Ms Mitchell? Your only regret?
**Mitchell** ( _stares at_ **Sherwood)**... is that I ever agreed to collect those prescriptions for him in the first place.
**Kersley** No more questions, My Lord.
**Judge** Your witness, Sir James.
**Barrington** Thank you, My Lord. Allow me to begin, Ms Mitchell, with the subject of the prescriptions. Can I confirm that you were aware that Potassium Chloride was classified as a controlled drug?
**Mitchell** Yes, of course I was.
**Barrington** So you would have had to sign for them?
**Mitchell** Yes, I signed for all six of them, and as the prescriptions are in the court's safe keeping, you can check for yourself.
**Barrington** I already have, from my photocopies. I just wanted you to confirm that it was your signature on the originals.
**Usher** _shows her the originals_.
**Mitchell** Yes, that is my signature.
**Barrington** Then perhaps you won't mind writing your name on the Usher's pad so that the jury can be left in no doubt. ( _She signs her signature with her left hand and the_ **Usher** _shows the pad to_ **Barrington**.) Yes, there is no doubt it is your signature. ( _He checks the prescriptions_.) You said in your statement to the police that Mr Sherwood only gave you those prescriptions on a Friday evening, just as you were about to leave for the weekend.
**Mitchell** Yes, that's correct.
**Barrington** Then perhaps you can explain why, of the six prescriptions ( _Holds them up_.) one is dated on a Tuesday, two on a Wednesday, two on a Thursday, leaving only one made out on a Friday, which was the one found in Mr Sherwood's bag. ( _He places five of them to one side_.)
**Mitchell** I said nothing about when they were made out, only when he handed them to me for collection and that was always on a Friday evening.
**Barrington** How convenient. Unless, of course, you held on to them until you went home for the weekend.
**Mitchell** Why should I do that?
**Barrington** Why indeed, unless, of course, you had your own reason for wanting to implicate Mr Sherwood, which brings me on to this mythical relationship you claim he began.
**Mitchell** It wasn't mythical and he certainly began it.
**Barrington** You say he showered you with presents, sent you flowers, took you out to dinner - even occasionally to the theatre.
**Mitchell** Yes, he did.
**Barrington** Do you have any proof that these presents ever existed?
**Mitchell** No, of course I don't. The flowers have died and I've eaten all the chocolates.
**Barrington** Now that is convenient. So you're asking the court to believe that all these presents were in one way or another, how can I put this, biodegradable? And were these dinners always at restaurants where he wouldn't be recognised?
**Mitchell** Yes, that was the idea. Patrick said it wouldn't look good if we were seen together while his wife was still alive.
**Barrington** Can you name any of these restaurants?
**Mitchell** ( _considers this_ ) Not off the top of my head, no. But then we never went back to the same one twice, although I do remember we once ate in Fulham.
**Barrington** And after you'd had dinner at these restaurants you can't name, you would sometimes go on to the theatre?
**Mitchell** Yes, on at least two occasions.
**Barrington** So you must be able to name at least two of the plays?
**Mitchell** ( _hesitates_ ) Not immediately. After all, it was over a year ago, but I remember one of them was by Tom Stoppard - and in any case I kept the programmes.
**Kersley** _whispers to_ **Ashton** , _who makes a note_.
**Barrington** And after he'd taken you to the theatre, would he then drive you home?
**Mitchell** Yes, he always took me home.
**Barrington** And on one occasion he asked if he could join you for coffee?
**Mitchell** Yes, that's right.
**Barrington** And until then, he hadn't tried to do anything that could be described as improper?
**Mitchell** No, but two weeks later he grabbed me at the staff party and started kissing me under the mistletoe, and then he put his hand on my...
**Barrington** Quite. Where was the mistletoe hanging, Ms Mitchell?
**Mitchell** ( _hesitates_ ) From a light in the middle of the room.
**Barrington** From a light in the middle of the room. Let me see if I can picture this, Ms Mitchell. He had one hand on your breast, while at the same time trying to kiss you, in the middle of a crowded room, in front of all his colleagues - and your friends.
**Mitchell** Yes. It was very embarrassing.
**Barrington** Not embarrassing enough to stop you letting him take you home and then end up in bed with him.
**Mitchell** I felt sorry for him.
**Barrington** Ah, yes, I remember, it was raining. Have you any idea, Ms Mitchell, how unlikely this story is sounding?
**Mitchell** It's the truth and he knows it.
**Barrington** Well, then let's continue with this morality fable, shall we? The following morning Mr Sherwood called you into his office, locked the door and started trying to undo your uniform?
**Mitchell** Yes, he did.
**Barrington** And then he made love to you on his couch?
**Mitchell** Yes, just as I described it.
**Barrington** What time of day was that?
**Mitchell** I don't remember exactly.
**Barrington Well** how about vaguely - eight o'clock, nine o'clock, ten o'clock?
**Mitchell** I think it must have been about nine thirty. Yes, it was just after we'd completed the early-morning rounds.
**Barrington** So you're asking the court to believe that having left you, exhausted, at three in the morning, he then made love to you again at nine thirty?
**Mitchell** Yes, and I wasn't the only nurse who'd been subjected to 'after-rounds' sex.
**Barrington** Strange, because I don't see any of those nurses on the list of Crown witnesses.
**Mitchell** They didn't want to know, once they realised the police were involved.
**Barrington** Strange how everybody and everything disappears whenever you're involved. And after this romp on the couch, he regularly made love to you in his office and sometimes spent the night with you back at your flat?
**Mitchell** No, he never stayed the night. He always left around three in the morning. He didn't want anyone to find out that we were having an affair.
**Barrington** And you say that this relationship went on for several weeks?
**Mitchell** Over three months.
**Barrington** And you would also have us believe that on one occasion, he asked you to marry him?
**Mitchell** Yes, he did.
**Barrington** Was this at a restaurant you can't remember, or after a play you don't recall?
**Mitchell** No, I remember the occasion well; it's not something a girl is likely to forget. You see, he proposed to me on the night his wife died.
**Barrington** ( _sharply_ ) How could you possibly have known it was the night his wife was going to die?
**Mitchell** Obviously I didn't know until, the following morning. But I'll never forget his words after we'd made love - 'Elizabeth won't be around for much longer, so we ought to start thinking about our future together.'
**Barrington** Elizabeth. Did you ever meet Mrs Sherwood?
**Mitchell** Only when she came to pick up her husband from the hospital.
**Barrington** So you never visited her at the flat?
**Mitchell** No, why should I do that?
**Barrington** To help her with her injections, perhaps?
**Mitchell** No, I've never been to Arcadia Mansions.
**Barrington** Arcadia Mansions. So it wasn't you who was at the apartment when Mrs Sherwood said, 'How did you get in?'
**Mitchell** No, I was at home all that evening.
**Barrington** Alone?
**Mitchell** No, Patrick joined me around eight o'clock. I cooked dinner for him and then we made love in front of the fire. Perhaps you'd like me to go into the details?
**Barrington** Yes, I would, Ms Mitchell, because if this relationship was as intimate as you say, you'll be able to tell the court if Mr Sherwood has any scars, birth marks or even unusual habits that only someone who was so intimate with him would be aware of.
**Mitchell** ( _triumphantly_ ) Yes, I can. Patrick has a small burn on his right arm ( _Touches her right forearm_.) which you can only see when he takes his shirt off.
**Barrington** 'A small burn on his right arm... Only when he takes his shirt off.' ( _He writes down as a quote_.) Anything else?
**Mitchell** ( _considers this_ ) Yes, whenever he took me out on a date, he would remove his wedding ring and when we made love he would always insist on putting out the light. And, oh yes, his great passion is sailing. He keeps a small boat at Burnham, which he takes out most weekends.
**Barrington** One could hardly call this list of mundane trivia intimate knowledge. You know, Ms Mitchell, this is beginning to sound less and less like an affair and more and more like wishful thinking.
**Mitchell** I can assure you it was real.
**Barrington** I have no doubt you wanted it to be real.
**Mitchell** What do you mean by that?
**Barrington** Simply that no one was better placed than you to take advantage of a man who was nursing a dying wife. But he rejected your advances, which made you bitter and resentful. And then, after his wife died, you come up with this cock-and-bull story about Wellingborough, only too aware of how hard it would be for Mr Sherwood to prove his innocence.
**Kersley** My Lord, I think I must have fallen asleep, because the last thing I remember was my learned friend cross-examining this witness, and I seem to have woken up in the middle of his closing speech. Can your Lordship advise me, have I missed any significant questions?
**Judge** No, but you have improperly interrupted defence counsel, just as - I suspect - he was about to ask one. Please continue, Sir James.
**Barrington** If it is a question my learned friend demands, then a question he will get. My Lord, could Ms Mitchell be shown exhibit twenty-three?
**Judge** Yes, Usher. ( _He nods the_ **Usher** _to hold up the glove_.)
**Barrington** Ms Mitchell, do you know what this is?
**Mitchell** Yes, of course I do, it's a rubber glove.
**Barrington** Used for hygienic purposes when administering an injection.
**Mitchell** Or for washing the dishes.
**Barrington** It was found on the floor of the Sherwoods' kitchen the night his wife died.
**Mitchell** So what does that prove?
**Barrington** It's a left-handed glove, Ms Mitchell - are you, by any chance, left-handed?
**Mitchell** No, I'm right-handed.
**Barrington** Strange, because when you signed your signature on the Usher's pad only a few moments ago, you did so with your left hand. Would you care to try on the glove, Ms Mitchell?
**Kersley** My Lord, this is outrageous, it's not Ms Mitchell who is on trial here.
**Judge** I agree, Mr Kersley. Sir James, your job is to defend Mr Sherwood, not to prosecute Ms Mitchell. Stick to your brief.
**Barrington** As you wish, My Lord. But I do hope my learned friend will not object to me asking Ms Mitchell about the phone calls she claims Mr Sherwood made to her after she had left St George's.
**Mitchell** He did call me, again and again.
**Barrington** Again and again. Then how is it that BT are unable to trace a single call to Wellingborough Cottage Hospital from either Mr Sherwood's office, his home or his mobile?
**Mitchell** He could have called me from a phone box.
**Barrington** Oh, I see, so several times a day he just popped out of the operating theatre to phone boxes all over London to plead with you to keep quiet about your affair.
**Mitchell** Yes, he did.
**Barrington** You know, Ms Mitchell, these calls are beginning to sound like the presents, the flowers, the restaurants and the theatre, absolutely no proof of anything actually taking place - which brings me on to the statement that you made to the police following Mrs Sherwood's death.
**Mitchell** Yes, I made a voluntary statement to Chief Inspector Payne.
**Barrington** You did indeed, Ms Mitchell, but what I want to know is what you mean by the word voluntary? Was it an unsolicited statement? Did you, for example, visit a police station and offer to assist them with their enquiries?
**Mitchell** It wasn't quite like that.
**Barrington** It wasn't anything like that, was it, Ms Mitchell? The only reason you volunteered a statement was because your father, Councillor Mitchell, had warned you that if you didn't, you might well be implicated yourself. And correct me if I'm wrong, you didn't make that statement until after Chief Inspector Payne had contacted you some weeks later?
**Mitchell** I volunteered a statement immediately he contacted me.
**Barrington** Yes, but why didn't you contact the police immediately following Mrs Sherwood's death? Why leave it until Inspector Payne had got in touch with you?
**Mitchell** ( _voice rising_ ) Because I didn't have any proof. It would only have been my word against his.
**Barrington** At last we come down to the reality of this case - you didn't have any proof. It's simply your word against his. The truth is, Ms Mitchell, that your word isn't worth the Usher's pad it's written on because there never was any relationship between you and the defendant.
**Mitchell** ( _breaking down sobbing_ ) Yes, there was. We were lovers and he even asked me to be his wife.
**Barrington** Did you accept his proposal?
**Mitchell** Yes, I did.
**Barrington** So you must have been in love with him at the time?
**Mitchell** Yes, I was at the time.
**Barrington** So how do you feel about him now, Ms Mitchell?
**Mitchell** I loathe him.
**Barrington** You loathe him?
**Mitchell** ( _voice rising_ ) Yes, I loathe him. When he needed me, nothing was too much trouble for him, but once I'd served my purpose he dumped me as if I had never existed.
**Barrington** Try not to raise your voice too much, Ms Mitchell. Otherwise the jury might begin to suspect that...
**Mitchell** ( _still sobbing_ ) I don't care what they...
**Kersley** My Lord, I must object. Do these attacks on Ms Mitchell have any real purpose other than to intimidate?
**Judge** Do they, Sir James?
**Barrington** They most certainly do, My Lord, their purpose is to ensure that an innocent man doesn't have to spend the rest of his life in gaol on the evidence of a jealous, vindictive woman who couldn't get her own way. Ms Mitchell, if you are going to persist in claiming that Mr Sherwood seduced you, I must remind you, before you answer my next question, that you are still under oath. ( _He pauses_.) Was Mr Sherwood the first person you had an affair with at St George's? ( **Jarvis** _hands him blank sheet of paper that the audience can see is blank_.)
**Mitchell** ( _hesitates_ ) There may have been one other.
**Barrington** ( _stares down at blank sheet_ ) Only one other, Ms Mitchell?
**Mitchell** Well, over a period of five years, perhaps two.
**Barrington** ( _continues to stare at sheet_ ) Two?
**Mitchell** ( _she hesitates_ ) Possibly three.
**Barrington** ( _slowly_ ) Or four, or five, or...
**Mitchell** No - three.
**Barrington** And were any or all three of these paramours also doctors or surgeons, by any chance?
**Mitchell** Yes, but the first one was years ago and didn't last that long.
**Barrington** Are you certain, of that Ms Mitchell?
**Mitchell** Yes, I am, but then I feel sure even you can remember when you lost your virginity, Sir James.
**Barrington** ( _continues to stare at the blank sheet of paper_ ) But your second and third affairs lasted a considerably longer time, didn't they?
**Mitchell** Yes, but they were over long before Patrick began courting me.
**Barrington** ( _voice rising_ ) But isn't it the truth, Ms Mitchell, that having failed to snare one doctor, you were willing to go to any lengths to catch another?
**Mitchell** No, that is not the truth. The truth is that Patrick told me that he loved me and asked me to be his wife, and I can prove it.
**Barrington** Like you can prove he gave you prescriptions only on a Friday evening, like you can prove that he showered you with presents that no longer exist, like you can prove which restaurants he took you to, but you can't remember their names, like you can prove which theatres you attended, but can't recall the titles of the plays, like you can prove he telephoned you again and again, but there's no trace of any calls, like you can prove...
**Mitchell** That he made me pregnant. ( **Barrington** _is stopped in his tracks_.) And when I told him I was going to have his baby, he begged me to have an abortion. ( _She looks defiantly at_ **Sherwood.)** And I only agreed to his demands because I wasn't willing to give birth to the bastard of a murderer.
CURTAIN
### **_Act Three_**
#### **Scene One**
_The following day_.
_The_ **Jury Bailiff** _walks through the door and addresses the audience. Clock shows 9.45_.
**Jury Bailiff** Good morning Ladies and Gentlemen of the Jury. Today it is the turn of Sir James Barrington to take up the cudgels on behalf of the Defence. I am unable to give you any guidance as to the witnesses he intends to call ( _He glances at his clipboard._ ) as no names appear on the list. This, of course, may be a ruse by Sir James to keep the Prosecution guessing. Or it may simply be that following Ms Mitchell's evidence yesterday, Sir James has not yet decided whether to put Mr Sherwood, in the witness box, where he would have to face cross-examination by Mr Kersley. It is, of course, not compulsory under English law for a defendant to take the stand. However, if he does not, inferences of guilt may be drawn. Ladies and Gentlemen of the Jury, if you are ready, we will return to Court Number One.
_He steps through the door; when the lights come up, we are back in Court Number One_ , _Everyone is in place awaiting the_ **Judge** _and jury_.
**Ashton** Do you think he'll put Sherwood in the box?
**Kersley** I wouldn't. Always quit while you're ahead.
**Ashton** You think he's ahead, after all you achieved yesterday?
**Kersley** Yes, I do. A trial can be going your way and then suddenly one piece of evidence will derail you. Did you see the faces of the jury when they discovered that Ms Mitchell was left-handed. Doubt must have crept into their minds. No, we must hope that he puts Sherwood in the witness box. And if he does, I'll need those two theatre programmes.
**Ashton** Ms Mitchell dropped them off in chambers this morning. They're both in the envelope.
**Kersley** Good.
**Barrington** Have you had a chance to speak to our client this morning?
**Jarvis** He was consuming gallons of black coffee, wondering if you had come to a decision about putting him in the witness box.
**Barrington** I haven't been left with a lot of choice after Mitchell's revelation yesterday. It was a bad mistake on my part, Andrew. What did you learn from it?
**Jarvis** Not to ask one question too many.
**Barrington** A barrister's worst nightmare. If I had stopped the moment she admitted to having affairs with three other doctors, the jury would not have been quite so sure about the rest of her evidence. Why in heaven's name didn't Sherwood tell us about the abortion in the first place?
**Jarvis** Perhaps he didn't know himself and I have a feeling that Kersley was as surprised as we were.
**Barrington** Don't you believe it. Kersley rehearsed her down to the last syllable.
**Jarvis** All the same, I thought he was impressive yesterday...
**Barrington** ( _He looks across at_ **Kersley** _who is talking to_ **Ashton)** Yes he was, damn the man - but that was yesterday. We still have one piece of evidence that Kersley doesn't know about. If he did, Mitchell would have revealed it, and that would have ended any chance of us winning this case.
_Three knocks to announce the arrival of the_ **Judge**.
**Usher** Be upstanding in the court. All persons having anything to do before my Lords, the Queen's Justices, draw near and give your attendance. God save the Queen.
_The_ **Judge** _enters and takes his place. All bow to the_ **Judge** _who returns the bow_.
**Usher** Bring up the prisoner.
**Judge** Sir James, are you ready to open the Defence case?
**Barrington** Yes, My Lord, I am. May it please your Lordship, Members of the Jury, I rose this morning with a heavy heart, only too aware of the responsibility that rests upon my shoulders. For it has been left for me to convince the Jury that Mr Sherwood, far from being consumed with murderous intent as Ms Mitchell would have us all believe, is in fact a simple decent man who has devoted his life to the service of others.
But Members of the Jury, for you to be convinced that this man is capable of murder, you have to ask yourself what was the motive, because all crimes must, in the end, have a motive. And, perhaps even more important, where is the evidence to convict Mr Sherwood? Because the evidence in this case has been at best circumstantial, and at worst, prejudicial.
Members of the Jury, English law does not demand that a defendant should appear in the witness box and it is right that it does not do so, but so determined is Mr Sherwood to clear his good name, that he is willing to face cross-examination and be judged by his peers.
My Lord, I call Mr Patrick Sherwood.
**Usher** Mr Sherwood
**Kersley** _smiles as_ **Sherwood** _leaves the dock, walks across the courtroom and enters the witness box_.
**Usher** Please take the testament in your right hand and read from the card.
**Sherwood** I swear by Almighty God that the evidence I shall give shall be the truth, the whole truth and nothing but the truth.
**Barrington** Your name is Patrick Hugh Sherwood and you reside at twenty-two Cadogan Villas in the county of London?
**Sherwood** Yes, I do.
**Barrington** You are presently a senior consultant, head of the Cardio Thoracic Unit at St George's, and a Fellow of the Royal College of Surgeons?
**Sherwood** That is correct.
**Barrington** Your life-long hobby has been sailing and until the age of forty-five you were a surgeon captain in the Royal Naval Volunteer Reserve?
**Sherwood** Yes, and I still keep a small boat at Burnham, which my wife... my late wife and I used to sail at weekends.
**Barrington** In 1982 you were called up as a reservist to serve in the Falklands, where you performed over a hundred operations in twenty-nine days.
**Sherwood** I have no idea how many operations I performed.
**Barrington** My Lord, over a hundred were the words mentioned in dispatches by the fleet commander. Now I'd like to begin, Mr Sherwood, with your relationship with your wife. How long were you married?
**Sherwood** Just over seventeen years.
**Barrington** And was your marriage a happy one?
**Sherwood** I adored Elizabeth; no one will ever be able to replace her.
**Barrington** When was it that you first discovered your wife had a heart problem?
**Sherwood** The first hint came in 1997, when Elizabeth complained of loss of breath, and of feeling pains in her chest, and left arm. These are the classic signs of a minor heart attack, so I took her into St George's and carried out some routine tests.
**Barrington** And what did those tests reveal?
**Sherwood** That her heartbeat was irregular and she was suffering from an arrhythmia.
**Barrington** Did you consider this curable at the time?
**Sherwood** Oh, yes, I deal with this sort of problem every day, and as long as a patient is willing to be disciplined with their diet, sensible about taking exercise and, if it applies, give up smoking, then there's no reason why they shouldn't live to an old age.
**Barrington** So what did you do next?
**Sherwood** I put her on a programme of medication that was approved by her GP and confirmed by the brightest young specialist on my staff.
**Barrington** And once she had begun that programme, did her health start to improve?
**Sherwood** No, she continued to complain about loss of breath and feeling tired, which caused me to doubt my own diagnosis.
**Barrington** So what did you do about it?
**Sherwood** I got in touch with Sir Magdi Yacoub at the Brompton Hospital. He's the leading authority in my field and I was keen to seek a second opinion.
**Barrington** And what was his judgement?
**Sherwood** He was puzzled. He could find no fault with my diagnosis and suggested that Elizabeth be put on a special fat-free diet.
**Barrington** And did you at last see some improvement?
**Sherwood** No, her health continued to deteriorate and during the last few months I couldn't get her to leave the flat, even to go for a short walk.
**Barrington** And she finally collapsed on the evening of March the twenty-first 1999, and was taken into hospital, where she died of cardiac arrest a few hours later.
**Sherwood** ( _bows his head_ ) Yes and I'll never forgive myself for not being by her side.
**Barrington** Looking back over that last year, do you think you could have done any more?
**Sherwood** I ask myself the same question a hundred times every day, and I think I can honestly say that I did everything in my power to prolong Elizabeth's life.
**Barrington** But the Crown would have us believe that over a period of three months, you instructed Ms Mitchell to pick up several ampoules of Potassium Chloride from a chemist in Wellingborough for the sole purpose of poisoning your wife.
**Sherwood** The Crown only has Ms Mitchell's word for that.
**Barrington** But your signature is on all the prescriptions.
**Sherwood** And hundreds of others like it. Sir James, but it's the first time I've been arrested and charged with murder.
**Barrington** Then why did you ask her to have them made up in Wellingborough?
**Sherwood** I didn't - she could have collected those prescriptions from the hospital pharmacy whenever it suited her.
**Barrington** And now I would like to address the Crown's suggestion that after your wife died, it was you who gave the instruction to have her body cremated. Is that true, Mr Sherwood?
**Sherwood** No. I'd been against the idea right from the start. If it hadn't been for a codicil in Elizabeth's will, she would have been buried in the family plot at Highgate cemetery. And if only she had been, I wouldn't be standing in front of you today.
**Barrington** Quite so. Which brings me on to Ms Mitchell, and the one question on which this whole case rests. Whether you did, or did not, have an affair with this lady - for the purpose of using her as part of a well-thought-out plan to poison your wife. So let me ask you straight away, Mr Sherwood, what was your relationship with Ms Mitchell?
**Sherwood** Entirely professional. On the rare occasions we met outside the hospital, it would have been at gatherings where other members of staff were present.
**Barrington** Did you ever flirt with Ms Mitchell?
**Sherwood** I flirt with all the nurses in my department, Sir James.
**Barrington** You flirt with all the nurses in your department?
**Sherwood** When you work on a cardiac unit you come into contact with death every day, which naturally causes relationships to be quite intense. One's moods swing from being morose to flippant and sometimes to just downright silly.
**Barrington** But Ms Mitchell claims that you gave her presents, sent her flowers, took her to restaurants and, on at least two occasions, accompanied her to the theatre?
**Sherwood** I think on one occasion I did pass on a box of chocolates to Ms Mitchell that had been given to me by a patient, but as for all her other suggestions, they are nothing more than fantasy.
**Barrington** She went on to tell the court that after having dinner together you would then drive her home. Is that also fantasy?
**Sherwood** It's not only fantasy, Sir James, it's simply not possible.
**Barrington** I'm not sure I understand, Mr Sherwood.
**Sherwood** It's quite hard to drive someone home when you don't own a car.
**Barrington** But you could have borrowed a car from the hospital, even hired one.
**Sherwood** Yes, I could have done, if I had a driving licence.
**Barrington** You don't have a driving licence?
**Sherwood** No and I've never had one. Elizabeth used to drive me everywhere.
**Barrington** But even if you didn't drive Ms Mitchell home, she claimed that you regularly joined her in her flat for coffee.
**Sherwood** I never drink coffee, Sir James, gallons of tea, but never coffee.
**Barrington** I must now ask you about Ms Mitchell's claims that you told her not to inform the police that you had been with her on the night your wife died, because you had come up with a more convincing alibi. How do you _answer_ that charge?
**Sherwood** There is no need to answer it, Sir James, because I don't even know where she lives.
**Barrington** But Mr Sherwood, you're on trial for murder. Why not admit to having an affair, rather than risk going to prison for the rest of your life?
**Sherwood** Because we didn't have an affair.
**Barrington** But you did go as far as kissing her on one occasion?
**Sherwood** Yes I did - it was at the staff Christmas party, I'd drunk a little too much, and regretted it immediately. I apologised and left soon afterwards.
**Barrington** So as far as you were concerned, that was an end of the matter?
**Sherwood** It would have been if I hadn't asked her to come to my office the following morning.
**Barrington** Why did you do that?
**Sherwood** I wanted to apologise more formally.
**Barrington** And what was her reaction?
**Sherwood** She said there was no need to apologise as she had enjoyed it. She then locked the door and started to unbutton her uniform.
**Barrington** How did you react?
**Sherwood** I went straight to the door, unlocked it, held it open and waited for her to leave.
**Barrington** Did she do so?
**Sherwood** Yes, she did, but I'll never forget her words as she stormed out of the room: 'Mr Sherwood, you will live to regret this.'
**Barrington** What did you imagine she meant by that?
**Sherwood** I thought she might report me to the hospital board, even to an industrial tribunal.
**Barrington** And did she do so?
**Sherwood** No, she was far more devious than that. But even I couldn't have imagined she would claim I was with her the night my wife died.
**Barrington** Where were you that night, Mr Sherwood?
**Sherwood** I was out on an emergency call visiting a patient in Westminster.
**Barrington** So why isn't that patient in court today to verify your story?
**Sherwood** Because he died later that night.
**Barrington** And what was the time entered on the death certificate?
**Sherwood** 10.27 p.m.
**Barrington** And who signed that death certificate?
**Sherwood** I did.
**Barrington** So if it was signed at 10.27 p.m., you would not have needed a more convincing alibi, as Ms Mitchell suggested.
**Sherwood** Ms Mitchell seems to think I could have been in two places at once.
**Barrington** Quite. So finally, let me ask you, Mr Sherwood, how do you answer Ms Mitchell's damning accusation that you used her as a courier over a period of three months, to collect ampoules of Potassium Chloride in order to poison your wife?
**Sherwood** Sir James, if I had wanted to poison my wife, I could have picked up any amount of drugs from the hospital pharmacy, without ever involving Ms Mitchell. No, the truth is that during that unhappy year I tended to my wife's every need, often neglecting other patients. I only wish I'd been as successful with Elizabeth as I have been with some of them.
**Barrington** No further questions, My Lord.
**Judge** Do you wish to cross-examine, Mr Kersley?
**Kersley** I most certainly do, My Lord. Mr Sherwood, much as we enjoyed the description of your wedded bliss, I suspect the time has now come for all of us in this courtroom to return to the real world. Let me begin by asking you when you first met Ms Mitchell?
**Sherwood** It must have been when she was transferred to the cardiac unit.
**Kersley** And did you find her attractive?
**Sherwood** No, I didn't think of her in that way.
**Kersley** So it wasn't love at first sight?
**Sherwood** Or second sight, Mr Kersley.
**Kersley** So it was some time later that you fell under her spell?
**Sherwood** I never fell under her spell.
**Kersley** Ah. So she was correct in suggesting that you were the pursuer and she the pursued in this relationship?
**Sherwood** There never was a relationship, Mr Kersley.
**Kersley** I shall return to the proof of that relationship later, Mr Sherwood.
**Sherwood** And I shall continue to deny it.
**Kersley** I'll look forward to that. So let me turn to something you can't deny. Where do you live?
**Sherwood** Twenty-two Cadogan Villas.
**Kersley** So when did you move out of your flat in Wimbledon?
**Sherwood** About a year ago.
**Kersley** And how much did it cost to purchase a penthouse in Chelsea?
**Sherwood** I don't remember exactly.
**Kersley** Come, come, Mr Sherwood, I think every one of us knows exactly what we paid for the home we live in, especially if we bought it less than a year ago. I repeat, how much did it cost to purchase a penthouse in Chelsea?
**Sherwood** Around eight hundred thousand pounds.
**Kersley** Wouldn't eight hundred and thirty-seven thousand be more accurate?
**Sherwood** Possibly.
**Kersley** So you must have won the lottery? Or did you inherit it?
**Sherwood** Neither. My wife left few assets in her will. By the time I had finished paying death duties I received less than twenty thousand pounds.
**Kersley** Well, that would just about cover the stamp duty on twenty-two Cadogan villas, so I must therefore ask you what the current salary is for a surgeon at St George's?
**Sherwood** Just over ninety thousand pounds a year.
**Kersley** How many people earning ninety thousand pounds a year can afford to purchase a penthouse in Chelsea for eight hundred and thirty-seven thousand pounds? I feel sure you're about to tell us that there's another simple explanation.
**Sherwood** Yes, there is. Some years ago I took out a joint life insurance policy on my wife and myself.
**Kersley** Some years ago. Wouldn't March 1997 be more precise?
**Sherwood** That is some years ago, Mr Kersley.
**Kersley** But it's only some weeks before your wife suffered her first heart attack - 'the first hint came in 1997, my wife complained of loss of breath, pains in her left arm and chest...' - And what was the value of the policy?
**Sherwood** A million pounds.
**Kersley** One million pounds. And would I be right in thinking that the life insurance company are refusing to settle the amount until the result of this trial is known?
**Barrington** My Lord, that can only be speculation.
**Judge** I agree, Sir James. Mr Kersley, unless you can provide evidence, move on.
**Kersley** Mr Sherwood, has the life insurance company settled the full amount?
**Judge** Mr Kersley.
**Kersley** I apologise, My Lord. However, I do hope my learned friend will consider one million pounds a large enough sum to constitute the motive he was searching for.
**Sherwood** No amount of money would constitute a motive for harming my wife.
**Kersley** Is that right? Then why did you tell Ms Mitchel that you were sick of the way she continually belittled you in front of the hospital staff and how you longed to be rid of her?
**Sherwood** My Lord, do I have to answer such a ridiculous suggestion?
**Judge** Yes, I'm afraid you do, Mr Sherwood.
**Sherwood** Of course I didn't want to be rid of my wife. She was the only woman I've ever cared for.
**Kersley** Then why did you seek solace in the arms of another?
**Sherwood** I did no such thing.
**Kersley** So when Ms Mitchell informed the court that your wife had refused to make love to you for several years, was that also untrue?
**Sherwood** How could she possibly know?
**Kersley** Oh, I see, you regularly made love to your wife, did you, Mr Sherwood? ( **Sherwood** _hesitates_.) Your silence speaks volumes and, more importantly, supplies us with yet another motive.
**Sherwood** It does no such thing. When will you work out that when it comes to my private life, Ms Mitchell simply made it up?
**Kersley** Well, then, let's consider something Ms Mitchell couldn't have made up - your wife's desire to be cremated. Mr Sherwood, your wife's will - was it written in her own hand?
**Sherwood** No, it was typed.
**Kersley** Typed by whom, may I ask?
**Sherwood** ( _hesitates_ ) I think I may have typed it, but it was signed and duly witnessed.
**Kersley** So who witnessed the document?
**Sherwood** Mr Webster, the porter at Arcadia Mansions.
**Kersley** A man who - by his own admission - could neither read nor write.
**Sherwood** It wasn't necessary for him to read or write, he was only witnessing my wife's signature.
**Kersley** So just a few days before your wife dies she suddenly adds a codicil to her will stating that she wishes to be cremated. No doubt my learned friend will once again casually dismiss this piece of evidence as circumstantial and coincidental, rather than using his favourite words, 'how convenient'.
**Barrington** My Lord, I'm enjoying this speech immensely, as I feel sure you are, but if it is to continue for much longer, perhaps my client might be allowed to sit down?
**Kersley** When I've finished, your client may well need to sit down. Mr Sherwood, you told the court that you consulted the eminent surgeon Sir Magdi Yacoub about your wife's condition and he put her on a special fat-free diet.
**Sherwood** Sir Magdi has written to the court confirming as much.
**Kersley** But wasn't that exactly what you were hoping to achieve, so that when your wife died, you would be able to show that she had been treated by the nation's leading specialist and therefore no awkward questions would be asked?
**Sherwood** That's ridiculous. How could I be expected to fool the leading authority in the land?
**Kersley** Because you neglected to inform him about the six ampoules of Potassium Chloride that later mysteriously disappeared. Or are you at last going to tell us what really happened to them?
**Sherwood** Mr Kersley, in a hospital as large as St George's, thousands of drugs are dispensed every week. How could I be expected to account for every one of them the following day, let alone a year later?
**Kersley** Let alone a year later. So why don't we turn to something you should be able to account for - even a year later - namely your actions on the night your wife died. Did you pour the glass of wine found by her side?
**Sherwood** Yes, I did, but that was before I left the flat.
**Kersley** So it must have been you who added the sedative that DCI Payne described as three times the normal dosage?
**Sherwood** No it was not. My wife was in the habit of taking sedatives, so it could have been a mistake.
**Kersley** Or part of your plan, like leaving the kitchen window open.
**Sherwood** Why should I do that?
**Kersley** So that you could return to your flat without being seen by the porter.
**Sherwood** How often do you have to be told? I didn't return to the flat. I was visiting a patient.
**Kersley** In that case, why did you forget to take your doctor's bag?
**Sherwood** I didn't forget it.
**Kersley** But Mr Webster testified that he saw you leaving the building without it.
**Sherwood** Only moments ago you dismissed Mr Webster as unfit to witness my wife's signature.
**Kersley** So are you now saying that you did have your doctor's bag with you when you left Arcadia Mansions?
**Sherwood** You don't need a doctor's bag if all you have to do is sign a death certificate.
**Kersley** And you certainly wouldn't have needed a doctor's bag if you were visiting your mistress in Tooting, unless, of course, it was full of condoms.
**Barrington** ( _leaps up_ ) My Lord...
**Judge** Yes, yes, Sir James. Mr Kersley, do try to remember that you are at the Old Bailey and not back in the House of Commons.
**Kersley** I do apologise, My Lord. So, wherever you were, Mr Sherwood - visiting a patient in Westminster without your doctor's bag, or making love to your mistress in Tooting...
**Sherwood** I don't have a mistress - in Tooting or anywhere else for that matter.
**Kersley** But Ms Mitchell told the court that after you made love, you left her at ten o'clock - without your doctor's bag - and returned an hour later in a nervous state.
**Sherwood** I wasn't in a nervous state.
**Kersley** Because you thought you'd got away with it.
**Sherwood** Because I wasn't with her in the first place.
**Kersley** Then where were you at 10.27?
**Sherwood** I was in Westminster signing a death certificate and that certificate is in the court's possession.
**Kersley** Indeed it is, but isn't it common practice to fill in a death certificate the following day?
**Sherwood** Not in my department, it isn't.
**Kersley** Then if it wasn't you who returned to your flat, how do you explain your wife's words - 'How did you get in?'
**Sherwood** ( _pause_ ) She could have been addressing Mr Webster.
**Kersley** But he walked in through the front door. I don't think so, Mr Sherwood.
**Sherwood** Then she must have been surprised by an intruder.
**Kersley** Rather familiar words for an intruder I would have thought. But singularly appropriate for a husband who had returned unexpectedly and not via the front door. 'How did you get in?'
**Sherwood** But I didn't return unexpectedly via any door.
**Kersley** Then via the fire escape, perhaps?
**Sherwood** Or the fire escape.
**Kersley** Then how about the kitchen window?
**Sherwood** Or the kitchen window.
**Kersley** Then who slipped back into the kitchen just before Mr Webster came in?
**Sherwood** It could have been the wind that caused the door to slam.
**Kersley** Then how do you explain the shouting, the quarrelling and the turning over of furniture?
**Sherwood** I accept Mr Webster's explanation - that there had been another burglary.
**Kersley** So how did this burglar enter the building, when there was no sign of a break-in?
**Sherwood** Via the fire escape and the kitchen window.
**Kersley** But the window had been opened from the inside?
**Sherwood** My wife often opened it on a warm evening.
**Kersley** In March? Ah, I see, so it was your wife who let the burglar in?
**Sherwood** That's a ridiculous suggestion, Mr Kersley, and you know it.
**Kersley** As ridiculous as suggesting that it was a burglar who caused the bruising on her arm.
**Sherwood** Not at all. The bruising could have been caused by a struggle with the burglar.
**Kersley** Rather than a struggle with you?
**Sherwood** Why should she be struggling with me?
**Kersley** Because when you returned to inject her, you were horrified to discover she hadn't drunk the glass of wine that contained the sedative you had prepared for her.
**Sherwood** Then how do you explain the unopened ampoule left in my bag?
**Kersley** Because you left it there, Mr Sherwood, having only managed to inject five of the ampoules before you were interrupted by Mr Webster, when you fled to the kitchen, dropping the rubber glove on the floor.
**Sherwood** Have you forgotten that the glove belonged to my wife?
**Kersley** Have you forgotten that it was soaked in Potassium Chloride?
**Sherwood** From a bottle of grapefruit juice - as Professor Forsyth confirmed.
**Kersley** From an injection, that induced a heart attack, as Professor Forsyth demonstrated.
**Sherwood** The heart attack was probably caused by the intruder.
**Kersley** Re-enter the intruder, who conveniently appears whenever you're in trouble. No, the truth is that there never was an intruder, Mr Sherwood, because it was you...
**Barrington** My Lord, I must object. My learned friend is putting words into the defendant's mouth.
**Judge** I agree with you, Sir James. Mr Kersley, you must stop attempting to be the witness as well as prosecuting council. If I were to allow this to continue much longer, you might well end up being the judge as well.
**Kersley** As Your Lordship pleases.
**Judge** ( _furious_ ) In future, Mr Kersley, allow the defendant to answer the questions and the jury to decide on the facts.
**Kersley** Is it a fact, Mr Sherwood, that you kissed Ms Mitchell at the staff Christmas party?
**Sherwood** ( _pause_ ) Yes - I've already admitted to that.
**Kersley** And then you left the party a few minutes later?
**Sherwood** Yes, I did.
**Kersley** Did Ms Mitchell leave with you?
**Sherwood** No, she did not.
**Kersley** Several people were present at that party, Mr Sherwood.
**Sherwood** I'm not saying she didn't leave at the same time. I'm simply pointing out that she didn't leave with me.
**Kersley** A nice distinction. But did you then offer to accompany her home?
**Sherwood** No, we went our separate ways.
**Kersley** Was it raining at the time?
**Sherwood** No, it began to rain later.
**Kersley** Just as you arrived back at her house?
**Sherwood** I've never been to her flat.
**Kersley** I didn't say flat, Mr Sherwood, I said house.
**Sherwood** I've never been to her flat, her appartment or her house. I don't even know where Oldfield Road is.
**Kersley** Oldfield Road? Who mentioned Oldfield Road?
**Sherwood** Ms Mitchel must have done when she gave her evidence.
**Kersley** I don't think so, Mr Sherwood, she referred to Tooting, but she made no mention of Oldfield Road. I could always call for the court transcript, just to be sure.
**Sherwood** Then I must have seen it written on her file somewhere, but I have no idea where it is.
**Kersley** Do you know where your office is by any chance, Mr Sherwood?
**Sherwood** Yes, of course I do.
**Kersley** And did Ms Mitchell lie when she told the court you asked to see her in your office the following morning?
**Sherwood** No, I wanted to apologise more formally for what had happened at the staff Christmas party.
**Kersley** And her response was to unbutton her uniform?
**Sherwood** Yes, that's exactly what she did.
**Kersley** Was that before or after you had locked the door?
**Sherwood** I didn't lock the door.
**Kersley** So you didn't have sex with her on the couch?
**Sherwood** Mr Kersley, I'm a surgeon not a film director.
**Kersley** So it's pure fantasy for her to suggest that you regularly visited her flat in the early hours of the morning?
**Sherwood** I never visited her flat at any hour of the night or day.
**Kersley** And you never took her out for dinner or to the theatre?
**Sherwood** No, I did not.
**Kersley** Have you seen the play, _The Real Thing_.
**Sherwood** ( _pause_ ) I may have done.
**Kersley** And whom did you see it with, Mr Sherwood?
**Sherwood** I don't recall - probably my wife.
**Kersley** Yet another person who is conveniently unable to confirm or deny your story. So allow me to try to refresh your memory, because we are now in possession of the two programmes for the plays Ms Mitchell says you took her to see. ( _He holds up two programmes.) An Inspector Calls_ , and _The Real Thing_.
**Sherwood** That doesn't prove I took Jennifer.
**Kersley** Jennifer. No it doesn't, but you will recall that Ms Mitchell - I do apologise - Jennifer, was also in the habit of making notes in her diary, and that diary has been in the court's safe keeping for several weeks. I wonder My Lord if you would allow the Usher to pass Ms Mitchell's diary - exhibit four ~ across to Mr Sherwood.
**Judge** Yes. Usher. ( **Usher** _hands the diary to the defendant_.)
**Kersley** Please turn to February the fifteenth, 1999 Mr Sherwood, and read to the court Jennifer's entry for that evening.
**Sherwood** 7.30 _The Real Thing_.
**Kersley** Yes, but do read on, Mr Sherwood. Isn't there another entry below that?
**Sherwood** ( _hesitates_ ) P.S. at the Albery Theatre.
**Kersley** P.S. Patrick Sherwood, I suspect - unless, of course, Jennifer had an assignation with Peter Stringfellow.
**Sherwood** Post script seems more likely, Mr Kersley.
**Kersley** But she stated unequivocally that it was you.
**Sherwood** She also stated unequivocally that I drove her home when I don't own a car or even have a driving licence.
**Kersley** No. No, Mr Sherwood, Ms Mitchell never claimed that you drove her home. Her exact words in reply to your learned counsel were, ( _Picks up a sheet of paper_.) 'He always _took_ me home' and if you would like her to explain what she meant by that, Mr Sherwood, we can always call her back as she would still be under oath.
**Sherwood** Why bother, when the oath means nothing to her?
**Kersley** Are you suggesting that Jennifer committed perjury?
**Sherwood** Your words, not mine, Mr Kersley.
**Kersley** But why should she do that?
**Sherwood** 'Mr Sherwood, you will live to regret this.'
**Kersley** Your words, not hers - because there's no proof she ever said them.
**Sherwood** Ask any doctor at St George's and they'll tell you about Ms Mitchell's reputation.
**Kersley** What a gallant fellow you are, Mr Sherwood, which would explain why you dumped her the moment she'd served her purpose.
**Sherwood** You can't dump someone with whom you've never had a relationship.
**Kersley** Wouldn't you describe having an affair for three months as a relationship?
**Sherwood** Yes, I would, but I wouldn't describe one drunken kiss as having an affair.
**Kersley** But if you weren't having an affair with Jennifer, why would she bother to go to Wellingborough to pick up those drugs for you?
**Sherwood** She didn't go to Wellingborough to pick up those drugs until after I'd thrown her out of my office.
**Kersley** Or was it after you'd had 'after-rounds sex' with her in your office?
**Sherwood** You've crawled back into the gutter, Mr Kersley.
**Kersley** I'm searching for you, Mr Sherwood.
**Sherwood** Then you won't find me there.
**Kersley** But I will find you in your flat pouring your wife a glass of wine. How did you get in? And then adding a sedative. How did you get in? And then leaving without your doctor's bag. How did you get in? And then returning via the fire escape. How did you get in? And then injecting her with five ampoules of Potassium Chloride. How did you get in? And then leaving her to die a slow, painful and terrible death.
**Sherwood** Nothing could be further from the truth.
**Kersley** I cannot think of a more accurate summing-up of your entire evidence. No more questions, My Lord.
**Judge** Sir James, do you wish to re-examine?
**Barrington** My Lord, it would be a travesty of justice were I not to do so. My learned friend has suggested that if the Jury conclude that Mr Sherwood did have an affair with Ms Mitchell, then they must dismiss the rest of his evidence as a tissue of lies. I now intend to prove beyond reasonable doubt that no such affair ever took place. But to do that I must return to your love of sailing, Mr Sherwood. When the Falklands armada was being assembled, were you called up to join the fleet?
**Sherwood** Yes, in 1982 I was still on the Reserve list.
**Barrington** And in what capacity were you asked to serve?
**Sherwood** As a surgeon captain to the fleet.
**Barrington** And to which ship were you assigned?
**Sherwood** HMS _Sheffield_.
**Barrington** And were you on board that gallant vessel when she was hit by an Exocet and went down in flames?
**Sherwood** Yes, I was, Sir James, and most fortunate to be among those who survived.
**Barrington** And while you were in the water, Mr Sherwood, were you burned - which would account for the small scar on your right forearm, which Ms Mitchell referred to as intimate knowledge only a lover could have been aware of?
**Sherwood** Yes, Ms Mitchell is quite right about the burn.
**Barrington** Would you show it to the jury? ( **Sherwood** _takes off his jacket and rolls up his sleeve to reveal a small burn_.) Now, she would have seen that every day in the operating theatre?
**Sherwood** Yes and whenever I was scrubbing up.
**Barrington** And whenever you were making love?
**Sherwood** We never made love.
**Barrington** Because if you had done so, you would have taken your shirt off?
**Sherwood** Yes, I would.
**Barrington** Would you please turn round now Mr Sherwood? Would you please take you shirt off? ( _He does so, to reveal a large bum that almost covers his back_.) 'A small burn on his right arm... Only when he takes his shirt off!' No more questions My Lord.
_The lights fade as everyone stares at_ **Sherwood** 's _back_.
#### **Scene Two**
_The following morning_.
**Jury Bailiff** _walks through the door and once again comes face to face with the audience. The clock shows 9.45 a.m._
**Jury Bailiff** Good morning Ladies and Gentlemen of the Jury, on what will be our last day together. When we return to the court in a few moments' time, Mr Justice Cartwright will begin his summing up. He will only give you guidance on points of law. He will not offer an opinion on the case, as that is solely your prerogative. After he has completed his summing up, I will accompany you back here to the jury room, where I will leave you to consider your verdict. ( _The_ **Jury Bailiff** _glances at the clock, which shows 9.50_.) The time has come for us to make our way back to Court Number One. Ladies and Gentlemen of the Jury, please follow me.
**Jury Bailiff** _steps through the door, so thai when the lights come up we are back in Court Number One. Everyone is in place awaiting the_ **Judge** _and jury_.
**Usher** Be upstanding in the court. All persons having anything to do before my Lords, the Queen's Justices, oyer and terminer, and general gaol delivery for the jurisdiction of the Central Criminal Court, draw near and give your attendance. God save the Queen.
_The_ **Judge** _enters. When he is in place, they all bow and he returns their bow_.
**Judge** Ladies and Gentlemen of the Jury. Having heard the arguments from both leading Counsel, it is now your solemn task to return a verdict as to whether Mr Patrick Sherwood be guilty or not guilty of murder.
Among the matters you will have to decide is did Mr Sherwood instruct Ms Mitchell to have six ampoules of Potassium Chloride made up outside London. Or was it compelling evidence - as Sir James suggested - that five out of the six prescriptions were not made out for a Friday - the day on which Ms Mitchell insists Mr Sherwood handed them over to her?
And then you will want to turn your attention to the open window leading to the Sherwoods' fire escape, the rubber glove dropped on the kitchen floor, the wineglass found by Mrs Sherwood's side and the doctor's bag left by the telephone. Are they simply four red herrings, or do they point to something far more sinister? And remember that when it comes to reaching your final decision, you must deal only with facts.
Yes, it is a fact that the rubber glove was found to have a deposit of Potassium on it - but was it concentrated grapefruit juice? Yes, it is a fact that an excess of Temazepam was discovered in Mrs Sherwood's wine - but who put it there? And at the same time, you will want to consider the victim's words, 'How did you get in?'. Were they addressed to the porter, Mr Webster, as he entered the room, or to someone else who quickly disappeared into the kitchen?
Did Mr Sherwood return to the flat that night, in order to inject his wife with Potassium Chloride, or was he, as he claims, signing a death certificate in Westminster? You are in possession of that death certificate which unquestionably bears his signature and is timed at 10.27 p.m., but was it signed that night or later the following morning? Unfortunately, no one has come forward to verify either account, so only you can decide.
Which takes me on to the significance of a life insurance policy for one million pounds, taken out only weeks before Mrs Sherwood suffered her first heart attack. Did you find this a compelling motive for murder, or just another red herring?
Now we come to the most crucial question in this whole case. Did Ms Mitchell - as she vehemently insisted - have an affair with Mr Sherwood that lasted for several months, or are you persuaded that, having been rejected by him, she made the whole story up? Were you affected by the fact that he obviously knew where she lived or were you more struck by seeing the scar on his back? Although you will recall Ms Mitchell told us that whenever they made love Mr Sherwood insisted that the lights were out. You have been able to observe both of them in the witness box, so you can draw your own conclusions as to which one you feel was telling the truth. For certainly, one of them is a brazen liar.
If you decide it was Ms Mitchell who was being deceitful about her relationship with Mr Sherwood, then you have every right to be sceptical about the rest of her evidence. But if, on the other hand, you are convinced she did have an affair with the accused, then you might feel that verifies the rest of her story.
And so the time has now come for you to return to the jury room and consider your verdict. If, when you have completed your deliberations, you feel the Defence have made their case, then it is your duty to return a verdict of Not Guilty. But, if you believe the Prosecution has proved its case beyond reasonable doubt, then it is nothing less than your duty to deliver a verdict of Guilty.
May God guide and assist you in your counsels.
_Lights dim in the court_. **Jury Bailiff** _steps through the jury door, into spot, as house lights go up_.
**Jury Bailiff** Members of the Jury, the time has come for you to make your decision, but do not do so until I instruct you.
It is your task to decide whether you believe Mr Patrick Sherwood is **guilty** or not guilty of the murder of Elizabeth Sherwood.
I will leave you for a few moments to talk among yourselves and consider your verdict. ( _He leaves the stage for 30 seconds_.)
**Usher** Silence in Court.
**Jury Bailiff** Please take the monitor from the back of the seat in front of you, and register your verdict, guilty or not guilty, now.
_Pause while the audience place their vote. This can be done with cards or a show of hands_.
**Jury Bailiff** Thank you, Ladies and Gentlemen of the Jury. We will now return to the court, in order that your verdict may be known.
**Jury Bailiff** _exits through the jury room doors, which split apart_.
**VERDIGT**
IF THE AUDIENCE DELIVERS THE VERDICT NOT GUILTY:
**Usher** Silence in court. Bring up the prisoner.
**Sherwood** _returns to the dock_.
**Judge** Will the foreman please rise. Mr Foreman, have you reached a verdict?
**Foreman** ( _on tape, or from the body of the audience_ ) Yes, we have, My Lord.
**Judge** Do you find the prisoner at the bar, Patrick Hugh Sherwood, guilty or not guilty of the murder of Elizabeth Sherwood?
**Foreman** Not guilty.
**Judge** ( _turns to face the defendant in the dock_ ) Patrick Hugh Sherwood, you have been found not guilty of the charges laid before you and I therefore discharge you. You are free to leave the court.
**Sherwood** _comes down from the dock and shakes hands with_ **Barrington** _and_ **Jarvis**.
**Barrington** Congratulations.
**Sherwood** Thank you.
**Jarvis** Well done.
**Sherwood** Thank you both. But Sir James, may I ask you a question?
**Barrington** Yes, of course.
**Sherwood** Am I right in thinking that when a verdict of not guilty has been reached, there can never be a retrial?
**Barrington** That is correct. In English law, once a jury has acquitted a defendant he cannot be put on trial again for the same offence.
**Sherwood** ( _smiles_ ) That's a relief.
**Jarvis** So tell me, Mr Sherwood, how do you intend to celebrate your great victory?
**Sherwood** I shall go back to work, Mr Jarvis, just as Sir James predicted I would.
_The attention moves to_ **Kersley** _and_ **Ashton,** _who are chatting on the other side of the stage_.
**Ashton** I have a feeling it must have been a close-run thing, so what do you imagine tipped the balance?
**Kersley** The burn on his back, would be my guess. Pity Ms Mitchell didn't stay to hear his evidence. I would like to have seen her face when the good doctor took off his shirt.
**Ashton** Perhaps the reality is as Sir James suggests, that having been rejected by Mr Sherwood she was simply seeking revenge.
**Kersley** I'm not convinced it's quite that simple. There's something about this case that doesn't ring true.
**Ashton** In what way?
**Kersley** ( _they start to leave the stage_ ) Well, to start with, why didn't Mitchell return to hear the verdict?
_As they depart the_ **Jury Bailiff** _walks quickly back on, as_ **Sherwood** _is leaving_.
**Jury Bailiff** Sir, sir. Can I suggest that you stay put for a few more minutes, as there is rather a large crowd gathering on the pavement outside. And once the corridors have been cleared, we can slip you out the back way.
**Sherwood** That's most thoughtful of you, Mr Pierce, but I'm quite happy to leave by the front. I've nothing to hide.
**Jury Bailiff** No, of course you haven't, Mr Sherwood indeed, if I may say so, sir, I never doubted for a moment that you were innocent. By the way, the woman Mitchell is hanging around in the corridor trying to get back in, but I didn't...
**Sherwood** No, that's fine, I'm only too happy to see her.
**Mitchell** _runs on_ to _the stage and the_ **Jury Bailiff** _exits_.
**Mitchell** I booked a table for us at the Caprice. No need to hide in Fulham any longer.
**Sherwood** You were always that confident of the verdict?
**Mitchell** Of course. Once the jury had seen that scar on your back they were never going to believe that I had slept with you and to quote Mr Kersley, 'A jury that has doubts will never send a man to gaol for the rest of his life.'
**Sherwood** You were brilliant in the witness box, in fact, Sir James only got the better of you once.
**Mitchell** And when was that?
**Sherwood** When he tricked you into writing your signature on the Usher's notepad, to show that you'd lied when you claimed you were right-handed.
**Mitchell** But I am right-handed. I worked out exactly what Sir James was up to and realised that the image of me signing that pad with my left hand would remain fixed in the jury's minds.
**Sherwood** Then why didn't you try on the rubber glove?
**Mitchell** I did, at the police station, and I just couldn't get it on, Elizabeth must have had very small hands.
**Sherwood** You thought of everything.
**Mitchell** Yes, and if only Webster hadn't walked in just before I' finished injecting her we could have been married a year ago.
**Sherwood** No, I'm afraid not, Jennifer. You see that was something else Mr Kersley was right about... I never intended to marry you in the first place. ( _He turns to walk away as the lights fade_.)
CURTAIN
IF THE AUDIENCE DELIVERS THE VERDICT GUILTY:
**Usher** Silence in court. Bring up the prisoner.
**Sherwood** _returns to the dock_.
**Judge Will** the foreman please rise. Mr Foreman, have you reached a verdict?
**Foreman** Yes, we have, My Lord.
**Judge** Do you find the prisoner at the bar, Patrick Hugh Sherwood, guilty or not guilty of the murder of Elizabeth Sherwood?
**Foreman** Guilty.
_The_ **Judge** _turns to face the prisoner in the dock_.
**Judge** Patrick Hugh Sherwood, you stand convicted of murder by a jury of your peers. You undoubtedly poisoned your young wife, taking advantage of your special knowledge and training, and in so doing dishonoured your profession. You murdered Elizabeth Sherwood in order to be rid of her and inherit a milion pounds. As the law requires, I sentence you to life imprisonment, with the recommendation that you serve a minimum of seventeen years. You may be curious, Mr Sherwood, to discover why I have recommended this particular sentence. Seventeen years is the period of time that was left on the life insurance policy, the profits of which you had planned to enjoy by spending the rest of your life in the luxury of a penthouse apartment in Chelsea with money to spare. You will instead spend those seventeen years in custody at Her Majesty's pleasure, and may God have mercy on your soul.
**Sherwood** 's _head drops into his hand. The attention moves to_ **Barrington** _and_ **Jarvis**.
**Barrington** God help the poor man.
**Jarvis** What do you imagine tipped the balance? It must have been a close-run thing.
**Barrington** We'll never know, but it can't have helped that we couldn't explain why he'd left his bag behind, when he went on insisting that he was with a patient in Westminster.
**Jarvis** ( _nods_ ) I know it's the jury's responsibility to decide the verdict, not ours, but I'm still not sure if the man is guilty.
**Barrington** Neither am I. ( _Watches as_ **Mitchell** _enters the courtroom and stares at_ **Sherwood**.) But I have a feeling that woman is the one person who knows the answer to your question.
**Guard** _approaches the dock_ , _motions_ **Sherwood** _to follow him. They walk halfway across the stage_.
**Ashton** Well, at least this time you beat the old buzzard.
**Kersley** Did I? ( _Stares at_ **Barrington**.) I'm not so sure. There's something about this case that doesn't ring true.
**Ashton** In what way?
**Kersley** Nothing I can prove, but I have a feeling we would discover what really took place on the night on March the twenty-first if we could only overhear...
**Sherwood** _leaves the witness box and begins to walk across the stage_.
**Guard** Wait there. ( _He goes over to another_ **Guard** _who completes the paperwork, as_ **Mitchell** _walks up behind_ **Sherwood**.)
**Sherwood** You know I didn't kill Elizabeth.
**Mitchell** Of course I do, but no one is ever going to believe that when you left my flat to go to Westminster and sign that death certificate it was _me_ who went to Arcadia Mansions.
**Sherwood** How did you get in?
**Mitchell** Via the fire escape and the kitchen window, and if Webster hadn't interrupted me I would have managed all six ampoules... ironic when you think about it. If you'd admitted to having an affair, you would have had the perfect alibi, but then, Patrick, I did warn you, you will live to regret this.
CURTAIN
**Bloomsbury Methuen Drama**
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**Bloomsbury is a registered trade mark of Bloomsbury Publishing Plc**
First published in 2000 by Methuen Publishing Limited
© 2000 Jeffrey Archer
Jeffrey Archer has asserted his right under the Copyright, Designs and
Patents Act, 1988, to be identified as the author of this work.
All rights reserved. No part of this publication may be reproduced or
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can be accepted by Bloomsbury or the author.
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**British Library Cataloguing-in-Publication Data**
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A catalog record for this book is available from the Library of Congress.
| {
"redpajama_set_name": "RedPajamaBook"
} | 23 |
namespace bond
{
BOND_NORETURN void NothingException();
namespace detail
{
/// Internal base class with shared implementation between the two maybe
/// variants. Consult the documentation for bond::maybe<T> for its public
/// interface.
template <typename T>
class maybe_common
{
public:
/// @brief The type of the value that may be inside the maybe.
using value_type = T;
maybe_common() = default;
#if defined(_MSC_VER) && _MSC_VER < 1900
// Using = default with MSVC 2013 on this function causes the compiler
// to make extra copies, which makes this type fail when it's holding a
// move-only type.
//
// Manually implementing this function works just fine, however.
maybe_common(const maybe_common& that)
: _value(that._value)
{ }
#else
maybe_common(const maybe_common&) = default;
#endif
template <typename... Args>
explicit maybe_common(const T& value, Args&&... args)
{
_value.emplace(value, std::forward<Args>(args)...);
}
template <typename... Args>
explicit maybe_common(T&& value, Args&&... args)
{
_value.emplace(std::move(value), std::forward<Args>(args)...);
}
maybe_common(maybe_common&& that) BOND_NOEXCEPT_IF(std::is_nothrow_move_constructible<boost::optional<T>>::value)
: _value(std::move(that._value))
{
// unlike std::optional/boost::optional, moved-from bond::maybe
// instances are guaranteed to be nothing.
//
// asigning boost::none is noexcept, but assigning { } is not
that._value = boost::none;
}
/// @brief Check if this object contains nothing.
/// @return true if this holds nothing; otherwise false.
bool is_nothing() const BOND_NOEXCEPT
{
return !static_cast<bool>(_value);
}
/// @brief Check if this object contains a value.
/// @return true if this object holds a value; otherwise false.
/// @since 8.0.0
explicit operator bool() const BOND_NOEXCEPT
{
return !is_nothing();
}
/// @brief Set to nothing.
void set_nothing() BOND_NOEXCEPT
{
// asigning boost::none is noexcept, but assigning { } is not
_value = boost::none;
}
/// @brief Construct a value in place.
///
/// @since 8.0.0
template<typename... Args>
T& emplace(Args&&... args)
{
_value.emplace(std::forward<Args>(args)...);
return *_value;
}
/// @brief Get a reference to the value.
/// @throw CoreException if the object contains nothing
T& value()
{
if (is_nothing())
{
NothingException();
}
return *_value;
}
/// @brief Get a constant reference to the value.
/// @throw CoreException if the object contains nothing
const T& value() const
{
if (is_nothing())
{
NothingException();
}
return *_value;
}
/// @brief Get a reference to the value.
///
/// Will never throw, but has undefined behavior if the object contains
/// nothing.
///
/// @since 8.0.0
T& value(const std::nothrow_t&) BOND_NOEXCEPT
{
BOOST_ASSERT(!is_nothing());
return *_value;
}
/// @brief Get a constant reference to the value.
///
/// Will never throw, but has undefined behavior if the object contains
/// nothing.
///
/// @since 8.0.0
const T& value(const std::nothrow_t&) const BOND_NOEXCEPT
{
BOOST_ASSERT(!is_nothing());
return *_value;
}
#if defined(_MSC_VER) && _MSC_VER < 1900
// Using = default with MSVC 2013 on this function causes the compiler
// to make extra copies, which makes this type fail when it's holding a
// move-only type.
//
// Manually implementing this function works just fine, however.
maybe_common& operator=(const maybe_common& that)
{
_value = that._value;
return *this;
}
// MSVC 2013 cannot = default rvalue asignment operators
maybe_common& operator=(maybe_common&& that)
BOND_NOEXCEPT_IF(std::is_nothrow_move_assignable<boost::optional<T>>::value)
{
_value = std::move(that._value);
// unlike std::optional/boost::optional, moved-from bond::maybe
// instances are guaranteed to be nothing.
//
// asigning boost::none is noexcept, but assigning { } is not
that._value = boost::none;
return *this;
}
#else
/// @brief Assign from another maybe.
maybe_common& operator=(const maybe_common&) = default;
/// @brief Move assign from another maybe.
maybe_common& operator=(maybe_common&&) = default;
#endif
/// @brief Compares two maybes for value equality.
///
/// @return true if both maybes hold nothing; returns false if one maybe
/// holds nothing and the other holds a values; otherwise, calls
/// operator== with the two values.
///
/// @since 8.0.0 (was a member function prior)
friend bool operator==(const maybe_common& lhs, const maybe_common& rhs)
{
return lhs._value == rhs._value;
}
/// @brief Compares two maybes for value inequality.
///
/// See operator==(const maybe_common&,const maybe_common&) for details
/// about how maybes holding nothing are handled.
///
/// @since 8.0.0 (was a member function prior)
friend bool operator!=(const maybe_common& lhs, const maybe_common& rhs)
{
return !(lhs == rhs);
}
/// @brief Compares a maybe and a value for equality.
///
/// @return false if the maybe holds nothing; otherwise, calls
/// operator== with the maybe's value and the provided value.
///
/// @since 8.0.0
friend bool operator==(const maybe_common& lhs, const T& rhs)
{
return lhs._value == rhs;
}
/// @brief Compares a maybe and a value for inequality.
///
/// See operator==(const maybe_common&,const T&) for details about how
/// maybes holding nothing are handled.
///
/// @since 8.0.0
friend bool operator!=(const maybe_common& lhs, const T& rhs)
{
return !(lhs == rhs);
}
/// @brief Compares a value and a maybe for equality.
///
/// @return false if the maybe holds nothing; otherwise, calls
/// operator== with the provided value and maybe's value.
///
/// @since 8.0.0
friend bool operator==(const T& lhs, const maybe_common& rhs)
{
return lhs == rhs._value;
}
/// @brief Compares and a value and a maybe for inequality.
///
/// See operator==(const T&,const maybe_common&) for details about how
/// maybes holding nothing are handled.
///
/// @since 8.0.0
friend bool operator!=(const T& lhs, const maybe_common& rhs)
{
return !(lhs == rhs);
}
protected:
boost::optional<T> _value;
};
template <typename T, typename Enable = void> struct
has_allocator
: std::false_type {};
template <typename T> struct
has_allocator<T, typename boost::enable_if<std::is_class<typename T::allocator_type> >::type>
: std::true_type {};
} // namespace detail
template <typename T, typename Enabled = void>
class maybe;
/// @brief Type used for fields with default values of \c nothing.
///
/// This specialization is used for instance of T without allocators.
///
/// See the [User's
/// Manual](../../manual/bond_cpp.html#default-value-of-nothing) for more
/// details about default values of \c nothing.
///
/// @see For details of %maybe's comparison operators, see
/// \li operator==(const detail::maybe_common&,const detail::maybe_common&)
/// \li operator!=(const detail::maybe_common&,const detail::maybe_common&)
/// \li operator==(const detail::maybe_common&,const T&)
/// \li operator==(const T&,const detail::maybe_common&)
/// \li operator!=(const detail::maybe_common&,const T&)
/// \li operator!=(const T&,const detail::maybe_common&)
template <typename T>
class maybe<T, typename boost::disable_if<detail::has_allocator<T> >::type>
: public detail::maybe_common<T>
{
public:
/// @brief Create a maybe that holds nothing.
maybe() = default;
#if defined(_MSC_VER) && _MSC_VER < 1900
// Using = default with MSVC 2013 on this function causes the compiler
// to make extra copies, which makes this type fail when it's holding a
// move-only type.
//
// Manually implementing this function works just fine, however.
maybe(const maybe& that)
: detail::maybe_common<T>(that)
{ }
// MSVC 2013 cannot = default ctors
maybe(maybe&& that) BOND_NOEXCEPT_IF(std::is_nothrow_move_constructible<typename detail::maybe_common<T>>::value)
: detail::maybe_common<T>(std::move(that))
{ }
#else
/// @brief Copy a maybe
maybe(const maybe&) = default;
/// @brief Move a maybe.
///
/// @note Unlike \c std::optional, a moved-from maybe holds nothing
/// (compared to a moved-from T).
maybe(maybe&&) = default;
#endif
/// @brief Create a maybe that holds a value by copying \c value.
explicit
maybe(const T& value)
: detail::maybe_common<T>(value)
{ }
/// @brief Create a maybe that holds a value by moving from \c value.
///
/// @since 8.0.0
explicit
maybe(T&& value)
: detail::maybe_common<T>(std::move(value))
{ }
#if defined(_MSC_VER) && _MSC_VER < 1900
// Using = default with MSVC 2013 on this function causes the compiler
// to make extra copies, which makes this type fail when it's holding a
// move-only type.
//
// Manually implementing this function works just fine, however.
maybe& operator=(const maybe& that)
{
detail::maybe_common<T>::operator=(that);
return *this;
}
// MSVC 2013 cannot = default rvalue asignment operators
maybe& operator=(maybe&& that) BOND_NOEXCEPT_IF(std::is_nothrow_move_assignable<maybe_common<T>>::value)
{
detail::maybe_common<T>::operator=(std::move(that));
return *this;
}
#else
maybe& operator=(const maybe&) = default;
maybe& operator=(maybe&&) = default;
#endif
/// @brief Assign by copying a value.
maybe& operator=(const T& value)
{
this->emplace(value);
return *this;
}
/// @brief Move-assign from a value.
/// @since 8.0.0
maybe& operator=(T&& value)
{
this->emplace(std::move(value));
return *this;
}
/// @brief Set the maybe to hold a value, if needed.
///
/// If this instance contains nothing, construct a default instance of
/// T; otherwise, preserve the existing value.
///
/// @return A reference to the value.
T& set_value()
{
if (this->is_nothing())
{
this->emplace();
}
return *this->_value;
}
/// @brief Swap this object with \c that.
void swap(maybe& that)
{
using std::swap;
swap(this->_value, that._value);
}
};
/// @brief Type used for fields with default values of \c nothing.
///
/// This specialization is used for instances of T with allocators.
///
/// See the [User's
/// Manual](../../manual/bond_cpp.html#default-value-of-nothing) for more
/// details about default values of \c nothing.
///
/// @see For details of %maybe's comparison operators, see
/// \li operator==(const detail::maybe_common&,const detail::maybe_common&)
/// \li operator!=(const detail::maybe_common&,const detail::maybe_common&)
/// \li operator==(const detail::maybe_common&,const T&)
/// \li operator==(const T&,const detail::maybe_common&)
/// \li operator!=(const detail::maybe_common&,const T&)
/// \li operator!=(const T&,const detail::maybe_common&)
template <typename T>
class maybe<T, typename boost::enable_if<detail::has_allocator<T> >::type>
: public detail::maybe_common<T>,
private detail::allocator_holder<typename T::allocator_type>
{
using alloc_holder = detail::allocator_holder<typename T::allocator_type>;
public:
// allocator_holder may inherit from an allocator that has its own
// value_type. Add an using to explicitly "export" the one from
// maybe_common
using typename detail::maybe_common<T>::value_type;
/// @brief The type of the allocator in use.
using allocator_type = typename T::allocator_type;
#if defined(_MSC_VER) && _MSC_VER < 1900
// = default fails on MSVC 2013 when the allocator is not default
// constructible.
maybe() { }
// Using = default with MSVC 2013 on this function causes the compiler
// to make extra copies, which makes this type fail when it's holding a
// move-only type.
//
// Manually implementing this function works just fine, however.
maybe(const maybe& that)
: detail::maybe_common<T>(that),
alloc_holder(that)
{ }
// MSVC 2013 cannot = default rvalue ctors
maybe(maybe&& that) BOND_NOEXCEPT_IF(
std::is_nothrow_move_constructible<typename detail::maybe_common<T>>::value
&& std::is_nothrow_move_constructible<alloc_holder>::value)
: detail::maybe_common<T>(std::move(that.base_common())),
alloc_holder(std::move(that.base_alloc_holder()))
{ }
#else
/// @brief Create a maybe that holds nothing.
maybe() = default;
/// @brief Copy a maybe
maybe(const maybe&) = default;
/// @brief Move a maybe.
///
/// @note Unlike \c std::optional, a moved-from maybe holds nothing
/// (compared to a moved-from T).
maybe(maybe&&) = default;
#endif
/// @brief Allocator-extended copy constructor. Uses alloc as the new
/// allocator, makes a copy of \c that.
maybe(const maybe& that, const allocator_type& alloc)
: detail::maybe_common<T>(),
alloc_holder(alloc)
{
if (!that.is_nothing())
{
this->emplace(*that._value, alloc);
}
}
/// @brief Allocator-extended move constructor. Uses alloc as the new
/// allocator, makes moved from \c that.
///
/// @note Unlike \c std::optional, a moved-from maybe holds nothing
/// (compared to a moved-from T).
maybe(maybe&& that, const allocator_type& alloc)
: detail::maybe_common<T>(),
alloc_holder(alloc)
{
if (!that.is_nothing())
{
this->emplace(std::move(*that._value), alloc);
// asigning boost::none is noexcept, but assigning { } is not
that._value = boost::none;
}
}
/// @brief Construct a maybe that holds nothing, but remember the
/// allocator so that it can be used to construct a T if needed.
///
/// @since 8.0.0
explicit maybe(const allocator_type& alloc)
: detail::maybe_common<T>(),
alloc_holder(alloc)
{ }
/// @brief Create a maybe that holds a copy of \c value.
explicit maybe(const T& value)
: detail::maybe_common<T>(value),
alloc_holder()
{ }
/// @brief Create a maybe that holds a value by moving from \c value.
///
/// @since 8.0.0
maybe(T&& value)
: detail::maybe_common<T>(std::move(value)),
alloc_holder()
{ }
#if defined(_MSC_VER) && _MSC_VER < 1900
// Using = default with MSVC 2013 on this function causes the compiler
// to make extra copies, which makes this type fail when it's holding a
// move-only type.
//
// Manually implementing this function works just fine, however.
maybe& operator=(const maybe& that)
{
base_common() = that.base_common();
base_alloc_holder() = that.base_alloc_holder();
return *this;
}
// MSVC 2013 cannot = default rvalue ctors
maybe& operator=(maybe&& that) BOND_NOEXCEPT_IF(
std::is_nothrow_move_assignable<detail::maybe_common<T>>::value
&& std::is_nothrow_move_assignable<alloc_holder>::value)
{
base_common() = std::move(that.base_common());
base_alloc_holder() = std::move(that.base_alloc_holder());
return *this;
}
#else
maybe& operator=(const maybe&) = default;
maybe& operator=(maybe&&) = default;
#endif
/// @brief Assign by copying \c value.
maybe& operator=(const T& value)
{
this->emplace(value);
return *this;
}
/// @brief Move-assign from \c value.
/// @since 8.0.0
maybe& operator=(T&& value)
{
this->emplace(std::move(value));
return *this;
}
// We need to get rid of any operator== that may come from the
// allocator_holder so the friend free functions from maybe_common don't
// have any competition.
bool operator==(const alloc_holder&) = delete;
/// @brief Set to non-empty, if needed.
///
/// If this object contains nothing, construct an instance of T, passing
/// it the saved allocator; otherwise, preserve the existing value.
///
/// @return A reference to the value.
T& set_value()
{
if (this->is_nothing())
{
this->emplace(base_alloc_holder().get());
}
return *this->_value;
}
/// @brief Swap this object with \c that.
void swap(maybe& that)
{
using std::swap;
swap(this->_value, that._value);
swap(base_alloc_holder(), that.base_alloc_holder());
}
/// @brief Get the allocator that this maybe uses.
/// @since 8.0.0
allocator_type get_allocator() const BOND_NOEXCEPT
{
return base_alloc_holder().get();
}
private:
detail::maybe_common<T>& base_common() BOND_NOEXCEPT { return *this; }
const detail::maybe_common<T>& base_common() const BOND_NOEXCEPT { return *this; }
alloc_holder& base_alloc_holder() BOND_NOEXCEPT { return *this; }
const alloc_holder& base_alloc_holder() const BOND_NOEXCEPT { return *this; }
};
/// @brief Swap two maybes.
template<typename T>
inline void swap(maybe<T>& x, maybe<T>& y)
{
x.swap(y);
}
} // namespace bond
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,332 |
Q: Using a PHP variable to aggregate in MongoDB I've been trying to use PHP Variables to get results from a mongoDB database.
This works:
//create Mongo connection
$m = new MongoClient();
//Select db
$db = $m->warehouse;
//create collection
$fact = $db->fact;
$res = $fact->aggregate([
array(
'$match'=>array(
'day'=>array('$gte'=>3,'$lte'=>28),
'month'=>array('$gte'=>3,'$lte'=>3),
'year'=>array('$gte'=>2017,'$lte'=>2017)
)
),
array(
'$group'=>array(
'_id'=>array(
'month'=>'$month',
'day'=>'$day',
'year'=>'$year'
),
'totalSales'=>array('$sum'=>'$sale'),
'totalCost'=>array('$sum'=>'$total_cost')
)
),
array(
'$project'=>array(
'_id'=>0,
'totalSales'=>1,
'totalCost'=>1,
'totalProfit'=>array(
'$subtract'=>['$totalSales','$totalCost']
)
)
)
]);
Output of var_dump($res):
array (size=3)
'waitedMS' => int 0
'result' =>
array (size=1)
0 =>
array (size=3)
'totalSales' => float 440
'totalCost' => float 350
'totalProfit' => float 90
'ok' => float 1
But When I try to replace the numbers with php variables like:
$res = $fact->aggregate([
array(
'$match'=>array(
'day'=>array('$gte'=>$dayfrom,'$lte'=>$dayto),
'month'=>array('$gte'=>$monthfrom,'$lte'=>$monthto),
'year'=>array('$gte'=>$yearfrom,'$lte'=>$yearto)
)
),
array(
'$group'=>array(
'_id'=>array(
'month'=>'$month',
'day'=>'$day',
'year'=>'$year'
),
'totalSales'=>array('$sum'=>'$sale'),
'totalCost'=>array('$sum'=>'$total_cost')
)
),
array(
'$project'=>array(
'_id'=>0,
'totalSales'=>1,
'totalCost'=>1,
'totalProfit'=>array(
'$subtract'=>['$totalSales','$totalCost']
)
)
)
]);
I don't get back any results:
array (size=3)
'waitedMS' => int 0
'result' =>
array (size=0)
empty
'ok' => float 1
What am I doing wrong here? I have tried several things like putting the PHP variables under single quotes or trying to put parts of the query into strings and using those strings to re-build the query but nothing seems to work excep for the hard-coded numbers into the aggregation. Any idea what's going on here?
EDIT: (This did the trick)
Tried initializing the variables in a PHP class constructor that explicitly specifies that the value is coming from a PHP variable:
class DateSet {
public $df;
public $dt;
public $mf;
public $mt;
public $yf;
public $yt;
function __construct( $df, $dt, $mf, $mt, $yf, $yt) {
$this->df = $df;
$this->dt = $dt;
$this->mf = $mf;
$this->mt = $mt;
$this->yf = $yf;
$this->yt = $yt;
}
};
$dateobj = new DateSet($dayfrom,$dayto,$monthfrom,$monthto,$yearfrom,$yearto);
$res = $fact->aggregate([
array(
'$match'=>array(
'day'=>array('$gte'=>$dateobj->df,'$lte'=>$dateobj->dt),
'month'=>array('$gte'=>$dateobj->mf,'$lte'=>$dateobj->mt),
'year'=>array('$gte'=>$dateobj->yf,'$lte'=>$dateobj->yt)
)
),
array(
'$group'=>array(
'_id'=>array(
'month'=>'$month',
'day'=>'$day',
'year'=>'$year'
),
'totalSales'=>array('$sum'=>'$sale'),
'totalCost'=>array('$sum'=>'$total_cost')
)
),
array(
'$project'=>array(
'_id'=>0,
'totalSales'=>1,
'totalCost'=>1,
'totalProfit'=>array(
'$subtract'=>['$totalSales','$totalCost']
)
)
)
]);
OUTPUT:
array (size=3)
'waitedMS' => int 0
'result' =>
array (size=7)
0 =>
array (size=3)
'totalSales' => float 640
'totalCost' => float 520
'totalProfit' => float 120
1 =>
array (size=3)
'totalSales' => float 8000
'totalCost' => float 6000
'totalProfit' => float 2000
2 =>
array (size=3)
'totalSales' => float 700
'totalCost' => float 600
'totalProfit' => float 100
3 =>
array (size=3)
'totalSales' => float 18000
'totalCost' => float 15000
'totalProfit' => float 3000
4 =>
array (size=3)
'totalSales' => float 6300
'totalCost' => float 6100
'totalProfit' => float 200
5 =>
array (size=3)
'totalSales' => float 68503
'totalCost' => float 55720
'totalProfit' => float 12783
6 =>
array (size=3)
'totalSales' => float 440
'totalCost' => float 350
'totalProfit' => float 90
'ok' => float 1
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,238 |
\section{Introduction}
Gravitational lensing studies have radically improved our understanding of the internal structure of galaxies and clusters of galaxies (e.g., \citealt{tre10b}; \citealt{bar10}). In particular, the combination of strong lensing with stellar dynamics or stellar population synthesis models has allowed to characterize some properties, previously almost unexplored, of the galaxy dark-matter haloes and sub-haloes. For example, it has become possible to measure the dark over total mass fraction and dark-matter halo density slope in the inner regions of galaxies (e.g., \citealt{gri09,gri10c,gri12}; \citealt{aug09}; \citealt{bar09,bar11}; \citealt{son12}; \citealt{eic12}), and to estimate the mass function of dark satellites (also called substructure) (e.g., \citealt{koc04}; \citealt{veg10,veg12}; \citealt{fad12}; \citealt{xu13}) and the spatial extent (e.g., \citealt{hal07}; \citealt{suy10}; \citealt{ric10}; \citealt{don11}; \citealt{eic13}) of galaxy dark-matter haloes. Moreover, the same combinations of mass diagnostics have enabled to investigate the total mass density profile (e.g., \citealt{rus03,rus05}; \citealt{koo06,koo09}; \citealt{ruf11}; \citealt{son13b}; \citealt{agn13}), the stellar Initial Mass Function (IMF; \citealt{gri08a,gri09}; \citealt{tre10}; \citealt{spi11,spi12}; \citealt{bar13}), and the origin of the tilt of the Fundamental Plane (e.g., \citealt{gri09,gri10b}; \citealt{aug10b}) of massive early-type galaxies and to use single lenses or statistical samples of them to infer cosmologically relevant quantities (e.g., \citealt{gri08b}; \citealt{sch10}; \citealt{suy10b,suy13}; \citealt{fad10}).
Taking advantage of the excellent data collected by the Lenses Structure and Dynamics (LSD; \citealt{tre04}), Sloan Lens ACS Survey (SLACS; \citealt{bol06}; \citealt{tre06}; \citealt{aug10b}), CFHT Strong Lensing in the Legacy Survey (SL2S; \citealt{mor12}; \citealt{gav12}; \citealt{son13a}), and the BOSS Emission-Line Lens Survey (BELLS; \citealt{bro12}; \citealt{bol12}), the analyses conducted so far have mainly examined the physical properties of isolated, massive early-type galaxies, acting as strong lenses on background sources. Only more recently, thanks also to the Cambridge And Sloan Survey Of Wide ARcs in the skY (CASSOWARY; \citealt{bel09}; \citealt{sta13}), growing interest has been shown in the study of early-type lens galaxies residing in galaxy groups and clusters (e.g., \citealt{gri08c,gri11,gri13}; \citealt{lim09}; \citealt{dea13}). Despite the large amount of results published to date, detailed strong lensing studies in ``small'' lens galaxies are still lacking, mostly because these systems are not observed frequently. One possibility to find more such systems is to look at clusters of galaxies, where, owing to the increase in the strong lensing cross section of a cluster member due to the presence of the extended mass distribution of the cluster, low-mass galaxies are more likely to produce strong lensing features than in less dense environments. Investigations of these objects are particularly useful, as they can provide the necessary piece of information to elucidate what is the amount and distribution of dark matter in astrophysical objects extending from the lowest to the highest ends of the galaxy mass function. The comparison of these observational measurements over a wide range of physical scales with the outcomes of cosmological simulations can give fundamental clues about the precise nature of dark matter and the role played by the interaction of baryons and dark matter during the mass assembly of cosmological structures.
The Cluster Lensing And Supernova survey with Hubble (CLASH; GO 12065, PI Postman) was awarded 524 orbits of Hubble Space Telescope (HST) time to observe 25 massive (virial mass $M_{\mathrm{vir}} \approx 5$-$30 \times 10^{14} M_{\odot}$, X-ray temperature $T_{X} \ge 5$ keV) galaxy clusters in 16 broadband filters, ranging from approximately 2000 to 17000 \AA $\,$ with the Wide Field Camera 3 (WFC3; \citealt{kim08}) and the Advanced Camera for Surveys (ACS; \citealt{for03}). The sample, spanning a wide redshift range ($z$ = 0.18-0.90), was carefully chosen to be largely free of lensing bias and representative of relaxed clusters, on the basis of their symmetric and smooth X-ray emission profiles (for a thorough overview, see \citealt{pos12}). CLASH has four main scientific goals: 1) measure the cluster total mass profiles over a wide radial range, by means of strong and weak lensing analyses (e.g., \citealt{zit11}; \citealt{coe12}; \citealt{med13}); 2) detect new Type Ia supernovae out to redshift z $\sim$ 2.5, to improve the constraints on the dark energy equation of state (e.g., \citealt{gra13}; \citealt{pat13}); 3) discover and study some of the first galaxies that formed after the Big Bang (z $>$ 7) (e.g., \citealt{zhe12}; \citealt{coe13}; \citealt{bow13}); 4) perform galaxy evolution analyses on cluster members and background galaxies. Ancillary science that can surely be carried out with the superb data set of CLASH is the analysis of several new strong lensing systems on galaxy scale.
A Large Programme (186.A-0798, PI Rosati) of 225 hours with the VIMOS instrument at the Very Large Telescope (VLT) has also been approved to perform a panoramic spectroscopic survey of the 14 CLASH clusters that are visible from ESO-Paranal (Rosati et al. 2014, in preparation). This observational campaign aims at measuring in each cluster the redshifts of 1) approximately 500 cluster members within a radius of more than 3 Mpc; 2) 10-30 lensed multiple images inside the HST field of view, including possible highly-magnified candidates out to z $\approx$ 7 (e.g., \citealt{mon13}; \citealt{bal13}); 3) possible supernova hosts. In one of the CLASH clusters (i.e., MACS J1206.2$-$0847, hereafter MACS 1206), the first spectroscopic redshifts have already been exploited to build robust strong lensing models (\citealt{zit12}; \citealt{ume12}), to obtain an independent total mass estimate from the spatial distribution and kinematics of the cluster members (\citealt{biv13}; \citealt{lem13}), and to confirm a source at $z = 5.703$ (\citealt{bra13}). Strong lensing (with spectroscopically confirmed systems) and cluster dynamics analyses are planned for all 14 southern clusters (e.g., in MACS J0416.1$-$2403, Grillo et al. 2014, in preparation; Balestra et al. 2014, in preparation).
Here, we focus on a rare strong lensing system in which two angularly close early-type galaxies, members of the galaxy cluster MACS 1206 at $z=0.44$, produce in total ten multiple images of a double source located at $z \approx 3.7$. This is the first example of the kind of strong lensing studies that can be conducted on galaxy cluster members, capitalizing on the extraordinary multi-band photometric observations obtained as part of the CLASH program and spectroscopic measurements of the VLT/VIMOS follow-up campaign.
This work is organized as follows. In Sect. 2, we introduce the photometric and spectroscopic observations used for this analysis. In Sect. 3, we present the strong lensing modeling performed to measure principally the total mass values of the lens galaxies and the magnification factors of the multiple images. In Sect. 4, we estimate the luminous mass values of the lens and lensed galaxies by means of stellar population synthesis models. In Sect. 5, we compare some physical quantities, related to the luminous and total masses, of the two lens galaxies with those of lower and higher-mass galaxies. In Sect. 6, we summarize our conclusions. All quoted errors are 68.3\% confidence limits (CL) unless otherwise stated. Throughout this work we assume $H_{0}=70$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_{m}=0.3$, and $\Omega_{\Lambda}=0.7$. In this model, 1\arcsec$\,$ corresponds to a linear size of 5.69 kpc at the cluster redshift of $z=0.44$.
\section{Observations}
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{cutout_lens_2c.ps}
\caption{Color composite image (9\arcsec$\times$15\arcsec) obtained by combining the F606W+F625W (blue channel), F814W+F850LP (green channel), and F140W+F160W (red channel) filters of HST/ACS and WFC3. Ten multiple images from a double source (at $z \approx 3.7$) are visible around two lens cluster galaxies (at $z=0.44$), G1, near the top, and G2, in the middle. More details on these objects are given in Tables \ref{tab1} and \ref{tab2} and the best-fitting strong lensing model is shown in Figure \ref{fi01}. North is top and East is left.}
\label{fi05}
\end{figure}
\begin{table*}
\centering
\caption{Photometric and spectroscopic properties of the lens galaxies.}
\begin{tabular}{ccccccccccc}
\hline\hline \noalign{\smallskip}
& R.A. & Decl. & $x$$^{\mathrm{a}}$ & $y$$^{\mathrm{a}}$ & $z_{\mathrm{ph}}$ & $z_{\mathrm{sp}}$ & $q_{L}$ & $\theta_{q_{L}}$$^{\mathrm{b}}$ & $\theta_{e}$ & $n$ \\
& (J2000) & (J2000) & (\arcsec) & (\arcsec) & & & & (deg) & (\arcsec) & \\
\noalign{\smallskip} \hline \noalign{\smallskip}
G1 & 12:06:16.01 & $-$08:48:17.3 & 0.00 & 0.00 & 0.42 & 0.436 & 0.96 & 113 & 0.43 & 3.7 \\
G2 & 12:06:15.67 & $-$08:48:22.0 & 5.00 & $-$4.68 & 0.50 & 0.439 & 0.84 & 25 & 1.18 & 3.0 \\
\noalign{\smallskip} \hline
\end{tabular}
\begin{list}{}{}
\item[$^{\mathrm{a}}$]With respect to the luminosity center of G1.
\item[$^{\mathrm{b}}$]Angles are measured East of North.
\end{list}
\label{tab1}
\end{table*}
MACS 1206 was observed as part of the CLASH program in HST Cycle 18 between April 3 and July 20 2011 to a total depth of 20 orbits in 16 broadband filters. The images were processed for debias, flats, superflats, and darks using standard techniques, and then co-aligned and combined using drizzle algorithms to a pixel scale of 0.065\arcsec$\,$ (for details, see \citealt{koe07,koe11}). By fitting the full UV to near-IR isophotal aperture magnitudes, photometric redshift estimates of all detected sources were measured through the BPZ (\citealt{ben00,ben04}; \citealt{coe06}) and LePhare (\citealt{arn99}; \citealt{ilb06}) codes.
A color composite image of the lensing system analyzed in this work is shown in Figure \ref{fi05}. The coordinates and photometric redshift measurements, $z_{\mathrm{ph}}$, of the two lenses G1 and G2 and of the multiple images are listed in Tables \ref{tab1} and \ref{tab2}. For the lensed source, we adopt the photometric redshift estimate of 3.7 presented by \citet{zit12} (see system number 7 in the cited paper; for more details on the photometric redshifts measured with the CLASH data, and particularly in MACS 1206, see \citealt{jou13}). This value was determined as an average estimate of the BPZ measurements of the three images (1, 3, and 5; see Figure \ref{fi01}) for which the photometry is less contaminated by the light distribution of the two lens galaxies. In passing, we mention that the BPZ probability distribution functions of the redshift of the three images were all unimodal and that the combined 95\% CL interval of the redshifts extended from 3.4 to 4.2 (this redshift uncertainty is not significant for the lensing analysis performed below; for example, it introduces a percentage error smaller than 1.5\% on the values of the lens effective velocity dispersions plotted in Figure \ref{fi02}). We used the public code \textsc{Galfit}\footnote{http://users.obs.carnegiescience.edu/peng/work/galfit/galfit.html} (\citealt{pen10}) to derive the luminosity structural parameters of the two lenses in the F160W band (see Table \ref{tab1}): the axis ratio, $q_{L}$, the position angle of the major axis, $\theta_{q_{L}}$, the half-light angle, $\theta_{e}$, and the Sersic index, $n$.
Spectroscopic follow-up observations were taken as part of the VLT/VIMOS Large Programme 186.A-0798. Four VIMOS pointings were used, keeping one of the four quadrants fixed to the cluster core in order to allow for long integrations on the strong lensing features. In each pointing, we took 45 to 60 minute exposure times. We used 1\arcsec-wide slits with either the low-resolution LR-Blue grism or the intermediate-resolution MR grism, covering a layout of $20'$-$25'$ across. MACS 1206 is the first of the 14 southern galaxy clusters in the CLASH sample targeted by this spectroscopic program for which the observational campaign was concluded. The measurements have resulted in a total integration time of approximately 11 hours, providing about 600 secure cluster members (see \citealt{biv13}) and 4 confirmed multiple image systems (see \citealt{zit12}). Additional spectroscopic measurements on some cluster members of MACS 1206 were obtained with the VLT/FORS2 in April 17 2012 (Programme ID 089.A-0879, PI Gobat). The observations were taken in good seeing conditions, with the medium resolution grism 600RI and 1\arcsec -wide slits, with a total exposure time of 60 minutes.
The flux-calibrated VIMOS MR and LR-Blue spectra of G1 and G2, respectively, are shown in Figure \ref{fi07}. The identification of the most prominent absorption lines, like CaII K and H and G-band, have provided spectroscopic redshift estimates, $z_{\mathrm{sp}}$, of 0.436 and 0.439 for these two galaxies. These values are included in the 95\% CL intervals of the estimated photometric redshifts, ranging from 0.36 to 0.46 and from 0.42 to 0.52 for, respectively, G1 and G2. In Figure \ref{fi07}, we also display several template emission and absorption features redshifted to the best-fitting values of the galaxy spectroscopic redshifts. The FORS2 spectrum of G2 is shown in Figure \ref{fi10} and will be discussed in the following section.
\begin{table}[h]
\centering
\caption{Astrometric and photometric measurements for the multiple images.}
\begin{tabular}{ccccc}
\hline\hline \noalign{\smallskip}
& $x$$^{\mathrm{a}}$ & $y$$^{\mathrm{a}}$ & $z_{\mathrm{ph}}$ & $\delta_{x,y}$ \\
& (\arcsec) & (\arcsec) & & (\arcsec) \\
\noalign{\smallskip} \hline \noalign{\smallskip}
A$_{1}$ & 0.26 & 1.36 & 3.7 & 0.065 \\
B$_{1}$ & 0.39 & 1.10 & 3.7 & 0.065 \\
B$_{2}$ & 0.72 & 0.20 & 3.7 & 0.065 \\
A$_{2}$ & 0.65 & $-$0.06 & 3.7 & 0.065 \\
A$_{3}$ & 0.58 & $-$1.10 & 3.7 & 0.065 \\
B$_{3}$ & 0.72 & $-$1.30 & 3.7 & 0.065 \\
B$_{4}$ & 1.36 & $-$5.52 & 3.7 & 0.065 \\
A$_{4}$ & 1.50 & $-$5.98 & 3.7 & 0.065 \\
A$_{5}$ & 3.51 & $-$10.01 & 3.7 & 0.065 \\
B$_{5}$ & 3.84 & $-$10.46 & 3.7 & 0.065 \\
\noalign{\smallskip} \hline
\end{tabular}
\begin{list}{}{}
\item[$^{\mathrm{a}}$]With respect to the luminosity center of G1.
\end{list}
\label{tab2}
\end{table}
\begin{figure*}
\centering
\includegraphics[width=0.84\textwidth]{G1_VIMOS_MR2.eps}
\includegraphics[width=0.83\textwidth]{G2_VIMOS_LRb_arch2.eps}
\caption{Flux-calibrated VLT/VIMOS spectra of G1 and G2 obtained with the MR and LR-Blue grisms, respectively. The exposure times are 45 and 6 minutes. The 1D spectra with several template emission and absorption lines shifted to the best-fitting redshift values and the 2D spectra are shown. \emph{On the top:} Spectra of G1 which provide a redshift value of 0.436. \emph{On the bottom:} Spectra of G2 which provide a redshift value of 0.439.}
\label{fi07}
\end{figure*}
\section{Strong lensing modeling}
In this section we present two different models of the strong gravitational lensing system. We use the public code \emph{gravlens}\footnote{http://redfive.rutgers.edu/$\sim$keeton/gravlens/} (\citealt{kee01a}) to reconstruct the total mass distribution of the lenses and to estimate the position and magnification factor of the sources. We stress the fact that the analysis presented below has a different perspective compared to the previous strong lensing models of MACS 1206 (\citealt{zit12}; \citealt{eic13}). We concentrate here on radial scales of a few kpc, where the total mass of the galaxies G1 and G2 is the main source of the gravitational potential and the mass of the cluster (extended over a typical radial scale of 100 kpc) is instead approximated and treated as a second-order term.
We keep the model complexity to a minimum and describe the three main lenses, i.e. the two galaxies G1 and G2 and the cluster, in terms of either three singular isothermal spheres (3SISs) or two singular isothermal spheres and a singular isothermal ellipsoid (2SISs+SIE). The mass components are fixed to the luminosity centroids of the galaxies G1 and G2 and of the BCG (the hypothesis of the center of mass of the cluster coinciding with that of the BCG is supported by the results of the studies cited in the previous paragraph) and parametrized by angular scales (labeled as $b_{\mathrm{G1}}$, $b_{\mathrm{G2}}$, and $b_{\mathrm{H}}$), which represent the strength of the lenses and are equal to the values of the Einstein angles in the spherical case. The SIE model requires two additional parameters: the values of the axis ratio, $q$, and of the major axis position angle $\theta_{q}$. The ten multiple images are approximated to point-like objects and associated to two close sources (A and B), each of which is lensed five times. The multiple images are identified with indices running from 1 to 5 (see Table \ref{tab2}). For each image we assume an observational error $\delta_{x,y}$ on the determination of its position of one image pixel (i.e., 0.065\arcsec). Although in each HST filter the luminosity centroids of the multiple images can be measured with positional errors of some fractions of a pixel, we have decided to consider a conservative uncertainty value of one pixel to take into account the point-like approximation, the centroid differences in the individual HST bands, and the contamination from the light of the lens galaxies in estimating the multiple image positions. To quantify the goodness of a model, we use a standard chi-square function, $\chi^{2}$, defined as the sum over all the sources $i$ and their multiple images $j$ of the squared ratios of the differences between the observed ($\bold{x}^{i,j}_{\mathrm{obs}}$) and model-predicted ($\bold{x}^{i,j}_{\mathrm{mod}}$) positions divided by the adopted positional uncertainties ($\sigma_{\bold{x}^{i,j}}$):
\begin{equation}
\chi^{2}(\bold{p}) := \sum_{i,j} \frac{|| \bold{x}^{i,j}_{\mathrm{obs}} - \bold{x}^{i,j}_{\mathrm{mod}} ||^{2}}{\sigma_{\bold{x}^{i,j}}^{2}}\, .
\label{eq:03}
\end{equation}
We minimize this $\chi^{2}$ estimator by varying the model parameters ($\bold{p}$) and compare the minimum value of the chi-square with the number of degrees of freedom (d.o.f.). These are the number of observables (twenty coordinates of the ten images) minus the number of the parameters of a model (four coordinates for the two sources, three lens angular scales, and, when present, two lens ellipticity parameters). The best-fitting (minimum chi-square) parameters are shown in Table \ref{tab3}. There, we also present the median ($\tilde{\Delta}$) and root mean square ($\Delta_{\mathrm{rms}}$) values of the Euclidean distances ($\Delta$) between the observed and model-predicted angular positions of the multiple images.
\begin{table*}
\centering
\caption{The parameters of the best-fitting models.}
\begin{tabular}{cccccccccc}
\hline\hline \noalign{\smallskip}
Model & $b_{\mathrm{G1}}$$^{a}$ & $b_{\mathrm{G2}}$$^{a}$ & $b_{\mathrm{H}}$ & $q$ & $\theta_{q}$$^{\mathrm{b}}$ & $\chi^2$ & d.o.f. & $\tilde{\Delta}$ & $\Delta_{\mathrm{rms}}$\\
& (\arcsec) & (\arcsec) & (\arcsec) & & (deg) & & & (\arcsec) & (\arcsec) \\
\noalign{\smallskip} \hline \noalign{\smallskip}
3SISs & 0.21 & 1.25 & 45.3 & & & 19.1 & 13 & 0.081 & 0.090 \\
2SISs+SIE & 0.21 & 1.25 & 37.1$^{c}$ & 0.70 & 14.1 & 13.3 & 11 & 0.059 & 0.075 \\
\noalign{\smallskip} \hline
\end{tabular}
\begin{list}{}{}
\item[$^{\mathrm{a}}$]See Eq. (\ref{eq:02}) for the interpretation of the lens strength $b$ in terms of effective stellar velocity dispersion $\sigma$.
\item[$^{\mathrm{b}}$]Angles are measured East of North.
\item[$^{\mathrm{c}}$]Note that Gravlens provides the value of the lens strength $b$ multiplied by a function $f(\cdot)$ of the minor-to-major axis ratio, $q$ (see \citealt{kee01a,kee01b}).
\end{list}
\label{tab3}
\end{table*}
The values of the best-fitting chi-square are very close to the number of degrees of freedom. This fact and the small values of $\tilde{\Delta}$ and $\Delta_{\mathrm{rms}}$ confirm that our relatively simple mass modeling choices are adequate to describe the lenses. More quantitatively, the probability that the value of a random variable extracted from the chi-square distribution with 11(13) degrees of freedom is greater than 13.3(19.1) is 0.27(0.12). It is not surprising that the spherical approximation used for describing the total mass distributions of the galaxies G1 and G2 is proved suitable, given the large values of the minor-to-major axis ratio $q_{L}$ of their luminous components (see Table \ref{tab1}). Looking at Table \ref{tab3}, we conclude that the values of $b_{\mathrm{G1}}$ and $b_{\mathrm{G2}}$, which characterize the total mass distribution of the two main lenses, are robust and not sensitive to the details of the cluster total mass modeling.
In Figure \ref{fi01}, we plot the best-fitting 2SISs+SIE model with the observed and model-predicted multiple images and the critical curves. We show also the values of the magnification factors in proximity to the reconstructed positions of each multiple image.
\begin{figure*}
\centering
\includegraphics[width=0.45\textwidth]{fig04c.ps}
\includegraphics[width=0.45\textwidth]{fig07c.ps}
\caption{The best-fitting 2SISs+SIE strong lensing model. \emph{On the left:} The model-predicted critical curves and multiple image positions (diamond and square symbols) of the two sources, $A$ and $B$, around the two main lenses, G1 and G2. The observed positions of the multiple images (cross and plus symbols) are shown for comparison. \emph{On the right:} Reconstructed values of the magnification factor at the positions of the model-predicted multiple images. Positive and negative values on the contour levels indicate, respectively, if the images have conserved or inverted parity with respect to the source. North is top and East is left.}
\label{fi01}
\end{figure*}
Interestingly, starting from the only lensing system analyzed in this work we measure that for the cluster component, when modeled as an SIE, the values of axis ratio and position angle are aligned with the prominent intracluster light (for more details on the properties of the intracluster light in MACS 1206, we refer to Presotto et al. 2014, in preparation). The values of these parameters are consistent with those obtained from the thorough lensing analyses performed on the cluster scale by \citet{zit12}, \citet{eic13}, and \citet{ume12}. The first two and the last studies exploit, respectively, the full strong lensing and strong plus weak lensing information in MACS 1206. Moreover, the best-fitting parameters for the SIE model associated to the cluster component provide a total mass estimate of about $4\times10^{14} M_{\odot}$ projected within a cylinder of radius equal to 320 kpc (the approximate average distance of the multiple images from the BCG luminosity center). Given our simplified assumptions on the cluster total mass distribution, it is remarkable that this estimate is only approximately $10\%$ higher than those obtained in the previously cited lensing works and in the cluster dynamical analysis by \citet{biv13}. If one used otherwise the very crude approximation of the Einstein radius of the cluster (for a source at redshift 3.7) given by the average projected distance between the strong lensing system and the BCG luminosity centre, this would translate into a total projected mass that is more than 1.5 times larger than what obtained from the other cluster total mass diagnostics. Several previous studies of strong lensing systems around galaxy cluster members (e.g., \citealt{gri08c}; \citealt{lim09}) have demonstrated that although these systems contain enough information to characterize reasonably well the cluster mass distribution, this last term is not the main focus of such studies and it can be safely modeled with an approximated convergence plus shear contribution.
In Table \ref{tab5}, we consider the two best-fitting models and list the values of the magnification factor for images A$_{5}$ and B$_{5}$, $\mu(\mathrm{A}_{5})$ and $\mu(\mathrm{B}_{5})$, and of the total magnification factor for the sources A and B, i.e., the sum of the magnification values over all the multiple images of each source:
\begin{eqnarray}
\mu_{\mathrm{tot}}(\mathrm{A}) &:=& \sum_{i=1}^{5} \mu(\mathrm{A}_{i}) \, ; \nonumber \\
\mu_{\mathrm{tot}}(\mathrm{B}) &:=& \sum_{i=1}^{5} \mu(\mathrm{B}_{i}) \, .
\label{eq:01}
\end{eqnarray}
We decide to concentrate on images A$_{5}$ and B$_{5}$ because they are the most distant objects from the luminosity (and mass) centers of the two lenses G1 and G2. For this reason, their photometry is less contaminated by the light distribution of the much brighter lens galaxies. In addition, their relatively large distance from the critical curves (see Figure \ref{fi01}) makes the measurements of their magnification factors less dependent on the modeling details. We conclude that A$_{5}$ and B$_{5}$ are magnified by a factor of approximately 9 and that each of the two sources is magnified in total by a factor of approximately 50.
\begin{table}
\centering
\caption{Values of the magnification factor for the best-fitting models.}
\begin{tabular}{ccccc}
\hline\hline \noalign{\smallskip}
Model & $\mu(\mathrm{A}_{5})$ & $\mu(\mathrm{B}_{5})$ & $\mu_{\mathrm{tot}}(\mathrm{A})$ & $\mu_{\mathrm{tot}}(\mathrm{B})$ \\
\noalign{\smallskip} \hline \noalign{\smallskip}
3SISs & 9.8 (0.4) & 8.5 (0.3) & 48 (1.2) & 50 (1.4) \\
2SISs+SIE & 9.3 (0.4) & 8.1 (0.3) & 47 (1.3) & 50 (1.5) \\
\noalign{\smallskip} \hline
\end{tabular}
\begin{list}{}{}
\item[Notes --]In parentheses, we show the 1$\sigma$ statistical uncertainties derived from the bootstrapping analysis.
\end{list}
\label{tab5}
\end{table}
To estimate the statistical uncertainties on the model parameters $b_{\mathrm{G1}}$ and $b_{\mathrm{G2}}$, we perform a bootstrapping analysis in the 2SISs+SIE case. We resample the position of the 10 multiple images by extracting random values from Gaussian distributions with average and standard deviation values equal to, respectively, the positions and positional uncertainties listed in Table \ref{tab2}. We simulate in this way $10^{4}$ data samples, minimize the positional $\chi^{2}$ shown in Eq. (\ref{eq:03}), and consider the best-fitting values of the lens strength. We recall that the value of the effective velocity dispersion $\sigma$ of an SIS model is related to that of the lens strength $b$ in the following way:
\begin{equation}
b = 4 \pi \bigg( \frac{\sigma}{c} \bigg)^2 \frac{D_{ls}}{D_{os}} \, ,
\label{eq:02}
\end{equation}
where $c$ is the speed of light and $D_{ls}$ and $D_{os}$ are, respectively, the angular diameter distances between the lens and the source and the observer and the source. We show the results of this analysis in Table \ref{tab4} and Figure \ref{fi02}.
\begin{table}
\centering
\caption{Median and statistical 1$\sigma$ error values of the SIS effective velocity dispersions of the two main lenses.}
\begin{tabular}{ccc}
\hline\hline \noalign{\smallskip}
Model & $\sigma_{\mathrm{G1}}$ & $\sigma_{\mathrm{G2}}$ \\
& (km s$^{-1}$) & (km s$^{-1}$) \\
\noalign{\smallskip} \hline \noalign{\smallskip}
3SISs & $100 \pm 3$ & $241 \pm 6$ \\
2SISs+SIE & $97 \pm 3$ & $240 \pm 6$ \\
\noalign{\smallskip} \hline
\end{tabular}
\label{tab4}
\end{table}
\begin{figure}
\centering
\includegraphics[width=0.44\textwidth]{fig02c.ps}
\caption{Values of the effective velocity dispersion of the two main lenses, G1 and G2, for the 2SISs+SIE model. The best-fitting values are represented by a cross. The 68\% and 95\% confidence regions (contour levels) and the 68\% confidence intervals (thick lines on the axes) are obtained from the $\chi^{2}$ minimization of $10^{4}$ resampled multiple image positions.}
\label{fi02}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.47\textwidth]{Gal2_plot_paper_v1_mult.ps}
\caption{VLT/FORS2 rest-frame spectrum of G2 from an aperture of 6 pixels (i.e., 1.5 arcsec) around the peak of emission. The total exposure time is 60 minutes. The data have been slightly smoothed with a box of 3 pixels for display purpose. Black and red solid lines show the data and the best-fitting model, respectively, and green dots display the difference of these last two spectra. The grey shaded areas correspond to the sky line regions which have been masked while performing the fit due to their high residuals. The main absorption lines and possible emission lines are labelled.}
\label{fi10}
\end{figure}
The values of $\sigma_{\mathrm{G1}}$ and $\sigma_{\mathrm{G2}}$ are positively correlated and measured with a small statistical uncertainty. The median values with 1$\sigma$ errors of $\sigma_{\mathrm{G1}}$ and $\sigma_{\mathrm{G2}}$ are $97 \pm 3$ and $240 \pm 6$ km s$^{-1}$, respectively. As already found in several strong lensing studies (e.g., \citealt{gri08c,gri11}), the total mass projected within the average distance of the multiple images from the lens center can be measured precisely. In this specific system, we confirm that an increase of the total mass component associated to the two lens galaxies (i.e., larger values of $\sigma_{\mathrm{G1}}$ and $\sigma_{\mathrm{G2}}$) is correlated to a decrease of the total mass contribution related to the cluster (i.e., smaller values of $\sigma_{\mathrm{H}}$), and viceversa, in order to keep their sum approximately constant.
Then, we repeat two more times the bootstrapping analysis in the 2SISs+SIE case, first allowing the center of the SIE component (i.e., the cluster dark-matter halo) to vary and then including the two candidate (based on the photometric redshift values) cluster members nearest in projection to G1 and G2. In the first case, we obtain $94 \pm 3$ and $227 \pm 8$ km s$^{-1}$ for the median and 1$\sigma$ error values of the SIS effective velocity dispersions of the two main lenses, $\sigma_{\mathrm{G1}}$ and $\sigma_{\mathrm{G2}}$, respectively. Compared to the previous estimates (see the last row in Table \ref{tab4}), we remark that our assumption on the center of mass of the cluster dark-matter halo does not affect significantly the measurements of the most relevant quantities of our lensing model, i.e. $\sigma_{\mathrm{G1}}$ and $\sigma_{\mathrm{G2}}$. Furthermore, the estimates of the lens strength of the halo, $b_{\mathrm{H}}$, change from $37 \pm 1$\arcsec$\,$ (in the previous analysis with the total mass center fixed) to $38 \pm 2$\arcsec$\,$ (in this new analysis with the total mass center free). We remind that the squared value of the strength of a lens is proportional to the mass of that lens projected within its Einstein radius. From this consideration and from the cited estimates of $b_{\mathrm{H}}$, we notice that the mass measurements of the cluster dark-matter halo with and without its mass center fixed are consistent, given their uncertainties. In the second case, we add two galaxies at a projected distance from G1 of approximately 6\arcsec$\,$ (to the North), with luminosity and size values not larger than those of G1. We include these two lenses in the model, fixing their total mass center and strength values (the former to the galaxy luminosity centroids and the latter to the upper limit given by the strength value of G1), and find $97 \pm 3$ and $249 \pm 6$ km s$^{-1}$, respectively, for $\sigma_{\mathrm{G1}}$ and $\sigma_{\mathrm{G2}}$. As from the previous test, comparing these new values with those of Table \ref{tab4}, we can exclude a significant effect of the two nearest candidate cluster members on our estimates of the total mass distributions of G1 and G2. Given their larger projected distances, the possible influence of other cluster members is expected to be even smaller that that of the two neighboring galaxies considered above. From these tests, we can confirm that our measurements of $\sigma_{\mathrm{G1}}$ and $\sigma_{\mathrm{G2}}$ are robust and that the possible systematic uncertainties, due to our specific modeling assumptions, are approximately on the same order of the statistical uncertainties. This means that, even considering both statistical and systematic errors, the values of $\sigma_{\mathrm{G1}}$ and $\sigma_{\mathrm{G2}}$ can be measured with relative errors of less than 10\% and, therefore, that the errors on these quantities are not dominating the error budget of the galaxy luminous over total mass fractions presented in Sect. 5. The errors on these last quantities are in fact mainly driven by the errors on the luminous mass values, estimated in the next section from the galaxy spectral energy distribution fitting.
We remark that our estimates of $\sigma_{\mathrm{G1}}$ and $\sigma_{\mathrm{G2}}$ are consistent, given the errors, with the values of 101 and 236 km s$^{-1}$, respectively, obtained by \citet{eic13}. They performed a strong lensing study of this cluster using the multiple images of 13 background sources and modeling the cluster total mass distribution with a combination of an extended NFW profile and several, smaller, truncated isothermal profiles (representing the candidate cluster member mass contribution), scaled according to the Faber-Jackson relation (\citealt{fab76}). We emphasize that the adoption of scaling relations to model the total mass distributions of candidate cluster members, necessary in order to reduce the number of parameters of a cluster strong lensing model, provides interesting results on the statistical ensemble of galaxies, but these results should be interpreted very carefully if the main focus is on the study of the mass properties of individual cluster members. We caution that the choice of particular scaling relations can drive the results on possible variations in the amount of dark matter present in the inner regions of different cluster members. For this reason, tailored strong lensing models, like the one presented above, are needed.
Furthermore, we measure the stellar velocity dispersion of G2, $\sigma_{*,\mathrm{G2}}$, within an aperture with diameter equal to 1.5\arcsec$\,$, from a fit of the VLT/FORS2 spectrum shown in Figure \ref{fi10}. We use the pixel-fitting method of \citet{cap04} and adopt the template stellar spectra of the MILES library (\citealt{san06,fal11}). We estimate a redshift of 0.4402, consistent with the VIMOS measurement, and a value of $\sigma_{*,\mathrm{G2}}$ of $(250 \pm 30)$ km s$^{-1}$, also consistent, given the uncertainties, with our strong lensing estimate. We notice that a good agreement between the values of the effective velocity dispersion of an isothermal model and of the central stellar velocity dispersion is common in galaxy-scale strong lensing systems where the multiple image geometry can be reconstructed well (e.g., \citealt{tre06,gri08b}).
By exploiting the optimized lens mass models obtained from our bootstrapping analysis, we can also estimate the statistical uncertainties on the values of the different magnification factors. We show the 1$\sigma$ errors (in parentheses) in Table \ref{tab5}. The values of only a few per cent for the magnification relative errors are not surprising, because of the large number of multiple images that provide detailed information about the lens total mass distributions. We remark that the small errors on the total magnification factors are a minor source of uncertainty on the unlensed luminous mass of the source presented in the next section.
Finally, assuming that an isothermal profile is a good description of the total mass distribution of the two lenses out to their effective radius (for these values in angular units, see $\theta_{e}$ in Table \ref{tab1}) and using the results of the bootstrapping analysis (see Table \ref{tab4}), we measure total mass values $M_{T}$ projected within $R_{e}$,
\begin{equation}
M_{T}(<R_{e}) = \frac{\pi \sigma^{2} R_{e}}{G} \, ,
\label{eq:04}
\end{equation}
(being $G$ the value of the gravitational constant) of $1.7_{-0.1}^{+0.1}\times10^{10}$ and $2.8_{-0.1}^{+0.2}\times10^{11}$ $M_{\odot}$ for $M_{T,\mathrm{G1}}(<2.4\, \mathrm{kpc})$ and $M_{T,\mathrm{G2}}(<6.7\, \mathrm{kpc})$, respectively.
\begin{table*}
\centering
\caption{Details of the composite stellar population models adopted to measure the luminous mass values of G1, G2, and $\mathrm{A}_{5}+\mathrm{B}_{5}$ (after correcting for the lensing magnification effect).}
\begin{tabular}{cccccc}
\hline\hline \noalign{\smallskip}
& IMF & SFH & $Z$ & dust & $M_{L}$ \\
\noalign{\smallskip} \hline \noalign{\smallskip}
G1 & Salpeter & delayed exponential & $Z_{\odot}$ & yes & $(1.7 \pm 0.7)\times10^{10}$ $M_{\odot}$ \\
G2 & Salpeter & delayed exponential & $Z_{\odot}$ & yes & $(4.5 \pm 1.8)\times10^{11}$ $M_{\odot}$ \\
$\mathrm{A}_{5}+\mathrm{B}_{5}$ & Salpeter & constant & $Z_{\odot}$ & yes & $(1.0 \pm 0.5)\times10^{9}$ $M_{\odot}$ \\
\noalign{\smallskip} \hline
\end{tabular}
\label{tab7}
\end{table*}
\section{Luminous mass estimates}
\begin{figure*}
\centering
\includegraphics[width=0.49\textwidth]{new_fig_g1.ps}
\includegraphics[width=0.49\textwidth]{new_fig_g2.ps}
\caption{Best-fitting composite stellar population models of the 16-band (the 4 bluest bands are not included in the plots nor in the modeling) HST photometry of the G1 (\emph{on the left}) and G2 (\emph{on the right}) lens galaxies. We use \citet{bru03} templates at solar metallicity and a Salpeter stellar IMF. Observed fluxes with 1$\sigma$ errors are represented with blue empty circles and bars, model-predicted fluxes are shown as orange filled circles.}
\label{fi06}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{new_fig_a5b5.ps}
\caption{Best-fitting composite stellar population models of the 16-band (the 4 bluest bands are not included in the plot nor in the modeling) HST photometry of the multiple images $\mathrm{A}_{5}$ and $\mathrm{B}_{5}$. We use \citet{bru03} templates at solar metallicity and a Salpeter stellar IMF. Observed fluxes with 1$\sigma$ errors are represented with blue empty circles and bars, model-predicted fluxes are shown as orange filled circles.}
\label{fi08}
\end{figure}
Here we model the multicolor photometry, composed of 16 HST bands, of the two lens galaxies G1 and G2 and of the sum of the multiple images $\mathrm{A}_{5}$ and $\mathrm{B}_{5}$. We concentrate on the measurement of the luminous mass of these objects, leaving the study of the physical properties of the source to a future work.
We use composite stellar population (CSP) models based on \citet{bru03} templates at solar metallicity and with a \citet{sal55} stellar IMF. We consider constant and delayed exponential (with a possible cut) Star Formation Histories (SFHs). We allow for the presence of dust, according to \citet{cal00}, and take into account the flux contribution of emission lines. For the two early-type galaxies and high-redshift source we choose, respectively, truncated delayed exponential and constant SFHs, which we believe are the most suitable SFHs for these classes of objects.
We summarize our modeling prescriptions and final results in Table \ref{tab7}. The best-fitting models for the two lens cluster members G1 and G2 are shown in Figure \ref{fi06} and for the lensed objects $A_{5}$ and $B_{5}$ in Figure \ref{fi08}. We have decided to exclude from the fitting and plots the 4 bluest bands because mostly affected by relevant contamination from very close objects. We have checked that removing these bands from the SED fitting does not change appreciably the results on the values of the luminous masses. In fact, photometric mass estimates are known to be more sensitive to the fluxes measured in the redder filters (e.g., \citealt{gri09}).
The best-fitting values of the luminous masses of G1, G2, and $\mathrm{A}_{5}+\mathrm{B}_{5}$ ($M_{L,G1}$, $M_{L,G2}$, and $M_{L,\mathrm{A}_{5}+\mathrm{B}_{5}}$), are, respectively, $1.7 \times 10^{10}$, $4.5 \times 10^{11}$, and $8.8 \times 10^{9}$ $M_{\odot}$. From the ranges of results obtained by considering the different photometric uncertainties, systematic errors associated to the several possible stellar population modeling assumptions (i.e., SFH, dust, emission lines), and rest-frame wavelength range covered by the HST observations, we estimate relative errors of 40\% on $M_{L,G1}$ and $M_{L,G2}$ and of 50\% on $M_{L,\mathrm{A}_{5}+\mathrm{B}_{5}}$. Furthermore, taking into account the value of the average magnification factor of approximately 9 at the positions where $\mathrm{A}_{5}$ and $\mathrm{B}_{5}$ are observed (see Figure \ref{fi01} and Table \ref{tab5}), we conclude that the measured luminous mass values and errors are $(1.7 \pm 0.7)\times10^{10}$ for G1, $(4.5 \pm 1.8)\times10^{11}$ for G2, and $(1.0 \pm 0.5)\times10^{9}$ $M_{\odot}$ for $\mathrm{A}_{5}+\mathrm{B}_{5}$.
We remark that to avoid possible artifacts in our following investigation we have explicitly omitted the recent results (e.g., \citealt{aug10a}; \citealt{tre10}; \citealt{cap12}; \citealt{bar13}) suggesting systematic variations in the stellar IMF of a galaxy as a function of its luminous mass or stellar velocity dispersion. Taking these results into account would probably result in luminous mass estimates approximately 2 times smaller for G1 and $\mathrm{A}_{5}+\mathrm{B}_{5}$.
\section{Discussion}
\begin{table}
\centering
\caption{Values and standard deviations of the luminous mass $M_{L}$, effective velocity dispersion $\sigma_{\mathrm{SIE}}$ (or $\sigma_{*}$), and luminous over total mass fraction projected within the effective radius $f_{\mathrm{L}}(<R_{e})$ of the G1, G2, and SLACS lens galaxies, SDSS massive early-type galaxies, and some dwarf spheroidals.}
\begin{tabular}{ccccc}
\hline\hline \noalign{\smallskip}
& $M_{L}$ & $\sigma_{\mathrm{SIE/*}}$ & $f_{\mathrm{L}}(<R_{e})$ & Ref.\\
& ($10^{10}\,M_{\odot}$) & (km s$^{-1}$) & & \\
\noalign{\smallskip} \hline \noalign{\smallskip}
G1 & $1.7 \pm 0.7$ & $97 \pm 3$ & $0.51 \pm 0.21$ & \\
G2 & $45 \pm 18$ & $240 \pm 6$ & $0.80 \pm 0.32$ & \\
SLACS & $46 \pm 28$ & $267 \pm 39$ & & 1,3 \\
SDSS & $31 \pm 21$ & & $0.64 \pm 0.21$ & 2 \\
dSph & & $6.1 \pm 2.2$ & $0.006 \pm 0.005$ & 4,5 \\
\noalign{\smallskip} \hline
\end{tabular}
\begin{list}{}{}
\item[References --] (1) \citet{gri09}; (2) \citet{gri10}; (3) \citet{tre09}; (4) \citet{mar08}; (5) \citet{wol10}
\end{list}
\label{tab6}
\end{table}
\begin{figure}[h]
\centering
\includegraphics[width=0.44\textwidth]{fig05c2.ps}
\caption{Values of the luminous mass $M_{L}$ and effective velocity dispersion $\sigma_{\mathrm{SIE}}$ of the SLACS (grey squares) and G1 and G2 (black diamonds with 1$\sigma$ error bars) lens galaxies. The solid line shows the best-fitting line based only on the values of the lenses of the SLACS sample.}
\label{fi03}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=0.44\textwidth]{fig06c2.ps}
\caption{Values with 1$\sigma$ error bars of the luminous over total mass projected within the effective radius $f_{\mathrm{L}}(<R_{e})$ and luminous mass $M_{L}$ of the G1 and G2 lens galaxies. The dotted and dashed lines show, respectively, the 68\% and 95\% confidence intervals of approximately 2$\times10^{5}$ SDSS early-type galaxies with luminous mass values between $5\times10^{10}$ and $2\times10^{12}M_{\odot}$.}
\label{fi04}
\end{figure}
In this section we compare the values of the luminous mass, effective velocity dispersion, and luminous over total mass fraction projected within the effective radius of G1 and G2 with those of three samples of SLACS lens galaxies, massive early-type galaxies from the SDSS, and dwarf spheroidals.
First, starting from the total and luminous mass estimates derived in the previous two sections, we measure for G1 and G2 the values of the fraction of luminous over total mass projected inside the effective radius, $f_{L}(<R_{e})$, in the following way:
\begin{equation}
f_{L}(<R_{e}) := \frac{M_{L}/2}{M_{T}(<R_{e})} \, .
\label{eq:05}
\end{equation}
We obtain $0.51 \pm 0.21$ and $0.80 \pm 0.32$ for G1 and G2, respectively.
Then, we consider early-type galaxies with physical properties similar to those of the lenses selected by the SLACS survey. For the SLACS galaxies, we use the luminous mass estimates by \citet{gri09}, that are obtained by fitting the galaxy SEDs with CSP models built on \citet{bru03} templates at solar metallicity and with a Salpeter stellar IMF, and the effective velocity dispersion measurements $\sigma_{\mathrm{SIE}}$, presented in \citet{tre09}, that are derived by modeling the total mass distribution of the lenses with SIE profiles. Several studies have shown that the SLACS lens galaxies are an unbiased subsample of the family of SDSS massive early-type galaxies (e.g., \citealt{bol06}; \citealt{gri10}; \citealt{aug10b}), as far as their luminous and mass properties are concerned. For this reason, we also use here the results on the luminous over total mass fractions of approximately 2$\times10^{5}$ SDSS early-type galaxies selected by \citet{gri10}. In this last study, the values of the galaxy luminous and total mass were measured under the same hypotheses adopted in this work. We show the results in Table \ref{tab6} and Figures \ref{fi03} and \ref{fi04}.
Looking at Table \ref{tab6} and Figure \ref{fi03}, we notice that G2 has values of $M_{L}$ and $\sigma_{\mathrm{SIE}}$ that are consistent with those of the galaxies in the SLACS sample. Interestingly, G1 has values of luminous mass and effective stellar velocity dispersion lower by approximately factors of 30 and 3, respectively, than the average SLACS lens galaxy, but these values are in good agreement with the extrapolation of the scaling relation based on the SLACS galaxies only. From the same Table and Figure \ref{fi04}, we observe that G2 has a value of luminous over total mass fraction that is typical of SDSS massive early-type galaxies. The value of $f_{L}(<R_{e})$ of G1 instead is smaller, but still consistent with the lower end of the distribution of the SDSS sample.
Considering the example provided by G1, we can conclude that in clusters of galaxies it is possible to study galaxy strong lensing on physical scales that are different from (i.e., smaller than) those characterizing isolated early-type galaxies (e.g., the SLACS lenses). This possibility is offered by the increase with the overdensity of the environment in the probability of one source to be strongly lensed by a (small) galaxy. In different words, the lensing cross section of a single (small) galaxy can be significantly enhanced by the presence of the mass distributions primarily on the cluster scale and secondarily on the scale of the neighboring cluster members. This opens the way to studies on the internal structure of lens galaxies over a more extended range of physical properties than done so far.
We remark that several studies (e.g., \citealt{hal07}; \citealt{lim07}; \citealt{eic13}) have found evidence for the truncation of the total mass profiles of early-type galaxies residing in galaxy clusters. Nonetheless, for a given galaxy, the value of its truncation radius is estimated to be significantly larger that that of its effective radius. This allows us to disregard, to a first approximation, the possible differences in the values of the stellar over total mass ratios projected within the effective radii of cluster and field early-type galaxies because of their different truncation radii. Moreover, we observe that the results of \citet{tre06} and \citet{gri10b} have shown that the SLACS (i.e., mainly field) and Coma (i.e., cluster) galaxies, with comparable stellar masses, do not differ appreciably as far as their inner total mass structure and stellar IMF are concerned. For these reasons, we consider appropriate to plot the values of the central luminous over total mass ratios of cluster and field galaxies in the same plots, as done in Figures \ref{fi03} and \ref{fi04}.
\begin{figure*}
\centering
\includegraphics[width=0.44\textwidth]{fig05d2.ps}
\includegraphics[width=0.44\textwidth]{fig06d2.ps}
\caption{Same quantities and symbols plotted in Figures \ref{fi03} and \ref{fi04} over ranges of values that include also 11 dwarf spheroidals (grey triangles).}
\label{fi09}
\end{figure*}
Following the previous results, we expand the intervals of physical scales plotted in Figures \ref{fi03} and \ref{fi04} to include a sample of 11 pressure-supported dwarf spheroidals (dSph) for which all the relevant quantities are available in the literature. We take the values of effective radius and luminous mass from \citet{mar08} and those of stellar velocity dispersion averaged along the line of sight $\sigma_{*}$ from \citet{wol10}. Simplistically, we decide to use the same stellar IMF (i.e., Salpeter) adopted to estimate the luminous mass of the previous galaxies and the expression given in Eq. (\ref{eq:04}) to measure the total mass projected within the effective radius. The results are plotted in Fig. \ref{fi09}.
The observed values of luminous mass and velocity dispersion of the dwarf spheroidals do not differ dramatically from the expected values obtained by extrapolating the SLACS scaling relation at several orders of magnitude difference. The projected fractions of luminous over total mass inside the effective radius show instead a clear variation from centrally luminous to dark-matter-dominated systems, moving from massive early-type galaxies to dwarf spheroidals. We speculate that the similarities and differences between these two classes of astrophysical objects might be explored effectively by extending strong lensing analyses to lenses with diverse physical scales, as started here with G1. The CLASH survey seems to be particularly well suited to this aim, as several other interesting systems of strong lensing on galaxy scale have already been discovered and are currently under investigation.
\section{Conclusions}
The combination of unprecedented HST multi-wavelength observations and VLT spectra has allowed us to perform a detailed strong lensing and stellar population analysis of an unusual system composed in total of ten multiple images of a double source, lensed by two early-type galaxies in the field of the CLASH galaxy cluster MACS 1206. Our main results can be summarized in the following points.
\begin{itemize}
\item[$\bullet$] Based on our 16-band photometry and low-resolution spectroscopy, we measure a photometric redshift of 3.7 for the source and spectroscopic redshifts of 0.436 and 0.439 for the two lens galaxies G1 and G2, respectively, thus confirming their membership to MACS 1206.
\item[$\bullet$] By modeling the total mass distribution of the cluster members and cluster in terms of singular isothermal profiles, we can reconstruct well the observed positions of the multiple images and predict a total magnification factor of approximately 50 for the source.
\item[$\bullet$] From the lensing modeling statistics, we estimate effective velocity dispersion values of $97 \pm 3$ and $240 \pm 6$ km s$^{-1}$, corresponding to total mass values projected within the effective radii of $1.7_{-0.1}^{+0.1}\times10^{10}$ and $2.8_{-0.1}^{+0.2}\times10^{11}$~$M_{\odot}$ for G1 and G2, respectively. Moreover, we obtain reasonable values for the distribution and amount of projected total mass in the galaxy cluster component.
\item[$\bullet$] Through composite stellar populations synthesis models (adopting a Salpeter stellar IMF), we infer luminous mass values of $(1.7 \pm 0.7)\times10^{10}$ and $(4.5 \pm 1.8)\times10^{11}$ $M_{\odot}$ for, respectively, G1 and G2, and $(1.0 \pm 0.5)\times10^{9}$ $M_{\odot}$ for the source, taking into account the estimated lensing magnification factor.
\item[$\bullet$] In G1 and G2, respectively, we derive luminous over total mass fractions of $0.51 \pm 0.21$ and $0.80 \pm 0.32$. We compare these values with those typical of massive early-type galaxies and dwarf spheroidals and conclude that more analyses in the CLASH fields of systems similar to that presented here will enable us to extend the investigation of the internal structure of galaxies in an important and still relatively unexplored region of the physical parameter space.
\end{itemize}
\acknowledgments
The CLASH Multi-Cycle Treasury Program is based on observations made with the NASA/ESA {\it Hubble Space Telescope}. The Space Telescope Science Institute is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. ACS was developed under NASA Contract NAS 5-32864. This research is supported in part by NASA Grant HST-GO-12065.01-A. We thank ESO for the continuous support of the Large Programme 186.A-0798. The Dark Cosmology Centre is funded by the DNRF. We acknowledge partial support by the DFG Cluster of Excellence Origin Structure of the Universe. V.P. acknowledges the grant PRIN INAF 2010 and ``Cofinanziamento di Ateneo 2010''. The work of L.A.M. was carried out at Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA. Support for A.Z. is provided by NASA through Hubble Fellowship grant HST-HF-51334.01-A awarded by STScI. Part of this work was also supported by contract research ``Internationale Spitzenforschung II/2-6'' of the Baden W\"urttemberg Stiftung.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,710 |
\subsection{A faster algorithm for binary trees}\label{sec:binary_upper}
For trees with degree at most two, the Edmonds-Matula procedure can be interpreted as follows. Let $H_L$ and $H_R$ be the left and right subtrees of $H$, and let $G_L$ and $G_R$ be the left and right subtrees of $G$. $H$ is isomorphic to a subtree of $G$ if and only if one of the following two conditions are true:
\begin{enumerate}
\item $H_L$ is isomorphic to a subtree of $G_L$, and $H_R$ is isomorphic to a subtree of $G_R$.
\item $H_L$ is isomorphic to a subtree of $G_R$, and $H_R$ is isomorphic to a subtree of $G_L$.
\end{enumerate}
Each case can be checked with two recursive calls, and checking whether $H$ is isomorphic to a subtree of $G$ can thus be done with at most four recursive calls, giving an $O(4^h)$ upper bound.
Observe that if $H_L$ is not isomorphic to a subtree of $G_L$, then there is no reason to check whether $H_R$ is isomorphic to a subtree of $G_R$. Similarly, if the algorithm concludes that the first condition is met, then there is no reason to check the second condition since we already know that $H$ is isomorphic to a subtree of $G$. Based on these observations, we introduce a simple randomized variant of the algorithm that achieves a significantly better running time by saving recursive calls: Swap $H_L$ and $H_R$ with probability $1/2$, and swap $G_L$ and $G_R$ with probability $1/2$. Then run the Edmonds-Matula algorithm, but do not perform unnecessary recursive calls. We give a formal description of the algorithm in Figure \ref{fig:alg}. We refer to the algorithm as $RandBinarySubIso$.
\begin{figure}[t]
\let\@latex@error\@gobble
\begin{center}
\parbox{\columnwidth}{
\SetAlgoFuncName{Algorithm}{anautorefname}
\begin{function}[H]
\caption{RandBinarySubIso($H,G$)}
\begin{enumerate}
\item If $|H|=0$, return \textbf{true};
\item If $|G|=0$, return \textbf{false};
\item With probability $1/2$ swap $H_L$ and $H_R$ in $H$;
\item With probability $1/2$ swap $G_L$ and $G_R$ in $G$;
\item If $RandBinarySubIso(H_L,G_L) = \textbf{false}$, then go to step 7;
\item If $RandBinarySubIso(H_R,G_R) = \textbf{true}$, then return \textbf{true};
\item If $RandBinarySubIso(H_L,G_R) = \textbf{false}$, then return \textbf{false};
\item If $RandBinarySubIso(H_R,G_L) = \textbf{true}$, then return \textbf{true}. \\Otherwise return \textbf{false};
\end{enumerate}
\vspace{-0.2cm}
\end{function}
}
\end{center}
\caption{A randomized, recursive algorithm for rooted Subtree Isomorphism on binary trees.}
\label{fig:alg}
\end{figure}
\begin{theorem}\label{heightthm_re}
The $RandBinarySubIso$ algorithm runs in expected time $O(\min\{2.8431^h,n^2\})$ for trees $H$ and $G$ of size $O(n)$ and height at most $h$. In particular, it runs in time $O(n^{1.507})$ for trees of height $(1+o(1))\cdot \log_2{n}$, and is strongly subquadratic for trees of height $h<1.3267\log_2{n}$.
\end{theorem}
Before proving Theorem \ref{heightthm_re} we first prove a useful lemma.
Let $T(h)$ be the maximum expected number of times $RandBinarySubIso(H,G)$ makes a recursive call with an empty tree when $H$ and $G$ are arbitrary rooted trees with height at most $h$. Let $T_{yes}(h)$ and $T_{no}(h)$ be defined similarly, but under the assumption that the algorithm returns \textbf{true} and \textbf{false}, respectively. Note that $T(0) = T_{yes}(0) = T_{no}(0) = 1$. Also note that $T(h) = \max\{T_{yes}(h),\; T_{no}(h)\}$.
\begin{lemma} \label{rec}
For all $h \geq 0$,
\begin{align*}
T_{yes}(h)&~\leq~ 2.25\cdot T_{yes}(h-1) + 0.5\cdot T_{no}(h-1)~,\\
T_{no}(h)&~\leq~ T_{yes}(h-1) + 2\cdot T_{no}(h-1)~.
\end{align*}
\end{lemma}
\begin{proof}
To simplify notation we write $H \subseteq G$ when $H$ is isomorphic to a subtree of $G$, and $H \not\subseteq G$ otherwise.
We first show that $T_{yes}(h)\leq 2.25\cdot T_{yes}(h-1) + 0.5\cdot T_{no}(h-1)$. Assume therefore that $H \subseteq G$. With probability $1/2$ we then have $H_L \subseteq G_L$ and $H_R \subseteq G_R$, such that the algorithm returns \textbf{true} in line 6 after spending $2\cdot T_{yes}(h-1)$ time in expectation. On the other hand, with probability $1/2$ the outcomes of lines 5 and 6 depend on the trees in question, and the recursive calls in lines 7 and 8 both return \textbf{true} if reached. More precisely, we get three cases that depend on the trees:
\begin{itemize}
\item[$(i)$]
$H_L \subseteq G_L$ and $H_R \subseteq G_R$: The recursive calls in lines 5 and 6 both return \textbf{true}, and the algorithm spends $2\cdot T_{yes}(h-1)$ time in expectation.
\item[$(ii)$]
$H_L \not\subseteq G_L$ and $H_R \not\subseteq G_R$: The recursive call in line 5 returns \textbf{false}, and the recursive calls in lines 7 and 8 both return \textbf{true}. The algorithm spends $T_{no}(h-1)+2\cdot T_{yes}(h-1)$ time in expectation.
\item[$(iii)$]
$H_L \subseteq G_L$ and $H_R \not\subseteq G_R$, or $H_L \not\subseteq G_L$ and $H_R \subseteq G_R$: The recursive call in line 5 returns \textbf{false} with probability $1/2$ and \textbf{true} with probability $1/2$. In the second case the recursive call in line 6 returns \textbf{false}. The recursive calls in lines 7 and 8 both return \textbf{true}. The algorithm spends $T_{no}(h-1)+2.5\cdot T_{yes}(h-1)$ time in expectation.
\end{itemize}
The third case thus dominates the two others, and we conclude that $T_{yes}(h)\leq 2.25\cdot T_{yes}(h-1) + 0.5\cdot T_{no}(h-1)$.
We next show that $T_{no}(h)\leq T_{yes}(h-1) + 2\cdot T_{no}(h-1)$. Assume therefore that $H \not\subseteq G$. We get the contribution $2\cdot T_{no}(h-1)$ as follows. In either line $5$ or $6$ we get the answer \textbf{false} from a recursive call, and in either line $7$ or $8$ we also get the answer \textbf{false} from a recursive call. This amounts to two ``no'' answers which cost $2\cdot T_{no}(h-1)$ in expectation. We get the contribution $T_{yes}(h-1)$ as follows. With probability at most $1/2$ we get the answer \textbf{true} in line $5$ (which means that we get \textbf{false} in line 6). Similarly, with probability at most $1/2$ we get the answer \textbf{true} in line $7$ (which means that we get \textbf{false} in line 8). In total, we get that $T_{no}(h)\leq 2\cdot T_{no}(h-1)+{1 \over 2}T_{yes}(h-1)+{1 \over 2}T_{yes}(h-1)$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{heightthm_re}]
Lemma \ref{rec} gives us that
\begin{align*}
\left( \begin{array}{c}
T_{yes}(h) \\
T_{no}(h)
\end{array}
\right)
&~\leq~
\left( \begin{array}{cc}
2.25 & 0.5 \\
1 & 2
\end{array}
\right)
\left( \begin{array}{c}
T_{yes}(h-1) \\
T_{no}(h-1)
\end{array}
\right)\\
&~\leq~
\left( \begin{array}{cc}
2.25 & 0.5 \\
1 & 2
\end{array}
\right)^h
\left( \begin{array}{c}
1 \\
1
\end{array}
\right) ~.
\end{align*}
A diagonalization of the matrix yields
$$
\left( \begin{array}{cc}
2.25 & 0.5 \\
1 & 2
\end{array}
\right)
~=~ Q^{-1}JQ~,
$$
where
\begin{align*}
Q^{-1} &~=~ \left( \begin{array}{cc}
{1 - \sqrt{33} \over 8} & {1 - \sqrt{33} \over 8} \\
1 & 1
\end{array}
\right)\\
J &~=~
\left( \begin{array}{cc}
{17 - \sqrt{33} \over 8} & 0 \\
0 & {17 + \sqrt{33} \over 8}
\end{array}
\right)\\
Q &~=~
\left( \begin{array}{cc}
-{4\over \sqrt{33}} & {1 \over 2}+{1 \over 2\sqrt{33}} \\
{4\over \sqrt{33}} & {1 \over 2}-{1 \over 2\sqrt{33}}
\end{array}
\right) ~,
\end{align*}
and therefore
\[
\left(\begin{array}{c}
\!T_{yes}(h)\! \\
\!T_{no}(h)\!
\end{array}\right)
\,\leq\,
\left(\begin{array}{c}
\!0.065\cdot 1.407^h+0.94 \cdot 2.8431^h\! \\
\!-0.109 \cdot 1.407^h+1.109 \cdot 2.8431^h\!
\end{array}\right)\,.
\]
Thus,
$T(h)=O(2.8431^h)$, which proves the theorem.
\end{proof}
\subsection{A Faster Algorithm for Ternary Trees}
\input{deg3case}
\subsection{An algorithm for any constant degree}\label{sec:any_upper}
In this section we describe a way to use randomization to save subtree comparisons in the Edmonds-Matula algorithm \cite{Matula1978} for all degrees $d > 2$. Recall that the algorithm works as follows.
Given two trees $H$ and $G$ of constant degree $d$, the goal is to decides whether $H$ is isomorphic to a subtree of $G$ by using recursion. If the roots of either $H$ or $G$ have less than $d$ children, we simply view the missing subtrees as being a special empty subtree.
\begin{enumerate}
\item Let $H_1,\ldots,H_d$ be the $d$ subtrees of $H$, and let $G_1,\ldots,G_d$ be the $d$ subtrees of $G$;
\item Build a bipartite graph $\mathcal{G}$ with $d$ vertices $\mathcal{U} = \{u_1,\ldots,u_d\}$ on the left and $d$ vertices $\mathcal{W} = \{v_1,\ldots,v_d\}$ on the right. For all $i,j \in [d]$, connect $u_i$ and $v_j$ if and only if $H_i$ is isomorphic to a subtree of $G_j$. We decide which edges appear in the graph recursively.
\item Output that $H$ is isomorphic to a subtree of $G$ if and only if there is a perfect matching in the bipartite graph $\mathcal{G}$.
\end{enumerate}
The runtime of the algorithm is $O(\min\{d^{2h},n^2\})$, where $h$ is the height.
Intuitively, we can improve the runtime of the algorithm as follows. Perform recursive calls corresponding to edges $(u_i,v_j)$ in a random order, and stop as soon as we either detect a perfect matching or rule out the existence of a perfect matching. It is not difficult to show that this randomized version of the algorithm performs $d^2-\Omega(1)$ recursive calls in expectation out of the $d^2$ possible calls. That is, in expectation, we save at least a constant number of recursive calls. This implies that the algorithm runs in $O((d^2-\Omega(1))^h)$ expected time, which is faster than the deterministic algorithm. However, we prove below that we can save $\Omega(d)$ recursive calls in expectation using a slightly different variant of the randomized algorithm.
\begin{lemma} \label{query}
Let $\mathcal{G}$ be a bipartite graph with $d$ vertices $\mathcal{U} = \{u_1,\ldots,u_d\}$ on the left and $d$ vertices $\mathcal{W} = \{v_1,\ldots,v_d\}$ on the right, and suppose we are given query access to the adjacency matrix of $\mathcal{G}$. There is a randomized query algorithm that decides whether $\mathcal{G}$ contains a perfect matching by making $d^2-{1 \over 3}d+{2 \over 3}$ queries in expectation, with probability 0 of making an error.
\end{lemma}
We use the following two claims to prove the lemma.
\begin{claim} \label{yescase}
Assume that $\mathcal{G}$ has a perfect matching. Then the following algorithm finds a perfect matching after making $d^2-d+2$ expected queries: Query edges $(u_i,v_j)$ in a random order, and stop when finding a perfect matching.
\end{claim}
\begin{proof}
Fix a perfect matching present in $\mathcal{G}$ and call its $d$ edges ``marked''.
We stop when all marked edges have been queried. There are $d^2-d$ unmarked edges. The probability that a given unmarked edge is not queried is $1 \over d+1$. Therefore, the expected number of unqueried, unmarked edges is ${d^2-d \over d+1}\geq d-2$.
\end{proof}
\begin{claim} \label{nocase}
Assume that $\mathcal{G}$ does not have a perfect matching. Then the following algorithm makes at most $d^2-{1 \over 2}d+1$ queries in expectation before determining that $\mathcal{G}$ does not contain a perfect matching.
\begin{enumerate}
\item With probability $1/2$ swap $\mathcal{U}$ and $\mathcal{W}$;
\item Randomly permute the vertices of $\mathcal{U}=\{u_1,u_2,\ldots,u_d\}$;
\item Query all edges adjacent to $u_i$ for $i$ going from 1 to $d$, but stop when ruling out the existence of a perfect matching, i.e., stop when the set of processed vertices $S=\{u_1,\ldots,u_i\}$ contains a subset $S'$ with a neighbourhood $N(S')$ that is smaller than the size of $S'$.
\end{enumerate}
\end{claim}
\begin{proof}
Consider the sets $\mathcal{U}$ and $\mathcal{W}$ prior to running the algorithm.
By Hall's theorem, the set $\mathcal{U}$ contains a set $S'$ such that $|N(S')|<|S'|$. We can assume that $|S'|=|N(S')|+1$, since otherwise we can iteratively remove a vertex from $S'$ until this condition is satisfied. Consider two cases.
\begin{itemize}
\item $d$ is even: If $|S'|\geq {d \over 2}+1$, we define $T'=\mathcal{W} \setminus N(S')$. Because $N(S')\geq d/2$, we get that $|T'|\leq {d \over 2}$. By our construction of $T'$, we have that $N(T') \subseteq \mathcal{U} \setminus S'$ and, as a result, $|N(T')|<|T'|$. Given the first step of the algorithm, with probability at least $1/2$ the set $\mathcal{U}$ therefore contains a set $S'$ such that $|N(S')|<|S'|\leq {d \over 2}$.
\item $d$ is odd: It follows from as similar argument that, with probability at least $1/2$, the set $\mathcal{U}$ contains a set $S'$ such that $|N(S')|<|S'|\leq {d+1 \over 2}$.
\end{itemize}
We now condition on the set $\mathcal{U}$ containing $S'$ with $|S'|\leq {d+1 \over 2}$ and $|N(S')|<|S'|$.
The algorithm stops once it queries all vertices from $S'$, since a perfect matching is then ruled out by Hall's theorem. The probability that we do not process a given vertex before processing all vertices in $S'$ is ${1/(|S'|+1)}$. Therefore the expected number of unprocessed vertices when the algorithm stops is at least
$$
(d-|S'|)\cdot {1 \over |S'|+1}~\geq~ {d-1 \over 2}\cdot {1 \over {d+1 \over 2}+1}~=~{d-1 \over d+3}~.
$$
Hence, with probability $1/2$, we query $d \left(d-{d-1 \over d+3}\right)$ edges, and overall the number of queried edges is
$$
{1 \over 2}\left[d \left(d-{d-1 \over d+3}\right)\right]+{1 \over 2}d^2~=~d^2\left(1-{1-{1 \over d} \over 2(d+3)}\right)
~\leq~ d^2-{1 \over 2}d+1~.
$$
In the last inequality we use that $d \geq 3$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{query}]
We prove the lemma by using claims \ref{yescase} and \ref{nocase}.
With probability $1/3$ we run the algorithm from Claim \ref{yescase} and with probability $2/3$ we run the algorithm from Claim \ref{nocase}.
Consider the case when $\mathcal{G}$ has a perfect matching. Then the expected number of edges queried is upper bounded by
$$
{1 \over 3}(d^2-d+2)+{2 \over 3}d^2~=~d^2-{1 \over 3}d+{2 \over 3}~.
$$
On the other hand, for the case when $\mathcal{G}$ does not contain a perfect matching, the expected number of edges queried is upper bounded by
$$
{1 \over 3}d^2+{2 \over 3}\left(d^2-{1 \over 2}d+1\right)~=~d^2-{1 \over 3}d+{2 \over 3}~.
$$
Overall, regardless of $\mathcal{G}$, we therefore query at most $d^2-{1 \over 3}d+{2 \over 3}$ edges in expectation.
\end{proof}
\begin{theorem} \label{constanddegree_re}
There is a randomized algorithm that solves Subtree Isomorphism on two rooted trees of size $O(n)$, constant degree $d$, and height at most $h$ in expected time $O\left(\left(d^2-\frac{1}{3} d+{2 \over 3}\right)^h \right)$.
In particular, the algorithm is strongly subquadratic for trees of height
$$
h~\leq~ \left(\frac{2 \log d}{\log(d^2-\frac{1}{3} d+{2 \over 3})}-\epsilon\right) \cdot \log_d{n}~,
$$
for any constant $\epsilon>0$.
\end{theorem}
\begin{proof}
We run the following randomized, recursive algorithm that decides whether $H$ is isomorphic to a subtree of $G$.
\begin{enumerate}
\item Let $H_1,\ldots,H_d$ be the $d$ subtrees of $H$, and let $G_1,\ldots,G_d$ be the $d$ subtrees of $G$;
\item Let $\mathcal{G}$ be a bipartite graph with $d$ vertices $\mathcal{U} = \{u_1,\ldots,u_d\}$ on the left and $d$ vertices $\mathcal{W} = \{v_1,\ldots,v_d\}$ on the right. For all $i,j \in [d]$, let $u_i$ and $v_j$ be connected if and only if $H_i$ is isomorphic to a subtree of $G_j$.
\item Decide whether the graph $\mathcal{G}$ has a perfect matching by running the algorithm from Lemma \ref{query}. Whenever we need to decide whether an edge $(u_i,v_j)$ is present in $\mathcal{G}$, do it recursively.
\end{enumerate}
By the proof of Lemma \ref{query}, it suffices to query $d^2-\frac{1}{3} d+{2 \over 3}$ edges for every level. Given that the height of the trees is upper bounded by $h$, we get the desired running time.
\end{proof}
\subsection{Hardness for Subtree Isomorphism}
\paragraph{A simpler reduction.}
We start with a ``warm-up" reduction that presents the high-level idea of our proofs.
In Theorem \ref{th:unbounded} below we reduce OV to Subtree isomorphism on trees with $n=O(ND)$ vertices, unbounded degree, and height $h=O(D)$. We later show how to change the construction to get trees with small constant degree and small height.
\begin{theorem} \label{th:unbounded}
Orthogonal Vectors on two lists of $N$ vectors in $\{0,1\}^D$ can be reduced to Subtree Isomorphism on two trees of size $O(ND)$ and depth $O(d)$.
\end{theorem}
\begin{proof}
Let us denote the vectors of the first list by $A=\{ \alpha_1,\ldots,\alpha_N \}$ and of the second list by $B = \{ \beta_1,\ldots,\beta_N\}$ and recall that our goal is to find a pair of vectors $\alpha\in A, \beta\in B$ such that for every coordinate $i \in [D]$ either $\alpha[i]=0$ or $\beta[i]=0$.
The first ingredient in the reduction is to construct \emph{vector gadgets}.
For every vector in the first list $\alpha \in A$ we create a vector gadget: a tree $H_\alpha$ of size $O(D)$ as follows.
First, add a path $u_0 \to u_1 \to u_2 \to \cdots \to u_{D+2}$ and let $u_0$ be the root of $H_\alpha$.
Then, for each coordinate $i \in [D]$ we consider $\alpha[i]$ and if it is a $1$ we add a node $u_{i,1}$ to the tree $H_\alpha$ as the child of the node $u_i$, i.e. we add the edge $u_i \to u_{i,1}$.
Otherwise, if $\alpha[i]=0$, the only child of $u_i$ will be $u_{i+1}$.
We now define the vector gadgets for the vectors in the second list.
For every $\beta \in B$ we create a vector gadget: a tree $G_\beta$ of size $O(D)$ as follows.
The first step is similar, we add a path $v_0 \to v_1 \to v_2 \to \cdots \to v_{D+2}$ and let $v_0$ be the root.
The difference is in the second step.
For each coordinate $i \in [D]$, we consider $\beta[i]$ and if it is a $0$ we add a node $v_{i,0}$ to $G_\beta$ as the child of the node $v_i$, i.e. we add the edge $v_i \to v_{i,0}$.
The following simple claim is the key to our reduction and explains our gadget constructions.
\begin{claim}
\label{cl1}
$H_\alpha$ is isomorphic to $G_\beta$ iff $\alpha,\beta$ are orthogonal.
\end{claim}
\begin{proof}
For the first direction, assume that $\alpha,\beta$ are orthogonal and therefore for every $i \in [D]$ we know that either $\alpha[i]=0$ or $\beta[i]=0$.
We will define a mapping $f$ from $H_\alpha$ to a subgraph of $G_\beta$ such that if $\{u,v\}$ is an edge in $H_\alpha$ then $\{ f(u),f(v)\}$ is an edge in $G_\beta$.
First, we map the roots and paths to each other, by setting $f(u_i)=v_i$ for all $i \in \{0,\ldots,D+2\}$.
Then, we consider every $i \in [D]$ for which $\alpha[i]=1$ and map $u_{i,1}$ to the node $v_{i,0}$ in $G_\beta$.
We are guaranteed that $v_{i,0}$ exists because if $\alpha[i]=1$ then $\beta[i]$ must be $0$, by the orthogonality of the vectors.
It is easy to check that two neighbours in $H_\alpha$ are mapped to two neighbours in $G_\beta$.
For the other direction, assume $H_\alpha$ is isomorphic to a subgraph of $G_\beta$, and let $f$ be the mapping.
First, note that $u_0$ must be mapped to $v_0$ since these are the roots of the two trees.
Then we observe that $u_{D+2}$ must be mapped to $v_{D+2}$ and the path $u_0\to \cdots \to u_{D+2}$ must be mapped to the path $v_0 \to \cdots \to v_{D+2}$ since these are the only paths of length at least $(D+2)$ in the trees.
Now, let $i \in [D]$ be such that $\alpha[i]=1$ and note that $u_i$ must have degree $3$ in this case, which implies that $f(u_i)=v_i$ must also have degree at least $3$ in $G_\beta$, which implies that the node $v_{i,0}$ must exist, and $\beta[i]=0$.
Thus, whenever $\alpha[i]=1$ it must be the case that $\beta[i]=0$, and the vectors are orthogonal.
\end{proof}
The final step is to combine the vector gadgets into two trees $H,G$ in a way such that $H$ is isomorphic to a subtree of $G$ if and only if there is a pair of orthogonal vectors within our two lists.
To this end, we define a special vector $\gamma = \vec{0}$ to be the all-zero vector in $D$ dimensions.
By Claim~\ref{cl1}, for any vector $\beta \in \{0,1\}^D$, we have that $H_\beta$ is isomorphic to a subtree of $G_\gamma$.
We are now ready to define the trees $H$ and $G$ of size $O(ND)$.
$G$ will be composed of a root node $g$ of degree $(2N-1)$ that has $G_{\beta_j}$ as a child for every $\beta_j \in B$, in addition to $(N-1)$ distinct $G_\gamma$ gadgets.
That is, first, for each $j \in [N]$ add the vector gadget $G_{\beta_j}$ to $G$ and add the edge $g \to v_0$ where $v_0$ is the root of $G_{\beta_j}$.
And then, we add $(N-1)$ trees $G_\gamma^{(1)},\ldots, G_\gamma^{(n-1)}$ to $G$ and for each $j \in [N-1]$ we add the edge $g\to v_0$ where $v_0$ is the root of $G_\gamma^{(j)}$.
$H$ will be constructed in a similar way, except we do not add the $\gamma$ vector gadgets.
Create a root node $h$ of degree $N$ that has $H_{\alpha_j}$ as a child for every $\alpha_j \in A$.
As in the definition of $G$, we add edges $h \to u_0$ where $u_0$ is the root of $H_{\alpha_j}$, for every $j \in [N]$.
Before proving the correctness of the reduction, note that the size of each tree is indeed $O(ND)$ since each gadget has size $O(D)$ and we are combining $O(N)$ gadgets into our trees $H,G$.
To conclude the proof, we claim that $H$ is isomorphic to a subgraph of $G$ iff there is a pair of orthogonal vectors.
\begin{claim}
In the above reduction, $H$ is isomorphic to a subtree of $G$ iff there is a pair $\alpha \in A, \beta \in B$ of orthogonal vectors.
\end{claim}
\begin{proof}
For the first direction, assume that there is a pair of orthogonal vectors $\alpha \in A, \beta \in B$ and we will show that $H$ is isomorphic to a subtree of $G$.
Consider the mapping which maps $H_\alpha$ to $G_\beta$ as in Claim~\ref{cl1}, and then for each of the $(N-1)$ $H_{\alpha'}$ subtrees, for $\alpha' \neq \alpha$, we map it to a different $G_\gamma$ subtree of $G$.
Finally, the root $h$ is mapped to $g$.
It is easy to check that neighbours in $H$ are mapped to neighbours in $G$.
For the other direction, assume that $H$ is isomorphic to a subgraph of $G$ and let $f$ be the corresponding mapping.
We know that $f(h)=g$ and for each vector gadget $H_{\alpha_j}$ in $H$, its image using our mapping $f$ must be entirely contained in exactly one vector gadget $G_{x}$ in $G$, where $x \in B \cup \{ \gamma \}$.
Moreover, two gadgets $H_{\alpha},H_{\alpha'}$ cannot be mapped to the same gadget $G_x$.
There are $N$ $H_\alpha$ gadgets but only $(N-1)$ $G_\gamma$ gadgets, thus, by the pigeonhole principle, there must be at least one $\alpha \in A$ for which $H_\alpha$ is mapped to a gadget $G_{x}$ for $x \neq \gamma$, i.e., $x = \beta$ for some $\beta \in B$.
We conclude that there is a mapping from $H_\alpha$ to $G_\beta$ in which every two neighbours are mapped to neighbours, that is, that $H_\alpha$ is isomorphic to a subgraph of $G_\beta$, which, by Claim~\ref{cl1}, implies that $\alpha \in A,\beta \in B$ are orthogonal.
\end{proof}
\end{proof}
\paragraph{Shorter Vector Gadgets.}
Next, we show how our reductions can be implemented with trees of smaller depth, by introducing a new construction of vector gadgets.
We will use these gadgets in our final reductions that prove Theorems~\ref{thm:lb} and~\ref{thm:LCST}.
\begin{lemma} \label{vector_gadgets}
Given two vectors $\alpha,\beta \in \{0,1\}^D$ we can construct two binary rooted trees $H_\alpha,G_\beta$ of depth $3\log_2(D)+O(1)$ in linear time, such that $H_\alpha$ is isomorphic to a subtree of $G_\beta$ if and only if $\alpha,\beta$ are orthogonal.
\end{lemma}
\begin{proof}
Our constructions will involve careful combinations of ``index gadgets", which are defined as follows.
For a sequence of $\ell$ binary values $b_1, b_2, \ldots, b_l$, we define a tree ``index gadget" $Q_{b_1, b_2, \ldots, b_l}$ (think of $\ell$ as being $\lceil \log_2(D+1)\rceil$ and think of $b_1, b_2, \ldots, b_l$ as bits representing an index in $[D]$) to be composed of a path $z_1 \to z_2 \to ... \to z_l$ of length $l$, in which $z_1$ is the root, and for all $i\in[l]$ we attach a child $z_{i,1}$ to $z_i$ if and only if $b_i=1$.
That is, our index gadget $Q_{b_1, b_2, \ldots, b_l}$ is representing the index in the natural way: the edge $z_i \to z_{i,1}$ will exist if and only if $b_i=1$.
Our first ``vector gadget" $H_\alpha$ is constructed as follows.
First, we build a complete binary tree with $D$ leaves $u_1, u_2, \ldots, u_D$ where the subtree at each leaf $u_i$ will encode the entry $\alpha[i]$ using our ``index gadgets".
We assume that every index $i\in [D]$ can be represented by $l=\lceil \log_2(D+1)\rceil$ bits and we let ${\bar i}$ denote this representation and let ${\bar i}^S$ denote the binary sequence obtained by flipping each bit of ${\bar i}$.
For each node $u_i$ we will attach three gadgets, one after the other: first we will attach the $Q_{\bar i}$ index gadget, then we follow it by the $Q_{{\bar i}^S}$ index gadget, and finally we append a path of length either $2$ or $3$ -- depending on $\alpha[i]$.
The necessity of this complicated encoding will become clear in the proof of correctness below.
More formally, we first attach $u_i \to Q_{\bar i}$, then we let $z_l'$ denote the node of $Q_{\bar i}$ corresponding to $z_l$ in the above construction (i.e. the last node on the path), and attach $z_l' \to Q_{{\bar i}^S}$.
Then, similarly, we let $z_l''$ be the node of $Q_{{\bar i}^S}$ which corresponds to $z_l$ in the above construction (i.e. the last node on the path), and we either attach three nodes $z_l''\to a_i \to b_i \to c_i$ if $\alpha[i]=1$, or we attach only two nodes $z_l'' \to a_i \to b_i$.
The second ``vector gadget" $G_\beta$ is constructed in the same way except that we attach a path of length $3$ if $\beta[i]=0$ (as opposed to $1$) and attach a path of length $2$ if $\beta[i]=1$.
By construction, the depth of both trees is $3\log_2(D)+O(1)$ as claimed.
To complete the proof we show that $H_\alpha$ is isomorphic to a subtree of $G_\beta$ iff $\alpha \cdot \beta=0$.
The first direction is easy: if the vectors are orthogonal then the natural mapping from $H_\alpha$ to $G_\beta$ that follows from our construction shows the isomorphism: map the binary trees on top to each other so that the $u_i$'s are mapped to each other, then map the attached $Q_{\bar i}\to Q_{{\bar i}^S}$ subtrees to each other, and finally, we can map the paths $a_i \to b_i \to c_i$ (if $\alpha[i]=1$) or $a_i \to b_i $ (if it is $0$) to each other since in the first case $\beta[i]$ must be zero and $c_i$ will also exist in $G_\beta$.
It remains to show that if $H_\alpha$ is isomorphic to a subtree of $G_\beta$, then $\alpha \cdot \beta=0$. Our index gadgets $Q_{\bar i}$ and $Q_{{\bar i}^S}$ will play a crucial role in this part, as they will show that in any mapping between the leaves of the complete tree we must map $u_i$ in $H_\alpha$ to $u_i$ in $G_\beta$ or else the index gadgets will not map into each other properly.
We claim that for any two indices $i,j \in [D]$ we have that $i=j$ if and only if both $Q_{\bar i}$ is contained in $Q_{\bar j}$ \emph{and} $Q_{{\bar i}^S}$ is contained in $Q_{{\bar j}^S}$.
This is true because of the following observation: $Q_{\bar x}$ is isomorphic to a subtree of $Q_{\bar y}$ iff the set of positions in $\bar x$ with $1$ is a subset of the set of positions of $\bar y$ with $1$.
Therefore, any mapping from $H_\alpha$ to a subtree of $G_\beta$ must map the path representing $\alpha[i]$ to the path representing $\beta[i]$, for all $i \in [D]$. By construction, this can only happen if $\alpha\cdot \beta =0$.
\end{proof}
\paragraph{Constant Degree Trees.}
Perhaps the most challenging element towards the proof of Theorem~\ref{thm:lb} is the combination of all the vector gadgets into two big trees, \emph{without using large degrees}.
To see the difficulty, recall the reduction in the proof of Theorem~\ref{th:unbounded}: in both trees, we added all $X$ vector gadgets as children of a root of degree $X$.
By doing so we have essentially allowed the isomorphism to pick \emph{any} matching between the gadgets.
Combined with the auxiliary gadgets that we added, this allowed us to show that the final two trees are a ``yes" instance of Subtree Isomorphism if and only if the original vectors contained an orthogonal pair.
However, when the trees have constant degree (say, binary) it is much harder to combine the vector gadgets into two trees such that any matching between the gadgets can be chosen by the isomorphism.
A natural approach would be to add the gadgets at the leaves of a complete binary tree.
One reason this does not work is that any isomorphism must map the first and second gadgets to adjacent gadgets in the second tree -- that is, only special kinds of matchings can be ``implemented".
We overcome this difficulty with a two-level construction that allows the isomorphism to pick exactly one gadget from each of the two trees and ``match" them, while all the other gadgets do not affect the outcome.
\begin{theorem} \label{th:bounded}
Given sets of vectors $A,B$, we can construct two rooted trees $H=H(A)$ and $G=G(B)$ such that the following properties hold.
\begin{enumerate}
\item The number of nodes in both trees and the construction time is upper bounded by $O(ND)$.
\item The degree of both trees is upper bounded by $d$.
\item The depth of both trees is upper bounded by $2\log_d(N)+O(\log D)$.
\item $H$ is isomorphic to a subtree of $G$ iff there are $\alpha \in A$ and $\beta \in B$ with $\alpha \cdot \beta=0$.
\end{enumerate}
\end{theorem}
\begin{proof}
Let $\{H_\alpha\}_{\alpha \in A}=\{H_{\alpha_i}\}_{i \in [N]}$ and $\{G_\beta\}_{\beta \in B}=\{G_{\beta_i}\}_{i \in [N]}$ be the two sets of vector gadgets corresponding to the vectors of $A$ and $B$ that are obtained by the construction in Lemma~\ref{vector_gadgets}.
We will now combine these vector gadgets into two big trees $H$ and $G$, which will be constructed quite differently from each other.
Assume that $\log_d(N)$ is an integer, otherwise add dummy vectors to increase $N$.
The first tree $H$ will be composed of a complete $d$-ary tree with $N$ leaves $u_1, u_2, \ldots u_N$, followed by a path of length $\log_d(N)+1$, followed by the vector gadgets $H_{\alpha_i}$.
More formally, for every $i \in [N]$ we add: $$ u_i \to h_{i,1} \to h_{i,2} \to \ldots \to h_{i,\log_d(N)+1} \to H_{\alpha_i}.$$
To construct the second tree $G$ we need to construct vector gadgets $G_\gamma$ corresponding to the all-zero vector $\gamma=\vec{0}$ of length $D$.
As before, we start with a complete $d$-ary tree with $N$ leaves $v_1, v_2, \ldots v_N$ and attach a path of length $\log_d(N)+1$ to each leaf, except for $v_N$ which will be treated differently. Then, we attach a copy of $G_\gamma$ at the end of each one of these paths, that is $N-1$ copies in total.
Formally, for every $i=1,\ldots,N-1$ we add:
$$
v_i \to h_{i,1} \to h_{i,2} \to \ldots \to h_{i,\log_d(N)+1} \to G_\gamma.
$$
Note that none of the vectors in the second list are encoded in this part of $G$ and they will appear now in the subtree rooted at $v_N$ which we construct next.
Rooted at $v_N$, we add another complete $d$-ary tree with $N$ leaves $v_1', v_2', \ldots v_N'$, and then attach the vector gadgets right after these leaves. That is, for every $i \in [N]$ we add: $v_i' \to G_{\beta_i}$.
This finishes the construction of $H$ and $G$ and the first two properties are immediate.
The third property follows from Lemma \ref{vector_gadgets}, and we now turn to proving the fourth property which is the correctness of our construction.
\begin{claim}
There is a pair of vectors $\alpha \in A$ and $\beta \in B$ with $\alpha\cdot \beta =0$ if and only if $H$ is isomorphic to a subtree of $G$.
\end{claim}
\begin{proof}
For the first direction, let $\alpha_i$ and $\beta_j$ be a pair of orthogonal vectors and we will show that $H$ is contained in $G$.
First, consider the rearrangement of $H$ so that the rightmost leaf of the complete $d$-ary tree (where $u_N$ used to be) is $u_i$, the node to which the vector gadget $H_{\alpha_i}$ is attached.
We claim that all vector gadgets in $H$ can now be properly mapped to subtrees of $G$, without rearranging the $v_i$ nodes in $G$.
To see this, first note that all vector gadgets $H_{\alpha_{x}}$ for $x \neq i$ will be paired up with the $G_\gamma$ vector gadgets, and by Lemma \ref{vector_gadgets} and the fact that $\gamma$ is orthogonal to any vector, we know that there is a proper mapping.
Then, it remains to show that the subtree of $H$ rooted at $u_i$ is contained in the subtree of $G$ rooted at $v_N$, which follows because we can map the vector gadget $H_{\alpha_i}$ to the vector gadget $G_{\beta_j}$ since $\alpha_i\cdot\beta_j=0$.
For the second direction, assume that there is a mapping from $H$ to a subtree of $G$ and we will show that there must exist a pair of orthogonal vectors.
First, note that under this mapping, there is some $i \in [N]$ such that $u_i$ is mapped to $v_N$.
By construction of the subtree rooted at $v_N$, this means that the vector gadget $H_{\alpha_i}$ must be mapped into one of the vector gadgets $G_{\beta_j}$ for some $j \in [N]$, and not into $G_\gamma$.
By Lemma~\ref{vector_gadgets}, this can only happen if $\alpha_i \cdot \beta_j=0$.
\end{proof}
\end{proof}
Theorem~\ref{th:bounded} and the connection between SETH and OV of Williams~\cite{W04} imply Theorem~\ref{thm:lb} from the introduction.
\subsection{Hardness for Largest Common Subtree}
Next, we prove a lower bound for the Largest Common Subtree (LCST) problem, which is a generalization of Subtree Isomorphism.
Although the reductions above already imply a quadratic lower bound for LCST, we will now optimize these reductions and prove a stronger hardness result: we will show that even on binary trees of depth $(1+o(1))\log{n}$ the LCST cannot be computed in truly subquadratic time.
This will show an interesting gap between LCST and Subtree Isomorphism, since the latter can be solved in truly subquadratic time on such trees - we present such upper bounds in Section~\ref{sec:UB}.
Our strengthened hardness result gives an explanation for why we are not able to extend our upper bounds to LCST: such extensions would refute SETH.
The next theorem implies Theorem~\ref{thm:LCST} from the introduction.
\begin{theorem}
If for some $\varepsilon>0$, the Largest Common Subtree problem on two trees size $n$ can be solved in $O(n^{2-\varepsilon})$ time, then Orthogonal Vectors on $N$ vectors in $\{0,1\}^D$ can be solved in $O(N^{2-\varepsilon}\cdot D^{O(1)})$ time. The trees produced in the reduction from the Orthogonal Vectors problem have degree $d$ and height at most $\log_d(N)+O(\log D)$ for arbitrary $d \geq 2$.
\end{theorem}
\begin{proof}
We note that the construction provided in Theorem \ref{th:bounded} is not sufficient for our purposes because the height of the produced trees is $2\log_d(N)+O(\log D)$, which is larger than what we want.
We will use the more expressive nature of LCST to implement our reduction with smaller height.
To achieve smaller height, we will try to implement vector gadgets such that the largest common subtree of two gadgets would be of a certain fixed size $E$ if the vectors are not orthogonal, while it will be of a larger size $E'>E$ if the vectors are orthogonal.
This trick was introduced by Backurs and Indyk in their reduction to Edit-Distance \cite{BI15} and later used in the reductions to LCS \cite{ABV15}.
Here, we carefully implement such gadgets with degree $d$ trees of small height instead of sequences. WLOG, we can assume that all vectors in $A$ start with $1$ and all vectors in $B$ start with $0$. If it is not so, we can add an extra coordinate at the beginning of every vector and set the entry accordingly. This does not change the answer to the problem (whether there are two orthogonal vectors). Also, we assume that all vectors in $A$ have the same number of entries equal to $1$. If it is not so, we can subdivide the set $A$ into smaller sets so that every set contain vectors with the same number of entries equal to $1$. Then we run the reduction on every subset of $A$ and $B$. This increase the runtime to solve the Orthogonal Vectors problem by a factor of $D+1$ but we are fine with that.
For each vector in the first list, $\alpha \in A$, we construct a vector gadget $H_\alpha$ as follows. Let $H_\alpha'$ be the vector gadget constructed in Lemma \ref{vector_gadgets} corresponding to vector $\alpha \in A$. Then $H_\alpha$ is equal to $r \to root(H_\alpha')$ for some vertex $r$, which is the root of $H_\alpha$.
\iffalse
Let $u_0$ be the root and create two paths that start at $u_0$.
First, add a path of length $3d$ and connect $u_0$ to it, i.e. $u_0 \to p_1 \to \cdots \to p_{3d}$.
Then, add a path of length $2d+2$ and connect $u_0$ to it, i.e. $u_0 \to u_1 \to \cdots \to u_{2d+1}$.
Finally, for each $i \in [d]$, if $\alpha[i]=0$ then we add two nodes $u_{2(i-1)+1,0}$ and $u_{2i,1}$ and add the edges $u_{2(i-1)+1} \to u_{2(i-1)+1,0}$ and $u_{2i} \to u_{2i,1}$, while if $\alpha[i]=1$ we add only one node $u_{2(i-1)+1,0}$ and connect it with the edge $u_{2(i-1)+1}\to u_{2(i-1)+1,0}$.
\fi
For each vector in the second list, $\beta \in B$, we construct a vector gadget $G_\beta$ as follows. Let $\delta$ be a vector with $D$ coordinates. The first entry is equal to $1$ and the rest of entries are equal to $0$. Let $G_\beta'$ be the vector gadget constructed in Lemma \ref{vector_gadgets} corresponding to vector $\beta \in B$. Then we obtain $G_\beta$ by choosing a vertex $r$ to be its root and adding $r \to G_\delta'$ and $r \to G_\beta'$.
\iffalse
First, create a path of length $(4d+2)$ $v_0 \to \cdots \to v_{4d+1}$ so that $v_0$ is the root of $H_\beta$.
Then, for each $i \in [d]$, if $\beta[i]=0$ then we add a node $v_{2(i-1)+1,0}$ and connect it with an edge from $v_{2(i-1)+1}$, while if $\beta[i]=1$ then we add a node $v_{2i,1}$ and connect it with an edge from $v_{2i}$.
\fi
The main idea behind this construction is that, when matching $H_\alpha$ and $G_\beta$, one has a choice: either match $H'_\alpha$ to $G'_\delta$ (giving a fixed score, independent of $\alpha$), or match it to $G'_\beta$ (and the score then depends on the orthogonality of $\alpha$ and $\beta$.)
We make this argument formal in the next lemma.
Let $E'$ denote the size of $H_\alpha$ for $\alpha \in A$, which is independent of $\alpha$ since all vectors in $A$ contain the same number of $1$'s.
Let $E=E'-1$.
\begin{lemma}
\label{cl:LCSTmain}
The largest common subtree of $H_\alpha$ and $G_\beta$ is of size $E'=|H_\alpha|$ if $\alpha,\beta$ are orthogonal and it is of size $E=E'-1$ otherwise. We have that the size of $H_\alpha$ and $H_{\alpha'}$ are equal $|H_\alpha|=|H_{\alpha'}|$ for all $\alpha,\alpha' \in A$.
\end{lemma}
\begin{proof}
First, if $\alpha,\beta$ are orthogonal, then by Lemma~\ref{vector_gadgets} we have that $H_\alpha$ is isomorphic to a subgraph of $G_\beta$ and the LCST has size $E'$.
For the second case, assume that $\alpha,\beta$ are not orthogonal.
We first remark that there is a common subtree of size $E'-1$: Let $\alpha'$ denote $\alpha$ where we set the first coordinate of $\alpha$ (which is equal to $1$) to $0$, then $H'_{\alpha'}$ is a subtree of $H'_{\alpha}$ of size $|H'_{\alpha'}|=E'-1$, and by Lemma \ref{vector_gadgets}, it is also a subtree of $G'_{\delta}$ because $\alpha' \cdot \delta=0$.
It remains to show that we cannot map the entire tree $H_{\alpha}$ to a subtree of $G_{\beta}$, which follows because $H'_{\alpha}$ is neither isomorphic to a subtree of $G'_{\delta}$ (since $\alpha \cdot \delta=1$) nor to a subtree of $G'_{\beta}$ (since $\alpha \cdot \beta\neq 0$).
\end{proof}
We are now ready to present the final trees $H,G$.
We construct $H$ as follows. First, we build a complete $d$-ary tree with $N$ leaves $h_1,\ldots,h_N$ at the lowest level. For every $j \in [N]$, we add $h_j \to H_{\alpha_j}$, where $A=\{\alpha_1, \ldots, \alpha_N\}$. Similarly we construct $G$. Take a complete $d$-ary tree with leaves $g_1, \ldots, g_N$ at the lower level. For every $j \in [N]$, we add $g_j \to G_{\beta_j}$, where $B=\{\beta_1, \ldots, \beta_N\}$.
\begin{theorem}
The Largest Common Subtree of $H$ and $G$ is of size at most $(2N-1)+(N \cdot E)$ if there is no pair of orthogonal vectors, and is at least $(2N-1)+(N \cdot E+1)$ otherwise.
\end{theorem}
\begin{proof}
We must map the nodes $h_i$ for every $i\in [N]$ to nodes $g_{\pi(i)}$, for some permutation $\pi:[N]\to[N]$.
Notice, however, that $\pi$ cannot be an arbitrary permutation since, e.g. $\pi(1)=\pi(2) \pm 1$ (the permutation must be implemented by swapping children in a complete binary tree.)
On the one hand, the total size of the common subtree can be upper bounded by the size of a complete binary tree with $N$ leaves, plus $\sum_{i=1}^N LCST(H_{\alpha_i}, G_{\beta_{\pi(i)}})$, for an arbitrary permutation $\pi$.
If there is no pair of orthogonal vectors, then by Lemma~\ref{cl:LCSTmain}, the latter sum is exactly $N \cdot E$, and the total size is bounded by $(2N-1)+N \cdot E$.
On the other hand, if there is an orthogonal pair $\alpha_i, \beta_j$, we can take any mapping in which $h_i$ is mapped to $g_j$ while the other $h_{x}$'s are mapped arbitrarily to different $g_{y}$'s.
This induces some permutation $\pi:[N]\to[N]$ so that $h_x$ is mapped to $g_{\pi(x)}$.
Since $\alpha_i\cdot\beta_j=0$, Lemma~\ref{cl:LCSTmain} implies that
this mapping can be completed to a mapping of score
$$
(2N-1) + \sum_{v=1}^N LCST(H_{\alpha_v}, G_{\beta_{\pi(v)}}) ~\geq
(2N-1)+(N-1)\cdot E + (E+1) ~=
(2N-1)+(N \cdot E+1)~.
$$
\end{proof}
\end{proof}
\section{Introduction}
\input{intro}
\section{SETH Lower Bounds}
\label{sec:LB}
\input{reductions}
\section{Algorithms}
\label{sec:UB}
\input{algs}
\medskip
\paragraph{Acknowledgements.}
We would like to thank Shiri Chechik, Piotr Indyk, Haim Kaplan, Michael Kapralov, Huacheng Yu, and Uri Zwick for many helpful discussions.
\bibliographystyle{abbrv}
| {
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} | 349 |
Norman Birkett went to immense trouble to get me a copy of "The Northern Muse" a selection of Scottish poetry made by John Buchan.
I saw Hilary at the Speakers and he told me he was doing a talk but was not allowed to mention horrors, we really are all mad. What do they think this trial was about? | {
"redpajama_set_name": "RedPajamaC4"
} | 6,832 |
\section{Introduction}
Morphogenesis is the biological mechanism that drives the development of cells, tissues, and organs, by directly impacting their shape and function.
This mechanism is tightly related to the various states cells undergo: proliferation, differentiation, apoptosis, migration and adhesion.
Despite recent advances in developmental biology (molecular biology, live-imaging and \textit{ex vivo} methodologies~\citep{STJOHNSTON_PLOS_13_2015}), we are mainly able to explain key mechanisms of morphogenesis in an isolated manner. In addition, obtaining quantitative results in a reproducible manner remains a tedious task, hence, the growing interest in computational methods~\citep{FLETCHER_BJ_106_2014,VANLIEDEKERKE_CPM_2_2015,TANAKA_PhD_2016,SHARPE_DEVELOPMENT_144_2017,MERZOUKI_PhD_2018,BUTTENSCHON_PLOSCB_16_2020}. \rcol{These methods obviously rely on empirical observations and ad-hoc parameters in order to compensate for our reduced understanding of active and mechanical properties of living cells. Yet, having access to these parameters allows researchers to (1) accurately quantify the impact of each mechanism of interest (elasticity, bending, internal pressure, chemical signaling, etc), and (2) simulate the morphogenesis of various types of living cells.}
Depending on the level of abstraction and the scale considered, several methodologies are available. At the largest scales,
cell tissues and organs can be approximated by a continuum material whose properties obey a given partial differential equation (PDE). This PDE is solved using standard numerical discretizations such as finite-element or finite-volume. More precisely, that approach requires the tissue/organ to be discretized in a collection of elements, each of them representing a group of cells. Continuum based models were applied to study various morphogenetic problems, such as bone~\citep{RODRIGUEZ_JoB_27_1994} and brain growth~\citep{TALLINEN_NP_12_2016}.
\rcol{Lattice-based approaches are another type of computational model that can simulate cell tissue dynamics over a wide range of scales~\citep{VANLIEDEKERKE_CPM_2_2015,METZCAR_JCOCCI_2_2019}. These methods rely on a fixed space discretization, whose lattice sites (or nodes) correspond to the position of cells. The number of cells represented by a lattice site can vary greatly depending on the considered approach. For cellular automata (CA), a lattice site can contain either a single cell~\citep{BLOCK_PRL_99_2007}, or many of them~\citep{RADSZUWEIT_PRE_79_2009}.
On the contrary, cellular Pott models (CPM) assume that a living cell occupies several lattice sites, hence, properly representing the evolution of the cell shape~\citep{GRANER_PRL_69_1992,GLAZIER_PRE_47_1993}.
Both CA and CPM simulate cellular processes (growth, mitosis, apoptosis, migration, etc) in a stochastic manner, e.g., Metropolis algorithm to minimize the energy given by a Hamiltonian, where the latter contains all biophysical constraints of interest.
Interestingly, these methods were shown to accurately simulate various biophysical problems. At the largest scales, CA were shown to be of particular interest for cancer modeling~\citep{ANDERSON_CELL_127_2006,ENDERLING_PBMB_106_2011}. At smaller scales, CPM
were used for cells sorting and cells migration~\citep{Scianna_Preziosi_Wolf_2013}, or even bio-tube morphogenesis~\citep{Hirashima_Rens_Merks_2017}. Nevertheless, due to the stochastic nature of lattice-based approaches, it is not always clear how to derive straightforward relationships between the model parameters and biophysical properties of living cells.}
At the scale of individual cells, several lattice-free approaches have also been proposed over the years, and each of them describes cells in a different way.
The centroid method models cells through their geometrical center (centroid), hence its name. The latter centroids obey Langevin equation of motion, and they can interact with one another as soon as the distance between the two cell centroids is smaller than a given radius of influence --in compliance with Johnson-Kendall-Roberts theory for adhesive spheres~\citep{CHU_PRL_94_2005}. This approach was used to simulate, e.g., cell tissue rheology and motility~\citep{PALSSON_PNAS_97_2000}, as well as, cell spreading/proliferation~\citep{DRASDO_PB_2_2005}. Interestingly, centroid-based methods were also hybridized with continuum approaches to create a multi-scale model that was used to investigate breast cancer~\citep{KIM_BMB_75_2013}.
Nevertheless, these models cannot provide any information on how cell deformation and morphology impact pattern formation and growth processes.
Vertex models are an alternative computational modeling of cells which represent cells through their membrane instead of their geometrical center~\citep{HONDA_JTB_84_1980,FLETCHER_PBMB_113_2013,FLETCHER_BJ_106_2014,MERZOUKI_PhD_2018}. More precisely, cell membranes are approximated with polygons, and their edges are in common with adjacent cells. Each membrane vertex follows Newton mechanics for which internal and external forces are derived through energy potentials. This approach is of particular interest for the simulation of 2D epithelial tissue, from either an apical or lateral viewpoint, and can even be extended to 3D simulations of single monolayers of cells in a straightforward manner~\citep{FLETCHER_BJ_106_2014}. Other application examples include the study of cell mechanical properties~\citep{MERZOUKI_SM_12_2016}, wound healing~\citep{FLETCHER_PBMB_113_2013}, and tissue buckling~\citep{MERZOUKI_NC_17_2018}.
Nevertheless, by sharing common vertices/edges with adjacent cells in order to avoid interpenetration issues, topological rearrangements do not naturally emerge from the interaction between cells due to the absence of interstitial gaps. Consequently, ad-hoc mechanisms must be added to the vertex model to account for these topological transitions.
A way to circumvent this issue is to move towards more accurate cell descriptions. The most detailed computational model~\citep{JAMALI_PO_5_2010} relies on a large collection of vertices that discretize both the cell membrane and nucleus. Contrarily to vertex models, this approach does not require vertices to be shared with adjacents cells, hence, properly modeling interstitial gaps between cells. Simple Voigt models (damper and spring coupled in parallel) are used to account for viscoeslastic constraints on the membrane, the cytoskeleton, and the nucleus. In addition, several cellular processes (growth, mitosis, motility, apoptosis, polarization) can be simulated by adapting spring and damper constants in either a static or dynamic manner. By accounting for external forces, this model can also include environmental effects such as cell-cell and cell-subtract interactions, or even external force fields (e.g., pushing/pulling and electromagnetic forces). Obviously, this model can reproduce most of cell developmental mechanisms, but it is extremely computationally demanding. Furthermore, mechanisms related to cell proliferation and death are rather complicated to implement.
One way to simplify the above approach is to give up on cell growth, mitosis and apoptosis mechanisms. Doing so, (1) only the cell membrane needs to be discretized, (2) the total amount of vertices remains constant over time, and (3) no dynamic recomputing of the model constants is required. Such an approach was proposed to study gastrulation~\citep{TAMULONIS_DB_351_2011,VANDERSANDE_DB_460_2020}. In that context, only springs are used to model interactions between the membrane vertices of q given cell. Springs between consecutive vertices model the membrane elasticity and tension, whereas those between opposite vertices represent the elastic behavior of the cytoskeleton. The inner structure of the cell is not accounted for. Only the cytoplasm is mimicked through a force depending on the cell area variations. Eventually, repulsive/adhesive contact fores are used to model the interaction between vertices belonging to adjacent cells.
If one still wants to simulate cell proliferation and apoptosis in a detailed and rather simple manner,
the immersed boundary methodology is an interesting alternative~\citep{REJNIAK_BMB_66_2004,DILLON_CM_466_2008,TANAKA_PhD_2016}.
The latter originates from Computational Fluid Dynamics, and it was originally introduced to study fluid-structure interactions~\citep{PESKIN_AN_11_2002}.
Contrarily to the simpler spring or Voigt models, forces are computed in an iterative manner depending on interactions with inner and outer fluids (cytoplasm and extra-cellular matrix, respectively). Similarly to the above simplified formulation, it only represents vertices on the cell membrane.
Interestingly, even if it does not explicitly represent the cell nucleus, this approach can be used to simulate cell proliferation and apoptosis in a simple manner. Indeed, when the distance between two consecutive vertices is greater/smaller than a predefined threshold, then vertices are added/removed to ensure an accurate and uniform discretization of the cell membrane.
Once the cell is large enough (usually twice its rest size), mitosis can be triggered by simply cutting the mother cell (along its minor axis) into two daughter cells, and adding vertices to close their membrane.
The main advantage of immersed-boundary formulations is that they self-adjust to the number of vertices, hence, no constant modification is required during cell proliferation and apoptosis. This particular approach was notably used to simulate, tissue bending~\citep{REJNIAK_BMB_66_2004}, pattern formation and necrosis~\citep{DILLON_CM_466_2008}, and morphogenic signaling~\citep{TANAKA_Bioinformatics_31_2015}.
Nevertheless, inner and outer fluids must be simulated to compute forces that apply on the membrane vertices, and these fluids generate spurious velocity currents at the cell membrane interface. In addition, the iterative computation of forces is expensive as compared to spring and Voigt models.
In order to simulate most cellular processes in a detailed, yet simple manner, we propose a general formulation that keeps most advantages of previous works~\citep{JAMALI_PO_5_2010,TAMULONIS_DB_351_2011,TANAKA_Bioinformatics_31_2015} while getting rid of most of their drawbacks. More precisely, we dynamically adjust the number of vertices used for the evolution of the cell membrane (as in~\citep{JAMALI_PO_5_2010}), but, similarly to ~\citep{TAMULONIS_DB_351_2011}, the cell nucleus, cytoskeleton and cytoplasm do not explicitly appear in our formulation. This is possible assuming the same cell division mechanism as in immersed boundary cell-based approaches~\citep{TANAKA_Bioinformatics_31_2015}. Discarding the cell inner structure further allows us to propose simple rules to adjust the cell mechanical properties, which are modelled through linear and torsion springs (elasticity and bending, respectively). Eventually, the present model does not require the simulation of inner and outer fluids. Nevertheless, the impact of the cytoskeleton, cytoplasm and extra-cellular matrix \rcol{is accounted for through internal and external forces.}
As a first study, we propose to present in details our new formalism, and to validate it in a qualitative manner through various benchmark tests. More precisely, choices made regarding vertex dynamics, single-cell proliferation mechanisms and chemical signaling are detailed in Section~\ref{sec:Model}. The minimal number of vertices required to accurately discretize the cell membrane is further identified through a simple convergence study. The incompressible, visco-elastic behavior of cells is investigated in Section~\ref{sec:Preliminary}, alongside with a first attempt to model cell motility inside a tissue. Section~\ref{sec:Validation} contains more complex validation cases, such as free and chemically piloted tissue growth in an unbounded domain, as well as, cell proliferation under external constraints. Eventually, conclusions are drawn in Section~\ref{sec:Conclusion}.
\section{Model description\label{sec:Model}}
\subsection{Cell description\label{subsec:cell_description}}
Vertex models are commonly used to describe the evolution of cell tissues in the context of morphogenesis~\citep{FLETCHER_BJ_106_2014}. These models describe cells as polygons composed of vertices, and which share edges with neighboring cells. This allows for an efficient modeling of phenomena occurring at the scale of tissues, whereas those emerging at the cellular level must be imposed through an ad-hoc mechanism -- the so called T1 and T2 transitions.
On the contrary, we propose hereafter to model a cell as a collection of vertices. In that context, the membrane of a cell is composed of (approximately one hundred) vertices whose \rcol{dynamic properties are} governed by mechanical and biological mechanisms (see Fig.~\ref{fig:resolution}).
As each cell is totally independent, more complex cell shapes and behaviors naturally emerge from the model without any ad-hoc mechanism.
As an example, cell migration within a tissue can be simulated through simple mechanisms (see Sec.~\ref{subsec:motility}), whereas, in approaches available in the literature, it is commonly imposed through consecutive T1 transitions within vertex models~\citep{FLETCHER_BJ_106_2014,MERZOUKI_PhD_2018}.
\begin{figure}[h]
\centering
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=0.49\textwidth]{multi_cells.png}
\hfill
\includegraphics[width=0.49\textwidth]{mul2_cells.png}
\caption{PalaCell}
\end{subfigure}
\hspace{0.05\textwidth}
\begin{subfigure}[b]{0.45\textwidth}
\includegraphics[width=0.49\textwidth]{multi_cells_vert.png}
\hfill
\includegraphics[width=0.49\textwidth]{mul2_cells_vert.png}
\caption{Vertex model}
\end{subfigure}
\hfill
\caption{Cells resolution, with the vertices in red and the membrane in black.}
\label{fig:resolution}
\end{figure}
In order to simulate the growth and division of cells, while keeping the membrane resolution constant, new vertices must be added. Similarly, when the perimeter of cells is reduced (e.g., apoptosis, external constraints, etc) extra vertices must be removed to avoid local vertex clusterings. Concretely speaking, the addition and the suppression of a vertex depend on the distance $d_{i,i+1}$ between two vertices $i$ and $i+1$, and a characteristic length $l_0$ that corresponds to the cell perimeter divided by a predefined number of vertices $n$ (see \ref{app:appendix} for its expression). In our formulation, both mechanisms are imposed through the following rules~\citep{TANAKA_PhD_2016}:
\begin{align*}
\text{adding vertex:} &\text{if } d_{i,i+1} > 2l_0, \\
\text{removing vertex:} &\text{if } d_{i,i+1} < l_0/2,
\end{align*}
and further illustrated in Fig.~\ref{fig:addrm_vertex}.
\begin{figure}[H]
\centering
\begin{subfigure}{0.3\textwidth}
\includegraphics[width=0.9\textwidth]{cell_add-rm.pdf}
\caption{Initial state: inhomogeneous discretization}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\includegraphics[width=0.9\textwidth]{cell_add-rm_1.pdf}
\caption{Vertices are added (blue) and removed (red).}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\includegraphics[width=0.9\textwidth]{cell_add-rm_2.pdf}
\caption{Final state: homogeneous discretization.}
\end{subfigure}
\caption{Example of addition/suppression mechanisms to keep a uniformly discretized membrane. In practice, the red vertex is removed because both its neighbors are at a distance lower than $l_0/2$.}
\label{fig:addrm_vertex}
\end{figure}
\subsection{Mechanical dynamics}
The evolution of vertices follows Newton's law of motion:
\begin{equation}
\bm{F} = \bm{F}_{P} + \bm{F}_{M} + \bm{F}_{A} + \bm{F}_{ext}.
\label{eq:all}
\end{equation}
The resulting force $\bm{F}$ (\ref{eq:all}) acts on a vertex through various mechanisms: pressure $\bm{F}_P$, membrane resistance to deformation $\bm{F}_M$, adherence between cells $\bm{F}_A$, and external force $\bm{F}_{ext}$. The latter force can either be local (e.g., interaction with a wall, migration) or global (e.g., gravity, interaction with the extra-cellular matrix), and is generally used to model an external constraint.
The force $\bm{F}_{M}$ accounts for both resistance to elongation and torsion, which allows us to control the membrane tension:
\begin{equation}
\bm{F}_{M} = \bm{F}_{B} + \bm{F}_{S}.
\label{eq:memb}
\end{equation}
$\bm{F}_B$ stands for the bending force which naturally appears when consecutive vertices are non-aligned.
Regarding the spring force $\bm{F}_S$, it connects two vertices and aims at balancing the distance between vertices, hence, ensuring a uniform discretization of the cell membrane.
While $\bm{F}_B$ is purely perpendicular to the membrane, $\bm{F}_S$ has both a tangential and a normal component. These membrane forces are illustrated in Fig.~{\ref{fig:memf}}.
\begin{figure}[!ht]
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[width=.9\linewidth]{cell_base_fb.pdf}
\caption{Bending force}
\end{subfigure}
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[width=.9\linewidth]{cell_base_fs.pdf}
\caption{Spring force}
\end{subfigure}
\caption{Illustration of the membrane force impact on the vertex dynamics. The normal vector $\bm{\hat{n}}$ is taken at vertex $i$.}
\label{fig:memf}
\end{figure}
Going into more details, the bending force $\bm{F}_{B}$ depends on the local curvature, i.e.,
when the cell is locally convex/concave, it points inward/outward the cell. At vertex $i$, it reads as~\citep{TANAKA_PhD_2016}
\begin{equation}
\bm{F}_B(\bm{x}_{i}) = \rcol{k_b} \left( \frac{\bm{x}_{i+1} - \bm{x}_i}{\lVert \bm{x}_{i+1} - \bm{x}_i \rVert} + \frac{\bm{x}_{i-1} - \bm{x}_i}{\lVert \bm{x}_{i-1} - \bm{x}_i \rVert} \right),
\label{eq:bend}
\end{equation}
with $\rcol{k_b}$ being the bending constant.
The spring force $\bm{F}_S$ depends on the distance with respect to neighbors vertices. In order to avoid local discrepancies due to, e.g., adding vertices to simulate proliferation, a global spring is used to model the spring force over the whole membrane (Fig.~\ref{fig:membrane-1}). A global spring constant $K_s$ is then assigned to this spring. Once the number of vertices $n$ is known, this global spring is divided into $n$ smaller springs (Fig.~\ref{fig:membrane-8}), whose spring constant is derived assuming springs are in series, i.e., $k_s = n K_s$. The latter condition also has the very nice property of leading to a cell radius that is independent of the membrane discretization (see \ref{app:appendix}). For a vertex $i$, the spring force reads as
\begin{equation}
\bm{F}_S(\bm{x}_i) = k_s \left[ (\bm{x}_{i+1} - \bm{x}_i) + (\bm{x}_{i-1} - \bm{x}_i) \right]
\label{eq:spring}
\end{equation}
with $k_s$ the local spring constant.
\begin{figure}[!ht]
\centering
\begin{subfigure}{.3\textwidth}
\centering
\includegraphics[width=\linewidth]{spring_1.pdf}
\caption{Global spring}
\label{fig:membrane-1}
\end{subfigure}
\hspace{2cm}
\begin{subfigure}{.3\textwidth}
\centering
\includegraphics[width=\linewidth]{spring_8.pdf}
\caption{Local springs}
\label{fig:membrane-8}
\end{subfigure}
\caption{Dual representation of the membrane springs. The global spring is used to ensure the cell elasticity will not depend on the number of vertices. Regarding local springs, they are dynamically adjusted when the number of vertices change, hence, ensuring a homogeneous representation of the membrane.}
\label{fig:membrane}
\end{figure}
The pressure force $\bm{F}_P$ can be decomposed into two parts
\begin{equation}
\bm{F}_{P} = \bm{F}_{IP} + \bm{F}_{CP}
\label{eq:press}
\end{equation}
where the internal pressure force $\bm{F}_{IP}$ allows us to discard the inner structure of the cell (cytoplasm, cytoskeleton and nucleus), by accounting for internal mechanisms, e.g., mitosis and apoptosis (see Section~\ref{subsec:growth}). On the contrary, the contact pressure force $\bm{F}_{CP}$ is purely mechanical, and it corresponds to the force imposed by vertices belonging to neighboring cell. The pressure force $\bm{F}_P$ (\ref{eq:press}) is used to balance the membrane force $\bm{F}_M$ (\ref{eq:memb}), and consequently, it can be used to trigger cell proliferation or death. More precisely, the internal pressure force is applied on a portion of the membrane centered around vertex $i$, i.e., $\mathrm{force} = \mathrm{pressure} \times \mathrm{surface}$. In our case, it is expressed as
\begin{equation}
\bm{F}_{{IP}}(\bm{x}_i) = p_{int} \frac{\left\lVert \bm{x}_{i+1} - \bm{x}_{i} \right\rVert + \left\lVert \bm{x}_{i-1} - \bm{x}_{i} \right\rVert}{2} \bm{\hat{n}},
\label{eq:ipress}
\end{equation}
with $p_{int}$ the pressure inside the cell (whose evolution is described in Eq.~(\ref{eq:internal_pressure})), and $\bm{\hat{n}}$ the local normal to the membrane, which reads as
\begin{equation*}
\bm{\hat{n}} = \frac{\bm{\hat{n}}_{i+1,i}+\bm{\hat{n}}_{i-1,i}}{\vert\vert\bm{\hat{n}}_{i+1,i}+\bm{\hat{n}}_{i-1,i}\vert\vert},
\end{equation*}
with
\begin{equation*}
\bm{\hat{n}}_{i+1,i}=\frac{\bm{x}_{i+1}-\bm{x}_{i}}{\vert\vert\bm{x}_{i+1} - \bm{x}_{i}\vert\vert}
\begin{pmatrix}
\phantom{-}0 & -1\phantom{-}\\
\phantom{-}1 & \phantom{-}0\phantom{-}
\end{pmatrix}, \quad
\bm{\hat{n}}_{i-1,i}= \frac{\bm{x}_{i-1}-\bm{x}_{i}}{\vert\vert\bm{x}_{i-1}- \bm{x}_{i}\vert\vert}
\begin{pmatrix}
\phantom{-}0 & \phantom{-}1\phantom{-}\\
-1 & \phantom{-}0\phantom{-}
\end{pmatrix},
\end{equation*}
and where matrices correspond to rotations of $\pi/2$ and $-\pi/2$ respectively.
Regarding the contact pressure, it only impacts the vertex dynamics when (at least) two cells are in contact. Vertices belonging to different cells are considered to be in contact, if their distance is smaller than a predefined maximal range of interaction $d_{max}$. In that case, the contact pressure force $\bm{F}_{CP}$ is the sum of internal pressure forces that belong to each cell in contact with the current one, weighted by the number of connections $w_j$:
\begin{equation}
\bm{F}_{CP}(\bm{x}_i) = \sum_{j \in CV_i} \frac{\bm{F}_{IP}(\bm{x}_j)}{w_{j}}.
\label{eq:cpress}
\end{equation}
with $CV_i$ the set of vertices belonging to other cells, and that are inside the range of interactions of vertex $i$. For the particular case illustrated in Fig.~\ref{subfig:5b}, we have $CV_i=\{j-1,j\}$, where the corresponding weights read are $w_{j}=w_{j-1}=2$ because $j$ and $j-1$ are connected to two vertices: $(i-1,i)$ and $(i,i+1)$ respectively.
\begin{figure}[!ht]
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{cell_base_fip.pdf}
\caption{Internal pressure force}
\end{subfigure}
\begin{subfigure}{.45\textwidth}
\centering
\includegraphics[width=.9\linewidth]{cell2_base_fcp.pdf}
\caption{Contact pressure force}
\label{subfig:5b}
\end{subfigure}
\caption{Representation of pressure forces. For the contact pressure force, solid lines corresponds to connections between the current vertex $i$ and vertices belonging to the other cell $(j,j-1)$, whereas dashed lines correspond to connections with adjacent vertices $(i-1,i+1)$.}
\label{fig:presf}
\end{figure}
Finally, the adherence force $\bm{F}_A$ is the force that keeps cells connected to each other. It also activates when vertices, belonging to different and adjacent cells, are within a given range of interaction $d_{max}$. The force used in our model is a adapted from the adherence model proposed in \citep{TANAKA_PhD_2016}. In our case, we replaced the linear approximation by a potential in order to impose a rest distance $d_0=d_{max}/2$. Doing so, the repulsive behavior of $\bm{F}_A$ is strengthen, and it prevents interpenetration issues.
This force is defined as follows
\begin{equation}
\bm{F}_{A}(\bm{x}_i) = k_a \sum_{j \in CV_i} \frac{\bm{x}_i - \bm{x}_j}{\lVert \bm{x}_i - \bm{x}_j \rVert} \left[ \lVert \bm{x}_i - \bm{x}_j \rVert - d_0\left(1 - \frac{d_0^2}{16\lVert \bm{x}_i - \bm{x}_j \rVert^2}\right) \right]
\label{eq:adh}
\end{equation}
with $k_a$ the adherence constant. The behavior of the adherence force $\bm{F}_A$ is sketched in Fig.~\ref{fig:adhf}, and further compared to~\citep{TANAKA_PhD_2016} in Fig.~\ref{fig:adh_comp}.
\begin{figure}[!ht]
\centering
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=0.9\linewidth]{cell2_base_fa.pdf}
\caption{Sketch of the adherence force}
\label{fig:adhf}
\end{subfigure}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=0.9\linewidth]{rl_2.pdf}
\caption{Adherence force modelling: \citep{TANAKA_PhD_2016} (linear) and our model (\ref{eq:adh})}
\label{fig:adh_comp}
\end{subfigure}
\caption{Sketch and comparison of adherence force modelling}
\end{figure}
\subsection{Biological dynamics}
\subsubsection{Cell growth, relaxation and apoptosis\label{subsec:growth}}
In our work, biological mechanisms are accounted for through the internal pressure $p_{int}$ that controls the cell size via
\begin{equation}\label{eq:internal_pressure}
p_{int} = \eta \left(\frac{m}{A} - \rho_0\right)
\end{equation}
with $m$ and $A$ the cell mass and area respectively. This biological mechanism corresponds to a relaxation of cells towards their target density $\rho_0$, and it is controlled by the pressure sensitivity $\eta$.
Cells have different states during their life cycle. To account for this natural cycle, the mass of a cell is modified depending on the cell state one wants to simulate (proliferation, rest or apoptosis).
As an example, by increasing the mass of the cell, the density of the cell increases accordingly. In turns, the cell starts to grow until the target area/density of the cell is reached (Fig.~\ref{fig:rho_high}). Alternatively, it is possible to reduce the cell size by reducing its mass (Fig.~\ref{fig:rho_low}).
\begin{figure}[!ht]
\centering
\begin{subfigure}{.3\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{rho_over.pdf}
\caption{Cell with high density (red)}
\label{fig:rho_high}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{rho_ok.pdf}
\caption{Cell with target density (grey)}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{rho_under.pdf}
\caption{Cell with low density (blue)}
\label{fig:rho_low}
\end{subfigure}
\caption{Cell internal mechanisms: growth, rest and relaxation (from left to right).}
\label{fig:cell_density}
\end{figure}
\noindent In addition, the cell state is also directly impacted by forces imposed by surrounding cells.
For instance, when the pressure on the cell membrane increases, the cell reduces its growth speed~\citep{ROUX_PRIVATE_2020}.
In our model, as the growth is controlled by the internal pressure, it is used to adapt the growth rate by adjusting the mass rate $dm$, which in turn increases the cell size according to the pressure exerted on the cell membrane.
Hereafter, these mechanisms are modelled through the following rules:
\begin{equation}\label{eq:growthRelax}
dm = \begin{cases}
\nu A \left( 1 - \frac{p_{ext}}{p_{max}}\right) dt &\text{ for proliferating cells}\\
- \nu_{relax} \left(A - A_0\right) \left(1-\frac{p_{ext}}{p_{max}} \right) dt &\text{ otherwise.}
\end{cases}
\end{equation}
where $\nu$ and $\nu_{relax}$ are mass growth rates related to proliferation and relaxation respectively state. $A_0$ is the target area, $dt$ the time step of the simulation, and $p_{max}$ a pressure threshold above which the cell proliferation is stopped. $p_{ext}$ is the external pressure acting on the cell, and is defined through
\begin{equation}
p_{ext} = \frac{2[\bm{F}_{{CP}}(\bm{x}_i) + \bm{F}_{ext}(\bm{x}_i)]\cdot\bm{\hat{n}}}{\left\lVert \bm{x}_{i+1} - \bm{x}_{i} \right\rVert + \left\lVert \bm{x}_{i-1} - \bm{x}_{i} \right\rVert},
\label{eq:pext}
\end{equation}
where ``$\cdot$'' is the scalar product.
All these mechanisms (growth, rest and relaxation) are illustrated in Fig.~(\ref{fig:cell_density}). It is worth noting that rules~(\ref{eq:growthRelax}) are one possibility among many others, and consequently, one could also choose, e.g., a power law of the type $(1-p_{ext}/p_{max})^{\alpha}$ with $\alpha>0$. The impact of such a choice, on the biophysical properties of cells, will be investigated in a future work.
Eventually, apoptosis is triggered when the number of vertices $n$, used to discretized the cell membrane, is below a predefined threshold $n_{a}=30$. This corresponds to $A\lesssim A_0/16$, assuming that $n=120$ (see Section~\ref{subsec:convergence}). In practice, the remaining points are simply removed from the simulation domain, and connections with vertices belonging to other cells are removed as well. Obviously, one could impose a minimal area $A_{min}$, instead of $n_a$, as a criterion to trigger apoptosis.
\subsubsection{Cell mitosis}
When a mother cell reaches a given size (here, $A\geq 2 A_0$), it divides into two daughter cells, according to the mitosis mechanism.
Similarly to \citep{TANAKA_PhD_2016}, this is numerically done by cutting the mother cell in the direction perpendicular to its major axis \citep{Thery_Racine_Pepin_Piel_Chen_Sibarita_Bornens_2005}, as sketched in Fig.~\ref{fig:cell_split_before}. Two new, independent membranes are then created along the cutting axis in order to obtain two distinct daughter cells. Vertices are finally added, to the newly created membrane, to keep the vertices density constant over the perimeter (see Fig.~\ref{fig:cell_split_after}).
\begin{figure}[!ht]
\centering
\begin{subfigure}{0.4\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{split_0.pdf}
\caption{Before division}
\label{fig:cell_split_before}
\end{subfigure}
~
\begin{subfigure}{0.4\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{split_1.pdf}
\caption{After division}
\label{fig:cell_split_after}
\end{subfigure}
\caption{Simplified cell division, with the major axis represented in blue, and the cutting axis in red. Only lines connecting adjacent vertices are shown here.}
\label{fig:cell_split}
\end{figure}
\subsection{Chemical signaling\label{subsec:signaling}}
\rcol{A simple way to move from one cell state to another one is to rely on chemical signaling. This type of signal is usually simulated as a scalar whose evolution is given by a reaction-diffusion equation. Hereafter, this equation is solved thanks to lattice Boltzmann (LB) methods, as proposed by~\citep{TANAKA_Bioinformatics_31_2015,TANAKA_PhD_2016}.
Contrarily to the cell-based approach dedicated to the evolution of the cell membrane, the LB scheme requires a Cartesian space discretization of the simulation domain to compute the evolution of the chemical signal. In the LB context, solving the reaction-diffusion equation is done through the D2Q5 lattice following instructions proposed, e.g.,} in Sec 5.3.5 of~\citep{KRUGER_Book_2017}. Concretely speaking, the evolution of population $f_k$ ($1\leq k \leq 5$) is given by the \rcol{LB} equation:
\begin{equation}
f_k(\bm{x}+\bm{c}_k dt, t + dt) - f_k(\bm{x},t) = -\frac{dt}{\tau_f}\left(f_k(\bm{x},t)-f_k^{eq}(\bm{x},t)\right) + \left(1-\dfrac{dt}{2\tau_f}\right) Q_k(\bm{x},t).
\end{equation}
The first right-hand-side (rhs) term \rcol{is the BGK collision term~\citep{BHATNAGAR_PR_94_1954} that} corresponds to a relaxation towards the equilibrium state $f_k^{eq}(\bm{x},t)=w_k \rho_{LB}(\bm{x},t)$, with $\rho_{LB}$ being the signal density assigned to one voxel. $w_k=(2/6,1/6,1/6,1/6,1/6)$ are the lattice weights, and the lattice velocities read as
\begin{equation}
\bm{c}_k =
\begin{pmatrix}
0 & \phantom{-}1 & \phantom{-}0 & -1 & \phantom{-}0 \\
0 & \phantom{-}0 & \phantom{-}1 & \phantom{-}0 &-1
\end{pmatrix}.
\end{equation}
The second rhs term is a source term defined as
\begin{equation}
Q_k(\bm{x},t) = w_k R(\bm{x},t),
\end{equation}
with $R(\bm{x},t)$ being the reaction coefficient. The prefactor $(1-dt/2\tau_f)$ is used to reduce discrete effects as proposed by~\citep{GUO_PRE_65_2002}.
In practice, the signal is produced inside a given cell at a rate $\tau_{p}/dt$, but its evolution is not bounded inside the cell membrane. On the contrary, it can freely propagate in all the simulation domain, even though its amplitude decays according to a predefined decay rate $\tau_{d}/dt$ outside the emitting cell. This decaying mechanism can be seen as a simplification of more realistic signaling processes, for which diffusive compounds are partially bloked by the cell membrane.
In any case, the reaction term locally modifies the lattice population $f_k$ through the reaction coefficient
\begin{equation}
R(\bm{x},t) = \begin{cases}
\tau_{p}/dt& \text{in the cell emitting the signal},\\
-(\tau_{d}/dt)\rho_{LB}(\bm{x},t) &\text{otherwise}.
\end{cases}
\end{equation}
Here, $\rho_{LB}$ is computed at each voxel via
\begin{equation}
\rho_{LB}(\bm{x},t) = \sum_{k=1}^5 \left(f_k + \dfrac{dt}{2}Q_k\right)(\bm{x},t).
\end{equation}
To switch from one cell state $s$ to another, the present model requires the signal density inside a cell to be large enough. In other words,
\begin{equation}
s = \begin{cases}
1 &\text{ if } \rho_{s} > \theta_{sig}\\
0 &\text{ otherwise }
\end{cases}
\end{equation}
with $s$ the state of the cell, $\rho_{s}$ the signal density inside the cell, and $\theta_{sig}$ the switching threshold.
The signal density $\rho_s$ is computed by integrating the density provided by the LBM $\rho_{LB}$ over the voxelized area of the cell $A_{LB}$:
\begin{equation}
\rho_{s} = \frac{1}{A_{LB}}\int_{\bm{x}\in A_{LB}}\rho_{LB}(\bm{x},t)\, \mathrm{d}A_{LB}.
\end{equation}
In our model, we assume that a lattice voxel belongs to $A_{LB}$ if at least half of its area is inside the membrane of the cell (see Fig.~\ref{fig:coupling_det} for an example). In order to account for topology changes of cells during a simulation, we adapted the lattice voxelizer that is available in Palabos library~\citep{LATT_CMA_81_2021}. This is illustraed in Fig.~\ref{fig:coupling_voxel}, where each cell has a flag corresponding to a given color, and white voxels represent interstitial gaps.
\begin{figure}[!ht]
\centering
\begin{subfigure}[B]{0.4\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{cell_chem.pdf}
\caption{In blue, the LBM lattice elements inside the cell.}
\label{fig:coupling_det}
\end{subfigure}
\begin{subfigure}[B]{0.4\textwidth}
\centering
\includegraphics[width=0.99\textwidth]{voxel_zoom.pdf}
\caption{Example of voxelization. Cell membranes are in black, and colored voxels are computed via Palabos library~\citep{LATT_CMA_81_2021}.}
\label{fig:coupling_voxel}
\end{subfigure}
\caption{Illustration of the coupling between the Eulerian lattice grid (LBM) and the Lagrangian representation of cells.}
\label{fig:coupling}
\end{figure}
\rcol{Before moving on to cell differentiation, it is worth noting a few shortcomings, as well as possible improvements, of the proposed model for chemical signaling simulation.
Firstly, the LB scheme can encounter stability issues when simulating steep gradients of chemical signaling with low values of $\tau_f$, i.e., low diffusion coefficient. A common way to tackle that problem is to adapt the numerical scheme by either replacing the BGK operator with a more sophisticated collision model, or by employing flux limiters.
A detailed discussion about these extensions is out of the scope of this work, and the interested reader may refer to, e.g.,~\citep{COREIXAS_PRE_100_2019} for a review about collision models,~\citep{COREIXAS_RSTA_378_2020} for a stability comparison in the context of fluid flow simulations, and~\citep{HOSSEINI_IJMPC_28_2017} for the use of total variation diminishing schemes to improve the accuracy and stability of LBM for the simulation of advection-diffusion equations with low diffusion coefficients.
Secondly, while LB schemes are particularly well-suited for high performance computing, they also require more memory storage than standard solvers for reaction-diffusion equations.
Even if several strategies have been proposed to reduce memory consumption while keeping good performance (see, e.g.,~\citep{LATT_ARXIV_11751_2020}), finite-difference or finite-volume methods might be a better choice when a large number of coupled reaction-diffusion equations are to be solved, as it is the case, e.g., for pattern formation induced by gene expression~\citep{LANDGE_DB_460_2020}.
Hereafter, a single equation is solved, hence, it is not necessary to rely on more advanced techniques than LBM to simulate chemical signaling.}
\subsection{Cell differentiation}
Starting from a 3D monolayer of cells (Fig.~\ref{fig:tissue_glob}), there are several ways to investigate its behavior in 2D. As an example, when simulating cell monolayers from an apical view, all vertices that compose the membrane share the very same properties (Fig.~\ref{fig:tissue_api}). On the contrary, adopting a lateral representation of cells means that membrane properties should be adjusted according to the apical, basal and lateral visco-elastic properties of the cell.
\begin{figure}[!ht]
\centering
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{tissue_proj.pdf}
\caption{Global 3D view}
\label{fig:tissue_glob}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{tissue_api.pdf}
\caption{2D apical view}
\label{fig:tissue_api}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=0.9\textwidth]{tissue_lat.pdf}
\caption{2D lateral view}
\label{fig:tissue_lat}
\end{subfigure}
\caption{2D representations of a simplified 3D tissue. The apical, basal and lateral sides are represented in red, blue and grey respectively.}
\label{fig:my_label}
\end{figure}
As done in the context of vertex models~\citep{MERZOUKI_PhD_2018,TRUSHKO_DC_54_2020}, this differentiation mechanism is included in our framework by simply tagging vertices (basal, lateral, apical), and fixing their properties (elasticity, bending, etc) depending on their tag.
Henceforth, the bio-physical properties of the proposed model will be investigated. All parameters, and their value are reported in~\ref{app:param}, and $dt=1$. Eventually, $20$ voxels are used to discretize the initial cell in the case of chemically-driven proliferation (Sec.~\ref{subsec:growth_w_wo_sig}).
\section{Single cell behavior\label{sec:Preliminary}}
\subsection{Convergence study: single cell relaxation \label{subsec:convergence}}
Before investigating the cell properties that naturally emerge from our framework, it is important to identify the minimal number of vertices required to accurately discretize the cell membrane. Doing so, the computational cost of our model can be reduced without decreasing the accuracy of the solver.
By definition, the cell membrane description should be independent of the number of vertices considered for its discretization. Hence, changing the number of vertices should not have any impact on the cell properties. To investigate that point, we consider the relaxation of a single cell, in an unbounded domain, toward its equilibrium state. In that particular case, vertices are only subjected to membrane and internal pressure forces, $\bm{F}_M$ and $\bm{F}_{IP}$ respectively. \rcol{Knowing that our model flows from Newton's law of motion, an analytical formulation for the equilibrium state can be derived by balancing both forces}. In the limit of small angle (high number of vertices), it is reached when the cell radius reads as
\begin{equation}
r_{eq} = \rcol{k_b} \frac{\sqrt{1 + \frac{2\eta m}{\pi \rcol{k_b}^2}\left(4\pi K_s + 2 \eta \rho_0 \right)}-1}{4\pi K_s + 2 \eta \rho_0}
\label{eq:radius}
\end{equation}
with $\rcol{k_b}$ the bending constant, $K_s$ the global spring constant, $\eta$ the pressure sensitivity, $\rho_0$ the target density, and $m$ the cell mass. The full derivation of the cell radius at equilibrium can be found in~\ref{app:appendix}.
\begin{figure}[!btp]
\centering
\includegraphics[width=0.5\textwidth]{mass_conv.pdf}
\caption{Impact of the number of vertices on the cell radius obtained after relaxation. The equilibrium radius corresponds to Eq.~(\ref{eq:radius}).}
\label{fig:conv_radius}
\end{figure}
To find the optimal value for $n$, a circular cell is initialized with a radius $r\neq r_{eq}$~(\ref{eq:radius}), and it is let free to relax toward its equilibrium state.
In that particular case, no mass is added to or removed from the system, thus, the cell can only change its radius to balance bending, spring and internal pressure forces.
It is expected that if $n$ is high enough, then we should end up with $r=r_{eq}$ at the end of the simulation. If not, $n$ must be increased to reduce numerical errors.
Corresponding results are reported in Fig.~\ref{fig:conv_radius} for a number of vertices that varies from 20 to 300, three different values of the cell mass $m$, while keeping other parameters fixed. The latter mass values correspond to three equilibrium radii, and a fortiori, three different cell states (see Fig.~\ref{fig:cell_density}).
Interestingly, results show a rapid convergence of our model, and it is sufficient to rely on $n\approx 100$ vertices to discretize the cell membrane in an accurate manner.
\rcol{It is worth noting that similar convergence studies were conducted for test cases shown hereafter, and they confirmed that converged results are achieved for $n=120$.
Nevertheless, such a number of vertices might be too optimistic in case cells encounter large deformations. In that case, vertices should locally be added/removed, in order to accurately capture modifications of the membrane, by following rules described in Sec.~\ref{subsec:cell_description}.
In this work, deformations remain small, and consequently, it is not necessary to dynamically adjust the number of vertices unless cells are in a proliferating state. Hence, $n=120$ in Secs.~\ref{subsec:cell_compression} and~\ref{subsec:motility}, whereas it is dynamically adjusted for cell proliferation studies in Sec.~\ref{sec:Validation}.}
\subsection{Cell compression\label{subsec:cell_compression}}
Cells can generally be assimilated to incompressible, visco-elastic materials~\citep{JAMALI_PO_5_2010}.
To test both properties, cell compression is performed for different compression force intensities, and the cell area is monitored over time.
Qualitatively speaking, the simulation is initialised with a cell at equilibrium (Fig.~\ref{fig:compr_init}). The cell is compressed between two walls moving along the horizontal axis (Fig.~\ref{fig:compr_compr}). The wall movement is imposed through a constant force, that is eventually balanced by the cell inner pressure force (Fig.~\ref{fig:compr_max}). The two walls are then removed, and the cell can relax toward is equilibrium state (Figs.~\ref{fig:compr_relax} and~\ref{fig:compr_end}).
\begin{figure}[!btp]
\centering
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{initial_state_0.pdf}
\caption{Initial state: t=0}
\label{fig:compr_init}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{comp_20.pdf}
\caption{Compression: t=0.05}
\label{fig:compr_compr}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{init_400.pdf}
\caption{Max compression: t=1}
\label{fig:compr_max}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{relax_420.pdf}
\caption{Relaxation: t=1.05}
\label{fig:compr_relax}
\end{subfigure}
\begin{subfigure}{0.19\textwidth}
\includegraphics[width=\textwidth]{end_800.pdf}
\caption{Final relaxation: t=2}
\label{fig:compr_end}
\end{subfigure}
\caption{Time evolution of a cell compression/relaxation, starting from a circular state (black circle). The time scale is normalized by the number of iterations required to reach maximal compression.}
\label{fig:compr_evo}
\end{figure}
Looking more in-depth at the cell deformation, it is clear that the compression induces a non-linear behavior of the cell elongation (Fig.~\ref{fig:deformation}). In fact, the latter follows a linear slope in the early compression/relaxation state, which is then rapidly damped, hence confirming the visco-elastic behavior of the simulated cell. In addition, we do recover a linear behavior for low force amplitudes, whereas the cell starts showing some resistance to compression after $~20\%$ of elongation along the $y$-axis (Fig.~\ref{fig:stressstrain}).
It is interesting to note that similar tendencies were obtained in the context of cell monolayer compression/elongation, via both experiments and numerical simulations (based on vertex models), as presented by~\citep{HARRIS_PNAS_41_2012} and~\citep{MERZOUKI_SM_12_2016} respectively.
Moreover, the cell always recovers its initial shape at $t=2$, and during the whole simulation deviation from the initial area remain negligible, i.e., $\vert A(t=0)-A(t)\vert / A(t=0) \leq 10^{-5}$. This means the cell does have an incompressible behavior even if its inner structure (cytoskeleton, cytoplasm and nucleus) is not explicitly accounted for in our formalism.
\begin{figure}[!btp]
\centering
\hspace*{\fill}
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{compression_2.pdf}
\caption{Evolution of the cell deformation along the $y$-axis during the compression/relaxation process.}
\label{fig:deformation}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{stressstrain2.pdf}
\caption{Membrane elongation along $y$-axis with respect to the applied compression force.}
\label{fig:stressstrain}
\end{subfigure}
\hspace*{\fill}
\caption{Compression of a single cell. Left: Time evolution for different compression force intensities (the time scale is normalized by the number of iterations required to reach maximal compression) Right: Maximal cell elongation evolution with respect to the applied force.}
\label{fig:cell_compression}
\end{figure}
\subsection{Simplified cell migration in a tissue \label{subsec:motility}}
In the context of vertex models, the simulation of active cell migration requires the inclusion of a mechanism that imposes topological (T1) transitions~\citep{FLETCHER_PBMB_113_2013,MERZOUKI_PhD_2018}.
Hereafter, we show that such a property can be simulated in a simple manner, because interstitial gaps are naturally accounted for in our formalism. While this test case does not reproduce a specific biological system, it is nevertheless the starting point to simulate ordered migration within epithelia~\citep{TRICHAS_PLOSB_10_2012,GAUQUELIN_SM_15_2019}.
To validate this mechanism, we reproduce the same experiment as in~\citep{FLETCHER_PBMB_113_2013}, starting from a tissue composed of hexagonal cells (Fig.~\ref{fig:moving}). The latter is an idealized description of a cell monolayer, as experimentally observed by~\citep{LEWIS_AR_39_1928}. From this, an external force is applied to a single (black) cell. This force points toward the right, and its amplitude is chosen so that it can balance the pressure exerted by surrounding cells. Interestingly, the active cell is able to migrate toward the right by sneaking in between other cells. While moving, it leaves a small trail in its wake that is rapidly occupied by other cells. In absence of cell proliferation, the behavior of surrounding cells is purely mechanical, and it aims at decreasing internal constraints of the tissue.
This methodology is currently being extended to a full tissue of cells, in order to investigate how cell motility impacts the proliferation of epithelial monolayers in a confined environment, as experimentally investigated by~\citep{GAUQUELIN_SM_15_2019}. Corresponding results will be presented in a future work.
\begin{figure}[!ht]
\centering
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{move_high.pdf}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{move_high_1.pdf}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{move_high_3.pdf}
\end{subfigure}
\caption{Cell migration inside an epithelial tissue. By applying an external force to the (black) cell, active migration can be triggered.}
\label{fig:moving}
\end{figure}
\section{Cell tissue behavior\label{sec:Validation}}
\subsection{Free and chemically controlled proliferation \label{subsec:growth_w_wo_sig}}
A common validation test consists in letting cells proliferate in an unbounded domain, while monitoring their number of neighbors. The latter quantity is well documented, notably, thanks to measurements from biological experiments \citep{LEWIS_AR_39_1928} as well as numerical approaches like the vertex model \citep{FARHADIFAR_CB_17_2007,MERZOUKI_SM_12_2016}. Two particular cases will be investigated, namely, free and chemically controlled proliferation. The latter requires the simulation of a chemical signal, as discussed in Section~\ref{subsec:signaling}.
Let us start with the setup for the freely proliferating cell tissue.
It consists of a single (circular) cell that is in a proliferating state, and which is located at the center of the simulation domain (Fig.~\ref{fig:tiss_prolif_start}). This cell is free to grow and divide since no external constraint is applied to it. Each daughter cell is in a proliferating state and continues to give birth to new cells (Fig.~\ref{fig:tiss_prolif_mid}). In fact, only the external pressure exerted by surrounding cells can reduce their growth rate, as explained in Section~\ref{subsec:growth}. As there is no way to break the symmetry of the system, the shape of the final tissue stays almost circular (Fig.~\ref{fig:tiss_prolif_end}).
\begin{figure}[!ht]
\centering
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{tissu_beg.pdf}
\caption{Initial state}
\label{fig:tiss_prolif_start}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{tissu_mid.pdf}
\caption{Intermediate state}
\label{fig:tiss_prolif_mid}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{tissu_end.pdf}
\caption{End of the simulation with $\sim 400$ cells}
\label{fig:tiss_prolif_end}
\end{subfigure}
\caption{Unbounded tissue growth with cells in constant proliferating mode.}
\label{fig:tiss_prolif}
\end{figure}
Continuing with chemically controlled cell proliferation, the setup is similar to the previous one, with the exception that the mother cell is now producing a chemical signal that diffuses over the domain (Fig.~\ref{fig:tiss_prolif_chem_start}). During the whole simulation, this initial cell will be the sole cell to produce a chemical signal that can trigger the proliferating mode of surrounding cells. As explained in Section~\ref{subsec:signaling}, if the signal density inside a cell is above the switching threshold, then the cell can proliferate, otherwise the cell is considered to be at rest.
As time is advancing, the cell producing the signal moves, and its motion depends on (1) the splitting axis used for mitosis, and (2) forces exerted by surrounding cells (Fig.~\ref{fig:tiss_prolif_chem_mid}). As a consequence, the tissue is not guaranteed to have a circular shape anymore (Fig.~\ref{fig:tiss_prolif_chem_end}).
\begin{figure}[!ht]
\centering
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{tissu_beg_chem.pdf}
\caption{Initial state}
\label{fig:tiss_prolif_chem_start}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{tissu_mid_chem.pdf}
\caption{Intermediate state}
\label{fig:tiss_prolif_chem_mid}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{tissu_end_chem.pdf}
\caption{End of the simulation with 300 cells}
\label{fig:tiss_prolif_chem_end}
\end{subfigure}
\caption{Tissue growth with signaling. Cells are either in proliferating (black) or resting state (white). Superimposed to the cell tissue is the density of the chemical signal created by the initial (red) cell.}
\label{fig:tiss_prolif_chem}
\end{figure}
Experiments conducted by~\citep{LEWIS_AR_39_1928} show that, in average, each cell have six neighboring cells, which corresponds to cells with a hexagonal shape. This is not surprising as cells inside a tissue tend to fill all the gaps, eventually leading to a \textit{regular} tiling of the Euclidean plane. Interestingly, simulations show very similar results despite the simplicity of our model (see Table~\ref{tab:comp}). In addition, the chemical signaling only have little impact on the averaged shape of cells, as already reported for immersed-boundary-based methods~\citep{TANAKA_PhD_2016}.
\begin{table}[!ht]
\centering
\begin{tabular}{|l|c|}
\hline
& mean number of neighboring cell \\
\hline
Experiment Drosophila~\citep{LEWIS_AR_39_1928} & $5.99$\\
\rcol{Experiment Allium~\citep{Korn_Spalding_1973}} & $5.89$\\
\rcol{Experiment Dryopteris~\citep{Korn_Spalding_1973}} & $6.03$\\
\rcol{Experiment Euonymus~\citep{Korn_Spalding_1973}} & $6.04$\\
Simulation with chemical signaling & $5.93$\\
Simulation without signaling & $5.90$ \\
\hline
\end{tabular}
\caption{Cell distributions obtained by several experiments~\citep{LEWIS_AR_39_1928,Korn_Spalding_1973} and our simulations.}
\label{tab:comp}
\end{table}
Going into more details, experimental results highlight an asymmetric distribution of the number of neighboring cells~\citep{Korn_Spalding_1973,Gibson_Patel_Nagpal_Perrimon_2006,FARHADIFAR_CB_17_2007,Gibson_Gibson_2009}. Indeed, cells are more likely to have five neighbors instead of seven. This is also observed in our simulations with and without chemical signaling (see Fig.~\ref{fig:dist}). \rcol{It is interesting to note that, while the distribution obtained without chemical signaling fits a large set of experimental data, further accounting for chemical signaling only recovers results obtained experimentally for the Anacharis leaf and Volvox green algae~\citep{Korn_Spalding_1973}. One possible explanation might be that, for the latter experiments, an underlying mechanism is biasing the cell growth~\citep{Korn_Spalding_1973}), as it is the case for simulations with chemical signaling. Yet, this hypothesis remains to be confirmed by a more in-depth comparison based on a larger set of experimental data.}
\begin{figure}[H]
\centering
\includegraphics[trim=0 0.5cm 0 1cm,width=0.9\textwidth]{nneig_comp.pdf}
\caption{Distribution of the number of neighbors $N$, obtained by experiments~\citep{LEWIS_AR_39_1928,Korn_Spalding_1973,Gibson_Patel_Nagpal_Perrimon_2006,FARHADIFAR_CB_17_2007} and our simulations.}
\label{fig:dist}
\end{figure}
\subsection{Growth under constraint \label{subsec:capsule}}
Instead of using a chemical signaling to drive the state of the cell, it is common to rely on a stochastic process to switch from the resting state to its proliferation counterpart. This probability to switch from one state to the other can be related to the external pressure applied to the cell, similarly to what was proposed in Section~\ref{subsec:growth} to reduce (or even stop) the cell growth. Here it reads as
\begin{equation}\label{eq:probSwtich}
P_{switch} = \begin{cases}
a_{prolif} \left(1 - \frac{p_{ext}}{p_{max}} \right) & \text{if } p_{ext}\leq p_{max} \\
0 & \text{otherwise}
\end{cases}
\end{equation}
with $a_{prolif}$ a parameter used to better control the probability $P_{switch}$~(\ref{eq:probSwtich}). Hence, if $p_{ext}\leq p_{max}$ then there is a probability $P_{switch}>0$ for the cell to start proliferating, but if $p_{ext}> p_{max}$, the cell will stop proliferating. In practice, this probability to switch between states is tested at each iteration, with $a_{prolif}=10^{-3}$.
As a preliminary validation of such a mechanism, it is proposed to investigate the proliferation of cells inside a circular capsule. In practice, the capsule acts as an elastic soft wall, which can be deformed. The force required to deform is assumed to increase with the capsule radius. Hence, cells can push the capsule when proliferating, but they are more likely to invade the empty space at the center of the domain, due to the force applied on cells by the capsule.
This is numerically investigated, starting from an initial configuration of twenty cells that are uniformly distributed over the inner part of the capsule (Fig.~\ref{fig:stopGrowth_0}). The capsule is modeled using an external force that acts on the vertices of the cell, and whose amplitude depends on the distance from the capsule center:
\begin{equation}
\bm{F}_{ext}(\bm{x}_i) = \begin{cases}
- k_{caps} \frac{\bm{x}_i - \bm{c}_{caps}}{\left \lVert \bm{x}_i - \bm{c}_{caps} \right \rVert}\left(\left \lVert \bm{x}_i - \bm{c}_{caps} \right \rVert - r_{caps}\right) & \text{if } \left \lVert \bm{x}_i - \bm{c}_{caps} \right \rVert \geq r_{caps}\\
0 & \text{otherwise}
\end{cases}
\end{equation}
with $r_{caps}$ the capsule radius, $k_{caps}$ the capsule spring constant, $\bm{c}_{caps}$ the center of the capsule.
Corresponding results are compiled in Fig.~\ref{fig:capsule_stop}. As anticipated, once cells have populated the inner surface of the capsule, they tend to invade the center part of the simulation domain. While proliferating toward the capsule center, the pressure inside the cell tissue rises. Soon, more and more cells can no longer withstand it, and their proliferation ends. A comparison of the initial and final capsule size is proposed in Fig.~\ref{fig:capsule}.
\begin{figure}[!ht]
\centering
\begin{subfigure}{0.3\textwidth}
\includegraphics[width=\textwidth]{stop_0.pdf}
\caption{t = 0}
\label{fig:stopGrowth_0}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\includegraphics[width=\textwidth]{stop_1.pdf}
\caption{t = 1}
\label{fig:stopGrowth_1}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\includegraphics[width=\textwidth]{stop_2.pdf}
\caption{t = 2}
\label{fig:stopGrowth_2}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\includegraphics[width=\textwidth]{stop_3.pdf}
\caption{t = 3}
\label{fig:stopGrowth_3}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\includegraphics[width=\textwidth]{stop_4.pdf}
\caption{t = 4}
\label{fig:stopGrowth_4}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\includegraphics[width=\textwidth]{stop_5.pdf}
\caption{t = 5}
\label{fig:stopGrowth_5}
\end{subfigure}
\caption{Simulation of the cells growth inside a capsule. Black cells are in proliferation mode, whereas grey cells are at rest.}
\label{fig:capsule_stop}
\end{figure}
\begin{figure}[!ht]
\centering
\begin{subfigure}{0.4\textwidth}
\centering
\includegraphics[scale=0.5]{capsule_init.pdf}
\caption{Initial state of the capsule}
\end{subfigure}
\begin{subfigure}{0.4\textwidth}
\centering
\includegraphics[scale=0.5]{capsule_final.pdf}
\caption{Final state of the capsule}
\end{subfigure}
\caption{Evolution of the capsule size, with the actual size in blue and the initial size in red}
\label{fig:capsule}
\end{figure}
Ongoing research shows that with this kind of configuration, the fraction of cells that are proliferating is assumed to follow two distinct steps~\citep{ROUX_PRIVATE_2020}. At the beginning of the simulation, a constant ratio of proliferating cells is observed. This corresponds to the phase where cells are progressively invading the center part of the capsule without encountering any trouble in terms of external pressure. Once a large number of cells have been created, the pressure inside the tissue increases which decreases the number of cells that can continue to proliferate. At some point, the pressure is so high that no cell can proliferate anymore. This obviously depends on the parameter $p_{max}$, but also on the force applied by the capsule on cells. To properly understand the impact of each term, comparisons with experimental data are being performed. Corresponding results will be presented in a future work.
\section{Conclusion\label{sec:Conclusion}}
This paper proposes a new framework for detailed tissue morphogenesis. It relies on the numerical discretization of the cell membrane through approximately one hundred vertices, whose space and time evolution is governed by Newton's law of motion. A number of forces can be accounted for in order to reproduce the mechanical properties of a cell (elasticity, bending, incompressibility). Interestingly, the present formulation can also simulate active processes such as mitosis, apoptosis, and (simplified) cell migration even though the inner structure of the cell (cyoplasm, cytoskeleton and nucleus) is not explicitly modelled. All of this is presented in details, and preliminary studies are conducted to validate the proposed approach. This includes benchmark tests at a single-cell scale (compression and migration), as well as, at full tissue scale (free, chemically-driven and geometrically constrained proliferation). First results are encouraging, and they will be consolidated through more comparisons with experimental data in a future work. \rcol{Eventually, the proposed methodology only requires a discretization of the cell membrane in terms of vertices. Hence, it can easily be extended to simulate 3D configurations by considering a 3D tessellation of the cell membrane, and then applying forces to the corresponding vertices. Such a 3D formulation will be presented in a future work by~\citep{RUNSEN_UnderPreparation_2021}.}
\section*{Acknowledgement}
Fruitful discussions with D. Iber and her team, as well as, A. Roux and F. Raynaud are gratefully acknowledged. This work was supported by the Swiss National Science Fund SNF through the Sinergia Project No. 170930, ``A 3D Cell-Based Simulation Framework for Morphogenetic Problems''.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,249 |
Alain Bordeleau (born 7 October 1956) is a Canadian long-distance runner. He competed in the marathon at the 1984 Summer Olympics.
References
1956 births
Living people
Athletes (track and field) at the 1984 Summer Olympics
Canadian male long-distance runners
Canadian male marathon runners
Olympic track and field athletes of Canada
People from Lachine, Quebec
Athletes from Montreal | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,417 |
\section{Introduction}
Robustness is a key challenge in learning to navigate diverse, real-world environments. A robotic learning system must be robust to the difference between an offline training dataset and the real world (i.e., it must generalize), be robust to non-stationary changes in the real world (i.e., it must ignore visual distractors), and be equipped with mechanisms to actively explore to gather information about traversability. Different environments may exhibit similar physical structures, and these similarities can be used to accelerate exploration of \textit{new} environments. Learning-based methods provide an appealing approach for learning a representation of this shared structure using prior experience.
\setcounter{figure}{0}
\begin{wrapfigure}[12]{r}{0.3\columnwidth}
\vspace{-0.75em}
\centering
\includegraphics[width=0.3\columnwidth]{jackal.jpg}
\caption{\small{We demonstrate \mbox{RECON~} on a Clearpath Jackal.}}
\label{fig:robot}
\end{wrapfigure}
In this work, we consider the problem of navigating to a user-specified goal in a previously unseen environment.
The robot has access to a large and diverse dataset of experience from \emph{other} environments, which it can use to learn general navigational affordances. Our approach to this problem uses an information bottleneck architecture to learn a compact representation of goals. Learned from prior data, this latent goal model encodes prior knowledge about perception, navigational affordances, and short-horizon control.
We use a non-parametric memory to incorporate experience from the new environment. Combined, these components enable our system to learn to navigate to goals in a new environment after only a few minutes of exploration.
The primary contribution of this work is a method for exploring novel environments to discover user-specified goals. Our method operates directly on a stream of image observations, without relying on structured sensors or geometric maps. An important part of our method is a compressed representation of goal images that simultaneously affords robustness while providing a simple mechanism for exploration. Such a representation allows us, for example, to specify a goal image at one time of day, and then navigate to that same place at a different time of day: despite variation in appearance, the latent goal representations must be sufficiently close that the model can produce the correct actions. Robustness of this kind is critical in real-world settings, where the appearance of landmarks can change significantly with times of day and seasons of the year.
We demonstrate our method, \textbf{R}apid \textbf{E}xploration \textbf{C}ontrollers for \textbf{O}utcome-driven \textbf{N}avigation~(RECON), on a mobile ground robot (Fig.~\ref{fig:robot}) and evaluate against 6 competitive baselines spanning over 100 hours of real-world experiments in 8 distinct open-world environments (Fig.~\ref{fig:overview}). Our method can discover user-specified goals up to 80m away after just 20 minutes of interaction in a new environment.
We also demonstrate robustness in the presence of visual distractors and novel obstacles. We make this dataset publicly available as a source of real-world interaction data for future resesarch.
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{teaser_v2.pdf}
\vspace{-0.05in}
\captionof{figure}{\textbf{System overview:} Given a goal image \emph{(a)}, RECON~ explores the environment \emph{(b)} by sampling prospective \emph{latent} goals and constructing a \changeD{topological map of images} (white dots), operating only on visual observations. After finding the goal \emph{(c)}, RECON~ can reuse the map to reach arbitrary goals in the environment (red path in \emph{(b)}). RECON~ uses data collected from diverse
training environments \emph{(d)} to learn navigational priors that enable it to quickly explore and learn to reach visual goals a variety of unseen environments \emph{(e)}.
}
\vspace{-0.21in}
\label{fig:overview}
\end{figure}
\section{Related Work}
\label{sec:prior-work}
Exploring a new environment is often framed as the problem of efficient mapping, posed in terms of information maximization to guide the robot to uncertain regions of the environment.
Some prior exploration methods use local strategies for generating control actions for the robots~\cite{kuipers1991exploration, bourgault2002exploration, kollar2008optimization, tabib2019real}, while others use use global strategies based on the frontier method~\cite{yamauchi1997frontier, charrow2015mapping, holz2011frontier}. However, building high-fidelity geometric maps can be hard without reliable depth information. Further, such maps do not encode semantic aspects of traversability, e.g., tall grass is traversable but a wire fence is not.
Inspired by prior work~\citep{savinov2018sptm, faust2018prm, eysenbach2019sorb, shah2020ving, meng2020scaling}, we construct a topological map by learning a distance function and a low-level policy. We estimate distances via supervised regression and learn a local control policy via goal-conditioned behavior cloning~\citep{ghosh2019learning, lynch2020learning}. However, these prior methods do not describe how to learn to navigate in \emph{new}, unseen environments. We equip RECON~ with an explicit mechanism for exploring new environments and transferring knowledge across environments.
Well-studied methods for exploration in reinforcement learning (RL) utilize a novelty-based bonus, computed from a predictive model~\citep{stadie2015incentivizing, pathak2017curiosity, bajcsy2018revisiting, burda2018exploration, satsangi2020maximizing, savinov2018episodic, chaplot2020ans}, information gain~\citep{houthooft2016vime,mirchev2018approximate}, or methods based on counts, densities, or distance from previously-visited states~\citep{bellemare2016unifying, hazan2019provably, lee2019efficient}. However, these methods learn to reason about the novelty of a state only after visiting it. Recent works~\citep{chaplot2020nts, chaplot2020semantic} improve upon this by predicting explorable areas for interesting parts of the environment to accelerate visual exploration.
While these methods can yield state-of-the-art results in simulated domains~\citep{savva2019habitat, Kolve2019AI2THOR}, they come at the cost of high sample complexity (over 1M samples) and are infeasible to train in open-world environments without a simulated counterpart. Instead, our method enables the robot to explore an environment from scratch in just 20 minutes, using prior experience from other environments.
The problem of reusing experience across tasks is studied in the context of meta-learning~\citep{duan2016rl, saemundsson2018meta, nagabandi2018learning} and transfer learning~\citep{taylor2007cross, lazaric2012transfer, parisotto2015actor, gupta2017learning, gamrian2019transfer}. Our method uses an information bottleneck~\citep{tishby2000information}, which serves a dual purpose: first, it provides a representation that can aid the generalization capabilities of RL algorithms~\citep{igl2019generalization, goyal2019infobot}, and second, it serves as a measure of task-relevant uncertainty~\citep{alemi2016deep}, allowing us to incorporate prior information for proposing goals that are functionally-relevant for learning control policies in the new environment.
The problem of learning goal-directed behavior has been studied extensively using RL~\citep{kaelbling1993learning, schaul2015universal, pong2018temporal, eysenbach2020c} and imitation learning (IL)~\citep{ding2019goal, ghosh2019learning, lynch2020learning, sun2019policy, srivastava2019training,Rhinehart2020Deep}. Our method builds upon prior goal-conditioned IL methods to solve a slightly different problem: how to reach goals in a \emph{new} environment. Once placed in a new environment, our method explores by carefully choosing which goals to visit, inspired by prior work~\citep{pong2019skew, colas2019curious, zhang2020world, pitis2020maximum, liu2020hallucinative}.
Unlike these prior methods, however, our method makes use of previous experience in \emph{different} environments to accelerate learning in the current environment.
\section{Problem Statement and System Overview}
\label{sec:overview}
We consider the problem of goal-directed exploration for visual navigation in novel environments: a robot is tasked with navigating to a goal location $G$, given an image observation $o_g$ taken at $G$. Broadly, this consists of three separate stages: (1) learning from offline data, (2) building a map in a new environment, and (3) navigating to goals in the new environment.
We model the task of navigation as a Markov decision process with time-indexed states $s_t \in {\mathcal{S}}$ and actions $a_t \in {\mathcal{A}}$. We \emph{do not assume} the robot has access to spatial localization or a map of the environment, or access to the system dynamics. We use videos of robot trajectories in a variety of environments to learn general navigational skills and build a compressed representation of the perceptual inputs, which can be used to guide the exploration of novel environments. We make no assumption on the nature of the trajectories: they may be obtained by human teleoperation, self-exploration, or as a result of a preset policy. These trajectories need not exhibit intelligent behavior. Since the robot only observes the world from a single on-board camera and does not run any state estimation, our system operates in a partially observed setting. Our system commands continuous linear and angular velocities.
\subsection{Mobile Robot Platform}
\label{sec:jackal_system}
We implement RECON~ on a Clearpath Jackal UGV platform
(see Fig.~\ref{fig:robot}). The default sensor suite consists of a 6-DoF IMU, a GPS unit for approximate global position estimates, and wheel encoders to estimate local odometry. In addition, we added a forward-facing $170^\circ$ field-of-view RGB camera and an RPLIDAR 2D laser scanner. Inside the Jackal is an NVIDIA Jetson TX2 computer. The GPS and laser scanner can become unreliable in some environments~\cite{kahn2020badgr}, so we use them solely as safety controllers during data collection. Our method operates only using images taken from the onboard RGB camera, without other sensors or ground-truth localization.
\subsection{Self-Supervised Data Collection \& Labeling}
\label{sec:data_collection}
Our aim is to leverage data collected in a wide range of different environments to enable the robot to discover and learn to navigate to novel goals in novel environments. We curate a dataset of self-supervised trajectories collected by a time-correlated random walk in diverse real-world environments (see Fig.~\ref{fig:overview} (d,e)). This data was collected over a span of $18$ months and exhibits significant variation in appearance due to seasonal and lighting changes. We make this dataset publicly available\footnote{Available for download at \href{https://sites.google.com/view/recon-robot/dataset}{\texttt{sites.google.com/view/recon-robot/dataset}}.} and provide further details in Appendix~\ref{appendix:dataset}.
\section{RECON~: A Method for Goal-Directed Exploration}
\label{sec:method}
Our objective is to design a robotic system that uses visual observations to efficiently discover and reliably reach a target image in a previously unseen environment. RECON~ consists of two components that enable it to explore new environments. The first is an uncertainty-aware, context-conditioned representation of goals that can quickly adapt to novel scenes. \changeD{The second component is a topological map, where nodes represent egocentric observations and edges are the predicted distance between them, constructed incrementally from frontier-based exploration, maintaining a compact memory of the target environment.}
\subsection{Learning to Represent Goals}
\label{sec:representation}
Our method learns a compact representation of goal images that is robust to task-irrelevant factors of variation. We learn this representation using a variant of the information bottleneck architecture~\cite{alemi2016deep, achille2018emergence}. We use a context-conditioned representation of goals to learn a control policy in the target environment (Fig.~\ref{fig:architecture} describes the graphical model). Letting $I(\cdot; \cdot)$ denote mutual information, the objective in Eq.~\ref{eq:bottleneck_objective} encourages the model to compress the incoming goal image $o_g$ into a representation $z_t^g$ conditioned on the current observation $o_t$ that is predictive of the best action $a_t^g$ and the temporal distance $d_t^g$ to the goal (upper-case denotes random variables):
\begin{align}
I\big((A_t^g, D_{t}^g); Z_t^g \mid o_t \big) - \beta I(Z_t^g; O_g\mid o_t) \label{eq:bottleneck_objective}
\end{align}
Following~\cite{alemi2016deep}, we approximate the intractable objective in Eq.~\ref{eq:bottleneck_objective} with a variational posterior and decoder (an upper bound), resulting in the maximization objective:
\begin{align}
L = \frac{1}{|\mathcal D|} \sum_{\substack{\\(o_t, o_g, a_t^g, d_t^g)\in \mathcal D}} & \mathbb E_{p_\phi(z_t^g\mid o_g, o_t)} \left[\log q_\theta\big(a_t^g, d_t^g \mid z_t^g, o_t\big)\right] -\beta \mathrm{KL}\left(p_\phi(\cdot \mid o_g, o_t) || r(\cdot) \right) \label{eq:learning_objective}
\end{align}
\begin{wrapfigure}{R}{0.4\textwidth}
\vspace{-2.5em}
\input{figures/bottleneck_graph_v2}
\centering
\resizebox{0.4\columnwidth}{!}{
\begin{tikzpicture}[->,>=stealth',scale=1, transform shape]
\bottleneckgraph
\end{tikzpicture}}
\caption{{\scriptsize Graphical model of actions and distances} }
\label{fig:architecture}
\end{wrapfigure}
where we define the prior $r(z_t^g)\triangleq \mathcal N(0, I)$ and $\mathcal{D}$ is \changeD{a dataset of trajectories characterized by $(o_t, o_g, a_t^g, d_t^g)$ quadruples}. The first term measures the model's ability to predict actions and distances from the encoded representation, and the second term measures the model's compression of incoming goal images.
As the encoder $p_\phi$ and decoder $q_\theta$ are conditioned on $o_t$, the representation $z_t^g$ only encodes information about \emph{relative} location of the goal from the context -- this allows the model to represent \emph{feasible} goals. If, instead, we had a typical VAE (in which the input images are autoencoded), the samples from the prior over these representations would not necessarily represent goals that are reachable from the current state. This distinction is crucial when exploring \emph{new} environments, where most states from the training environments are not valid goals.
\subsection{Goal-Directed Exploration with Topological Memory}
\label{sec:exploration_alg}
\begin{algorithm}[b]
\caption{\textit{RECON~ for Exploration:}
RECON~ takes as input an encoder $p_\phi$, a decoder $q_\theta$, prior $r$, the current observation $o_t$ and goal observation $o_g$. $\delta_1,\delta_2,\epsilon, \beta \in \mathbb R_{+};H,\gamma \in \mathbb N$ are hyperparameters.}\label{alg:exploration}
\begin{algorithmic}[1]
\Function{RECON~}
{$q_\theta,p_\phi,r,o_t,o_g; \delta_1,\delta_2, \epsilon, \beta, \gamma, H$}
\State ${\mathcal{G}} \gets \emptyset, {\mathcal{D}} \gets \emptyset$ \Comment{\emph{Initialize graph and data}}
\While{not reached goal $[\bar{d}_t^g < \delta_1]$} \Comment{\emph{Continue while not at goal}}
\State $o_n \gets \mathrm{LeastExploredNeighbor}({\mathcal{G}}, o_t; \delta_2)$
\State $z_{t}^g \sim p_\phi(z \mid o_t, o_g)$ \Comment{\emph{Encode relative goal}}
\If{goal is feasible $[r(z_t^g) > \epsilon]$}
\State $z_t^w \gets z_t^g$ \label{algline:feasible} \Comment{{\em Will go to the goal}}
\ElsIf{robot at frontier $[\bar{d}_t^n < \delta_1]$}
\State $z_{t}^w \sim r(z)$ \label{algline:from_frontier}\Comment{\emph{Will explore from frontier}}
\Else
\State $z_{t}^w \sim p_\phi(z \mid o_t, o_n)$ \label{algline:to_frontier}\Comment{{\em Will go to frontier}}
\EndIf
\State $\mathcal D_w, o_t \gets \mathrm{SubgoalNavigate}(z_t^w; H)$ \label{algline:subgoal_navigate}
\State ${\mathcal{D}} \gets {\mathcal{D}} \cup \mathcal D_w$
\State \changeD{$\mathrm{ExpandGraph}({\mathcal{G}}, o_t)$} \label{algline:expand_graph}
\State Step $L(\phi, \theta; \mathcal D, \beta)$ for $\gamma$ epochs \Comment{Eq.~\ref{eq:learning_objective}} \label{algline:finetune}
\EndWhile
\State \textbf{return} networks $p_\phi, q_\theta$ and graph ${\mathcal{G}}$
\EndFunction
\end{algorithmic}
\end{algorithm}
The second component of our system is a topological memory constructed incrementally as the robot explores a new environment. It provides an estimate of the exploration frontier as well as a map that the robot can use to later navigate to arbitrary goals. To build this memory, the robot uses the model from the previous section to propose \emph{subgoals} for data collection.
Note that this is done in the exploration phase and have a latent goal model pre-trained on the offline dataset. Given a subgoal, our algorithm (Alg.~\ref{alg:exploration}) proceeds by executing actions towards the subgoal for a fixed number of timesteps (Alg.~\ref{alg:exploration} L\ref{algline:subgoal_navigate}). The data collected during subgoal navigation expands the topological memory (Alg.~\ref{alg:exploration} L\ref{algline:expand_graph}) and is used to fine-tune the model (Alg.~\ref{alg:exploration} L\ref{algline:finetune}). Thus, the task of efficient exploration is reduced to the task of choosing subgoals.
\begin{wrapfigure}{R}{0.5\columnwidth}
\vspace*{-0.5em}
\begin{minipage}[t]{.5\columnwidth}
\begin{algorithm}[H]
\caption{\emph{RECON~ for Goal-Reaching:} After exploration, RECON~ uses the topological graph ${\mathcal{G}}$ to quickly navigate towards the goal $o_g$.}\label{alg:postexplore_rollout}
\begin{algorithmic}[1]
\Procedure{$\mathrm{GoalNavigate}$}{${\mathcal{G}}, o_t, o_g; H$}
\State $v_t \gets \mathrm{AssociateToVertex}({\mathcal{G}}, o_t)$
\State $v_g \gets \mathrm{AssociateToVertex}({\mathcal{G}}, o_g)$
\State $(v_t, \dots, v_g) \gets \mathrm{ShortestPath}({\mathcal{G}}, v_t, v_g)$
\For{$v \in (v_t, \dots, v_g)$}
\State $z \gets p_\phi(z \mid o_t, o_g=v\mathrm{.o})$
\State $\mathcal D_w, o_t \gets \mathrm{SubgoalNavigate}(z; H)$
\EndFor
\EndProcedure
\end{algorithmic}
\end{algorithm}
\end{minipage}
\vspace*{-0.5em}
\end{wrapfigure}
Subgoals are represented by latent variables in our model, which may either come from the posterior $p_\phi(z|o_t,o_g)$, or from the prior $r(z)$. Given a subgoal $z$ and observation~$o_t$, the model decodes it into an action and distance pair $q(a_t^g, d_t^g|z, o_t)$; the action is used to control the robot towards the goal, and the distance is used to construct edges in the topological graph. The choice of intermediate subgoal to navigate toward at any step is based on the robot's estimate of the goal reachability and its proximity to the frontier. To determine the frontier of the graph, we track the number of times each node in the graph was selected as the navigation goal; nodes with low counts are considered to be on the frontier. In the following, we use $\bar{z}_t^g$ to denote the mean of the encoder $p_\phi(z \mid o_t, o_g)$, and $\bar{d}_t^g$ to denote the distance component of the mean of the decoder $q_\theta(a_t, d_t \mid \bar{z}_t^g, o_t)$ (i.e., the predicted number of time steps from $o_t$ to $\bar{z}_t^g$). The choice of subgoal at each step is made as follows:
\noindent \textbf{(i) Feasible Goal:} The robot believes it can reach the goal and adopts the representation of the goal image as the subgoal (Alg.~\ref{alg:exploration} L\ref{algline:feasible}). The robot's confidence in reaching the goal is based on the probability of the current goal embedding $z_t^g$ under the prior $r(z)$. \changeD{Large $r(z_t^g)$ implies the relationship between the observation $o_t$ and the goal $o_g$ is \emph{in-distribution}, suggesting that the model's estimates of the distances is reliable -- intuitively, this means that the model is confident about the distance to $o_g$ and can reach it.}
\noindent \textbf{(ii) Explore at Frontier:} The robot is at \changeA{the ``least-explored node'' (frontier) $o_n$} and explores by sampling a random conditional subgoal latent $z_t^w$ from the prior (Alg.~\ref{alg:exploration} L\ref{algline:from_frontier}). \changeD{The robot determines whether it is at the frontier based on the distance (estimated by querying the model) to its ``least explored neighbor'' $\bar{d}_t^n$} -- \changeA{the node in the graph within a distance threshold ($\delta_2$) of the current observation that has the lowest visitation count.} \changeD{If the distance to this node $\bar{d}_t^n$ is low (threshold $\delta_1$), then the robot is at the frontier.}
\noindent \textbf{(iii) Go to Frontier:} The robot adopts its \changeA{``least-explored neighbor'' $o_n$ as a subgoal} (Alg.~\ref{alg:exploration} L\ref{algline:to_frontier}).
\changeA{The $\mathrm{SubgoalNavigate}$ function rolls out the learned policy for a fixed time horizon $H$ to navigate to the desired subgoal latent $z^w_t$, by querying the decoder $q_\theta(a_t, d_t | z_t^w, o_\tau)$ with a fixed subgoal latent.} The endpoint of such a rollout is used to update the visitation counts in the graph ${\mathcal{G}}$. At the end of each trajectory, the $\mathrm{ExpandGraph}$ subroutine is used to update the edge and node sets $\{{\mathcal{E}}, {\mathcal{V}}\}$ of the graph ${\mathcal{G}}$ to update the representation of the environment. We provide the pseudocode for these subroutines in Appendix~\ref{appendix:alg}. We also share broader implementation details including choice of hyperparameters, model architectures and training details in Appendix~\ref{appendix:params}.
\subsection{System Summary}
RECON~ uses the latent goal model and topological graph to quickly explore new environments and discovers user-specified goals. Our complete system consists of three stages:
\begin{enumerate}[leftmargin=14pt, label=\Alph*)]
\item \emph{Prior Experience:} The goal-conditioned distance and action model (Sec.~\ref{sec:representation}) is trained using experience from previously visited environments. Supervision for training our model is obtained by using time steps as a proxy for distances and \changeC{a relabeling scheme (Appendix~\ref{appendix:dataset})}.
\item \emph{Exploring a Novel Environment:} When placed in a new environment, RECON~ uses a combination of frontier-based exploration and latent goal-sampling with the learned model. The learned model is also fine-tuned to this environment. These steps are summarized in Alg.~\ref{alg:exploration} and Sec.~\ref{sec:exploration_alg}.
\item \emph{Navigating an Explored Environment:} Given an explored environment (represented by a topological graph ${\mathcal{G}}$) and the model, RECON~ uses ${\mathcal{G}}$ to navigate to a goal image by planning a path of subgoals through the graph. This process is summarized in Alg.~\ref{alg:postexplore_rollout}.
\end{enumerate}
\iffalse
\begin{figure*}[!ht]
\centering
\begin{subfigure}{0.475\textwidth}
\centering
\includegraphics[width=0.95\columnwidth]{quantitative_explore.png}
\caption{}
\end{subfigure}
\begin{subfigure}{0.475\textwidth}
\centering
\includegraphics[width=0.95\columnwidth]{quantitative_nav.png}
\caption{}
\end{subfigure}
\caption{\textbf{Exploring and Learning to Reach Goals:} \emph{(left)} Amount of time needed for each method to search for the goals in a new environment ($\downarrow$ is better; hashed out bars represent failure).
\emph{(right)} Amount of time needed to reach the goal a second time, after reaching the goal once and constructing the map, in seconds ($\downarrow$ is better).}
\label{fig:quantitative}
\end{figure*}
\fi
\definecolor{OurDarkGrey}{cmyk}{0,0,0,0.1}
\begin{table}
\centering {\footnotesize
\begin{tabular}{l ccc}
\toprule
Method & Expl. Time (mm:ss) $\downarrow$ & Nav. Time (mm:ss) $\downarrow$ & {SCT~\citep{yokoyama2021success}} $\uparrow$ \\
\midrule
PPO + RND~\citep{burda2018exploration} & 21:18 & 00:47& 0.22 \\
InfoBot~\citep{goyal2019infobot} & 23:36 & 00:48& 0.21 \\
{Active Neural SLAM (ANS)~\citep{chaplot2020ans}} & 21:00 & 00:45 & 0.33 \\
ViNG~\citep{shah2020ving} & 19:48 & 00:34 & 0.60 \\
\hline
\rowcolor{OurDarkGrey}Ours + Episodic Curiosity (ECR)~\citep{savinov2018episodic} & 14:54 & 00:31& 0.73 \\
\rowcolor{OurDarkGrey}\textbf{RECON~ (Ours)} & \textbf{09:54} & \textbf{00:26}& \textbf{0.92}\\
\bottomrule \\
\end{tabular}}
\caption{{\textbf{Exploration and goal reaching performance:}
Exploring 8 real-world environments, RECON~ reaches the goal 50\% faster than the best baseline (ECR).
ANS takes up to 2x longer to find the goal and NTS~\cite{chaplot2020nts} fails to find the goal in every environment.
On subsequent traversals, RECON~ navigates to the goal 20--85\% faster than other baselines, and exhibits $>\!30\%$ higher weighted success.}} \label{tab:result_averages}
\vspace{-1.5em}
\end{table}
\section{Experimental Evaluation}
\label{sec:experiments}
We designed our experiments to answer four questions:
\begin{enumerate}[label={\bf Q\arabic{*}.}, leftmargin=24pt]
\item How does RECON~ compare to prior work for visual goal discovery in novel environments?
\item After exploration, can RECON~ leverage its experience to navigate to the goal efficiently?
\item What is the range of perturbations and non-stationary elements to which RECON~ is robust?
\item How important are the various components of RECON, such as sampling from an information bottleneck and non-parametric memory, to its performance?
\end{enumerate}
\setlist[enumerate]{leftmargin=0.6cm}
\subsection{Goal-Directed Exploration in Novel Environments}
\label{sec:main_results}
\begin{figure*}[h]
\centering
\vspace{-1em}
\begin{subfigure}{0.54\textwidth}
\includegraphics[width=\columnwidth]{generalization_on_qualitative_v4.jpg}
\end{subfigure}%
~\;~
\begin{subfigure}{0.44\textwidth}
\includegraphics[width=\columnwidth]{discovered_all.jpg}
\end{subfigure}
\caption{{\textbf{Visualizing goal-reaching behavior of the system:}
\emph{(left)} Example trajectories to goals discovered by RECON~ in \textit{previously unseen} environments.
\emph{(right)} Policies learned by the different methods in one such environment. Only RECON~ and ECR reach the goal successfully, and RECON~ takes the shorter route.}}
\vspace{-0.2em}
\label{fig:discovered_all}
\end{figure*}
We perform our evaluation in a diverse variety of outdoor environments (examples in Fig.~\ref{fig:overview}), including parking lots, suburban housing, sidewalks, and cafeterias. We train our self-supervised navigation model using an offline navigation dataset (Sec.\ref{sec:data_collection}) collected in a distinct set of training environments, and evaluate our system's ability to discover user-specified goals in previously unseen environments. {We compare RECON~ to five baselines, each trained on the same 20 hours of offline data as our method, and finetuned in the target environment with online interaction.
\begin{enumerate}
\item {\bf PPO + RND:} Random Network Distillation (RND) is a widely used prediction bonus-based exploration strategy in RL~\cite{burda2018exploration}, which we use with PPO~\cite{schulman2017ppo, wijmans2020ddppo}. This comparison is representative of a frequently used approach for exploration in RL using a novelty-based bonus.
\item {\bf InfoBot:} An offline variant of InfoBot~\cite{goyal2019infobot}, which uses goal-conditioned information bottleneck, analogous to our method, but does not use the non-parametric memory.
\item {\bf Active Neural SLAM (ANS):} A popular indoor navigation approach based on metric spatial maps proposed for coverage-maximizing exploration~\citep{chaplot2020ans}. We adapt it to the goal-directed task by using the distance function from RECON~ to detect when the goal is nearby.
\item {\bf Visual Navigation with Goals (ViNG):} A method that uses random action sequences to explore and incrementally build a topological graph without reasoning about visitation counts~\cite{shah2020ving}.
\item {\bf Episodic Curiosity (ECR):} A method that executes random action sequences at the frontier of a topological graph for exploration~\citep{savinov2018episodic}. We implement this as an \textit{ablation} of our method that samples random action sequences, rather than rollouts to sampled goals (Alg.~\ref{alg:exploration} Line 7).
\end{enumerate}}
We evaluate the ability of RECON~ to discover visually-indicated goals in $8$ \emph{unseen} environments and navigate to them repeatedly. For each trial, we provide an RGB image of the desired target (one per environment) to the robot and report the time taken by each method to \emph{(i)} discover the desired goal ({\bf Q1}), and \emph{(ii)} reliably navigate to the discovered goal a second time using prior exploration ({\bf Q2}). Additionally, we quantify navigation performance using Success weighed by Completion Time (SCT), a success metric that takes into account the agent's dynamics~\cite{yokoyama2021success}. We show quantitative results in Table~\ref{tab:result_averages}, and visualize sample trajectories of RECON~ and the baselines in Fig.~\ref{fig:discovered_all}.
\begin{wrapfigure}{R}{0.45\columnwidth}
\vspace*{-1em}
\centering
\includegraphics[width=0.45\columnwidth]{nonstationary.jpg}
\caption{\textbf{Exploring non-stationary environments:} The learned representation and topological graph is robust to visual distractors, enabling reliable navigation to the goal under novel obstacles \emph{(c--e)} and appearance changes \emph{(f--h)}. }
\label{fig:nonstationary}
\end{wrapfigure}
RECON~ outperforms all the baselines, discovering goals that are up to 80m away in under 20 minutes, including instances where no other baseline can reach the goal successfully. RECON+ECR and ViNG discover the goal in only the easier environments, and take up to 80\% more time to discover the goal in those environments.
RND, InfoBot and ANS are able to discover goals that are up to $25$m away but fails to discover more distant goals, likely because using reinforcement learning for fine-tuning is data-inefficient. We exclude reporting metrics on NTS, which fails to successfully explore any environment, likely due to overfitting to the offline trajectories. Indeed, the simulation experiments reported in each of these online algorithms require upwards of $1$M timesteps to adapt to new environments~\cite{goyal2019infobot, chaplot2020ans, chaplot2020nts}. We attribute RECON's success to the context-conditioned sampling strategy (described in Sec.~\ref{sec:representation}), which proposes
goals that can accelerate the exploration of new environments.
We then study RECON's ability to quickly reach goals after initial discovery. Table~\ref{tab:result_averages} shows that RECON~ variants are able to quickly recall a feasible path to the goal. These methods create a compact topological map from experience in the target environment, allowing them to quickly reach previously-seen states. The other baselines are unsuccessful at recalling previously seen goals for all but the simplest environments. Fig.~\ref{fig:discovered_all} shows an aerial view of the paths recalled by various methods in one of the environments. Only the RECON~ variants are successfully able to navigate to the checkerboard goal; all other baselines result in collisions in the environment. Further, RECON~ discovers a shorter path to the goal and takes $30\%$ less time to navigate to it than ECR ablation.
\subsection{Exploring Non-Stationary Environments}
Outdoor environments exhibit non-stationarity due to dynamic obstacles, such as automobiles and people, as well as changes in appearance due to seasons and time of day. Successful exploration and navigation in such environments requires learning a representation that is invariant to such distractors. This capability is of central interest when using a non-parametric memory: for the topological map to remain valid when such distractors are presented, we must ensure the invariance of the learned representation to such factors (\textbf{Q3}).
\changeA{To test the robustness of RECON to unseen obstacles and appearance changes}, we first had RECON~ explore in a new ``junkyard'' to learn to reach a goal image containing a blue dumpster (Fig.~\ref{fig:nonstationary}-a). Then, without any more exploration, we evaluated the learned goal-reaching policy when presented with \emph{previously unseen} obstacles (trash cans, traffic cones, and a car) and and weather conditions (sunny, overcast, and twilight).
Fig.~\ref{fig:nonstationary} shows trajectories taken by the robot as it successfully navigates to the goal in scenarios with varying obstacles and lighting conditions.
These results suggest that the learned representations are invariant to visual distractors that do not affect robot's decisions to reach a goal (e.g., changes in lighting conditions do not affect the trajectory to goal, and hence, are discarded by the bottleneck).
\subsection{Dissecting RECON}
RECON~ explores by sampling goals from the prior distribution over state-goal representations.
To quantify the importance of this exploration strategy (\textbf{Q4}), we deploy RECON~ to perform undirected exploration in a novel target environment \emph{without building a graph} of the environment. We compare the coverage of trajectories of the robot over $5$ minutes of exploration when: \emph{(a)} it executes random action sequences~\cite{savinov2018episodic}, and \emph{(b)} it performs rollouts towards sampled goals. We see that performing rollouts to sampled goals results in $5$x faster exploration in novel environments (see Fig.~\ref{fig:sampling}).
We also evaluate several variants of RECON~ that ablate its goal sampling and non-parametric memory on the end-to-end task of visual goal discovery in novel environments:
\begin{enumerate}
\item[-] \emph{Reactive:} our method deployed \emph{without} the topological graph for memory.
\item[-] \emph{Random Actions:} a variant of our method that executes random action sequences at the frontier rather than rollouts to sampled goals. This is identical to the ECR baseline described in Sec.~\ref{sec:main_results}.
\item[-] \emph{Vanilla Sampling:} a variant of our method which learns a goal-conditioned policy and distances \emph{without} an information bottleneck to obtain compressed representations.
\end{enumerate}
\begin{figure}[t!]
\centering
\begin{minipage}{0.4\textwidth}
\centering
\includegraphics[width=\columnwidth]{figures/exploration.jpg}
\caption{\textbf{Exploration via sampling} from our context-conditioned prior (\emph{right}) allows the robot to explore 5 times faster than using random actions, e.g. in ECR~\citep{savinov2018episodic} (\emph{left}).}
\label{fig:sampling}
\end{minipage}\hspace{0.5em}
\begin{minipage}{0.58\textwidth}
\centering
{\footnotesize
\begin{tabular}{l m{1.3cm} m{1.3cm} m{0.8cm}}
\toprule
Method & Expl. Time (mm:ss) $\downarrow$ & Nav. Time (mm:ss) $\downarrow$ & {SCT \citep{yokoyama2021success}} $\uparrow$ \\
\midrule
{Reactive} & 11:54 & 00:37.4 & 0.63\\
Random Actions & 14:54 & 00:31.4 & 0.73 \\
Vanilla Sampling & 14:06 & 00:28.7 & 0.83 \\
\textbf{Ours} & \textbf{09:56} &\textbf{00:25.8} & \textbf{0.92} \\
\bottomrule \\
\end{tabular}}
\captionof{table}{\textbf{Ablation experiments} confirm the importance of using an information bottleneck and a non-parametric memory.} \label{tab:ablations}
\end{minipage}
\vspace*{-1em}
\end{figure}
We deploy these variants in a subset of the unseen test environments and summarize their performance in Table~\ref{tab:ablations}. These results corroborate the observations in Fig.~\ref{fig:sampling}: learning a compressed goal representation is key to the performance of RECON. ``Vanilla Sampling'', despite sampling from a joint prior, performs poorly and is unable to discover distant goals. We hypothesize that our method is more robust because the information bottleneck helps learn a representation that ignores task-irrelevant information.
We also observe that ``Reactive'' experiences a smaller degradation in exploration performance, suggesting that goal-sampling can help with the exploration problem even without the graph. However, we find a massive degradation in its ability to recall previously discovered goals, suggesting that the memory is key to the navigation performance of RECON.
\section{Discussion}
\label{sec:conclusion}
We proposed a system for efficiently learning goal-directed policies in new open-world environments. The key idea behind our method is to use a learned goal-conditioned distance model with a latent variable model representing visual goals for rapid goal-directed exploration. The problem setup studied in this paper, using past experience to accelerate learning in a \textit{new} environment, is reflective of real-world robotics scenarios: collecting new experience at deployment time is costly, but experience from prior environments can provide useful guidance to solve new tasks.
In future work, we aim to provide theoretical guarantees for when and where we can expect stochastic policies and the information bottleneck to provide efficient exploration. One limitation of the current method is that it does not explicitly account for the value of information. Accounting for such states can generate a better goal-reaching policy.
\section*{Acknowledgments}
This research was supported by ARL DCIST CRA W911NF-17-2-0181, DARPA Assured Autonomy, and the Office of Naval Research. The authors would like to thank Suraj Nair and Brian Ichter for useful discussions, and Gregory Kahn for setting up the infrastructure used for autonomous collection of real-world data.
\bibliographystyle{IEEEtran}
| {
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package org.apereo.cas.oidc.dynareg;
import com.fasterxml.jackson.annotation.JsonIgnoreProperties;
import com.fasterxml.jackson.annotation.JsonProperty;
import org.apache.commons.lang3.builder.ToStringBuilder;
import org.apache.commons.lang3.builder.ToStringStyle;
import java.io.Serializable;
import java.util.List;
/**
* This is {@link OidcClientRegistrationRequest}.
*
* @author Misagh Moayyed
* @since 5.1.0
*/
@JsonIgnoreProperties(ignoreUnknown = true)
public class OidcClientRegistrationRequest implements Serializable {
private static final long serialVersionUID = 1832102135613155844L;
@JsonProperty("redirect_uris")
private List<String> redirectUris;
@JsonProperty("client_name")
private String clientName;
@JsonProperty("token_endpoint_auth_method")
private String tokenEndpointAuthMethod;
@JsonProperty("scope")
private String scope;
@JsonProperty("grant_types")
private List<String> grantTypes;
@JsonProperty("response_types")
private List<String> responseTypes;
@JsonProperty("jwks_uri")
private String jwksUri;
@JsonProperty("request_object_signing_alg")
private String requestObjectSigningAlg;
public OidcClientRegistrationRequest() {
}
public String getScope() {
return scope;
}
public List<String> getRedirectUris() {
return redirectUris;
}
public String getClientName() {
return clientName;
}
public String getTokenEndpointAuthMethod() {
return tokenEndpointAuthMethod;
}
public List<String> getGrantTypes() {
return grantTypes;
}
public List<String> getResponseTypes() {
return responseTypes;
}
public String getJwksUri() {
return jwksUri;
}
public String getRequestObjectSigningAlg() {
return requestObjectSigningAlg;
}
@Override
public String toString() {
return new ToStringBuilder(this, ToStringStyle.NO_CLASS_NAME_STYLE)
.append("redirectUris", redirectUris)
.append("clientName", clientName)
.append("tokenEndpointAuthMethod", tokenEndpointAuthMethod)
.append("scope", scope)
.append("grantTypes", grantTypes)
.append("responseTypes", responseTypes)
.append("jwksUri", jwksUri)
.append("requestObjectSigningAlg", requestObjectSigningAlg)
.toString();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,981 |
SEASONAL, Relocation, Short or Long Term Furnished Single Level Executive Home with private Pool & Spa. Sunny & Warm! Highly upgraded and Beautifully appointed in Pottery Barn style just 10 minutes from the Beach, Dana Point Harbor and World Class Hotels. All in one level, open floor-plan and fireplace in the formal living room. Entertainers back yard with outdoor cabana, BBQ, eating area and cozy nights around the fire pit. Private sunny, Warm back yard! Close to shopping, restaurants, train station and local entertainment. The perfect home away from home. Whether a temporary home for relocation, or long term lease. This is like your own private Villa. A must see! Home is also available for long term rental 12+ month furnished or unfurnished. Rate may vary per season and availability, call listing agent. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,379 |
Corydalis turtschaninovii, Corydalis yanhusuo
Corydalis is an herb native to the Chinese province of Zhejiang. The portion of the plant that is used medicinally is the tuberous rhizome.1
Corydalis contains several ingredients, one of which has been shown to influence the nervous system, providing pain relief and promoting relaxation.
Corydalis contains several ingredients, one of which has been shown to influence the nervous system, providing pain relief and promoting relaxation. People with insomnia were able to fall sleep more easily after taking 100 to 200 mg per day of a corydalis extract (called dl-tetrahydropalmatine, or DHP), according to a preliminary report.People taking the extract reported no drug hangover symptoms, such as dizziness or vertigo .
Take an amount supplying 75 mg per day of tetrahydropalmatine (THP)
Preliminary reports indicate that THP (an alkaloid from the plant corydalis) may be effective in reducing nerve pain.
Preliminary reports from Chinese researchers also note that 75 mg per day of THP (an alkaloid from the plant corydalis) was effective in reducing nerve pain in 78% of those tested.
An active constituent in corydalis, dl-tetrahydropalmatine, may have an anti-arrhythmic effect on the heart.
An active constituent in corydalis, dl-tetrahydropalmatine (dl-THP), may exert an anti-arrhythmic action on the heart. This action was observed in a preliminary trial with 33 patients suffering from a specific type of arrhythmia called supraventricular premature beat or SVPB. Each patient took 300 to 600 mg of dl-THP per day in tablet form, and the dl-THP was found to be significantly more effective than placebo in reducing arrhythmia.
A constituent of corydalis called tetrahydropalmatine appears to heave pain-relieving and sedative effects. It has shown to be effective for painful menstruation.
Corydalis contains several alkaloids, and one called tetrahydropalmatine (THP) is considered to be the most potent. In laboratory research, THP has been shown to exhibit a wide number of pharmacological actions on the central nervous system, including pain-relieving and sedative effects. According to a secondary reference, painful menstruation responded favorably to the administration of THP. For a pain-relieving effect, the recommended amount for the crude dried rhizome is 5–10 grams per day. Alternatively, one can take 10–20 ml per day of a 1:2 extract.
Peptic Ulcer
Corydalis extracts are useful in relieving pain and in treating stomach ulcers.
Extracts of the herb corydalis are not only helpful as pain-relief agents but also may be useful in the treatment of stomach ulcers. In a study of people with stomach and intestinal ulcers or chronic inflammation of the stomach lining, 90 to 120 mg of corydalis extract per day (equal to 5 to 10 grams of the crude herb) was found to be effective in 76% of the participants.
In Traditional Chinese Medicine, corydalis is said to invigorate the blood, move qi (energy that travels through the body), and alleviate pain , including menstrual, abdominal, and hernial.2
Scientists have isolated a number of alkaloids from the tuber of corydalis, including corydaline, tetrahydropalmatine (THP), dl-Tetrahydropalmatine (dl-THP), protopine, tetrahydrocoptisine, tetrahydrocolumbamine, and corybulbine.3 Of the full range of 20 alkaloids found in the plant, THP is considered to be the most potent. In laboratory research, it has been shown to exhibit a wide number of pharmacological actions on the central nervous system, including analgesic and sedative effects.4 dl-THP has been found to exhibit a tranquilizing action in mice. Scientists have suggested that dl-THP blocks certain receptor sites (e.g., dopamine) in the brain to cause sedation.5
In addition to its central nervous system effects, studies in the laboratory have shown the alkaloids from corydalis also have cardiovascular actions. For example, dl-THP has been shown to both decrease the stickiness of platelets and protect against stroke ,6 as well as lower blood pressure and heart rate in animal studies.7 Additionally, it seems to exert an anti-arrhythmic action on the heart. This was found in a small double-blind clinical trial with patients suffering from a specific type of heart arrhythmia (e.g., supra-ventricular premature beat or SVPB).8 People taking 300–600 mg of dl-THP per day in tablet form, had a significantly greater improvement than those taking placebo pills.
Other human clinical trials on dl-THP have shown the ability to fall asleep was improved in people suffering from insomnia after taking 100–200 mg of dl-THP at bedtime. No drug hangover symptoms such as morning grogginess, dizziness or vertigo were reported by people taking the alkaloid extract.9
Reports from Chinese researchers also note that 75 mg of THP daily was effective in reducing nerve pain in 78% of the patients tested.10 Painful menstruation ( dysmenorrhea ), abdominal pain after childbirth, and headache have also been reported to be successfully treated with THP.11
Extracts of the herb may also be useful in the treatment of stomach ulcers . In a large sample of patients with stomach and intestinal ulcers or chronic inflammation of the stomach lining, a 90–120 mg extract of the herb per day (equal to 5–10 grams of the crude herb) was found to improve healing and symptoms in 76% of the patients.12
For an analgesic or sedative effect, the crude, dried rhizome is usually recommended at 5–10 grams per day.13 Alternatively, one can take 10–20 ml per day of a 1:2 extract.14
As of the last update, we found no reported interactions between this supplement and medicines. It is possible that unknown interactions exist. If you take medication, always discuss the potential risks and benefits of adding a new supplement with your doctor or pharmacist.
Corydalis should not be taken by pregnant or nursing women.15 There have been several reports in Western journals of THP toxicity, including acute hepatitis .16 , 17 , 18 In addition, people taking corydalis can experience vertigo , fatigue, and nausea.19
1. Zhu YP. Chinese Materia Media: Chemistry, Pharmacology, and Applications. Australia: Harwood Academic Publishers, 1998, 445-8.
2. Bensky D, Gamble A, Kaptchuk T. Chinese Herbal Medicine Materia Medica. Vista, CA: Eastland Press, 1993, 270.
3. Hsu HY. Oriental Materia Medica: A Concise Guide. Long Beach, CA: Oriental Healing Arts Institute, 1986, 448-50.
6. Xing JF, Wang MN, Ma XY, et al. Effects of dl-tetrahydropalmatine on rabbit platelet aggregation and experimental thrombosis in rats. Chin Pharm Bull 1997;13:258-60.
7. Lin MT, Chueh FY, Hsieh MT, et al. Antihypertensive effects of dl-tetrahydropalmatine: an active principle isolated from corydalis. Clin Exper Pharm Physiol 1996;23:738-42.
8. Xiaolin N, Zhenhua H, Xin M, et al. Clinical and experimental study of dl-tetrahydropalmatine effect in the treatment of supraventricular arrhythmia. J Xi'An Med Univ 1998;10:150-3.
9. Chang HM, But PPH. Pharmacology and Applications of Chinese Materia Medica vol 1. Singapore: World Scientific Inc., 1986, 521.
10. Lin DZ, Fang YS. Modern Study and Application of Materia Medica. Hong Kong: China Ocean Press, 1990, 323-5.
11. Zhu YP. Chinese Materia Media: Chemistry, Pharmacology, and Applications. Australia: Harwood Academic Publishers, 1998, 445-8.
12. Chang HM, But PPH. Pharmacology and Applications of Chinese Materia Medica vol 1. Singapore: World Scientific Inc., 1986, 521.
13. Bone K. Clinical Applications of Ayurvedic and Chinese Herbs. Warwick, Queensland, Australia: Phytotherapy Press, 1996, 25-8.
15. Bensky D, Gamble A, Kaptchuk T. Chinese Herbal Medicine Materia Medica. Vista, CA: Eastland Press, 1993, 270.
16. Horowitz RS, Feldhaus K, Dart RC, et al. The clinical spectrum of Jin Bu Huan toxicity. Arch Int Med 1996;156:899-903.
17. Kaptchuk TJ, Woolf GM, Vierling JM. Acute hepatitis associated with Jin Bu Huan. Ann Int Med 1995;122:636.
18. Anonymous. Jin Bu Huan toxicity in adults—Los Angeles, 1993. JAMA 1994;271:423-4. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,929 |
Q: Should the return value of every function be checked? I have seen code like this
hdc = BeginPaint(hwnd, &ps);
TextOut(hdc, x, y, L"Hello", 5);
EndPaint(hwnd, &ps);
What if BeginPaint() fails? Should the return value of every function called be checked in important apps like for example commercial apps?
A: The BeginPaint function can fail, in which case hdc will be set to NULL. The Microsoft documentation doesn't specify what happens when a NULL hdc is passed to TextOut, but it won't be something good. Thus checking the return value of BeginPaint is needed if the code is to be considered reliable.
When in doubt, you should check the return value of functions. I've seen way too much code that assumes that memory allocation always succeeds, file systems don't fill up, and networking functions always succeed.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,202 |
Home https://server7.kproxy.com/servlet/redirect.srv/sruj/smyrwpoii/p2/ US https://server7.kproxy.com/servlet/redirect.srv/sruj/smyrwpoii/p2/ Warren has a point bow that met face to face
Warren has a point bow that met face to face
US 5 Views
Massachusetts Senator and Democrat 2020 Warren Elizabeth Warren was seen double on Monday after meeting with her Minnesota simulator.
Stephanie Oyen shared short blond hair, glasses and a blue blazer at the McAlester College Town Hall event, knowing that her outfit would be similar to the presidential candidate's style. But she clearly underestimated how good the Doppelganger is.
"I thought there would be some giggles," Oyen tells the Star Tribune. "Then people started yelling, 'Senator Warren! "People were clapping and running towards me to take pictures. I kept saying "I'm not her!", But I looked at hundreds of people staring at me. "
In the end, she had enough confusion to cause during the event, and took off her blazer and glasses.
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"It became very strange," she explained. looks more like Elizabeth Warren. I said, "I'm not her!", But I could say "Medicare for all!"
She continued, "I'm not a joker. I really thought people would know, but then they started running and saying, "You're my hero!" And took pictures. I felt so bad. "
However, things turned weird when Oyen met Warren face to face.
" I couldn't find out if she admits that I'm dressed like her or if she thinks, "Here's a weird lady dressed as a stereotypical politician "Oyen recalled.
Click here to download the application of FOX NEWS
Warren told her that "We need to talk!", Pointing to Oyen
The candidate for 2020 later joked on Twitter that "it would have been even better if she had brought her own Bailey," also known as the golden retriever and campaign partner of Warren.
The Ann Coulter event at UC Berkeley attracts masked protesters; numerous arrests have been reported
Colorado school district closes more than 40 schools after highly contagious viral disease | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,661 |
{"url":"https:\/\/www.physicsforums.com\/threads\/calculating-distance-angle-bet-velocity-and-acceleration.920484\/","text":"# Calculating distance, angle bet. velocity and acceleration\n\nTags:\n1. Jul 18, 2017\n\n### Pushoam\n\n1. The problem statement, all variables and given\/known data\n\n2. Relevant equations\n\n3. The attempt at a solution\nA)\n\ns = \u221a[(x2) + (y2 ) ]= a\u221a[2(1- cos (\u03c9t) ) ]|t= \u0393\n\nThe book says, s = a\u03c9\u0393\n\nWhat I can do is ,\nFor very small \u0393 i.e. \u03c9\u0393<<1 , cos (\u03c9\u0393) \u2248 1 - {(\u03c9\u0393)2}\/2\n\nThen , I get,\ns = a\u03c9\u0393\n\nBut, in question it is not given that \u03c9\u0393<<1. So, is it correct to do this approximation?\n\nB)\n\nIn many questions, Irodov asks to find out the angle between velocity and acceleration.\nDoes this angle have any physical significance?\nI mean if I know this angle what can I tell about the motion?\nWhy is one supposed to know this angle?\n\nLast edited: Jul 19, 2017\n2. Jul 19, 2017\n\n### haruspex\n\nYour equation for distance traversed is wrong. It should read ds2=dx2+dy2, not s2=x2+y2.\n\n3. Jul 19, 2017\n\n### Pushoam\n\nThanks.\n\nLearning , I should be careful while calculating magnitude of the displacement or distance. There is a tendency to get confused between the two.\n\nO.K. So, s is the magnitude of the displacement traveled in time \u0393.\n\nds2 = dx2+dy2\n\ndx = a\u03c9 cos(\u03c9t) dt\n\ndy = a\u03c9 sin(\u03c9t) dt\n\ndx2 = (dx) (dx) = [ a\u03c9 cos(\u03c9t)]2 (dt)2\ndy2 = (dy) (dy) = [ a\u03c9 sin(\u03c9t)]2 (dt)2\n\nds2 = [ a\u03c9 ]2 (dt)2\n\nds = a\u03c9 dt\n\n0s ds = a\u03c9 \u222b0\u0393 dt\n\ns = a\u03c9\u0393\nAnd for \u03c9\u0393 <<1,the distance is approximately equal to the displacement.\n\nIs this solution correct?\n\n4. Jul 19, 2017\n\n### haruspex\n\nYes. (But the original uses $\\tau$, Greek lowercase tau, which you are writing as $\\Gamma$, Greek uppercase gamma.)\n\n5. Jul 19, 2017\n\n### Pushoam\n\nThanks for this, too, as earlier I thought both \u0393 and \u03c4 are tau's.\n\n6. Jul 19, 2017\n\n### haruspex\n\nUppercase tau is indistinguishable from T.\n\n7. Jul 19, 2017","date":"2017-12-18 07:19:14","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.869066059589386, \"perplexity\": 8176.332461118721}, \"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-2017-51\/segments\/1512948609934.85\/warc\/CC-MAIN-20171218063927-20171218085927-00730.warc.gz\"}"} | null | null |
Пол Кінґсмен (, 15 червня 1967) — новозеландський плавець.
Призер Олімпійських Ігор 1988 року, учасник 1984 року.
Призер Пантихоокеанського чемпіонату з плавання 1987, 1989 років.
Призер Ігор Співдружності 1986, 1990 років.
Посилання
Пол Кінґсмен на Olympedia.org
Пол Кінґсмен на Sports-Reference.com
Примітки
Народились 1967
Новозеландські плавці
Новозеландські бронзові олімпійські медалісти
Плавці та плавчині на літніх Олімпійських іграх 1984
Плавці та плавчині на літніх Олімпійських іграх 1988
Бронзові олімпійські медалісти з плавання
Випускники Університету Каліфорнії у Берклі
Призери літніх Олімпійських ігор 1988
Призери Ігор Співдружності з плавання | {
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In yesterday's "10 minutes to getting connected" post, we asked you to get a Twitter account so that you could start learning from some of the best conversations and sharing that are happening online for educators.
If signing up for Twitter in 10 minutes was difficult, please do not hesitate to ask for help in the comments for this blog. #OSSEMOOC is here to help. We are a community of education leaders who believe strongly in the importance of collaboration and connecting to build the supportive team we all need to continue to learn in this ever-changing world.
Last year, on the last day of the #BIT13 conference, ECOO President Mark Carbone emphasized the importance of going out and nurturing other educators.
Nurturing is a powerful word.
Change is fast and we can often feel that it is too hard to keep up. We sometimes feel overwhelmed with all that there is to know and understand to stay on top of change. If we all take the stance that we are here to nurture each other in getting connected and learning together, we will create a much friendlier environment for learning and risk-taking. This is the culture we need for innovative thinking to thrive.
Building a supportive Professional Learning Network is a critical component in continuing to grow as educators.
Today, we will focus on following other educators on Twitter. As you watch your Twitter feed today, think about who is sharing information that interests you. Who is pushing your thinking? What is resonating with you?
Below, we have listed a few educators who are committed to sharing online. Feel free to explore their work, keeping in mind that a PLN is very personal, and you need to build the network that works for you.
This is a starting point, but it takes a daily commitment to continue to grow and nurture a PLN that can transform your professional practice.
We all look forward to learning and sharing with you!
Thanks for the mention. Thinking together is one the best parts of my day!
We would love to hear who some of your favourite follows are. Can you share a few with new users? Thanks! | {
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Q: Changing column names in list elements I have a huge list with data.frames (same number of columns, different number of rows).
I succeeded to use apply - instead of the for loops I learned to avoid - to create a mean value over specific columns in each list element with
t2<-lapply(t1, function(x) cbind(x,rowMeans(x[,c("ColumnX","ColumnY","ColumnZ")])))
The problem I am stuck with now is the new columns name. It is "rowMeans(x[,c("ColumnX","ColumnY","ColumnZ")])".
How can I change this for all list elements? My poor "lapply"-knowledge was not sufficient for this task.
A: There's two ways to do this, and it actually has to do with the cbind function and not the lapply function:
cbind(x,DesiredName = rowMeans(x[,...]))
Or after you've cbind'ed:
> names(x)
[1] "Column X" "Column Y" "Column Z" "rowMeans(x[,...])"
> names(x)[4]
"rowMeans(x[,...])"
> names(x)[4] <- "DesiredName" ###Assign a name to the fourth column
> names(x)
[1] "Column X" "Column Y" "Column Z" "DesiredName"
That's obviously the long way, but it useful for if you forget to name something during the apply or cbind process.
A: Just add a colname, for example RowMeans
t2 <-lapply(t1, function(x) cbind(x,RowMeans=rowMeans(x[,c("ColumnX","ColumnY","ColumnZ")])))
Actually you can accomplish your goal using this alternative:
lapply(t1, function(x) transform(x,RowMeans=rowMeans(x[,c("ColumnX","ColumnY","ColumnZ")])))
Here RowMeans is the name of the new variable containing each row mean.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,508 |
HD Portable DVR with 2.5 TFT LCD Screen Driving Recorder is a device installed in car windshield to record video, audio data related to vehicle impact or accidents. It records everything that occurs while driving. It's a real road safety guard!
►3-months warranty upon manufacturing defect from date of receipt.
►STEP 2: Upon confirmation, a mailing address will be provided.
►Delivery costs for warranty claim / return / repair covered by customer.
►Voxtera will cover delivery costs of replacement or repaired product to customer.
►Printed receipt will be required as proof of purchase.
►Warranty claims which do not fulfil the steps above will not be processed due to missing information. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,748 |
Ommatius queenslandi är en tvåvingeart som beskrevs av Ricardo 1913. Ommatius queenslandi ingår i släktet Ommatius och familjen rovflugor. Inga underarter finns listade i Catalogue of Life.
Källor
Rovflugor
queenslandi | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 169 |
Acoma sexfoliata är en skalbaggsart som beskrevs av Saylor 1948. Acoma sexfoliata ingår i släktet Acoma och familjen Pleocomidae.
Källor
Skalbaggar
sexfoliata
Insekter i nearktiska regionen | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,060 |
from ccxt.async.base.exchange import Exchange
import json
from ccxt.base.errors import ExchangeError
class virwox (Exchange):
def describe(self):
return self.deep_extend(super(virwox, self).describe(), {
'id': 'virwox',
'name': 'VirWoX',
'countries': ['AT', 'EU'],
'rateLimit': 1000,
'hasCORS': True,
'urls': {
'logo': 'https://user-images.githubusercontent.com/1294454/27766894-6da9d360-5eea-11e7-90aa-41f2711b7405.jpg',
'api': {
'public': 'http://api.virwox.com/api/json.php',
'private': 'https://www.virwox.com/api/trading.php',
},
'www': 'https://www.virwox.com',
'doc': 'https://www.virwox.com/developers.php',
},
'requiredCredentials': {
'apiKey': True,
'secret': False,
'login': True,
'password': True
},
'api': {
'public': {
'get': [
'getInstruments',
'getBestPrices',
'getMarketDepth',
'estimateMarketOrder',
'getTradedPriceVolume',
'getRawTradeData',
'getStatistics',
'getTerminalList',
'getGridList',
'getGridStatistics',
],
'post': [
'getInstruments',
'getBestPrices',
'getMarketDepth',
'estimateMarketOrder',
'getTradedPriceVolume',
'getRawTradeData',
'getStatistics',
'getTerminalList',
'getGridList',
'getGridStatistics',
],
},
'private': {
'get': [
'cancelOrder',
'getBalances',
'getCommissionDiscount',
'getOrders',
'getTransactions',
'placeOrder',
],
'post': [
'cancelOrder',
'getBalances',
'getCommissionDiscount',
'getOrders',
'getTransactions',
'placeOrder',
],
},
},
})
async def fetch_markets(self):
markets = await self.publicGetInstruments()
keys = list(markets['result'].keys())
result = []
for p in range(0, len(keys)):
market = markets['result'][keys[p]]
id = market['instrumentID']
symbol = market['symbol']
base = market['longCurrency']
quote = market['shortCurrency']
result.append({
'id': id,
'symbol': symbol,
'base': base,
'quote': quote,
'info': market,
})
return result
async def fetch_balance(self, params={}):
await self.load_markets()
response = await self.privatePostGetBalances()
balances = response['result']['accountList']
result = {'info': balances}
for b in range(0, len(balances)):
balance = balances[b]
currency = balance['currency']
total = balance['balance']
account = {
'free': total,
'used': 0.0,
'total': total,
}
result[currency] = account
return self.parse_balance(result)
async def fetch_market_price(self, symbol, params={}):
await self.load_markets()
response = await self.publicPostGetBestPrices(self.extend({
'symbols': [symbol],
}, params))
result = response['result']
return {
'bid': self.safe_float(result[0], 'bestBuyPrice'),
'ask': self.safe_float(result[0], 'bestSellPrice'),
}
async def fetch_order_book(self, symbol, params={}):
await self.load_markets()
response = await self.publicPostGetMarketDepth(self.extend({
'symbols': [symbol],
'buyDepth': 100,
'sellDepth': 100,
}, params))
orderbook = response['result'][0]
return self.parse_order_book(orderbook, None, 'buy', 'sell', 'price', 'volume')
async def fetch_ticker(self, symbol, params={}):
await self.load_markets()
end = self.milliseconds()
start = end - 86400000
response = await self.publicGetTradedPriceVolume(self.extend({
'instrument': symbol,
'endDate': self.YmdHMS(end),
'startDate': self.YmdHMS(start),
'HLOC': 1,
}, params))
marketPrice = await self.fetch_market_price(symbol, params)
tickers = response['result']['priceVolumeList']
keys = list(tickers.keys())
length = len(keys)
lastKey = keys[length - 1]
ticker = tickers[lastKey]
timestamp = self.milliseconds()
return {
'symbol': symbol,
'timestamp': timestamp,
'datetime': self.iso8601(timestamp),
'high': float(ticker['high']),
'low': float(ticker['low']),
'bid': marketPrice['bid'],
'ask': marketPrice['ask'],
'vwap': None,
'open': float(ticker['open']),
'close': float(ticker['close']),
'first': None,
'last': None,
'change': None,
'percentage': None,
'average': None,
'baseVolume': float(ticker['longVolume']),
'quoteVolume': float(ticker['shortVolume']),
'info': ticker,
}
def parse_trade(self, trade, symbol=None):
sec = self.safe_integer(trade, 'time')
timestamp = sec * 1000
return {
'id': trade['tid'],
'timestamp': timestamp,
'datetime': self.iso8601(timestamp),
'order': None,
'symbol': symbol,
'type': None,
'side': None,
'price': self.safe_float(trade, 'price'),
'amount': self.safe_float(trade, 'vol'),
'fee': None,
'info': trade,
}
async def fetch_trades(self, symbol, since=None, limit=None, params={}):
await self.load_markets()
response = await self.publicGetRawTradeData(self.extend({
'instrument': symbol,
'timespan': 3600,
}, params))
result = response['result']
trades = result['data']
return self.parse_trades(trades, symbol)
async def create_order(self, market, type, side, amount, price=None, params={}):
await self.load_markets()
order = {
'instrument': self.symbol(market),
'orderType': side.upper(),
'amount': amount,
}
if type == 'limit':
order['price'] = price
response = await self.privatePostPlaceOrder(self.extend(order, params))
return {
'info': response,
'id': str(response['orderID']),
}
async def cancel_order(self, id, symbol=None, params={}):
return await self.privatePostCancelOrder(self.extend({
'orderID': id,
}, params))
def sign(self, path, api='public', method='GET', params={}, headers=None, body=None):
url = self.urls['api'][api]
auth = {}
if api == 'private':
self.check_required_credentials()
auth['key'] = self.apiKey
auth['user'] = self.login
auth['pass'] = self.password
nonce = self.nonce()
if method == 'GET':
url += '?' + self.urlencode(self.extend({
'method': path,
'id': nonce,
}, auth, params))
else:
headers = {'Content-Type': 'application/json'}
body = self.json({
'method': path,
'params': self.extend(auth, params),
'id': nonce,
})
return {'url': url, 'method': method, 'body': body, 'headers': headers}
def handle_errors(self, code, reason, url, method, headers, body):
if code == 200:
if (body[0] == '{') or (body[0] == '['):
response = json.loads(body)
if 'result' in response:
result = response['result']
if 'errorCode' in result:
errorCode = result['errorCode']
if errorCode != 'OK':
raise ExchangeError(self.id + ' error returned: ' + body)
else:
raise ExchangeError(self.id + ' malformed response: no result in response: ' + body)
else:
# if not a JSON response
raise ExchangeError(self.id + ' returned a non-JSON reply: ' + body)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 461 |
class Upload < ActiveRecord::Base
attr_accessible :upload, :user, :campaign, :cause
has_attached_file :upload
#validates_attachment :upload,
# :content_type => {:content_type => ['image/jpg', 'image/gif', 'image/png']}
#
belongs_to :user
belongs_to :campaign
belongs_to :cause
include Rails.application.routes.url_helpers
def url
upload.url(:original)
end
def content_type
upload.content_type
end
def to_jq_upload
{
'name' => read_attribute(:upload_file_name),
'size' => read_attribute(:upload_file_size),
'url' => upload.url(:original),
'delete_url' => upload_path(self),
'delete_type' => 'DELETE'
}
end
end | {
"redpajama_set_name": "RedPajamaGithub"
} | 2,664 |
On 25th March, 2014 Senator John Madigan, the Victorian representative of the DLP Party, made the following speech in the Australian Senate… He is not the first in making a speech along these lines. On March 3, 2014, George Christensen of the National Party made a similar speech… We at Mens Rights Sydney do not align ourselves…
18/04/2014 in family law.
Australian MP George Christensen speaks out about family law/child support system reform
This news item was first published by Robert O'Hara on A Voice for Men and is republished here in its entirety. link Last week Australian MP George Christensen delivered an impassioned speech to Parliament regarding the state of the Family Law and Child Support systems and the effect they have on relationships between fathers and…
04/03/2014 in family law, News. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,297 |
The scent of jasmine carried by the breeze on a hot June night overwhelms the fading aroma of garlic bread and barbecued meat. The midges dance in anger outside an invisible curtain of citronella smoke released from a circle of burning tea-lights. Pipistrelle bats prey on insects hovering around the solar lamps in the hanging baskets. An illuminated display of orange, yellow and crimson dangling like chariots of fire in my own personal Jerusalem. An old lurcher lies on his bed on the decking, watching the bats through a half-raised eyelid. His old master is watching too, while scribbling notes for yet another project. A wonderful evening and the world here is good.
Can't say the same across the world can we? Many dead after a radical attack in Tunisia. Other so called 'terrorist' attacks in other countries. You're not safe abroad or at home it seems. As always, in the name of religion or creed. Madness. Why do folk dedicate their life or their death to a series of myths and fables (written or fabricated) in ancient scribblings? And why do they insist on impressing their 'belief' on others? What a waste of intelligence and life.
More importantly, why are we tolerating this insult to humanity? This 'I don't like your beliefs so I want to kill you' mentality? The second world war was fought and won by liberal thinking nations to oppose such outrage. Yet those same nations are now inviting within their borders thousands of immigrants who bring with them a hatred of our values, our creeds, our liberalism and our tolerance. Why are we allowing this? Why is our governance constantly apologising for and legislating against our rights as natural, indigenous citizens to voice our objections to the arrogant, subversive demands of people who asked to be homed here and were allowed here? They should be reminded that they here at our behest and on our terms … not theirs!
Everyone expected World War Three to be a nuclear conflagration. It hasn't worked out that way, has it? It is happening, now. Right in front of us. The human race eating itself alive on a menu of hatred, religious intolerance, greed, politics and charity.
Me? I'm sitting here smelling the jasmine, watching the bats and thinking that I'm glad I'm approaching the twilight years. I feel sorry for my son and the generations beyond them.
A few millennia from now, an intelligent life form will land on Earth and say that an intelligent life form once lived on this planet … but it destroyed itself while arguing about the reason it came to exist. How sad is that?
Copyright, Ian Barnett, Wildscribbler, June 2015. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,318 |
<p>You've told us you intend in to be in business. All your answers indicate you are similar to someone who is <strong>undertaking a hobby</strong>.</p>
<p>Please come back and use the tool again if things change. For example, if you:</p>
<ul>
<li>aim to make a profit from selling your work</li>
<li>increase the time you spend on your activity</li>
<li>increase the size of your operation.</li>
</ul>
<div class="panel callout radius">
<h2>Did you know?</h2>
<p>You can't claim a deduction for any losses from your creative work if it is a hobby.</p>
<p>It's a myth that there is a dollar threshold to be in a business. Some people can have very expensive hobbies! What matters is whether, as a whole, your activity is commercial with an aim to make a profit. <strong>Once</strong> you're in business, there are dollar thresholds that can affect <strong>your tax obligations</strong>.</p>
</div>
<p>Based on what you've told us, these are your government obligations.</p>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,847 |
Larry the Vegan Cannibal likes vegetables, people, and movie reviews.
The Age of Elitism
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Featured quote (not written by me)
Cultural critic James Wolcott, on the new film critic:
"Film critics today have become these rabid completists... They feel like that with festivals, they have to see everything, no matter how minor. Part of it is bragging rights. The other part is that the only thing that feeds into their movie writing is other movies."
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Categories Select Category Action/Adventure Comedy Documentary Drama Horror Sci-Fi Thriller Uncategorized Western
By Adam Lippe
The initial financial failure of Disney's Tron (directed by Steven Lisberger) was attributed to the enormous expectations based on the egregious expense of the project because of the complicated computer effects. Tron was simplistic story-wise, and the replication of the feel of a video game would become commonplace not long after its 1982 release. The $20 million had been spent on the technological advances which ironically succeeded in developing a following for a video game based on the movie. There were several versions of the game; one had the player in the film's colored lightcycles*, trying to avoid the other bikes on the screen.
The colored motorcycles resemble an image from another $20 million sci-fi film from 1982, Hal Needham's Megaforce, where real life motorcycles emit colored smoke behind them. And that's pretty much where Megaforce's visual creativity ends which makes the financial expenditure rather mysterious. Tron and Megaforce both owe a lot to Star Wars, both were clearly envisioned in sci-fi western terms, but Megaforce is far more interested in explosions, car stunts, and gunfire than actual ingenuity. That's because Needham, who appears to be represented in Megaforce as Michael Beck's** redneck character (though Needham has a cameo himself), was a stuntman who used his connections as the guy who lived in Burt Reynolds' poolhouse to develop into a film director.
Needham's most famous films were the first two Smokey and the Bandit films, and to a lesser extent, the first two Cannonball Run films (the third is called Speed Zone) and it was this momentum that got Needham the Megaforce job. Other factors probably had to with Jackie Chan, who Megaforce producer Raymond Chow forced on Needham when he was making the Cannonball Run films, in an effort to make Chan a star in America too. In Hong Kong it worked, as Chan was top-billed (as opposed to 10th billed in the US), but Chan wouldn't break through until 1996's Rumble in the Bronx.
Needham, making his cameo
Now Chan is not in Megaforce, it's difficult to imagine him in any role apart from the token Asian among the cast of cornpone warriors, played by Evan Kim, who also played Bruce Lee in the Enter the Dragon parody in The Kentucky Fried Movie. No, Needham was more interested in pure white-bread jingoism; hence the villains in Megaforce are all Communist stereotypes or vague foreigners with Spanish accents, like lead henchman Henry Silva. Barry Bostwick is the ethnically cleansed Ace Hunter, sporting a blue headband and perfectly coiffed blond hair, and his compatriots are all juvenile males, but honorable to a fault. That's why when Edward Mulhare (in a role eerily similar to the one he had in Knight Rider, which premiered later that same year) asks for help from the Megaforce*** to goad Silva and his troops into crossing into Mulhare's country's territory so they can attack, Mulhare is the one who ends up as the bad guy. He has ulterior motives, and nothing underhanded can be accepted in Megaforce. How do we know that? Why, when the Megaforce attacks a base they give themselves four minutes to get in and out and Needham makes sure we're aware of this by literally having a clock on screen the entire time, counting down. The Megaforce can't spare a second, and they don't think the enemy will ever fire back, delaying them.
These sorts of mild bits of macho aren't why Megaforce has a cult following though, going as far as to clearly inspire Team America: World Police, with several sequences even duplicated. No, that has to do with the rampant homoerotic costuming (from the costumers of Batman and Robin) and the absolute pointlessness of what appears on screen. The Megaforce wear motorcycle helmets with arrows painted on them, arrows that seem to be pointing at the brains they don't have.
This lack of awareness comes full center in scenes like one where Ace Hunter lets Major Zara (played by Indian beauty queen Persis Khambatta, from Star Trek: The Motion Picture) know that no matter how proficient she is at military games, she can't come with them to fight Silva and his gang because it would upset the team's morale, or some such nonsense. Her response to the decision? "That's what makes you what you are, a great leader." Their romance is based on such respect, or that's all we can believe, because when they embrace in the next scene as if they were longtime lovers, there's either 15 minutes of development missing or Needham didn't think women had a choice in who they would fall in love with and why.
Despite an early scene where we learn that women don't like cartoon pigs, I'm going with the deleted scenes option, because Megaforce is such a mess structurally and pacingwise. Bostwick doesn't appear for the first 15 minutes and he's the hero. Silva is in one or two shots in the opening scene and then we don't see him for 45 minutes. The movie appears to be set in the future, but Ace Hunter has a very important putdown on Silva, "The good guys win, even in the '80s." I guess that explains the electronic score Megaforce has, but a time period rarely seems relevant to the proceedings. Besides, you don't want to pretend your movie is set in the future when they appear to use balloons made out of stone for target practice.
One can only speculate where the money went (up their nose?), because it certainly isn't in the shoddy miniatures, or the fantastically abysmal blue-screen work that isn't the least bit convincing (using the already antiquated Introvision process, also used by Knight Rider). Maybe all the money went to a wrangler for Bostwick's headband?
* Ok, it was really just moving lines on the screen, but you could easily imagine yourself in the lightcycle.
** In an odd twist, Beck was the lead in another 1982 war movie, Battletruck, where he played a character named Hunter.
*** The Megaforce is an elite group of military stragglers from armies all over the world, presumed dead or missing. Or at least that's what the hilarious opening crawl says, read to us in the utmost of seriousness. A crawl only slightly shorter and more convoluted than the legendary one that opens Uwe Boll's Alone in the Dark.
P.S. Enjoy the theme from Megaforce.
Tags: 1980's, Ace Hunter, Adam Lippe, Alone in the Dark, Asian, Barry Bostwick, Batman and Robin, Battletruck, big budget, Bruce Lee, Burt Reynolds, camp, Cannonball Run, Cannonball Run II, CGI, Cliché, cocaine, communism, Communist, Disney, Edward Mulhare, Enter the Dragon, Evan Kim, exploitation, Hal Needham, Henry Silva, Hollywood, homoerotic, homophobia, Hong Kong, Introvision, Jackie Chan, Knight Rider, Michael Beck, Persis Khambatta, PG, Raymond Chow, redneck, Rumble in the Bronx, Speed Zone, Star Trek, Star Trek: The Motion Picture, Star Wars, Steven Lisberger, Team America: World Police, The Kentucky Fried Movie, The Warriors, Tron, Uwe Boll, Western
This entry was posted on Sunday, September 25th, 2011 at 12:00 am and is filed under Action/Adventure, Sci-Fi. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.
1 comment on "Megaforce"
Maria T. Kelly says:
I just had a weird thought. Do you think that Megaforce is in some strange way a really bad Classic Mission: Impossible spoof?
Now on DVD and Blu-Ray
Whenever there's a genre parody or ode to a specific era of films, such as Black Dynamite's mocking of Blaxploitation films or Quentin Tarantino's Death Proof, the second half of Grindhouse, the danger is that the film might fall into the trap of either being condescending without any particular insight, or so faithful that it becomes the very flawed thing it is emulating.
Black Dynamite has nothing new to say about Blaxploitation films, it just does a decent job of copying what an inept [...]
A radio interview with the person who wrote this sentence, Part IV: Comfort and Joy and Dream Lover
Escape From the Bronx [aka Bronx Warriors 2]
A podcast with Tim League, CEO of The Alamo Drafthouse, Fantastic Fest, and Drafthouse Films
A podcast with Summer Qing [Qing Xu], co-star of Looper: Mandarin and English friendly version.
Luke Wilson's Hands Across America
Swimming to Cambodia
Winner: BEST ONLINE FILM CRITIC, 2010 National Veegie Awards (Vegan Themed Entertainment)
Nominee: BEST NEW PRODUCT, 2011 National Veegie Awards: The Vegan Condom
House of the Dead
Naked Killer
The Forbidden Kingdom
Leave the Cults to the Scientologists
Uwe Boll's Heart of America
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Monique Mccullough on Monster's Ball
Emilian Moreno on Surf II: The End of the Trilogy
kate on Monster's Ball
Edward Sullivan on Burn! – Uncut
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Featured Quote (written by me)
On Cold Fish:
Though the 16 year old me described the 1994 weepie Angie, starring Geena Davis as a Brooklyn mother raising her new baby alone, as "maudlin and melodramatic," Roger Ebert, during his TV review, referring to the multitude of soap-operaish problems piling up on the titular character, suggested that it was only in Hollywood where Angie would get a happy ending. "If they made this movie in France, Angie would have shot herself."
Well Cold Fish was made in Japan, where Angie would have shot herself and that would have been the happy ending.
Copyright © 2014 by Adam Lippe. All Rights Reserved. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,902 |
\section{Introduction}\label{sec:introduction}}
\else
\section{Introduction}
\label{sec:introduction}
\fi
\IEEEPARstart{T}{his} demo file is intended to serve as a ``starter file''
for IEEE Computer Society journal papers produced under \LaTeX\ using
IEEEtran.cls version 1.8b and later.
I wish you the best of success.
\hfill mds
\hfill August 26, 2015
\subsection{Subsection Heading Here}
Subsection text here.
\subsubsection{Subsubsection Heading Here}
Subsubsection text here.
\section{Conclusion}
The conclusion goes here.
\appendices
\section{Proof of the First Zonklar Equation}
Appendix one text goes here.
\section{}
Appendix two text goes here.
\ifCLASSOPTIONcompsoc
%
\section*{Acknowledgments}
\else
%
\section*{Acknowledgment}
\fi
The authors would like to thank...
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\section{Introduction}
This demo file is intended to serve as a ``starter file''
for IEEE Computer Society conference papers produced under \LaTeX\ using
IEEEtran.cls version 1.8b and later.
I wish you the best of success.
\hfill mds
\hfill August 26, 2015
\subsection{Subsection Heading Here}
Subsection text here.
\subsubsection{Subsubsection Heading Here}
Subsubsection text here.
\section{Conclusion}
The conclusion goes here.
\ifCLASSOPTIONcompsoc
%
\section*{Acknowledgments}
\else
%
\section*{Acknowledgment}
\fi
The authors would like to thank...
\section{Introduction}\label{sec:introduction}}
\IEEEPARstart{T}{his} demo file is intended to serve as a ``starter file''
for IEEE Computer Society journal papers produced under \LaTeX\ using
IEEEtran.cls version 1.8b and later.
I wish you the best of success.
\hfill mds
\hfill August 26, 2015
\subsection{Subsection Heading Here}
Subsection text here.
\subsubsection{Subsubsection Heading Here}
Subsubsection text here.
\section{Conclusion}
The conclusion goes here.
\appendices
\section{Proof of the First Zonklar Equation}
Appendix one text goes here.
\section{}
Appendix two text goes here.
\ifCLASSOPTIONcompsoc
%
\section*{Acknowledgments}
\else
%
\section*{Acknowledgment}
\fi
The authors would like to thank...
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\section{Introduction}
\label{sec:prodpoly_introduction}
Deep Convolutional Neural Networks (DCNNs)~\cite{lecun1998gradient, krizhevsky2012imagenet} have demonstrated impressive results in a number of tasks the last few years~\cite{krizhevsky2012imagenet, huang2017densely, miyato2018spectral}. Arguably, the careful selection of architectural pipelines, e.g. skip connections~\cite{he2016deep}, normalization schemes~\cite{ioffe2015batch} etc., is significant, however the core structure relies on compositional functions of linear and nonlinear operators. Both theoretical~\cite{arora2018convergence, ji2018minimax} and empirical studies reveal the limitations of the existing structure.
Recent empirical~\cite{karras2018style} and theoretical~\cite{jayakumar2020Multiplicative} results support that multiplicative interactions expand the classes of functions that can be approximated. Motivated by these findings, we study a new class of function approximators, which we coin $\Pi-$nets, where the output is a polynomial function of the input. Specifically, we model a vector-valued function $\bm{G}(\bm{z}): \realnum^{d} \to \realnum^{o}$ by a high-order multivariate polynomial of the input $\bm{z}$, whose unknown parameters are naturally represented by high-order tensors. The number of parameters required to accommodate all higher-order correlations of the input explodes with the desired order of the polynomial. To that end, we cast polynomial parameters estimation as a coupled tensor factorization~\citep{sidiropoulos2017tensor} that jointly factorizes all the polynomial parameters tensors. We introduce three joint decompositions with shared factors and exhibit the resulting hierarchical structures (i.e., architectures of neural networks).
\begin{figure}[!t]
\centering
\includegraphics[width=1\linewidth]{schematics-model_intro.pdf}
\caption{In this paper we introduce a class of networks called $\Pi-$nets, where the output is a polynomial of the input. The input in this case, $\binvar$, can be either the latent space of Generative Adversarial Network for a generative task or an image in the case of a discriminative task. Our polynomial networks can be easily implemented.}
\label{fig:prodpoly_model_intro_schematic}
\end{figure}
In our preliminary works~\cite{chrysos2019polygan, chrysos2019newton, chrysos2020poly}, we introduced the concept of higher-order expansions for both generative and discriminative networks. In this work, our improvements are threefold. The concepts and the motivation behind each model are elaborated; the new intuitions will enable practitioners to devise new models tailored to their specific tasks. In addition, we extend the experimental results, e.g. include experiment in the challenging task of face verification and identification. Lastly, we conduct a thorough discussion on several challenging topics that require further work on this new class of neural networks.
In particular, the paper bears the following contributions:
\begin{itemize}
\item A new family of neural networks (called $\Pi-$nets) where the output is a high-order polynomial of the input is introduced. To avoid the combinatorial explosion in the number of parameters of polynomial activation functions~\citep{kileel2019expressive}, our $\Pi-$nets cast polynomial parameters estimation as a coupled tensor factorization with shared factors (please see Fig.~\ref{fig:prodpoly_model_intro_schematic} for an indicative schematic representation).
\item The proposed architectures are applied in a) generative models such as GANs, and b) discriminative networks. Additionally, the polynomial architectures are used to learn high-dimensional distributions without non-linear activation functions.
\item We convert state-of-the-art baselines using the proposed $\Pi-$nets and show how they can largely improve the performance of the baseline. We demonstrate it conclusively in a battery of tasks (i.e., generation, classification and face verification/identification). Our architectures are applicable to many different signals (e.g. images, meshes, and audio) and outperform the prior art.
\end{itemize}
The rest of the paper is organized as follows: Sec.~\ref{sec:prodpoly_related} summarizes the related work. In Sec.~\ref{sec:prodpoly_method} we introduce the polynomial networks and showcase the resulting architectures for three decompositions. \rebuttal{The core experimental evaluation is conducted in Sec.~\ref{sec:prodpoly_experiments}, while a number of experiments are deferred to the supplementary.} The existing limitations and future directions of the polynomial networks are discussed in Sec.~\ref{sec:prodpoly_discussion}, while Sec.~\ref{sec:prodpoly_conclusion} concludes the paper.
\section{Related work and notation}
\label{sec:prodpoly_related}
\textbf{Expressivity of (deep) neural networks}: The last few years, (deep) neural networks have been applied to a wide range of applications with impressive results. The performance boost can be attributed to a host of factors including: a) the availability of massive datasets~\cite{deng2009imagenet, liu2015deep}, b) the machine learning libraries~\cite{chainer_learningsys2015, paszke2017automatic} running on massively parallel hardware, c) training improvements. The training improvements include a) optimizer improvement~\cite{kingma2014adam, reddi2019convergence}, b) augmented capacity of the network~\cite{simonyan2014very}, c) regularization tricks~\cite{glorot2010understanding, saxe2013exact, ioffe2015batch, ulyanov2016instance}. However, the paradigm for each layer remains largely unchanged for several decades: each layer is composed of a linear transformation and an element-wise activation function. Despite the variety of linear transformations~\cite{fukushima1980neocognitron, lecun1998gradient, krizhevsky2012imagenet} and activation functions~\cite{ramachandran2017searching, nair2010rectified} being used, the effort to extend this paradigm has not drawn much attention to date.
Recently, hierarchical models have exhibited stellar performance in learning expressive generative models~\cite{brock2019large, karras2018style, zhao2017learning}. For instance, the recent BigGAN \citep{brock2019large} performs a hierarchical composition through skip connections from the noise $\bm{z}$ to multiple resolutions of the generator. A similar idea emerged in StyleGAN~\citep{karras2018style}, which is an improvement over the Progressive Growing of GANs (ProGAN)~\citep{karras2017progressive}. As ProGAN, StyleGAN is a highly-engineered network that achieves compelling results on synthesized 2D images. In order to provide an explanation on the improvements of StyleGAN over ProGAN, the authors adopt arguments from the style transfer literature \citep{huang2017arbitrary}. We believe that these improvements can be better explained under the light of our proposed polynomial function approximation. Despite the hierarchical composition proposed in these works, we present an intuitive and mathematically elaborate method to achieve a more precise approximation with a polynomial expansion. We also demonstrate that such a polynomial expansion can be used in both image generation (as in \cite{karras2018style, brock2019large}), image classification, and graph representation learning.
\begin{table*}[h]
\begin{minipage}{.7\linewidth}
\caption{Nomenclature}
\label{tbl:prodpoly_primary_symbols}
\centering
\begin{tabular}{|c | c | c|}
\toprule
Symbol & Dimension(s) & Definition \\
\midrule
$n, N$ & $\naturalnum$ & Polynomial term order, total approximation order. \\
$k$ & $\naturalnum$ & Rank of the decompositions. \\
$\binvar$ & $\realnum^d$ & Input to the polynomial approximator. \\
$\bm{C}, \bm{\beta}$ & $\realnum^{o\times k}, \realnum^{o}$ & Parameters in all decompositions. \\
$\bm{A}\matnot{n}, \bm{S}\matnot{n}, \bm{B}\matnot{n}$ & $\realnum^{d\times k}, \realnum^{k\times k}, \realnum^{\omega\times k}$ & Matrix parameters in the hierarchical decomposition.\\
$\odot, *$ & - & Khatri-Rao product, Hadamard product. \\
\hline
\end{tabular}
\end{minipage}
\begin{minipage}{.2\linewidth}
\caption{Single polynomial models (Sec.~\ref{ssec:prodpoly_single_poly})}
\label{tbl:prodpoly_single_polynomial_models}
\centering
\begin{tabular}{|c | c | c |}
\toprule
Name & Schematic & Recursive eq.\\
\midrule
CCP & Fig.~\ref{fig:prodpoly_model1_schematic} & \eqref{eq:prodpoly_model1}\\
NCP & Fig.~\ref{fig:prodpoly_model2_schematic} & \eqref{eq:prodpoly_model2}\\
NCP-Skip & Fig.~\ref{fig:prodpoly_model3_schematic} & \eqref{eq:prodpoly_model3} \\
\hline
\end{tabular}
\end{minipage}
\end{table*}
\textbf{Polynomial networks}: Polynomial relationships have been investigated in two specific categories of networks: a) self-organizing networks with hard-coded feature selection, b) pi-sigma networks.
The idea of learnable polynomial features can be traced back to Group Method of Data Handling (GMDH)~\cite{ivakhnenko1971polynomial}\footnote{This is often referred to as the first deep neural network~\cite{schmidhuber2015deep}.}. GMDH learns partial descriptors that capture quadratic correlations between two predefined input elements. In \cite{oh2003polynomial}, more input elements are allowed, while higher-order polynomials are used. The input to each partial descriptor is predefined (subset of the input elements), which does not allow the method to scale to high-dimensional data with complex correlations.
Shin \etal~\cite{shin1991pi} introduce the pi-sigma network, which is a neural network with a single hidden layer. Multiple affine transformations of the data are learned; a product unit multiplies all the features to obtain the output. Improvements in the pi-sigma network include regularization for training in~\cite{xiong2007training} or using multiple product units to obtain the output in~\cite{voutriaridis2003ridge}.
The pi-sigma network is extended in sigma-pi-sigma neural network (SPSNN)~\cite{li2003sigma}. The idea of SPSNN relies on summing different pi-sigma networks to obtain each output. SPSNN also uses a predefined basis (overlapping rectangular pulses) on each pi-sigma sub-network to filter the input features. Even though such networks use polynomial features or products, they do not scale well in high-dimensional signals. In addition, their experimental evaluation is conducted only on signals with known ground-truth distributions (and with up to 3 dimensional input/output), unlike the modern generative models where only a finite number of samples from high-dimensional ground-truth distributions is available.
\rebuttal{Convolutional arithmetic circuits (ConvACs) are also related to our work. Arithmetic circuits are networks with two types of nodes: sum nodes (weighted sum of their inputs), and product nodes (computing the product of their inputs). Those two types of nodes are sufficient to express a polynomial expansion. On \cite{cohen2016expressive}, the authors want to characterize the depth efficiency of (deep) convolutional neural networks. The CP decomposition is used to factorize the weights of a shallow convolutional network, while the hierarchical Tucker decomposition is used for the deep network. In \cite{cohen2016convolutional}, the authors generalize their previous results by inserting nonlinear activation functions (only linear activation functions were considered in \cite{cohen2016expressive}). The aforementioned works have a complementary role to our work, since their intent is to characterize (the depth efficiency of) convolutional neural networks, while our goal is to use polynomial expansion to approximate the target function.}
Another instance of such polynomial networks is through multiplicative interactions. Recently, there is a surge of methods~\citep{bahdanau2014neural, srivastava2015highway, reed2014learning} reporting superior performance through multiplicative interactions. The work of \citet{jayakumar2020Multiplicative} provides a theoretical understanding on why such connections might be beneficial. The aforementioned works model interactions of second or third order. Polynomial networks can be seen as high-order generalizations of such multiplicative interactions~\citet{jayakumar2020Multiplicative, bahdanau2014neural, srivastava2015highway}.
\subsection{\rebuttal{Notation}}
\rebuttal{Tensors\footnote{Further details on the tensor notation are deferred to the supplementary.} are symbolized by calligraphic letters, e.g., $\bmcal{X}$, while matrices (vectors) are denoted by uppercase (lowercase) boldface letters e.g., $\bm{X}$, ($\bm{x}$). A set of $M$ real matrices (vectors) of varying dimensions is denoted by $\{\bm{X}\matnot{m} \in \realnum^{I_m \times N} \}_{m=1}^M$ $( \{ \bm {x}\matnot{m} \in \realnum^{I_m} \}_{m=1}^M )$. }
\rebuttal{\textbf{Products}: The \textit{Hadamard} product of $\bm{A} \in \realnum^{I \times N}$
and $\bm{B} \in \realnum^{I \times N}$ is defined as $\bm{A} * \bm{B}$ and is equal to ${A}_{(i, j)} {B}_{(i, j)}$ for the $(i, j)$ element. The \textit{Khatri-Rao} product of matrices $\bm{A} \in \realnum^{I \times N}$
and $\bm{B} \in \realnum^{J \times N}$ is
denoted by $\bm{A} \odot \bm{B}$ and yields a matrix of
dimensions $(IJ)\times N$. For a set of matrices $\{\bm{A}\matnot{m} \in \realnum^{I_m \times N} \}_{m=1}^M$ the Khatri-Rao product is denoted by:}
\begin{equation}
\bm{A}\matnot{1} \odot \bm{A}\matnot{2} \odot \cdots \odot \bm{A}\matnot{M} \doteq \bigodot_{m=1}^M \bm{A}\matnot{m}
\end{equation}
\rebuttal{\textbf{Tensors}: Each element of an $M^{th}$ order tensor $\bmcal{X}$ is addressed by $M$ indices, i.e., $(\bmcal{X})_{i_{1}, i_{2}, \ldots, i_{M}} \doteq x_{i_{1}, i_{2}, \ldots, i_{M}}$. An $M^{th}$-order real-valued tensor $\bmcal{X}$ is defined over the
tensor space $\realnum^{I_{1} \times I_{2} \times \cdots \times
I_{M}}$, where $I_{m} \in \mathbb{Z}$ for $m=1,2,\ldots,M$.
The \textit{mode-$m$ vector product} of $\bmcal{X}$ with a
vector $\bm{u} \in \realnum^{I_m}$, denoted by
$\bmcal{X} \times_{m} \bm{u} \in \realnum^{I_{1}\times
I_{2}\times\cdots\times I_{m-1} \times I_{m+1} \times
\cdots \times I_{M}} $, results in a tensor of order $M-1$:}
\begin{equation}\label{E:Tensor_Mode_n}
(\bmcal{X} \times_{m} \bm{u})_{i_1, \ldots, i_{m-1}, i_{m+1},
\ldots, i_{M}} = \sum_{i_m=1}^{I_m} x_{i_1, i_2, \ldots, i_{M}} u_{i_m}.
\end{equation}
\rebuttal{Furthermore, we denote
$\bmcal{X} \times_{1} \bm{u}^{(1)} \times_{2} \bm{u}^{(2)} \times_{3} \cdots \times_{M} \bm{u}^{(M)} \doteq
\bmcal{X} \prod_{m=1}^m \times_{m} \bm{u}^{(m)}$.}
A core tool in our analysis is the CP decomposition that factorizes a tensor into a sum of component rank-one tensors~\cite{kolda2009tensor}. By considering the mode-$1$ unfolding of an $M^{th}$-order tensor $\bmcal{X}$, the CP decomposition can be written in matrix form as \citep{kolda2009tensor}: $ \bm{X}_{(1)} \doteq \bm{U}\matnot{1} \bigg( \bigodot_{m = M}^{2} \bm{U}\matnot{m}\bigg)^T$ where $\{ \bm{U}\matnot{m}\}_{m=1}^{M}$ are the factor matrices.
\section{Method}
\label{sec:prodpoly_method}
We want to learn a function approximator where each element of the output $\outvar_j$, with $j\in [1, o]$, is expressed as a polynomial\footnote{The theorem of \citep{stone1948generalized} guarantees that any smooth function can be approximated by a polynomial. The approximation of multivariate functions is covered by an extension of the Weierstrass theorem, e.g., in \cite{nikol2013analysis} (pg 19).} of all the input elements $\invar_i$, with $i\in [1, d]$. That is, we want to learn a function $G: \realnum^{d} \to \realnum^{o}$ of order $N \in \naturalnum$, such that:
\begin{equation}
\begin{split}
\outvar_j = G(\binvar)_j = \beta_j + {\bm{w}_j^{[1]}}^T\binvar + \binvar^T \bm{W}_j^{[2]}\binvar + \\
\bmcal{W}_j^{[3]}\times_1\binvar\times_2\binvar\times_3\binvar + \cdots + \bmcal{W}_j^{[N]}\prod_{n=1}^N \times_{n} \binvar
\end{split}
\label{eq:prodpoly_starting_poly_eq_element}
\end{equation}
where $\beta_j \in \realnum$, and $\big\{\bmcal{W}_j^{[n]} \in \realnum^{\prod_{m=1}^n\times_m d}\big\}_{n=1}^N$ are parameters for approximating the output $\outvar_j$. The correlations (of the input elements $\invar_i$) up to $N^{th}$ order emerge in \eqref{eq:prodpoly_starting_poly_eq_element}.
A more compact expression of \eqref{eq:prodpoly_starting_poly_eq_element} is obtained by vectorizing the outputs:
\begin{equation}
\boutvar = G(\binvar) = \sum_{n=1}^N \bigg(\bmcal{W}^{[n]} \prod_{j=2}^{n+1} \times_{j} \binvar\bigg) + \bm{\beta}
\label{eq:prodpoly_poly_general_eq}
\end{equation}
where $\bm{\beta} \in \realnum^o$ and $\big\{\bmcal{W}^{[n]} \in \realnum^{o\times \prod_{m=1}^{n}\times_m d}\big\}_{n=1}^N$ are the learnable parameters. This form of \eqref{eq:prodpoly_poly_general_eq} allows us to approximate any smooth function (for large $N$), however the parameters grow with $\mathcal{O}(d^N)$.
A variety of methods, such as pruning~\cite{frankle2018lottery, han2015learning}, special linear operators~\cite{ding2017c} with reduced parameters, parameter sharing/prediction~\cite{yunpeng2017sharing, denil2013predicting}, can be employed to reduce the parameters. The aforementioned approaches are post-processing techniques, i.e., given a (pre-trained) network, they reduce the parameters of the specific network. Instead, we design two principled ways which allow an efficient implementation. The first method relies on performing an off-the-shelf tensor decomposition on \eqref{eq:prodpoly_poly_general_eq}, while the second considers the final polynomial as the product of lower-degree polynomials.
\subsection{Single polynomial}
\label{ssec:prodpoly_single_poly}
A tensor decomposition on the parameters is a natural way to reduce the parameters and to implement \eqref{eq:prodpoly_poly_general_eq} with a neural network. Below, we demonstrate how three such decompositions result in novel architectures for a neural network training. The main symbols are summarized in Table~\ref{tbl:prodpoly_primary_symbols}, while the equivalence between the recursive relationship and the polynomial is analyzed in the supplementary.
\vspace{3mm}
\noindent \textbf{Model 1: CCP{} (Coupled CP decomposition)}
Instead of factorizing each parameter tensor $\bmcal{W}^{[n]}$ individually we propose to jointly factorize all the parameter tensors using a coupled CP decomposition~\cite{kolda2009tensor} with a specific pattern of factor sharing. To illustrate the factorization, we assume a third order approximation ($N=3$), and then provide the recursive relationship that can scale to arbitrary expansion.
Let us assume that the parameter tensors admit the following coupled CP decomposition with the factors corresponding to lower-order levels of approximation being shared across all parameters tensors. That is:
\begin{itemize}
\item Let $\bm{W}^{[1]} = \bm{C}\bm{U}\matnot{1}^T$, be the parameters for first level of approximation.
\item Let $\bmcal{W}^{[2]}$ being a superposition of of two weights tensors, namely $\bmcal{W}^{[2]} = \bmcal{W}^{[2]}_{1:2} + \bmcal{W}^{[2]}_{1:3}$, with $\bmcal{W}^{[2]}_{i:j}$ denoting parameters associated with the second order interactions across the $i^{th}$ and $j^{th}$ order of approximation. By enforcing the CP decomposition of the above tensors to share the factor with tensors corresponding to lower-order of approximation we obtain in matrix form: $\bm{W}^{[2]}_{(1)} = \bm{C}(\bm{U}\matnot{3} \odot \bm{U}\matnot{1})^T + \bm{C}(\bm{U}\matnot{2} \odot \bm{U}\matnot{1})^T$.
\item Similarly, we enforce the third-order parameters tensor to admit the following CP decomposition (in matrix form) $\bm{W}^{[3]}_{(1)} = \bm{C}(\bm{U}\matnot{3} \odot \bm{U}\matnot{2} \odot \bm{U}\matnot{1})^T $. Note that all but the $\bm{U}\matnot{3}$ factor matrices are shared in the factorization of tensors capturing polynomial parameters for the first and second order of approximation.
\end{itemize}
The parameters are $\bm{C} \in \realnum^{o\times k}, \bm{U}\matnot{m} \in \realnum^{d\times k}$ for $m=1,2,3$. Then, (\ref{eq:prodpoly_poly_general_eq}) for $N=3$ is written as:
\begin{equation}
\begin{split}
& G(\bm{z}) = \bm{\beta} + \bm{C}\bm{U}\matnot{1}^T\bm{z} + \bm{C}\Big(\bm{U}\matnot{3} \odot \bm{U}\matnot{1}\Big)^T(\bm{z} \odot \bm{z}) + \\
& \bm{C}\Big(\bm{U}\matnot{2} \odot \bm{U}\matnot{1}\Big)^T(\bm{z} \odot \bm{z}) + \\
& \bm{C}\Big(\bm{U}\matnot{3} \odot \bm{U}\matnot{2} \odot \bm{U}\matnot{1}\Big)^T(\bm{z} \odot \bm{z} \odot \bm{z})
\label{eq:polygan_recursive_gen_third_order}
\end{split}
\end{equation}
Using the Lemma 1 (provided in the supplementary), we can transform the \eqref{eq:polygan_recursive_gen_third_order} into a neural network as depicted in Fig.~\ref{fig:prodpoly_model1_schematic}.
The CCP{} factorization generalizes to $N^{th}$ order expansion. The recursive relationship for the $N^{th}$ order approximation is:
\begin{equation}
\boutvar_{n} = \Big(\bm{U}\matnot{n}^T \binvar \Big)* \boutvar_{n-1} + \boutvar_{n-1}
\label{eq:prodpoly_model1}
\end{equation}
for $n=2,\ldots,N$ with $\boutvar_{1} = \bm{U}\matnot{1}^T \binvar$ and $\boutvar = \bm{C}\boutvar_{N} + \bm{\beta}$. The parameters $\bm{C} \in \realnum^{o\times k}, \bm{U}\matnot{n} \in \realnum^{d\times k}$ for $n=1,\ldots,N$ are learnable.
\begin{figure}[!h]
\centering
\includegraphics[width=1\linewidth]{schematics-model1.pdf}
\caption{Schematic illustration of the CCP{} (for third order approximation). Symbol $*$ refers to the Hadamard product.}
\label{fig:prodpoly_model1_schematic}
\end{figure}
\noindent\textbf{Model 2: NCP{} (Nested coupled CP decomposition)}
Instead of explicitly separating the interactions between layers, we can utilize a joint hierarchical decomposition on the polynomial parameters. Let us first introduce learnable hyper-parameters $\big\{\bm{b}\matnot{n} \in \realnum^\omega\big\}_{n=1}^N$, which act as scaling factors for each parameter tensor. Therefore, we modify (\ref{eq:prodpoly_poly_general_eq}) to:
\begin{equation}
G(\bm{z}) = \sum_{n=1}^N \bigg(\bmcal{W}^{[n]} \times_2 \bm{b}\matnot{N+1-n} \prod_{j=3}^{n+2} \times_{j} \bm{z}\bigg) + \bm{\beta},
\label{eq:prodpoly_general_polynomial_with_b}
\end{equation}
with $\big\{\bmcal{W}^{[n]} \in \realnum^{o\times \omega \times \prod_{m=1}^{n}\times_m d}\big\}_{n=1}^N$. Similarly to CCP{}, we demonstrate the decomposition assuming a third order approximation ($N=3$), and then provide the general recursive relationship.
To estimate the parameters (in $N=3$ expansion) we jointly factorize all parameter tensors by employing nested CP decomposition with parameter sharing as follows (in matrix form):
\begin{itemize}
\item First order parameters : $\bm{W}^{[1]}_{(1)} = \bm{C} (\bm{A}\matnot{3} \odot \bm{B}\matnot{3})^T$.
\item Second order parameters: \\
$\bm{W}^{[2]}_{(1)} = \bm{C} \bigg\{\bm{A}\matnot{3} \odot \bigg[\Big(\bm{A}\matnot{2} \odot \bm{B}\matnot{2}\Big) \bm{S}\matnot{3}\bigg]\bigg\}^T$.
\item Third order parameters: \\
$\bm{W}^{[3]}_{(1)} = \bm{C} \bigg\{\bm{A}\matnot{3} \odot \bigg[\bigg(\bm{A}\matnot{2} \odot \Big\{\Big(\bm{A}\matnot{1} \odot \bm{B}\matnot{1}\Big) \bm{S}\matnot{2}\Big\} \bigg)\bm{S}\matnot{3} \bigg]\bigg\}^T$
\end{itemize}
with $\bm{C} \in \realnum^{o\times k}, \bm{A}\matnot{n} \in \realnum^{d\times k}, \bm{S}\matnot{n} \in \realnum^{k\times k}, \bm{B}\matnot{n} \in \realnum^{\omega\times k}$ for $n=1,\ldots,N$.
Altogether, (\ref{eq:prodpoly_general_polynomial_with_b}) for $N=3$ is written as:
\begin{equation}
\begin{split}
& G(\bm{z}) = \bm{\beta} + \bm{C} (\bm{A}\matnot{3} \odot \bm{B}\matnot{3})^T (\bm{z} \odot \bm{b}\matnot{3}) + \\
& \bm{C} \bigg\{\bm{A}\matnot{3} \odot \bigg[\Big(\bm{A}\matnot{2} \odot \bm{B}\matnot{2}\Big) \bm{S}\matnot{3}\bigg]\bigg\}^T \Big(\bm{z} \odot \bm{z} \odot \bm{b}\matnot{2} \Big) + \\
& \bm{C} \bigg\{\bm{A}\matnot{3} \odot \bigg[\bigg(\bm{A}\matnot{2} \odot \Big\{\Big(\bm{A}\matnot{1} \odot \bm{B}\matnot{1}\Big) \bm{S}\matnot{2}\Big\} \bigg)\bm{S}\matnot{3} \bigg]\bigg\}^T \bm{\mu}
\label{eq:polygan_third_order_decomp_init}
\end{split}
\end{equation}
with $\bm{\mu} = \Big(\bm{z} \odot \bm{z} \odot \bm{z} \odot \bm{b}\matnot{1}\Big)$. Using Lemma1 and further algebraic operations (see Sec. 3.2 in the supplementary), \eqref{eq:polygan_third_order_decomp_init} can be implemented by a neural network as depicted in Fig.~\ref{fig:prodpoly_model2_schematic}.
The recursive relationship for $N^{th}$ order approximation is defined as:
\begin{equation}
\boutvar_{n} = \Big(\bm{A}\matnot{n}^T\binvar\Big) * \Big(\bm{S}\matnot{n}^T \boutvar_{n-1} + \bm{B}\matnot{n}^T\bm{b}\matnot{n}\Big)
\label{eq:prodpoly_model2}
\end{equation}
for $n=2,\ldots,N$ with $\boutvar_{1} = \Big(\bm{A}\matnot{1}^T\binvar\Big) * \Big( \bm{B}\matnot{1}^T\bm{b}\matnot{1} \Big)$ and $\boutvar = \bm{C}\boutvar_{N} + \bm{\beta}$. The parameters $\bm{C} \in \realnum^{o\times k}, \bm{A}\matnot{n} \in \realnum^{d\times k}, \bm{S}\matnot{n} \in \realnum^{k\times k}, \bm{B}\matnot{n} \in \realnum^{\omega\times k}$, $\bm{b}\matnot{n} \in \realnum^{\omega}$ for $n=1,\ldots,N$, are learnable.
\begin{figure}[!h]
\centering
\includegraphics[width=1\linewidth]{schematics-model2.pdf}
\caption{Schematic illustration of the NCP{} (for third order approximation). Symbol $*$ refers to the Hadamard product.}
\label{fig:prodpoly_model2_schematic}
\end{figure}
\noindent\textbf{Model 3: NCP-Skip{} (Nested coupled CP decomposition with skip)}
The expressiveness of NCP{} can be further extended using a skip connection (motivated by CCP). The new model uses a nested coupled decomposition and has the following recursive expression:
\begin{equation}
\boutvar_{n} = \Big(\bm{A}\matnot{n}^T\binvar\Big) * \Big(\bm{S}\matnot{n}^T \boutvar_{n-1} + \bm{B}\matnot{n}^T\bm{b}\matnot{n}\Big) + \bm{V}\matnot{n}\boutvar_{n-1}
\label{eq:prodpoly_model3}
\end{equation}
for $n=2,\ldots,N$ with $\boutvar_{1} = \Big(\bm{A}\matnot{1}^T\binvar\Big) * \Big( \bm{B}\matnot{1}^T\bm{b}\matnot{1} \Big)$ and $\boutvar = \bm{C}\boutvar_{N} + \bm{\beta}$. \rebuttal{The parameters $\bm{V}\matnot{n} \in \realnum^{k\times k}$ are learnable}, while the rest parameters are the same as in NCP. The difference in the recursive form results in a different polynomial expansion and thus architecture.
\begin{figure}[!h]
\centering
\includegraphics[width=1\linewidth]{schematics-model3.pdf}
\caption{Schematic illustration of the NCP-Skip{} (for third order approximation). The difference from Fig.~\ref{fig:prodpoly_model2_schematic} is the skip connections added in this model.}
\label{fig:prodpoly_model3_schematic}
\end{figure}
\noindent \textbf{Comparison between the models}
All three models (see Table~\ref{tbl:prodpoly_single_polynomial_models} for names and schematics) are based on a polynomial expansion, however their recursive forms and employed decompositions differ.
CCP{} is a straightforward coupled decomposition and is a proof of concept that polynomials can learn high-dimensional distributions. NCP{} illustrates how to convert a popular CNN/linear model of the form $\boutvar_{k} = \bm{S}\matnot{k}^T \boutvar_{k-1} + \bm{b}\matnot{k}$ to a polynomial (i.e., $\boutvar_{k} = (\bm{A}\matnot{k}^T\binvar) * (\bm{S}\matnot{k}^T \boutvar_{k-1} + \bm{b}\matnot{k})$). Similarly, NCP-Skip{} demonstrates how a residual network can be transformed into a polynomial.
\rebuttal{An illustrative comparison of the three decompositions is conducted below. The challenging task of synthesizing images is selected. Each model is implemented using the respective decomposition, i.e., CCP, NCP, NCP-Skip. Following the derivations of the previous few paragraphs, we train a generator without activation functions between the blocks; a single hyperbolic tangent is used in the output space as a normalization\footnote{Further experiments without activation functions are deferred to the supplementary.}. The generator is trained with an adversarial loss, i.e., using Generative Adversarial Nets (GANs)~\cite{goodfellow2014generative}. GANs typically consist of two deep networks, namely a generator $G$ and a discriminator $D$. $G$ is a decoder, which receives as input a random noise vector $\bm{z} \in \realnum^{d}$ and outputs a sample $\bm{x} = G(\bm{z})$. $D$ receives as input both $G(\bm{z})$ and real samples and tries to differentiate the fake and the real samples. During training, both $G$ and $D$ compete against each other.}
\rebuttal{The three models are originally compared in fashion image generation. The outcomes, visualized in Fig.~\ref{fig:prodpoly_linear_fashionmnist_model_comparison}, demonstrate similar generation properties.}
\begin{figure*}[htb]
\subfloat[GT]{\includegraphics[width=0.24\linewidth]{linear_generation_image-fashionmnist_gt.png}\hspace{1mm}}
\subfloat[ProdPoly{} - CCP]{\includegraphics[width=0.24\linewidth]{linear_generation_image-fashionmnist_prod_model1.png}\hspace{1mm}}
\subfloat[ProdPoly{} - NCP]{\includegraphics[width=0.24\linewidth]{linear_generation_image-fashionmnist_prod.png}\hspace{1mm}}
\subfloat[ProdPoly{} - NCP-Skip]{\includegraphics[width=0.24\linewidth]{linear_generation_image-fashionmnist_prod_model3.png}\hspace{1mm}}
\caption{Comparison of the proposed models in fashion image~\cite{xiao2017fashion} generation without activation functions.}
\label{fig:prodpoly_linear_fashionmnist_model_comparison}
\end{figure*}
\begin{figure*}[htb]
\subfloat[GT]{\includegraphics[width=0.24\linewidth]{linear_generation_image-yaleb_gt.png}\hspace{1mm}}
\subfloat[ProdPoly{} - CCP]{\includegraphics[width=0.24\linewidth]{linear_generation_image-yaleb_prod_model1.png}\hspace{1mm}}
\subfloat[ProdPoly{} - NCP]{\includegraphics[width=0.24\linewidth]{linear_generation_image-yaleb_prod.png}\hspace{1mm}}
\subfloat[ProdPoly{} - NCP-Skip]{\includegraphics[width=0.24\linewidth]{linear_generation_image-yaleb_prod_model3.png}\hspace{1mm}}
\caption{Comparison of the proposed models in facial image~\cite{georghiades2001few} generation without activation functions.}
\label{fig:prodpoly_linear_yaleb_model_comparison}
\end{figure*}
\rebuttal{In Fig.~\ref{fig:prodpoly_linear_yaleb_model_comparison}, samples of the three models are synthesized when trained facial images. All three models can generate faces without activation functions between the layers, while the three models share similar generation quality.}
In the remainder of the paper, for comparison purposes we use the NCP{} by default for the image generation and NCP-Skip{} for the image classification. In all cases, to mitigate stability issues that might emerge during training, we employ certain normalization schemes that constrain the magnitude of the gradients.
\subsection{Product of polynomials}
\label{ssec:prodpoly_product_poly}
Instead of using a single polynomial, we express the function approximation as a product of polynomials. The product is implemented as successive polynomials where the output of the $i^{th}$ polynomial is used as the input for the $(i+1)^{th}$ polynomial. The concept is visually depicted in Fig.~\ref{fig:prodpoly_prod_schematic}; each polynomial expresses a second order expansion. Stacking $N$ such polynomials results in an overall order of $2^N$. Trivially, if the approximation of each polynomial is $B$ and we stack $N$ such polynomials, the total order is $B^N$. The product does not necessarily demand the same order in each polynomial, the model and the expansion order of each polynomial can be different and dependent on the task. For instance, for generative tasks that the resolution increases progressively, the expansion order could increase in the last polynomials. In all cases, the final order will be the product of each polynomial power.
There are two main benefits of the product over the single polynomial: a) it allows using different decompositions (e.g., like in Sec.~\ref{ssec:prodpoly_single_poly}) and expansion order for each polynomial; b) it requires much fewer parameters for achieving the same order of approximation. Given the benefits of the product of polynomials, we assume below that a product of polynomials is used, unless explicitly mentioned otherwise. The respective model of product polynomials is called ProdPoly.
\begin{figure}[!h]
\centering
\includegraphics[width=1\linewidth]{schematics-model4.pdf}
\caption{Abstract illustration of the ProdPoly. The input variable $\binvar$ on the left is the input to a $2^{nd}$ order expansion; the output of this is used as the input for the next polynomial (also with a $2^{nd}$ order expansion) and so on. If we use $N$ such polynomials, the final output $G(\binvar)$ expresses a $2^N$ order expansion. In addition to the high order of approximation, the benefit of using the product of polynomials is that the model is flexible, in the sense that each polynomial can be implemented as a different decomposition of Sec.~\ref{ssec:prodpoly_single_poly}.}
\label{fig:prodpoly_prod_schematic}
\end{figure}
\subsection{Task-dependent input/output}
\label{ssec:prodpoly_method_details}
The aforementioned polynomials are a function $\boutvar = G(\binvar)$, where the input/output are task-dependent. For a generative task, e.g., learning a decoder, the input $\binvar$ is typically some low-dimensional noise, while the output is a high-dimensional signal, e.g., an image. For a discriminative task the input $\binvar$ is an image; for a domain adaptation task the signal $\binvar$ denotes the source domain and $\boutvar$ the target domain.
\section{Experiments}
\label{sec:prodpoly_experiments}
We conduct four experiments against state-of-the-art models in three diverse tasks\footnote{The source code is available in \url{https://github.com/grigorisg9gr/polynomial_nets}.}: image generation, image classification, face verification/identification and graph representation learning. In each case, the baseline considered is converted into an instance of our family of $\Pi$-nets and the two models are compared. \rebuttal{Experiments on image generation and image classification without using activation functions between the layers are deferred to the supplementary.}
\subsection{Image generation}
\label{ssec:prodpoly_experiment_stylegan}
The robustness of ProdPoly{} in image generation is assessed in two different architectures/datasets below.
\textbf{SNGAN on CIFAR10}: In the first experiment, the architecture of SNGAN~\cite{miyato2018spectral} is selected as a strong baseline on CIFAR10~\cite{krizhevsky2014cifar}. The baseline includes $3$ residual blocks in the generator and the discriminator.
The generator is converted into a $\Pi$-net, where each residual block is a single order of the polynomial. We implement two versions, one with a single polynomial (NCP) and one with product of polynomials (where each polynomial uses NCP). In our implementation $\bm{A}_{[n]}$ is a thin FC layer, $(\bm{B}\matnot{n})^T \bm{b}\matnot{n}$ is a bias vector and $\bm{S}\matnot{n}$ is the transformation of the residual block. Other than the aforementioned modifications, the hyper-parameters (e.g., discriminator, learning rate, optimization details) are kept the same as in SNGAN~\cite{miyato2018spectral}.
Each network was run for 10 times and the mean and variance are reported. The popular Inception Score (IS)~\citep{salimans2016improved} and the Frechet Inception Distance (FID)~\citep{heusel2017gans} are used for quantitative evaluation. Both scores extract feature representations from a pre-trained classifier (the Inception network~\citep{szegedy2015going}).
The quantitative results are summarized in Table~\ref{tab:prodpoly_exper_cifar10_sota}. In addition to SNGAN and our two variations with polynomials, we have added the scores of \cite{grinblat2017class,gulrajani2017improved,du2019implicit, hoshen2019non,lucas2019adversarial} as reported in the respective papers. Note that the single polynomial already outperforms the baseline, while the ProdPoly{} boosts the performance further and achieves a substantial improvement over the original SNGAN.
\begin{table}
\caption{IS/FID scores on CIFAR10~\citep{krizhevsky2014cifar} generation. The scores of \cite{grinblat2017class,gulrajani2017improved} are added from the respective papers as using similar residual based generators. The scores of \cite{du2019implicit, hoshen2019non, lucas2019adversarial} represent alternative generative models. ProdPoly\ outperforms the compared methods in both metrics.}
\begin{tabular}{|c | c | c|}
\hline
\multicolumn{3}{|c|}{Image generation on CIFAR10}\\
\hline
Model & IS ($\uparrow$) & FID ($\downarrow$)\\
\hline
SNGAN & $8.06\pm 0.10$ & $19.06\pm 0.50$\\
\hline
NCP (Sec.~\ref{ssec:prodpoly_single_poly}) & $8.30\pm 0.09$ & $17.65\pm 0.76$\\
\hline
ProdPoly & $\bm{8.49\pm 0.11}$ & $\bm{16.79\pm 0.81}$\\
\cmidrule[\heavyrulewidth](){1 - 3}
CSGAN-\cite{grinblat2017class} & $7.90\pm0.09$ & -\\
\hline
WGAN-GP-\cite{gulrajani2017improved} & $7.86\pm0.08$ & -\\
\hline
CQFG-\cite{lucas2019adversarial} & $8.10$ & $18.60$\\
\hline
EBM~\cite{du2019implicit} & $6.78$ & $38.2$ \\
\hline
GLANN~\cite{hoshen2019non} & - & $46.5\pm0.20$\\
\hline
\end{tabular}
\label{tab:prodpoly_exper_cifar10_sota}
\end{table}
\textbf{StyleGAN on FFHQ}: StyleGAN~\cite{karras2018style} is the state-of-the-art architecture in image generation. The generator is composed of two parts, namely: (a) the mapping network, composed of 8 FC layers, and (b) the synthesis network, which is based on ProGAN~\cite{karras2017progressive} and progressively learns to synthesize high quality images. The sampled noise is transformed by the mapping network and the resulting vector is then used for the synthesis network. As discussed in the introduction, StyleGAN is already an instance of the $\Pi$-net family, due to AdaIN.
Specifically, the $k^{th}$ AdaIN layer is $\bm{h}_k = (\bm{A}_k^T\bm{w}) * n(c(\bm{h}_{k-1}))$, where $n$ is a normalization, $c$ the convolution operator and $\bm{w}$ is the transformed noise $\bm{w} = MLP(\bm{z})$ (mapping network).
This is equivalent to our NCP model by setting $\bm{S}\matnot{k}^T$ as the convolution operator.
In this experiment we illustrate how simple modifications, using our family of products of polynomials, further improve the representation power. We make a minimal modification in the mapping network, while fixing the rest of the hyper-parameters. In particular, we convert the mapping network into a polynomial (specifically a NCP), which makes the generator a product of two polynomials.
The Flickr-Faces-HQ Dataset (FFHQ) dataset~\cite{karras2018style} which includes $70,000$ images of high-resolution faces is used. All the images are resized to $256\times256$. The best FID scores of the two methods (in $256\times 256$ resolution) are $\bm{6.82}$ for ours and $7.15$ for the original StyleGAN, respectively. That is, our method improves the results by $5\%$. Synthesized samples of our approach are visualized in Fig.~\ref{fig:prodpoly_stylegan_visual}.
\begin{figure*}
\centering
\centering
\includegraphics[width=0.123\linewidth]{stylegan-000000_ours.png}\hspace{-0.5mm}
\includegraphics[width=0.123\linewidth]{stylegan-000001_ours.png}\hspace{-0.5mm}
\includegraphics[width=0.123\linewidth]{stylegan-000002_ours.png}\hspace{-0.5mm}
\includegraphics[width=0.123\linewidth]{stylegan-392.png}\hspace{-0.5mm}
\includegraphics[width=0.123\linewidth]{stylegan-423.png}\hspace{-0.5mm}
\includegraphics[width=0.123\linewidth]{stylegan-754.png}\hspace{-0.5mm}
\includegraphics[width=0.123\linewidth]{stylegan-000003_ours.png}\hspace{-0.5mm}
\includegraphics[width=0.123\linewidth]{stylegan-689.png}
\caption{Samples synthesized from ProdPoly{} (trained on FFHQ).}
\label{fig:prodpoly_stylegan_visual}
\end{figure*}
\subsection{Classification}
We perform two experiments on classification: a) audio classification, b) image classification. Residual Network (ResNet)~\cite{he2016deep,srivastava2015highway} and its variants~\cite{huang2017densely, wang2018mixed, xie2017aggregated, zhang2017residual, zagoruyko2016wide} have been applied to diverse tasks including object detection and image generation~\cite{grinblat2017class,gulrajani2017improved,miyato2018spectral}.
The core component of ResNet{} is the residual block; the $t^{th}$ residual block is expressed as $\bm{z}_{t+1} = \bm{z}_t + \bm{C} \bm{z}_t$ for input $\bm{z}_t$. Each residual block is adapted (using NCP-Skip) to express a higher-order expansion; \rebuttal{this is achieved by using $\bm{V}\matnot{n} = \bm{I} + \bm{S}\matnot{n}$ in \eqref{eq:prodpoly_model3}, where $\bm{I}$ is an identity matrix.} The output of each residual block is the input for the next residual block, which makes our ResNet{} a product of polynomials.
\textbf{Audio classification}: \rebuttal{The goal of this experiment is to reduce the number of residual blocks (of higher-order polynomial expansion) without sacrificing the performance of the original ResNet, while validating the performance of the proposed method in a distribution that differs from that of natural images.}
The performance of ResNet{} is evaluated on the Speech Commands dataset~\cite{warden2018speech}.
The dataset includes $60,000$ audio files; each audio contains a single word of a duration of one second. There are $35$ different words (classes) with each word having $1,500 - 4,100$ recordings. Every audio file is converted into a mel-spectrogram of resolution $32\times32$. Each method is trained for $70$ epochs with SGD and initial learning rate of $0.01$. The learning rate is reduced if the validation accuracy does not improve for two consecutive epochs.
The baseline is a \resnet34 architecture; we use second-order residual blocks to build the Prodpoly-\resnet{} to match the performance of the baseline.
The quantitative results are added in Table~\ref{tab:prodpoly_resnet_speech_command}. The two models share the same accuracy, however Prodpoly-\resnet{} includes $38\%$ fewer parameters. \rebuttal{This result validates our assumption that $\Pi$-nets can achieve the same performance with less parameters than the baseline.}
\begin{table}[h]
\caption{Speech classification with ResNet. The accuracy of the compared methods is similar, but Prodpoly-\resnet{} has $38\%$ fewer parameters. The symbol `\# par' abbreviates the number of parameters (in millions). }
\centering
\begin{tabular}{|c | c | c | c|}
\hline
\multicolumn{4}{|c|}{Speech Commands classification with ResNet}\\
\hline
Model & \# blocks & \# par & Accuracy\\
\hline
\resnet34 & $[3, 4, 6, 3]$ & $21.3$ & $0.951 \pm 0.002$\\
\hline
Prodpoly-\resnet & $[3, 3, 3, 2]$ & $\bm{13.2}$ & $0.951 \pm 0.002$\\
\hline
\end{tabular}
\label{tab:prodpoly_resnet_speech_command}
\end{table}
\textbf{Classification on CIFAR}: \rebuttal{The performance of the polynomial networks is also assessed on CIFAR10 and CIFAR100 classification. The goal is to reduce the number of residual blocks (of higher-order polynomial expansion) without sacrificing the performance of the original ResNet.}
We select the \resnet18 and \resnet34 as baselines.
Each method is trained for $120$ epochs with batch size $128$. The SGD optimizer is used with initial learning rate of $0.1$. The learning rate is multiplied with a factor of $0.1$ in epochs $40, 60, 80, 100$.
\begin{table}[h]
\caption{Image classification on CIFAR10 with ResNet. The \# abbreviates `number of', while the parameters are measured in millions. The term `block' abbreviates a `residual block'. Note that each baseline, e.g. \resnet18, has the same performance with the respective Prodpoly-\resnet, but significantly more parameters.}
\centering
\begin{tabular}{|c | c | c | c|}
\hline
\multicolumn{4}{|c|}{CIFAR10 classification with ResNet}\\
\hline
Model & \# blocks & \# params (M) & Accuracy\\
\hline
\resnet18 & $[2, 2, 2, 2]$ & $11.2$ & $0.945 \pm 0.000$\\
\hline
Prodpoly-\resnet & $[2, 2, 1, 1]$ & $\bm{6.0}$ & $0.945 \pm 0.001$\\
\hline\hline
\resnet34 & $[3, 4, 6, 3]$ & $21.3$ & $0.948 \pm 0.001$\\
\hline
Prodpoly-\resnet & $[3, 3, 2, 2]$ & $\bm{13.0}$ & $0.949 \pm 0.002$\\
\hline
\end{tabular}
\label{tab:prodpoly_resnet_cifar10}
\end{table}
\begin{figure}[h]
\centering
\subfloat[\resnet18]{\includegraphics[width=0.48\linewidth]{accuracy_bars_resnet-resnet18.pdf}\hspace{-0.1mm}}
\quad
\subfloat[Prodpoly-\resnet]{\includegraphics[width=0.48\linewidth]{accuracy_bars_resnet-resnet18_ours.pdf}\hspace{-1.9mm}}
\caption{The test accuracy of (a) \resnet18 and (b) the respective Prodpoly-\resnet{} are plotted (CIFAR10 training). The two models perform similarly throughout the training, while ours has $46\%$ less parameters. The width of the highlighted region denotes the standard deviation of each model.}
\label{fig:prodpoly_resnet_accuracy_bars_resnet18}
\end{figure}
In Table~\ref{tab:prodpoly_resnet_cifar10} the two different ResNet{} baselines are compared against Prodpoly-\resnet\ on CIFAR10; the respective Prodpoly-\resnet{} models have the same accuracy. However, each Prodpoly-\resnet{} has $\sim 40\%$ less parameters than the respective baseline. In addition, we visualize the test accuracy for \resnet18 and the respective Prodpoly-\resnet{} in Fig.~\ref{fig:prodpoly_resnet_accuracy_bars_resnet18}. The test error of the two models is similar throughout the training.
The same experiment is repeated on CIFAR100 with \resnet34 as the baseline. Table~\ref{tab:prodpoly_resnet_cifar100} exhibits a similar pattern. That is, the test accuracy of \resnet34 and Prodpoly-\resnet{} is similar, however Prodpoly-\resnet{} has $\sim 30\%$ less parameters.
\begin{table}[h]
\caption{CIFAR100 classification with ResNet. The accuracy of the compared methods is similar, but Prodpoly-\resnet{} has $30\%$ less parameters.}
\centering
\begin{tabular}{|c | c | c | c|}
\hline
\multicolumn{4}{|c|}{CIFAR100 classification with ResNet}\\
\hline
Model & \# blocks & \# params (M) & Accuracy\\
\hline
\resnet34 & $[3, 4, 6, 3]$ & $21.3$ & $0.769 \pm 0.003$\\
\hline
Prodpoly-\resnet & $[3, 4, 3, 2]$ & $\bm{14.7}$ & $0.769 \pm 0.001$\\
\hline
\end{tabular}
\label{tab:prodpoly_resnet_cifar100}
\end{table}
\textbf{Classification on ImageNet}:
We perform a large-scale classification experiment on ImageNet~\cite{russakovsky2015imagenet}. Models are trained on a DGX station with 4 Tesla V100 (32GB) GPUs.
To stabilize the training, the second order of each residual block is normalized with a hyperbolic tangent unit. SGD with momentum $0.9$, weight decay $10^{-4}$ and a mini-batch size of $512$ is used. The initial learning rate is set to $0.2$ and decreased by a factor of $10$ at $30, 60$, and $80$ epochs. Models are trained for $90$ epochs from scratch, using linear warm-up of the learning rate during first five epochs according to \citet{goyal2017accurate}.
The Top-1 and Top-5 error throughout the training is visualized in Fig.~\ref{pic:imagenet}, while the validation results are added in Table~\ref{tab:prodpoly_resnet_imagenetfp32}. For a fair comparison, we report the results from our training in both the original ResNet{} and Prodpoly-\resnet{}\footnote{The performance of the original ResNet~\citet{he2016deep} is inferior to the one reported here and in \citet{hu2018squeeze}.}. Prodpoly-\resnet{} consistently improves the performance with a negligible increase in computational complexity. Remarkably, \modelres50 achieves a single-crop Top-1 validation error of $22.827\%$ and Top-5 validation error of $6.431\%$, exceeding \resnet50 by $0.719\%$ and $0.473\%$, respectively.
\begin{figure*}[h!]
\centering
\subfloat[Top-1 Error]{
\label{pic:imagenettop1}
\includegraphics[width=0.5\textwidth]{figures/imagenet/pami2020top1.pdf}}
\subfloat[Top-5 Error]{
\label{pic:imagenettop5}
\includegraphics[width=0.5\textwidth]{figures/imagenet/pami2020top5.pdf}}
\caption{Top-1 and Top-5 error curves on the ImageNet dataset.}
\label{pic:imagenet}
\end{figure*}
\begin{table}[h!]
\caption{ImageNet classification results of ResNet50 and the proposed \modelres50. ``Throughput'' denotes the total Images Per Second (IPS) during training.}
\label{tab:prodpoly_resnet_imagenetfp32}
\centering
\begin{tabular}{c | c | c | c}
\hline
Model & Top-1 error ($\%$) & Top-5 error ($\%$) & Throughput\\
\hline
\resnet50 & 23.546 & 6.904 & 1625 \\
\modelres50 & 22.827 \tiny{($\downarrow 0.719$)} & 6.431 \tiny{($\downarrow 0.473$)} & 1531\\
\hline
\end{tabular}
\end{table}
\subsection{Face verification and identification}
We scrutinize the performance of the $\Pi$-nets on the challenging task of face recognition. The architecture of the current state-of-the-art method of ArcFace~\cite{deng2019arcface} is a ResNet, which we can convert into a polynomial network using the NCP-Skip.
\noindent{\bf Training Data}: The data of MS1M-RetinaFace dataset~\cite{guo2016ms,deng2019lightweight} consist the training images; all face images inside MS1M-RetinaFace are pre-processed to the size of $112\times112$ based on the five facial landmarks predicted by RetinaFace~\cite{deng2019retinaface}. In total, there are 5.1M images of 93K identities.
\noindent{\bf Testing Data}: The performance is compared on widely used face verification data-sets (e.g., LFW~\cite{huang2008labeled}, CFP~\cite{sengupta2016frontal}, AgeDB~\cite{Moschoglou2017AgeDB}, CPLFW~\cite{zheng2018cross}, CALFW~\cite{zheng2017cross} and RFW~\cite{Wang2019racial}).
Besides, we also extensively test the proposed method on large-scale benchmarks (e.g., IJB-B \cite{whitelam2017iarpa}, IJB-C \cite{maze2018iarpa} and MegaFace \cite{kemelmacher2016megaface}); the fundamental statistics of all the datasets are summarized in Table~\ref{table:dataset}. To get the embedding features for templates (e.g., IJB-B \cite{whitelam2017iarpa} and IJB-C \cite{maze2018iarpa}), we simply calculate the feature center of all images from the template or all frames from the video.
\noindent{\bf Training Details}: For the baseline embedding network, we employ the widely used CNN architecture, \resnet50.
Specifically, we follow \cite{deng2019arcface} to set the feature scale $s$ to 64 and choose the angular margin $m$ at $0.5$.
The batch size is set to $512$ with momentum $0.9$ and weight decay $5e-4$, while we decrease the learning rate in iterations 100K, 160K, 220K. The training finishes after 30 epochs. The implementation is by MXNet \cite{chen2015mxnet,deng2019arcface} and the models are trained on 8 NVIDIA 2080ti (11GB) GPUs.
The baseline residual block is converted into a second-order residual block to build the Prodpoly-\resnet{}, while we keep all the other settings exactly the same as the baseline. After training, we only keep the feature embedding network without the fully connected layer (174.5MB for \resnet50 and 181.8MB for \modelres50) and extract the $512$-$D$ features ($5.76$ ms/face for \resnet50 and $6.02$ ms/face for \modelres50) for each normalised face. Compared to \resnet50, \modelres50 obviously boosts the performance only by a negligible increase in model size and latency.
\begin{table}[t!]
\begin{center}
\caption{Face datasets for training and testing. ``(P)'' and ``(G)'' refer to the probe and gallery set, respectively.}
\label{table:dataset}
\begin{tabular}{c|c|c}
\hline
Datasets & \#Identity & \#Image\\
\hline
MS1MV2 & 93K & 5.1M \\
\hline
LFW \cite{huang2008labeled} & 5,749 & 13,233 \\
CFP \cite{sengupta2016frontal} & 500 & 7,000\\
AgeDB \cite{Moschoglou2017AgeDB} & 568 & 16,488 \\
CPLFW \cite{zheng2018cross} & 5,749 & 11,652 \\
CALFW \cite{zheng2017cross} & 5,749 & 12,174 \\
RFW \cite{Wang2019racial} & 11,430 & 40,607 \\
RFW-Caucasian \cite{Wang2019racial} & 2,959 & 10,196\\
RFW-Indian \cite{Wang2019racial} & 2,984 & 10,308\\
RFW-Asian \cite{Wang2019racial} & 2,492 & 9,688 \\
RFW-African \cite{Wang2019racial} & 2,995 & 10,415\\
\hline
MegaFace \cite{kemelmacher2016megaface} & 530 (P) & 1M (G) \\
IJB-B \cite{whitelam2017iarpa} & 1,845 & 76.8K \\
IJB-C \cite{maze2018iarpa} & 3,531 & 148.8K \\
\hline
\end{tabular}
\end{center}
\end{table}
\noindent {\bf Results on LFW, CFP-FF, CFP-FP, CPLFW, AgeDB-30, CALFW and RFW.}
LFW \cite{huang2008labeled} contains 13,233 web-collected images from 5,749 different identities, with limited variations in pose, age, expression and illuminations.
CFP \cite{sengupta2016frontal} consists of collected images of celebrities in frontal and profile views.
On CFP, there are two evaluation protocols: CFP-Frontal-Frontal and CFP-Frontal-Profile. CFP-Frontal-Profile is very challenging as the pose gap within positive pairs is around $90^{\circ}$.
CPLFW \cite{zheng2018cross} was collected by crowd-sourcing efforts to seek the pictures of people in LFW with pose gap as large as possible from the Internet.
CALFW \cite{zheng2017cross} is similar to CALFW, but from the perspective of age difference.
AgeDB \cite{Moschoglou2017AgeDB} contains manually annotated images. In this paper, we use the evaluation protocol with 30 years gap \cite{deng2019arcface}.
RFW \cite{Wang2019racial} is a benchmark for measuring racial bias, which consists of four test subsets, namely Caucasian, Indian, Asian and
African.
The quantitative results of the comparisons are exhibited in Table \ref{table:verification11}.
On LFW and CFP-FP, the results of \resnet50 and \modelres50 are similar to face verification on semi-frontal faces is saturated.
Nevertheless, \modelres50 significantly outperforms \resnet50 on CFP-FP, CPLFW, AgeDB-30, CALFW and RFW, indicating that the proposed method can enhance the robustness of the embedding features under pose variations, age variations and racial variations.
\begin{table}[t!]
\begin{center}
\caption{Verification performance ($\%$) of \resnet50 and the proposed \modelres50 on LFW, CFP-FF, CFP-FP, CPLFW, AgeDB-30, CALFW and RFW (Caucasian, Indian, Asian and African).}
\label{table:verification11}
\begin{tabular}{c|c|c}
\hline
Method & \resnet50 & \modelres50 \\
\hline
LFW & $ 99.733\pm0.309$ & {\bf 99.833$\pm0.211$} \tiny{($\uparrow 0.100$)} \\
CFP-FF & $ 99.871\pm0.135$ & {\bf 99.886$\pm0.178$} \tiny{($\uparrow 0.015$)}\\
\hline
CFP-FP & $ 98.800\pm0.249$ & {\bf 98.986$\pm0.274$} \tiny{($\uparrow 0.186$)} \\
CPLFW & $ 92.433\pm1.245$ & {\bf 93.317$\pm1.343$} \tiny{($\uparrow 0.884$)} \\
\hline
AgeDB-30 & $ 98.233\pm0.655$ & {\bf 98.467$\pm0.623$} \tiny{($\uparrow 0.234$)}\\
CALFW & $ 95.917\pm1.209$ & {\bf 96.233$\pm1.114 $} \tiny{($\uparrow 0.316$)}\\
\hline
RFW-Caucasian & $ 99.333\pm0.307$ & {\bf 99.700$\pm0.100 $} \tiny{($\uparrow 0.367$)}\\
RFW-Indian & $ 98.567\pm0.507$ & {\bf 99.300$\pm0.296 $} \tiny{($\uparrow 0.733$)}\\
RFW-Asian & $ 98.333\pm0.435$ & {\bf 98.950$\pm0.350 $} \tiny{($\uparrow 0.617$)}\\
RFW-African & $ 98.650\pm0.329$ & {\bf 99.417$\pm0.227 $} \tiny{($\uparrow 0.767$)}\\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table*}[t!]
\begin{center}
\caption{\textbf{$1$:$1$ verification TAR} on the IJB-B and IJB-C datasets.}
\label{tab:comp_ijb1vs1}
\begin{tabular}{c|cccc|cccc}
\hline
\multirow{2}{*}{Methods (\%) } & \multicolumn{4}{c}{IJB-B} & \multicolumn{4}{c}{IJB-C}\\
\cmidrule(r){2-5} \cmidrule{6-9}
&FAR=$1e$$-$$6$ &FAR=$1e$$-$$5$ & FAR=$1e$$-$$4$ & FAR=$1e$$-$$3$
&FAR=$1e$$-$$6$ &FAR=$1e$$-$$5$ & FAR=$1e$$-$$4$ & FAR=$1e$$-$$3$ \\
\hline
ResNet50 & 37.28 & 90.73 & 94.73 & 96.63
& 90.47 & 94.28 & 96.17 & 97.57 \\
Prodpoly-ResNet50 & {\bf 43.46} \tiny{($\uparrow 6.18$)} & {\bf 91.95} \tiny{($\uparrow 1.22$)} & {\bf 95.19} \tiny{($\uparrow 0.46$)} & {\bf 96.67} \tiny{($\uparrow 0.04$)}
& {\bf 90.77} \tiny{($\uparrow 0.30$)}& {\bf 95.16} \tiny{($\uparrow 0.88$)}& {\bf 96.58} \tiny{($\uparrow 0.41$)} & {\bf 97.66} \tiny{($\uparrow 0.09$)} \\\hline
\end{tabular}
\end{center}
\end{table*}
\begin{table*}[t!]
\begin{center}
\caption{\textbf{1:N (mixed media) Identification} on the IJB-B and IJB-C datasets. False positive identification rate (FPIR) is the proportion of non-mated searches returning any (1 or more) candidates at or above a threshold.}
\label{tab:comp_ijb1vsN}
\begin{tabular}{c|cccc|cccc}
\hline
\multirow{2}{*}{Methods (\%) } & \multicolumn{4}{c}{IJB-B} & \multicolumn{4}{c}{IJB-C} \\
\cmidrule{2-5} \cmidrule{6-9}
&FPIR=$0.01$ & FPIR=$0.1$ & Rank~$1$ & Rank~$5$
&FPIR=$0.01$ & FPIR=$0.1$ & Rank~$1$ & Rank~$5$ \\\hline
ResNet50 & 84.70 & 94.01 & 95.29 & 97.14
& 92.87 & 95.28 & 96.52 & 97.69 \\
Prodpoly-ResNet50 & {\bf 85.58} \tiny{($\uparrow 0.88$)} & {\bf 94.69} \tiny{($\uparrow 0.68$)} & {\bf 95.52} \tiny{($\uparrow 0.23$)} & {\bf 97.16} \tiny{($\uparrow 0.02$)}
& {\bf 93.60} \tiny{($\uparrow 0.73$)} & {\bf 95.93} \tiny{($\uparrow 0.65$)} & {\bf 96.86} \tiny{($\uparrow 0.34$)} & {\bf 97.79} \tiny{($\uparrow 0.10$)} \\\hline
\end{tabular}
\end{center}
\end{table*}
\noindent {\bf Results on IJB-B and IJB-C.} The IJB-B dataset \cite{whitelam2017iarpa} contains $1,845$ subjects with $21.8$K still images and $55$K frames from $7,011$ videos. The IJB-C dataset \cite{whitelam2017iarpa} is a further extension of IJB-B, having $3,531$ subjects with $31.3$K still images and $117.5$K frames from $11,779$ videos. On IJB-B and IJB-C datasets, there are two evaluation protocols, 1:1 verification and 1:N identification.
In Figure \ref{pic:ijb}, ROC curves of \resnet50 and \modelres50 under 1:1 verification protocol on IJB-B and IJB-C is plotted.
On IJB-B, there are $12,115$ templates with $10,270$ genuine matches and $8$M impostor matches. On IJB-C, there are $23,124$ templates with $19,557$ genuine matches and $15,639$K impostor matches. The proposed method
surpasses the baseline by a clear margin. The comparison of TAR in Table \ref{tab:comp_ijb1vs1} illustrates that \modelres50 improves the TAR (@FAR=1e-4) by $0.46\%$ and $0.41\%$ on IJB-B and IJB-C, respectively.
Table \ref{tab:comp_ijb1vsN} compares \resnet50 and \modelres50 under the 1:N end-to-end mixed protocol, which contains both still images and full-motion videos. On IJB-B, there are $10,270$ probe templates containing $60,758$ still images and video frames.
On IJB-C, there are $19,593$ probe templates containing $127,152$ still images and video frames.
\modelres50 outperforms \resnet50 by $0.23\%$ and $0.34\%$ on IJB-B and IJB-C rank-1 face identification.
\begin{figure}[h!]
\centering
\subfloat[ROC for IJB-B]{
\label{pic:ijbb_roc}
\includegraphics[width=0.228\textwidth]{figures/facerecognition/IJBB.pdf}}
\subfloat[ROC for IJB-C]{
\label{pic:ijbc_roc}
\includegraphics[width=0.228\textwidth]{figures/facerecognition/IJBC.pdf}}
\caption{ROC curves of ResNet50 and Prodpoly-ResNet50 under 1:1 verification protocol on the IJB-B and IJB-C dataset.}
\label{pic:ijb}
\end{figure}
\noindent {\bf Results on MegaFace.} The MegaFace dataset \cite{kemelmacher2016megaface} includes 1M images of 690K different individuals as the gallery set and 100K photos of $530$ unique individuals from FaceScrub \cite{ng2014data} as the probe set. On MegaFace, there are two testing protocols (e.g., identification and verification). Table \ref{table:megaface} show the identification and verification results on
MegaFace dataset. In particular, the proposed Prodpoly-ResNet50
achieve $0.50\%$ improvement at the Rank-1@1e6 identification rate and $0.31\%$ improvement at the verification
TPR@FAR=1e-6 rate over the baseline ResNet50. In Figure \ref{fig:megafacecmcroc}, Prodpoly-ResNet50 shows superiority over ResNet50 and forms an upper envelope under both identification and verification scenarios.
\begin{table}[t!]
\begin{center}
\caption{Face identification and verification evaluation of ResNet50 and the proposed Prodpoly-ResNet50 on MegaFace Challenge1 using FaceScrub as the probe set. ``Id'' refers to the rank-1 face identification accuracy with 1M distractors, and ``Ver'' refers to the face verification TAR at $10^{-6}$ FAR. Results are evaluated on the refined MegaFace dataset~\cite{deng2019arcface}.}
\label{table:megaface}
\begin{tabular}{c|c|c}
\hline
Methods & Id ($\%$) & Ver ($\%$) \\
\hline
ResNet50 & 98.28 & 98.64\\
Prodpoly-ResNet50 & {\bf 98.78} (\tiny{$\uparrow 0.50$)} & {\bf 98.95} \tiny{($\uparrow 0.31$)} \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}[h!]
\centering
\subfloat[CMC]{
\label{fig:megafacecmc}
\includegraphics[width=0.226\textwidth]{figures/facerecognition/refinefacescrubCMCPAMI2020.pdf}}
\subfloat[ROC]{
\label{fig:megafaceroc}
\includegraphics[width=0.226\textwidth]{figures/facerecognition/refinefacescrubROCPAMI2020.pdf}}
\caption{CMC and ROC curves of ResNet50 and the proposed Prodpoly-ResNet50 on MegaFace. Results are evaluated on the refined MegaFace dataset~\cite{deng2019arcface}.}
\label{fig:megafacecmcroc}
\end{figure}
\subsection{3D Mesh representation learning}
\label{ssec:prodpoly_mesh_representation_learning_experiment}
\begin{figure}
\centering
\centering
\includegraphics[width=\linewidth]{figures/graphs/fig1.png}
\includegraphics[width=\linewidth]{figures/graphs/fig2.png}
\caption{ProdPoly{} vs $1^{st}$ order graph learnable operators for mesh autoencoding. Note that even without using activation functions the proposed methods significantly improve upon the state-of-the-art.}
\label{fig:prodpoly_graphs}
\end{figure}
Below, we evaluate higher order correlations in graph related tasks. We experiment with 3D deformable meshes of fixed topology~\cite{ranjan2018generating}, i.e., the connectivity of the graph $\mathcal{G}= \{\mathcal{V},\mathcal{E}\}$ remains the same and each different shape is defined as a different signal $\bm{x}$ on the vertices of the graph: $\bm{x}:\mathcal{V}\to\mathbb{R}^d$. As in the previous experiments, we extend a state-of-the-art operator, namely spiral convolutions \cite{Bouritsas_2019_ICCV}, with the ProdPoly{} formulation and test our method on the task of autoencoding 3D shapes. We use the existing architecture and hyper-parameters of \cite{Bouritsas_2019_ICCV}, thus showing that ProdPoly{} can be used as a plug-and-play operator to existing models, turning the aforementioned one into a Spiral $\Pi$-Net.
Our implementation uses a product of polynomials (referred as \textit{ProdPoly{} full}), where each layer is a $N^{th}$ order polynomial instantiated as a specific case of \eqref{eq:prodpoly_model2} or \eqref{eq:prodpoly_model3}:
\newline
\textbf{NCP}: ${\boutvar_{n} = \Big(\bm{A}\matnot{n}^T \boutvar_{1}\Big) * \Big(\bm{S}\matnot{n}^T \boutvar_{n-1}\Big) + \bm{A}\matnot{n}^T\boutvar_{1}}$
\newline
\textbf{NCP-Skip}: ${\boutvar_{n} = \Big(\bm{A}\matnot{n}^T \boutvar_{1}\Big) * \Big(\bm{S}\matnot{n}^T \boutvar_{n-1}\Big) + \bm{A}\matnot{n}^T\boutvar_{1} + \boutvar_{n-1}}$,
${\boutvar = \boutvar_{N} + \bm{\beta}}$
, where $\bm{A}\matnot{n}, \bm{S}\matnot{n}$ are spiral convolutions written in matrix form, $\bm{\beta}$ is a bias vector, $\boutvar_1$, $\boutvar$ is the input (which is equal to the output of the previous layer) and the output of the layer respectively. Stability of the optimization is ensured by applying vertex-wise instance normalization on the $2^{nd}$ order term of the recursive formulation.
Additionally, we evaluate our formulation with a simpler model (\textit{ProdPoly{} simple}) that allows for an attractive trade-off between increased expressivity and constrained parameter budget. In specific, we can create higher-order polynomials without adding new blocks in the original architecture as follows:
\newline
$\boutvar_{N} =
\sum_{n=2}^N \underbrace{\Big(\bm{S}^T\boutvar_{1}\Big)*\Big(\bm{S}^T\boutvar_{1}\Big)\cdots\Big(\bm{S}^T\boutvar_{1}\Big)}_\text{n times} + \bm{S}^T\boutvar_{1} + \bm{\beta}$. We use the same normalization scheme as before, by independently normalizing each higher order term. Note that here we only use one learnable operator $\bm{S}$ (spiral convolution) per layer. It is interesting to notice that this model can be also re-interpreted as a learnable polynomial activation function as in \cite{kileel2019expressive}, which is a specific case of ProdPoly{}.
Polynomial activation functions lead to increased expressivity per se, but are less expressive when compared to richer multiplicative interactions as introduced by our NCP and NCP-skip models. \rebuttal{In addition, as can be seen in Fig. \ref{fig:prodpoly_graphs}, experimental evidence suggests that such interactions also lead to improved empirical performance.}
\begin{figure}
\centering
\centering
\includegraphics[width=\linewidth]{figures/graphs/final_dfaust.png}
\caption{Color coding of the per vertex reconstruction error on an exemplary human body mesh. From left to right: ground truth mesh, 1st order SpiralGNN, $2^{nd}$, $3^{rd}$ and $4^{th}$ order Spiral ProdPoly{}.}
\label{fig:color_coding}
\end{figure}
In Fig.~\ref{fig:prodpoly_graphs}, we compare the reconstruction error of the proposed method to the baseline spiral convolutions along with other popular graph learnable operators, i.e., the Graph Attention Network (GAT) \cite{velickovic2018graph}, FeastNet \cite{verma2018feastnet}, Mixture model CNNs (MoNet) \cite{monti2017geometric}, Convolutional Mesh Autoencoders (COMA) \cite{ranjan2018generating} which are based on the spectral graph filters of ChebNet \cite{defferrard2016convolutional}, as well as with Principal Component Analysis (PCA), which is quite popular in shape analysis applications \cite{blanz1999morphable}. The evaluation is performed on two popular 3D deformable shape benchmarks, COMA \cite{ranjan2018generating} and DFAUST \cite{bogo2017dynamic}, that depict facial expressions and body poses respectively. $\Pi$-nets outperform all published methods even when discarding the activation functions across the entire network. Similar patterns emerge in both datasets: NCP{} and NCP-Skip{} behave similarly regardless of the absence of activation functions or not, leading to an increased performance when the order of the polynomial increases, i.e. \rebuttal{empirical performance} improves along with expressivity. Moreover, the simple model provides a boost in performance as well, although we observe a decrease for the $3^{rd}$ and $4^{th}$ order of the linear model, which might be attributed to overfitting (similarly to the linear experiments in \rebuttal{Sec. 5 in the supplementary material}). Overall, we showcase that \rebuttal{performance} may seamlessly improve by converting the existing architecture to a polynomial, without having to increase the depth or width of the architecture as frequently done by ML practitioners, and with small sacrifices in terms of inference time (\rebuttal{see Sec. 6, supplementary}) and parameter count.
Finally, in Fig.~\ref{fig:color_coding} we assess how the order of the polynomial qualitatively reflects in the reconstruction of an exemplary mesh. In particular, we color code the per vertex reconstruction error on the reconstructed meshes (right) and compare them with the input (left). Notice that the overall shape resembles the input more as we increase the order of the polynomial (especially in the head), while body parts with strong articulations (e.g. hands) are reconstructed with higher fidelity.
\section{Future directions}
\label{sec:prodpoly_discussion}
The new class of $\Pi-$nets has strong experimental results and few empirical theoretical results already. We expect in the following years new works that improve our results and extend our formulation. To that end, we summarize below several fundamental topics that are open for interested practitioners.
The generalization of the $\Pi-$nets is a crucial topic. In our evaluation without activation functions, we noticed that polynomials might be prone to overfitting (e.g., in the classification setting without activation functions in the supplementary). When we add the non-linear activation functions we did not observe such a consistent pattern.
In this work, we created a link between different decompositions and the resulting architectures (the three decompositions resulted in three different architectures). The relationship between neural architecture search and the tensor decomposition can be further nurtured.
Reducing the network redundancy is also an exciting topic. The theoretical properties of multiplicative interactions along with our experiments, exhibit how polynomial neural networks can be used to reduce the network redundancy. Additional post-processing techniques, such as pruning, or exploiting tools from the tensor methods, such as low-rank constraints, might be beneficial in this context.
\rebuttal{Lastly, $\Pi-$nets inherit the properties of polynomials, e.g., higher-order terms might result in unbounded gradients. That makes studying normalization schemes of paramount significance. There might be normalization techniques that obtain a superior performance to the batch/instance normalization we employed.}
\section{Conclusion}
\label{sec:prodpoly_conclusion}
In this work, we have introduced a new class of DCNNs, called $\Pi$-Nets, that perform function approximation using a polynomial neural network. Our $\Pi$-Nets can be efficiently implemented via a special kind of skip connections that lead to high-order polynomials, naturally expressed with tensorial factors. The proposed formulation extends the standard compositional paradigm of overlaying linear operations with activation functions. We motivate our method by a sequence of experiments without activation functions that showcase the expressive power of polynomials, and demonstrate that $\Pi$-Nets are effective in both discriminative, as well as generative tasks. Trivially modifying state-of-the-art architectures in image generation, image and audio classification, face verification/identification as well as mesh representation learning, the performance consistently improves.
\section{Acknowledgements}
\label{sec:prodpoly_acks}
GC conducted this work while at Imperial College London.
The work of SM, and GB was partially funded by an Imperial College DTA. The work of JD was partially funded by Imperial President's PhD Scholarship. The work of SZ was partially funded by the EPSRC Fellowship DEFORM: Large Scale Shape Analysis of Deformable Models of Humans (EP/S010203/1) and a Google Faculty Award.
{\small
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,484 |
Home » News » Guber poll: Kwankwaso congratulates Obaseki, Edo people
Guber poll: Kwankwaso congratulates Obaseki, Edo people
Governor Godwin Obaseki of Edo State
By Joseph Erunke – Abuja
Former governor of Kano State and Chieftain of the Peoples Democratic Party, PDP, Senator Rabiu Musa Kwankwaso, has congratulated Governor Godwin Obaseki for his reelection in Saturday's governorship election.
He described the reelection of the governor as a "well-deserved victory."
Kwankwaso, in a statement through his Principal Private Secretary, Muhammad Inuwa Ali, Monday, in Abuja, said he particularly congratulated the people of Edo State for the governor's victory because their action "symbolizes hope for every true democrat in our country."
READ ALSO: Okowa's aide commends S'South Govs over Obaseki's victory
He said Obaseki "worked very hard for the people of Edo State and they have reciprocated by coming out in droves not only to cast their votes for you but also to defend their votes and ensure that the multitude of election riggers mobilized by agents of doom, from different states, to steal the Edo election was made ineffective."
"The good people of Edo State have spoken. And the true choice of the people had prevailed.
"I, therefore, join all citizens of goodwill to congratulate the people of Edo State on the successful and peaceful conclusion of the gubernatorial election which was held on Saturday, September 19, 2020," he said.
He added:" I am especially congratulating the Edo people for their foresight in re-electing their hardworking and effective Governor, Gov. Godwin Obaseki to continue with the excellent work of developing Edo State."
According to him,"Beyond this success, the action of the good people of Edo State symbolizes hope for every true democrat in our country."
"Without a doubt, our victory in Edo State was hugely facilitated by the outstanding commitment, resilience, courage, and the doggedness of the chairman of our Campaign Council, His Excellency Gov. Nyesom Wike of Rivers State and the entire members of the Campaign Council including all PDP governors. I commend you all as we celebrate the success-for-democracy in Nigeria.)," he added.
"Let me also thank the representatives of the international community in Nigeria, especially the United State embassy in Nigeria and the United Kingdom embassy in Nigeria for their timely advice to merchants of election malpractice and perpetrators of electoral violence," he further said.
READ ALSO: ONDO 2020: Jegede congratulates Obaseki, asks Ondo voters to choose good governance
The statement read further: "Your advisory to election riggers and promoters of election violence to know that there are consequences, beyond the borders of Nigeria, for their ignominious action was very effective. We thank you for your interest in deepening democracy in our country.
"There is no gainsaying that the Edo Victory is a victory for our great party, the PDP. I, therefore, congratulate our amiable Chairman Chief Uche Secondus and his entire team. This success is a testimony to your dexterity, political savvy, and focused leadership. Congratulations.
"Finally, I wish to express my appreciation to numerous interest groups in Edo State, including the business community, traditional institution, the clergy, the academics, women groups, Civil Society groups, Arewa community, Edo Kwankwasiyya, the youth, the media, and students, most of whom I met with during my campaign visit to Edo State, for their support and cooperation. Edo success is your success!" | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,319 |
Q: How to achieve Available zone aware for neutron-server I have read the manual.
http://docs.openstack.org/mitaka/networking-guide/adv-config-availability-zone.html
it tell us how to support multi AZ for dhcp/l3 router service.
But I still not know where the neutron server and qpid I will place in multi AZ environment.
Obviously, if we start neutron-server/qpid processes in only one available zone, it is not a good solution, there's still a single point failure problem.
But I did not find any solution from official site of openstack, till now.
Anyone know that?
Regards,
Joe
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,436 |
package com.fnklabs.draenei.orm;
import org.jetbrains.annotations.NotNull;
import org.jetbrains.annotations.Nullable;
import java.nio.ByteBuffer;
import java.util.*;
import java.util.function.Function;
import java.util.stream.Collectors;
import java.util.stream.Stream;
class EnumeratedMetadata implements ColumnMetadata {
@NotNull
private final Class enumType;
/**
* Enum types
*/
private final Map<String, Object> enumValues = new HashMap<>();
private final ColumnMetadata columnMetadata;
/**
* @param columnMetadata
* @param enumType
*/
protected EnumeratedMetadata(@NotNull ColumnMetadata columnMetadata, @NotNull Class enumType) {
this.columnMetadata = columnMetadata;
this.enumType = enumType;
/**
* Need for deserialization, because enum saved as string
*/
if (enumType.isEnum()) { // todo unsafe operation. use #name() to retrieve enum name
for (Object constant : enumType.getEnumConstants()) {
enumValues.put(constant.toString(), constant);
}
}
}
@Override
public ByteBuffer serialize(final Object value) {
if (value == null) {
return null;
}
if (value instanceof Collection) {
Collection<Object> collection = (Collection<Object>) value;
Stream<String> enumStream = collection.stream()
.map(new MapEnumToString());
if (value instanceof Set) {
Set<String> setValue = enumStream.collect(Collectors.toSet());
return columnMetadata.serialize(setValue);
} else if (value instanceof List) {
List<String> listValue = enumStream.collect(Collectors.toList());
return columnMetadata.serialize(listValue);
}
}
return columnMetadata.serialize(value.toString());
}
private static class MapEnumToString implements Function<Object, String> {
@Override
public String apply(Object o) {
try {
return o.toString();
} catch (NullPointerException e) {
e.printStackTrace();
}
return null;
}
}
@Override
public <T> T deserialize(@Nullable ByteBuffer data) {
Object deserializedValue = columnMetadata.deserialize(data);
if (deserializedValue == null) {
return null;
}
if (deserializedValue instanceof Collection) {
Collection<String> collection = (Collection<String>) deserializedValue;
Stream<Object> enumStream = collection.stream()
.map(value -> enumValues.get(value));
if (deserializedValue instanceof Set) {
return (T) enumStream.collect(Collectors.toSet());
} else if (deserializedValue instanceof List) {
return (T) enumStream.collect(Collectors.toList());
}
}
return (T) enumValues.get(deserializedValue);
}
@NotNull
@Override
public String getName() {
return columnMetadata.getName();
}
@NotNull
@Override
public Class getFieldType() {
return columnMetadata.getFieldType();
}
@Override
public void writeValue(@NotNull Object entity, @Nullable Object value) {
columnMetadata.writeValue(entity, value);
}
@Nullable
@Override
public <FieldType> FieldType readValue(@NotNull Object object) {
return columnMetadata.readValue(object);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,073 |
Q: Error de validación de sesiones con php Al haber creado mis datos y tener que iniciar sesión necesito que me lleve al a sesion.php pero algo falla y no sé qué lo que hace es que me lleve es al salir php adjunto acá abajo el código, requiero ayuda puesto que no logro entender bien la lógica y el funcionamiento correcto.
Abajo adjunte mi conexion.php mi sesion.php y el loguear.php que es el que va conectado al login para verificar los datos, como se dan cuenta el primero es logear.php luego adjunté la conexión y por último el sesion.php que es la página a la que se supone que debería ir.
Este es mi código:
loguear.php
<?php
include ("conexion.php");
session_start();
$usuario = $_POST['usuario'];
$password =md5($_POST['password']);
$proceso = mysqli_query($conectar,"SELECT * FROM registro WHERE username = '$usuario' and pw = '$password' ");
if($resultado = mysqli_fetch_array($proceso)){
$_SESSION['usuario'] = $usuario;
header("Location: sesion.php");
}else{
header("Location: salir.php");
}
?>
conexion.php
<?php
$dbhost = "localhost";
$dbuser = "root";
$dbpass = "12345";
$dbname = "darkbins";
//pa crear la conexion a la db y unirla a php
$conectar = mysqli_connect($dbhost, $dbuser, $dbpass, $dbname);
if(! $conectar){
echo "error";
}
?>
sesion.php
<!DOCTYPE html>
<html lang="en">
<head>
</head>
<body>
<?php
session_start();
if(isset($_SESSION['usuario'])){
echo"Session exitosa Bienvenido";
echo"<a href='salir.php'> cerrar sesion </a>";
echo"Aqui se coloca toda la informacion que se requiera";
}else{
//header("Location: index.php");
//te manda al index para volver a colocar los datos correctamente
}
?>
</body>
</html>
A: Como bien menciona D.Bulten no se recomienda guardar en MD5, mira este codigo que utilizo para el proceso del Login y para hashear la contraseña.
<?php
session_start();
include ('conexion.php');
//Verificamos que este enviando desde el formulario
if(isset($_POST)){
$email = filter_var($_POST['txtEmail'], FILTER_SANITIZE_STRING);
$pass = filter_var($_POST['txtPass'], FILTER_SANITIZE_STRING);
//Verificamos que exista el usuario
$proceso = mysqli_query($conectar, "SELECT * FROM registro WHERE username = '$usuario'");
$resultado = mysqli_fetch_array($proceso);
//Si no existe, matamos la operación.
if(!$resultado){
//Credenciales incorrectas (email no existe)
//Redireccionamos al login
}
//Validamos contraseñas
if(password_verify($pass, $resultado['contrasenia'])){
$_SESSION['LOGUEADO'] = true;
$_SESSION['USUARIO'] = $resultado;
//Redireccionamos al perfil
}else{
//Credenciales incorrectas(contraseña incorrecta)
//Redireccionamos al login
}
}else{
//Redireccionamos al login por que no se esta haciendo desde POST
}
Y para hashear la contraseña hazlo de esta manera en tu registro.
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| {
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\section{ Introduction}
The Englert-Brout-Higgs (EBH) mechanism holds the key to understanding the electroweak interaction \cite{eng}\cite{hig}\cite{kib}. Recently, a Higgs-like particle, which has a relatively small mass ($m_H=$125 GeV), is found \cite{atl}\cite{cms}.
On the nature of the Higgs particle, it has been intensely debated whether it is an elementary particle or a composite one made of other constituents of the system.
However, whether the Higgs particle is elementary or composite, it is no doubt that the Glashaw-Weinberg-Salam (GWS) model using the EBH model \cite{gla}\cite{wei}\cite{sal} (from now, we abbreviate it as Higgs model) is a simple and successful model with which almost no experimental results conflict up to now.
In this paper, we explore the physical mechanism lying behind the phenomenological aspect of the Higgs model.
To avoid the complications due to non-Abelian gauge invariance, we consider as a warmup example an Abelian-gauge field $B_{\mu}(x)$ coupled to a massless fermion and antifermion $ \varphi (x)$ in the non-chiral manner. The basic Lagrangian density $L_0(x)$ is
\begin{equation}
L_0(x)= -\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\bar{\varphi}(i\partial_{\mu}+gB_{\mu})\gamma^{\mu}\varphi ,
\label{eq:1}
\end{equation}
where $F_{\mu\nu}= \partial _{\mu}B_{\nu} - \partial _{\nu}B_{\mu} $. The operator of massless fermi field is written as
\begin{eqnarray}
\varphi(x)&=& \frac{1}{\sqrt{V}}\sum_{p,s} \frac{1}{\sqrt{2\epsilon_p}}
\nonumber \\
& \times& \left[ a^s(\mbox{\boldmath $p$})u^s(p)e^{-ipx}+ b^{s\dagger }(-\mbox{\boldmath $p$})v^s(-p)e^{-ipx} \right] ,
\label{eq:2} \\
\bar{\varphi}(x)&=& \frac{1}{\sqrt{V}}\sum_{p,s} \frac{1}{\sqrt{2\epsilon_p}}
\nonumber \\
& \times& \left[ a^{s\dagger }(\mbox{\boldmath $p$})\bar{u}^s(p)e^{ipx}+b^s(-\mbox{\boldmath $p$})\bar{v}^s(-p)e^{ipx} \right] ,
\label{eq:3}
\end{eqnarray}
where the four-component spinors are normalized as
\begin{equation}
\bar{u}^r(p) u^s(p)= -\bar{v}^r(p)v^s(p) = 2 \epsilon_p \delta ^{rs} ,
\label{eq:4}
\end{equation}
\begin{equation}
\bar{u}^r(p) v^s(-p)= \bar{v}^r(-p)u^s(p) = 0 .
\label{eq:5}
\end{equation}
The creation and annihilation operators with no dimension obey the anti-commutation rules
\begin{equation}
[a^s(\mbox{\boldmath $p$}), a^{s'\dagger}(\mbox{\boldmath $p$}')]_+
= [b^s(\mbox{\boldmath $p$}), b^{s'\dagger}(\mbox{\boldmath $p$}')]_+= \delta_{ss'}\delta^3(\mbox{\boldmath $p$}-\mbox{\boldmath $p$}') .
\label{eq:6}
\end{equation}
In the Higgs model, the mass of gauge boson is derived from the following Lagrangian density
\begin{equation}
L_h(x)= |(i\partial_{\mu}+gB_{\mu})(v_h+h)|^2 +\mu ^2|v_h+h|^2 -\lambda |v_h+h|^4 ,
\label{eq:01}
\end{equation}
Furthermore, the Yukawa interaction $(m_f/v_h)\bar{\varphi} \varphi h$ between the Higgs field $h$ and the massless fermi particle $\varphi$ is introduced. After the Higgs field condenses in the vacuum $\langle 0 |h| 0\rangle =v_h$, a mass $m_f$ arises in the fermion. The purpose of such a procedure is to give a mass to the fermion in the gauge-invariant manner when it is chirally coupled to the gauge field.
The recent experiments show that the amplitude of the Higgs decay to quarks are proportional to the quark mass $m_f$. This result agrees with the prediction by the Yukawa coupling in the Higgs model. Despite these success, we have not yet understood the meaning behind it.
The Higgs model is a powerful theory in the accessible energy region now. If we regard the Higgs particle as an elementary one, however, this model possesses a strange feature. The Higgs potential $ -\mu^2 |h |^2$ plays a double role, as the potential energy to stabilize the vacuum condensate, and as the mass term of the Higgs particle. If we regard it as a phenomenology, it is the simplest assumption. But if we regard it as the fundamental theory, it is questionable that the large-scale property such as the stabilization of the vacuum, and the one-particle property such as the mass of the Higgs boson, are expressed by the same term. Furthermore, $\lambda |v_h+h|^4$ is necessary to stabilize the symmetry-broken vacuum, and therefore the quadratic divergence appears inevitably in the perturbation calculation. Hence, the fine tuning to cancel this divergence is necessary.
This situation requires us to obtain a deeper understanding of it.
In this paper, using the Abelian-gauge system $L_0(x)$, we derive spontaneous symmetry breaking without using $L_h(x)$, and the Yukawa coupling.
In the real world, all fermions we know have their own masses, including neutrinos. Hence, the massless fermion in $L_0(x)$, in which symmetry breaking has not yet occurred, is not a real particle. In dealing with $L_0(x)$, we must reconsider some assumptions usually made for the massive particle.
(1) There is no reason to expect that such massless fermions have the properties that the physical law does not necessarily require, such as the excess of fermions over antifermions in the real world, even though the massive fermion after symmetry breaking have these properties.
(2) The vacuum is electrically neutral. Hence, when the massless fermion becomes a massive one by symmetry breaking, the massless antifermion must be {\it equally involved } in this process.
(3) The process of spontaneous symmetry breaking does not depend on the specific inertial-system. Rather, it must be formulated as if it is looked from the {\it space-time viewpoint}, because it is related to {\it the definition of antiparticle}.
These views can be formulated in a simple way when we use the path-integral formalism.
In Section 2, using the path-integral formalism, the physical vacuum of massless fermion is derived from the requirement of relativity and superposition principle. In Section 3, the generation of mass of the massless fermion in this physical vacuum is explained. In Section 4, the generation of mass of the gauge boson in the physical vacuum is explained. In Section 5, we derive the Higgs-like boson as a local excitation propagating in the physical vacuum, the mass of which is calculated as an excitation energy. In Section 6, we perform one-loop perturbation calculation for the effective coupling of the Higgs-like mode to the gauge field $B_{\mu}$, and the self interaction of this mode. In Section 7, as an example of the reaction through this Higgs-like mode, we calculate the total cross section of the pair-annihilation of massive fermion and antifermon to massive gauge bosons. In Section 8, renormalizability of this model is examined. In Section 9, we briefly discuss some generalization of the present model to the electroweak interaction, which will be explained in details in the next paper.
\section{A space-time view of spontaneous symmetry breaking}
\subsection{Possible paths and antiparticle}
The mechanism of spontaneous symmetry breaking will be explained in this paper using the path-integral picture. The path-integral framework will be used in an intuitive rather than a formal way. (In what follows, there is no need to regard $\varphi (x)$ in $L_0(x)$ as a Grassmann number.) In this formalism, the propagation of particle is given by a weighted average over all possible classical paths.
The classical orbit and an example of possible paths in space-time are shown in Figure.\ref{fig.1}, illustrated only for the $x$-axis in space. The classical orbit is represented by a solid straight line $AC$, and other possible paths are illustrated as thin solid curves connecting $A$ and $C$. The weight associated for a given set of paths include a factor $\exp \left(i\int^{t_c}_{t_a} {\cal L}[\dot{x}(t), x(t)] dt \right) \equiv \exp (iS)$. The total amplitude for traveling from $A$ to $C$ is
\begin{equation}
\int^C_A {\cal D} x(t)\exp \left(i\int^{t_c}_{t_a} {\cal L}[\dot{x}(t), x(t)] dt \right) \equiv \int^C_A {\cal D} x(t)\exp (iS)
\label{eq:6145}
\end{equation}
This total amplitude satisfies relativistic invariance. The one-point-dotted lines represent the light cones relative to $A$, $B_1$ and $B_2$, respectively. The classical orbit from $A$ to $C$ must remain within the absolute future relative to $A$ satisfying $(ct)^2-x^2>0$. However, the possible paths from $A$ to $C$ in $\exp(iS)$ include not only the timelike path, but also the spacelke path, as illustrated for example by the path $AB_1C$ or $AB_2C$. Since the total amplitude in Eq.(\ref{eq:6145}) must be zero outside of the light cone relative to $A$, the variable $x(t)$ in the path integral cannot be confined to the timelike paths. Rather, by cancellation of timelike and spacelike paths, zero outside of the light cone is realized.
\begin{figure}
\begin{center}
\includegraphics [scale=0.38]{fig390.eps}
\caption{ The motion of particle from $A$ to $C$ in space-time. The solid straight line represents a classical orbit. Various thin solid curves represent various possible paths from $A$ to $C$ contributing to the action $S$ in the path integral. The one-point-dotted lines represent the light cones relative to $A$, $B_1$ and $B_2$, respectively. }
\label{fig.1}
\end{center}
\end{figure}
When the particle travels at a large velocity, that is, when the de Broglie wavelength of the particle is small in comparison with the characteristic length of a given problem, $\exp (iS)$ coming from the spacelike path will vary periodically with high frequency about zero, and it will practically vanish in the path integral. Only the timelike paths near the classical orbit $AC$ contribute to $\exp (iS)$. However, when the particle travels from $A$ to $C$ at a small velocity and its de Broglie wavelength is large, it implies that the fermion travels in a narrow space, and the interference between possible paths has a significant effect on $\exp(iS)$. In such a case, a comparatively large variation of path from the classical orbit, even if it includes the spacelike path, produces only a small variation in $\exp(iS)$. Hence, in addition to the timelike paths, the spacelike paths will make non negligible contribution to total amplitude.
When we view the propagation of particle in Figure.\ref{fig.1} from other inertial systems, how does it look to us? To study it, we consider a path $AB$ in Figure.\ref{fig.2}. A particle with a negative charge and momentum $\mbox{\boldmath $p$}$ departs from $x_1$ at $t_1$ and arrives at $x_2$ at $t_2(>t_1)$ in an inertial system represented by an orthogonal coordinate $(\mbox{\boldmath $x$},t)$. When we observe this motion from another inertial system moving along the $x$-axis at a relative velocity $v$ to the first system, it follows the Lorentz transformation in an oblique coordinates $(\mbox{\boldmath $x'$},t')$. The time difference $t_2-t_1$ between $A$ and $B$ are Lorentz transformed to
\begin{equation}
t_2'-t_1'=\frac{1}{\sqrt{1-(v/c)^2}}\left[ t_2-t_1-\frac{v}{c^2}(x_2-x_1) \right] .
\label{eq:7}
\end{equation}
\begin{figure}
\begin{center}
\includegraphics [scale=0.40]{fig398.eps}
\caption{ The Lorentz transformation applied to (a) the timelike path $AB$, and (b) the spacelike path $AB$. The relative inclination of Ot' to Ot corresponds to the velocity $v$ of the observer. }
\label{fig.2}
\end{center}
\end{figure}
When the particle travels along the timelike path as in Figure.\ref{fig.2}(a), we will not observe strange phenomenon.
One of the remarkable feature of the Lorentz transformation is that when the particle travels along the spacelike path as in Figure.\ref{fig.2}(b), the order of events is not left invariant. When the velocity $v$ of an observer is sufficiently large relative to the initial frame, or when the spacelike interval between two events is sufficiently large as $c(t_2-t_1) <(v/c)(x_2-x_1) $ in Eq.(\ref{eq:7}), the order of two events separated by the spacelike interval is reversed as $t_2'<t_1'$. A natural interpretation of this observation is that an antiparticle with an opposite charge moves in the opposite direction, that is, from $B$ to $A$. Extending the idea of Stueckelberg on this feature \cite{stu}, Feynman made an intuitive explanation for the raison d'etre of antiparticle using the virtual processes \cite{fey}.
\subsection{Massless fermion and antifermion}
In Figure.\ref{fig.1}, when the massless fermion travels at a small velocity from $A$ to $C$, not only the timelike but also the spacelike path such as the path $AB_1C$ makes a significant effect on $\exp(iS)$. When we view such a path from another inertial system, how does it look to us? When the spacelike interval between $A$ and $B_1$ is large enough to satisfy $c(t_2-t_1) <(v/c)(x_2-x_1) $ (such a $B_1$ always exists near $A$ in Figure.\ref{fig.1}), two different interpretations of the phenomenon at $A$ are possible.
At the starting point $A$ of the timelike path $AC$ in Figure.\ref{fig.1}, we find the annihilation of massless fermion. and at the end point $A$ of the spacelike path $AB_1$, we find the creation of massless antifermion.
As for the path from $A$ to $C$ via $B_2$, the interval between $B_2$ and $C$ is spacelike as well. Hence, at the end point $C$ of the timelike path $AC$, we find the creation of massless fermion, and at the starting point $C$ of the spacelike path $B_2C$, we find the annihilation of massless antfermion.
When the massless fermion travels at a large velocity, only the timelike paths near the classical orbit $AC$ contribute to $\exp(iS)$ in the path integral. Hence, the weight of these two interpretations depends on the relative momentum between the observer and the object.
In the real world after symmetry breaking, the influence of antifermion is not so large, because the massive antifermion such as positrons appears only in particular phenomena in the real world. But, when it is {\it before symmetry breaking}, the massless fermion and antifermion must be equally dealt with, and these two interpretations at $A$ or $C$ in Figure.\ref{fig.1} have equal validity.
To formulate such a relationship between massless fermion and antifermion at A in Figure.\ref{fig.1}, a new annihilation operator $A^s(\mbox{\boldmath $p$})$ which is a superposition of $a^s(\mbox{\boldmath $p$})$ and $b^{s\dagger}(-\mbox{\boldmath $p$})$
\begin{equation}
A^s(\mbox{\boldmath $p$})= \cos \theta_{\mbox{\boldmath $p$}} a^s(\mbox{\boldmath $p$})
+ \sin \theta_{\mbox{\boldmath $p$}}b^{s\dagger }(-\mbox{\boldmath $p$}) ,
\label{eq:8}
\end{equation}
is introduced. The ratio $ \theta_{\mbox{\boldmath $p$}}$ in Eq.(\ref{eq:8}) that determines the relative importance of antifermion depends on the relative momentum $\mbox{\boldmath $p$}$ between the object and the observer. $A^s(\mbox{\boldmath $p$})$ is reduced to $a^s(\mbox{\boldmath $p$})$ at $\mbox{\boldmath $p$} \rightarrow \infty$ limit, and therefore $ \theta_{\mbox{\boldmath $p$}}$ in Eq.(\ref{eq:8}) must approach zero at $\mbox{\boldmath $p$} \rightarrow \infty$.
Similar interpretation is possible at C in Figure.\ref{fig.1} as well. Two different interpretations; the annihilation $b^s(-\mbox{\boldmath $p$})$ of massless antifermion at C, and the creation $a^{s\dagger }(\mbox{\boldmath $p$})$ of massless fermion at C are both possible. (The spin direction of $a^s(\mbox{\boldmath $p$})$ is opposite to that of $b^{s}(-\mbox{\boldmath $p$})$ with the same $s$. Hence, a similar argument holds for the flow of spin as well.) Hence, a new annihilation operator $B^s(-\mbox{\boldmath $p$})$
\begin{equation}
B^s(-\mbox{\boldmath $p$})= \cos \theta_{\mbox{\boldmath $p$}} b^s(-\mbox{\boldmath $p$})
- \sin \theta_{\mbox{\boldmath $p$}}a^{s\dagger }(\mbox{\boldmath $p$}) ,
\label{eq:9}
\end{equation}
is introduced, which is orthogonal to $A^s(-\mbox{\boldmath $p$})$. (Equations (\ref{eq:8}) and (\ref{eq:9}) have the same form as the Bogoliubov transformation in superconductivity, but they have different physical meaning.)
\subsection{Physical vacuum of massless fermion and antifermion}
The vacuum is a state with the lowest-possible energy to all observers. For the massive object, the free vacuum $|0\rangle$ is the starting point of all models. However, for the massless fermion in which symmetry breaking has not yet occurred, the free vacuum $|0\rangle$ satisfying $a^s(\mbox{\boldmath $p$})| 0 \rangle =b^s(-\mbox{\boldmath $p$})| 0 \rangle =0$ is a state regarded as a vacuum only by an observer who sees the massless particle moving at a large velocity.
For all observers on different inertial-systems, we define a new state $|\widetilde{0}\rangle$ which satisfies the condition of vacuum of the new annihilation operator $A^s(\mbox{\boldmath $p$})$ and $B^s(-\mbox{\boldmath $p$})$, so that it satisfies $A^s(\mbox{\boldmath $p$})|\widetilde{0}\rangle =B^s(-\mbox{\boldmath $p$})|\widetilde{0}\rangle =0$ to all observers. We call such a $|\widetilde{0}\rangle$ {\it physical vacuum of massless fermion and antifermion }.
The explicit form of $|\widetilde{0}\rangle$ can be guessed as follows. When the momentum of the fermion is large, $\exp(iS)$ is determined, in effect, only by the timelike paths, and $ \cos \theta_{\mbox{\boldmath $p$}} \rightarrow 1$ is expected at $\mbox{\boldmath $p$} \rightarrow \infty$ in Eqs.(\ref{eq:8}) and (\ref{eq:9}). This implies that the physical vacuum becomes the free vacuum, and $|\widetilde{0}\rangle$ includes $ \cos \theta_{\mbox{\boldmath $p$}} |0\rangle$. On the other hand, when $\mbox{\boldmath $p$} \rightarrow 0$, various possible paths extending into the wider region in space-time contribute to the total amplitude. At the point $A$ or $C$ in Figure.\ref{fig.1}, massless fermion and antifermion are both possible in the interpretation of $A^s(\mbox{\boldmath $p$})$ and $B^s(-\mbox{\boldmath $p$})$. Hence, $\cos \theta_{\mbox{\boldmath $p$}}= \sin \theta_{\mbox{\boldmath $p$}}$ is expected at $\mbox{\boldmath $p$} = 0$ in Eqs.(\ref{eq:8}) and (\ref{eq:9}). This suggests that $|\widetilde{0}\rangle$ includes $ \sin \theta_{\mbox{\boldmath $p$}} b^{s\dagger}(-\mbox{\boldmath $p$}) a^{s\dagger}(\mbox{\boldmath $p$}) |0\rangle$. The simplest possible form of $|\widetilde{0}\rangle$ is a superposition of these two limits for each $\mbox{\boldmath $p$}$
\begin{equation}
|\widetilde{0}\rangle =\prod_{p,s} \left[ \cos \theta_{\mbox{\boldmath $p$}}
+ \sin \theta_{\mbox{\boldmath $p$}} b^{s\dagger}(-\mbox{\boldmath $p$}) a^{s\dagger}(\mbox{\boldmath $p$}) \right]|0\rangle .
\label{eq:135}
\end{equation}
The proof of $A^s(\mbox{\boldmath $p$})|\widetilde{0}\rangle =B^s(-\mbox{\boldmath $p$})|\widetilde{0}\rangle =0$ for Eq.(\ref{eq:135}) is as follows. Defining an operator
\begin{equation}
G= i \sum_{p,s} \theta_{\mbox{\boldmath $p$}} [b^{s\dagger}(-\mbox{\boldmath $p$})a^{s\dagger}(\mbox{\boldmath $p$})-a^s(\mbox{\boldmath $p$})b^s(-\mbox{\boldmath $p$})] ,
\label{eq:951}
\end{equation}
and applying the following expansion
\begin{equation}
e^{-iG} F e^{iG} = F + [-iG,F]+ \frac{1}{2!} \left[-iG, [-iG,F] \right] + \cdots ,
\label{eq:10}
\end{equation}
to the operators $a^s(\mbox{\boldmath $p$})$ and $b^s(-\mbox{\boldmath $p$})$ for $F$, and Eq.(\ref{eq:951}) for $G$ , we rewrite Eqs.(\ref{eq:8}) and (\ref{eq:9}) in the following compact form
\begin{equation}
A^s(\mbox{\boldmath $p$})= e^{-iG}a^s(\mbox{\boldmath $p$})e^{iG} ,
\label{eq:11}
\end{equation}
\begin{equation}
B^s(-\mbox{\boldmath $p$})= e^{-iG}b^s(-\mbox{\boldmath $p$})e^{iG} .
\label{eq:12}
\end{equation}
The vacuum $|\widetilde{0}\rangle$ in $A^s(\mbox{\boldmath $p$})|\widetilde{0}\rangle =B^s(-\mbox{\boldmath $p$})|\widetilde{0}\rangle =0$ is expressed as
\begin{eqnarray}
|\widetilde{0}\rangle&=& e^{-iG}|0\rangle
\nonumber \\
&=&\exp\left( \sum_{p,s}\theta_{\mbox{\boldmath $p$}} [b^{s\dagger}(-\mbox{\boldmath $p$})a^{s\dagger}(\mbox{\boldmath $p$})
-a^s(\mbox{\boldmath $p$})b^s(-\mbox{\boldmath $p$}) ]\right ) |0\rangle .
\nonumber \\
&=& \prod_{p,s} \left[ \sum_{n} \frac{1}{n!} \theta_{\mbox{\boldmath $p$}}^n [b^{s\dagger}(-\mbox{\boldmath $p$})a^{s\dagger}(\mbox{\boldmath $p$})
-a^s(\mbox{\boldmath $p$})b^s(-\mbox{\boldmath $p$})]^n \right] |0\rangle.
\nonumber \\
\label{eq:13}
\end{eqnarray}
Since massless fermions and antifermions obey Fermi statistics, only a single particle can occupy each state on the hyperboloid ($p^2=0$) set in $ |0\rangle$ at each point in space-time, and we obtain for each $\mbox{\boldmath $p$}$
\begin{eqnarray}
\lefteqn{\sum_{n} \frac{\theta^n}{n!} (b^{\dagger}a^{\dagger}-ab)^n |0\rangle }
\nonumber \\
&&= |0\rangle + \theta b^{\dagger} a^{\dagger} |0\rangle - \frac{\theta^2}{2!} abb^{\dagger} a^{\dagger} |0\rangle
- \frac{\theta^3}{3!} b^{\dagger} a^{\dagger} abb^{\dagger} a^{\dagger} |0\rangle
\nonumber \\
&&+ \frac{\theta^4}{4!} abb^{\dagger} a^{\dagger} abb^{\dagger} a^{\dagger} |0\rangle + \cdots .
\label{eq:1333}
\end{eqnarray}
In this expansion, $\cos \theta_{\mbox{\boldmath $p$}}$ appears from the sum of even-order terms of $\theta$, and $\sin \theta_{\mbox{\boldmath $p$}}$ appears from the sum of odd-order terms, and then Eq.(\ref{eq:135}) is yielded.
(1) This $|\widetilde{0}\rangle$ has the same structure as that of Nambu and Jona-Lasinio on analogy with superconductivity \cite{nam}, but this $|\widetilde{0}\rangle$ does not come from the attractive interaction. There is the following reason for deriving $|\widetilde{0}\rangle$ kinematically. In superconductivity, the BCS ground state is expressed using the relative momentum between the electron and the center-of-mass of the crystal \cite{sup}. If a naive analogy between the BCS ground state and the physical vacuum was made, the center-of-mass of the crystal would be replaced by the center-of-mass of the universe. Such an absolute rest-frame is not appropriate to the relativistic model, and the vacuum
$|\widetilde{0}\rangle$ must be derived using {\it the relative motion of the particle to the observer}.
(2) When the operators in $L_0(x)$ act on $|\widetilde{0}\rangle$, the massless fermion and antifermion annihilate to a gauge boson in $B_{\mu}$, and this massless boson annihilates to another massless fermion-antifermion pair. There is no threshold energy in such a $s$-channel process between massless objects. Hence, this $|\widetilde{0}\rangle$ is an equilibrium state between the massless fermion-antifermion pairs and gauge bosons.
(3) The gauge boson, produced by the pair annihilation of massless fermion and antifermion with opposite momentum in Eq.(\ref{eq:135}), has a 4-momentum
$(2p_0, \mbox{\boldmath $0$})$ in the center-of-mass system of this pair. On each point in space, the influence of these gauge bosons created at other points
accumulate, and the field energy of $B_{\mu}$, which is a Lorentz- and gauge-invariant scalar quantity with the dimension of mass
\begin{equation}
\int T^{00}(x)d^3x= \frac{1}{4\pi} \int \left (-F^{0\mu}F_{0\mu}+\frac{1}{4}F^{\mu\nu}F_{\mu\nu} \right ) d^3x \equiv \widehat{ \beta},
\label{eq:14}
\end{equation}
condenses at each point in space. To incorporate such a $\widehat{ \beta}$, the free vacuum $|0 \rangle$ in the right-hand side of Eq.(\ref{eq:135}) is replaced by a condensed vacuum $|0_r \rangle$ satisfying $ \langle 0_r | \int T_c^{00}(x)d^3x | 0_r \rangle \equiv \langle \widehat{ \beta} \rangle \neq 0 .$ To incorporate such a $|0_r\rangle$, we redefine Eq.(\ref{eq:135}) as follows
\begin{equation}
|\widetilde{0}\rangle=\prod_{p,s} \left[ \cos \theta_{\mbox{\boldmath $p$}}
+\sin \theta_{\mbox{\boldmath $p$}} e^{i\alpha(x)}b^{s\dagger}(-\mbox{\boldmath $p$}) a^{s\dagger}(\mbox{\boldmath $p$})\right]|0_r\rangle ,
\label{eq:16}
\end{equation}
where an arbitrary phase $\alpha(x)$ of $U(1)$ symmetry appears at each point in space-time. The Higgs potential in $L_h(x)$ is not necessary for deriving the symmetry-broken vacuum.
\section{Massive fermions}
The physical vacuum in Eq.(\ref{eq:16}) is a consequence of general properties. However, the specific form of $\sin \theta_{\mbox{\boldmath $p$}}$ is determined by the interaction in $L_0(x)$. In this physical vacuum $|\widetilde{0}\rangle$, $\varphi(x)$ is coupled not only to $B_{\mu}$, but also to the gauge field included in
$\widehat{ \beta} $ in Eq.(\ref{eq:14}). Hence, the condensed field-energy $\widehat{ \beta} $ acts on the massless fermion as if it is a mean field as follows
\begin{equation}
- \frac{1}{4}F^{\mu\nu}F_{\mu\nu} +\bar{\varphi}(x) \left( i \slash {\!\!\!}{\partial } + g\left[ \widehat{ \beta} + B_{\mu} \gamma ^{\mu} \right]\right) \varphi(x) ,
\label{eq:17}
\end{equation}
This mean field $\widehat{ \beta}$ is an average over many degrees of freedom, and therefore it does not easily change through the creation or annihilation of each fermion. Hence, we approximate $\widehat{ \beta}$ as a constant $\langle \widehat{ \beta} \rangle$. The substitution of Eqs.(\ref{eq:2}) and (\ref{eq:3}) into Eq.(\ref{eq:17}) yields
\begin{eqnarray}
&\frac{1}{V}& \sum_{p, s}\epsilon_p[a^{s\dagger}(\mbox{\boldmath $p$})a^s(\mbox{\boldmath $p$})
+b^{s\dagger}(-\mbox{\boldmath $p$})b^s(-\mbox{\boldmath $p$}) ]
\nonumber \\
+&\frac{1}{V}& g \langle \widehat{ \beta} \rangle \sum_{p, s} [a^{s\dagger}(\mbox{\boldmath $p$})b^{s\dagger}(-\mbox{\boldmath $p$})
+b^s(-\mbox{\boldmath $p$}) a^s(\mbox{\boldmath $p$})] ,
\label{eq:18}
\end{eqnarray}
plus $-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$ and the coupling term to $B_{\mu}$. By inverting Eqs.(\ref{eq:8}) and (\ref{eq:9}), we find
\begin{equation}
a^s(\mbox{\boldmath $p$})
=\cos \theta_{\mbox{\boldmath $p$}}A^s(\mbox{\boldmath $p$})-\sin \theta_{\mbox{\boldmath $p$}}B^{s\dagger}(-\mbox{\boldmath $p$}) ,
\label{eq:19}
\end{equation}
\begin{equation}
b^s(-\mbox{\boldmath $p$})
=\cos \theta_{\mbox{\boldmath $p$}}B^s(-\mbox{\boldmath $p$})+\sin \theta_{\mbox{\boldmath $p$}}A^{s\dagger}(\mbox{\boldmath $p$}) .
\label{eq:20}
\end{equation}
Using these $a^s(\mbox{\boldmath $p$})$ and $b^s(-\mbox{\boldmath $p$})$ in Eq.(\ref{eq:18}), we obtain
\begin{eqnarray}
& \frac{1}{V}& \sum _{p, s} [\epsilon_p( \cos ^2\theta_{\mbox{\boldmath $p$}} -\sin ^2\theta_{\mbox{\boldmath $p$}})
-2g \langle \widehat{ \beta} \rangle \cos \theta_{\mbox{\boldmath $p$}}\sin \theta_{\mbox{\boldmath $p$}} ]
\nonumber \\
& \times & [A^{s\dagger}(\mbox{\boldmath $p$})A^s(\mbox{\boldmath $p$})
+B^{s\dagger}(-\mbox{\boldmath $p$})B^s(-\mbox{\boldmath $p$}) ]
\nonumber \\
+& \frac{1}{V}& \sum _{p, s} [2\epsilon_p\cos \theta_{\mbox{\boldmath $p$}}\sin \theta_{\mbox{\boldmath $p$}}
+g \langle \widehat{ \beta} \rangle ( \cos ^2\theta_{\mbox{\boldmath $p$}} -\sin ^2\theta_{\mbox{\boldmath $p$}} ) ]
\nonumber \\
& \times& [A^{s\dagger}(\mbox{\boldmath $p$})B^{s\dagger}(-\mbox{\boldmath $p$})
+B^s(-\mbox{\boldmath $p$}) A^s(\mbox{\boldmath $p$})] .
\label{eq:21}
\end{eqnarray}
The stability of the physical vacuum $|\widetilde{0}\rangle$ requires that the coefficient of $A^{s\dagger}(\mbox{\boldmath $p$})B^{s\dagger}(-\mbox{\boldmath $p$}) +B^s(-\mbox{\boldmath $p$}) A^s(\mbox{\boldmath $p$})$ vanishes, with the result that
\begin{equation}
\cos ^2\theta_{\mbox{\boldmath $p$}} =\frac{1}{2}\left(1+\frac{\epsilon_p}{\sqrt{\epsilon_p^2+(g \langle \widehat{ \beta} \rangle )^2} }\right) ,
\label{eq:22}
\end{equation}
\begin{equation}
\sin ^2\theta_{\mbox{\boldmath $p$}} =\frac{1}{2}\left(1-\frac{\epsilon_p}{\sqrt{\epsilon_p^2+(g \langle \widehat{ \beta} \rangle )^2} }\right) ,
\label{eq:23}
\end{equation}
which satisfies $\cos \theta_{\mbox{\boldmath $p$}} \rightarrow 1$ at $\mbox{\boldmath $p$} \rightarrow \infty$, and
$\cos \theta_{\mbox{\boldmath $p$}} = \sin \theta_{\mbox{\boldmath $p$}}$ at $\mbox{\boldmath $p$} =0$ as expected.
Substitution of Eqs.(\ref{eq:22}) and (\ref{eq:23}) into the first term of Eq.(\ref{eq:21}) yields
\begin{equation}
\frac{1}{V}\sum _{p,s}\sqrt{\epsilon_p^2+(g \langle \widehat{ \beta} \rangle )^2} [A^{s\dagger}(\mbox{\boldmath $p$})A^s(\mbox{\boldmath $p$})
+B^{s\dagger}(-\mbox{\boldmath $p$})B^s(-\mbox{\boldmath $p$}) ] . \label{eq:24}
\end{equation}
The mass $m_f$ of the real fermion is defined by the condensed field-energy of the massless $B_{\mu}$
\begin{equation}
m_f= g \langle \widehat{ \beta} \rangle =g \langle \widetilde{0} | \int T_c^{00}(x)d^3x |\widetilde{0}\rangle .
\label{eq:245}
\end{equation}
The above derivation is irrelevant to whether the effective interaction between massless fermions $\varphi(x)$ is an attractive or a repulsive one \cite{gap}.
The factors represented by Eqs.(\ref{eq:22}) and (\ref{eq:23}) has the following meaning.
(1) When the momentum of the massive particle increases, it becomes difficult to distinguish the massive particle from the massless one. The factor $\sin \theta_{\mbox{\boldmath $p$}}$ in Eq.(\ref{eq:23}) is a decreasing function of the momentum $\mbox{\boldmath $p$}$. Hence, $A^s(\mbox{\boldmath $p$})$ and $B^s(-\mbox{\boldmath $p$})$ in Eqs.(\ref{eq:8}) and (\ref{eq:9}) changes to $a^s(\mbox{\boldmath $p$})$ and $b^s(-\mbox{\boldmath $p$})$ as $\mbox{\boldmath $p$} \rightarrow \infty$, which is consistent to the explanation using Figure.\ref{fig.1}.
(2) When the momentum of particle is large, its vacuum approaches $|0\rangle$ in the physical vacuum. When its momentum is small, its vacuum approaches
$(1/\sqrt{2})(1+ b^{\dagger }a^{\dagger } )|0\rangle$. The physical vacuum (\ref{eq:16}) with Eqs.(\ref{eq:22}) and (\ref{eq:23}) is consistent to this picture.
With the transition of vacuum from $|0\rangle$ to $|\widetilde{0}\rangle$, new four-component spinors $U^s(p)$ and $V(p)$ satisfying
\begin{equation}
\bar{U}^r(p) U^s(p)= -\bar{V}^r(p)V^s(p) = 2 \sqrt{\epsilon_p^2+m_f^2} \delta ^{rs} ,
\label{eq:251}
\end{equation}
are defined. Hence, new field operators $\psi(x)$ and $\bar{\psi}(x)$ are defined as
\begin{eqnarray}
\psi(x)&=& \frac{1}{\sqrt{V}}\sum_{p,s} \frac{1}{\sqrt{2E_p}}
\nonumber \\
& \times& \left[ A^s(\mbox{\boldmath $p$})U^s(p)e^{-ipx}+ B^{s\dagger }(-\mbox{\boldmath $p$})V^s(-p)e^{-ipx} \right] ,\nonumber \\
\label{eq:26} \\
\bar{\psi}(x)&=& \frac{1}{\sqrt{V}}\sum_{p,s} \frac{1}{\sqrt{2E_p}}
\nonumber \\
& \times& \left[ A^{s\dagger }(\mbox{\boldmath $p$})\bar{U}^s(p)e^{ipx}+B^s(-\mbox{\boldmath $p$})\bar{V}^s(-p)e^{ipx} \right] .
\label{eq:27}
\end{eqnarray}
The relations between the old $u(p), v(p)$ and the new $U(p), V(p)$ are given by
\begin{equation}
\bar{U}^r(p) u^r(p)= \bar{V}^r(p)v^r(p) = \cos \theta_{\mbox{\boldmath $p$}} ,
\label{eq:28}
\end{equation}
\begin{equation}
\bar{U}^r(p) v^r(-p)= -\bar{V}^r(p)u^r(-p) = \sin \theta_{\mbox{\boldmath $p$}} .
\label{eq:29}
\end{equation}
In place of $L_0(x)$, a Lagrangian density expressed by $\psi (x)$ is needed. We obtain such a $L_1(x)$ as
\begin{equation}
L_1(x)= -\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\bar{\psi}(i\partial_{\mu}+gB_{\mu})\gamma^{\mu}\psi - m_f\bar{\psi}\psi .
\label{eq:30}
\end{equation}
In place of $a^s$ and $b^{s\dagger}$ in $\varphi(x)$, $A^s$ and $B^{s\dagger}$ in $\psi(x)$ are directly coupled to $B_{\mu}$, hence, $g \bar{\psi}B_{\mu}\gamma^{\mu}\psi $ appears. This $L_1(x)$ has the same form as that of QED, and the physical vacuum $|\widetilde{0}\rangle$ exists as if it is the free vacuum of massive fermion $\psi(x)$. However, this Lagrangian density includes qualitatively different phenomena, such as the massive gauge boson and Higgs-like mode.
\section{Massive gauge boson}
In the Higgs model, the mass of gauge boson is derived from $|(i\partial_{\mu}+gB_{\mu})(v_h+h)|^2$ in $L_h(x)$. In the present model, instead of such a phenomenological coupling, we derive the massive gauge boson from $L_0(x)$ acting on the physical vacuum $|\widetilde{0}\rangle$.
In $L_0(x)$, the interaction of the massless gauge boson in $B_{\mu}(x)$ with the massless fermion $\varphi(x)$ is as follows
\begin{equation}
{\cal H} _I(x)=g \bar{\varphi}(x)\gamma ^{\mu} \varphi(x) B_{\mu}(x) \equiv gj^{\mu}(x) B_{\mu}(x) .
\label{eq:32}
\end{equation}
Since the physical vacuum is not a simple system, the response of $|\widetilde{0}\rangle$ to $B_{\mu}$ causes a non-linear behavior of $B_{\mu}$. Owing to this ${\cal H} _I(x)$, the kinematic term $L_0^{kin}(x)=\bar{\varphi}(x)(i\partial_{\mu}+gB_{\mu})\gamma^{\mu}\varphi(x) $ itself changes its form as follows. Let us consider a perturbation expansion of $L_0^{kin}(x)$ in powers of $g$ in ${\cal H} _I(x)$
\begin{eqnarray}
\lefteqn{ \langle \widetilde{0}|_p \int d^4x L^{kin} _0(x) |\widetilde{0}\rangle_p }
\nonumber \\
&&= \langle \widetilde{0}| T\left[ \int d^4 x_1 L_0^{kin}(x_1)
exp\left( i\int {\cal H} _I(x_2) d^4x_2 \right) \right] |\widetilde{0}\rangle
\nonumber \\
&&= \langle \widetilde{0}| \int d^4x_1 \bar{\varphi}(x_1) \gamma ^{\mu}[i\partial_{\mu} + gB_{\mu}(x_1)] \varphi(x_1) |\widetilde{0}\rangle
\nonumber \\
&& + \langle \widetilde{0}| T\left[ \int d^4x_1 L_0^{kin}(x_1)
ig\int d^4x_2 j^{\nu}(x_2) B_{\nu}(x_2) \right] |\widetilde{0}\rangle
\nonumber \\
&& + \cdots ,
\label{eq:34}
\end{eqnarray}
where $ |\widetilde{0}\rangle _p$ in the first line represents a vacuum $|\widetilde{0}\rangle$ perturbed by ${\cal H}_I(x)$. The structure of the physical vacuum comes, not from ${\cal H}_I(x)$, but from the general properties, and therefore the perturbation of $|\widetilde{0}\rangle$ by ${\cal H}_I(x)$ is not a double counting.
\subsection{Mass of the gauge boson}
In the last line of Eq.(\ref{eq:34}), the second-order term of $B_{\mu}$ is the following two-point-correlation function
\begin{equation}
\langle \widetilde{0}| T\left[ \int d^4x_1 {\cal H} _I(x_1)
\int d^4x_2 {\cal H} _I(x_2)
\right] |\widetilde{0}\rangle .
\label{eq:35}
\end{equation}
This correlation implies that a massless fermion-antifermion pair $\bar{\varphi}(x_1)\gamma ^{\mu} \varphi(x_1)$ annihilates to $B_{\mu}$ through Eq.(\ref{eq:17}), and this $B_{\mu}$ reversely annihilate to another massless fermion-antifermion pairs $\bar{\varphi}(x_2)\gamma ^{\mu} \varphi(x_2)$. As a result, the correlation between $ \bar{\varphi}(x_1) \gamma ^{\mu} \varphi (x_1)$ and $ \bar{\varphi}(x_2)\gamma ^{\nu} \varphi(x_2)$ appears in Eq.(\ref{eq:35}) when $\mu=\nu$. To obtain the coefficient of $B_{\mu}(x_1)B_{\mu}(x_2)$, we transform Eq.(\ref{eq:35}) as follows
\begin{eqnarray}
&g^2& \int \langle \widetilde{0}| T\left [ \int j_{\mu}(x_1) d^2x_1
\int j^{\mu}(x_2) d^2x_2 \right] |\widetilde{0}\rangle
\nonumber \\
& \quad \times& B^{\mu}(x_1)B_{\mu}(x_2) d^2x_1d^2x_2
\nonumber \\
&=& g^2 \int \langle \widetilde{0}| T\left [ \int \partial_{\mu} j_{\mu}(x_1) d^3x_1
\int \partial_{\nu} j_{\mu}(x_2) d^3x_2 \right] |\widetilde{0}\rangle
\nonumber \\
&\quad \times& B^{\mu}(x_1)B_{\mu}(x_2) d^2x_1d^2x_2 .
\nonumber \\
\label{eq:361}
\end{eqnarray}
These $x_1$ and $x_2$ are microscopically separated points in space-time, and if we observe this phenomenon from a far distant point, it looks like a local phenomenon at
$X=(x_1+x_2)/2$, and the relative motion corresponding to $x_2-x_1$ is not directly observed, but contained in the coefficient corresponding to a mass. To such an observer, it is appropriate to assume $x_1 \rightarrow X$ and $x_2 \rightarrow X$ in $B_{\mu}(x)$, and the gauge boson behaves as a massive boson as
\begin{equation}
m_B^2 \int B^{\mu}(X) B_{\mu}(X) d^4X .
\label{eq:31.8}
\end{equation}
The mass $m_B$ is given by the following two-point correlation function
\begin{equation}
m_B= g \left [ \langle \widetilde{0}| tr \left( \frac{1}{2} \int \partial_{\mu} j^{\mu}(x_1) d^3x_1
\int \partial_{\nu}j^{\nu}(x_2) d^3x_2 \right) |\widetilde{0}\rangle \right ]^{1/2} .
\label{eq:36}
\end{equation}
where $\frac{1}{2}$ comes from the time-ordered product in Eq.(\ref{eq:361}).
Since the physical vacuum $|\widetilde{0}\rangle$ is filled with the kinetic energy of massless fermion, the right-hand side of Eq.(\ref{eq:36}) has a finite Lorentz-invariant value.
(Since $\sin^2 \theta_{\mbox{\boldmath $p$}} \rightarrow 0$ at $\mbox{\boldmath $p$} \rightarrow \infty$ in Eq.(\ref{eq:23}), the integral over infinite space in Eq.(\ref{eq:36}) is finite, as long as a similar integral for the free vacuum $|0\rangle$ is finite.)
\subsection{Goldstone mode}
The Goldstone mode comes from the phase factor of the condensate. In the Higgs model, the phase $\theta$ is assumed in the vacuum condensate $|v_h|\exp (i\theta)$. The present model gives it a concrete meaning as the phase $\alpha (x)$ in the physical vacuum $|\widetilde{0}\rangle$.
(1) In the last line of Eq.(\ref{eq:34}), there is a first-order term of $B_{\nu}(x_2)$. Let us partially integrate this term over $x_1$ in $\bar{\varphi}(x_1)i\partial^{\mu}\gamma _{\mu} \varphi(x_1)$ included in $L^{kin}_0(x_1)$, in which $ \varphi(x_1)$ vanishes at $x_1 \rightarrow \infty$. The physical vacuum $ |\widetilde{0}\rangle$ in Eq.(\ref{eq:16}) has an explicit $x$-dependence in the phase $\alpha(x)$. As a result of partial integration, we obtain a term including $i\partial^{\mu}\bar{\varphi}(x_1)\gamma _{\mu} \varphi(x_1)$, and that including $\partial^{\mu} |\widetilde{0}\rangle$. The latter term is given by
\begin{eqnarray}
& g& \langle \widetilde{0}| T\left[ \int d^4x_1 j_{\mu}(x_1)
\int d^4x_2 j^{\nu}(x_2) B_{\nu}(x_2) \right] \partial_{\mu} |\widetilde{0}\rangle
\nonumber \\
+& g& \partial_{\mu} \langle \widetilde{0}| T\left[ \int d^4x_1 j_{\mu}(x_1)
\int d^4x_2 j^{\nu}(x_2) B_{\nu}(x_2) \right] |\widetilde{0}\rangle .
\nonumber \\
\label{eq:374}
\end{eqnarray}
in which $\partial_{\mu} |\widetilde{0}\rangle|$ is a product of $ \partial_{\mu}\alpha (x)$ and $ \prod_{p,s}\sin \theta_{\mbox{\boldmath $p$}} e^{i\alpha(x)}b^{s\dagger}(-\mbox{\boldmath $p$}) a^{s\dagger}(\mbox{\boldmath $p$}) |0_r\rangle$.
To an observer at a distant point in space-time, it is appropriate to rewrite $d^4x_1d^4x_2$ using the variable $X=(x_1+x_2)/2$. Hence, we obtain
$d^4x_1d^4x_2 \rightarrow \partial^{\mu}d^5x_1 \partial_{\nu}d^5x_2 \rightarrow \partial^{\mu}d^3x_1 \partial_{\nu}d^3x_2 d^4X$. With the definition of $m_B$ in Eq.(\ref{eq:36}), we transform Eq.(\ref{eq:374}) in $\mu=\nu$ as
\begin{equation}
\frac{2i }{g} m^2_B \int B_{\mu}(X)\partial ^{\mu} \alpha (X) d^4X \equiv m_B \int B_{\mu}(X) \partial^{\mu} \phi(X) d^4X ,
\label{eq:38}
\end{equation}
where the Goldstone mode $\phi (X)$ is defined as $\phi (X)= 2 i g^{-1}m_B \alpha (X)$. Since the global rotation of phase requires no energy, the propagator of the Goldstone mode is given by
\begin{equation}
F(q^2) \equiv \int \frac{dX^4}{(2\pi)^4} \langle \widetilde{0}| T[ \phi (X) \phi (0) ] |\widetilde{0}\rangle e^{iqX}= \frac {i} { q^2 } .
\label{eq:381}
\end{equation}
The Fourier transform of Eqs.(\ref{eq:31.8}) and (\ref{eq:38}) are given by $m_B^2B^{\mu}(q)B_{\mu}(q)$ and $m_Bq^{\mu} \phi(q) B_{\mu}(q)$, respectively. Regarding the latter as a perturbation to the former, its second-order perturbation is obtained
\begin{eqnarray}
&B^{\mu}(q)& \left [im_B^2g^{\mu\nu} - m_Bq^{\mu} F(q^2) m_Bq^{\nu} \right] B_{\nu}(q)
\nonumber \\
&=& im_B^2 \left( g^{\mu\nu} - \frac{q^{\mu}q^{\nu}}{q^2} \right) B^{\mu}(q)B^{\nu}(q).
\label{eq:382}
\end{eqnarray}
Adding this term to the Fourier transform of $-\frac{1}{4} F^{\mu\nu}F_{\mu\nu}$, we obtain
\begin{equation}
i B^{\mu}(q) [q^2-m_B^2] \left( g^{\mu\nu} - \frac{q^{\mu}q^{\nu}}{q^2} \right) B^{\nu}(q).
\label{eq:383}
\end{equation}
The inversion of this matrix yields
\begin{eqnarray}
D^{\mu\nu}(q) &=& \frac{-i}{q^2-m_B^2} \left( g^{\mu\nu} - \frac{q^{\mu}q^{\nu}}{q^2} \right)
\nonumber \\
&\equiv &i D(q^2) \left( g^{\mu\nu} - \frac{q^{\mu}q^{\nu}}{q^2} \right),
\label{eq:384}
\end{eqnarray}
which is the propagator of the massive gauge boson in the Landau gauge \cite{trans}. The massless gauge boson becomes massive by eating the phase of the condensed fermion-antifermion pairs in the physical vacuum.
(2) The Goldstone mode directly couples to fermions. There is a zeroth-order term of $B_{\mu}$ in the third line of Eq.(\ref{eq:34}). We partially integrate it over $x_1$, and obtain a
term including $i\partial^{\mu}\bar{\varphi}(x_1)\gamma _{\mu} \varphi(x_1)$, and that including $\partial^{\mu} |\widetilde{0}\rangle$. The latter term is given by
\begin{equation}
i \langle \widetilde{0}| \int d^4x_1 j^{\mu}(x_1) \partial_{\mu} |\widetilde{0}\rangle + i \partial_{\mu} \langle \widetilde{0}| \int d^4x_1 j^{\mu}(x_1) |\widetilde{0}\rangle .
\label{eq:385}
\end{equation}
Expressing $ \partial_{\mu} |\widetilde{0}\rangle$ in terms of $ \partial_{\mu}\alpha(x)$, and with the definition $\phi(x)= 2i g^{-1}m_B\alpha(x)$, we rewrite it as
\begin{equation}
\frac{g}{m_B} \langle \widetilde{0}| \int d^4x_1 \bar{\varphi} (x)\gamma^{\mu} \varphi (x) \partial_{\mu}\phi (x) |\widetilde{0}\rangle .
\label{eq:3851}
\end{equation}
As a result, an additional term $gm_B^{-1} \partial_{\mu}\phi (x)\bar{\varphi} (x)\gamma^{\mu} \varphi (x)$ appears. This coupling to fermions is different from $ g(m_f/m_B) \phi (x)\bar{\varphi} (x) \varphi (x) $ in the Higgs model, because it is not derived from the Yukawa coupling $(m_f/v_h) [v_h+h(x)+ i\phi(x)] \bar{\varphi} (x) \varphi (x)$, but from the response of the physical vacuum.
In place of $L_1(x)$, a new Lagrangian density describing the massive gauge boson is needed. We obtain such a $L_2(x)$ including Eqs.(\ref{eq:31.8}), (\ref{eq:38}) and (\ref{eq:3851})
\begin{eqnarray}
L_2(x)= & -& \frac{1}{4}F^{\mu\nu}F_{\mu\nu}+ m_B^2 B^{\mu}B_{\mu} + m_B\partial_{\mu} \phi B_{\mu}
\nonumber \\
&+& \frac{g}{m_B} \bar{\psi} \gamma^{\mu} \psi \partial_{\mu} \phi
\nonumber \\
&+& \bar{\psi}(i\partial_{\mu}+gB_{\mu})\gamma^{\mu}\psi - m_f\bar{\psi}\psi .
\label{eq:39}
\end{eqnarray}
\section{Higgs-like boson as a local excitation propagating in vacuum}
\begin{figure}
\begin{center}
\includegraphics [scale=0.4]{fig-9_edited.eps}
\caption{Chain of pair creation and annihilation of virtual massless fermion-antifermion pairs}
\label{fig.15}
\end{center}
\end{figure}
The coupling of the massless fermions $\varphi(x)$ to the mean field $\widehat{\beta}$ in Eq.(\ref{eq:17}) is accompanied by the creation of a local excitation $\beta(x)$ from
$\widehat{ \beta}$ through the annihilation of these fermion and antifermion at $x$. This excitation then annihilate to another massless fermion-antifermion pair at $x'$.
Hence, it propagates through a chain of pair creations and annihilations of massless fermion-antifermion pairs as illustrated in Figure \ref{fig.15}. Since this mode is a local excitation from the physical vacuum, it plays a similar role to the Higgs particle that causes a deviation from the minimum point of the Higgs potential, and therefore we can call it a {\it Higgs-like excitation mode} $H_{\beta}(x)$. The excitation does not have any specific direction in space,, and therefore it is appropriate to regard $H_{\beta}(x)$ as a scalar field.
(1) The static field $\widehat{ \beta}$ determines the fermion mass as $m_f=g\langle \widehat{ \beta}\rangle$, and therefore the propagating field $H_{\beta}(x)$ of the local excitation $\beta(x)$ is also coupled to the massless fermion with a strength proportional to $m_f$.
(2) For the coupling of $\varphi $ to $H_{\beta}(x)$, a dimensionless coupling constant $m_f/M$ is necessary, where $M$ is a constant with a dimension of mass, and we asume
\begin{equation}
{\cal V}_I(x)= \frac{m_f}{M} [ \bar{\varphi}_L(x) \varphi _R(x) H_{\beta}(x) + \bar{\varphi}_R(x) \varphi _L(x) H^*_{\beta}(x)] .
\label{eq:43}
\end{equation}
This ${\cal V}_I(x)$ is not directly derived from $L_0(x)$, but derived physically.
(At first sight, this ${\cal V}_I(x)$ looks like the Yukawa coupling, but it contributes neither to the mass generation of the fermion nor to its coupling to the Goldstone mode. ) The reason why the mass of the Higgs particle has been an unknown parameter in the electroweak model is that it is not a quantity which we can guess using symmetry principle, but a result of the many-body phenomenon.
Figure \ref{fig.15} shows the interaction between the massless fermions by the exchange of the Higgs-like mode as
\begin{equation}
\bar{\varphi }(k) \varphi (k+q) \left( \frac{m_{f}}{M} \right) \frac{1}{q^2[1-J(q^2)]} \left( \frac{m_{f}}{M} \right) \bar{\varphi }(p)\varphi (p+q) ,
\label{eq:45}
\end{equation}
where $J(q^2)$ represents the creation and annihilation of the fermion-antifermion pair
\begin{equation}
q^2J(q^2)= \frac{m_f}{M} \int_{0}^{\Lambda}\frac{d^4p}{(2\pi)^4} tr\left[ \frac{i}{\slash {\!\!\!}{p}-m_f} \frac{i}{\slash {\!\!\!}{p}+\slash {\!\!\!}{q}-m_f} \right] .
\label{eq:46}
\end{equation}
This $J(q^2)$ has following features.
(1) Although the fermion $\varphi (x)$ in the bubble diagram is massless, its excitation from $|\widetilde{0}\rangle$ can be simply written by the massive fermion operator $\psi(x)$. The resulting $J(q^2)$ has a similar form to the vacuum polarization in QED, but an important difference is that $\gamma ^{\mu}$ and $\gamma ^{\nu}$ are absent in the trace.
(2) $\Lambda$ in Eq.(\ref{eq:46}) is not a cutoff for the Pauli-Villars regularization of the divergent integral, but an upper end of energy-momentum of massless fermion-antifermion pairs in the excitation. We can calculate Eq.(\ref{eq:46}) as if $\Lambda$ is such a cutoff for regularization. But, since the upper end $\Lambda$ is a dynamical variable, it is evaluated, not as $p_E^2$ in the Euclidian space, but as $p^2=(ip_0)^2-\mbox{\boldmath $p$}^2= -p_E^2$ in the Minkowski space. Hence, after a 4-momentum integration, the square of upper end appears as $-\Lambda ^2$ in the result.
Using the ordinary rule for diagrams, we obtain
\begin{eqnarray}
J(q^2)&=& \frac{1}{4\pi^2} \frac{m_f}{M}
\nonumber \\
&\times & \int_{0}^{1}dx \left[ x(x-1) +\frac{m_f^2}{q^2} \right] \ln \left(\frac{x(1-x)\Lambda^2+m_f^2}{x(x-1)q^2+m_f^2}\right) .
\nonumber \\
\label{eq:471}
\end{eqnarray}
A peculiar feature of this $J(q^2)$ is that $m_f^2/q^2$ appears in the integrand. With this $J(q^2)$, the propagator of the Higgs-like excitation mode $H_{\beta}(x)$ is given by
\begin{equation}
\int \frac{d^4x}{(2\pi)^4} \langle \widetilde{0}| T H_{\beta}(x) H_{\beta}(0) | \widetilde{0}\rangle e^{iqx} =\frac{1}{q^2\left[1- J(q^2)\right]} .
\label{eq:48}
\end{equation}
\begin{figure}
\begin{center}
\includegraphics [scale=0.6]{vac-25.eps}
\caption{Propagator of the Higgs-like collective mode $1/(q^2[1-J(q^2)])$ with $m_f/M=0.64$ and $\Lambda /m_f=800$. }
\label{fig.21}
\end{center}
\end{figure}
Figure \ref{fig.21} shows $1/[q^2(1-J(q^2))]$ with parameters $m_f/M=0.64$ and $\Lambda /m_f=800$ as an example. A single pole at $\sqrt{q^2}= 0.42$ appears as if it is represented by
\begin{equation}
\frac{1}{q^2-m_H^2} .
\label{eq:49}
\end{equation}
with $m_H=0.42m_f$.
{\it The reason for the pole to appear is the existence of $m_f^2/q^2$ in the integrand of $J(q^2)$ in Eq.(\ref{eq:471}). For this $m_f^2/q^2$ to appear, the absence of $\gamma$ matrix in the interaction energy density ${\cal \widetilde{H}}_I(x)$ in Eq.(\ref{eq:43}) plays an important role. \/} If there is $\gamma^{\mu}$ and $\gamma^{\nu}$ in the trace of Eq.(\ref{eq:46}), such a $m^2_f/q^2$ disappears, and then the pole structure is not yielded in Eq.(\ref{eq:48}) \cite{nam2}.
\begin{figure}
\begin{center}
\includegraphics [scale=0.65]{vac-3.eps}
\caption{The rate of Higgs-like collective mode mass $m_H$ to fermion mass $m_f$ as a function of $m_f/M$ and $\Lambda/m_f$.}
\label{fig.3}
\end{center}
\end{figure}
The Higgs-like boson's mass $m_H$ is strongly depends on the fermion's mass $m_f$. But their relationship also depends on the parameter $M$ and $\Lambda$. Figure \ref{fig.3} shows $m_H/m_f$, obtained by solving $J(m_H^2)=1$, as a function of $m_f/M$ and $\Lambda/m_f$. As these variables increase, it takes much energy $m_H$ to excite the Higgs-like collective mode.
This Higgs-like mode is unstable with respect to the decay into fermion-antifermion pairs at $q^2>(2m_f)^2$. In Eq.(\ref{eq:471}), the logarithm function includes $x(x-1)q^2+m_f^2$ in the denominator, in which $x(x-1)$ is at most $-1/4$ at $x=1/2$. Hence, $x(x-1)+m_f^2/q^2$ becomes negative at $q^2>(2m_f)^2$, which leads to an imaginary energy in Eq.(\ref{eq:471}). For any fixed $q^2$ at $q^2>(2m_f)^2$, the $x$-value that can contribute to the imaginary energy in Eq.(\ref{eq:471}) satisfies $x(x-1)q^2+m_f^2<0$, which lies in a region between the points $x=\frac{1}{2} \pm \frac{1}{2} \delta$, where $\delta = \sqrt{1-4m_f^2/q^2 }$. Using $Im[-X\pm i\epsilon ]=\pm \pi$, and $y=x-\frac{1}{2}$, we obtain the imaginary part
\begin{eqnarray}
&-& \frac{1}{4\pi^2} \frac{m_f}{M} (\pm\pi) \int^{(1+\delta)/2}_{(1-\delta)/2} dx \left[x(x-1)+\frac{m_f^2}{q^2} \right]
\nonumber \\
&=& \pm \frac{1}{4\pi} \frac{m_f}{M} \int^{\delta/2}_{-\delta/2} dy \left[ (y^2-\frac{1}{4} )+\frac{m_f^2}{q^2} \right].
\label{eq:493}
\end{eqnarray}
Finally, we obtain the propagator of $H_{\beta}(x)$ at $q^2>(2m_f)^2$
\begin{eqnarray}
&\int & \frac{d^4x}{(2\pi)^4} \langle \widetilde{0}| T H_{\beta}(x) H_{\beta}(0) | \widetilde{0}\rangle e^{iqx}
\nonumber \\
&=& \frac{1}{q^2 - m_H^2 \pm i \displaystyle{ \frac{1}{24\pi} \frac{m_f}{M} \sqrt {1-\frac{4m_f^2}{q^2} } (q^2 - 4m_f^2) }} .
\label{eq:495}
\end{eqnarray}
in which $m_H^2$ is used for the real part of the logarithm function in Eq.(\ref{eq:471}).
The imaginary part of the self energy increases with increasing $q^2$, and finally the excitation mode becomes unstable. For the electroweak interaction, however, we know $m_H=$125 GeV $<2m_t=$346 GeV. Since $m_H<2m_f$, the structure of the pole-mass around $q^2=m_H^2$ is not affected by the onset of damping at $q^2>(2m_f)^2$.
In place of $L_2(x)$, the final Lagrangian density including the Higgs-like excitation mode is needed. We obtain such a $L_3(x)$ as
\begin{eqnarray}
L_3(x) &=& -\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+ m_B^2 B^{\mu}B_{\mu} + m_B B_{\mu} \partial_{\mu} \phi
\nonumber \\
&+& \bar{\psi}(i\partial_{\mu}+gB_{\mu})\gamma^{\mu}\psi - m_f\bar{\psi}\psi
\nonumber \\
&+& \frac{g}{m_B} \bar{\psi} \gamma^{\mu} \psi \partial_{\mu} \phi
\nonumber \\
&+& |\partial_{\mu} H_{\beta}|^2 - m_H^2 H_{\beta}^{\dagger} H_{\beta} + \left(\frac{m_f}{M}\right) \bar{\psi}\psi H_{\beta} .
\label{eq:50}
\end{eqnarray}
where the excitation mode $H_{\beta}$ is simply formulated in the last three terms.
This $L_3(x)$ does not include the direct coupling of $H_{\beta}$ to $B_{\mu}$, and the Higgs potential. In the Higgs model, the quadratic divergence due to $L_h(x)$ appears in the perturbation calculation of the properties of the Higgs particle. In contrast, due to the absence of $L_h(x)$, the perturbation calculation based on this $L_3(x)$ includes only logarithmic divergence.
\section{Effective coupling of the Higgs-like mode}
Recently the decay of the Higgs-like particle to two gauge bosons $W^{+}$ and $W^{-}$ are observed \cite{atl}\cite{cms}. In the Higgs model applied to $B_{\mu}$, the coupling responsible for such a decay is simply derived from $|(i\partial _{\mu} + gB_{\mu})(v_h+h) |^2$. As a result, the following direct coupling
\begin{equation}
g^2v_h g^{\mu\nu} \times B_{\mu}(p)B_{\nu}(k) h(p+k) ,
\label{eq:88}
\end{equation}
appears in the tree processes. Furthermore, by minimizing the Higgs potential $-\mu^2|v_h+h|^2+\lambda |v_h+h|^4$ with respect to $v_h$, the direct self-coupling of the Higgs particle
\begin{equation}
\lambda v_h \times h(p) h(k) h(p+k) ,
\label{eq:90}
\end{equation}
appear at $v_h=\sqrt{\mu^2/2\lambda}$. The quadratic divergence comes from Eqs.(\ref{eq:88}) and (\ref{eq:90}).
In the present model, the effective coupling of the Higgs-like mode $H_{\beta}(x)$ comes from one-loop processes illustrated in Figure \ref{fig.85}. As a warmup example, we derive such an effective coupling in the case of $U(1)$ gauge field, and use it in the calculation of the cross section in Section.7.
The result in Section.6 and 7 is obtained using the computing algorithm {\it Package-X} \cite{pat}.
The future precise measurement of the electroweak interaction will determine whether the deviation from the simple Higgs model exists or not.
\begin{figure}
\begin{center}
\includegraphics [scale=0.35]{fig-2_edited-2.eps}
\caption{ The effective coupling induced by the one-loop diagram (a) $\Gamma^{\mu\nu} _{BBH}(q, p, k, m_B, m_f) $ in Eq.(\ref{eq:922}), and (b) $F_{H}(q^2, p^2, k^2, m_f)$ in Eq.(\ref{eq:93}). Wavy and dotted lines represent $B_{\mu}$ and $H_{\beta}$, respectively. }
\label{fig.85}
\end{center}
\end{figure}
\subsection{Coupling to massive gauge bosons}
The effective coupling term responsible for the $H_{\beta}$ decay into gauge bosons in Figure \ref{fig.85}(a) is composed of $-(m_f/M) \bar{\psi}\psi H_{\beta} $ and $ \bar{\psi}gB_{\mu} \gamma^{\mu}\psi$ in Eq.(\ref{eq:50}). In coordinate space, such a coupling takes a form
\begin{eqnarray}
\lefteqn{g^2 \frac{m_f}{M} \times B^{\mu}(x_1)B^{\nu}(x_2) H_{\beta}(x_3)}
\nonumber \\
&&\times \langle \widetilde{0}| T\left [ \int j_{\mu} (x_1) d^4x_1
\int j _{\nu} (x_2) d^4x_2 \int \bar{\varphi}(x_3) \varphi(x_3) d^4x_3\right] |\widetilde{0}\rangle .
\nonumber \\
\label{eq:3611}
\end{eqnarray}
(A) When this coupling is viewed from a distant point in space-time, it looks like a local phenomenon at $X=(x_1+x_2+x_3)/3$ as
\begin{eqnarray}
\lefteqn{g^2 \frac{m_f}{M} \int B^{\mu}(X)B^{\nu}(X) H_{\beta}(X) d^4X}
\nonumber \\
&&\times \langle \widetilde{0}| T\left [ \int j _{\mu} (x_1)
\int j _{\nu}(x_2) \int t(x_3) d^3x_1 d^3x_1d^3x_3\right] |\widetilde{0}\rangle ,
\nonumber \\
\label{eq:362}
\end{eqnarray}
where $t(x)= \bar{\varphi}(x) \slash {\!\!\!}{\partial } \varphi(x)$. The coefficient of $B^{\mu}B^{\nu}H_{\beta}$ is a three-point correlation function with a dimension of mass.
Since the physical vacuum $|\widetilde{0}\rangle$ is filled with massless fermion-antifermion pairs, the coefficient of $g_{\mu\nu}B^{\mu}B^{\nu}H_{\beta}$ in Eq.(\ref{eq:362}) has a finite value $V_h$
\begin{equation}
V_h= \langle \widetilde{0}| T\left [ \int j _{\mu} (x_1)
\int j^{\mu} (x_2) \int t (x_3) d^3x_1 d^3x_1d^3x_3\right] |\widetilde{0}\rangle,
\label{eq:363}
\end{equation}
because of $\prod_{p,s}\sin \theta_{\mbox{\boldmath $p$}} e^{i\alpha(x)}b^{s\dagger}(-\mbox{\boldmath $p$}) a^{s\dagger}(\mbox{\boldmath $p$}) |0_r\rangle $ in $ |\widetilde{0}\rangle$.
This $V_h$ plays the role of $v_h$ in Eq.(\ref{eq:88}).
(B) When this coupling is viewed in high resolution, anomalous momentum-dependent coupling is observed.
We consider an amplitude $ {\cal M} [ H_{\beta}(p+k)\rightarrow B_{\mu}(p)B_{\nu}(k)] $ in Figure \ref{fig.85}(a) \cite{effe}
\begin{eqnarray}
\lefteqn{g^2 i {\cal M} [ H_{\beta}(p+k)\rightarrow B_{\mu}(p)B_{\nu}(k)] } \nonumber \\
&& \propto (-ig)^2(-i\frac{m_f}{M})
\nonumber \\
&& \quad \times \int\frac{d^4l}{(2\pi)^4} tr\left[\gamma^{\mu}\frac{i}{\slash {\!\!\!}{l}-\slash {\!\!\!}{p}-m_f} \gamma^{\nu}\frac{i}{\slash {\!\!\!}{l}-\slash {\!\!\!}{q}-m_f}\frac{i}{\slash {\!\!\!}{l}-m_f}\right]
\nonumber \\
&& \quad + (p \leftrightarrow k, \mu \leftrightarrow \nu) .
\label{eq:51}
\end{eqnarray}
Following the standard procedure, the analytic result is obtained as follows.
The general form of the interaction between $B_{\mu}$ and $H_{\beta}$
\begin{eqnarray}
&g^2& (\frac{m_f}{M}) \left[ V_hg^{\mu\nu}+m_f\Gamma^{\mu\nu} _{BBH}(q, p, k, m_B, m_f) \right]
\nonumber \\
&\times & B_{\mu}(p)B_{\nu}(k) H_{\beta}(p+k) ,
\label{eq:922}
\end{eqnarray}
has the following anomalous momentum-dependence
\begin{eqnarray}
\lefteqn{ \Gamma^{\mu\nu} _{BBH}(q, p, k, m_B, m_f) }
\nonumber \\
&& = F_1(q^2,p^2,k^2,m_f)g^{\mu\nu}
\nonumber \\
&&+ F_2(q^2,p^2,k^2,m_f) \frac{p^{\nu}k^{\mu}+p^{\mu}k^{\nu}}{m_B^2}
\nonumber \\
&& + F_3(q^2,p^2,k^2,m_f) \frac{p^{\nu}p^{\mu}+k^{\mu}k^{\nu}}{m_B^2} .
\label{eq:925}
\end{eqnarray}
In these $F_i$, the divergences coming from the triangle-loop integral are cancelled to each other in Eq.(\ref{eq:51}).
(a) We obtain the leading anomalous couplings $ F_1(q^2,p^2,k^2,m_f)$ in Eq.(\ref{eq:925}) \cite{shi}
\begin{eqnarray}
\lefteqn{F_1(q^2,p^2,k^2, m_f) = }
\nonumber \\
&& \frac{8}{(4\pi)^2} \left[ 1+ \frac{(p^2+k^2)q^2 -8p^2k^2}{(q^2-4p^2) (q^2-4k^2) } f \left(\frac{q}{2m_f} \right) \right]
\nonumber \\
&& - \frac{8}{(4\pi)^2} \left[ \frac{p^2}{q^2-4p^2} f \left(\frac{p}{2m_f} \right) + (p \leftrightarrow k)\right]
\nonumber \\
&& - \frac{2}{(4\pi)^2} \frac{ q^4 -(6p^2+4m_f^2)q^2 +4p^4 +16p^2m_f^2 }{q^2-4p^2}
\nonumber \\
&& \quad \times C_0(p^2, p^2, q^2, m_f^2, m_f^2, m_f^2)
\nonumber \\
&& - (p \leftrightarrow k),
\label{eq:565}
\end{eqnarray}
where
\begin{figure}
\begin{center}
\includegraphics [scale=0.65]{vac-fff.eps}
\caption{Schematic view of $f(x)$. $Re f(x)$ and $Im f(x)$ are represented by a solid, and a one-point-dotted curve, respectively. }
\label{fig.945}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics [scale=0.65]{vac-AAA.eps}
\caption{Schematic view of $A(x)$. $Re A(x)$ and $Im A(x)$ are represented by a solid, and a one-point-dotted curve, respectively. }
\label{fig.946}
\end{center}
\end{figure}
(1) The function $f(x)$
\begin{equation}
f(x)= \sqrt{1-\frac{1}{x^2}} \ln \left[ 1-2x^2+2x^2 \sqrt{1-\frac{1}{x^2}} \right] ,
\label{eq:571}
\end{equation}
is illustrated in Figure.\ref{fig.945}.
(2) $C_0(p^2, p^2, q^2, m_f^2, m_f^2, m_f^2) $ is the scalar $C_0$ function in the Passarino-Veltman integrals \cite{pas}\cite{bar},
\begin{eqnarray}
\lefteqn{C_0(p^2, p^2, q^2, m_f^2, m_f^2, m_f^2) }
\nonumber \\
&& = \frac{2}{q^2} A(\frac{q}{2m_f})+ \left( \frac{1}{m_f^2q^2} + \frac{1}{q^4} A(\frac{q}{2m_f}) \right) p^2 + \cdots ,
\nonumber \\
\label{eq:5672}
\end{eqnarray}
where
\begin{equation}
A(x)= - \arcsin ^2 x
\label{eq:56741}
\end{equation}
for for $x<1$, and
\begin{equation}
A(x)= \frac{1}{4} \left(\ln \displaystyle{ \left[ 1-2x^2+2x^2 \sqrt{1-\frac{1}{x^2}} \right] }-i\pi \right)^2,
\label{eq:56742}
\end{equation}
for $x>1$, which is illustrated in Figure.\ref{fig.946}. Since $f(x) \rightarrow -2$ and $ \arcsin x \rightarrow x$ at $x\rightarrow 0$, $F_1(0,0,0,m_f)=0$ is obtained in Eq.(\ref{eq:565}).
\begin{figure}
\begin{center}
\includegraphics [scale=0.65]{vac-405.eps}
\caption{ The overall $q^2$-dependence of the anomalous effective coupling $ F_1(q^2,m_B^2, m_B^2,m_f)$; the real part $ReF_1$ (solid curve), and its imaginary part $ImF_1$ (one-point-dotted curve). The mass of the gauge boson $B_{\mu}$ is $m_B=80$ GeV. The mass $m_H$ of $H_{\beta}$ and the mass $m_f$ of the fermion in the triangle loop are assumed as $m_H=123$ GeV and $m_f=160 $ GeV, respectively. The condition of $q^2=m_H^2$ corresponds to $\sqrt{q^2}/m_f=0.76$, from which $F_1(q^2)$ begins to exist. The solid curve at $\sqrt{q^2}/m_f<2$ is magnified in Figure.\ref{fig.9}. }
\label{fig.95}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics [scale=0.65]{vac-21.eps}
\caption{Three types of the real part of anomalous effective coupling at $\sqrt{q^2}<2m_f$, $ F_1(q^2,m_B^2, m_B^2,m_f)$ (solid curve), $ F_2(q^2,m_B^2,m_B^2,m_f) $ (dotted curve), and $ F_3(q^2,m_B^2,m_B^2,m_f) $ (short dotted curve), under the same condition as Figure \ref{fig.95}. }
\label{fig.9}
\end{center}
\end{figure}
(b) For the decay $ H_{\beta}(p+k)\rightarrow B_{\mu}(p)B_{\nu}(k)$ illustrated in Figure \ref{fig.85}(a), we numerically calculate the anomalous effective coupling at $p^2=k^2=m_B^2$. Figure \ref{fig.95} shows the overall $q^2$-dependence of $F_1(q^2,m_B^2, m_B^2,m_f)$: its real part $Re F_1$ (solid curve), and its imaginary part $Im F_1$ (one-point-dotted curve), in which $m_B=80$ GeV, $m_H=123$ GeV, and $m_f=160 $ GeV are used. The real part $ Re F_1(q^2,m_B^2, m_B^2,m_f)$ increases with increasing $\sqrt{q^2}$ to $2m_f$, and decreases at $\sqrt{q^2}>2m_f$, then changing its sign. The amplitude of imaginary part $Im F_1(q^2,m_B^2, m_B^2,m_f)$ remains zero at $\sqrt{q^2}<2m_f$, but gradually increases at $\sqrt{q^2}>2m_f$.
(c) In the present model, other effective anomalous couplings $F_2(q^2, p^2,k^2, m_f)$ and $F_3(q^2, p^2,k^2, m_f)$ inevitably appears.
(See Appendix). Figure \ref{fig.9} shows $Re F_1$, $Re F_2$ and $Re F_3$ at $\sqrt{q^2}<2m_f$, in the case of same $m_f$, $m_B$, $m_H $ as in Figure \ref{fig.95}. $Re F_2$ and $Re F_3$ are much smaller than $Re F_1$. The imaginary parts $Im F_1$, $Im F_2$ and $Im F_3$ are zero in $\sqrt{q^2}< 2m_f$.
\subsection{Self coupling}
The effective self-coupling of the Higgs-like mode is created by the one-loop process illustrated in Figure \ref{fig.85}(b). In coordinate space, this coupling is expressed by another three-point correlation $\widetilde{V}_h$
\begin{equation}
\widetilde{V}_h= \langle \widetilde{0}| T\left [ \int \widetilde{t} (x_1)
\int \widetilde{t} (x_2) \int t (x_3) d^3x_1 d^3x_1d^3x_3\right] |\widetilde{0}\rangle,
\label{eq:36314}
\end{equation}
where $\widetilde{t}(x)= \bar{\varphi}(x) \varphi(x)$. The anomalous momentum-dependent self-coupling is obtained by the following amplitude
\begin{eqnarray}
\lefteqn{ i {\cal M} [H_{\beta}(p+k)\rightarrow H_{\beta}(p) H_{\beta}(k)] } \nonumber \\
&& \propto (-i\frac{m_f}{M})^3
\int\frac{d^4l}{(2\pi)^4} tr\left[\frac{i}{\slash {\!\!\!}{l}-\slash {\!\!\!}{p}-m_f} \frac{i}{\slash {\!\!\!}{l}-\slash {\!\!\!}{q}-m_f}\frac{i}{\slash {\!\!\!}{l}-m_f}\right]
\nonumber \\
&& + (p \leftrightarrow k) .
\label{eq:58}
\end{eqnarray}
\begin{figure}
\begin{center}
\includegraphics [scale=0.65]{vac-30.eps}
\caption{The renormalized anomalous effective self-coupling $ F^{ren}_{H}(q^2, m_H^2,m_H^2, m_f)$ of Higgs-like mode, for the decay of off-shell such a mode into two on-shell such modes, in the case of same $m_H$ and $m_f $ as in Figure \ref{fig.95}. The condition of $q^2=m_H^2$ corresponds to $\sqrt{q^2}/m_f=0.76$.}
\label{fig.10}
\end{center}
\end{figure}
Hence, the effective self-coupling of the Higgs-like mode has a form such as
\begin{equation}
(\frac{m_f}{M})^3 [ \widetilde{V}_h+ m_f F_{H}(q^2, p^2, k^2, m_f) ] H_{\beta}(p) H_{\beta}(k) H_{\beta}(p+k) ,
\label{eq:93}
\end{equation}
which corresponds to $ \lambda v_h$ in Eq.(\ref{eq:90}). This $ (m_f/M)^3 \widetilde{V}_h$ no longer plays the role of stabilizing the symmetry-broken vacuum as in the Higgs potential in $L_h(x)$.
Following the standard procedure, we calculate the anomalous self-coupling, and find that the divergence $1/\epsilon$ coming from the triangular-loop integrals does not cancel each other in Eq.(\ref{eq:58}). We set a condition that $F_{H}(q^2, p^2,k^2, m_f)$ is zero at the on-shell level, and has a finite value only at the off-shell level $q^2>m_H^2$. Hence, $F_{H}(q^2, p^2, k^2, m_f)$ must vanish at $q^2=m_H^2$. (For the divergence that is independent of momentum, it is to be renormalized to $\widetilde{V}_h$.) We define the renormalized one as $F^{ren}_{H}(q^2, p^2,k^2, m_f) = F_{H}(q^2, p^2,k^2, m_f)- F_{H}(m_H^2, p^2,k^2, m_f)$, and obtain
\begin{eqnarray}
\lefteqn{ F^{ren}_{H}(q^2, p^2,k^2, m_f)
= \frac{8 }{(4\pi)^2} \left[ f \left(\frac{q}{2m_f}\right)- f \left(\frac{m_H}{2m_f}\right) \right] }
\nonumber \\
&&+ \frac{2 }{(4\pi)^2} (8m_f^2 - 2p^2 - q^2) C_0(p^2, p^2, q^2, m_f^2, m_f^2, m_f^2)
\nonumber \\
&&- \frac{2 }{(4\pi)^2} (8m_f^2 - 2p^2 - m_H^2) C_0(p^2, p^2, m_H^2, m_f^2, m_f^2, m_f^2)
\nonumber \\
&& +(p \leftrightarrow k) .
\label{eq:61}
\end{eqnarray}
For the decay $H_{\beta}(p+k)\rightarrow H_{\beta}(p) H_{\beta}(k)$ in Figure \ref{fig.85}(b), we numerically calculate $ F^{ren}_{H}(q^2, m_H^2,m_H^2, m_f)$. Figure \ref{fig.10} shows the result under the same $m_H$ and $m_f $ as in Figure \ref{fig.95}. This effective coupling gradually increases from zero with increasing $\sqrt{q^2}$ to $2m_f$. The overall $q^2$-dependence of $F^{ren}_{H}(q^2, m_H^2,m_H^2, m_f)$ is quite similar to $F_1(q^2, m_B^2,m_B^2, m_f)$, except for its absolute value.
\section{Pair annihilation of fermion and antifermion to gauge boson pair}
As an example of the reaction including the production and decay of the Higgs-like mode, we consider a $s$-channel process: fermion $\psi(p_1)$ + antifermion $\bar{\psi}(k_1)$ $\rightarrow H_{\beta}(q)$ $ \rightarrow B_{\mu} (p)$ +$ B_{\nu}(k)$ illustrated in Figure.\ref{fig.104}. This reaction occurs together with the background reactions, such as the tree process through the $t$- and $u$-channel exchanges of fermion. The cross section of such a tree process is a slowly and monotonously decreasing function of the center-of-mass energy $E_{cm}$. In the total cross section, the effective coupling of the Higgs-like mode will appear above this almost constant background cross section. In view of Figure \ref{fig.95} and \ref{fig.9}, we use only $F_1(q^2,m_B^2,m_B^2,m_f)$ as the first approximation of the total effective coupling. We obtain the amplitude of the process in Figure.\ref{fig.104}
\begin{eqnarray}
\lefteqn{ i {\cal M}= \frac{m_f}{M} \bar{V}(k_1)U(p_1)
g^2\frac{m_f}{M}}
\nonumber \\
&& \times \frac{ \displaystyle{ \left[V_h+m_f[ReF_1(q^2)+iImF_1(q^2)]\right] }}{q^2 - m_H^2\pm i \displaystyle{ \frac{1}{24\pi} \frac{m_f}{M} \sqrt {1-\frac{4m_f^2}{q^2} } (q^2-4m_f^2) }}
\epsilon^{\mu} (p)\epsilon_{\mu} (k) ,
\nonumber \\
\label{eq:611}
\end{eqnarray}
where $ \epsilon^{\mu} (p)$ is one of polarization vectors of the massive gauge boson, satisfying $ \epsilon^{\mu} (p)\epsilon_{\mu} (p)=-1$ for longitudinal and transverse polarizations.
\begin{figure}
\begin{center}
\includegraphics [scale=0.5]{fig46.eps}
\caption{ Fermion-antifermion pair annihilation through the intermediate Higgs-like mode to gauge boson pair. }
\label{fig.104}
\end{center}
\end{figure}
Since the incident fermion and antifermion are lighter than the particles in the triangle loop, the former are assumed to be massless. Using $ \sum_{s} U(p_1)\bar{U}(p_1) = \slash {\!\!\!}{p_1}$ and $\sum_{s} V(k_1)\bar{V}(k_1) = \slash {\!\!\!}{k_1}$, we obtain the squared amplitude
\begin{eqnarray}
\lefteqn{\frac{1}{4} \sum_{s,s'} | {\cal M} |^2 = g^4 \left(\frac{m_f}{M}\right)^4 12 (p_1\cdot k_1) }
\nonumber \\
&& \times \frac{\displaystyle{ [V_h+ m_f ReF_1(q)]^2+ m_f^2 [ImF_1 (q)]^2}}
{(q^2 - m_H^2)^2 + \displaystyle{ \left(\frac{1}{24\pi} \frac{m_f}{M}\right)^2 \left(1-\frac{4m_f^2}{q^2} \right) (q^2-4m_f^2)^2 }} .
\nonumber \\
\label{eq:615}
\end{eqnarray}
\begin{figure}
\begin{center}
\includegraphics [scale=0.65]{vac-4230.eps}
\caption{The rate of total cross section $\sigma(E_{cm})$ to $\sigma(2m_f)$ in the fermion-antifermion annihilation to gauge boson pair through the intermediate Higgs-like collective mode, in the case of $m_B=80$ GeV, $m_f=160$ GeV, $m_H=123$ GeV and $V_h/m_f= 0.02$. }
\label{fig.105}
\end{center}
\end{figure}
In the center-of-mass frame where $p_1=(E_{cm}/2, \mbox{\boldmath $p$}_1 )$ and $k_1=(E_{cm}/2, -\mbox{\boldmath $p$}_1 )$, we can use $q^2=(p_1+k_1)^2=E_{cm}^2$ and
$p_1\cdot k_1=E_{cm}^2/2$. The total cross section $\sigma (E_{cm})$ is given by
\begin{equation}
\sigma (E_{cm})= \frac{\pi}{E_{cm}^2}\sqrt{1- \frac{4m_B^2}{E_{cm}^2}} \sum_{s,s'} | {\cal M} |^2 .
\label{eq:616}
\end{equation}
Figure \ref{fig.105} shows the rate of total cross section $\sigma (E_{cm})$ to $\sigma (2m_f)$ in the case of $V_h/m_f= 0.02$, and same $m_f$, $m_B$, $m_H $ as in Figure \ref{fig.95}. The steep rise of $\sigma$ occurs at $E_{cm}=2m_B$. In contrast to the Higgs model, $\sigma$ gradually increases below $E_{cm}= 2.3m_f$, and later decreases at $E_{cm}> 2.3m_f$, which are peculiar feature of the Higgs-like excitation mode. The smaller the rate $V_h/m_f$ is in Eq.(\ref{eq:615}), this gradual increase and decrease of $ \sigma (E_{cm})$ is is more clearly observed around $E_{cm}=2.3m_f$.
(If $m_H>2m_B$, a sharp resonance peak representing a pole at $q^2=m_H^2$ also appears.) This $E_{cm}$-dependence mainly comes from $ReF_1$ and $ImF_1$. It is not affected by the damping of the Higgs-like mode $H_{\beta}$, because the effect of damping is weakened by the small factor $(1/24\pi)^2$ in Eq.(\ref{eq:615}).
{\it The gradual increase and decrease of the cross section around $E_{cm}=2.3 m_f$ is a key feature for the experimental confirmation of the physical vacuum}, which is absent in $L_h(x)$.
\section{Renormalizability}
In the Higgs model, symmetry breaking is caused simply by changing the sign of the coefficient $\mu ^2$ in the Higgs potential $-\mu^2 |v_h+h|^2+ \lambda |v_h+h|^4$.
Renormalizability of the Higgs model is systematically proved using the generating functional with this Higgs potential \cite{abe}.
(1) In the symmetric vacuum, this generating functional is expanded in powers of the Higgs field $h(x)$. In the symmetry-broken vacuum, it is expanded in powers of $h(x)-v_h$. The algebraic relationship between these two types of expansion can be obtained, which assures that the renormalizability of the theory in the symmetric vacuum can be transferred to that in the symmetry-broken one. (2) When the Bogoliubov-Parasuik-Hepp-Zimmermann (BPHZ) method is applied to the Ward-Takahashi identities derived from this generating functional, renormalizability is systematically proved without reference to the symmetric model \cite{lee}.
In the present model using $L_3(x)$ in Eq.(\ref{eq:50}), however, the Higgs potential is not assumed. Hence, we can not immediately apply this systematic method. Rather, we go back to the inductive proof of renormalizability originated by Dyson in QED \cite{dys}. Let us consider the following Green's functions: $S(p)$ for the massive fermion $\psi$,
\begin{equation}
[ S(p) ] ^{-1} = - \gamma \cdot p +m_f - \Sigma ^* (p) ,
\label{eq:62}
\end{equation}
$D(p^2)$ for the massive gauge boson $B_{\mu}$ in Eq.(\ref{eq:384}),
\begin{equation}
[ D (p^2) ] ^{-1} = -p^2 + m_B^2 - \Pi ^*(p^2) ,
\label{eq:63}
\end{equation}
and $G(p^2)$ for the Higgs-like mode $H_{\beta}$
\begin{equation}
[ G (p^2) ] ^{-1} = -p^2 +m_H^2 - \Pi_H ^*(p^2) .
\label{eq:64}
\end{equation}
\begin{figure}
\begin{center}
\includegraphics [scale=0.34]{vfig5_edited.eps}
\caption{(a) Self energy $\Sigma (p)$ of the fermion $\psi$, (b) Vacuum polarization $\Pi_{\mu\nu}(q)$ of the gauge field $B_{\mu}$, (c) Vacuum polarization
$\Pi_H(q)$ of the Higgs-like mode $H_{\beta}$. Wavy and dotted lines represent $B_{\mu}$ and $H_{\beta}$, respectively. A white triangle denotes a vertex $\Gamma _{\mu}$ between $B_{\mu}$ and $\psi$, and a black triangle denotes $\widehat{\Gamma}_H$ between $H_{\beta}$ and $\psi$. }
\label{fig.5}
\end{center}
\end{figure}
The self energy $ \Sigma ^* (p)$ of fermions satisfies the following Dyson equations illustrated in Figure \ref{fig.5},
\begin{eqnarray}
i \Sigma ^* (p)&= & \frac{g^2}{(2\pi)^4} \int d^4k \gamma _{\nu}S(p-k) \Gamma _{\mu}(p-k,p) D^{\mu\nu}(k)
\nonumber \\
&+& \frac{1}{(2\pi)^4} \left( \frac{m_f}{M} \right )^2 \int d^4k S(p-k) \widehat{\Gamma}_H(p-k,p) G(k)
\nonumber \\
&\equiv & i\Sigma _1(p) + i\Sigma _2 (p)
\label{eq:65}
\end{eqnarray}
where the vertex $\Gamma _{\mu}$ illustrated by a white triangle is the vertex function between $B_{\mu}$ and $\psi$, and another vertex $\widehat{\Gamma} _H$ illusrated by a black triangle is the vertex function between $H_{\beta}$ and $\psi$. Correspondingly, the self energy $\Sigma (p)$ of fermion is composed of $\Sigma _1 (p)$ due to the coupling to $B_{\mu}$, and $\Sigma _2 (p)$ due to the coupling to $H_{\beta}$ \cite{gold}.
Similarly, the vacuum polarization $ \Pi_{\mu\nu} ^*(p^2)$ of the massive gauge boson satisfies
\begin{equation}
i \Pi_{\mu\nu} ^*(p^2)= \frac{g^2}{(2\pi)^4} \int d^4k tr[S(p+k) \gamma _{\mu}S(k) \Gamma _{\nu}(k,p+k)] ,
\label{eq:66}
\end{equation}
and the vacuum polarization $ \Pi_{H} ^*(p^2)$ of the Higgs-like mode satisfies
\begin{equation}
i \Pi_{H} ^*(p^2)= \frac{g^2}{(2\pi)^4} \int d^4k tr[S(p+k) S(k) \widehat{\Gamma} _{H}(k,p+k)] .
\label{eq:67}
\end{equation}
\subsection{The vertex $\Gamma_{\mu} (p,p')$ between $B_{\mu}$ and $\psi$ }
For the vertex $\Gamma_{\mu} (p,p')= \gamma_{\mu} + \Lambda _{\mu}^* (p,p')$ between $B_{\mu}$ and $\psi$ in Eq.(\ref{eq:65}), its proper vertex $ \Lambda _{\mu} ^* (p,p')$ satisfies the following Dyson equation as illustrated in Figure \ref{fig.6}(a)
\begin{eqnarray}
\lefteqn{ \Lambda _{\mu}^* (p,p') }
\nonumber \\
&&= \frac{g^2}{(2\pi)^4} \int d^4k \Gamma _{\nu }(p+k, k) S(p+k) D^{\nu\rho}(k)
\nonumber \\
&& \quad \times \Gamma _{\rho}(k, k+p') S(-p'+k)
\Gamma _{\mu}(p+k, -p'+k)
\nonumber \\
&&+ \frac{1}{(2\pi)^4} \left( \frac{m_f}{M} \right )^2 \int d^4k \widehat{\Gamma }_{H }(p+k, k) S(p+k) G(k)
\nonumber \\
&& \quad \times \widehat{\Gamma} _{H} (k, k+p') S(-p'+k)
\Gamma _{\mu}(p+k, -p'+k)
\nonumber \\
&&+ \cdots .
\label{eq:68}
\end{eqnarray}
\begin{figure}
\begin{center}
\includegraphics [scale=0.4]{fig66_edited.eps}
\caption{(a) Proper vertex $\Lambda _{\mu}$ between the fermion $\psi$ and the gauge field $B_{\mu}$, (b) Proper vertex $\Lambda _{H}$ between the fermion $\psi$ and the Higgs-like mode $H_{\beta}$.}
\label{fig.6}
\end{center}
\end{figure}
This $ \Lambda _{\mu}^* (p,p')$ is related to the fermion self-energy $\Sigma _1^* (p)$ (due to the coupling to $B_{\mu}$) as \cite{war}
\begin{equation}
\frac{\partial}{\partial p_{\mu}} \Sigma _1^* (p) = \Lambda _{\mu} (p,p) .
\label{eq:69}
\end{equation}
Similarly, if we consider a proper vertex $\Lambda _H(p,p')$ for $\widehat{\Gamma} _H(p,p') = 1+\Lambda _H(p,p')$ between $H_{\beta}$ and $\psi $ in Eq.(\ref{eq:65}), it satisfies the Dyson equation illustrated in Figure \ref{fig.6}(b). The other fermion self-energy $\Sigma _2(p)$ (due to the coupling to $H_{\beta}$) is related to this $\Lambda _H$ as
\begin{equation}
\frac{\partial}{\partial p} \Sigma _2 ^* (p) = \Lambda_{H} (p,p) .
\label{eq:825}
\end{equation}
With these $ \Lambda _{\mu} (p,p)$ and $ \Lambda _{H} (p,p)$, we obtain the Green function of fermions
\begin{equation}
[ S(p) ] ^{-1} = - \gamma \cdot p +m_f - \int_{p_0}^{p} dk^{\mu} \Lambda _{\mu}(k,k) - \int_{p_0}^{p} dk \Lambda _{H}(k,k) .
\label{eq:70}
\end{equation}
We will introduce the following renormalization-constants $Z_i$ ($i=1\sim 5$ )
\begin{equation}
S(p) = Z_2 S^{ren} (p) ,
\label{eq:71}
\end{equation}
\begin{equation}
D_{\mu\nu}(p^2) = Z_3 D_{\mu\nu}^{ren} (p^2) ,
\label{eq:72}
\end{equation}
\begin{equation}
G(p^2) = Z_5 G^{ren} (p^2) ,
\label{eq:725}
\end{equation}
\begin{equation}
\Gamma _{\mu}(p,p') = Z_1^{-1} \Gamma_{\mu} ^{ren} (p,p') ,
\label{eq:73}
\end{equation}
\begin{equation}
\widehat{ \Gamma} _{H}(p,p') = Z_4^{-1} \widehat{\Gamma}_{H} ^{ren} (p,p') .
\label{eq:735}
\end{equation}
Let us estimate the divergence appearing successively in the perturbation expansion in Eq.(\ref{eq:68}).
(1) In the expansion of $\Lambda _{\mu}(p,p')$, we focus on $\Lambda _{\mu}$ of the order of $g^{2j}$ obtained by $j$ times of iteration. These $\Lambda _{\mu}$ contain
$2j$ $S(p)$, $j$ $D(p^2)$, and $2j$ $\Gamma _{\mu} (p,p')$, in addition to the original $\Gamma _{\mu}$.
When $S(p)$, $D(p^2)$, $\Gamma_{\mu}$ and $\widehat{\Gamma} _H$ are replaced by their counterparts in Eqs.(\ref{eq:71}) $\sim$ (\ref{eq:735}), this $\Lambda _{\mu} (p,p')$ is renormalized as
\begin{equation}
\Lambda _{\mu} (p,p') = (Z_1^{-1} Z_2 Z_3^{1/2})^{2j} Z_1^{-1} \Lambda _{\mu}^{ren} (p,p') ,
\label{eq:736}
\end{equation}
(2) Next, we focus on each $\Lambda _{\mu}$ of the order of $(m_f/M)^{2j}$. In addition to the original $\Gamma _{\mu}$, they contain $2j$ $S(p)$, $j$ $G(p^2)$, and $2j$ $\widehat{\Gamma} _{H} (p,p')$. Hence, this $\Lambda _{\mu} (p,p')$ is renormalized as
\begin{equation}
\Lambda _{\mu} (p,p') = (Z_4^{-1} Z_2 Z_5^{1/2})^{2j} Z_1^{-1} \Lambda _{\mu}^{ren} (p,p') ,
\label{eq:737}
\end{equation}
If $g$ is replaced by
\begin{equation}
g^{ren} = Z_1^{-1} Z_2 Z_3^{1/2} g ,
\label{eq:74}
\end{equation}
and $M$ by
\begin{equation}
M_{ren} ^{-1}= Z_4^{-1} Z_2 Z_5^{1/2} M ^{-1 } ,
\label{eq:75}
\end{equation}
the total proper vertex is renormalized as
\begin{equation}
\Lambda _{\mu} (p,p') = Z_1^{-1} \Lambda _{\mu}^{ren} (p,p', g^{ren}, M_{ren}) ,
\label{eq:76}
\end{equation}
For the higher-order terms with various combination of $g^2$ and $(m_f/M)^2$, similar renormalization is possible.
\subsection{The vertex $\widehat{\Gamma} _H(p,p')$ between $H_{\beta}$ and $\psi $}
For the vertex $\widehat{\Gamma} _H(p,p') = 1+\Lambda _H(p,p')$ between $H_{\beta}$ and $\psi $ in Figure \ref{fig.6}(b), the proper vertex $\Lambda _H(p,p')$ has a similar structure to $\Lambda _{\mu}(p,p')$ if $\Gamma _{\mu}$ for the external $B_{\mu}$ is replaced by $\widehat{\Gamma} _H$ for the external $H_{\beta}$. Hence, using Eqs.(\ref{eq:74}) and (\ref{eq:75}),
$\Lambda _{H} (p,p')$ is renormalized as
\begin{equation}
\Lambda _{H} (p,p') = Z_4^{-1} \Lambda _{H}^{ren} (p,p', g^{ren}, M_{ren}) ,
\label{eq:82}
\end{equation}
and $Z_1=Z_4$. Since Eqs.(\ref{eq:69}) and (\ref{eq:825}) also hold for the renormalized quantities, $Z_2=Z_1=Z_4$ is obtained. Using Eqs.(\ref{eq:76}) and Eq.(\ref{eq:82}) in (\ref{eq:70}), we obtain the total renormalized fermion self-energy $\Sigma^{ren} (p)=\Sigma^* _1 (p)+\Sigma^* _2 (p)$, and the renormalized $S^{ren}(p)$.
\subsection{The vacuum polarization $\Pi(p)$ of the massive gauge field $B_{\mu}$}
For the vacuum polarization $\Pi^*(p)$ of $B_{\mu}$ in Eq.(\ref{eq:63}), the proper vertex $\Delta _{\mu} (p,p')$ satisfying
\begin{equation}
\frac{\partial}{\partial p_{\mu}} \Pi ^* (p) = \Delta _{\mu} (p,p) ,
\label{eq:77}
\end{equation}
is defined. This $\Delta _{\mu} (p,p)$ is useful, because the Green's function $D (p^2) $ of $B_{\mu}$ is expressed as
\begin{equation}
[ D (p^2) ] ^{-1} = -p^2 + m_B^2 - \int _{p'_0}^{p} dk^{\mu} \Delta _{\mu} (k, k) .
\label{eq:78}
\end{equation}
\begin{figure}
\begin{center}
\includegraphics [scale=0.4]{fig62_edited.eps}
\caption{Proper vertex $\Delta _{\mu}$ between the gauge fields $B_{\mu}$.}
\label{fig.7}
\end{center}
\end{figure}
This $ \Delta _{\mu} (p, p')$ satisfies the following Dyson equation illustrated in Figure \ref{fig.7}
\begin{eqnarray}
\lefteqn{ \Delta _{\mu} ^* (p,p')= \frac{g^2}{(2\pi)^4} \int d^4k
\Gamma _{\nu }(p+k, k) S(p+k) S(k)}
\nonumber \\
&& \mbox{ } \times \Gamma _{\nu}(k, k+p') S(-p'+k) \Gamma _{\mu}(p+k, -p'+k)
\nonumber \\
&&+ \frac{g^4}{(2\pi)^8} \int d^4k d^4k' \Gamma _{\nu }(p+k, k) S(p+k) \Gamma _{\sigma}(p+k,p'+k)
\nonumber \\
&& \mbox{ } \times S(p+k') S(k) \Gamma _{\tau} (k, k') D^{\sigma\tau} (k- k') S(k')
\nonumber \\
&& \mbox{ } \times \Gamma _{\nu}(k',-p'+k') S(k'-p') \Gamma _{\mu}(p+k',-p'+k')
\nonumber \\
&&+ \frac{1}{(2\pi)^8} g^2 \left( \frac{m_f}{M} \right )^2 \int d^4k d^4k' \Gamma _{\nu }(p+k, k) S(p+k)
\nonumber \\
&& \mbox{ } \times \widehat{\Gamma} _{H}(p+k,p'+k) S(p+k')
\nonumber \\
&& \mbox{ } \times S(k) \widehat{\Gamma} _{H} (k, k') G (k- k') S(k') \Gamma ^{\nu}(k',-p'+k')
\nonumber \\
&& \mbox{ } \times S(k'-p') \Gamma _{\mu}(p+k',-p'+k')
\nonumber \\
&&+ \cdots .
\label{eq:79}
\end{eqnarray}
Only few terms are written in the above expansion, but it extends to the higher-order terms of $g^2$ and $(m_f/M)^2$.
(1) Let us consider $ \Delta _{\mu}$ of the order of $g^{2j}$ in the above expansion. They contain $(2j+1)$ $S(p)$, $(j-1)$ $D_{\mu\nu}(p^2)$, and $(2j+1)$ $\Gamma _{\mu} (p,p')$ \cite{re1}
\begin{eqnarray}
\Delta _{\mu} (p,p')&=& Z_2^{2j+1} Z_3^{j-1} Z_1^{-(2j+1)} \Delta^{ren} _{\mu} (p,p')
\nonumber \\
&=& Z_2^{2j+1} Z_3^{j-1} Z_1^{-2j} Z_2^{-1} \Delta^{ren} _{\mu} (p,p')
\nonumber \\
&=& (Z_1^{-1} Z_2 Z_3^{1/2} )^{2j} Z_3^{-1} \Delta^{ren} _{\mu} (p,p') .
\nonumber \\
\label{eq:795}
\end{eqnarray}
Hence, if we replace $g$ by $g^{ren}$ defined in Eq.(\ref{eq:74}), each $\Delta _{\mu}$ with $g^{2j}$ is renormalized.
(2) Similarly, for $ \Delta _{\mu}$ with the coefficient $g^{i}(m_f/M)^j$, it contains $(i+j+1)$ $S(p)$, $(i/2-1)D_{\mu\nu}(p)$, $j/2$ $G(p^2)$, $(i+1)$ $\Gamma _{\mu} (p,p')$, and $j$ $\widehat{\Gamma} _{H} (p,p')$ \cite{re2}
\begin{eqnarray}
\Delta _{\mu} (p,p') &=& Z_2^{i+j+1} Z_3^{i/2-1} Z_5^{j/2} Z_1^{-(i+1)} Z_4^{-j} \Delta^{ren} _{\mu} (p,p')
\nonumber \\
&=& (Z_1^{-1} Z_2Z_3^{1/2} )^i (Z_4^{-1} Z_2 Z_5^{1/2} )^jZ_3^{-1} \Delta^{ren} _{\mu} (p,p') .
\nonumber \\
\label{eq:80}
\end{eqnarray}
Hence, if we replace $g$ by $g^{ren}$ in Eq.(\ref{eq:74}), and $M$ by $M_{ren}$ in Eq.(\ref{eq:75}), each $\Delta _{\mu}$ with $g^i(m_f/M)^j$ is renormalized.
(3) As a result, the total proper vertex in Figure.\ref{fig.7} is renormalized as
\begin{equation}
\Delta _{\mu} (p,p') = Z_3^{-1} \Delta _{\mu}^{ren} (p,p', g^{ren}, M_{ren}) .
\label{eq:81}
\end{equation}
Using Eq.(\ref{eq:81}) in Eq.(\ref{eq:78}), we obtain the renormalized $D^{ren}(p^2)$.
For the higher terms, similar renormalization is possible.
\begin{figure}
\begin{center}
\includegraphics [scale=0.4]{fig81_edited.eps}
\caption{Proper vertex $\Delta _{H}$ between the Higgs-like modes $H_{\beta}$.}
\label{fig.8}
\end{center}
\end{figure}
\subsection{The vacuum polarization $\Pi _H(p)$ of the Higgs-like mode $H_{\beta}$}
For the vacuum polarization $\Pi _H(p)$ of the Higgs-like mode $H_{\beta}$ in Eq.(\ref{eq:64}), the proper vertex $\Delta _{H} (p,p')$ satisfying
\begin{equation}
\frac{\partial}{\partial p} \Pi ^* _H(p) = \Delta _{H} (p,p) ,
\label{eq:83}
\end{equation}
is defined. This $\Delta _{H} (p,p)$ is useful, because the Green's function $ G (p^2) $ of the Higgs-like mode $H_{\beta}$ is expressed as
\begin{equation}
[ G (p^2) ] ^{-1} = -p^2 + m_H^2 - \int _{p''_0}^{p} dk \Delta _{H} (k, k) .
\label{eq:835}
\end{equation}
The proper vertex $\Delta_H(p,p')$ satisfies the Dyson equation shown in Figure \ref{fig.8}. If $\widehat{\Gamma} _H$ (black triangle) is replaced by $\Gamma _{\mu}$ (white triangle), and $\Gamma _{\mu}$ by $\widehat{\Gamma} _H$, it shows a similar structure to Figure \ref{fig.7} for $\Delta _{\mu}$. Hence, in analogy with Eq.(\ref{eq:81}), the proper vertex is renormalized as
\begin{equation}
\Delta _{H} (p,p') = Z_5^{-1} \Delta _{H}^{ren} (p,p', g^{ren}, M_{ren}) .
\label{eq:84}
\end{equation}
where $Z_5=Z_3$. Using Eq.(\ref{eq:84}) in Eq.(\ref{eq:835}), we obtain the renormalized $G^{ren}(p^2)$.
\subsection{The renormalized masses of $\psi$, $B_{\mu}$ and $H_{\beta}$}
The renormalized mass of the fermion $\psi$ is a solution of $[S(p)]^{-1}=0$, in which $\Lambda ^{ren}_{\mu}(p,p', g^{ren}, M^{ren})$ and $\Lambda ^{ren}_{H}(p,p', g^{ren}, M^{ren})$ are used in Eq.(\ref{eq:70}),
Similarly, the renormalized masses of $B_{\mu}$ and $H_{\beta}$ are solutions of $[D(p^2)]^{-1}=0$ in Eq.(\ref{eq:78}), and $[ G (p^2) ] ^{-1} =0$ in Eq.(\ref{eq:835}), respectively. (When such masses are calculated, $\Delta ^{ren}_{\mu}(p,p', g^{ren}, M^{ren})$ and $\Delta ^{ren}_{H}(p,p', g^{ren}, M^{ren})$ must be used, respectively.)
In view of Eqs.(\ref{eq:62}) $\sim$ (\ref{eq:64}), we obtain the first approximation of such renormalized masses as follows
\begin{equation}
m_f^{ren}= m_{f} - \Sigma ^{ren}(m_f) ,
\label{eq:85}
\end{equation}
\begin{equation}
(m_B^{ren })^2= m^2_{B} - \Pi ^{ren}(m_B^2) ,
\label{eq:86}
\end{equation}
\begin{equation}
(m_H^{ren })^2= m^2_{H} - \Pi _H^{ren}(m_H^2) ,
\label{eq:87}
\end{equation}
in which $\Sigma ^{ren}$, $\Pi ^{ren}$ and $ \Pi _H^{ren}$ are self energy and vacuum polarizations in the right-hand side of Eqs.(\ref{eq:70}), (\ref{eq:78}) and (\ref{eq:835}) using the renormalized quantities.
Since the Higgs Lagrangian density $L_h(x)$ does not exist in our model, all renormalization constants $Z_i$ ($i=$ 1 $\sim$ 5) are determined so as to absorb only the logarithmic divergence, not the quadratic one.
\section{Discussion}
\subsection{Implications for the Higgs Lagrangian}
The power of the Higgs Lagrangian density $L_h(x)$ in providing experimental predictions comes from its simple structure. Many quantities are derived from a single quantity $v_h$, such as the gauge boson's mass in $m_B= g v_h$, fermion's mass in $(m_f/v_h)\bar{\varphi} \varphi h$, and the coupling of Higgs boson to gauge boson pair in $gv_h g^{\mu\nu}B_{\mu}B_{\nu} h$. If $L_h(x)$ is replaced by a more microscopic description, these $v_h$'s appearing in the different quantities may not have the same value. The precise measurement of the properties of the Higgs-like particle on this point will have a crucial importance for future development.
In the Higgs model, there are three parameters: two coefficients $\mu$ and $\lambda$ in the Higgs potential $-\mu^2|h|^2+\lambda |h|^4$, and the fermion masses $m_f$. In the present model, there are six parameters: four type of condensed energies in the physical vacuum, (a) the two-point correlation in the physical vacuum leading to $m_B$ in Eq.(\ref{eq:36}), (b) the field energy of massless gauge boson, leading to $m_f$ in Eq.(\ref{eq:245}), (c) the three-point correlations in the physical vacuum leading to $V_h$ and $\widetilde{V}_h$ in Eqs.(\ref{eq:363}) and (\ref{eq:36314}). In addition to them, (d) the upper end of energy-momentum $\Lambda$ of fermions involved in the excitation in Eq.(\ref{eq:46}), and (e) $M$ in the coupling of the Higgs-like mode to fermions $m_f/M$ in Eq.(\ref{eq:43}).
When the precise measurement of the Higgs-like particle is performed, the above degree of freedom will turn out to be important.
\subsection{Extension to the electroweak interaction}
When we apply the present model to the electroweak interaction, it is appropriate to begin with the third generation, in which fermions with large masses, such as the top and bottom quarks (plus $\tau$ lepton and $\tau$ neutrino), are included. Such a Lagrangian density without the Higgs field is given by
\begin{eqnarray}
L_0(x) &=& -\frac{1}{4} B^{\mu\nu}B_{\mu\nu} -\frac{1}{4} W^{a\mu\nu}W^a_{\mu\nu}
\nonumber \\
&+& \bar{q}(i\partial_{\mu} +g\tau_aW_{\mu}^a +g'Y_QB_{\mu}) \gamma ^{\mu}q
\nonumber \\
& +& \bar{l}(i\partial_{\mu} +g\tau_aW_{\mu}^a +g'Y_LB_{\mu}) \gamma ^{\mu}l
\nonumber \\
&+& \bar{r} \left( i\partial_{\mu} +g'
\left [ \begin{array}{cc} Y_t& 0 \\ 0&Y_b \end{array} \right ] B_{\mu} \right) \gamma ^{\mu}r
\nonumber \\
& +& \bar{l}_r \left( i\partial_{\mu} +g'
\left [ \begin{array}{cc} 0 & 0 \\ 0&Y_ {\tau} \end{array} \right ] B_{\mu} \right) \gamma ^{\mu}l_r ,
\label{eq:622}
\end{eqnarray}
where
\begin{eqnarray}
B^{\mu\nu} &=& \partial_{\mu}B^{\nu}-\partial_{\nu}B^{\mu},
\nonumber \\
W_{\mu\nu}^{a} &=& \partial_{\mu}W^a_{\nu}-\partial_{\nu}W^a_{\mu} - gf^{abc}W^b_{\mu}W^c_{\nu} ,
\label{eq:633}
\end{eqnarray}
and massless fermions (top and bottom quarks, $\tau$ lepton and $\tau$ neutrino) make up left-handed doublets $q$ and $l$, and right-handed doublets $r$ and $l_r$,
\begin{equation}
q= \left( \begin{array}{cc} \varphi _ t \\ \varphi _b \end{array} \right)_L,
l= \left( \begin{array}{cc} \varphi _{\nu_{\tau}} \\ \varphi _ {\tau} \end{array} \right)_L,
r= \left( \begin{array}{cc} \varphi _ t \\ \varphi _b \end{array} \right)_R,
l_r= \left( \begin{array}{cc} 0 \\ \varphi _{\tau} \end{array} \right)_R
\nonumber \\
\label{eq:634}
\end{equation}
For the quarks in the third generation, the physical vacuum in Eq.(\ref{eq:16}) is generalized so that it reflects $SU(2) \times U(1)$ symmetry of the top and bottom quarks as
\begin{eqnarray}
\lefteqn{ | \widetilde{0} \rangle= e^{i\tau_3 \alpha _3(x)} }
\nonumber \\
&& \mbox{} \times
\left( \begin{array}{cc} \prod_{p,s} \left( \cos \theta^{t}_{\mbox{\boldmath $p$}}
+ \sin \theta^{t}_{\mbox{\boldmath $p$}} e^{i\alpha_3(x)}b_{t}^{s\dagger}(-\mbox{\boldmath $p$}) a_{t}^{s\dagger}(\mbox{\boldmath $p$}) \right) |0_r\rangle \\
\prod_{p,s} \left( \cos \theta^{b}_{\mbox{\boldmath $p$}}
+ \sin \theta^{b}_{\mbox{\boldmath $p$}} e^{i\alpha_3(x)} b_{b}^{s\dagger}(-\mbox{\boldmath $p$}) a_{b}^{s\dagger}(\mbox{\boldmath $p$}) \right) |0_r\rangle
\end{array} \right)
\nonumber \\
\label{eq:645}
\end{eqnarray}
In this case, the following condensed-energy accumulates in the physical vacuum. The field energy of the massless gauge boson condenses in the vacuum as
\begin{eqnarray}
&g& \langle \widetilde{0}| \int T^{00}_{W,(t)}(x)d^3x |\widetilde{0}\rangle +g' Y_Q \langle \widetilde{0}| \int T^{00}_{B,(t)} (x)d^3x |\widetilde{0}\rangle
\nonumber \\
= &g' &Y_t \langle \widetilde{0}| \int T^{00}_{B,(t)} (x)d^3x |\widetilde{0}\rangle = m_t ,
\label{eq:646}
\end{eqnarray}
then producing the mass of the top quark. Similarly, the kinetic energy of the left-handed and right-handed quarks separately condense in vacuum
\begin{eqnarray}
& g^2& \frac{1}{2} \int d^3x_1 d^3x_2 \langle \widetilde{0} | tr \left[ \bar{q}(x_1) \tau _a \slash {\!\!\!}{ \partial} q (x_1)
\bar{q}(x_2)\tau _a \slash {\!\!\!}{ \partial} q (x_2) \right] |\widetilde{0} \rangle
\nonumber \\
&=& m_W^2 ,
\label{eq:66}
\end{eqnarray}
and
\begin{eqnarray}
\lefteqn{2g'^2 \int d^3x_1d^3x_2 }
\nonumber \\
&& \mbox{} \times \langle \widetilde{0} |
\left [\bar{q}(x_1)Y_Q \slash {\!\!\!}{ \partial } q(x_1) + \bar{r}(x_1) \left( \begin{array}{cc} Y_t& 0 \\ 0&Y_b \end{array} \right)
\slash {\!\!\!}{ \partial } r (x_1)\right ]
\nonumber \\
&& \mbox{} \times \left [\bar{q}(x_2)Y_Q \slash {\!\!\!}{ \partial } q(x_2) + \bar{r}(x_2) \left( \begin{array}{cc} Y_t& 0 \\ 0&Y_b \end{array} \right) \slash {\!\!\!}{\partial} r (x_2)\right ]
|\widetilde{0} \rangle
\nonumber \\
&& =m_B^2 ,
\label{eq:67}
\end{eqnarray}
leading to masses of real gauge bosons.
The Higgs-like mode is defined as a local excitation of all quarks and leptons.
By this generalization, the following possibilities are expected.
(a) In the GWS model, $W_{\mu} ^a$ and $B_{\mu}$ are reorganized to $W_{\mu}^{\pm}, Z_{\mu}$ and $A_{\mu}$. When $L_h(x)$ in Eq.(\ref{eq:01}) is extended to include $W_{\mu}^a$ and $B_{\mu}$, the couplings of $B_{\mu}$ to $W^1_{\mu}$ and $W^2_{\mu}$ inevitably appear. In order to eliminate such a couplings, the vacuum condensate of Higgs particle is phenomenologically assumed to have a structure $\langle 0 | h | 0 \rangle =(0,v_h)$ . The extension of the present model has a possibility of deriving the vanishing of $B_{\mu}W^1_{\mu}$ and $B_{\mu}W^2_{\mu}$, not from $(0,v_h)$, but from the dynamical reason.
(b) When the argument in Section 3 is extended across different generations, a microscopic explanation of CKM matrix is expected. If $a(p)$ and $b(-p)$ in the second term of Eq.(\ref{eq:18}) represent the down and strange quarks respectively, an orthogonal transformation giving rise to the mixing of different generations is introduced for the state coupled to the gauge field, and the Cabibbo angle is defined as an angle of such a transformation.
(c) In the electroweak version of the present model, two types of condensed kinetic energy of massless quarks in Eqs.(\ref{eq:66}) and (\ref{eq:67}) are assumed, and therefore the definition of Weinberg angle $\theta _W$, which determines the mixing of $W^3_{\mu}$ and $B_{\mu}$, will be slightly changed.
(d) Intensely examined by experiments now are the custodial symmetry in the coupling of the observed Higgs-like particle to $A_{\mu}$, $Z_{\mu}$ and $W_{\mu}^a$, and the existence of the anomalous coupling in it \cite{ano}. In the Higgs model, the coupling of the Higgs field $h$ to $A_{\mu}$, $Z_{\mu}$ and $W_{\mu}^a$ comes from the common $v_h$. In the electroweak version of the present model, the effective coupling of the Higgs-like mode to $A_{\mu}$, $Z_{\mu}$ and $W_{\mu}^a$ have a variety of strength and $q^2$-dependence. The anomalous effective coupling such as Eq.(\ref{eq:925}) is worth precise measurements.
(e) The present model predicts the total cross section $\sigma(q^2)$ of the fermion-antifermion annihilation to gauge boson pair, in a different way from the Higgs model. In addition to the rise at $q^2=(2m_B)^2$, the gradual increase and decrease of $\sigma(q^2)$ around $q^2=(2m_f)^2$ is expected.
When the present model is generalized to the electroweak interaction, the Higgs model and the present model will give different predictions in the intermediate- and high-energy processes.
The precise measurements of properties of the recently discovered Higgs-like particle are expected in future experiment. \newline
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,388 |
{"url":"https:\/\/www.di-mgt.com.au\/public-key-crypto-discrete-logs-0.html","text":"DI\u00a0Management Home > Cryptography > Public key cryptography using discrete logarithms\n\n# Public key cryptography using discrete logarithms\n\nThis is an introduction to a series of pages that look at public key cryptography using the properties of discrete logarithms.\n\nWe outline some of the important cryptographic systems that use discrete logarithms; explain the mathematics behind them; and give simple examples, using small numbers to illustrate the mechanics. We assume you understand arithmetic modulo $p$ and maybe a bit of group theory.\n\nAll the systems use the properties of the multiplicative group modulo p, denoted $\\mathbb{Z}_p^*$, for a prime $p$. Their security ultimately depends on the intractability of solving the Discrete Logarithm Problem (DLP): namely, if you are given $g \\in \\mathbb{Z}_p^*$ and $g^n\\pmod{p}$ then find $n$.\n\nThis page looks at the basic Diffie-Hellman key exchange system introduced in 1976. The following pages look at improvements and derivatives.\n\n1. Diffie-Hellman Key Exchange - with a large cyclical subgroup\n2. MQV Key Agreement - an improvement on Diffie-Hellman\n3. ElGamal Encryption\n4. Digital Signature Algorithm (DSA)\n next: Diffie-Hellman with a large cyclical subgroup >>\n\n## Basic Diffie-Hellman\n\nThe Diffie-Hellman Key Exchange method was first published in 1976 [DH76]. It enables two parties independently to compute a shared secret that cannot be obtained by an eavesdropper.\n\nIn its most basic form, two parties agree on and make public a large prime $p$ (the modulus) and a generator $g$ that generates the group $\\mathbb{Z}_p^*$ or a large subgroup of it. Party A (Alice) chooses a private key $a$ in the range $1 \\lt a \\lt p-1$, and computes her public key $A = g^a \\mod p$, which is sent to party B (Bob). Similarly, Bob chooses a private key $b$ in the range $1 \\lt b \\lt p-1$, and computes his public key $B = g^b \\mod p$, which is sent to Alice. Both parties can independently compute a shared secret $Z=g^{ab}\\mod p$, as follows\n\nAlice computes $(B)^a = (g^b)^a = g^{ab} \\mod p = Z$\n\nBob computes $(A)^b = (g^a)^b = g^{ab} \\mod p = Z$\n\nAn eavesdropper only sees $p$, $g$, $g^a$ and $g^b$. Given these values, the Diffie-Hellman Problem (DHP) is to find $g^{ab}\\mod p$. This is considered to be an intractable problem for a large enough prime $p$, say, over 1000 bits long. Obviously if we could solve the Discrete Logarithm Problem (DLP) we could also solve the DHP. It is conjectured that the DHP is at least as hard as the DLP.\n\nThe basic Diffie-Hellman method is secure against a passive eavesdropper, but it is not secure against an active \"man-in-the-middle\" attacker. It is also vulnerable if an attacker can trick users into using parameters $(p,g)$ where $g$ only generates a small subgroup. The attacker can then do a brute force search of this much smaller set.\n\n## Improvements and derivatives\n\nIn the following pages we look at improvements to the basic method.\n\nDiffie-Hellman with a large cyclical subgroup\nTo prevent \"small-group\" attacks we make sure that the order $p-1$ of the group has a large prime divisor $q$ and that $g$ is a generator of the unique cyclical subgroup of order $q\\mod p$. The improved Diffie-Hellman Key Exchange method uses the domain parameters $(p,q,g)$. These parameters are used by all the derivatives we discuss below.\nMQV Key Agreement\nThe original Diffie-Hellman method is still susceptible to a man-in-the-middle attack. MQV key agreement is a protocol that attempts to eliminate this problem.\nElGamal Encryption\nElGamal encryption uses the Diffie-Hellman domain parameters $(p,q,g)$ and private\/public key pair $(b,B) for recipient party B$ to encrypt a message encoded as an integer $m$ in the range $[1,p-2]$. The resulting ciphertext is a pair of integers $(c_1,c_2)$ both about $p$ bits long.\nDigital Signature Algorithm (DSA)\nThe Digital Signature Algorithm also uses the Diffie-Hellman domain parameters $(p,q,g)$ and private\/public key pair $(a,A)$ for a signing party A. It creates a digital signature of a message digest as two integers $(r,s)$ both about $q$ bits long. This is very compact compared to the size of the modulus $p$.\n\n## Notation\n\n1. Strictly speaking, $x = y\\mod n$ means $x$ is equal to the remainder on dividing $y$ by $n$ where $x$ is a unique value in the range $0\\leq x\\lt n$; and $x\\equiv y\\pmod{n}$ means $x$ and $y$ have the same remainder when divided by $n$. In the latter case $x$ has an infinite range of possible values of the form $x=y+kn$. We are usually interested in the unique value of $x$ in the range $0\\leq x\\lt n$ anyway, so we might be a bit sloppy in their use.\n\nThese expressions are related by the equivalence $x\\mod n = y\\mod n \\quad\\Leftrightarrow\\quad x\\equiv y\\pmod{n}$.\n\n2. It is conventional when describing Diffie-Hellman algorithms to use a notation that is completely different from the notation used in any other publication. Extra credit can be obtained by using the same symbols as others use, but swopping the variables they represent :-).\n\n## References\n\n\u2022 [DH76] W. Diffie and M. Hellman (1976). New directions in cryptography, IEEE Transactions on Information Theory 22, 644-654. <pdf-link>","date":"2017-12-17 04:25:47","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\": 0.6197478771209717, \"perplexity\": 504.04740714836805}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-51\/segments\/1512948592972.60\/warc\/CC-MAIN-20171217035328-20171217061328-00294.warc.gz\"}"} | null | null |
\section{Introdu\c{c}\~ao}
\hspace{.5cm} Nos dias atuais a Teoria Qu\^antica de Campos \'e largamente
empregada em diversas \'areas da f\'{\i}sica, tais como, altas energias,
mec\^anica estat\'{\i}stica, mat\'eria condensada, etc. Sendo a Teoria
Qu\^antica de Campos fundamentalmente de aspectos perturbativos, ela sofre
de pesados problemas de diverg\^encias. O tratamento destas diverg\^encias
tem sido um enorme desafio para os f\'{\i}sicos. A natureza matem\'atica do
problema \'e bem conhecida. Diverg\^encias ocorrem nos c\'alculos
perturbativos porque duas distribui\c{c}\~oes n\~ao podem ser multiplicadas
em um mesmo ponto. V\'arios m\'etodos tem sido propostos para solucionar
este problema. Entretanto somente \'e poss\'{\i}vel eliminar estes infinitos
de uma maneira f\'{\i}sica e consistente por absorv\^e-los nos par\^ametros
livres da teoria (massa e constante de acoplamento).
O procedimento usual para sanar o problema das diverg\^encias \'e empregar
um m\'etodo de regulariza\c{c}\~ao (cut-off, dimensional, zeta, etc ),
tornando a teoria finita atrav\'es do uso de um regulador (par\^ametro de
regulariza\c{c}\~ao) a fim de isolar as diverg\^encias e, ent\~ao,
restabelecer a teoria original com a elimina\c{c}\~ao do regulador usando
uma prescri\c{c}\~ao de renormaliza\c{c}\~ao, subtra\c{c}\~ao dos p\'olos ou
adi\c{c}\~ao de contra-termos.
De maneira geral o entendimento do procedimento de renormaliza\c{c}\~ao
empregado fica prejudicado devido a complexidade da Teoria Qu\^antica de
Campos. A fim de contornar esta dificuldade, vamos tratar aqui de dois
problemas simples e bem conhecidos por qualquer aluno de gradua\c{c}\~ao em
f\'{\i}sica e possivelmente dos demais cursos da \'area de Ci\^encias Exatas.
Os problemas aos quais nos referimos \'e o da determina\c{c}\~ao do
potencial escalar el\'etrico e do potencial vetor magn\'etico de um fio
infinito de carga e de corrente, respectivamente. Tais problemas, de um modo
geral parecem amb\'{\i}guos para os alunos, pois escondido neles existe um
procedimento de renormaliza\c{c}\~ao, como apontou Hans em seu artigo [1].
Uma maneira encontrada para se evitar diretamente as diverg\^encias nos
c\'alculos dos potenciais, \'e primeiramente determinar os campos
el\'etricos e magn\'etico e em seguida calcular os potenciais escalar
el\'etrico e vetorial magn\'etico do fio infinito.
O artigo est\'a organizado com segue. Na se\c{c}\~ao-2 tratamos do c\'alculo
do potencial escalar el\'etrico de um fio infinito com densidade linear de
carga $\lambda$ e do potencial vetor magn\'etico de um fio infinito de
corrente constante, que nos conduzir\'a a uma integral divergente. Nas se\c{c%
}\~oes 3, 4 e 5 n\'os regularizamos a integral divergente obtida na se\c{c}%
\~ao anterior usando os m\'etodos, cut-off [3], dimensional [4] e fun\c{c}%
\~ao zeta [5] respectivamente. Na se\c{c}\~ao-6 usando as prescri\c{c}\~oes
de renormaliza\c{c}\~ao, determinamos os potenciais renormalizados,
discutimos o par\^ametro de escala e apresentamos as id\'eias b\'asicas da
Teoria de Renormaliza\c{c}\~ao em Teoria Qu\^antica de Campos.
\section{Potencial Escalar El\'etrico e Potencial Vetor Magn\'etico}
\hspace{.5cm} O potencial escalar el\'etrico $\Phi(\vec{r})$ gerado por um
fio infinito com densidade linear de carga $\lambda$ em um ponto qualquer do
espa\c{c}o exceto no fio \'e dado por [2-3]
\begin{eqnarray}
\Phi(\vec{r})=\frac{\lambda}{4\pi\varepsilon_{0}}\int_{-\infty}^{\infty}%
\frac{dz} {\sqrt{z^{2}+\rho^{2}}},
\end{eqnarray}
onde temos colocado o fio sobre o eixo z e $\rho$ \'e a dist\^ancia do ponto
ao fio, coordenada radial cil\'{\i}ndrica.
O potencial vetor magn\'etico $\vec{A}(\vec{r})$ produzido por um fio
infinito de corrente el\'etrica constante $i$, \'e dado por [3]
\begin{eqnarray}
\vec{A}(\vec{r})=\frac{\mu_{0}i}{4\pi}\int_{-\infty}^{\infty}\frac{dz}{\sqrt{%
z^{2}+\rho^{2}}}\hat{k},
\end{eqnarray}
onde temos usando a mesma geometria anterior.
Uma an\'alise dimensional da integral
\begin{eqnarray}
I=\int_{-\infty}^{\infty}\frac{dz}{\sqrt{z^{2}+\rho^{2}}},
\end{eqnarray}
que aparece nas equa\c{c}\~oes dos potenciais, mostra que ela \'e
adimensional e portanto sofre de uma diverg\^encia logar\'{\i}tmica.
Assim, vemos que para estes dois problemas simples devemos empregar um
procedimento de renormaliza\c{c}\~ao a fim de obtermos os potenciais
renormalizados, isto \'e, "observados" (a difer\^en\c{c}a de potencial entre
dois pontos, pois ele \'e uma grandeza relativa e n\~ao absoluta).
A fim de tornar a teoria finita e assim manuze\'avel, devemos empregar um
m\'etodo de regulariza\c{c}\~ao. Isto vai nos permitir separarmos a parte
finita da divergente. Por\'em, a teoria fica dependente de um par\^ametro de
regulariza\c{c}\~ao e uma prescri\c{c}\~ao de renormaliza\c{c}\~ao dever\'a
se empregada para restabelecermos a teoria original. Vamos utilizar
diferentes m\'etodos de regulariza\c{c}\~ao e mostrar que, embora cada um
forne\c{c}a um resultado diferente, a teoria final, isto \'e, renormalizada
(f\'{\i}sica) \'e independente do m\'etodo de regulariza\c{c}\~ao usado.
\section{Cut-off}
\hspace{.5cm} Esse m\'etodo de regulariza\c{c}\~ao se baseia no emprego de
um corte nos limites da integral, isto \'e, trocamos o limite infinito por
um valor finito $\Lambda$ (par\^ametro regularizador).
Com a inclus\~ao do corte tornamos a teoria finita, por\'em dependente de $%
\Lambda$. Portanto, para restabelecermos a teoria original, devemos ao final
tomar o limite com $\Lambda$ tendendo a infinito.
Na integral da eq.(3) vamos introduzir um corte
\begin{eqnarray}
I_{\Lambda}=\int_{0}^{\Lambda}\frac{dz}{\sqrt{z^{2}+\rho^{2}}}.
\end{eqnarray}
Uma vez que tomaremos o limite, \'e conveniente obtermos o resultado da
integral da eq.(4) em pot\^encias de $\Lambda$ e de $\frac{1}{\Lambda}$ de
forma a permitir a separa\c{c}\~ao do(s) p\'olo(s) da parte finita. Vamos
dividir a integral da eq.(4) em duas partes
\begin{eqnarray}
I_{\Lambda}=\int_{0}^{\rho}\frac{dz}{\rho\sqrt{\frac{z^{2}}{\rho^{2}}+1}} +
\int_{\rho}^{\Lambda}\frac{dz}{z\sqrt{\frac{\rho^{2}}{z^{2}}+1}},
\end{eqnarray}
para considerarmos os casos em que $z<\rho$ e $z>\rho$. Realizando as
expans\~oes em s\'erie de Taylor dos integrandos da eq. (5) e depois
integrando termo a termo obtemos
\[
I_{\Lambda }=C+\ln \left( \frac{\Lambda }{\rho }\right) +O\left( \frac{1}{%
\Lambda ^{2}}\right) ,
\]
onde $C$ \'{e} uma constante.
Podemos observar que quando tentamos restabelecer a teoria original, ou
seja, tomamos o limite de $\Lambda$ tendento a infinito, presenciamos uma
diverg\^encia logar\'{\i}tmica, como j\'a esperavamos.
\section{Regulariza\c{c}\~ao Dimensional}
\hspace{.5cm} Este m\'etodo de regulariza\c{c}\~ao consiste em modificar a
dimens\~ao da integral atrav\'es de uma continua\c{c}\~ao anal\'{\i}tica de
forma a torn\'a-la finita. Consegue-se isto trocando a dimens\~ao do
diferenciando por uma outra complexa, atrav\'es da inclus\~ao de um
par\^ametro regularizador complexo, $\omega$
\begin{eqnarray}
I(\rho,\omega)=\int_{-\infty}^{\infty}\frac{d^{1-\omega}z}{\sqrt{z^{2}+
\rho^{2}}}.
\end{eqnarray}
A integral (7) agora \'e finita e pode ser realizada usando a rela\c{c}\~ao
[4]
\[
\int_{-\infty }^{\infty }\left( k^{2}+a^{2}\right) ^{-\alpha }d^{m}k=\pi ^{%
\frac{m}{2}}\frac{\Gamma (\alpha -\frac{m}{2})}{\Gamma (\alpha )}\left(
a^{2}\right) ^{\frac{m}{2}-\alpha },
\]
obtendo
\[
I(\rho ,\omega )=\pi ^{\frac{-\omega }{2}}\Gamma \left( \frac{\omega }{2}%
\right) (\rho )^{-\omega }.
\]
Para separarmos a parte finita da diverg\^{e}nte quando $\omega $ vai a
zero, vamos fazer uma expans\~{a}o em pot\^{e}ncias de $\omega $ da eq.(9),
para isto usamos para $|\omega |\ll 1$ as seguintes rela\c{c}\~{o}es
\[
\Gamma \left( \frac{\omega }{2}\right) =\frac{2}{\omega }-\gamma +O(\omega )
\]
e
\[
\rho ^{-\omega }=1-\frac{\omega }{2}\ln (\rho ^{2})+O(\omega ^{2}),
\]
onde $\gamma $ \'{e} o n\'{u}mero de Euler. Ent\~{a}o temos
\[
I(\rho ,\omega )=\pi ^{-\frac{\omega }{2}}\left[ \frac{2}{\omega }-\gamma
-\ln \left( \frac{\rho ^{2}}{\mu ^{2}}\right) +O(\omega )\right] ,
\]
onde temos inclu\'{i}do um par\^{a}metro de escala $\mu $ com dimens\~{a}o
de comprimento, a fim de tornar o logaritmando adimensional.
\section{Regulariza\c{c}\~ao por Fun\c{c}\~ao Zeta}
\hspace{.5cm} A fun\c{c}\~ao zeta generalizada associada a um operador $M$,
\'e definida como
\begin{eqnarray}
\zeta_{M}(s)=\sum_{i}\lambda_{i}^{-s},
\end{eqnarray}
onde $\lambda_i$, s\~ao os auto-valores do operador $M$ e $s$ um par\^ametro
complexo
Definimos, para o nosso caso, a fun\c{c}\~ao zeta como
\[
\zeta (s+1/2)=\int_{-\infty }^{\infty }\left( \frac{z^{2}}{\mu ^{2}}+\frac{%
\rho ^{2}}{\mu ^{2}}\right) ^{-s-1/2}d\left( \frac{z}{\mu }\right)
\]
e a integral (3) fica
\[
I(\rho ,s)=\zeta (s+1/2).
\]
O par\^{a}metro de escala $\mu $, com dimens\~{a}o de comprimento foi
inclu\'{i}do para tornar a fun\c{c}\~{a}o zeta admensional para todo $s$.%
\newline
Usando a rela\c{c}\~{a}o (8) obtemos
\[
\zeta (s+1/2)=\sqrt{\pi }\frac{\Gamma (s)}{\Gamma (s+1/2)}\left( \frac{\rho
^{2}}{\mu ^{2}}\right) ^{-s}
\]
que com a aproxima\c{c}\~{a}o
\[
2\sqrt{\pi }\frac{\Gamma (s)}{\Gamma (s-1/2)}\approx -\frac{1}{s},
\]
para $|s|\ll 1$, temos
\[
\zeta (s+1/2)=-\frac{\left( \frac{\rho ^{2}}{\mu ^{2}}\right) ^{-s}}{%
2s(s-1/2)}.
\]
A continua\c{c}\~{a}o anal\'{i}tica para s igual a zero da eq.(18) \'{e}
obtida multiplicando a equa\c{c}\~{a}o por s e em seguida derivando em $s=0$
[5]. Assim
\[
\Phi (\vec{r})=\frac{\lambda }{2\pi \varepsilon _{0}}-\frac{\lambda }{2\pi
\varepsilon _{0}}\ln \left( \frac{\rho }{\mu }\right) ,
\]
\[
\vec{A}(\vec{r})=\frac{\mu _{0}i}{2\pi }\hat{k}-\frac{\mu _{0}i}{2\pi }\ln
\left( \frac{\rho }{\mu }\right) \hat{k}.
\]
\section{Condi\c{c}\~oes de Renormaliza\c{c}\~ao}
\hspace{.5cm} Como podemos observar os potenciais obtidos atrav\'es dos
resultados dados pelas eq.(6) e (12) s\~ao ainda divergentes, portanto,
devemos lan\c{c}ar m\~ao de uma prescri\c{c}\~ao de renormaliza\c{c}\~ao a
fim de eliminar a parte divergente (p\'olo).
Como prescri\c{c}\~{a}o de renormaliza\c{c}\~{a}o usaremos a condi\c{c}%
\~{a}o f\'{i}sica, de que os potenciais n\~{a}o s\~{a}o grandezas absolutas
e sim relativas, isto \'{e}, somente diferen\c{c}as de potenciais podem ser
observadas. Assim, usando as eq.(6) e (12) obtemos
\[
\Phi (\vec{r})-\Phi (\vec{r_{0}})=\frac{\lambda }{2\pi \varepsilon _{0}}\ln
\left( \frac{\rho _{0}}{\rho }\right)
\]
e
\[
\vec{A}(\vec{r})-\vec{A}(\vec{r_{0}})=\frac{\mu _{0}i}{2\pi }\ln \left(
\frac{\rho _{0}}{\rho }\right) \hat{k}
\]
Agora tomando o potencial nulo no ponto de refer\^{e}ncia $\vec{r_{0}}$,
temos
\[
\Phi _{R}(\vec{r})=\frac{\lambda }{2\pi \varepsilon _{0}}\ln \left( \frac{%
\rho _{0}}{\rho }\right)
\]
e
\[
\vec{A}_{R}(\vec{r})=\frac{\mu _{0}i}{2\pi }\ln \left( \frac{\rho _{0}}{\rho
}\right) \hat{k}.
\]
Note que o ponto de refer\^encia $\vec{r_{0}}$ \'e completamente
arbitr\'ario.
Embora os resultados obtidos nas eq.(19) e (20) sejam finitos, eles ainda
n\~ao representam os resultados f\'{\i}sicos, pois n\~ao sabemos se o que
retiramos da parte divergente foi mais que o necess\'ario. Uma renormaliza\c{%
c}\~ao finita deve ser realizada para que os potenciais obtidos sejam
aqueles que representem a f\'{\i}sica do problema.
Novamente usando a diferen\c{c}a de potencial como condi\c{c}\~ao de
renormaliza\c{c}\~ao, obtemos das eq.(19) e (20) os mesmos resultados
obtidos nas eq.(23) e (24)
\'E importante comentarmos a presen\c{c}a do par\^ametro de escala $\mu$ nas
eq.(12), (19) e (20).
A prescri\c{c}\~{a}o de renormaliza\c{c}\~{a}o usada aqui fornece
imediatamente o resultado f\'{i}sico, isto \'{e}, o potencial no ponto $\vec{%
r}$ medido em rela\c{c}\~{a}o aquele medido no ponto de refer\^{e}ncia $\vec{%
r_{0}}$. Se desejassemos como primeira etapa obter um resultado finito para
as eq.(6) e (12) poder\'{i}amos usar como prescri\c{c}\~{a}o a subtra\c{c}%
\~{a}o do termo divergente (p\'{o}lo). Na eq.(6) a fim de separarmos a parte
divergente da finita devemos multiplicar e dividir o logaritimando por um
par\^{a}metro arbitr\'{a}rio finito, o par\^{a}metro de escala $\mu $.
\[
I(\rho ,\mu ,\Lambda )=C-\left[ \ln \left( \frac{\rho }{\mu }\right) -\ln
\left( \frac{\Lambda }{\mu }\right) \right] +O\left( \frac{1}{\Lambda ^{2}}%
\right) .
\]
Agora usando como prescri\c{c}\~{a}o a subtra\c{c}\~{a}o do p\'{o}lo,
obtemos, para o cut-off
\[
\Phi (\vec{r})=\frac{\lambda }{2\pi \varepsilon _{0}}\ln \left( \frac{\rho }{%
\mu }\right) +\frac{\lambda }{2\pi \varepsilon _{0}}C,
\]
e para a dimensional
\[
\Phi (\vec{r})=\frac{\lambda }{2\pi \varepsilon _{0}}\ln \left( \frac{\rho }{%
\mu }\right) +\frac{\gamma }{2\pi \varepsilon _{0}}.
\]
Ent\~ao, notamos que no caso da regulariza\c{c}\~ao dimensional e zeta, esta
separa\c{c}\~ao j\'a foi realizada de alguma forma escondida dentro dos
procedimento usados.
Uma maneira mais elegante e formal de introduzimos o par\^{a}metro de
escalar \'{e} fazendo com que a integral inicial (3) seja adimensional, isto
\'{e},
\[
I=\int_{-\infty }^{\infty }\frac{d\left( \frac{z}{\mu }\right) }{\sqrt{\frac{%
z^{2}}{\mu ^{2}}+\frac{\rho ^{2}}{\mu ^{2}}}}.
\]
E desta forma tornando a eq.(7) adimensional para qualquer $\omega $.
\'{E} claro que a continua\c{c}\~{a}o anal\'{i}tica usada no m\'{e}todo da
fun\c{c}\~{a}o zeta \'{e} a prescri\c{c}\~{a}o de renormaliza\c{c}\~{a}o
necess\'{a}ria para se obter o resultado finito e \'{e} equivalente a subtra%
\c{c}\~{a}o do p\'{o}lo. Isso fica claro se tivessemos realizado a
expans\~{a}o em s\'{e}rie de Laurent da eq.(18)
\[
I(\rho ,s)=\frac{a_{-1}}{s}+\ln \left( \frac{\rho }{\mu }\right) +O(s),
\]
onde $a_{-1}$ \'{e} o res\'{i}duo.
Note que os resultados das eq.(19),(26) e (27) diferem por uma constante e
s\~ao dependentes do par\^ametro de escala. Como j\'a dissemos, embora os
resultados destas equa\c{c}\~oes sejam finitos eles ainda n\~ao representam
a f\'{\i}sica da teoria. Isto \'e obvio, pois, n\~ao podemos ter os
resultados f\'{\i}sicos (observados) dependentes do m\'etodo de regulariza\c{%
c}\~ao. Uma renormaliza\c{c}\~ao finita deve ser feita para ajustar os
potenciais obtidos aqueles observados (diferen\c{c}as). Esta condi\c{c}\~ao
de renormaliza\c{c}\~ao nos permite escrever os potenciais em fun\c{c}\~ao
daqueles observados em um determinado ponto. Ela tamb\'em permite que o
par\^ametro de escala seja escrito em fun\c{c}\~ao do ponto de refer\^encia $%
\rho_{0}$. \'E claro que o ponto de refer\^encia \'e arbitr\'ario e portanto
tamb\'em o par\^ametro de escala.
Agora estamos aptos a sintetizar como funciona a renormaliza\c{c}\~ao. Os
potenciais dados pelas eq.(6), (12) e (19), n\~ao s\~ao aqueles f\'{\i}sicos
(observ\'aveis) sendo at\'e mesmo divergentes. Para torn\'a-los aqueles
observados devemos ajust\'a-los. Assim, medimos (na verdade aqui definimos
um valor qualquer, em geral zero) o potencial em um ponto de refer\^encial
qualquer $\vec{r_{0}}$ que no caso da Teoria Qu\^antica de Campos \'e
chamado ponto de renormaliza\c{c}\~ao ou subtra\c{c}\~ao. Por fim escrevemos
o potencial f\'{\i}sico (observado) como fun\c{c}\~ao daquele medido no
ponto de refer\^encia (ponto de renormaliza\c{c}\~ao). Este procedimento
ent\~ao absorve a diverg\^encia do potencial original n\~ao f\'{\i}sico.
Em resumo:
i) Potencial original n\~ao f\'{\i}sico
\begin{eqnarray}
\Phi_d(\vec{r})=D + C + \Phi_F(\vec{r}),
\end{eqnarray}
onde D \'e o termo divergente separado por um m\'etodo qualquer de regulariza%
\c{c}\~ao, e C \'e uma constante que depende do m\'etodo de regulariza\c{c}%
\~ao e $\Phi_F(\vec{r})$ \'e o potencial.
ii) Potencial medido no ponto de refer\^encia (renormaliza\c{c}\~ao)
\begin{eqnarray}
\Phi_{0}=D + C + \Phi_F(\vec{r_{0}}).
\end{eqnarray}
Neste caso para $\Phi_{0}$ \'e determinado um valor arbitr\'ario e n\~ao
realmente medido
Agora escrevemos
\begin{eqnarray}
D + C = \Phi_{0}-\Phi_F(\vec{r_{0}})
\end{eqnarray}
e substituindo na eq.(31), fica
\begin{eqnarray}
\Phi_R(\vec{r})=\Phi(\vec{r})-\Phi(\vec{r_{0}})+\Phi_{0},
\end{eqnarray}
onde $\Phi_R(\vec{r})$ \'e o potencial renormalizado.
Note que mesmo no caso de um m\'etodo de regulariza\c{c}\~ao que forne\c{c}a
um resultado finito, ainda temos de ajustar este resultado aquele
f\'{\i}sico.
Finalmente, podemos analizar como funciona a renormaliza\c{c}\~ao na Teoria
Qu\^antica de Campos. A teoria original depende de alguns par\^ametros em
geral divergentes, tais como $m$ e $\lambda$. Tais par\^ametro n\~ao
representam a massa $(m)$ e a constante de acoplamento $\lambda$ observados
da teoria e sim s\~ao ajustando atrav\'es das condi\c{c}\~oes de renormaliza%
\c{c}\~ao a estas quantidades f\'{\i}sicas renormalizadas, medidas em caso
de teorias realistas, ou definidas no caso de teorias n\~ao realistas, em um
determinado ponto, chamado ponto de renormaliza\c{c}\~ao ou subtra\c{c}\~ao.
Este ponto, pode ser o quadri-momento da teoria ou um determinado estado do
sistema, em geral o de menor energia, ou estado de v\'acuo, embora qualquer
ponto seja t\~ao bom quanto outro, isto \'e, o ponto de renormaliza\c{c}\~ao
\'e arbitr\'ario.
Escrevendo agora a teoria original em fun\c{c}\~ao n\~ao mais dos
par\^ametros originais $m$ e $\lambda$ e sim das quantidades f\'{\i}sicas
renormalizadas ("observadas") $m_{R}$ e $\lambda_{R}$, as diverg\^encias
s\~ao absorvidas de forma semelhante ao que ocorreu com o potencial.
Uma maneira alternativa usada \'e tomar os par\^ametros $m$ e $\lambda$ da
teoria original como sendo realmente aquele observados (renormalizados) e
absorver as diverg\^encias da teoria em contra-termos $\delta$$m$ e $%
\delta\lambda$ inclu\'{\i}dos na teoria. Tais contra-termos, \'e claro,
devem ser de termos de mesma pot\^encia nos campos que aqueles de $m$ e $%
\lambda$. Ent\~ao, usando as condi\c{c}\~oes de renormaliza\c{c}\~ao os
contra-termos s\~ao determinados de forma a anular as diverg\^encias e
fornecer a f\'{\i}sica da teoria.
\section{Conclus\~ao}
\hspace{.5cm} Atrav\'es de um exemplo simples do c\'alculo dos potenciais
escalar e vetorial de um fio infinito de carga e de corrente,
respectivamente, podemos apresentar as diverg\^encias que sofrem algumas
teorias, os m\'etodos usados para lidar com estas diverg\^encias
(separ\'a-los da parte finita) e o procedimento usado para tornar tais
teorias em teorias f\'{\i}sicas (renormaliza\c{c}\~ao).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,048 |
Welcome to boadata's documentation!
===================================
Contents:
.. toctree::
:maxdepth: 2
api
Indices and tables
==================
* :ref:`genindex`
* :ref:`modindex`
* :ref:`search`
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,807 |
You are here: Home>People>Tribute to John Joe Maher R.I.P.
By Mary Kyne
Frank and I were so sorry to learn of John Joe's death in August 2013. While we knew he wasn't well yet when death comes knocking it takes us all by surprise. Many memories of times past come flooding back. One always thought that John Joe would always be with us as he was always in control – organising events and golf outings as only he could, paying attention to every little detail. It seemed so effortless to him.
His greatest delight was bringing pleasure to his many friends and acquaintances. He was jovial, generous, upright and honourable. He was truly a man of great integrity, a good man who used his talents wisely for the benefit of his fellow man. The Good Lord will indeed reward him for his goodness.
We in Oughterard Golf Club – in fact in the village of Oughterard are indebted to him as he brought his first group of Fruit Traders here in 1976 and continued to do so well into the 90's. He put us on the map with other Dublin Clubs and this brought great business to the village. We had some hilarious nights in the Golf Club enjoying barbeques, barn dances, sing songs and dancing into the early hours of the morning. We enjoyed ourselves while John Joe looked on quietly amused at the antics around him. Anyone who knew John Joe and was counted among his friends was privileged. We will miss him.
Death ends a life but relationships live on and for as long as we live John Joe will be part of what we are and we will remember him. The bonds of love can never be broken.
Members of Oughterard Golf Club and Forrest Little, Dublin and his many friends shared many happy experiences and adventures with John Joe and we will carry with us our own individual memories. The late Seamus Heaney, the poet, said, "Memories are the biscuits of life."
John Joe's life's work is now completed and we now pray that the Good Lord will welcome him into his kingdom and reward him for his goodness.
Suaimhneas síoraí dó.
Comments about this page
A Tribute to John Joe Maher RIP, delivered by Ted Corcoran, Captain, Forrest Little, in Forrest Little G.C., August 10th President Joe, Forrest Little, Captain Pierce, Oughterard, Nora and Alice and John Joe's extended family, Members of the Golfing Union of Ireland, Visitors, Friends, Ladies and Gentlemen. I would like to welcome you all to Forrest Little today, as we gather here as relatives and friends of the late, dearly beloved, John Joe. On behalf of all the members of the Club, I would like to express our sincere sympathy to Nora and Alice and the extended family members on their sad loss. John Joe was a special human being, widely known, not only in golfing circles all around the country, but also in many other facets of Irish life. He was such a special figure here in Forrest Little, that, when word spread of his passing early on Wed last, nobody, but nobody, said John Joe who? Everyone knew who John Joe was and was saddened by the news. An air of gloom hung over the club all day, as person after person whispered to one another, "did you hear about John Joe?" Henry W Longfellow, an American poet, in his well known poem, The Psalm of Life, wrote as follows: "Lives of great men all remind us We can make our lives sublime, And, departing, leave behind us Footprints on the sands of time." John Joe was a great man in so many ways and his footprints will be forever etched on the memories of all the very many people who knew and loved him. John Joe became a member of Forrest Little G.C. in its early years and went on to serve as Captain in 1984, before serving as President in 1996 and 1997. During the intervening years, Oughterard became his home from home, mainly because of his fishing interests and where he formed lifelong friendships. It's wonderful to see so many Oughterard members here today at his funeral, a fact, no doubt, much appreciated by Nora and Alice and all his Dublin friends. As Oughterard grew in tandem with Forrest Little, John Joe set up a twinning arrangement between both clubs. Inter club visits were arranged twice yearly and these proved wonderful social occasions, always ending well before midnight! Sorry my mistake! Before morning! Both Forrest Little and Oughterard Golf clubs, in recognition of his huge input and contribution to both clubs, afforded him Honorary Life Membership, awards he greatly appreciated and savoured with great pride. John Joe always showed a keen interest in promoting juvenile golf, so it must have been particularly satisfying for him to learn of his grand niece, Julie McCarthy, winning the Irish Girls Close Championship in Ballyliffin G. C. just a few weeks ago and her subsequent selection on the Irish Girls team for the Home Internationals. In 1997, John Joe was elected to a position in the Golfing Union of Ireland and travelled the length and breadth of the country, officiating at various competitions. He was a familiar figure at first tees, from early in the morning till late in the evening, carrying out his duties in is his usual fastidious and efficient manner. As a member of the GUI, John Joe captained a winning Leinster Boys Interprovincial side, in 1999, and would have been delighted to hear that the Leinster Boys team has once again won the Interprovincial title, a week or so ago, this time captained by another Forrest Little and GUI member, his very close friend, Kevin Raftery. Someone unknown once said, "We do not remember days. We remember moments in our lives." Everyone here can remember such moments, involving John Joe. My most recent moment happened on the morning of July 6th, my Captain's Prize, Final Day. John Joe phoned the bar and asked to speak to me. He sounded tired and weak, but he still found the energy to call me and wish me every success on the day. I'm told he did this religiously for every captain on his final day every year. It was typical of John Joe, always thinking of other people. You have all experienced such a moment, or moments, involving John Joe; moments of kindness, moments of generosity, moments of friendship, moments of camaraderie, moments of helpfulness, moments of solace, moments of support, moments of happiness, moments of enjoyment. While I pause for a second, or two, I ask you to recall such a moment. I thank you for coming here today in such large numbers. John Joe has received the send off he deserves. I once again sympathise with Nora and Alice and reassure you that his legacy will live on, because he truly was a special person. As the late American sports writer, Grantland Rice, put it so beautifully, writing about the day when all of us meet our maker, as we must; "For when the One Great Scorer comes To write against your name, He marks – not that you won or lost- But how you played the Game." John Joe never cared whether he won or whether he lost – he was more concerned with living a good life and helping everyone he could. Thanks John Joe. We will miss you. May you rest in peace. Ar dheis lamh De go raibh a anam dilis.
By Ted Corcoran (08/02/2014) | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,452 |
Call Us : +1-866-332-2152
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,297 |
Pocketful of Pebbles is a new work for children inspired by three lesser known traditional tales.
Featuring Tammy Weller, Zachary Boulton, Anna Straker, Lisa Smith and Louise Brehmer.
The Van Huon Pine Family invites you to share in an evening of nostalgia, cocktail onions and kabana as they take a stroll down memory lane. Dim the lights, grab the popcorn … This is Slide Night.
Created through the collating of real memories and images shared by locals and visitors from summer's past, this piece is both a celebration and an excavation of our iconic city, the Gold Coast, and the Memory Keepers that call it home.
The Grand is the story of hotel proprietress Matilda and her apprentice Eugenia, taking us behind the closed doors of a once grand hotel. The women struggle against each other and the insidious creeping of the world outside the doors, relying increasingly on games and routine in the face of something unknown.
Directed by Lisa Smith. Featuring Anna Mowry and Tammy Weller.
Genevieve's life is perfect - at least from the outside. Little does she know that her path is about to cross with an old school chum, Lucy, who's bluntness and honesty will unravel everything Gen thought she knew about her past, her fiancé, and her life.
A collection of monologues exploring love and dating, and White Rabbit Theatre Ensemble's first official production in February 2012. Presented by White Rabbit Theatre Ensemble and collaborating artists at The Courtyard Theatre, Café Lumbini. Works by Ziella Bryars, Lisa Keddie, Lisa Smith, Fred Quillemby, Jolyon Coy and Maev Mac Coille. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,279 |
Fortunately, now there is a standard way to handle different errors uniformly.
First of all we should fix the usage of GetLastError(). GetLastError() should be called before anything else. At least now we are creating some strings which involves memory allocation, which can modify the result of GetLastError().
We have to have custom error_category (C++11) for GetLastError(). Since we are not going to create the insane enum for all Windows error codes and properly implement the equivalence, so only implement partially (name, message (based on FormatMessage)) error_category and the factory method (which should not be available using ADL, something like std::error_code makeWinErrorCode(int code);). | {
"redpajama_set_name": "RedPajamaC4"
} | 2,859 |
Zonitoschema is een geslacht van kevers uit de familie oliekevers (Meloidae).
Het geslacht is voor het eerst wetenschappelijk beschreven in 1909 door Péringuey.
Soorten
Het geslacht omvat de volgende soorten:
Zonitoschema alluaudi Pic, 1913
Zonitoschema atrimembris (Fairmaire, 1894)
Zonitoschema bivittipennis Kaszab, 1981
Zonitoschema bradleyi Mohamedsaid, 1981
Zonitoschema burgeoni Pic, 1931
Zonitoschema capensis Kaszab, 1961
Zonitoschema celebensis (Pic, 1911)
Zonitoschema coccinea (Fabricius, 1801)
Zonitoschema corporaali Borchmann, 1929
Zonitoschema cothurnata (Marseul, 1873)
Zonitoschema dollei (Fairmaire, 1888)
Zonitoschema eborina (Fåhraeus, 1870)
Zonitoschema elongaticeps Pic, 1935
Zonitoschema elongatipennis (Pic, 1915)
Zonitoschema ennsi Kaszab, 1958
Zonitoschema gibdoana (Kaszab, 1956)
Zonitoschema gigantea (Fairmaire, 1894)
Zonitoschema gombakiensis Mohamedsaid, 1977
Zonitoschema griseohirta Pic, 1914
Zonitoschema hebridesiensis Mohamedsaid, 1981
Zonitoschema iranica Kaszab, 1959
Zonitoschema japonica (Pic, 1910)
Zonitoschema kimi (Kôno, 1936)
Zonitoschema klapperichi Borchmann, 1941
Zonitoschema krombeini Mohamedsaid, 1979
Zonitoschema latipennis Pic, 1909
Zonitoschema leleupi Kaszab, 1960
Zonitoschema macroxantha (Fairmaire, 1887)
Zonitoschema megalops (Fairmaire, 1883)
Zonitoschema miwai (Kôno, 1936)
Zonitoschema nigrimembris Pic, 1913
Zonitoschema nigroapicalis Pic, 1948
Zonitoschema nitidissima (Pic, 1911)
Zonitoschema oculatissima Peyerimhoff, 1929
Zonitoschema okinawensis (Miwa, 1928)
Zonitoschema pallida (Fabricius, 1794)
Zonitoschema pallidissima (Reitter, 1908)
Zonitoschema posticalis (Péringuey, 1892)
Zonitoschema rhobompana Pic, 1911
Zonitoschema rubricolor Pic, 1924
Zonitoschema saga (Péringuey, 1899)
Zonitoschema seminigra (Pic, 1909)
Zonitoschema squalida (Fairmaire, 1889)
Zonitoschema tenuicollis (Fabricius, 1811)
Zonitoschema testaceiventris Pic, 1930
Zonitoschema vonhayekae Mohamedsaid, 1981
Oliekevers | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 850 |
#
The Sound of the Mountain
Yasunari Kawabata (1899-1972), the first Japanese to win the Nobel Prize for Literature, in 1968, was born in Osaka, the son of a wealthy and prominent physician, but was raised in the country by his grandfather after his parents' early deaths. Interested in both painting and writing from young—his rich visualization forms the heart of his writing—he published his first stories while he was still in high school. After graduating from Tokyo Imperial University in 1924, Kawabata quickly rose to the forefront of modern Japanese writing, and after the war built an immense international literary reputation. He took his own life at the age of seventy-three.
Edward G. Seidensticker was born in Castle Rock, Colorado, in 1921. He received his BA from the University of Colorado, his MA from Columbia University, and did graduate work at both Harvard and Tokyo universities. In June 1942, Seidensricker enrolled at the US Navy Japanese Language School in Boulder, graduating with a burgeoning command of Japanese that would become the basis of his academic career as a translator of Japanese literature, including works by Tanizaki, Kawabata, Mishima and Kafu, and the daunting The Tale of Genji. In the 1950s, he taught at Sophia University in Tokyo for four years, returning to the US at the invitation of Stanford in 1962. After Stanford, Seidensticker taught at Michigan and Columbia, from which he retired in 1985.
Yasunari Kawabata
The Sound of
the Mountain
Translated by Edward G. Seidensticker
TUTTLE PUBLISHlNG
Boston • Rutland, Vermont • Tokyo
#
UNESCO COLLECTION OF REPRESENTATIVE WORKS
Japanese Series
This book has been accepted in the Japanese Series of the Translations Collection of the United Nations Educational, Scientific and Cultural Organization (UNESCO).
Originally published in Japanese as Yama no Oto
Published by Tuttle Publishing,
an imprint of Periplus Editions (HK) Ltd.,
by special arrangement with Alfred A, Knopf Inc., New York
Copyright © 1970 by Alfred A, Knopf, Inc., New York. All rights reserved. No part of this book may be reproduced in any form without permission in writing from the publishers, except by a reviewer, who may quote brief passages in a review to be printed in a magazine or newspaper.
ISBN: 978-1-4629-1238-4 (ebook)
First Tuttle edition, 1971
Twelfth printing, 2002
Printed in Singapore
Distributed by:
Japan & Korea
Tuttle Publishing Japan
RK Building 2nd Floor
Yaekari Building 3rd Floor, 5-4-12
Osaki Shinagawa-ku,
Tokyo 141-0032
Tel: (03) 5437 0171
Fax: (03) 5437 0755
Asia Pacific
Berkeley Books Pte, Ltd.
61 Tai Seng Avenue, #02-12
Singapore 534167
Tel: (65) 280 1330
Fax: (65) 280 6290
Email: inquiries@periplus.com.sg
Web site: www.periplus.com
#
Contents
The Sound of the Mountain
The Wings of the Locust
A Blaze of Clouds
The Chestnuts
A Dream of Islands
The Cherry in the Winter
Water in the Morning
The Voice in the Night
The Bell in Spring
The Kite's House
A Garden in the Capital
The Scar
In the Rain
The Cluster of Mosquitoes
The Snake's Egg
Fish in Autumn
Note on the Pronunciation of Japanese Names
Consonants are pronounced approximately as in English, except that "g" is always hard, as in Gilbert. Vowels are pronounced as in Italian. Also as in Italian, the final e is always sounded. Thus the name Kaname is pronounced Kah-nah-meh. There is no heavy penultimate accent as in English; it is adequate to accent each syllable equally.
The Japanese name order has been followed throughout this translation, with the family name first.
The Sound of the Mountain
#
The Sound of the Mountain
Ogata Shingo, his brow slightly furrowed, his lips slightly parted, wore an air of thought. Perhaps to a stranger it would not have appeared so. It might have seemed rather that something had saddened him.
His son Shuichi knew what was happening. It happened so frequently that he gave it little thought.
Indeed, more was apparent to him than the simple fact that his father was thinking. He knew that his father was trying to remember something.
Shingo took off his hat and, absently holding it in his right hand, set it on his knee. Shuichi put it on the rack above them.
"Let me see. What was it, I wonder?" At such times Shingo found speech difficult. "What was the name of the maid that left the other day?"
"You mean Kayo?"
"Kayo. That was it. And when was it that she left?"
"Last Thursday, That would make it five days ago."
"Five days ago? Just five days ago she quit, and I can't remember anything about her."
To Shuichi his father's performance seemed a trifle exaggerated.
"That Kayo—I think it must have been two or three days before she quit. When I went out for a walk I had a blister on my foot, and I said I thought I had picked up ringworm. 'Footsore,' she said. I liked that. It had a gentle, old-fashioned ring to it. I liked it very much. But now that I think about it I'm sure she said I had a boot sore. There was something wrong with the way she said it. Say 'footsore.'"
"Footsore."
"And now say 'boot sore.'"
"Boot sore."
"I thought so. Her accent was wrong."
Of provincial origins, Shingo was never very confident about standard Tokyo pronunciation. Shuichi had grown up in Tokyo.
"It had a very pleasant sound to it, very gentle and elegant, when I thought she said 'footsore.' She was there in the hallway. And now it occurs to me what she really said, and I can't even think of her name. I can't remember her clothes or her face. I imagine she was with us six months or so?"
"Something of the sort." Used to these problems, Shuichi offered his father no sympathy.
Shingo was accustomed enough to them himself, and yet he felt a twinge of something like fear. However hard he tried to remember the girl, he could not summon her up. There were times when such futile searchings were leavened by sentimentality.
So it was now. It had seemed to him that Kayo, leaning slightly forward there in the hallway, was consoling him for being footsore.
She had been with them six months, and he could call up only the memory of that single word. He felt that a life was being lost.
2
Yasuko, Shingo's wife, was sixty-three, a year older than he.
They had a son, a daughter, and two grandchildren, daughters of the girl, Fusako.
Yasuko was young for her age. One would not have taken her to be older than her husband. Not that Shingo himself seemed particularly old. They seemed natural together, he just enough older than she to make them a most ordinary couple. Though diminutive, she was in robust health.
Yasuko was no beauty. In their younger years she had looked older than he, and had disliked being seen in public with him.
Shingo could not have said at what age she had begun to look the younger of the two. Probably it had been somewhere toward their mid-fifties. Women generally age faster than men, but in their case the reverse had been true.
The year before, the year he had entered his second cycle of sixty years, Shingo had spat up blood—from his lungs, it had seemed. He had not had a medical examination, however, and presently the affliction had gone away. It had not come back.
Nor had it meant that he grew suddenly older. His skin had seemed firmer since, and in the two weeks or so that he had been in bed the color of his eyes and lips had improved.
Shingo had not detected symptoms of tuberculosis in himself, and to spit blood at his age gave him the darkest forebodings. Partly because of them he refused to be examined. To Shuichi such behavior was no more than the stubborn refusal of the aged to face facts. Shingo was not able to agree.
Yasuko was a good sleeper. Sometimes, in the middle of the night, Shingo would be tempted to blame her snoring for having awakened him. She had snored, it seemed, as a girl of fifteen or sixteen, and her parents had been at great pains to correct the habit; it had stopped when she married. Then, when she passed fifty, it had begun again.
When she snored Shingo would twist her nose in an effort to stop her. If the twisting had no effect, he would take her by the throat and shake her. On nights when he was not in good spirits he would be repelled by the sight of the aged flesh with which he had lived for so long.
Tonight he was not in good spirits. Turning on the light, he looked at her profile and took her by the throat. She was a little sweaty.
Only when she snored did he reach out to touch her. The fact seemed to him infinitely saddening.
He took up a magazine lying at his pillow. Then, the room being sultry, he got up, opened a shutter, and sat down beside it.
The moon was bright.
One of his daughter-in-law's dresses was hanging outside, unpleasantly gray. Perhaps she had forgotten to take in her laundry, or perhaps she had left a sweat-soaked garment to take the dew of night.
A screeching of insects came from the garden. There were locusts on the trunk of the cherry tree to the left. He had not known that locusts could make such a rasping sound; but locusts indeed they were.
He wondered if locusts might sometimes be troubled with nightmares.
A locust flew in and lit on the skirt of the mosquito net. It made no sound as he picked it up,
"A mute." It would not be one of the locusts he had heard at the tree.
Lest it fly back in, attracted by the light, he threw it with all his strength toward the top of the tree. He felt nothing against his hand as he released it.
Gripping the shutter, he looked toward the tree. He could not tell whether the locust had lodged there or flown on. There was a vast depth to the moonlit night, stretching far on either side.
Though August had only begun, autumn insects were already singing.
He thought he could detect a dripping of dew from leaf to leaf.
Then he heard the sound of the mountain.
It was a windless night. The moon was near full, but in the moist, sultry air the fringe of trees that outlined the mountain was blurred. They were motionless, however.
Not a leaf on the fern by the veranda was stirring.
In these mountain recesses of Kamakura the sea could sometimes be heard at night. Shingo wondered if he might have heard the sound of the sea. But no—it was the mountain.
It was like wind, far away, but with a depth like a rumbling of the earth. Thinking that it might be in himself, a ringing in his ears, Shingo shook his head.
The sound stopped, and he was suddenly afraid, A chill passed over him, as if he had been notified that death was approaching. He wanted to question himself, calmly and deliberately, to ask whether it had been the sound of the wind, the sound of the sea, or a sound in his ears. But he had heard no such sound, he was sure. He had heard the mountain.
It was as if a demon had passed, making the mountain sound out.
The steep slope, wrapped in the damp shades of night, was like a dark wall So small a mound of a mountain, that it was all in Shingo's garden; it was like an egg cut in half.
There were other mountains behind it and around it, but the sound did seem to have come from that particular mountain to the rear of Shingo's house.
Stars were shining through the trees at its crest.
As he closed the shutter, a strange memory came to him.
Some ten days before, he had been awaiting a guest at a newly built restaurant. A single geisha was with him. The guest was late, and so were the other geisha.
"Why don't you take off your tie?" she said. "You must be warm."
Shingo nodded, and let her take it off for him.
She was not a geisha with whom he was particularly familiar, but when she had folded the tie and put it into the pocket of his coat, which lay beside the alcove, the conversation moved on to personal matters.
Some two months before, she said, she had been on the point of committing suicide with the carpenter who had built the restaurant. But as they had prepared to take poison, doubts had overtaken her. Were the portions in fact lethal?
"He said there was plenty. The doses were all measured out, his and mine, he said, and that proved it."
But she could not believe him. Her doubts only grew.
"I asked him who did the measuring. Someone might have measured out just enough to make us sick and teach us a lesson. I asked him who the druggist or doctor was that gave it to him, but he wouldn't say. Isn't that strange? There we were, going to die together. Why wouldn't he answer me? After all, who was to know afterwards?"
"A good yarn," Shingo had wanted to say.
And so she had insisted, she went on, that they try again after she had found someone to do the measuring.
"I have it here with me."
Shingo thought the story an odd one. All that had really stayed with him was the fact that the man was a carpenter and had built the restaurant.
The geisha had taken two packets from her purse and opened them for him.
He had only glanced at them. He had had no way of knowing whether or not they were poison.
As he closed the shutter, he thought of the geisha.
He went back to bed. He did not wake his wife to tell her of the fear that had come over him on hearing the sound of the mountain.
3
Shuichi and Shingo worked for the same firm. The son served as a sort of prompter for the father.
There were other prompters too, Yasuko and Kikuko, Shuichi's wife. The three of them worked together, a team supplementing Shingo's powers of memory. The girl in the office was yet another prompter.
Coming into Shingo's office, Shuichi took a book from the small stand in one corner and began leafing through it.
"Well, well," he said. He went over to the girl's desk and pointed to an open page.
"What is it?" asked Shingo, smiling. Shuichi brought the book to him,
"One is not to understand that the sense of chastity has here been lost," said the passage in question. "We have but a device for loving longer. A man unable to bear the pain of loving a woman, a woman unable to bear the pain of loving a man—they should go happily out in search of other partners, and so find a way to make their hearts more steadfast."
"Where is 'here'?"
"Paris. It's a novelist's account of his trip to Europe."
Shingo's mind was no longer as alive as it had once been to aphorism and paradox. This seemed to him, however, neither of the two. It seemed, more simply, penetrating insight.
Shuichi had probably not been moved by the passage. He had found a way, on the spur of the moment, for signaling to the girl that he wanted her to go out with him after work.
As he got off the train in Kamakura, Shingo found himself wishing that he had come home with Shuichi, or perhaps later.
The bus was crowded with commuters. He decided to walk.
The fishmonger nodded a greeting as Shingo stood outside the shop. He went in. The water in the tub of prawns was a cloudy white. He prodded a lobster, It should have been alive, but it did not move. He decided on whelks, of which there was a good supply.
When asked how many he wanted, however, he was perplexed.
"Well, make it three. Three of the biggest ones."
"Shall I dress them for you, sir?"
The fishmonger and his son dug out the meat with butcher knives. Shingo disliked the sound of scraping against the shell.
As the man washed and cut the meat, two girls stopped in front of the shop.
"What will you have?" he asked, going on with the dicing.
"Herring."
"How many?"
"One."
"One?"
"Yes."
"Just one?"
The herring were not the smallest possible, but they were little larger than minnows. The girl did not seem to be especially put off by this show of disapproval, however.
The man took up the herring in a bit of paper and handed it to her.
"But we didn't need any fish," said the second girl, hanging over the other and prodding her elbow.
"I wonder if they'll still be here on Saturday," said the other. She was looking at the lobsters. "My boy friend sort of likes them."
The second girl did not answer.
Startled, Shingo ventured a glance.
Prostitutes of the new sort, they had bare backs, cloth shoes, and good figures.
The fishmonger collected the diced meat at the center of his board and, dividing it in three parts, began to put it back into the shells.
"We're getting more and more of their kind. Even here in Kamakura."
His asperity struck Shingo as most odd. "But I thought they were behaving rather well," he said, protesting against he hardly knew what.
Casually, the man was putting the meat back into the shells, so mixed together, thought Shingo, that it was unlikely to be reassembled in the particular shells from which it had come. He was aware of very small niceties.
Today was Thursday, Two more days until Saturday—but then, he told himself, there were plenty of lobsters to be had these days. He wondered how the uncouth maiden would prepare lobster for her American friend. A lobster made a simple, uncouth dish, however, fried or boiled or roasted.
Shingo had felt well disposed toward the girls, and yet afterwards he was taken with vague feelings of despondency.
There were four in his family, but he had bought only three whelks. He had not acted precisely out of consideration for Kikuko, although he had of course known that Shuichi would not be home for dinner. He had simply deleted Shuichi.
At a grocery farther on he bought gingko nuts.
4
It was unusual for Shingo to buy food on his way home, but neither Yasuko nor Kikuko showed surprise.
Perhaps they wished to hide their thoughts about the fact that Shuichi, who should have been with him, was not.
Handing his purchases to Kikuko, he followed her into the kitchen.
"Some water please, with a little sugar in it." He went to the faucet himself.
In the sink were prawns and lobsters. He was struck by the coincidence. He had seen both at the fishmonger's, but had not thought of buying both.
"A good color," he said. The prawns had a fresh luster.
Kikuko cracked a gingko nut with the back of a knife.
"It was a nice thought, but I'm afraid they're no good."
"Oh? I did think they were a little out of season."
"I'll call the grocery and tell them."
"Don't bother. But all these shellfish—my contribution doesn't add much."
"We might open a seaside restaurant." Kikuko showed the tip of her tongue, in mild derision. "Let's see, now. We can boil these, shell and all. So maybe we should roast the lobsters and fry the prawns. I'll go buy some mushrooms. While I'm at it would you mind going out to get eggplant from the garden?"
"I'd be delighted."
"Little ones. And bring in some sage, too. I wonder if the prawns might be enough by themselves."
Kikuko brought only two whelks to the table.
"But there should be another," said Shingo, a little puzzled.
"Oh, dear. But the two of you have such bad teeth, Grandpa—I thought you might want to share one nicely between you."
"I don't see any grandchildren around."
Yasuko looked down and snickered.
"I'm sorry." Kikuko got up lightly and went to the kitchen for the third.
"We should do as Kikuko says," said Yasuko. "Share one nicely between us."
Shingo thought Kikuko's words beautifully apt. It was as though his own problem, whether to buy three or four, had thus been brushed away. Her tact and skill were not to be underestimated.
She might have been expected to say that she would leave one for Shuichi, or that she and Yasuko would share one. Perhaps she had considered these possibilities.
"But were there only three in the store?" asked Yasuko, not alive to such subtleties. "You only brought three, and there are four of us."
"We didn't need another. Shuichi didn't come home."
Yasuko smiled what should have been a wry smile, but, perhaps because of her age, it ended up as something less than that.
No trace of a shadow passed over Kikuko's face, nor did she ask what might have happened to Shuichi.
She was the youngest of eight children.
The other seven were also married, and all had numerous progeny. Shingo sometimes thought of the fecundity she had inherited from her parents.
She would complain that he had not yet learned the names of her brothers and sisters. He was even further from remembering the names of her nieces and nephews. She had been born at a time when her mother no longer wanted children or thought herself capable of having them. Indeed, her mother had felt rather ashamed, at her age, and had considered abortion. It had been a difficult birth. Forceps had been applied to Kikuko's head.
Kikuko had told Shingo of having heard these facts from her mother.
It was difficult for him to understand a mother who would speak of such things to her daughter, or a girl who would reveal them to her father-in-law.
Kikuko had held back her hair to show a faint scar on her forehead.
The scar, whenever he chanced to glimpse it afterwards, somehow drew him to her.
Still, Kikuko had been reared as the pet of the family, it seemed. She was not spoiled, precisely, but she seemed to expect affection. And there was something a little weak about her.
When she had first come as a bride, Shingo had noted the slight but beautiful way she had of moving her shoulders. In it, for him, there was a bright, fresh coquetry.
Something about the delicate figure made him think of Yasuko's sister.
Shingo had as a boy been strongly attracted to the sister. After her death Yasuko had gone to take care of the children. Yasuko had quite immersed herself in the work, as if wishing to supplant her sister. It was true that she had been fond of the brother-in-law, a handsome man, but she had also been in love with her sister, so beautiful a woman as to make it difficult to believe that the two could have had the same mother. To Yasuko her sister and brother-in-law had been like inhabitants of a dream world.
She worked hard for her brother-in-law and the children, but the man behaved as if he were quite indifferent to her feelings. He lost himself in pleasure, and for Yasuko self-immolation became a career.
And so Shingo had married her.
Now more than thirty years had passed, and Shingo did not think the marriage a mistake. A long marriage was not necessarily governed by its origins.
Yet the image of the sister remained with both of them. Neither spoke of her, and neither had forgotten her.
There was nothing especially unhealthy about the fact that, after Kikuko came into the house, Shingo's memories were pierced by moments of brightness, like flashes of lightning.
Married to her less than two years, Shuichi had already found another woman, a source of some surprise for Shingo.
Unlike Shingo himself, reared in the provinces, Shuichi showed no evidence of deprivation in matters of love and desire. Shingo could not have said when his son had had his first woman.
Shingo was certain that whoever now held Shuichi's attention was a business woman, perhaps a prostitute of sorts.
He suspected that affairs with women in the office meant no more than dancing after work, and might be only for purposes of distracting his father's attention.
She would not in any case be a sheltered girl like the one before him. Somehow Shingo had sensed as much from Kikuko herself. Since the beginning of the affair there had been a ripening in the relations between Kikuko and Shuichi. There had been a change in Kikuko's body.
Waking in the night—it was the night they had had the shellfish—Shingo heard Kikuko's voice as he had not heard it before.
He suspected that she knew nothing of Shuichi's mistress.
"And so Father has made the apologies, with a shellfish," he muttered to himself.
How was it that, although she knew nothing of the other woman, she should feel emanations come drifting toward her?
Shingo drowsed off, and suddenly it was dawn. He went for the paper. The moon was still high. After glancing over the news he fell asleep once more.
5
Shuichi pushed his way aboard the train and surrendered his seat to Shingo when the latter followed after.
He then handed over the evening paper and took Shingo's bifocals from his pocket. Shingo had a pair of his own, but he was much given to forgetting them. Shuichi was entrusted with a spare set.
Shuichi leaned over the paper. "Tanizaki said today that a classmate of hers was looking for work. We do need a maid, you know. So I said we'd take her."
"Don't you think it might be a little dangerous, having a friend of Tanizaki's around?"
"Dangerous?"
"She might hear things from Tanizaki and pass them on to Kikuko."
"What would she have to pass on?"
"Well, I suppose it will be good to have a maid with proper introductions." Shingo turned back to the paper.
"Has Tanizaki been talking about me?" asked Shuichi as they got off in Kamakura.
"She hasn't said a thing, I would have imagined that you had silenced her."
"Oh, fine. Suppose something actually were going on between me and your secretary. You'd be the joke of the office."
"Of course. But make sure, if you don't mind, that Kikuko doesn't find out."
Shuichi did not seem inclined toward secretiveness. "So Tanizaki has been talking."
"She knows you have a girl friend. And so I imagine she wants to go out with you herself."
"Maybe. Half of it might be jealousy."
"Splendid."
"I'm going to break it off. I'm trying to break it off."
"I don't understand you. Well, let me hear all about it some time."
"After I've broken it off."
"Don't let Kikuko know."
"She may already know."
Shingo lapsed into disgruntled silence.
It continued through dinner. He got up abruptly from the table and went to his room.
Kikuko brought him watermelon.
"You forgot the salt," said Yasuko, coming after her. The two sat down on the veranda. "Kikuko kept calling and calling. Didn't you hear her?"
"No. I did know that there was watermelon in the icebox."
"He didn't hear you," said Yasuko. "And you called and called."
It's because he's annoyed about something." Kikuko turned to her mother-in-law.
Shingo was silent for a moment. "There's been something wrong with my ears these last few days, I think. The other night I opened the shutter to let in a little air, and I heard the mountain rumbling. And you were snoring away."
Yasuko and Kikuko both looked toward the mountain.
"Do mountains roar?" asked Kikuko. "But you did say something once, Mother—remember? You said that just before your sister died Father heard the mountain roar."
Shingo was startled. He could not forgive himself for not remembering. He had heard the sound of the mountain, and why had the memory not come to him?
Apparently Kikuko regretted having made the remark. Her beautiful shoulders were motionless.
#
The Wings of the Locust
Fusako, the daughter, came home with her two children.
"Might another be on the way?" asked Shingo casually, although he knew that with the older girl four and the younger barely past her first birthday the spacing would not call for another quite yet.
"You asked the same question just the other day." She laid the younger child on its back and started to unswad-dle it. "And what about Kikuko?"
Her question was also a casual one, but Kikuko's face, as she looked down at the baby, was suddenly tense.
"Leave it as it is for a while," said Shingo.
"Her name is Kuniko, not 'it'. Didn't you name her yourself?"
Only Shingo, it seemed, was aware of the expression on Kikuko's face. He did not let it worry him, however. He was much taken with the movements of the emancipated little legs.
"Yes, leave her," said Yasuko. "She looks very happy. It must have been warm." She half tickled, half slapped the baby's stomach and thighs. "Why don't we send your mother and sister off, now, to freshen themselves up a bit?"
"Shall I get towels?" Kikuko started for the door.
"We've brought our own," said Fusako. It appeared that she meant to stay for some time.
Fusako took towels and clothes from a kerchief. The older child, Satoko, stood behind her, clinging sullenly to her. Satoko had not said a word since their arrival. Her thick black hair caught the eye.
Shingo had seen the kerchief before, but all he remembered was that it had been in the house. He did not know when.
Fusako had walked from the station with Kuniko on her back, Satoko tugging on one hand, the kerchief in the other. It must have been a pleasing sight, thought Shingo.
Satoko was not an easy child to lead. She had a way of being particularly difficult when matters were already complicated enough for her mother.
Did it trouble Yasuko, Shingo wondered, that of the two young women it was Kikuko who kept herself in good trim?
Yasuko sat rubbing a reddish spot on the inside of the baby's thigh. Fusako had gone to bathe. "I don't know, she somehow seems more manageable than Satoko."
"She was born after things started going bad with her father," said Shingo. "It all happened after Satoko was born, and it had an effect on her."
"Would a four-year-old child understand?"
"She would indeed. And it would influence her."
"I think she was born the way she is."
After elaborate contortions the baby turned over on its stomach, crawled off, and, catching hold of the door, stood up.
"Let's go have a walk, just the two of us," said Kikuko, taking the child by the hands and walking it to the next room.
Yasuko promptly went over to the purse beside Fu-sako's belongings and opened it.
"And what the devil do you think you're doing?" Shingo kept his voice low, but he was almost quivering with annoyance. "Stop it. Stop it, I tell you."
"And why should I?" Yasuko was calm.
"I told you to stop. What do you think you're up to?" His hands were trembling.
"I don't intend to steal anything."
"It's worse than stealing."
Yasuko replaced the purse. She was still sitting beside it, however. "And what is wrong with being interested in the affairs of your own daughter? Maybe she's come to us without enough money to buy the children candy. I want to know how things are with her. That's all."
Shingo glared at her.
Fusako came back from the bath.
"I looked inside your purse, Fusako," said Yasuko the moment her daughter stepped into the room, "and so I got a scolding from your father. If it was wrong I apologize."
"If it was wrong!" snorted Shingo.
This way of taking Fusako into her confidence only irritated him more.
He asked himself whether it might be true, as Yasu-ko's manner suggested, that such incidents were routine between mother and daughter. He was shaking with anger, and the fatigue of his years came flooding over him.
Fusako looked at him. It was possible that she was less surprised at her mother's behavior than at her father's.
"Please. Go ahead and look! Help yourself!" she said, half flinging the words out and slapping the purse down at her mother's knee.
Her manner did nothing to lessen his irritation.
Yasuko did not take up the purse.
"Without any money I wouldn't be able to run away, Aihara thought. I couldn't run away if I didn't have any money. So of course there's nothing in it. Go ahead and look."
Kuniko, her hands still in Kikuko's, suddenly collapsed. Kikuko picked her up.
Fusako lifted her blouse and presented her breast. She was not a beautiful woman, but she had a good figure. Her carriage was erect and the milk-swollen breast was firm.
"Is Shuichi away somewhere?" she asked, "Even on Sunday?"
She seemed to feel that she must do something to relieve the tension.
2
Almost home, Shingo looked up at the sunflowers blooming beside a neighboring house.
He was directly beneath the blossoms, which hung down over the gate.
The daughter of the house paused behind him. She could have pushed past him and gone into the house, but because she knew him, she waited there.
"What big flowers," he said, noticing her. "Remarkable flowers."
She smiled, a little shyly. "We pinched them back to one flower for each plant."
"Oh? That's why they're so big, then. Have they been blooming long?"
"Yes."
"How many days now?"
The girl—she was perhaps twelve or thirteen—did not answer. Apparently lost in silent calculation, she looked at Shingo, and then, with him, at the flowers again. Her face was round and sunburned, but her arms and legs were thin.
Thinking to make way for her, Shingo looked down the street. Two or three doors further on there were more sunflowers, three to each plant. The blossoms were only half the size of these,
As he started off he looked up again.
Kikuko was calling him. Indeed, she was standing right behind him. Stalks of green soybeans protruded from her market bag.
"You've been admiring the sunflowers?"
Of more concern to her, no doubt, than the fact that he admired the sunflowers, was the fact that he had come home without Shuichi. Almost home, he was viewing sunflowers by himself.
"They're fine specimens," he said. "Like heads of famous people."
Kikuko nodded, her manner casual.
Shingo had put no thought into the words. The comparison had simply occurred to him. He had not been searching for one.
With the remark, however, he felt in all its immediacy the strength of the great, heavy, flowering heads. He felt the regularity and order with which they were put together. The petals were like crowns, and the greater part of the central discs was taken up by stamens, clusters of them, which seemed to thrust their way up by main strength. There was no suggestion that they were fighting one another, however. They were quietly systematic, and strength seemed to flow from them.
The flowers were larger in circumference than a human head. It was perhaps the formal arrangement of volume that had made Shingo think of a brain.
The power of nature within them made him think of a giant symbol of masculinity. He did not know whether they were male or not, but somehow he thought them so.
The summer sun was fading, and the evening air was calm.
The petals were golden, like women.
He walked away from the sunflowers, wondering whether it was Kikuko's coming that had set him to thinking strange thoughts.
"My head hasn't been very clear these last few days. I suppose that's why sunflowers made me think of heads, I wish mine could be as clean as they are. I was thinking on the train—if only there were some way to get your head cleaned and refinished, Just chop it off—well, maybe that would be a little violent. Just detach it and hand it over to some university hospital as if you were handing over a bundle of laundry. 'Do this up for me, please,' you'd say. And the rest of you would be quietly asleep for three or four days or a week while the hospital was busy cleaning your head and getting rid of the garbage. No tossing and no dreaming."
"You must be tired," said Kikuko, a shadow passing over her face.
"I am. Today someone came to see me in the office. I took a puff on a cigarette and laid it down and lighted another and laid it down, and I saw that there were three of them, lighted and almost unsmoked. It was very embarrassing."
He had thought on the train of sending his head to a laundry, it was true, but he had been drawn not so much to the idea of the laundered head as to that of the sleeping body. A very pleasant sleep, with head detached. There could be no doubt of it: he was tired.
He had had two dreams toward dawn this morning and the dead had figured in both.
Aren't you taking a vacation this summer?"
"I'd thought of going to Kamikochi. There's no one I can leave my head with, and so I think I'd like to go have a look at the mountains."
"Oh, go, by all means," said Kikuko, a little too gaily.
"But we have Fusako with us now. She's come for a rest too. What do you think? Would it be better for her with me in the house, or away?"
"I envy her, having such a good father." Kikuko did not seem wholly at ease.
Had he hoped, he wondered, to badger her, throw her off the scent, distract her from the image of his own solitary figure, coming home without his son? Such had not been his conscious intention; and yet he wondered.
"Are you being sarcastic?" he asked.
He spoke lightly, but Kikuko seemed surprised.
"Take a look at Fusako and then tell me whether I've been a good father."
She flushed to the ears. "It wasn't your fault about Fusako," she said, and he felt consolation in her voice.
3
Shingo disliked cold drinks even in hot weather. Yasuko did not give them to him, and the habit of not taking them had formed over the years.
In the morning when he got up and in the evening when he came home he would have a brimming cup of tea, Kikuko always saw to supplying it.
When they got home from viewing the sunflowers she hurried for his tea. He drank about half of it, changed to a cotton kimono, and took his cup out to the veranda, sipping as he went. Kikuko came after him with a cold towel and cigarettes and poured more tea. Then she went for his glasses and the evening paper.
He looked out at the garden. It seemed too much of an effort, when he had wiped his face, to put on his glasses.
The grass was rough and untended. On the far side was a clump of bush clover and pampas grass, so tall that it almost looked wild.
There were butterflies beyond. Shingo could see them flickering past gaps in the leaves, more than one butterfly, surely. He waited to see whether they would alight on the bush clover or come out from behind it. They went on fluttering through the leaves, however.
He began to feel that there was some sort of special little world apart over behind the shrubbery. The butterfly wings beyond the leaves of bush clover seemed to him extraordinarily beautiful.
He thought of the stars he had seen through the trees on the hilltop, that night a month earlier, when the moon had been near full.
Yasuko came out and sat down beside him.
"Shuichi will be late again?" she asked, fanning herself.
Shingo nodded and continued to look at the garden. "There are butterflies behind the shrubbery."
But as if they disliked being seen by Yasuko, the butterflies flew up over the bush clover. There were three of them.
"Swallowtails."
For swallowtails they were small, and their color was somehow muddy.
They cut a line diagonally across the board fence and emerged against the pine next door. Forming a vertical column, they proceeded, without breaking the column or changing the distance that separated them, up the middle of the tree to the top. It had grown untended, and did not have the shaped look of a garden tree.
A moment later another swallowtail butterfly appeared from an unexpected quarter and, describing a horizontal line across the garden, skimmed the top of the bush clover.
"This morning I had two dreams about dead people. The old man at the Tatsumiya treated me to noodles."
"You didn't eat them, did you?"
"Shouldn't I have?" Shingo wondered if eating food offered in a dream by a dead person meant that the dreamer himself would die. "I don't really remember. I don't think I did. I do remember that they were cold." He thought he must have awakened before eating.
He could remember even the color of the noodles, laid on bamboo, in a frame lacquered black on the outside and red on the inside.
He did not know, however, whether he had seen the color in the dream or assigned it upon awakening. In any case, the noodles were clear in his mind, though everything else was blurred.
One helping of noodles had been laid on the floor, and it seemed that Shingo had been standing beside it. The shopkeeper and his family, it seemed, had been sitting down. It seemed that no one had had a cushion to sit on. It seemed, strangely, that Shingo alone had been standing. So much he could remember, but only vaguely.
Awakening from the dream, he had remembered it clearly. After going back to sleep and getting up in the morning, he had remembered it even more clearly. Now, however, it was almost gone. The picture centering on the noodles had stayed in his mind, but he could not remember the plot, what had gone before and followed after.
The man in the dream was a cabinetmaker who had died in his seventies some three or four years before. Because he was an artisan of the old school, Shingo had taken a great liking to him and given him considerable work. Yet he had not been such a close friend as to figure in a dream so long after his death.
It seemed to Shingo that the noodles had appeared in the family quarters, at the back of the shop. Even though he might on occasion have stood outside talking to the old man, he could not remember having gone into the back rooms. He was puzzled to know why he should have had a dream in which noodles figured.
The old man had had six children, all daughters.
Shingo had slept with a girl in the dream, but now, in the evening, he could not remember whether or not it had been one of the daughters.
He remembered clearly having touched someone, but he had no notion who she might have been. He could remember nothing that even gave him a hint.
He felt that he had known who it was when he woke, and when, after going to sleep again, he had again awakened, he had perhaps still known. But now, in the evening, he could remember nothing at all.
Since the dream was a continuation of the one about the old cabinetmaker, he tried to decide whether the girl he had slept with might have been one of the man's daughters. No sort of awareness came to him. He could not even call up the faces of the Tatsumi daughters.
It was a continuation, that much was clear; but he did not know what had gone before and come after the noodles. It now seemed likely that they had been the clearest image in his mind when he woke. Yet would it not be true to the laws of dreams if he bad awakened at the shock of contact with the girl?
Not, of course, that it had been a sharp enough sensation to wake him.
Here, too, nothing definite of the dream remained. The figure had gone, and he could not bring it back; all that remained was a sense of physical disparity, a failure of physical contact.
Shingo had not, in actuality, experienced such a woman. He had not recognized her, but because she had been a mere girl, the meeting could not have happened in real life.
At sixty-two, an absence of sensual dreams would not be unusual, but what puzzled him now was the positive insipidity of it all.
He had promptly gone back to sleep and had another dream.
Fat old Aida had come around, a half-gallon bottle of sake in his band. He had, it seemed, drunk a good bit already. The pores on his red face were agape.
Shingo could remember no more of the dream. He did not know whether the house had been this one or a house he had lived in earlier.
Aida had, until ten years or so before, been a director of Shingo's company. He had died of apoplexy toward the end of the previous year. In his last years he had grown thin.
"And then I had another dream. This time Aida came around to the house with a bottle."
"Mr. Aida? But that's strange. Mr. Aida didn't drink."
"That's true. He had asthma, and when he had his stroke it was the mucus that killed him. But he didn't drink. He was always wandering around with a medicine bottle in his hand."
And yet he had strode into the dream like a brave roisterer. The image floated up vividly in Shingo's mind.
"And did you and Mr. Aida have a drinking party?"
"I didn't have a drop. Aida was walking toward me, but I woke up before he had a chance to sit down."
"It's not very pleasant, dreaming of dead people."
"Maybe they've come for me."
He had reached an age when most of his friends were dead. It was perhaps natural that he should dream of the dead.
Neither the old cabinetmaker nor Aida had appeared to him as dead, however. They had come into his dreams as living people.
And the figures of both, as they had come into the dreams, were still vivid in his mind. They were much clearer than his usual memories of the two men. Aida's face, red from drink, was of a sort that the living Aida had never presented; and yet Shingo remembered such details as the distended pores.
Why should it be that, remembering the other two so clearly, he could not call up the face of the girl who had touched him, could not remember who she might be?
He asked whether, from feelings of guilt, he had managed to forget. But such did not seem to be the case. He had not been awake long enough for more than a certain sensual disappointment.
He was not especially interested in the fact that it had come to him in a dream.
He did not describe this part of the dream to Yasuko.
Kikuko and Fusako were getting dinner. He could hear their voices in the kitchen. They seemed a trifle too loud.
4
Every night locusts would come flying in from the cherry tree.
Shingo walked over to the trunk of the tree.
Engulfed by the sound of whirring wings, he looked up. He was astonished at the number of locusts, and astonished too at the noise of their wings. It was as if a flock of sparrows had started up.
Locusts were flying off as he looked into the great tree.
All the clouds in the sky were racing toward the east. The weather forecast had said that that most ominous of days, the two-hundred-tenth after the beginning of spring,* was likely to pass without incident, but Shingo suspected that there would be winds and showers to bring down the temperature.
"Has something happened?" Kikuko came up. "I heard locusts and wondered."
"They do make you think there might have been an accident, don't they? You hear about the wings of ducks and geese, but these are just as impressive."
Kikuko was holding a needle and red thread. "It wasn't the wings. It was the screeching all of a sudden, as if something might be threatening them."
"I hadn't noticed that so much."
He looked into the room from which she had come. Spread out in it were the makings of a child's dress, the cloth from an ancient singlet of Yasuko's. "Does Satoko still play with locusts?"
Kikuko nodded. A faint motion of her lips seemed to shape the word "yes."
Locusts were strange and interesting creatures to Satoko, a child of the city; and there was something in her nature that responded to the sport. At first she had been afraid when Fusako had given her one to play with. Then Fusako had cut off the wings, and afterwards whenever the child caught a locust she would come running up to anyone nearby, Kikuko or Yasuko or whomever, to have the wings clipped.
Yasuko hated the practice,
Fusako had not always been that sort of girl, she grumbled. Her husband had ruined her.
Yasuko had blanched when she found a swarm of red ants dragging off a wingless locust.
She was not, on the whole, a person to be moved by such matters, Shingo was both amused and disturbed.
Her recoil, as from a poisonous vapor, was perhaps a sign of some evil foreboding. Shingo suspected that locusts were not the problem.
Satoko was an obdurate child, and when the adult in question had surrendered and cut the wings she would still be dawdling about. Then, with somber, shadowy eyes, she would throw the insect, its wings freshly cut, out into the garden, as if to hide it. She knew that adults would be watching her.
Fusako apparently poured forth her complaints to Yasuko every day, but it seemed, from the fact that she never touched upon the question of when she would be leaving, that she had not yet brought herself to the heart of the matter.
When they were in bed Yasuko would pass the day's complaints on to Shingo. Though he did not pay a great deal of attention, he would feel that something had been left out.
He knew that as her father he should step forward to give Fusako advice; but she was thirty and married, and matters are not simple for fathers in such cases. It would not be easy to accommodate a woman with two children. A decision was postponed from day to day, as if the principals were all waiting for nature to take its course.
"Isn't Father nice to Kikuko," said Fusako.
Kikuko and Shuichi were both at the dinner table.
"Yes, of course," said Yasuko. "I try to be good to her myself."
Fusako's manner had not suggested that she required an answer. There was laughter in the tone of Yasuko's gratuitous answer, but it was meant to quell Fusako all the same.
"After all, she's good to us."
Kikuko turned crimson.
Yasuko's second remark was uncomplicated enough. It contained something like a thrust at her daughter, however.
It seemed to suggest that she liked her happy daughter-in-Iaw and disliked her unhappy daughter. One might have suspected cruelty and malice. Shingo sensed something like self-loathing too. He detected a similar vein in himself. Yet it seemed strange to him that Yasuko, woman and aging mother, should have given way to it in the presence of her daughter.
"I don't agree that she's all that kind," said Shuichi. "She's not to her husband." The joke was not successful.
It should have been clear to all of them, to Shuichi and Yasuko as well as to Kikuko herself, that Shingo was particularly gentle toward Kikuko. The fact scarcely needed mentioning, and somehow mention of it saddened him.
Kikuko was for him a window looking out of a gloomy house. His blood kin were not as he would wish them to be, and if they were not able to live as they themselves wished to live, then the impact of the blood relation became leaden and oppressive. His daughter-in-law brought relief.
Kindness toward her was a beam lighting isolation. It was a way of pampering himself, of bringing a touch of mellowness into his life.
For her part, Kikuko did not indulge in dark conjectures on the psychology of the aged, nor did she seem afraid of him.
Fusako's remark, he felt, brushed against his secret.
It had been made at dinner some three or four evenings before.
Under the cherry tree, Shingo thought of it, and of Satoko and the locust wings.
"Is Fusako having a nap?"
'Yes." Kikuko looked into his face. "She's giving Kuniko hers."
She's a funny child, Satoko. Whenever Fusako gives the baby its nap she goes along and lies there clinging to her mother's back. That's when she behaves."
"It's sweet, really."
"Yasuko can't stand the child. But when she gets to be fourteen or fifteen she'll be snoring away, the image of her grandmother."
Kikuko did not seem to understand.
She called after Shingo as he turned to go off.
"You went dancing?"
"What?" Shingo looked around. "You know about it, do you?"
Two nights earlier he had gone to a dance hall with the girl from his office.
Today was Sunday; so it would appear that the girl, Tanizaki Eiko, had told Shuichi the day before, and Shuichi had passed the news on to Kikuko.
Shingo had not been dancing in years. The girl had clearly been surprised at his invitation. She had said that if she went out with him troublesome rumors would spread through the office, and he had said that she only needed to keep quiet. And it seemed that she had promptly told Shuichi.
Shuichi, for his part, had neither yesterday nor today given Shingo a hint that he knew.
Eiko evidently went dancing with Shuichi from time to time. Shingo had asked her out because he had thought he might see Shuichi's mistress at the hall the two frequented.
He bad not, however, been able to find a likely girl, and he had not been up to asking Eiko for an identification.
Apparently the surprise had made the girl a bit giddy. The note of discord struck Shingo as dangerous and touching.
Although in her early twenties, she had tiny breasts, barely enough to fill one's cupped hands. Shingo was put in mind of an erotic print by Harunobu.
Given the noisy surroundings, he was somehow amused by the association.
"Next time let me take you," he said to Kikuko.
"Yes. Please do."
She had been blushing from the time she called him back.
Had she guessed that he had gone in the hope of seeing Shuichi's mistress?
He had no particular reason to keep the incident a secret, but the thought of the other women left him a little flustered.
He went from the front door to Shuichi's room. "Tanizaki told you?" He did not sit down.
"Ah, yes. That she did. Important news having to do with our household."
"I wouldn't have thought it all that newsworthy. But when you take her dancing next time, buy her a decent summer dress."
"You were ashamed of her, were you?"
"The blouse and skirt didn't seem to match very well."
"Oh, she has clothes enough. It's your fault for not Warning her. Just make your dates in advance, and she'll come dressed for the occasion," He turned away.
Skirting the room where Fusako and the two children were sleeping, Shingo looked up at the clock.
Five," he muttered, as if to confirm an important fact.
## Footnote
* Early in September.
#
A Blaze of Clouds
Although the newspaper had predicted that the two-hundred-tenth day would pass uneventfully that year, there was a typhoon the night before.
Shingo could not remember how many days earlier he had seen the article, and so it could not perhaps have been called a weather forecast. There were of course forecasts and warnings as the day approached.
"I suppose you'll be coming home early tonight?" said Shingo to Shuichi. It was more a suggestion than a query.
Having helped Shingo with his preparations for departing, the girl Eiko hurried to go home herself. Through the transparent white raincoat, her breasts seemed even smaller.
He had taken greater notice of them since, on the night he had gone dancing with her, he had noticed how meager they were.
Eiko came running down the stairs after them and stood with them in the entrance. Because of the downpour, she had apparently not taken time to repowder her face.
"And where is it you live?" But Shingo did not finish the question. He must have asked it twenty times already, and he did not remember the answer.
At Kamakura Station passengers stood under the eaves trying to judge the violence of the wind and rain.
As Shingo and Shuichi passed the house with sunflowers at the gate, the theme song from Quatorze Juillet came through the wind and rain.
"She doesn't seem very worried," said Shuichi.
They knew that it would be Kikuko playing the Lys Gauty record.
When it was over she started it again.
Midway through they heard shutters being closed.
And they heard Kikuko singing to the record as she closed them.
Through the storm and the music, she did not hear the two come in from the gate.
"My shoes are flooded." Shuichi took off his stockings in the doorway.
Shingo went in, wet stockings and all.
So you're back." Kikuko came toward them, her face glowing with pleasure.
Shuichi handed her his stockings.
"Father's must be wet too," said Kikuko. Starting the record again, she went off with their wet clothes.
"They can hear you all over town, Kikuko," said Shuichi as he wound an obi around his waist. "You might try to seem a little more worried."
"But I was playing it because I was worried. I couldn't sit still, thinking about the two of you."
But her frolicsome manner suggested that she found the storm exhilarating.
She was still humming to herself as she went off for Shingo's tea.
Shuichi, fond of the Parisian chanson, had bought the collection for her.
He knew French. Kikuko did not, but, with lessons in pronunciation, she had become fairly proficient at imitating the record. Not, of course, that she could give, as could Gauty, a sense of having struggled and somehow lived on. All the same, her delicate, hesitant delivery was most pleasing.
Kikuko's wedding present from her seminary classmates had been a collection of nursery songs from the world over. In the early months of her marriage she had been very fond of it. When she was alone, she would quietly join in the singing; it gave Shingo a sense of warm repose.
A most womanly kind of observance, thought Shingo. And he felt that, listening to the nursery songs, she was sunk in memories of her girlhood.
"Shall I ask you to play them at my funeral?" Shingo had once said to her. "Then I won't need any prayers." He had not been serious, but then suddenly he was on the edge of tears.
But Kikuko was still childless, and it seemed, since he had not heard it recently, that she had tired of the collection.
As the chanson was nearing its end, it suddenly faded away.
"The electricity has gone off," said Yasuko from the breakfast room.
"It won't go back on tonight," said Kikuko, switching off the phonograph. "Let's have dinner early, Mother."
At dinner, the thin candles went out three or four times as the wind blew through cracks in the shutters.
The ocean seemed to be shouting above the wind. It was as if the sea were doing more than the wind to heighten the terror.
2
The scent of the candle that he had just blown out was still in Shingo's nostrils.
Each time the house would shake, Yasuko would reach for the matchbox on the bed and rattle it, as if to reassure herself and to let Shingo know.
And she would reach for his hand, and gently touch it.
"Will we be all right?"
Of course. And if something does blow over the fence, we can't very well go out and look."
"Will it be all right at Fusako's?"
At Fusako's?" He had not thought of Fusako. "I imagine so. On a night like this they ought to go off to sleep early like a good married couple, whatever they do on other nights."
"How could they sleep?" Turning away his remark, she fell silent.
They heard Shuichi's voice and Kikuko's. There was a soft coaxing quality in Kikuko's.
"She has two small children," said Yasuko after a time. "Things are not as easy as they are with us."
"And he has a crippled mother. How is her arthritis?"
"There's that too. If they were to run away Aihara would have to carry the old lady on his back."
"Can't she walk?"
"She can move around, I believe. But in this storm? Gives you the blues, doesn't it."
"Gives you the blues?" The word "blues" from the sixty-three-year-old Yasuko struck Shingo as comical.
"It said in the paper that a woman changes her hair style any number of times in the course of her life. I liked that."
"What was it in?"
It was, according to Yasuko, in the opening words of the eulogy of a painter in the old style, a specialist in portraits of women, to a recently deceased woman painter, also of old-style beauties.
But in the eulogy proper it came out that with the woman artist the case had been the opposite. For a good fifty years, from her twenties to her death at seventy-five, she had worn her hair straight back and held in place by a comb.
Yasuko apparently found it admirable that a woman could make her way through life with her hair pulled straight back; but the thought that through her life a woman wore her hair in many ways also seemed to appeal to her.
Yasuko was in the habit of saving the newspapers she read every day and looking again through several days' accumulation. One could not be sure how old an article she would suddenly come up with. And since she always listened carefully to the nine-o'clock news commentary as well, she would launch forth on the most improbable topics.
"And so you mean that Fusako will do her hair all sorts of ways?"
"She's a woman, after all. But there won't be as many changes with her as there were with us who did it the old way. And it would be more fun if she were as good-looking as Kikuko."
"You weren't kind to her when she came home. She was desperate."
"Don't you suppose I was under your influence? You only care about Kikuko."
"That's not true. An invention of yours."
"It is true. You never liked Fusako—Shuichi was always your favorite. That's the way you are. Even now that he has another woman you can't say anything to him. And you really show too much affection for Kikuko. It amounts to cruelty. She can't give a sign of her jealousy because she's afraid of what it might do to you. It really gives me the blues. I hope the typhoon blows us all away."
Shingo was startled, "A typhoon," he said, thinking of the rising fury of his wife's observations.
Yes, it is a typhoon. And Fusako, trying to have her Parents get a divorce for her, at her age, in this day and age. It's cowardly."
Not really. But has there been talk of a separation?"
"More important is what I can see right ahead of me, your scowling face when she comes back and you have to take care of her and those two children."
"You've been outspoken enough yourself."
"That is because we have Kikuko, whom Father is so fond of. But Kikuko aside, I have to admit I don't like it. Sometimes Kikuko says or does something that takes a load off my mind, but when Fusako says something the load only gets heavier. It wasn't so bad before she got married. I know perfectly well that it's my own daughter and grandchildren I'm talking about, and I can still feel that way? Frightening, that's what it is. It's your influence."
"You're more of a coward than Fusako."
"I was joking. You couldn't see me stick out my tongue."
"The old woman is good with her tongue. Remarkably."
"But I do feel sorry for her. Don't you?"
"We can take her in if you want." Then, as if he remembered something: "The kerchief she brought with her."
"The kerchief?"
"The kerchief. I've seen it before, but can't remember where. Is it ours?"
"The big cotton one? She took her mirror in it when she got married. It was a very big mirror."
"So that was it."
"I didn't like that bundle. She could perfectly well have put her things in the suitcase she took on her honeymoon."
"A suitcase would have been heavy, and she had those two children. And I don't suppose she cared very much at that point how she looked."
"But we have Kikuko to think of. That kerchief—I brought something wrapped in it when we were married."
"Oh?"
"It's even older. It was my sister's. When she died they sent it home with a dwarf tree tied up in it. A fine maple."
"Oh?" said Shingo again, softly. His head was full of the red glow of that remarkable maple.
Back in the country, his father-in-law's chief extravagance had been dwarf trees. He gave particular attention, it seemed, to maples. Yasuko's elder sister was his assistant.
In bed with the storm roaring about him, Shingo could see her among the shelves of dwarf trees.
Probably her father had given her one when she married. Perhaps she had asked for it. And when she died her husband's family had sent it back, because it was so important to her father, and because they had no one to look after it. Or possibly her father had gone for it.
The maple that now filled Shingo's head had been on the family altar.
Had her sister died in autumn then? Autumn came early to Shinano.
But would they have sent it back immediately upon her death? That it should have been red and on the altar made everything seem a little too neatly arranged. Was not a nostalgic syndrome working upon his imagination? had no confidence.
Shingo could not remember the anniversary of his sister-in-law's death. Yet he did not ask Yasuko.
That was because Yasuko had once said: "Father never let me help him with his trees, I suppose it had something to do with my nature, but he felt much closer to my sister. I couldn't stand up to her myself. I wasn't just jealous, I was ashamed. She did everything so much better than I did."
That was the sort of remark she could make when the talk touched upon Shingo's preference for Shuichi, and she would add: "I suppose I was rather like Fusako myself."
Shingo was surprised to learn that the kerchief was a memento of Yasuko's sister. He fell silent, now that the sister had come into the conversation.
"Suppose we go to sleep," said Yasuko. "They'll think we old ones have trouble sleeping too. Kikuko laughed away through the storm, and she put on one record after another, I am sorry for her."
"There was a contradiction even in those few words."
"There always is."
"That was for me to say. I go to bed early for a change, and see what happens to me."
The dwarf maple was still with Shingo.
And in another part of his mind he asked whether, even now that he had been married to Yasuko for more than thirty years, his boyhood yearning for her sister was still with him, an old wound.
He went to sleep an hour or so after Yasuko, A violent crash awoke him.
"What is it?"
He heard Kikuko groping her way along the veranda.
"Are you awake? They say a sheet of tin from the shrine blew over onto our roof."
3
The tin roof of the mikoshi* shed had quite blown away.
The caretaker came early in the morning to collect seven or eight sheets from Shingo's roof and garden.
The Yokosuka line was running. Shingo left for work,
"How was it? Could you sleep?" Shingo asked Eiko as she brought tea.
"Not a wink." Eiko described the wake of the storm as she had seen it from the train window.
"I don't suppose we can go dancing today," said Shingo after he had had a cigarette or two.
Eiko looked up smiling.
"The morning after the other time, my hips were stiff. It's my age."
She smiled mischievously, from her eyes down toward her nose. "Don't you suppose it's because of the way you arch your back?"
"Arch my back? Do I? Do I bend from the hips?"
"You arch your back and keep your distance. As if it might be against the law to touch me."
"That can't be true."
"Oh, but it is."
"Was I trying to make myself look good? I wasn't aware of it."
"No?"
"You young people hang on to each other so when you dance. It's all in very bad taste."
"That's not kind of you."
Shingo had thought, the time before, that Eiko had been a little off balance, a little giddy perhaps; and that had been too sanguine a view. The point was rather that he had been stiff and clumsy himself.
"Well, let's go again. This time I'll lean forward and hang on to you."
She looked down and laughed. "I'd be delighted. But not tonight. Not in this dress."
"No, not tonight."
She was wearing a white blouse and had a white ribbon in her hair.
It was not unusual for her to wear a white blouse. Possibly the white ribbon, a fairly broad one, made it look whiter. Her hair was knotted tightly at the back. She was, one might say, dressed for a storm.
The hairline was fresh and clean, tracing a curve behind her ears. The hair stood out cleanly against the fair skin it normally covered.
She had on a thin wool skirt of navy blue. It was somewhat worn.
When she was so dressed, the smallness of her breasts did not matter.
"Has Shuichi asked you out again?"
"No."
"What a pity. The young man keeps his distance because you go dancing with his father."
"I'll have to ask him to go out."
"And so I needn't worry?"
"If you insist on making fun of me, I'll have to refuse to go dancing with you."
"I'm not making fun of you. But I haven't been able to look you in the eye since you began noticing him."
She reacted with silence,
"I suppose you know Shuichi's woman."
This time she registered confusion.
"A dancer?"
There was no reply.
"Is she older?"
"Older? She's older than his wife."
"And good-looking?"
"Yes, very good-looking." She stumbled over the words, but continued: "She has a husky voice. No, not so much husky as broken, you might say. In two parts. He finds it very erotic."
"Well!"
She seemed about to go on. He did not want to listen.
He felt ashamed for himself, and he felt a revulsion, as if the true nature of Shuichi's woman and of Eiko herself were about to emerge.
He was taken aback by this initial observation, about the eroticism in the woman's voice. There had been bad taste on Shuichi's part, of course, but what about Eiko herself?
Noting the displeasure on his face, Eiko fell silent.
That night too Shuichi went home with Shingo, When they had closed the shutters the four of them went out to see a movie version of the Kabuki play Kanjincho.
As Shuichi took off his undershirt, changing to go to the movie, Shingo saw red marks high on his chest and shoulder. Had Kikuko left them there during the storm?
The principal actors in the film, Koshiro and Uzaemon and Kikugoro, were all dead.
Shingo's feelings were different from those of Kikuko and Shuichi.
"I wonder how many times we saw Koshiro do Benkei," said Yasuko.
"I forget."
"Yes, you always forget."
The town was bright in the moonlight. Shingo looked up at the sky.
The moon was in a blaze. Or so, just then, it seemed to Shingo.
The clouds around the moon made him think of the flames behind Acala in a painting, or a painting of a fox-spirit. They were coiling, twisted clouds.
But the clouds, and the moon too, were cold and faintly white. Shingo felt autumn come over him.
The moon, high in the east, was almost full. It lay in a blaze of clouds, it was dimmed by them.
There were no other clouds near the blaze in which the moon lay. In a single night after the storm the sky had turned a deep black.
The shops were shuttered. The town too had taken on a melancholy aspect in the course of the night. People were on their way home from the movie through silent, deserted streets.
"I couldn't sleep last night. I'm going to bed early." Shingo felt a lonely chill pass over him, and a yearning for human warmth.
And it was as if a crucial moment had come, as if a decision were forcing itself upon him.
## Footnote
* The portable shrine used in Shinto festivals.
#
The Chestnuts
The gingko is sending out shoots again," said Kikuko.
"You've only just noticed?" said Shingo. "I've been watching it for some time now."
"But you always sit facing it, Father."
Kikuko, who sat so that Shingo saw her in profile, was looking at the gingko behind her.
The places of the four as they took their meals had in the course of time become fixed.
Shingo sat facing east. On his left was Yasuko, facing south, and on his right Shuichi, who faced north. Kikuko, facing west, sat opposite Shingo.
Since the garden was to the south and east, it might be said that the old people occupied the better places. And the women's places were the convenient ones for serving.
At times other than meals, they had come to occupy the same fixed places.
So it was that Kikuko always had the gingko behind her.
Yet Shingo was troubled: that she had not noticed unseasonal buds on the great tree suggested a certain emptiness.
"But you ought to notice when you open the shutters or go out to clean the veranda," he said.
"I suppose that's true."
"Of course it is. And you're facing it when you come in the gate. You have to look at it whether you want to or not. Do you have so much on your mind that you come in looking at the ground?"
"This will never do." Kikuko gave her shoulders that slight, beautiful shrug. "I'll be very careful from now on to notice everything you do and imitate it."
For Shingo, there was a touch of sadness in the remark. "This won't do either."
In all his life no woman had so loved him as to want him to notice everything she did.
Kikuko continued to gaze in the direction of the gingko. "And some of the trees up the mountain are putting out new leaves."
"So they are. I wonder if they lost their leaves in the typhoon."
The mountain in Shingo's garden was cut off by the shrine precincts, a level stretch just above. The gingko lay at the boundary, but from Shingo's breakfast room it looked as if it were yet higher.
It had been stripped bare on the night of the storm.
The gingko and the cherry were the trees left bare by the wind.
Since they were the larger of the trees around the house, they were perhaps good targets for the storm. Or was it that their leaves were especially vulnerable?
The cherry had had a few drooping leaves even after the storm, but it had shed them since, and now stood quite naked.
The leaves of the bamboo up the mountain had withered, perhaps because, with the ocean so near, the wind had brought in salt spray. Stalks of bamboo had broken off and blown into the garden.
The great gingko was again sending out buds.
Shingo faced it as he turned up the lane from the main street, and every day on his way home he looked at it. He also saw it from the breakfast room.
"The gingko has a sort of strength that the cherry doesn't," he said. "I've been thinking the ones that live long are different from the others. It must take a great deal of strength for an old tree like that to put out leaves in the fall."
"But there's something sad about them."
"I've been wondering whether they'd be as big as the leaves that came out in the spring, but they refuse to grow."
Besides being small, the leaves were scattered, too few to hide the branches. They seemed thin, and they were a pale yellowish color, insufficiently green.
It was as if the autumn sun fell on a gingko that was, after all, naked.
The trees in the shrine precincts were mostly evergreen. They seemed to be strong against wind and rain, and were quite undamaged. Above the luxuriant evergreens was the pale green of new leaves. Kikuko had just discovered them.
Yasuko had come in through the back gate. He heard running water. She said something, but, over the sound of water, he could not make out what.
"What did you say?" he shouted.
Kikuko helped him. "She says that the bush clover is blooming very nicely."
"Oh?"
Kikuko passed on another message. "And she says that the pampas grass is putting out plumes."
"Oh?"
Yasuko had something more to say.
"Oh, be quiet. I can't hear you."
"I'll be happy to interpret." On the edge of laughter, Kikuko looked down.
"Interpret? It's just an old woman talking to herself."
"She says she dreamed last night that the house in Shinano was going to pieces."
"Oh?"
"And what is your answer?"
"I said 'Oh,' and that's all I have to say."
The sound of water stopped. Yasuko called Kikuko.
"Put these in water, please, Kikuko. They were so beautiful that I had to break some off. But you take care of them, please."
"Let me show them to Father first."
She came in with an armful of bush clover and pampas grass.
Yasuko had evidently washed her hands and then moistened a Shigaraki vase, which she brought in.
"The amaranth next door is a beautiful color too," she said as she sat down.
"There is amaranth by the house with the sunflowers," said Shingo, remembering that those remarkable sunflowers had been knocked down in the storm.
Blossoms had lain in the street, broken off with six inches or so of stem. They had been there for several days, like severed human heads.
First the petals withered, and then the stems dried and turned dirty and gray.
Shingo had to step over them on his way to and from work. He did not like to look at them.
The bases of the stems stood leafless by the gate.
Beside them, five or six stalks of amaranth were taking on color,
"But there aren't any around here like the ones next door," said Yasuko.
2
It was her family house that Yasuko had dreamed of.
It had been unoccupied for several years now, since her parents' death.
Apparently meaning Yasuko to succeed to the family name,* her father had sent his older daughter out in marriage. It should have been the opposite for a father who favored his older daughter, but, with so many men asking for the hand of her beautiful sister, he had probably felt sorry for Yasuko.
Perhaps, therefore, he gave up hope for Yasuko when, after her sister's death, she went to work in the house into which the sister had married, and seemed intent upon taking her place. Perhaps he felt a certain guilt because parents and family had made her feel so inclined,
Yasuko's marriage to Shingo seemed to please him.
He decided to live out his years with no family heir.
Shingo was now older than the father had been when he gave Yasuko in marriage,
Yasuko's mother had died first, and the fields had all been sold when the father died, leaving only the house and a modest amount of forest land. There were no heirlooms of any importance.
The remaining property was in Yasuko's name, but the management had been turned over to a country relative, The forests had probably been cut down to pay taxes. It had been many years since Yasuko had last had either income or expenses related to the country place.
There was a prospective buyer when, during the war, the countryside was crowded with refugees, but Yasuko felt nostalgic about the house, and Shingo did not press her.
It was in that house that they had been married. In return for giving his only surviving daughter in marriage, the father had asked that the ceremony be held in his house.
A chestnut fell as they were exchanging marriage cups. It struck a large stone in the garden, and, because of the angle, rebounded a very long way and fell into a brook. The rebound was so extraordinary that Shingo was on the point of calling out in surprise. He looked around the room.
No one else seemed to have noticed.
The next day Shingo went down to hunt for it. He found several chestnuts at the edge of the water. He could not be sure he had the one that had fallen during the ceremony; but he picked one up, thinking to tell Yasuko of it.
But then he decided that he was being childish. And would Yasuko, and others to whom he might speak of it believe him?
He threw it into a clump of grass by the water.
It was less fear that Yasuko would not believe him than shyness before her brother-in-law that kept him from speaking.
Had the brother-in-law not been present, Shingo might have spoken of it at the ceremony the day before In the presence of her brother-in-law, he felt a constraint very like shame.
He had certain feelings of guilt for having continued to be drawn to the sister even after she was married and the sister's death and Yasuko's marriage had disturbed her brother-in-law.
For Yasuko, the feelings of shame must have been even stronger. One might say that, pretending not to know her real feelings, her sister's widower had used her a convenient substitute for a maid.
It was natural that, as a relative, he should be invited Yasuko's wedding. Very uncomfortable all the same Shingo found it difficult to look at him.
The brother-in-law was a handsome man who quite outshone the bride. It seemed to Shingo that there was a peculiar radiance in his part of the room.
To Yasuko, her sister and brother-in-law were inhabitants of a dream world. In marrying her, Shingo had tacitly descended to her own lower rank.
He felt as if her brother-in-law were coldly looking down on the wedding from an elevation.
And the blank left by his failure to speak of so small a thing as the falling chestnut probably stayed on in their marriage.
When Fusako was born, Shingo secretly hoped that she might be a beauty like her aunt. He could not speak of this hope to his wife. But Fusako proved to be even homelier than Yasuko.
As Shingo would have put it, the blood of the older sister had failed to flow through the younger. He was disappointed in Yasuko.
Three or four days after Yasuko dreamed of the house in the country, a telegram came from a relative saying that Fusako had arrived with her two children.
Kikuko signed for the telegram and passed it on to Yasuko, who waited for Shingo to come home from the office.
"Was something warning me in that dream?" She was remarkably calm as she watched Shingo read the telegram.
"Back to the country, is it?"
So she won't kill herself—that was the first thought that came to him.
"But why didn't she come here?"
"She probably thought Aihara would find out and be after her."
"Has anything come from Aihara?"
"No."
"I suppose it's finished, then, with Fusako taking the children, and not a word from him."
"But she came home the other time, and maybe she told him she was coming home again for a while It wouldn't be easy for him to show his face."
"It's all over, whatever you say."
"I'm surprised that she should have had the nerve to go back to the country."
"Couldn't she just as well have come here?"
"Couldn't she just as well—that's not a very warm way to put it. We have to feel sorry for her, when she cant come back to her own home. We're parents and child, and this is what we've come to. I've been very unhappy."
Frowning, Shingo raised his chin to untie his necktie.
"Where's my kimono?"
Kikuko brought a kimono, and went off silently with his suit.
Yasuko sat with bowed head while he was changing "It's not at all impossible that Kikuko will run out on us," she muttered, looking at the door Kikuko had closed behind her.
"Do parents have to be responsible forever for their children's marriages?"
You don't understand women. It's different when women are sad."
And do you think a woman can understand every-thing about every other woman?"
Shuichi is away again tonight. Why can't the two of come home together? You come home by yourself and here is Kikuko to take care of your clothes. Is that right?"
Shingo did not answer.
"Won't we want to talk to him about Fusako?"
"Shall we send him off to the country? We'll probably have to send him for her."
"She might not want him to come for her. He's always made a fool of her."
"There's no point in talking about that now. We'll send him on Saturday."
"We look good before the rest of the family, I must say. And here we stay away as if we never meant to have another thing to do with them. It's strange that she should pick them to run off to, when they've meant so little to her."
"Who is taking care of her?"
"Maybe she means to stay in the old house. She can't stay on forever with my aunt."
Yasuko's aunt would be in her eighties. Yasuko had had very little to do with her or with her son, the present head of the family. Shingo could not even remember how many brothers and sisters there were.
It was unsettling to think that Fusako had fled to the house seen ruined in his dream.
3
On Saturday morning, Shingo and Shuichi left the house together. There was still some time before Shuichi's train.
Shuichi came into Shingo's office. "I'll leave this with you," he said, handing his umbrella to Eiko.
She cocked her head inquiringly. "You're off on a business trip?"
"Yes."
Putting down his bag, Shuichi took a seat by Shingo's desk.
Eiko's eye followed him. "Take care of yourself. It will probably be cold."
"Oh, yes." Shuichi spoke to Shingo, though he was looking at Eiko. "I was supposed to go dancing with the young lady this evening."
"Oh?"
"Get the old man to take you."
Eiko flushed.
Shingo did not feel inclined to comment.
Eiko picked up the bag as if she were going to see Shuichi off.
"Please. That's not for a lady to do." He snatched the bag and disappeared through the door.
Eiko made an unobtrusive little motion toward the door, and returned disconsolately to her desk. Shingo could not tell whether the gesture had been from confusion or calculation; but it had had in it a touch of the feminine that pleased him.
"What a shame, when he promised you."
"I don't put much stock in his promises these days."
"Shall I be a substitute?"
"If you like."
"Are there complications?"
"What?" She looked up, startled.
Does Shuichi's woman come to the dance hall?"
"No!"
Shingo had learned from Eiko that the woman's husky voice was erotic. He had not asked for further details.
It was not perhaps remarkable that his secretary should be acquainted with the woman when his own family was not; but he found that fact hard to accept.
It was particularly hard to accept when he had Eiko here before him.
One knew that she was a person of no consequence, and yet on such occasions she seemed to hang heavily before him, like the curtain of life itself. He could not guess what was passing through her mind.
"Did you meet her when he took you dancing?" he asked lightly.
"Yes."
"Many times?"
"No."
"Did he introduce you?"
"It wasn't an introduction, really."
"I don't understand. He took you to meet her—he wanted to make her jealous?"
"I'm no one to be jealous of." Eiko shrugged her shoulders very slightly.
Shingo could see that she was drawn to Shuichi, and that she was jealous.
"Then be someone to be jealous of."
"Really!" She looked down and laughed. "There were two of them too."
"What? She had a man with her?"
"Not a man. A woman."
"I was worried."
"Worried?" She looked at him. "The woman she lives with."
"They have a room together?"
"A house. It's small, but very nice."
"You mean you've been to the house?"
"Yes." Eiko half swallowed the word.
Once more Shingo was surprised. "Where is it?" he asked, somewhat abruptly.
"I shouldn't tell you," she said softly, a shadow crossing her face.
Shingo fell silent.
"In Hongo, near the University."
"Oh?"
She continued as if the pressure had been relieved, "It's up a dark narrow lane, but the house itself is nice. And the other lady is beautiful. I'm very fond of her."
"You mean the one that's not Shuichi's?"
"Yes. She's a very pleasant person."
"Oh? And what do they do? Are they both single?"
"Yes—I don't know, really."
"Two women living together."
Eiko nodded. "I've never known a pleasanter person. I'd like to see her every day." There was a certain coyness in her manner. She spoke as if the pleasantness of the woman made it possible for her to be forgiven something in herself.
All very strange, thought Shingo.
It did occur to him that, in praising the other woman, she might be indirectly reprimanding Shuichi's woman; but he had trouble guessing her real intentions.
Eiko looked out of the window. "It's clearing."
"Suppose you open it a little."
"I was a little worried when he left his umbrella. It's nice that he has good weather for his trip."
She stood for a time with her hand at the open window. Her skirt was askew, higher on one side than the other. Her stance suggested confusion.
She went back to her desk, head bowed.
A boy brought in three or four letters. Eiko put them on Shingo's desk.
"Another funeral," muttered Shingo. "Too many of them, Toriyama this time? At two this afternoon. I wonder what's happened to that wife of his."
Used to the way he talked to himself, Eiko only looked at him.
"I can't go dancing tonight. There's a funeral." His mouth slightly open, he was staring absently before him. "He was persecuted. She really tormented him when she was going through the change of life. She wouldn't feed him. She really wouldn't feed him. He would manage to have breakfast at home somehow, but she would get nothing at all ready for him. There would be food for the children, and he would have some of it when she wasn't watching. He was so afraid of her that he couldn't go home at night. Every night he would wander around or go to a movie or a variety show or something, and stay away until they were all safe in bed. The children all sided with her and helped persecute him."
"I wonder why."
"That's the way it was. The change is a terrible thing."
Eiko seemed to think that she was being made fun of. "Might it have been his fault?"
"He was important in the government, and then he joined a private firm. They've rented a temple for the funeral, so I suppose he did fairly well. He had only good habits when he was in the government."
"I suppose he took care of his family?"
"Naturally."
"It's not easy to understand."
"No, I don't suppose it is. But there are plenty of fine gentlemen in their fifties and sixties who spend their nights wandering around because they're afraid of their wives."
Shingo tried to remember Toriyama's face, but it refused to come to him. They had not met in ten years.
He wondered whether Toriyama had died at home.
4
Shingo thought he might meet university classmates at the funeral. He stood by the temple gate after he had offered incense, but he saw no one he knew.
There was no one his age at the funeral. Perhaps he had come too late.
He looked inside. The line by the door of the main hall was beginning to break up and move away.
The family seemed to be inside.
The widow survived, as Shingo had supposed she would. The thin woman directly in front of the coffin would be she.
She evidently dyed her hair, but had not dyed it in some time. It was white at the roots.
He thought, as he bowed to her, that she had not been able to dye it because Toriyama's long illness had kept her busy. But then as he turned to light incense before the coffin he felt like muttering to himself that a person could never be sure.
As he had come up the stairs and paid his respects to the family, he had quite forgotten the dead man had been persecuted; and then as he turned to pay his resects to the dead man, he remembered again. He was astonished at himself.
Making his way out, he turned so as not to have to look at the window.
He had been startled not by the widow but by his own strange forgetfulness. He felt somehow repelled as made his way back own the flagstone walk.
And as he walked away, he felt as if forgetfulness and loss lay pressing against the nape of his neck.
There were no longer many people who knew about Toriyama and his wife. Even though a few might survive, the relationship had been left to the the wife, to remember as she pleased. There were no third parties to look back upon it intently.
At a gathering of six or seven classmates, including Shingo, there had been no one to give it serious thought when Toriyama's name came up. They only laughed. The man who mentioned it coated his remarks with derision and exaggeration.
Two of the men at the gathering had died before Toriyama.
It was now possible for Shingo to think that not even Toriyama and his wife had known why the wife had persecuted him, or why he had come to be persecuted.
Toriyama was being taken the grave, not knowing. For the wife, left behind, it was all in the past. Without Toriyama, it had gone into the past. Probably she too would go to the grave unknowing.
The man who, at the gathering of classmates, had mentioned Toriyama, had as family heirlooms four or five old No masks. Toriyama had come calling, he said, and had stayed on and on when the masks were brought out. Since they could hardly have been of such great interest to someone seeing them for the first time, the man went on, he had probably been killing time until his wife would be safely in bed.
But it seemed to Shingo today that a man in his fifties, the head of a household, walking the streets each night, would be sunk in thoughts so deep they could not be shared.
The photograph at the funeral had evidently been taken on New Year's Day or some other holiday before Toriyama left the government. He was in formal dress, his face round and tranquil. The photographer had touched away the shadows.
The quiet face in the picture was too young for the widow by the coffin. One was made to think that she was the persecuted one, old before her time.
She was a short woman, and Shingo looked down at her hair and the white at its roots. One shoulder drooped a little, giving an impression of weariness and emaciation.
The sons and daughters and people who seemed to be their spouses were ranged beside the widow, but Shingo did not really look at them.
"And how are things with you?" he meant to ask if he met an old acquaintance. He waited at the temple gate.
He thought he would reply, if asked the same question, "I've managed somehow to come through; but there's been trouble in my son's family and my daughter's." And it seemed to him that he meant to tell of his problems.
To make such revelations would be of no help to either of them, nor would there be any thought of intercession. They would but walk to the street-car stop and say good-bye.
That much Shingo wanted to do.
"Now that Toriyama is dead, nothing is left of his torment."
"Are Toriyama and his wife to be called successes if their children's families are happy?"
"How much responsibility must a parent take these days for his children's marriages?"
Such mutterings came to Shingo one after another as the sort of things he would like to say were he to meet an old friend.
Sparrows were chirping away on the roof of the temple gate.
They cut arcs along the eaves, and then cut the same arcs again.
5
Two callers were awaiting him when he got back to his office. He had whiskey brought from the cabinet behind him and poured it into black tea. It was a small help to his memory.
As he received the callers, he remembered the sparrows he had seen in the garden the morning before.
At the foot of the mountain, they were pecking at plumes of pampas grass. Were they after the seeds or after insects? Then he saw that in what he had taken to be a flock of sparrows there were also buntings. He looked more carefully.
Six or seven birds jumped from plume to plume. The plumes waved violently.
There were three buntings, quieter than the sparrows. They did not have the nervous energy of the sparrows, and they were less given to jumping.
The glow of their wings and the fresh color of their breasts made them seem like birds new this year. The sparrows seemed coated with dust.
Shingo of course preferred the buntings. Their call was unlike that of the sparrows, and there was a similar difference in their motions.
He gazed on for a time, wondering whether the sparrows and buntings would quarrel.
But sparrows called to and flew with sparrows, and buntings flocked together.
When occasionally they mingled, there was no sign of a quarrel.
At his morning ablutions, Shingo looked on with admiration.
It was probably because of the sparrows on the temple gate that the scene had come back to him.
When he had seen the callers out, he turned and said to Eiko: "Show me where Shuichi's woman lives."
He had been thinking the possibility over as he talked to the callers. Eiko was taken by surprise.
With a gesture as of resistance, she frowned briefly; then she seemed to wilt. Yet she answered coolly, her voice restrained and distant. "And what will you do if I take you there?"
"Nothing that will embarrass you."
"Do you mean to see her?"
Shingo had not gone so far as to think of seeing her today.
"Can't you wait and have Shuichi take you?" Still she spoke calmly.
Shingo felt a certain contempt in her voice.
She remained silent even after they were in the cab.
He was unhappy with himself for having imposed upon her, and he felt that he was shaming both himself and his son.
He had imagined himself settling matters while Shuichi was away; but he suspected that he would stop at imagining.
"I think that if you are to talk to someone it should be the other lady."
"The one you say is so pleasant?"
"Yes. Shall I have her come to the office?"
"I wonder."
"He has much too much to drink at their house, and he gets violent and orders the other lady to sing. She has a very good voice. And then Kinu* starts crying. If it makes so much difference to her, then I imagine she listens to what the other lady says."
It was a somewhat confused way of expressing herself, Kinu must be Shuichi's woman.
Shingo had not known that Shuichi had taken to drink.
They got out by the University and turned up a narrow lane.
"If Shuichi hears about this, I'll have to leave the office," said Eiko softly. "I'll have to ask to be let go."
A chill passed over Shingo.
Eiko had stopped. "You turn by the stone fence there and it's the fourth house. You'll see the name Ikeda on the gate. They'll see me. I can't go any farther."
"Let's give it up, then, if it embarrasses you so."
"Why, when you've come this far? You have to go ahead. It means peace in your family."
He felt a certain malice in this defiance.
Eiko had called it a stone fence, but it was actually concrete. He turned past a large maple. There was nothing remarkable about the house, small and old, that carried the name Ikeda. The entrance faced north and was dark. The glass doors upstairs were closed. The house was silent.
There was nothing further to catch his eye.
Disconsolately, he walked on.
What sort of life did his son live behind that door? He Was not ready to put in an unannounced appearance.
He turned up another street.
Eiko was not where he had left her. Nor was she to be seen on the main street from which they had turned up the lane.
Back at home, he avoided Kikuko's eye. "Shuichi came the office for a few minutes and then left," he said. "I'm glad he has good weather."
Exhausted, he went to bed early.
"How many days did he take off?" Yasuko was in the breakfast room.
"I didn't ask," he answered from bed. "But all he has to do is bring Fusako back. I imagine it will be two or three days."
"I helped Kikuko change the wadding in the quilts today."
Fusako would be coming home with two children. Shingo thought how difficult things would be now for Kikuko.
Shuichi should take a separate house, he said to himself. He thought of the house in Hongo.
And he thought of the defiant Eiko. He was with her every day, and he had not until today witnessed such an outburst.
He had never seen Kikuko give vent to her emotions. Yasuko had said that she controlled her jealousy out of consideration for Shingo himself.
He was soon asleep. Awakened by Yasuko's snoring, took her nose between his fingers.
"Do you suppose Fusako will have that kerchief again?" said Yasuko, as if she had been awake all the time.
"I wouldn't be surprised."
They had nothing more to say to each other.
## Footnotes
* It is common for a husband to take his wife's name when her family is without male heirs.
* Translator's note: The name is Kinuko in the original. It has here been shortened, with Mr. Kawabata's permission, to avoid confusion with Kikuko,
#
A Dream of Islands
Astray bitch dropped puppies under the floor of Shingo's house.
"Dropped puppies" is a somewhat brusque way of Putting the matter; but for Shingo and his family it was Just so: suddenly, there was a litter under the veranda.
"We didn't see Teru yesterday, Mother," Kikuko had remarked in the kitchen a week or so before, "and she isn't here today either. Do you suppose she's having Puppies?"
"She hasn't been around, now that you mention it," Yasuko, with no great show of interest.
Shingo was in the kotatsu* making tea. He had since autumn been in the habit of having the most expensive of teas in the morning, and he made it for himself.
Kikuko had mentioned Teru while she was getting breakfast. Nothing more had been said.
"Have a cup," said Shingo, pouring tea, as Kikuko brought him his breakfast.
"Thank you very much." This had not happened before. Kikuko's manner was most ceremonious.
There were chrysanthemums on her obi and cloak. "And the season for chrysanthemums is past. With all the stir over Fusako, we forgot about your birthday."
"The pattern on the obi is 'The Four Princes.' You can wear it the year round."
"'The Four Princes'?"
"Orchid and bamboo and plum and chrysanthemum," said Kikuko briskly. "You must have seen it somewhere. It's always being used in paintings and on kimonos."
"A greedy sort of pattern."
"It was delicious," said Kikuko, putting down the tea bowl.
"Who was it that gave us the gyokuro?† In return for a funeral offering, I think. That was when we started drinking it again. We used to drink it all the time, and never bancha."
Shuichi had already left for the office.
As he put on his shoes in the doorway, Shingo was still trying to remember the name of the friend because of whom they had had the gyokuro. He could have asked Kikuko, but did not. The friend had taken a young girl to a hot-spring resort and died there suddenly.
"It's true that we don't see Teru," said Shingo.
"Not yesterday, and not today either," said Kikuko.
Sometimes Teru, hearing Shingo prepare for his departure, would come around to the doorway and follow him out the gate.
He had recently seen Kikuko in the doorway feeling Teru's belly.
"All puffy and bloated," said Kikuko, frowning But she went on feeling for the puppies all the same.
"How many are there?"
Teru looked up quizzically at Kikuko, showing the whites of her eyes. Then she rolled over, belly up.
It was not so swollen as to be repulsive. Toward the tail, where the skin seemed thinner, it was a faint pink there was dirt around the nipples.
"Ten of them?" said Kikuko. Shingo counted with his eyes. The pair farthest forward was small as if withered.
Teru had a master and a license, but it appeared that the master did not often feed her. She had become a stray. She made the rounds of the kitchens in the neighborhood. She had been spending more time at Shingo's since Kikuko had taken to giving her leftovers morning and evening, with something special added for Teru herself. Frequently, at night, they heard her barking in the garden. It seemed that she had attached herself to them, but not even Kikuko had come to think her their own.
Teru always went home to have puppies.
Her absence yesterday and today, Kikuko had intended to say, meant that she had again gone home to have puppies.
It seemed sad that she should go home for that purpose.
But this time the puppies had been born under the floor of Shingo's house. It was ten days or so before anyone noticed.
"Teru has had her puppies here, Father," said Kikuko when Shingo and Shuichi came home from the office.
"Oh? Where?"
"Under the maid's room."
"Oh?"
Since they had no maid, the maid's room, small and narrow, was used as a storeroom.
"Teru is always going in under the maid's room. So I looked, and there do seem to be puppies."
"How many?"
"It's too dark to tell. They're back in under."
"So she had them here."
"Mother said that Teru was behaving very strangely, going around and around the tool-shed and pawing at the ground. She was looking for a place to have puppies. I imagine if we had put out straw she would have had them in the shed."
"They'll be one fine problem when they grow up," said Shuichi.
Shingo was pleased that Teru had had her puppies here; but the unpleasant thought also came to him of the day when, unable otherwise to dispose of mongrel puppies, they would have to abandon them.
"I'm told that Teru had puppies here," said Yasuko.
"So I'm told."
"I'm told that she had them under the maid's room. The only room in the house with no one in it. Teru thought things out nicely."
Still in The kotatsu, Yasuko frowned slightly as she looked up at Shingo.
Shingo too got into the kotatsu. When he had had his cup of tea, he said to Shuichi: "What happened to the maid Tanizaki was to get for us?" He poured a second cup.
"That's an ashtray, Father."
He had poured his second cup into the ashtray.
2
"I am an aged man, and I have not yet climbed Mount Fuji." Shingo was in his office.
They were words that came out of nothing, but they seemed to him somehow significant. He muttered them over again.
Last night he had dreamed of Matsushima Bay and its islands. That was perhaps why the words had come to him.
This morning it had seemed odd to him that he should have dreamed of Matsushita, since he had never been there.
And it occurred to him that at his age he had been to only one of the "three great sights of Japan." He had seen neither Matsushima nor the strand at Amanohashih. date. Once, on his return from a business trip to Kyushu, he had had a look at the Miyajima Shrine. It had been winter, not the proper season.
In the morning, he could remember only fragments of the dream; but the color of the pine on the islands and of the water remained clear and fresh and he was certain that the dream had been of Matsushima.
On a grassy meadow in the shade of the pines, he had a woman in his arms. They were hiding, in fear. They seemed to have left their companions. The woman was very young, a mere girl. He did not know how old he himself was. He must have been young, however, to judge from the vigor with which they ran among the pines. He did not seem to feel a difference in their ages as he held her in his arms. He embraced her as a young man would. Yet he did not think of himself as rejuvenated, nor did it seem to be a dream of long ago. It was as if, at sixty-two, he were still in his twenties. In that fact lay the strangeness.
The motorboat in which they had come went off across the sea. A woman stood in the boat, waving and waving her handkerchief. The white handkerchief against the sea was vivid in his mind even after he woke. The two were left alone on the island, but there was none of the apprehension that they should have felt. He just told himself that they could see the boat out at sea, and that their hiding place would not be discovered.
Watching the white of the handkerchief, he woke.
He did not know, after he woke, who the woman had been. He could remember neither face nor figure. Nor did any tactile impression remain. Only the colors of the landscape were clear. He knew neither why he was sure that it had been Matsushima nor why he should have dreamed of Matsushima.
He had not been to Matsushima, nor had he crossed by boat to an uninhabited island.
He thought of asking someone in the house whether to see colors in a dream was a sign of nervous exhaustion, but in the end remained silent. He did not find it pleasing to think that he had dreamed of embracing a woman. It seemed altogether reasonable that, at his present age, he should have been his young self.
The contradiction was somehow a comfort to him.
He felt that the strangeness would vanish were he to know who the woman was. As he sat smoking, there was a tap on the door.
"Good morning."
Suzumoto came in. "I thought you wouldn't be here yet."
Suzumoto hung up his hat. Tanizaki came up in some haste to take his coat, but he sat down without removing it. His bald head seemed comical to Shingo. The discoloration of age was to be seen above his ears. The aged skin was muddy.
"What brings you here so early?" Restraining a laugh, Shingo looked at his own hands. A faint discoloration would appear from the back of his hand down over the wrist, and then go away again.
"Mizuta. He had such a pleasant death."
"Ah, yes, Mizuta." Shingo remembered. "They sent gyokuro after the funeral, and I got into the habit of drinking it again. Very good it was, too."
"I don't know about the gyokuro, but I envy him the way he died. I've heard about such things. But Mizuta of all people."
Shingo snorted.
"Don't you envy him?"
"You're bald and fat, and there's hope for you."
"But I don't have all that much blood pressure. I've been told that Mizuta was so afraid of a stroke that he refused to spend a night alone."
Mizuta had died in a hot-spring hotel. At the funeral his old friends whispered of what Suzumoto called his pleasant death. It seemed a little strange afterwards to have concluded that, by virtue of the fact that he had had a young woman with him, it had been such a death. They were curious to know whether the woman might be at the funeral. There were those who said that she would carry unpleasant memories through her life, and those who said that, if she loved him, she would be grateful for what had happened.
To Shingo, the fact that because they were university classmates these men in their sixties should toss out student jargon seemed another of the ugly marks of old age. They still addressed one another by the nicknames and affectionate diminutives of their student days. They had known all about one another when they were young, and the knowledge brought intimacy and nostalgia; but the moss-grown shell of the ego resented it. The death of Mizuta, who had made a joke of Toriyama's death, had now become a joke.
Suzumoto had insisted, at the funeral, upon speaking of the pleasant death; but the thought of it brought a wave of revulsion over Shingo.
"It's not very good form for an old man," he said.
"No. We don't even dream of women anymore." Suzumoto's tone too was dispassionate.
"Have you ever climbed Fuji?"
"Fuji?" Suzumoto seemed puzzled. "Why Fuji? No, I haven't. Why do you ask?"
"Neither have I. I am an aged man, and I have not yet climbed Mount Fuji."
"What? Is that some sort of dirty joke?"
Shingo let out a guffaw.
At work over an abacus near the door, Eiko snickered.
"When you think about it, there must be a surprising number of people who go to their graves without climbing Fuji or seeing the three great sights. What percentage of Japanese do you suppose climb Fuji?"
"Not one per cent, I'd say." Suzumoto returned to the earlier subject. "I doubt if one person in tens of thousands, in hundreds of thousands, has the good luck of Mizuta."
"He won a lottery? But it must not be pleasant for his family."
"Yes, the family. As a matter of fact his wife came," said Suzumoto, with an air as of entering upon his real business, "and asked me about this." He put a cloth-wrapped parcel on the table. "Masks. No masks. She asked me to buy them. I thought I'd ask you to look them over."
"I know nothing about masks. They're like the three great sights. I know they're in Japan, but I've never been to see them."
There were two boxes. Suzumoto took the masks from their pouches.
"This one is the jido mask, I'm told, and this the kasshiki. They're both children."
"This one is a child?" Shingo took up the kasshiki mask by the paper cord that passed from ear to ear.
"It has hair painted on it. See? In the shape of a gingko leaf. That's the mark of a boy who hasn't come of age. And there are dimples."
"Oh?" Shingo held the mask at arm's length. "Tanizaki. My glasses, there, please."
"No, you have it right as it is. They say you're supposed to hold a No mask a little above eye level with your arms stretched out. It's actually better for old men like us. And turn it down a little to cloud it."
"It looks like someone I know. Very realistic."
Turning a No mask slightly downward is known as "clouding," explained Suzumoto, because the mask takes on a melancholy aspect; and turning it up is known as "shining" because the expression becomes bright and happy. Turning it to the left or the right, he added, is known as "using" or "cutting" or something of the sort.
"It looks like someone I know," said Shingo again. "It's hard to think of it as a child. More like a young man."
"Children were precocious in those days. And a real child's face would be wrong for the No. But look at it carefully. It's a boy. I'm told that the jido is a sprite of some sort. Probably a symbol of eternal youth."
Shingo turned the jido mask this way and that as Suzumoto directed. The hair was in childish bangs.
"Why not keep them company?" said Suzumoto.
Shingo put the mask on the table. "You buy them. You were the one she asked."
"She actually had five. I bought two women's masks, and forced one on Unno. I thought you might take the others."
"So I get the leftovers? You took good care of yourself, buying the women's masks first."
"You'd rather have the women's?"
"What does it matter, now that they're gone?"
"I can bring them if you want. I'll save money if you take them. It's just that I felt sorry for her because of the way Mizuta died, and couldn't refuse. But she said that these are better made than the women's masks. And don't you like the idea of eternal youth?"
"Mizuta is dead, and Toriyama—he looked at them for such a long time at Mizuta's—Toriyama is dead too. Your masks don't make a person feel very comfortable."
"But the jido mask is a symbol of eternal youth. Don't you like the idea?"
"Did you go to Toriyama's funeral?"
"I don't remember why, but I couldn't." Suzumoto stood up. "Well, I'll leave them with you. Take a good look at them. If you don't like them, find someone who does."
"Whether I like them or not is beside the point. They are nothing to me. I don't doubt that they're good masks, and doesn't it mean that if I cut them off from the No, I'll be killing them when I die?"
"You needn't worry."
"Are they expensive?" Shingo asked, as if chasing after him.
"Yes. I was afraid I might forget, and had her write it down. There on the cord. It seems to be about what they're worth, but I'm sure you can bargain."
Shingo put on his glasses and started to untwist the cord; and the moment he could see them clearly the hair and lips of the jido mask struck him as so beautiful that he wanted to cry out in surprise.
When Suzumoto had left, Eiko came to his desk.
"Isn't it beautiful."
Eiko nodded silently.
"put it on for a minute."
"But that would be all wrong. Here I am in foreign clothes." When Shingo handed her the mask, however, she put it on and tied the cord.
"Move your head, very gently."
Standing before him. Eiko moved her head this way and that.
"Good. Very good." The words came of their own accord. Even with so little movement, the mask quite came to life. Eiko had on a russet dress, and her hair sent waves cascading at the sides of the mask, but she had taken on a charm that held him captive.
"Is that enough?"
"Yes." Shingo immediately sent Eiko out to buy a reference work on No masks.
3
The masks carried the names of the makers. The reference book reported that they did not fall into the category of "old masks," from the Muromachi Period, but they were the work of masters of the next age. Even a novice like Shingo sensed, as he took them in his hands, that they were not forgeries.
"Give you the creeps," said Yasuko, putting on his bifocals.
Kikuko laughed softly. "Can you see with Fathers glasses?"
"Bifocals are very promiscuous," Shingo answered for his wife. "Almost anyone's will work for almost anyone."
She was using the glasses he had taken from his pocket.
"In most houses the husband wears them sooner, but in this one the old woman is a year older." In high spirits, Shingo had sat down in the kotatsu without taking off his coat. "The chief trouble is that you can't see when you're eating. You can't see the food set before you. If it's in fine pieces, there are times when you can't even make out what it is. You first start wearing them, and take up a bowl of rice like this, and the kernels all blur into one another, and you can't separate them. It's very inconvenient at first." Shingo was gazing at the masks.
But then it came to him that Kikuko, a kimono in front of her, was waiting for him to change. And it came to him that this evening again Shuichi was away from home.
He continued to look at the kotatsu as he stood up to change. In part he was avoiding Kikuko's face.
He felt a heaviness in his chest. Probably it was because Shuichi had not come home that Kikuko had come to look at the masks. She set about putting away his clothes as if nothing of importance had happened.
"Like heads from the chopping block. They really give you the creeps," said Yasuko.
Shingo came back to the kotatsu. "Which do you like best?"
"This." Yasuko answered without hesitation, taking up the kasshiki mask.
"Oh?" Shingo was somewhat intimidated by Yasuko's decisiveness. "They're by different makers, but from the same period. About the time of Toyotomi Hideyoshi." He brought his face to the jido mask from directly above.
The kasshiki was masculine, the eyebrows those of a man; but the jido was neuter. There was a wide space between eyebrows and eyes, and the gently arched eyebrows were those of a girl.
As he brought his face toward it from above, the skin, smooth and lustrous as that of a girl, softened in his aging eyes, and the mask came to life, warm and smiling.
He caught his breath. Three or four inches before his eyes, a live girl was smiling at him, cleanly, beautifully.
The eyes and the mouth were truly alive. In the empty sockets were black pupils. The red lips were sensuously moist. Holding his breath, he came so close as almost to touch his nose to that of the mask, and the blackish pupils came floating up at him, and the flesh of the lower lip swelled. He was on the point of kissing it. Heaving a sigh, he pulled away.
He felt, from a distance, as if it had lied to him. He breathed heavily for a time.
Glumly, he put the jido mask back into its pouch of gold brocade on a red ground. He handed the pouch for the kasshiki mask to Yasuko.
"Put it away."
He felt as if he had looked behind the lower lip of the jido, to where the antique red faded away inside the mouth. The mouth was slightly open, but there were no teeth ranged behind the lower lip. It was like a flower in bud upon a bank of snow.
To bring one's face so near as to touch it was probably, for a No mask, an inexcusable perversion. It was probably a way of viewing the mask not intended by the maker. Shingo felt the secret of the maker's own love in the fact that the mask, most alive when viewed at a proper distance from the No stage, should all the same be most alive when, as now, viewed from no distance at all.
For Shingo had felt a pulsing as of heaven's own perverse love. Yet he sought to laugh at it, telling himself that his ancient eyes had made the skin more alluring than that of a real woman.
He wondered whether this sequence of strange occurrences-he had embraced a girl in a dream, he had thought Eiko quite captivating in the mask, he had almost kissed the jido-meant that something was about to shake the foundations of his house.
He had not brought his face to that of a young girl since he had begun wearing bifocals. Would such a face in his aged eyes, be faintly softer?
"They belonged to Mizuta. You know, the one we got gyokuro from. The one who died at a hot spring."
Give you the creeps," said Yasuko again.
Shingo put whiskey in his tea. In the kitchen, Kikuko was dicing onions for a fish chowder.
4
On the morning of the twenty-ninth of December as he was washing his face, Shingo saw Teru out sunning herself with all her puppies.
Even when the puppies had begun to come out from under the maid's room, he had not known whether there ere lour or five of them. Kikuko would pounce upon a Puppy and bring it into the house. In her arms the Puppies were docile enough, but they would flee back under the house when they saw someone approaching At no one time had they all been out together. Kikuko had said that there were four, and at another time that there were five.
He saw that there were five puppies out in the morning sunlight.
They were at the foot of the mountain, where he had seen the buntings mixed in among the sparrows. It was where earth was piled up from a cave they had dug as an air-raid shelter, and where, during the war, they had had a vegetable patch. It now seemed to be a place where animals sunned themselves.
The pampas grass at which the sparrows and buntings had been so busy had withered, but the powerful stalks, still upright, covered the side of the mound. The earth above was covered with soft weeds, Shingo was filled with admiration at Teru's sagacity in having chosen it.
Teru had taken her puppies out to a good place before people were up, or while their attention was on getting breakfast, and she lay nursing them and letting them warm themselves in the morning sun. They were quietly enjoying a moment when there was no one to bother them. So he thought at first, and smiled at the scene that presented itself in the warm sunlight. It was late December, but in Kamakura the sun was as warm as in autumn.
But as he looked more closely he saw that the five were shoving and jostling one another in a competition for nipples. Their front paws pumped at Teru's belly like pistons, and they were giving free rein to their young animal strength; and Teru, perhaps because they were now strong enough to climb the slope, seemed reluctant to let them nurse. She twisted and turned, and lay on her belly. It was red from the threshing paws.
Finally she got up and shook the puppies away, and came running down the slope. A black puppy that had clung to a nipple with particular stubbornness was sent tumbling from the mound.
It was a three-foot drop. Shingo caught his breath in alarm. The puppy got up as if nothing had happened and, after standing there blockishly for a second or two, walked off sniffing at the earth.
"What is it?" He felt that he was seeing the pose for the first time, and that he had seen exactly that pose before. He thought for a moment.
"That's it. The Sotatsu painting," he muttered. "Remarkable."
Shingo had glanced at Sotatsu's ink painting of a puppy, and had thought it altogether stylized, like a toy; and now he was astonished to see it reproduced in life. The dignity and elegance of the black puppy were exactly like the Sotatsu.
He thought again of how realistic the kasshiki mask was, and of how it had reminded him of someone.
Sotatsu and the mask-maker were of the same period.
Sotatsu had painted what would today be called a mongrel puppy.
"Come and look. All the puppies are out."
Hugging the ground in fright, the other four puppies came down the slope.
He watched expectantly, but none of the other four struck the Sotatsu pose.
He had seen the puppy become the Sotatsu picture, and the jido mask had become a living woman; and had he also had a fleeting glimpse of the two in reverse?
He had hung the kasshiki mask on a wall, but he had Put the jido mask far back in a drawer, like some esoteric object.
Yasuko and Kikuko came to the wash stand to see the puppies.
"You didn't notice while you were washing?"
Kikuko, looking out from behind them, put her hand lightly on Yasuko's shoulder. "A woman is too rushed in the morning. Isn't that so, Mother?"
"It is. And Teru?"
"Where will she have gone? She's left them to wander around like strays," said Shingo. "I hate to think of throwing them away."
"I've already married off two of them," said Kikuko.
"You've found someone to take them?"
"Yes. Teru's owner wants one. He says he wants a female."
"Really? He wants to change her for a puppy now that she's gone astray?"
"So it would seem." Kikuko turned to Yasuko: "Teru has gone off to eat somewhere." Then, leaving her earlier answer to speak for itself, she amplified upon this last remark for Shingo: "Everyone in the neighborhood is amazed at how clever Teru is. She knows when everyone eats, and shows up exactly on time."
"Really?" Shingo was a little disappointed. He had thought that, taking her morning and evening meals here, Teru had made this her home; was she still walking the neighborhood with an eye on all the leftovers?
"To be more precise," added Kikuko, "it's not the mealtimes she knows but the times when people are cleaning up afterwards. Everyone in the neighborhood is talking about how Teru had puppies here, and I get all sorts of reports on her activities. And when you're away, Father, children come and ask to see the puppies."
"She seems very popular."
"Oh, yes," said Yasuko. "One lady said something interesting. She said that now that Teru had had puppies here we would be having a baby. She said that Teru was urging us on. Aren't we to be congratulated?"
"Really, Mother." Kikuko flushed, and took her hand from Yasuko's shoulder.
"I'm just reporting what a lady in the neighborhood said."
"You mean there is someone who puts people and dogs in the same category?" It came to Shingo that the remark had not been very tactful.
But Kikuko looked up. "Grandfather Amamiya is very worried about Teru. He came and asked if we wouldn't take her in. He spoke of her as if she were a child, and I didn't know what to say."
"Why not take her in?" said Shingo. "She's here all the time anyway."
Amamiya had lived next door to Teru's master, but, failing in business, he had sold his house and moved to Tokyo. His old mother and father had lived with him and done odd jobs around the house, and, the Tokyo place being small, they had been left behind in a rented room. The old man was known in the neighborhood as "Grandfather Amamiya."
He was the one Teru was fondest of. Even after he moved into the rented room, he came inquiring after her.
"I'll run and tell him so," said Kikuko, going back into the house. "He'll be very relieved."
His eye on the black puppy, Shingo noticed a broken thistle under the window. The flower had fallen, but the stem, bent from its base, was still a fresh green.
"Thistles are very strong plants," said Shingo.
## Footnotes
* A quilt-covered frame over a sunken brazier for warming the extremities.
† With bancha, three lines below:
Two varieties of tea. Shingo and Kikuko are drinking the former.
#
The Cherry in the Winter
It began raining on New Year's Eve, and New Year's Day was rainy.
On New Year's Day the occidental way of reckoning ages became official. Shingo was therefore sixty-one, Yasuko sixty-two.
New Year's Day was a day for late sleepers; but Shingo was aroused early by Satoko. The child was scampering up and down the veranda.
"Come here, Satoko." Kikuko also seemed to be up. "I've got a New Year's pudding for you. You can help me heat it."
Apparently she wanted to lure the child to the kitchen, away from Shingo's room. Satoko seemed indifferent, however. The scampering went on.
"Satoko," Fusako called from bed. "Come here, Satoko." Satoko was no swifter to answer her mother.
"A rainy New Year," said Yasuko, also awake.
Shingo grunted.
"With Satoko up, Kikuko has to be up and around. Fusako manages to stay in bed, I see." She stumbled over the last words. Shingo was amused. "It's been a long time since I last had a child to wake me up on New Year's morning."
"You'll have plenty of other mornings, too."
"Oh, I don't think so, really. It's just that there aren't any verandas in Aihara's house. Once she gets used to them she'll stop running around."
"I wonder. Don't most children her age like running up and down verandas? Why do her feet sound as if they were sticking to the floor?"
"Because they're so soft." Yasuko listened. "It gives you a strange feeling, doesn't it? She should be five this year, and all of a sudden she's three. It doesn't make all that much difference for me, shifting from sixty-four to sixty-two."
"But there's something you haven't thought of. My birthday comes before yours, and for a while then we'll be the same age. From my birthday to yours."
Yasuko seemed aware of the fact for the first time.
"Quite a discovery. Once in a lifetime."
"Maybe so," muttered Yasuko. "But it doesn't do much good to start being the same age this late in life."
"Satoko," Fusako called again. "Satoko." Apparently tired of running, Satoko went to her mother. "Just feel how cold your feet are."
Shingo closed his eyes,
"It would be good if she'd do all that running while we're there to see it," said Yasuko after a time. "But when we are she starts sulking and banging onto her mother."
Perhaps each of them was trying to detect in the other signs of affection for the child.
It seemed to Shingo, in any case, that he was being probed by Yasuko.
Or perhaps he was probing himself.
The sound of the feet clinging to the floor had not been pleasant, for he had not had enough sleep; but, on the other hand, it had not particularly irritated him.
Yet he had not felt the tenderness the footsteps of a grandchild ought to bring. There was no doubt that he was wanting in affection.
No sense of the darkness, out there on the veranda with the shutters still closed, came to him. It had apparently come to Yasuko immediately. In such ways the child was capable of arousing her compassion.
2
Fusako's unhappy marriage had left a scar on Satoko. It aroused a certain compassion in Shingo too, but more frequently it was a source of irritation. For nothing could be done about it.
He was astonished at the extent of his helplessness.
No parent could do a great deal about the married life of his children, of course; but what was truly striking, now that matters had reached a point where divorce seemed the only solution, was the helplessness of the daughter herself.
For her parents to take her and the children in after the divorce would solve nothing. It would be no cure, and it would bring her no life of her own.
Was there then no answer at all for a woman whose marriage had failed?
When, in the autumn, Fusako had left her husband, she had gone not to her parents' house but rather to the family seat in Shinano. It was from there that she had Set them know by telegram of her departure.
Shuichi had gone to bring her back.
She had then left again, after a month in Kamakura, saying that she was going to make a clean and final break with Aihara.
Perhaps it would be better for Shingo or Shuichi to go have a talk with him, they had said; but she had not listened. She must go herself.
"But that's exactly the point, what to do with the children," she said when Yasuko suggested that she leave at least them behind. She leaped on Yasuko in a manner almost hysterical. "I don't know whether I'm to have them or Aihara is." And so she had gone out and not come back.
It was after all a matter between husband and wife, and Shingo and his family were worried, not knowing how long they should wait in silence. And so the uneasy days passed.
There came no word from Fusako.
Had she settled down with Aihara again?
"Things are just going to drag on?" said Yasuko.
"Well, it's we who are letting them," replied Shingo. Both faces were clouded.
Then, suddenly, on New Year's Eve, Fusako had come back.
"What's happened?" Yasuko seemed frightened as she looked down at her daughter and grandchildren.
Her hands trembling, Fusako tried to close her umbrella. One or two of the ribs seemed to be broken.
"Is it raining?" asked Yasuko.
Kikuko stepped down into the doorway and took Satoko in her arms.
She had been helping Yasuko with the New Year's food.
Fusako had come in through the kitchen.
Shingo suspected that she had come for money, but such did not seem to be the case.
Yasuko wiped her hands and went into the living room. "A fine thing, sending you away on New Year's Eve!" She stood gazing at her daughter.
Fusako was weeping silently.
"It's better this way," said Shingo. "A clean break."
"Oh? But I wouldn't have thought it possible for anyone to be turned out of the house on New Year's Eve."
"I came of my own accord." Fusako was choked with tears.
"Oh? Well, that's different, I suppose. You've just come to spend the New Year with your family. I shouldn't have put it the way I did. I apologize. But let's not talk about it now. We'll have a good talk during the holiday," Yasuko went to the kitchen.
Shingo was taken somewhat aback by his wife's tone; but it had in it a certain echo of maternal affection.
Yasuko was moved, naturally enough, both by the sight of her daughter coming home through the kitchen door on New Year's Eve, and by the sound of the child's footsteps on the dark veranda; but Shingo sensed an element of deference toward himself.
Fusako slept later than the others on New Year's morning.
They could hear her gargling as they sat at the table. Her ablutions went on and on.
"Let's have one while we're waiting," said Shuichi, pouring sake for his father. "You're getting a good number of gray hairs these days."
"Of course. At my age you get more of them every day. Every day—sometimes you see them turn gray right in front of your eyes."
"Ridiculous."
"No. Just watch." Shingo leaned forward. Yasuko and Shuichi looked at his head, and Kikuko was gazing intently.
She had Fusako's younger child on her lap.
3
A second kotatsu had been brought out for Fusako and the children. Kikuko joined them in another room.
Yasuko sat to one side as Shingo and Shuichi faced each other over wine cups.
Shuichi seldom drank at home; but today, perhaps having been led beyond his capacity by the rainy New Year's Day, he poured cup after cup for himself, almost ignoring his father. His expression was not his usual one. Shingo had heard how, at the establishment of his mistress, Shuichi had gotten violently drunk, and how be had made the woman cry by insisting that her friend sing to him.
"Kikuko," called Yasuko. "Would you mind bringing us some oranges too?" Kikuko slid open the door. "Come and sit over here. I have a pair of silent drinkers on my hands."
Kikuko glanced at Shuichi. "I don't think Father is drinking all that much."
"I've been thinking a little," muttered Shuichi. "About Father's life."
"About my life?"
"Oh, nothing very definite. But if I had to summarize my speculations, I suppose they would go something like this: has Father been a success or a failure?"
"You think you would be able to judge?" Shingo was silent for a moment. "Well, the food this New Year has a little of the taste it had before the war. In that sense you can say I'm a success."
"The food—did you say?"
"I did. And wouldn't that be about it? If you say you've been giving a little thought to your father's life.
"A little."
"An ordinary, mediocre life that's come as far as it has, and now it runs into good food at New Year's. Lots of people have died, you know."
"True."
"But whether or not a parent is a success would seem to have something to do with whether or not his children's marriages are successful. There I haven't done too well."
"That's your feeling, is it?"
"Oh, stop it, you two." Yasuko looked up. "You aren't getting the year off to a very good start." She lowered her voice. "And don't forget that Fusako is here. Where is she, by the way?"
"Asleep," said Kikuko.
"Satoko?"
"Satoko and the baby too."
"Well, now. All three of them sleeping away?" Yasuko's eyes were round and on her face was something of the innocence that comes with old age.
The gate opened. Kikuko went out. Tanizaki Eiko had come to make her New Year's call.
"Well, now. And in this rain." Shingo was indeed surprised, but that "Well, now" he had borrowed from Yasuko.
"She says she won't come in," said Kikuko.
"Oh?" Shingo went to the door.
Eiko was standing with her coat over her arm. She had on a black velvet dress. Her makeup was heavy despite the fact that she seemed to have shaved away the fuzz. Bowing from the hips, she looked even smaller.
Her greeting was a little stiff.
"It was good of you to come in this downpour. I hadn't expected callers, and I hadn't thought of going out. Come in and warm yourself."
"Thank you."
Eiko had come through the cold of the wind and rain. He had trouble knowing whether she had come to register a protest, or whether she really had something to talk about.
He felt, in any case, that it had been brave of her.
Eiko seemed reluctant to come inside.
"In that case I'll pull myself together and go out with you. Why don't you wait inside while I get ready? I always go at least to see Mr. Itakura. The old president of the company."
Itakura had been on his mind all morning, and Eiko's arrival had made the decision for him. He hurried to change clothes.
Shuichi had apparently lain back, his feet in the kotatsu, when Shingo had gone out. He got up again as Shingo started to change.
"Tanizaki is here," said Shingo.
"Yes." Shuichi spoke as if the matter were no concern of his, and did not seem disposed to greet her.
As Shingo went out, Shuichi looked up and followed with his eyes. "Don't stay out after dark."
"I'll be back early."
Teru was at the gate.
A black puppy came running out and, imitating its mother, cut across in front of Shingo toward the gate. It staggered and fell, wetting one side of its body.
"What a shame," said Eiko. She seemed about to kneel down beside it.
"We had five of them. We've given four away, and this is the only one left. It's promised too."
The train on the Yokosuka Line was empty.
Looking at the rain, driven horizontal by the wind, Shingo felt somehow happy that she had braved it.
"There are generally swarms of people from the Hachiman Shrine."
Eiko nodded.
"Yes, it's true—you always do come on New Year's Day."
"Yes." Eiko looked down for a time. "I'd like to keep coming even after I've quit work."
"You won't be able to once you're married. Did you have something on your mind?"
"No."
"You needn't feel shy. I'm a little dull and absent-minded these days."
"Stop pretending." It was an odd remark. "But I think I'll have to ask you to let me quit work."
This announcement had not been wholly unexpected, but Shingo was troubled for an answer.
"I didn't come especially on New Year's Day to tell you," Her manner seemed old beyond her years. "We'll talk about it later."
"Oh?" Shingo no longer felt as happy.
She had been in his office for three years, and now, suddenly, she seemed a different woman. She was not her usual self.
Not that he had paid a great deal of attention to her. She had only been his secretary.
He felt, of course, that he would like to keep her. Yet she was in no sense his captive.
"But I think it's my fault that you want to quit. I had you show me that house, and I made things unpleasant for you; and I imagine it isn't easy to have to see Shuichi."
"It has been hard." Her answer was unequivocal. "But when I thought about it all afterwards, it seemed the natural thing for a father. And I saw that I had been wrong, too. I was very proud of myself when he took me dancing, and I went to Kinu's house. It was depraved of me."
"That seems a little strong."
"But I did get worse." Her eyes were half closed, in sorrow. "If I quit work, I'll ask Kinu to give him up. To pay you back for all you've done."
Shingo was startled. It was as if something had brushed against a ticklish spot.
"That was his wife at the door?"
"Kikuko?"
"Yes. It was very hard for me. I decided I really had to speak to Kinu."
He felt a certain lightness in her, and a lightening of his own spirits.
It was not impossible, the thought came to him, that by even such light devices the problem might be solved, and with unexpected dispatch.
"I can't really ask you to do that."
"I'm doing it of my own free will, to pay you back for all you've done." That such a grand statement should have come from Eiko's small lips made Shingo once again feel aware of the ticklish spot.
And he thought of telling her not to rush into affairs that were no concern of hers.
But Eiko seemed much affected by her own "decision."
"I can't understand him, when he has such a good wife. I don't like watching him with Kinu, but I couldn't be jealous of his wife, I don't care how close they might seem to be. Or is it that men are dissatisfied with women who don't make other women jealous?"
Shingo smiled wryly.
"He was always saying what a child she is."
"To you?" There was sharpness in the words.
"Yes, and to Kinu. He said you were fond of her because she was a child."
"The fool!" Shingo looked at her.
"But he doesn't any more," said Eiko in some confusion. "He doesn't talk about her anymore."
Shingo was almost trembling with anger.
He sensed that Shuichi had referred to her body.
Had he wanted to find a prostitute in his bride? There was astonishing ignorance in the fact, and Shingo felt in it too a frightening paralysis of the soul.
Did the immodesty with which he spoke of his wife to Kinu and even to Eiko arise from that same paralysis?
He sensed cruelty in Shuichi. And not only in him: in Kinu and Eiko too he sensed cruelty toward Kikuko.
Did Shuichi not feel the cleanness in her?
The pale, delicate, childlike face of Kikuko, baby of her family, floated before him.
It was a little abnormal, Shingo could see, for him to feel a sensual resentment toward his son because of his son's wife; but he could not help himself.
There was an undercurrent running through his life, the abnormality that made Shingo, drawn to Yasuko's sister, marry Yasuko, a year his senior, upon the sister's death; was it exacerbated by Kikuko?
When Shuichi had found another woman so remarkably early in their marriage, Kikuko had seemed at a loss to control her jealousy; and yet it seemed that, in the presence of Shuichi's cruelty and moral paralysis, indeed because of them, she had awakened as a woman.
He remembered that Eiko was less well developed Physically than Kikuko.
Shingo fell silent, seeking somehow to control his anger through his sadness.
Eiko too was silent. Taking off her gloves, she smoothed her hair.
4
Shingo was in Atami. In the garden of the inn a cherry tree was in full bloom. It was January.
Winter cherries, he had been told, had been blooming from before the end of the year; but he felt as if he had come upon spring in a wholly different world.
He mistook the red plum blossoms for peaches, and wondered if the white might be apricots.
Attracted to the cherry blossoms as they were reflected by the pond, he went over to stand on the bank. He had not yet been shown to his room.
He crossed the bridge to the opposite bank, there to look at a plum tree shaped like an umbrella and covered with red blossoms.
Several ducks came running out from under the tree. In their yellow bills and the slightly deeper yellow of their feet he again felt spring.
Tomorrow the firm would be entertaining guests, and Shingo had come to make the arrangements. His business was over once he had conferred with the innkeeper.
He sat on the veranda and looked out at the garden.
There were also white azaleas.
Heavy rain clouds were bearing down from Jikkoku Pass, however, and he went inside.
On the desk were a pocket watch and a wristwatch. The wrist watch was two minutes the faster.
It was seldom that the two were exactly together, which fact sometimes bothered him.
"But if they worry you so, why don't you just carry one?" said Yasuko.
She had a point, to be sure. But the habit had formed over the years.
Already before dinner there were heavy rains and strong winds.
The lights failed and he went to bed early.
He awoke to the howling of a dog in the garden, and the sound of wind and rain, like a raging sea.
There were drops of perspiration on his forehead. The room had a heaviness about it, like the beginning of a spring storm beside the sea. The air was tepid, and seemed to press down upon his chest.
Taking a deep breath, he felt a surge of disquiet, as if he were about to spit blood.
It's not in my chest," he muttered to himself. He was only having an attack of nausea.
An unpleasant tightness in his ears moved through his temples to gather at his forehead. He rubbed his forehead and throat.
The sound like a raging sea was a mountain downpour and above it the sharp rasp of the wind came nearer.
In the depths of the storm there was a roaring.
A train was passing through the Tanna Tunnel, he thought. Such was no doubt the case. A whistle blew as the train emerged.
Shingo was suddenly afraid; he was now wide awake.
The roaring had gone on and on. The tunnel being some five miles long, the train would have taken perhaps seven or eight minutes to pass through. His impression was that he had heard it entering the far mouth, beyond Kannami. But was it possible that, a half mile from the Atami exit, he could have heard it at such a distance?
He had somehow felt the presence of the train in the tunnel as if it were inside his head. He had felt it all the way to the near mouth, and heaved a sigh of relief as it came out.
But he was perplexed. He would make inquiries of the inn people the next morning, he decided, and he would telephone the station.
For a time he was unable to sleep.
"Shingo-o-oh! Shingo-o-oh!" Half asleep and half awake, he heard someone calling him.
The only person who called with that particular lilt was Yasuko's sister.
For Shingo it was a piercingly sweet awakening.
"Shingo-o-oh! Shingo-o-oh! Shingo-o-oh!"
The voice had stolen into the back garden and was calling from under the window.
Shingo was awake. The sound of the brook behind the inn had become a roar. There were children's voices.
He got up and opened the back shutters.
The morning sun was bright. It had the warm brightness of a winter sun that was damp with the rain of spring.
On the path beyond the brook seven or eight children had gathered, on their way to grammar school.
Had he then heard them calling one another?
But Shingo leaned out of the window and searched through the bamboo thickets on the near side of the stream.
#
Water in the Morning
Told by his son on New Year's Bay that his hair was getting white, Shingo had replied that at his age a person had more white hairs every day, indeed that he could see hairs growing white before his eyes. He had remembered Kitamoto.
His schoolmates were now in their sixties. Among them were considerable numbers whose luck, from the middle of the war on into the defeat, had not been good. Since they were already then in their late fifties, the fall was cruel and recovery difficult. And they were of an age to lose sons in the war.
Kitamoto had lost three sons. When his company turned to war production, he was a technician whose services were no longer needed.
"They say it happened while he was sitting in front of a mirror pulling out white hairs," said an old friend who, visiting Shingo's office, told him of Kitamoto. "He was at home with nothing to do, and at first his family didn't take it too seriously. They thought he was just pulling out white hairs to keep himself busy. It was nothing to be all that worried about. But every day he would squat in front of the mirror. Where he thought he had pulled them all out the day before there would be white hairs again. I imagine there were actually too many for him to get them all. Every day he would spend more time in front of the mirror. They would wonder where he was, and there he would be in front of the mirror pulling out hair. He'd be nervous and jumpy if he was away from the mirror for even a minute, and rush back to it again. Finally he was spending all his time there."
"It's a wonder he didn't lose all his hair." Shingo was on the point of laughter.
"It's no laughing matter. He did. He pulled out every last hair."
This time Shingo laughed openly.
"But it's no lie," said the friend, looking into Shingo's face. "They say that even while he was pulling out white hair his hair would get whiter. He'd pull out one white hair, and two or three hairs next to it would be white. He would look at himself in the mirror with a sort of desperate expression on his face, and he would be getting whiter as he pulled out white hairs. His hair got thinner and thinner."
Shingo restrained his laughter. "And his wife let him go on pulling?"
But the friend went on as if the question needed no answer. "Finally he had almost no hair left, and what was left was white."
"It must have hurt."
"When he was pulling it out? No, it didn't hurt. He didn't want to lose any black hair, and he was careful to pull out the white hairs one by one. But when he had finished, the skin was drawn and shriveled. It hurt when you ran your hand over it, the doctor said. It didn't bleed, but it was raw and red. Finally he was put in a mental hospital. They say it was in the hospital that he pulled out what little was left. But think of the willpower and the concentration. They almost scare you. He didn't want to be old, he wanted to be young again. No one seems to know whether he started pulling it out because he had lost his mind, or he lost his mind because he pulled out too much."
"But I suppose he's better?"
"Yes. And there was a miracle. A fine crop of black hair came out on his naked head."
"You can't mean it!" Shingo was laughing again.
"But it's true," said the friend, unsmiling. "Lunatics have no age. If we were crazy, you and I, we might be a great deal younger." He looked at Shingo's hair. "There's still hope for you. For me it's too late."
The friend had lost most of his hair.
"Shall I pull out one of my own?" muttered Shingo.
"Have a try at it. But I doubt if you have the willpower to pull them all out."
"I doubt it too. And white hair doesn't worry me. I have no mad desire for black hair."
"You've had security. You calmly swam through while everyone else was going under."
"You make it seem so easy. You might as well have said to Kitamoto that he would save himself trouble by dyeing his hair."
"Dyeing is cheating. If we're going to let ourselves think of cheating, then I doubt if we can hope for miracles like Kitamoto's."
"But isn't Kitamoto dead? Even though there was a miracle."
"Did you go to the funeral?"
"I didn't know of it at the time. I didn't hear of it till the war was over and things had settled down a little. I doubt if I would have gone into Tokyo anyway. It was during the air raids."
"You can't hold on to miracles for very long. Kitamoto may have pulled out his white hair and fought against the years, but life goes its own way. You don't live longer just because your hair goes dark again. It might even be the opposite. It might be that he used up all his energy growing that crop of dark hair, and his life was actually shortened. But don't think the struggle means nothing to you and me." He nodded to emphasize this conclusion. Hair was combed across his bald crown like the strips of a blind.
"Everyone I meet these days has white hair," said Shingo. "It wasn't so bad with me during the war, but I've gotten whiter and whiter since."
Shingo did not believe all the details of the story. He suspected embroidering.
That Kitamoto had died was a fact, however. He had learned of it from someone else.
As Shingo turned the story over in his mind, his thoughts took a strange turn. If it was true that Kitamoto was dead, then it must also be true that his white hair had grown out black. If it was true that he had lost his mind, then it must also be true that he had pulled out all his hair. If it was true that he had pulled out his hair, then it must also be true that it had grown white as he sat before the mirror. Was not the whole of the story true? Shingo was surprised at his own conclusion.
"I forgot to ask whether Kitamoto's hair was white or black when he died," he said, laughing. But neither the words nor the laughter were audible. They were for him alone.
Even if the story he had heard was true and without embroidery, there had probably been an element of parody in the manner of its telling. One old man had told of the death of another old man with derision and not without cruelty. The taste left by the encounter was not pleasant,
Among the friends of his student days, Kitamoto and Mizuta had been the ones to die strange deaths, Mizuta had died suddenly at a hot-spring resort. He had gone there with a young girl. Shingo had been importuned late the year before to buy his No masks. It had been because of Kitamoto that he had hired Tanizaki Eiko.
Mizuta having died since the war, Shingo had been able to go to the funeral. He did not hear until later of Kitamoto's death, which occurred during the air raids; and when Tanizaki Eiko came with her introduction, Kitamoto's wife and children were still in Gifu Prefecture, where they had taken refuge from the raids.
Eiko was a schoolfriend of Kitamoto's daughter. But it seemed altogether too unceremonious that he should be asked this favor by the daughter. He had not met her, and Eiko said that she had not seen her since the war. It seemed too precipitous on the part of both girls. If Kitamoto's widow, at the daughter's prompting, had remembered Shingo. then she should have written herself.
Shingo felt no obligation toward the daughter and her letter of introduction.
As for Eiko, who brought it, she seemed slight in body and frivolous in mind.
Yet he hired her, and took her into his own office. She had been working there for three years.
The three years had gone by swiftly, but it seemed odd, now, that she had lasted so long, It was perhaps not surprising that she had, in the course of the three years, gone dancing with Shuichi, but she had even been in the house of Shuichi's woman And Shingo himself, under her guidance, had gone to see it.
Eiko seemed to feel intimidated by these events. She had come to dislike her work.
Shingo had not spoken to Eiko of Kitamoto. Probably she did not know that he had lost his mind. She and the daughter were probably not such close friends as to frequent each other's houses.
He had thought her Frivolous, but now that she had quit work he sensed certain traces of conscience and benevolence in her. And with them a purity, for she. was not yet married.
2
"You're up early, Father," Pouring out the water with which she was about to wash her face, Kikuko drew water for him.
Drops of blood fell into it, and spread and thinned.
Remembering how he himself had coughed up a small amount of blood, and thinking how much cleaner was Kikuko's, he was afraid that she too might be spitting up blood; but it was a nosebleed.
She held a cloth to her nose. The blood traced a line from her wrist down to her elbow.
"Look up, look up." He put his arm over her shoulders. She fell slightly forward, as if avoiding him. He pulled her back by the shoulders, and, taking her forehead, made her look up.
"I'm all right, Father. I'm sorry."
"Keep quiet and kneel down. Lie down."
Supported by Shingo, Kikuko leaned against the wall.
"Lie down," he said again.
But she remained in the same position, her eyes closed. On her face, white as if she had fainted, there was an innocent quality, as of a child who has quit resisting. He saw the small scar on her forehead.
"Has it Stopped? If it has, go in and lie down."
'Yes. I'm ail right." She wiped her nose with the cloth. "The basin is dirty. I'll wash it for you."
"Please don't bother."
Shingo poured out the water in some haste. Faintly, melting away, there were traces of blood at the bottom of the basin.
Shingo did not use it. He washed his face directly from the faucet.
He thought of rousing Yasuko and sending her to help.
But then he decided not to. Kikuko might not want to reveal her discomfort to her mother-in-law.
The blood had fallen as from a bursting pod. To him it had been as if pain itself were bursting forth.
Kikuko passed while he was combing his hair.
"Kikuko."
"Yes?" She looked over her shoulder at him, but went on to the kitchen. She came back with charcoal in a firepan. He saw it send off sparks. She had lighted charcoal for the kotatsu over the kitchen gas.
Shingo was startled at himself. He had quite forgotten that his own daughter, Fusako, had come home. The breakfast room was dark because Fusako and her two children were asleep in the next room. The shutters had not yet been opened.
Rather than his old wife, he could have roused Fusako to help Kikuko. It was odd that Fusako had not come into his mind when he had thought to call Yasuko.
At the kotatsu, Kikuko poured lea for him.
"Are you dizzy?"
"Just a little."
"It's still early. Why don't you rest this morning?"
"It's time I was up and around." Kikuko spoke as of a triviality. "The cold wind was good for me when I went to get the paper. And I've always been told that a woman's nosebleed is nothing to worry about. Why are you up so early yourself? It's cold again this morning."
"I wonder. I was awake before the temple bell rang. It rings at six, summer and winter, the whole year round."
Shingo was up earlier than Shuichi, but later in starting for the office. Such was their way in the winter.
He look Shuichi to lunch at a nearby occidental restaurant.
"You know about the scar on Kikuko's forehead?" asked.
"Yes."
"The mark of the forceps, I imagine. It was a difficult birth. You couldn't exactly call it the remains of suitering at birth, I suppose, but it stands out when she's in pain."
"You mean this morning?"
"Yes."
"It was probably because of the nosebleed. It stands out when her color is bad."
Shingo felt somehow forestalled. When had kikuko told Shuichi?
"But she didn't sleep last night."
Shuichi frowned. After a moment of silence he said: "You needn't behave so properly with an outsider.
"An outsider? Isn't she your own wife?"
"That's what I'm saying. You needn't behave yourself so properly with your son's wife."
"What do you mean?"
Shuichi did not answer.
3
When Shingo returned to his office, Eiko was seated in the reception room. Another woman was standing beside her.
Eiko too stood up. She offered the usual sort of greeting about the weather and her remissness.
It's been a long time Two months."
Eiko seemed to have put on a little weight, and her face was more heavily made up. He remembered how, when he had gone dancing with her, her breasts had seemed just enough to fill his hands.
"This is Mrs. Ikeda. You will remember that I spoke of her." Eiko's eyes were most appealing, as if she might be on the point of tears. So it was with her on solemn occasions.
"How do you do." Shingo could not bring himself to thank the Ikeda woman, as ceremony required, for her ministrations to Shuichi.
"I dragged Mrs. Ikeda here. She said she didn't want to come. She said there was no point in coming."
"Oh? Shall we talk here, or would you rather go out?"
Eiko looked inquiringly at the other woman.
"This will do nicely, as far as I am concerned," she said curtly.
Shingo was confused. Eiko had said, he seemed to remember, that she would introduce him to the woman who was living with Shuichi's woman. He had not pursued the matter.
It seemed to him very odd indeed that two months after she had quit work Eiko should acquit herself of the undertaking.
Had Shuichi and his woman at length agreed on a Separation? Shingo waited for Eiko or the Ikeda woman to speak.
"Eiko pestered me into coming. But it won't do any good." Her manner was hostile. "I've been telling Kinu that she ought to leave Shuichi. I thought if I came I might get your help."
"I see."
"Eiko is in your debt, and she sympathizes with his wife."
"A very nice lady," put in Eiko.
"Eiko has said that to Kinu too. But there aren't many women these days who will withdraw just because a man has a nice wife. Kinu says if she is to give another man back, then let her have her own husband back. He was killed in the war. Just bring him back to me alive she says, and I'll let him do exactly what he wants, He can have as many affairs with other women and as many mistresses as he wants. She asks if I don't agree. Anyone who lost her husband in the war has to agree. Didn't we send them off to war? And what are we to do now that they're dead? He's in no danger of getting killed when he comes to see me, she says. I send him back undamaged."
Shingo smiled wryly.
"I don't care how good a wife she is, she isn't a war widow."
"That's a blunt way to put it."
"Yes. That's what she says when she's in her cups. She and Shuichi are ugly drinkers. She says to tell his wife she's never had to wait for someone to come home from a war. She waits for someone who's sure to come back. All right, he shouts back. He wilt tell her. I'm a war widow, too. Doesn't it always go bad when a war widow falls in love?"
"What do you mean?"
Shuichi too—he's an ugly drinker. He's been very rough with her. He told her to sing for him. She doesn't like to sing, and there was nothing for me to do but sing in her place. I sang in a very small voice If I hadn't done something to quiet him down we'd have been the scandal of the neighborhood, I felt so insulted myself that I could hardly go on But I wonder if it's really because of drink. Mightn't it be because of the war? Don't you Suppose he had women that, way somewhere? When I saw him out of control, I thought I was seeing my own husband during the war, I went dizzy and I could hardly breathe, and it seemed to me that I was the woman he was having. I cried and I sang some songs that weren't very proper. I said to Kinu that I wanted to think of my husband as the only exception; but I suppose it was that way with him too. Afterwards, when Shuichi made me sing, Kinu would be crying with me."
Shingo's face clouded over. It was a morbid story,
"The best thing would be to put. an end to it as soon as possible."
"I agree. After he's gone, she's always saying that this sort of thing means complete ruin. If that's how she feels, then of course she ought to leave him. But I suspect she's afraid that what would come afterwards would really be ruin. A woman. . . ."
"She needn't worry," put in Eiko.
"You're right I suppose. She has her work. You've seen how it is."
"Yes."
"She did this for me." The Ikeda woman gestured toward her own suit. "I suppose she's about the most important after the chief cutter. They think very highly of her. They took in Eiko on the spot because of her."
"You're working in the same shop?" Shingo looked at Eiko in surprise.
"Yes." Eiko nodded, and flushed slightly,
He found it hard to understand her. First she had Shuichi's mistress get her a job in the same shop, and now she brought the Ikeda woman to see him.
"And so I doubt if she costs Shuichi much money, said the latter.
"It's not a matter of money." Shingo was irritated, but he controlled himself.
"There's something I often say to her after he's been bad to her." She sat with bowed head. Her hands were folded on her knees. "He goes home wounded, too, I say. He goes home a wounded soldier." She looked up. "Can't he and his wife live away from you? I often think that if he and his wife were alone together, he would leave Kinu. I've thought about it a great deal."
"Possibly so. I'll give it some thought myself."
He considered her forward, but he had to agree with her.
4
Shingo had had no intention of asking anything of the woman named Ikeda, and he had had nothing to say to her. He had just let her talk on.
To the woman the visit must have seemed pointless. Without going so far as to seem a suppliant, Shingo should have discussed the matter candidly with her. She had done well to say as much as she had. It had been as if she were apologizing for Kinu, and doing yet something more.
Shingo felt that he should be grateful to both Eiko and Ikeda.
The visit had aroused neither doubts nor suspicions.
But, perhaps because his self-respect was on trial, he answered irritably when, going into a business dinner, a geisha whispered something in his ear.
"What? I'm deaf, damn it. I can't hear you."
He clutched at her shoulder. He took his hand away immediately, but the geisha frowned with pain and rubbed at the shoulder.
"Come out here for a minute," she said, reading the irritation on his face. She pressed her shoulder and led him to the veranda.
He was back in Kamakura at about eleven. Shuichi was not yet home.
In her room, next to the breakfast room, Fusako raised herself on an elbow and looked up at him. She was nursing her younger child.
"Is Satoko asleep?"
"She just now went to sleep. Mother, she said, which is bigger, a thousand yen or a million yen? Which is bagger? We laughed and laughed. Ask Grandfather when he comes home, I said. She went to sleep while she was waiting for you."
"If she was asking about a thousand yen before the war and a million yen since, it was a good question," laughed Shingo. "May I have a glass of water, please, Kikuko?"
"Water? A glass of water?" Kikuko got up, but she spoke as if it were an odd request.
"From the well, I don't want all those chemicals."
"Yes."
"Satoko wasn't born before the war," said Fusako, still in bed. "I wasn't married yet."
"It would be better if you hadn't married at all, before the war or after," said Yasuko. They heard water being drawn at the well. "The pump doesn't sound cold any more. In the winter when Kikuko goes out early to get water for your tea, that squeaking makes me shiver even when I'm warm in bed."
"I'm thinking of having them live away from us," he said in a low voice,
"Away from us?"
"Don't you think that would be better?"
"Maybe. If Fusako is going to stay on."
"I'll leave, Mother, if it's a question of living away from you." Fusako got out of bed. "I'll move out. Isn't that the thing to do?"
"It has nothing to do with you," Shingo half snarled at her.
"It does have something to do with me. A great deal, in fact. When Aihara said that you made me what I am by not liking me, I almost choked. I've never been so hurt in my life."
"Control yourself, control yourself. Here you are in your thirties."
"I can't control myself because I have no place to control myself in."
Fusako brought together her night kimono over her rich breasts,
Shingo got up wearily. "Let's go to bed, Granny."
Kikuko brought his glass of water. In her other hand she had a large leaf.
"What is it?" he asked, drinking the water down in one breath.
"A loquat leaf. There's a new moon, and there was a white blur by the well. I wondered what it might be. A new loquat leaf, already this big."
"Very school-girlish of you," said Fusako sardonically.
#
The Voice in the Night
Shingo woke to a sound as of a man's groaning.
He was not sure whether it was a dog or a man. At first it sounded like the moaning of a dog. It would be Teru, in her death agonies. Had she been poisoned?
His heart was racing.
He held his hand to his chest. It was as though he had had a seizure.
But when he was fully awake, he knew that it was not a dog but a man. He was being throttled; his voice was thick. Shingo was in a cold sweat. Someone was being attacked.
"Kiko-o-oh. Kiko-o-oh," the voice seemed to say. "Tell me, tell me." There was pain in it, the words caught in the throat and refused to take shape.
"Kiko-o-oh. Kiko-o-oh."
About to be killed, would he be asking his assailant's reasons or demands?
Shingo heard someone fall against the gate. He hunched his shoulders, preparing to get up.
"Kikuko-o-oh. Kikuko-o-oh."
It was Shuichi calling Kikuko. His speech was muddled, and the second syllable was lost. He was dead drunk.
Exhausted, Shingo sank back on the pillow. His heart was still racing. He rubbed his chest and breathed deeply and regularly.
"Kikuko-o-oh, Kikuko-o-oh."
Shuichi seemed not to be knocking on the gate but falling against it.
Shingo thought he would go out after he had rested a moment.
But then he decided that that might not be the best thing to do. Shuichi seemed to be calling out in heartbroken love and in sorrow. It was the voice of one for whom there is nothing else. The groaning was like a child calling out for its mother in a moment of pain and sorrow, or of mortal fear. And it seemed to come from depths of guilt. Shuichi was calling out to Kikuko, seeking to endear himself to her, with a heart that lay cruelly naked. Perhaps, his drunkenness his excuse, he called out in a voice that begged for affection, thinking he Would not be heard. And it was as if he were doing reverence to her.
"Kikuko-o-oh, Kikuko-o-oh."
The sadness came across to Shingo.
Had he ever himself, even once, called out to his wife in a voice filled with such hopeless love? Perhaps, unconsciously, it had in it the hopelessness of a certain moment on a foreign battlefield.
He listened on, wishing that Kikuko would awaken. At the same time he felt a little embarrassed to have his daughter-in-law hear that misery-ridden voice. He thought he would rouse Yasuko if Kikuko did not get up soon; but it would be better for Kikuko to go.
He pushed the hot-water bottle to the foot of the bed. Was it because he still had a hot-water bottle, even now in spring, that his heart raced so?
Kikuko was in charge of the bottle. He would occasionally ask her for one. The water stayed warm longer when she heated it, and the lid was secure.
Perhaps because Yasuko was stubborn, perhaps because she was healthy, she disliked hot-water bottles even at her age. She had warm feet. As late as his fifties, Shingo had still taken warmth from his wife, but now they slept apart.
She never moved to touch his hot-water bottle.
"Kikuko-o-oh, Kikuko-o-oh." Again the voice came from the gate.
Shingo turned on the light by his pillow. It was almost two-thirty.
The last train on the Yokosuka line got to Kamakura before one. Shuichi had evidently held out in one of the taverns by the station.
Shingo thought, from the tone of the voice, that the end was in sight for Shuichi and the woman in Tokyo.
Kikuko went out through the kitchen.
Much relieved, Shingo turned off the light.
"Forgive him," he muttered, addressing the words to Kikuko.
She seemed to be holding Shuichi up.
"Please. You're hurting me." It was Kikuko. "You're pulling my hair with your left hand."
"Am I?"
The two of them fell down in the kitchen.
"Steady, now. On my knees. Your legs swell when you're drunk."
"My legs swell? You're a liar."
Kikuko seemed to be taking off his socks, his legs on her knees.
She had forgiven him. Perhaps he need not have worried. Perhaps, as his wife, she took pleasure in sometimes being able to forgive. And perhaps she had listened well to the voice.
His legs on her knees, she pulled off the socks of a husband drunk and back from visiting another woman. Shingo felt the gentleness in her.
When she had put Shuichi to bed, she went out to lock the back gate and the kitchen door.
Shuichi's snoring was so loud that even Shingo could hear it.
Here was Shuichi, put to bed by his wife and promptly asleep; and what would be the position of the woman Kinu who, until just now, had been made his companion in ugly drunkenness? Had Shingo not heard that he drank and resorted to violence and made her Weep?
And Kikuko: she had sometimes been pale and drawn because of Kinu, but the flesh at her hips had drawn richer.
2
The snoring soon stopped, but Shingo could not get back to sleep.
He wondered if Yasuko's snoring had been handed on to their son.
Probably not. Probably he snored tonight because he had had so much to drink.
Yasuko did not seem to snore these days. She seemed to sleep even better in cold weather.
Shingo disliked mornings after he had slept badly because his memory was worse than usual, and he was overtaken by fits of sentimentality.
It might have been sentimentality that had made him hear Shuichi's voice as he had. Possibly it had been a voice thick from drink, nothing more. Had Shuichi hidden his discomfiture behind his drunkenness?
It also seemed to Shingo that the love and the sadness he had sensed in that fuddled voice had only been what he hoped for in his son.
Because of that voice, Shingo himself had forgiven Shuichi. And he thought that Kikuko had forgiven him. The selfishness of blood ties bore itself in on Shingo.
He thought himself good to his daughter-in-law, and yet he seemed in some respects to side with his son.
It was an ugly picture. Shuichi had drunk too much at the house of the woman in Tokyo, and come home to fall against the gate.
If Shingo himself had gone to open the gate, he probably would have glared at Shuichi, and Shuichi would have sobered up. It was better that Kikuko had gone. Shuichi had thus been able to come in hanging on her shoulder.
Kikuko the injured party was Kikuko the absolver.
How many times would Kikuko, now in her early twenties, have to forgive Shuichi before she had lived with him to the ages of Shingo and Yasuko? Would there be no limit to her forgiving?
A marriage was like a dangerous marsh, sucking in endlessly the misdeeds of the partners. Kinu's love for Shuichi, Shingo's love for Kikuko—would they disappear without trace in the swamp that was Shuichi's and Kikuko's marriage?
It seemed to Shingo quite proper that in postwar domestic law the basic unit had been changed from parent and child to husband and wife.
"In other words, the husband-wife marsh," he muttered to himself. "They'll have to have their own house."
It was because of his age that he had this way of muttering what came into his mind.
The expression "husband-wife marsh" meant only that a husband and wife alone, putting up with each other's misdeeds, deepened the marsh with the years.
That was probably because the wife awoke to herself in confrontation with the husband's misdeeds.
Shingo rubbed at an itching eyebrow.
Spring was near.
He did not dislike awakening in the night as he had during the winter.
He had awakened from a dream before Shuichi's voice had awakened him. At the time he remembered it well. but when he was awakened the second time he had almost forgotten it.
Perhaps it was the pounding of his heart that had erased it.
He remembered only the fact that a girl fourteen or fifteen years old had an abortion, and the words. "And she has become a holy child forever."
He had been reading a novel. Those were the concluding words.
He had read the novel as words, and seen the plot as a movie or play. He had not appeared in it himself. He had been completely the onlooker.
A girl who had an abortion at fourteen or fifteen and was at the same time a holy child was something of an oddity; but there had been a long story. Shingo's dream had read a masterpiece about pure love between a boy and a girl. His feelings were still with him when he woke at the end of the reading.
Had it been that the girl did not know she was pregnant and did not think of it as an abortion, and went on longing for the boy from whom she had been separated? But such a twist in the dream would be unnatural and unclean.
A forgotten dream could not be Put together again. And his feelings upon reading the novel were a dream.
The girl must have had a name, and he must have seen her face, but only her size, or more properly her smallness, remained vaguely in his mind. She seemed to have been in Japanese dress.
He asked himself whether it had been a vision of Yasuko's beautiful sister, but decided that it could not have been.
The source of the dream was no more than an article in last night's paper.
"Girl Has Twins. Misguided Awakening of Spring in Aomori." Under the large headline was this article: "According to a survey by the Aomori Prefectural Public Health Service of legal abortions under the Eugenics Law, five girls fifteen years old, three girls fourteen, and one girl thirteen have undergone abortions. There have been four hundred cases of abortion among girls of high-school age, sixteen to eighteen, and of these twenty per cent have been high-school students. There has been one middle-school pregnancy in Hirosaki and one in Aomori, and there have been four in South Tsugaru District and one in North Tsugaru District, Though the girls have gone to specialists, a lack of sexual knowledge has produced the horrifying results of death in .2 per cent of the cases and serious illness in 2.5 per cent. The thought that others, in secret, go to their deaths at the hands of unlicensed doctors makes one tremble for 'young mothers.'"
Four actual cases were listed. A second-year middle-school student, fourteen years old, in North Tsugaru District, had, in February of the year before, suddenly felt the coming on of birth pangs, and borne twins. Mother and children were healthy, and the girl was back in school, now a third-year student. Her parents had not known of her pregnancy.
A high-school student in Aomori, having promised herself to a classmate, became pregnant the summer before. The parents of the two, on the grounds that they were still in school, decided upon an abortion. But the boy said: "We weren't playing. We're going to get married soon."
The article had been a shock to Shingo; and so he had gone to bed and dreamed of an abortion.
But his dream had seen nothing ugly in the boy and girl. It had told a story of pure love, and made the girl "a holy child." He had not so viewed the matter before going to sleep.
The shock had become something beautiful. Why should there have been such a transformation?
Perhaps, in the dream, he had rescued the girl, and himself, too.
In any case, benevolence had emerged from the dream.
Shingo reflected upon himself, wondering whether for him benevolence woke in dreams.
And he became somewhat sentimental. Had a flicker of youth given him a dream of pure love in old age?
The sentimentality had remained after the dream, and perhaps made him greet Shuichi's voice, like a loud moaning, with benevolence, made him feel in it love and sadness.
3
Still in bed, Shingo heard Kikuko arousing Shuichi.
He woke too early these days. Yasuko, a late sleeper, had reprimanded him. "Old people aren't popular when they make fools of themselves and get up too early."
He too thought it improper to be up before Kikuko, and so he would go quietly to the front door for the paper and read it in bed.
Shuichi seemed to have gone to wash.
There was a sound of vomiting. He had evidently gagged while brushing his teeth.
Kikuko ran to the kitchen.
Shingo got up. On the veranda he met Kikuko coming back from the kitchen.
"Father!"
She stopped, almost running into him, and flushed. Something spilled from the cup in her hand. It seemed to be a cold sake, a remedy for Shuichi's hangover.
Shingo thought her very beautiful, a flush on the somewhat pale face, without cosmetics, shyness in the still sleepy eyes, the beautiful teeth showing between plain, unpainted lips upon which floated a smile of embarrassment.
Was there still this childlike quality in her? Shingo thought of his dream.
But it was not so very strange that girls no older than those in the article should marry and bear children. In ancient times that had been the ordinary thing.
And when he was no older than the boys, Shingo himself had been strongly drawn to Yasuko's sister.
Seeing that he had come into the breakfast room, Kikuko opened the shutters in some haste.
Spring sunlight flooded in.
Kikuko seemed startled by the brightness. Shingo was watching her from behind. She brought both hands to her head and pulled at her hair, still tangled from sleep.
The great gingko in the shrine precincts was not yet sending out new shoots. Yet somehow, in the morning light and the morning nostrils, there was something akin to a scent of buds.
Quickly putting herself in order, Kikuko brought him gyokuro.
"Here you are, Father. I'm being very slow this morning."
Always when he got up in the morning, Shingo had gyokuro in very hot water. The hotter the water the more difficult the steeping, and Kikuko did it best.
Shingo wondered whether gyokuro might not be even better if it came from an unmarried girl.
"You're very busy," he said cheerfully. "Sake for the drunkard, gyokuro for the dotard."
"You knew about it?"
"He woke me up. At first I thought it was Teru."
"You did?" Kikuko sat with bowed head, as if unable to move.
"I was awake before you were, Kikuko," said Fusako, in the next room. "It wasn't at all pleasant. I knew it was Shuichi, because Teru is a quieter sort."
Still in her nightgrown, her younger child at her breast, Fusako came into the breakfast room. Her features were bad, but her breasts were white and remarkably full.
"You're a mess," said Shingo. "Put something on."
"Aihara was a mess, and so naturally I'm a mess too. There's nothing else for you to be when you're married to a man who's a mess." Fusako shifted Kuniko from her right breast to her left. "If you didn't want it that way, then it would have been a very good idea for you to look into things before you married me off."
"Men and women are different."
"They are the same. Look at Shuichi."
She started for the washstand.
Kikuko reached for Kuniko. Fusako handed her over so roughly that she began to cry.
Unconcerned, Fusako walked off.
Yasuko, back from washing her face, took the child. "What do you suppose her father means to do? It was New Year's Eve when Fusako came back. More than two months ago. He says that Fusako is a mess, but I think that Father here is even messier in the matter that is most important. You said on New Year's Eve that it was good to have a clean break, and since then you've done exactly nothing. And there's not been a word from Aihara." She was looking down at the baby as she spoke. "The Tanizaki girl you had in your office—Shuichi says she's half a widow. I suppose Fusako is half a divorcee."
"What does 'half a widow' mean?"
"She wasn't married, but the man she was in love with was killed in the war."
"But Tanizaki would have been a mere child."
"She was sixteen or seventeen by the old count. Old enough to have a man you can't forget."
Shingo thought the expression "a man you can't forget" an odd one to come from Yasuko.
Shuichi left without breakfast. He was late, and probably not feeling well.
Shingo killed time until the morning mail came. Among the letters Kikuko brought was one addressed to her.
He handed it to her.
She had apparently brought them in without looking at them. She rarely got letters. Nor did it seem that she expected them.
She read the letter in the breakfast room.
"It's from a friend. She had an abortion and hasn't been well since She's in the University Hospital in Hongo."
"Oh?" He took off his glasses and looked into her face. Did she fall into the clutches of some unlicensed old midwife? Very dangerous."
The newspaper article last night and Kikuko's letter—Shingo was struck by the coincidence. And he had dreamed of an abortion.
He was tempted to tell Kikuko of the dream.
But, as he gazed at her, unable to speak, he felt in himself a flicker of something youthful, and was startled as another thought flashed across his mind, that Kikuko was pregnant and was thinking of an abortion.
4
"See how the plums are blooming," said Kikuko wonderingly as the train passed through the North Kamakura valley.
In North Kamakura there were large numbers of plums very near the train window. Shingo saw them every day but paid no particular attention to them.
The white blossoms were past their prime. In the warm sunlight, they were beginning to look dirty.
"But our plums are in bloom too," said Shingo. There were only two or three of them, however, and perhaps this was the first real display that Kikuko had seen.
It was rare for her to get letters, and it was rare for her to go out, save to shop in Kamakura.
On her way to pay a sick call on her friend in the University Hospital, she had left with Shingo.
The house of Shuichi's woman was near the University. That fact troubled Shingo.
He wanted, along the way, to ask whether Kikuko was pregnant.
The question was not such a difficult one, and yet it seemed quite possible that he would let the opportunity pass.
How many years had it been since he had stopped asking Yasuko about her physiological processes? Since the change of life, Yasuko herself had said nothing. Had it become a question not of vigor but of decay?
Shingo had forgotten a matter of which Yasuko had stopped speaking.
As he thought to ask Kikuko the question, Yasuko came into his mind.
Perhaps if Yasuko had known that Kikuko was going to an obstetric ward, she would have suggested an examination.
Yasuko sometimes spoke to Kikuko of children. To Shingo it seemed that Kikuko found the subject forbidding.
Kikuko had without doubt said something to Shuichi. Long ago Shingo had listened with admiration to a friend's theory that for a woman the man to whom she made the revelation was everything. If she had another man, she kept the secret of her condition to herself.
A daughter did not make it to her own father.
Shingo seemed to avoid talking to Kikuko of Shuichi's Woman, and she to him.
If she was pregnant, it might be because of a ripening brought on by Shuichi's woman. An unpleasant thought, but a part of being human; and it seemed to him that there was cruelty hidden in speaking to Kikuko of children.
"Did Mother tell you that Grandfather Amamiya came around yesterday?"
"No."
"He came to say that he was being taken into the Tokyo house. And he brought two big sacks of cookies and asked us to be good to Teru."
"Are the cookies for Teru?"
"Mother thinks so. Or maybe one is for us. Grandfather Amamiya was very happy. He said that young Mr. Amamiya's business was doing well, and that he had built on to the house."
"That's how it is. A good businessman sells his house right away and starts all over again, and before you know it he is building on to a new house. With people like me, ten years go by like a day. Even this train ride gets to seem like too much trouble. The other day we had dinner together, all of us old men. It's remarkable how we go on year after year, doing the same old things. We get tired and bored, and ask when they'll come for us."
It did not seem that Kikuko quite understood the last remark.
"Someone said that when we go before the judge we should tell him that spare parts commit no sins. That's what we are, life's spare parts. And while we're alive, shouldn't life at least be kind to us?"
"But. . . ."
"It's true. I doubt whether anyone in any age can say he has really lived life for everything in it. Think of the man who checks your shoes at the restaurant. All he does day after day is put away shoes and take out shoes. One of us old men had a theory all his own—that things are actually easier for that kind of spare part. But the waitress didn't agree. The old man who takes care of shoes has a hard life, she said, He has to work in a hole with shoe shelves all around him, and there he sits hugging a charcoal fire and shining shoes. It's cold in the winter there in the doorway and hot in the summer. You've noticed how our own granny likes to talk about old people's homes?"
"Mother? But with Mother it's not that serious. It's like young people who keep saying they wish they were dead."
"That's true, I suppose. She assumes she's going to outlive me. But what young people are you talking about?"
"Young people. . . ." Kikuko hesitated. "In my friend's letter."
"The letter this morning?"
"Yes. She's not married."
"Well!"
He fell silent. Kikuko could not go on.
It was as the train left Totsuka. Hodogaya, the next stop, was some distance away.
"Kikuko. I've been thinking. Wouldn't you and Shuichi like to live away from us?"
Kikuko looked at him, waiting for him to say more. Then, a pleading note in her voice: "Why, Father? Because Fusako has come back?"
"It has nothing to do with Fusako. I know it's hard for you, having a half-divorcee with us; but even if she divorces Aihara, she probably won't be with us long. No, it has nothing to do with her. It has to do with the two of you. Don't you think it would be better?"
"No. You are good to me, and I would rather be with you. I don't think you can imagine how lonely I'd be away from you."
"You're being very kind."
"Oh, no. I'm taking advantage of you. I'm the baby, the spoiled child of the family. I was always my father's favorite and I like being with you."
"I can understand why your father favored you, and it's good having you with us. I wouldn't be happy to see you go. But Shuichi is the way he is, and I haven't once talked the problem over with you. A useless sort of parent to be living with. If the two of you were by yourselves, mightn't you come up with your own solution?"
"No. You don't say anything, but I know that you're worrying about me and sympathizing with me. That's how I manage to go on." There were tears in the large eyes. "I think I'd be afraid if you made us live away. I don't think I could stand to wait at home alone. I'd be too lonely, and frightened."
"I see—waiting for him by yourself. But this isn't the sort of thing to talk about on a train. Think it over."
She did seem to be frightened. Her shoulders were trembling.
He saw her to Hongo in a taxi.
Perhaps because she had been pampered by her father, or perhaps because she was upset, she did not seem to think the service unnatural.
It was most unlikely that Shuichi's woman would be out walking, and yet he was concerned. He waited until Kikuko was safely inside the hospital.
#
The Bell in Spring
In Kamakura in the season of cherry blossoms, the seven-hundredth anniversary of the Buddhist capital was being celebrated. The temple bell rang all through the day.
There were times when Shingo could not hear it. Kikuko heard it, apparently, even when she was working or talking; but Shingo had to listen carefully.
"There," Kikuko would inform him. "There. It rang again."
"Oh?" said Shingo, cocking his head to one side. "And how is it with Granny?"
Yasuko was no comfort. "Of course I can hear it. It's practically deafening."
She was reading at her own pace through the five days' accumulation of newspapers before her.
"There it goes, there it goes," said Shingo. Once he had caught the sound, it was easy to follow succeeding strokes.
"You seem very pleased." Yasuko took off her glasses and looked at him. "The priests must get tired, ringing away at it day after day."
"No, they have the pilgrims ring, at ten yen a stroke," explained Kikuko. "It's not the priests."
"A clever idea," said Shingo.
"They call it the bell for the dead, or something of the sort. The angle is to have a hundred thousand people or a million people or something of the sort ring the bell."
"The angle?" Her choice of words struck Shingo as amusing.
"It has a dark sound to it." said Kikuko. "I don't really like it."
"You think it's dark?"
Shingo himself had been thinking how pleasantly quiet and relaxed it was, sitting in the breakfast room on an April Sunday, looking at the cherry blossoms and listening to the bell.
"What is this the seven-hundredth anniversary of, anyway?" asked Yasuko. "Some say it has to do with the Great Buddha, and some say it's Nichiren."
Shingo could not answer.
"Do you know, Kikuko?"
"No."
"Very odd. And here we all are living in Kamakura."
"Isn't there anything in your newspapers, Mother?"
"There might be." Yasuko passed them on to Kikuko. They were neatly folded and stacked. Yasuko kept one for herself. "I believe I did see something. But I was so struck by the piece about the old couple who left home that I forgot everything else. You saw it, I suppose?" she asked Shingo.
"Yes."
"A great benefactor of Japanese boat racing. The vice-chairman of the Japanese Rowing Association." She began to read the article, and then went on in her own words. "He was the president of a company that makes boats and yachts. He was sixty-nine, and she was sixty-eight."
"And what was there about it that struck you so?"
"He left behind notes to their daughter and son-in-law and grandchildren. Here it is in the paper." Yasuko began reading again. "'Miserable old creatures, living our leftover lives, forgotten by the world? No, we have decided that we do not want to live so long. We quite understand the feelings of Viscount Takagi.* People should go away while they are still loved. We shall go now, still in the embrace of family affection, blessed with numbers of comrades and colleagues and schoolmates.' That's to the daughter and son-in-law. And this is the one to the grandchildren: 'The day of Japanese independence is approaching, but the way ahead is dark. If young students who know the horrors of war really want Peace, then they must persist to the end in the nonviolent methods of Gandhi. We have lived too long and no longer have the strength to lead and to pursue the way we believe right. Were we to live idly on into "The Spiteful Years"* then we would have made meaningless the years we have lived thus far. We want to leave behind memories of a good grandfather and grandmother. We do not know where we are going. We but go quietly off.'"
Yasuko fell silent.
Shingo turned aside, to look at the cherries in the garden.
Yasuko was still looking at the newspaper. "They left their house in Tokyo and disappeared after they had visited his sister in Osaka. The sister is in her eighties."
"Did the wife leave a note?"
"What?" Yasuko looked up in surprise.
"Didn't the wife leave a note?"
"The wife? The old woman?"
"Of course. If they went off together, it would have been natural for the wife to leave a note too. Suppose you and I were to commit suicide. You'd have something you wanted to say, and write it down."
"That wouldn't be necessary," said Yasuko briskly. "It's when young people commit suicide that they both leave notes. They want to talk about the tragedy of being kept apart. What would I have to say? With a husband and wife it's enough for the husband to leave a note."
"You really think so?"
"It would be different if I were to die by myself."
"I suppose you'd have mountains of pains and regrets."
"They wouldn't matter. Not at my age."
Shingo laughed, "Comfortable remarks from an old woman who doesn't plan to die and isn't about to die. And Kikuko?"
"Me?" It was a low, hesitant voice.
"Supposing you were to commit suicide with Shuichi. Would you want to leave a note?"
He knew immediately that it had not been the thing to say.
"I don't know. I wonder how it would be." She looked at Shingo. The forefinger of her right hand was in her obi, as if to loosen it. "I have a feeling I'd want to say something to you, Father." Her eyes had a youthful moistness, and then there were tears in them.
Yasuko had no intimations of death, thought Shingo, but Kikuko was not without them.
Kikuko leaned forward. He thought she was about to collapse in tears, and then she got up.
Yasuko looked after her, "Odd. She has nothing to cry about. Hysteria, that's what it is. Plain hysteria."
Shingo unbuttoned his shirt and put his hand to his chest.
"Is your heart pounding?"
"No. The nipple itches. It's hard and itches."
"Like a teen-age girl."
Shingo rubbed his left nipple with his forefinger.
When a couple committed suicide together the husband left a note and the wife did not. Did the wife have the husband substitute for her, or act for the two in concert? The question had puzzled and interested Shingo as Yasuko read from the newspaper.
Living together for long years, had the two become one? Had the old wife lost her identity, was she without a testament to leave behind?
Was it that the woman, with no compulsion to die, went in attendance upon her husband, had her part in the husband's testament, without bitterness, regrets, hesitation? It all seemed very odd to Shingo.
But his own old wife had in fact said that if they were to commit suicide she would not need to leave a note. It would be enough for him to.
A woman who went uncomplainingly to death with a man—there were times when the opposite was the case, but usually the woman followed the man. Somehow it startled Shingo that such a woman, grown old, should be here beside him.
Kikuko and Shuichi had not been together long, and already they were having their troubles.
It might be that he had been cruel to Kikuko, that he had injured her when he asked whether she would want to leave a note.
He knew that she stood on a dangerous brink.
"You're too good to her. That's why she cries over such silly things." said Yasuko. "You pamper her, and you've done not one single thing about the most important problem. It's the same with Fusako."
Shingo was looking at the cherry tree, heavy with blossoms.
Under the great cherry was a rich growth of yatsude.*
Disliking yatsude, Shingo had meant to cut it away before the cherry bloomed; but there had been heavy snowfalls in March, and now the blossoms had come.
Though he had cut it down three years ago, it had come back all the more luxuriantly. He had thought then that he should dig up the roots, and he had been right.
Yasuko's observations made him dislike the heavy green of the leaves all the more. Without the yatsude the cherry would stand alone, spreading until its branches fell in the four directions. It had spread well even with the yatsude crowding it.
And it was so laden with blossoms that one wondered how a tree could bear so many.
The blossoms floated up grandly in the light of the afternoon sky. Neither the shape of the tree nor its color was particularly strong, but one fell that it quite filled the sky. The blossoms were at their best. It was hard to think that they would fall.
But two or three petals were constantly falling, and the ground was carpeted with them.
"When you read that a young person has been killed or committed suicide, you just say to yourself it's happened again," mused Yasuko. "But with old people it really hits you. 'People should go away while they are still loved.'" She had evidently read the article over two and three times. "The other day there was a story about a sixty-one-year-old man who brought his grandson down from Tochigi to put him in St. Luke's Hospital. The boy was seventeen and had had infantile paralysis. The grandfather carried the boy around on his back to show him Tokyo. But the boy absolutely refused to go to the hospital, and so finally the grandfather strangled him with a towel, It was in the paper the other day."
"Oh? I didn't notice." Shingo's answer was indifferent, but he remembered how deeply he had been impressed by the article about young girls and their abortions, how he had even dreamed of it.
The difference between him and the old woman who Was his wife was very considerable.
2
"Kikuko," called Fusako. "This sewing machine is always breaking the thread. Is something wrong with it? Come and have a look. It's a Singer and ought to be a good machine. I wonder if I've lost my touch. I wonder if maybe I'm hysterical."
"It might be going to pieces. I've had it since I was in school." Kikuko came into the room. "But it listens when I speak to it. Let me have a try."
"Oh? I get so nervous with Satoko hanging onto me. I'm always sewing her hand. Of course I don't sew her hand, but she puts it up here like this, and while I'm watching the seam everything gets blurry and she and the cloth run together."
"You're tired."
"Just as I said. Hysterical. You're tired yourself. The only ones in this house who aren't tired are Grandfather and Grandmother. Here Grandfather is in his sixties and he complains about an itching nipple. Ridiculous."
On her way back from the sick call in Tokyo, Kikuko had bought material for the two children.
Fusako was at work on dresses, and well disposed toward Kikuko.
Displeasure was plain on Satoko's face, however, as Kikuko changed places with Fusako.
"Aunt Kikuko bought the material, and now you're going to have her sew it?"
"Don't listen to her, Kikuko. She's exactly like Aihara," Apologies did not come easily from Fusako.
Kikuko put her hand on Satoko's shoulder. "Get Grandfather to take you down to the Buddha. There will be a procession with little princes and all, and there will be dancing."
At Fusako's urging, Shingo went out with Fusako and Satoko.
As they walked down the main street of the Hase district, Shingo's eye was caught by a dwarf camellia before a tobacco shop. Buying a package of Hikari cigarettes, he offered a word of praise. The blossoms, of which there were five or six, were double, with crinkly petals.
No, replied the tobacconist. Double blossoms were not right for dwarf trees. One should stay with the single-petalled wild camellia. He led them to the garden behind. Dwarf trees in pots were lined up by a vegetable patch some four or five yards square. The wild camellia was an old tree with a powerful trunk.
"I've pinched off the buds," the man said. "It wouldn't do to wear the tree out."
"It does have buds?"
Plenty of them. But I leave very few. The one out in front must have had twenty or thirty."
The man talked of the techniques of dwarfing, and of People in Kamakura with a fondness for dwarf trees. Shingo had frequently seen them in shop windows.
Thank you very much," he said, leaving the shop. "I envy you."
I don't have any really good ones, but the wild camellia does have its points. You get yourself a tree, and then you re responsible for seeing that it doesn't die or lose its shape. It's good medicine for laziness."
Shingo lit one of the cigarettes he had just bought, "It has the Buddha on it," he said, handing the package to Fusako. "Especially made for Kamakura."
"Let me see." Satoko stretched for the cigarettes.
"You remember last fall when you ran away from home and went off to Shinano?"
"I did not run away from home."
"Were there any dwarf trees in the old house?"
"I didn't see any."
"Probably not. It must be forty years ago. The old man was addicted to dwarf trees. Yasuko's father. But you know how Yasuko is, and he preferred her sister. It was her sister he had help him with the trees. She was such a beauty that you'd never have dreamed she was Yasuko's sister. I can see her now, dressed in a red kimono, bangs on her forehead, going down of a morning when snow was piled on the shelves to brush it away from the branches. I can see it right here in front of me, all fresh and clean. Shinano is cold, and her breath was white."
The white breath was scented with the softness of the young girl.
Sunk in memories, Shingo was taking advantage of the fact that Fusako, of a different generation, was not interested.
"I imagine that camellia has been at it for more than forty years." It seemed to be of a venerable age. How many years would it take for a dwarfed trunk to become like flexed biceps?
The maple that had glowed red on the altar after Yasuko's sister died—would it, in someone's hands, still be alive?
3
When the three came to the temple precincts, the "procession of little princes" was weaving its way up the flagstone walk before the Great Buddha. The little boys had walked a considerable distance, it seemed. Some of them were exhausted.
Fusako lifted Satoko to see over the wall of people, Satoko gazed at the boys in their flowery kimonos.
Having heard that there was a stone in the precincts bearing a poem by Yosano Akiko,* they went behind the statue to look for it. It seemed to be in Akiko's own hand, enlarged, and carved on the stone.
"I see it has Sakyamuni," said Shingo.
He was astounded that Fusako did not know this most famous of poems.
Akiko had written:
> A summer grove, Kamakura; a Buddha he may be.
>
> But a handsome man he also is, Lord Sakyamuni.
"But the Great Buddha isn't a Sakyamuni. He's actually an Amitabha. Seeing that she had made a mistake, she rewrote the poem, but by that time the Sakyamuni version was too well known, and to change it to the Great Buddha or something of the sort would spoil the rhythm—and bring in Buddha twice. But it is a mistake. A mistake carved on stone, right here in front of us."
Ceremonial tea was being served in a curtained-off space near the stone, Kikuko had given Fusako tickets.
The tea in the open sunlight had its own special color. Shingo wondered whether Satoko too would drink it. Satoko clutched at the edge of the bowl with one hand. It was a most ordinary bowl, but Shingo reached to help her.
"It's bitter."
"Bitter?"
Even before she had tasted the tea, Satoko's face announced that it was bitter.
The little dancing girls came inside the curtain. Perhaps half sat down on stools by the door. The others crowded in front of them, one against another. They were all heavily made up and had on long-sleeved festive kimonos.
Behind them, two or three young cherries were in full bloom. Defeated by the powerful colors of the girls' kimonos, they seemed pale and wan. The sun was shining on the green of the tall trees beyond.
"Water, Mother, water," said Satoko, glaring at the dancers.
"There is no water. You can have some when we get home."
Suddenly Shingo too wanted water.
One day in March, from the Yokosuka train, Shingo had seen a girl about Satoko's age drinking at a fountain. She laughed in surprise when, as she turned it on, the water shot high in the air. The laughing face was very pretty. Her mother adjusted the fountain for her. Watching her drink as if it were the world's most delicious water, Shingo thought to himself that this year too spring had come. The scene returned to him now.
He wondered what it was about the cluster of little girls dressed for dancing that had made both him and Satoko want water. Satoko was grumbling again. "Buy me a kimono, Mother, buy me a kimono."
Fusako got up.
Among the girls was a most appealing one a year or two older than Satoko. Her eyebrows were painted on in thick, short, sloping lines, and at the edges of her eyes, round as bells, there was rouge.
Satoko stared at the girl as Fusako led her off, and as they started out through the curtain, lunged in her direction.
"A kimono," she persisted. "A kimono."
"Grandfather said he'll buy you one for Three-five-seven Day,"* said Fusako insinuatingly. "She hasn't once worn a kimono. Only diapers from an old cotton kimono, an outcast of a kimono."
They went into a tea stall, and Shingo asked for water. Satoko gulped down two glasses.
They had left the precincts of the Great Buddha and were walking toward home when a girl in a dancing kimono hurried past on her mother's hand, apparently also on the way home. This would not do, thought Shingo, taking Satoko by the shoulder; but he was too late.
"A kimono," said Satoko, reaching for the girl's sleeve.
"Don't!" Pulling away, the girl tripped over her long sleeve and fell.
Shingo gasped and brought his hands to his face.
The child was being run over. Shingo heard only his own gasp, but it seemed that numbers of other people had cried out.
The automobile screeched to a stop. Three or four ran forward from among the horrified outlookers.
The girl jumped up. Clinging to her mother's skirt, she began screaming as if set afire.
"Good, good," someone said. "The brakes worked. An expensive car."
"If it had been a broken-down wreck you wouldn't be alive."
Satoko was terrified. Her eyes were rolled back into her head as if she were having a convulsion.
Was the girl hurt, had she torn her kimono, asked Fusako, apologizing profusely to the girl's mother. The mother was looking absently into space.
When the girl had finished screaming, her thick powder had run; but her eyes were shining, as if washed clean.
Shingo had little to say the rest of the way home.
They heard the baby wailing.
Singing a lullaby, Kikuko came out to greet them.
"I'm sorry," she said to Fusako. "I have her crying. I'm a failure."
Perhaps led on by her sister, perhaps surrendering now that she was safe at home, Satoko too was wailing.
Ignoring Satoko, Fusako pulled her kimono open and took the baby from Kikuko.
"Just look, will you. I'm all in a cold sweat here in the hollow between."
Shingo glanced up at a framed inscription that purported to be a Ryokan:* "In the heavens, a high wind. He had bought it when Ryokans were still cheap, but it was a forgery all the same. A friend having pointed this fact out, he could see that it must be true.
"We had a look at the Akiko stone," he said to Kikuko. "It's in Akiko's own hand, and it says 'Sakyamuni.'"
"It does, does it?"
4
After dinner Shingo went out alone to look through the new and used kimono shops.
But he found nothing that seemed appropriate for Satoko.
The matter weighed even more heavily on his mind.
He felt a dark foreboding.
Did even so young a girl covet another's bright kimono?
Was it only that Satoko's envy and greed were somewhat stronger than the usual? Or were they quite extraordinarily powerful? In either case, the outburst had struck Shingo as lunatic.
What would be happening now if the girl in the dancing clothes had been run over and killed? The pattern of the girl's kimono came up vividly before him. There was seldom anything so festive in the shop windows.
But the thought of returning empty-handed made the street seem dark.
Had Yasuko given Satoko only old cotton kimonos to be made into diapers? Or was Fusako lying? There had been poison in the remark. Had Yasuko not given the girl a swaddling kimono, or a kimono for her first visit to a shrine? Had Fusako perhaps asked for western clothes?
"I forget," he muttered to himself.
He had forgotten whether or not Yasuko had consulted with him in the matter; but if they, he and Yasuko, had paid more attention to Fusako, they might have been given a pretty grandchild even by so ill-favored a daughter. Feelings of inescapable guilt dragged at him.
"Because I know how it was before birth, because I know how it was before birth, I have no parent to love, Because I have no parent, neither have I child to be loved by."
A passage from a No play came to Shingo, but that alone scarcely brought the enlightenment of the darkcloaked sage.
"The former Buddha has gone, the later not yet come. I am born in a dream, what shall I think real? I have chanced to receive this human flesh, so difficult of receiving."
Had Satoko, about to pounce upon the dancing girl, inherited her violence and malice from Fusako? Or did she have them from Aihara? If from Fusako, then did Fusako have them from Yasuko or from Shingo?
If Shingo had married Yasuko's sister, then probably he would have had neither a daughter like Fusako nor a granddaughter like Satoko.
This was hardly a proper occasion to stir in him so intense a yearning for a person long dead that he wanted to rush into her arms.
He was sixty-three, and the girl who had died in her twenties had been older than he.
When he got home, Fusako was in bed, the baby in her arms. The door between her room and the breakfast room was open.
"She's asleep," said Yasuko. "Her heart was pounding and pounding, and Fusako gave her sleeping medicine.
She went right off to sleep." Shingo nodded. "Suppose you pull the door shut.
"Yes." Kikuko got up.
Satoko was tight against Fusako's back. But her eyes seemed to be open. She had a way of staring at a person. silently and rigidly. Shingo said nothing about having gone out to buy her a kimono.
It appeared that Fusako had not told her mother of the crisis that had arisen from Satoko's desire for a kimono.
He went into his room, Kikuko brought charcoal.
"Have a seat," he said to her.
"In just a second." She went out, and came back with a pitcher on a tray. One did not need a tray for a pitcher; but there seemed to be flowers beside it. "What are they?" He took a flower in his hand. "Kikyo,* maybe?"
"Black lilies, I'm told."
"Black lilies?"
"Yes. A friend I take tea lessons with gave them to me a little while ago," She opened the closet door behind Shingo and took out a little vase.
"Black lilies, are they?"
"She said that on the anniversary of Rikyu's death this year the head of the Enshu School arranged a tea ceremony in the museum tea cottage. There was an old narrow-necked bronze vase in the alcove with black lilies and white hyacinths in it. A very interesting combination, she said."
"Oh?"
Shingo gazed at the black lilies. There were two of them, with two flowers on each stem.
"It must have snowed eleven or thirteen times this spring."
"We did have a lot of snow."
"She said that there were four or five inches of snow on the anniversary of Rikyu's death. It was very early in the spring, and black lilies seemed even more unusual, They're mountain flowers, you know."
"The color is a little like a black camellia."
"Yes." Kikuko poured water into the vase. "She said that Rikyu's testament was on display, and the dagger he committed suicide with."
"Oh? Your friend gives tea lessons?"
"Yes. She's a war widow. She worked hard, and now the returns are coming in."
"What school?"
"Kankyuan. The Mushanokoji family."
This meant nothing to Shingo, who knew little about tea.
Kikuko waited, ready to put the flowers in the vase, but Shingo still had one in his hand.
"It seems to droop a little. I don't suppose it's wilting?
"No. I had them in water."
"Do kikyo droop too?"
"I beg your pardon?"
"It seems a little smaller than kikyo."
"I believe so."
"At first it looks black, but it isn't. It's like dark I'll purple, but not that either—touched with crimson. I'll have to have a good look at it tomorrow in daylight." "In the sun it's a transparent purple touched with ed."
The flowers, fully opened, would be scarcely an inch across. There were six petals. The tips of the pistils parted in three directions, and there were four or five stamens. The leaves spread in the four directions at stages about an inch apart. They were small for lily leaves, not two inches long.
Finally Shingo sniffed at the flower.
"The smell of a dirty woman." It was a badly chosen remark.
He had not meant to suggest wantonness, but Kikuko looked down and flushed slightly around the eyes.
"The smell is a disappointment," he corrected himself. "Here. Try it."
"I think I'll not investigate as thoroughly as you, Father." She started to put the flowers into the vase. "Four is too many for a tea ceremony. But shall I leave them as they are?"
"Yes, do."
Kikuko set the vase in the alcove.
"The masks are in that closet, the one you took the Vase from. Would you mind getting them out?"
He had thought of the No masks when that passage from a No play had come to him.
He took up the jido. "This one is a sprite. A symbol of eternal youth. Did I tell you about it when I bought it?"
"No."
"Tanizaki, the girl who was in the office. When I bought it I had her put it on. She was charming. A great surprise."
Kikuko put the mask to her face. "Do you tie it behind?"
No doubt, deep behind the eyes of the mask, Kikuko's eyes were fixed on him.
"It has no expression unless you move it."
The day he had brought it home, Shingo had been on the point of kissing the scarlet lips. He had felt a flash like heaven's own wayward love.
"It may be lost in the undergrowth, but while it still has the flower of the heart. . . ."
Those too seemed to be words from a No play.
Shingo could not look at Kikuko as she moved the glowing young mask this way and that.
She had a small face, and the tip of her chin was almost hidden behind the mask. Tears were flowing from the scarcely visible chin down over her throat. They flowed on, drawing two lines, then three.
"Kikuko," said Shingo. "Kikuko. You thought if you were to leave Shuichi you might give tea lessons, and that was why you went to see your friend?"
The jido Kikuko nodded.
"I think I'd like to stay on with you here and give lessons." The words were distinct even from behind the mask.
A piercing wail came from Satoko.
Teru barked noisily in the garden.
Shingo felt something ominous in it all. Kikuko seemed to be listening for a sign at the gate that Shuichi, who evidently went to visit the woman even on Sunday, had come home.
## Footnotes
* Father-in-law of the present Emperor's youngest brother. His death, in 1948, is commonly believed to have been at his own hand.
* A reference to a story (1947) by Niwa Fumio.
* FATSIA JAPONICA.
* 1878-1942.
* November 15. Children those ages are presented at shrines.
* Poet, 1757-1831.
* Sometimes called bell flowers.
#
The Kite's House
In summer and in winter, the bell in the temple rang at six; and in summer and winter, Shingo told himself, when he heard it, that he was awake too soon.
This did not necessarily mean that he got out of bed.
Six o'clock was of course not in summer what it was in winter. Because the bell rang at the same time, he could tell himself that it was six; but in summer the sun Was already up.
He had a large pocket watch at his pillow. He had to turn on the light and put on his glasses, however, and so he seldom looked at it. Without his glasses, he had trouble distinguishing the hour hand from the minute hand.
He had no worries about oversleeping. The trouble was the reverse, that he woke too early.
Six of a winter morning was very early, but, unable to stay in bed, Shingo would go for the paper.
Since the maid had left them, Kikuko had been getting up to do the morning work.
"You're early, Father," she would say.
"I'll sleep a little longer," Shingo would reply, embarrassed.
"Yes, do. I don't even have hot water yet."
With Kikuko up, Shingo would feel that he had company.
At what age had it been that he had begun to feel lonely, waking before the winter sun was up?
With spring, the waking became warmer.
Mid-May had passed, and after the bell he heard the cry of a kite.
"So it's here again," he muttered to himself, listening from bed.
The kite seemed to be strolling grandly over the roof, and then it flew off toward the sea.
Shingo got up.
He scanned the sky as he brushed his teeth, but the kite was not to be seen.
But it was as if a fresh young voice had departed and left the sky over the roof serene.
"Kikuko. You heard our kite, I suppose?" he called to the kitchen.
"No, I didn't. Careless of me." Kikuko was transferring rice, hot and steaming, from the pot to the serving cask.
"It makes its home with us. Wouldn't you say so?"
"Yes, I suppose so."
"We heard a lot from it last year, too. What month was it, I wonder? About now? My memory isn't what it ought to be."
With Shingo looking at her, Kikuko untied the ribbon around her hair.
It would seem that she sometimes slept with her hair tied up.
Leaving the cask uncovered, she hurried to make Shingo's tea.
"If our kite is here, then our buntings ought to be here too."
"Yes. And crows."
"Crows?" Shingo laughed. If it was "our" kite, then it should also be "our" crows. "We think of it as a house for human beings, but all sorts of birds live here too."
"And the fleas and mosquitoes will be coming out."
"That's a nice thought. But fleas and mosquitoes don't live here. They don't live over from one year to the next."
"I imagine the fleas do. We have them in the winter."
"I've no idea how long fleas live, but I doubt if this year's fleas are last year's."
Kikuko looked at him and laughed. "That snake will be coming out one of these days."
The aodaisho* that scared you so?"
"Yes."
He's the master of the place."
Back from shopping, one day last summer, Kikuko had seen the snake at the kitchen door, and come in trembling with fright.
Teru ran up at Kikuko's scream and raised a mad barking. Teru would lower her head as if to bite at the snake, jump back four or five feet, and come in for the attack again. The process was repeated over and over.
The snake raised its head and put out a red tongue, then turned and slithered off past the kitchen doorsill.
From Kikuko's description, it stretched more than twice the width of the door, or more than two yards; and it was thicker than her wrist.
Kikuko was greatly agitated, but Yasuko was calm. "It's the master of the place," she said. "It was here I don't know how many years before you came."
"What would have happened if Teru had bitten it?"
"Teru would have lost. She would have gotten all tangled up in it. She knew it well enough, and that's why she only barked."
Kikuko was still trembling. For a time she avoided the kitchen door, and went in and out through the front door.
It bothered her to think that there was such a monster under the floor.
But it probably lived on the mountain behind and rarely came down.
The land behind the house did not belong to Shingo. He did not know whose it was.
The mountain pressed down in a steep slope upon Shingo's house, and for animals there seemed to be no boundary marking off his garden, into which leaves and flowers from the mountain fell liberally.
"It's back again," he muttered to himself. And then, cheerfully: "Kikuko, the kite seems to be back."
"Yes. This time I hear it." Kikuko glanced at the ceiling.
The crying of the kite went on for a time.
"It flew off to the sea a few minutes ago?"
"So it seemed."
"It went off for something to eat and then came back."
Now that Kikuko had said so, that seemed a most likely possibility. "Suppose we put fish out where it will see them."
"Teru would eat them."
"Some high place."
It had been the same last year and the year before: Shingo felt a surge of affection when, on waking, he heard the call of the kite.
He was not alone, it would seem. The expression "our kite" was current throughout the house.
Yet he did not know for certain whether it was one kite or two. It seemed to him that he had, one year or another, seen two kites dancing on the roof.
And was it the same kite whose voice they heard year after year? Had a new generation taken the place of the old? Had the parent kite perhaps died, and was its young now calling out in its place? The thought came to Shingo this morning for the first time.
It seemed to him an interesting thought, that the old kite had died last year, and that, without knowing it, half asleep and half awake, they should be listening to a new kite this year, and thinking it their own.
And it seemed strange that, with all the mountains in Kamakura, the kite should have chosen to live on the mountain behind Shingo's house.
"I have met with what is difficult of meeting, I have heard what is difficult of hearing."* Perhaps it was so with the kite.
If the kite was living with them, it let them have the pleasure of its voice.
2
Because Shingo and Kikuko were the early risers, they could say what they had to say to each other early in the morning. Shingo talked alone with Shuichi only when the two of them happened to be on the same train.
"We're almost there," he would say to himself as they crossed the railway bridge into Tokyo and the Ikegami grove came in sight. It was his habit to look out the window of the morning train at the grove.
But, for all the years he had taken the same train, he had but recently discovered two pine trees in the grove.
The pine trees stood out above the grove. They leaned toward each other, as if about to embrace. The branches came so near that it was as if they might embrace at any moment.
Since they so stood out, the only tall trees in the grove, they should have caught his eye immediately. Now that he had noticed them, it was always the two pines he saw first.
This morning they were blurred by wind and rain.
"Shuichi," he said. "What's wrong with Kikuko?"
"Nothing in particular." Shuichi was reading a weekly magazine.
He had bought two in Kamakura station and had handed one to his father. Shingo's lay unread.
"What's wrong with her?" Shingo repeated quietly.
"She complains of headaches."
"Oh? The old woman says she was in Tokyo yesterday and went to bed when she got back last night. She's not her usual self. Something happened in Tokyo, the old woman thinks. She didn't have dinner last night, and when you got home and went to your room, it must have been about nine, we heard her crying. She tried to smother it, but we could hear."
"She'll be all right in a few days. I don't think it's anything to worry about."
"Oh? She wouldn't cry if it were only a headache. And wasn't she crying again early this morning?"
"Yes."
"Fusako says that when she went in with breakfast, Kikuko refused to look at her. Fusako was very unhappy about it. I thought I might ask you to tell me what's wrong."
"All the eyes in the family seem to be on Kikuko." Shuichi cocked an eye up at his father. "She gets sick occasionally, like everyone else."
"And what is the ailment?" he asked irritably.
"An abortion." Shuichi flung out the words.
Shingo was aghast. He looked at the seat ahead of them. It was occupied by two American soldiers. He had started the conversation on the assumption that they Would not understand.
He lowered his voice. "She went to a doctor?"
"Yes."
"Yesterday?" It was a hollow mutter.
Shuichi had laid down his magazine. "Yes."
"And came back the same day?"
"Yes."
"You had her do it."
"She wanted it done, and wouldn't listen to anything said."
"Kikuko wanted to? You're lying."
"It's the truth."
"Why? What could make her feel that way?"
Shuichi was silent.
"Don't you think it's your fault?"
"I suppose so. But she said she didn't want it now and that was that."
"You could have stopped her if you'd tried."
"Not this time, I think."
"What do you mean, this time?"
"You know what I mean. She won't have a baby with me as I am."
"While you have the other woman?"
"I'd say so."
"You'd say so!" Shingo's chest was tight with anger. "It was half a suicide. Don't you think so? It wasn't so much that she was getting back at you as that she was half killing herself." Shuichi fell back before the assault. "You've destroyed her spirit and the damage can't be undone."
"I'd say her spirit is still pretty strong."
"But isn't she a woman? Your wife? If you'd done one thing to comfort her she'd have been happy to have the baby. Quite aside from the other woman."
"Oh, but it isn't quite aside."
"Kikuko knows how much Yasuko wants grandchildren. So much that she feels guilty about taking so long. She doesn't have the baby she wants to have, and that's because you've murdered her spiritually."
"It's a little different, actually. She has her own squeamishness."
"Squeamishness?"
"Resents being put in that situation."
"Oh?" It was a matter between husband and wife. He wondered whether Shuichi had in fact made Kikuko feel so debased and insulted. "I don't believe it. She may have talked and acted as if she felt that way, but I doubt if she really did. For a husband to make so much of his wife's squeamishness is a sign that he's short on affection. Does a husband take a fit of pouting so seriously?" Shingo had somewhat lost his momentum. "I wonder what Yasuko will say when she hears she's lost a grandchild."
"I'd think she'd feel relieved. She knows now that Kikuko can have children."
"What's that? You guarantee that she will have children later?"
"I'm prepared to guarantee it."
"Anyone who can say that has no fear of heaven and no human affection."
A difficult way to put it. Isn't it a simple enough matter?"
It's not simple at all. Think about it a minute. Think about the way she was crying."
"It's not that I don't want children myself. But with things between us as they are now, I doubt if it would be a very superior child."
"I don't know about things with you, but there is nothing wrong about things with Kikuko. It's only you that things are wrong with. She's not that way. You do nothing to help her get rid of her jealousy. That's why she lost the child. And maybe more than the child, too." Shuichi was looking at him in surprise. "Suppose you have a try at getting blind drunk with that woman and coming into the house with your dirty shoes on and putting them on Kikuko's knee and having her take them off for you."
3
Shingo went to the bank that morning on company business, and had lunch with a friend who worked there, They talked until about two-thirty. After telephoning the office from the restaurant, he started for home.
Kikuko was sitting on the veranda with Kuniko on her lap.
She got up hastily, surprised at his early return.
"No, please." He came out to the veranda. "Shouldn't you be in bed?"
"I was about to change her diaper."
"Fusako?"
"She's gone to the post office with Satoko.''
"What business does she have at the post office? Leaving the baby behind."
"Just a minute," said Kikuko to the baby. "I'll get out Grandfather's kimono first."
"No, please. Get her changed first."
Kikuko looked up smiling. Her small teeth showed between her lips.
"He says I'm to get you changed first." She was in déshabille, her bright silk kimono tied with a narrow obi. "Has it stopped raining in Tokyo?"
"Raining? It was raining when I got on the train, but clear when I got off. I didn't notice where it stopped."
"It was raining here until just a few minutes ago. Fusako went out when it stopped."
"It's still wet up the hill."
Laid face up on the veranda, the baby raised her bare feet and took her toes in her hands. The feet moved more freely than the hands. "Yes, have a look up the mountain," said Kikuko, wiping the baby's rear.
Two American military planes flew low overhead. Startled by the noise, the baby looked up at the mountain. They did not see the planes, but great shadows passed over the slope. Probably the baby saw them too.
Shingo was touched by the gleam of surprise in the innocent eyes.
"She doesn't know about air raids. There are all sorts of babies who don't know about war." He looked down at the baby. The gleam had already faded. "I wish I had a picture of her eyes just now. With the shadow of the airplanes in it. And the next picture. . . ."
Of a dead baby, shot from an airplane, he was about to say; but he held himself back, remembering that Kikuko had the day before had an abortion.
In fact, there were numberless babies like Kuniko as he had seen her in the two pictures.
Kuniko in her arms and a rolled-up diaper in one hand, Kikuko went off to the bath.
Shingo had come home early out of concern for Kikuko. He went into the breakfast room.
"What brings you back so soon?" said Yasuko, joining him.
"Where were you?"
"I was washing my hair. It stopped raining and the sun came blazing out, and my head got to feeling all itchy. An old person's head seems to itch for no reason at all."
"Mine doesn't."
"Probably because it's such a good head," she laughed. "I knew you were back, but I thought if I came in with my hair all which-way I'd get a scolding."
"The old woman's hair all undone—why not cut it off and make a tea whisk out of it?"
"Not at all a bad idea. Men have their whisks too. It used to be, you know, that both men and women cut their hair short and pulled it back like a tea whisk. You see it in the Kabuki."
"I don't mean hair tied up. I mean hair cut off."
"I wouldn't mind. We both have too much hair."
"Kikuko is up and around?'' he asked in a low voice.
"She's been having a try at it. She doesn't look at all well."
"She shouldn't be taking care of the baby."
"Take care of her for a minute, please, Fusako said, and dumped her by Kikuko's bed. The baby was sound asleep."
"Why didn't you take her?"
"I was washing my hair when she started crying. Yasuko went for his kimono. "I wondered if something might be wrong with you, too, you got home so early."
Shingo called to Kikuko, who seemed to be going from the bath to her room.
"Bring Kuniko in here."
"We'll be there in a minute."
Her hand in Kikuko's, Kuniko was having a walk. Kikuko had put on a more formal obi.
Kuniko clutched at Yasuko's shoulder Yasuko, who was brushing Shingo's trousers, took the baby on her knee.
Kikuko went off with Shingo's suit.
Having put it. away in the next room, she slowly closed the doors of the wardrobe.
She seemed taken aback by her face in the wardrobe mirror, and she wavered between going to her room and returning to the breakfast room.
"Wouldn't you be better off in bed?" said Shingo.
"Yes." A spasm passed across Kikuko's shoulders. She went off to her room without looking back.
"Doesn't she seem strange to you?" Yasuko frowned.
Shingo did not answer.
"And it's not at all clear what's the matter She gets up and walks around, and then starts breaking down again. I'm very worried."
"So am I."
"You have to do something about Shuichi and that affair of his."
Shingo nodded.
"Suppose you have a good talk with Kikuko, I'll take the baby out to Fusako and while I'm about it do Some shopping for dinner. That Fusako—she's another one."
Yasuko got up, the baby in her arms.
"What business does she have at the post: office?"
Yasuko looked back "I wondered myself. Do you suppose she's written to Aihara? They've been separated for six months. It's almost six months since she came back. It was New Year's Eve."
"If it was just a letter she could have put it in the mailbox down the street."
"I imagine she thinks it will be quicker and safer if she sends it from the post office. Maybe the thought of Aihara comes into her head and she can't sit still a minute."
Shingo smiled wryly. He sensed optimism in Yasuko.
It would seem that optimism put down deep roots in a woman who had been given charge of a household on into old age.
He took up the heap of newspapers, four or five days worth of them, that Yasuko had been reading. Though he was not really interested, his eye fell on a remarkable headline: "Lotus in Bloom, Two Thousand Years Old."
The spring before, in the course of a Yayoi excavation in the Kemigawa district of Chiba, three lotus seeds had been found in a dugout canoe. They were judged to be two thousand years old. A certain "doctor of lotuses succeeded in making them sprout, and in April of this year the shoots were planted in three places, the Chiba Agricultural Experimental Station, the pond of a Chiba park, and the house of a sake brewer in Hatake -machi, Chiba. The brewer apparently had been among the sponsors of the excavation. He had put his shoot in a water cauldron and set it out in the garden, and his was first to bloom. The lotus doctor rushed to the spot upon hearing the news. "It's in bloom, it's in bloom," he said, stroking the handsome flower. It would go from the "vase shape" to the "cup shape" to the "bowl shape," the newspaper reported, and finally, at the "tray shape," shed its petals. There were twenty-four petals, it was further reported.
Below the article was a picture of the bespectacled, apparently graying doctor, the stem of the opening lotus in his hand. Glancing back over the article, Shingo saw that he was sixty-nine.
Shingo looked for a time at the photograph of the lotus, then took the paper into Kikuko's room.
It was her room and Shuichi's. On the desk, which was part of her dowry, lay Shuichi's felt hat. There was stationery beside it—perhaps she thought of writing to someone. A piece of embroidery hung over the drawer.
He seemed to catch the scent of perfume.
"How are you? You shouldn't be jumping out of bed all the time." He sat down by the desk.
Opening her eyes, she gazed at him. She seemed embarrassed that he should have ordered her to stay in bed. Her cheeks were faintly flushed. Her forehead was a wan white, however, and her eyebrows stood out cleanly.
"Did you see in the paper that a lotus two thousand years old has come into bloom?"
"Yes."
"Oh, you did," he muttered. "If you had only told us, you wouldn't have had to overdo it. You shouldn't have come back the same day."
Kikuko looked up in surprise.
"It was last month, wasn't it, that we talked about a baby? I suppose you already knew."
Kikuko shook her head. "No. If I had known, I would have been too embarrassed to say anything."
Oh? Shuichi said it was squeamishness." Seeing tears in her eyes, he dropped the subject. "You won't have to go to the doctor again?"
"I'll look in on him tomorrow."
When he came back from work the next day, Yasuko was waiting impatiently.
"Kikuko's gone back to her family. They say she's in bed. There was a call from the Sagaras, it must have been at about two. Fusako took it. They said that Kikuko had come by and wasn't feeling well, and had gone to bed, and they wondered if they might let her stay and rest for two or three days."
"Oh?"
"I told Fusako to say we'd send Shuichi around tomorrow. It was Kikuko's mother, Fusako said. Do you suppose Kikuko went to Tokyo especially for that?"
"No."
"What can be the matter with her?"
Shingo had taken off his coat and, lifting his chin, was slowly untying his tie.
"She had an abortion."
"What!" Yasuko was stunned. "Without telling us. Kikuko could do that? People these days are too much for me."
"You're very unobservant, Mother," said Fusako, coming into the breakfast room with Kuniko in her arms. "I knew all about it."
"And how did you know?" The question came of its own accord.
"That I can hardly tell you. But there's cleaning up afterwards, you know."
Shingo could think of nothing more to say.
## Footnotes
* A large, harmless snake.
* A common saying in pietist Buddhism.
#
A Garden in the Capital
Father is a very interesting man, isn't he, Mother?" said Fusako, noisily loading the dinner dishes onto a tray. "He's more reserved with his daughter than with the girl who came in from outside."
"Please, Fusako."
"But it's true. If the spinach was overdone, why didn't he come out and say so? It wasn't as if I'd cooked it to a pulp. You could still see the shape of spinach. Maybe he should have it done in a hot spring."
"A hot spring?"
"They cook eggs and dumplings in hot springs, don't they? I remember you once gave me something called radium eggs, from somewhere or other, with the whites hard and the yolks soft. And didn't you say they could cook a fine egg at the Squash House in Kyoto?"
"Squash House?"
"Oh, the Gourd House. Every beggar knows that much. I'm just saying you can squash your ideas about good and bad cooking for all the difference they make to me."
Yasuko laughed.
But Fusako went on unsmiling. "If he takes it to a raduim spring and watches the time and the temperature very, very closely, he'll be as healthy as Popeye, even without Kikuko to look after him. Myself, I've had enough of all this moping." Pushing herself up from her knees, she went off with the heavy tray, "Dinner doesn't seem to taste the same without a handsome son and a beautiful daughter-in-law."
Shingo looked up. His eyes met Yasuko's. "She does talk."
"Yes. She's been holding back both the talk and the tears because of Kikuko."
"You can't keep children from crying," muttered Shingo.
His mouth was slightly open, as if he meant to say more, but Fusako, staggering off toward the kitchen, spoke first. "It's not the children. It's me. Of course children cry."
They heard her flinging dishes into the sink.
Yasuko half stood up. They heard sniffling in the kitchen.
Rolling her eyes up at Yasuko, Satoko ran off after her mother.
A most unpleasing expression, thought Shingo.
Yasuko put Kuniko on Shingo's knee. "Watch her for a minute," she said, following them to the kitchen.
The baby was soft in his arms. He pulled her close to him. He took her feet in his hand. The hollow of the ankles and the swelling of the calves were also in his hand.
"Does it tickle?" But Kuniko evidently did not think so.
It seemed to Shingo that when Fusako, still a babe in arms, had lain naked, having a change of clothes, and he had tickled her armpits, she had wrinkled her nose and waved her arms at him, but he could not really remember.
Shingo had seldom spoken of what a homely baby she was. To speak of the matter would have been to bring back the image of Yasuko's beautiful sister.
His hope that Fusako would change faces several times before she grew up had not been realized, and the hope itself had faded with the years.
His granddaughter Satoko seemed somewhat better favored than her mother, and there was hope for the baby.
Was he searching for the image of Yasuko's sister even in his grandchildren? The thought made Shingo dislike himself.
And even while disliking himself, he was lost in fantasy: would not the child Kikuko had done away with, his lost grandchild, have been Yasuko's sister, reborn, was she not a beauty refused life in this world? He was even more dissatisfied with himself.
As he loosened his grip on her feet, Kuniko climbed from his knee and started off toward the kitchen. Her arms were bent in front of her, and her legs were unsteady.
"You'll fall," said Shingo. Bui the baby had already fallen.
She fell forward and rolled to her side, and for a time did not cry.
The four of them came back into the breakfast room, Satoko clinging to Fusako's sleeve, Yasuko with Kuniko in her arms.
"Father is very absent-minded these days, Mother," said Fusako, wiping the table. "When he was changing clothes this evening, he was quite a sight. He was starting to put on an obi, and he had his kimono and juban* with the right side pulled over the left. Can you imagine it? I don't suppose he's ever done that before. He must be getting senile."
"I did it once before. I had the right side over the left, and Kikuko said that in Okinawa it didn't matter Whether you had the left side or the right side over."
"In Okinawa? I wonder if that's true."
Fusako was scowling again. "Kikuko knows how to please you. That was very clever of her. In Okinawa, was it?"
Shingo controlled his irritation. "The word juban comes from Portuguese. I don't know whether they wear the left or the right side on top in Portugal."
"Another piece of information from Kikuko?"
Yasuko sought to intercede. "Father is always putting on summer kimonos inside out."
"There is a difference between accidentally putting a kimono on inside out and standing there like a fool bringing the right side over the left."
"Let Kuniko have a try at putting on a kimono. You can't be sure which side will come out in front."
"It's early for second childhood, Father," said Fusako, unflagging. "Isn't it a little too much, Mother? So his daughter-in-law does go home for a day or two, that's no excuse for losing track of which side of his kimono goes in front. Hasn't it been six months now since his own daughter came home to Mother?"
It was true: six months had passed since that rainy New Year's Eve, There had been no word from her husband, Aihara, nor had Shingo seen Aihara.
"Six months," nodded Yasuko. "Not that there's any relation between that and Kikuko."
"No relation? I think both have some relation to Father."
"You are his children. It would be nice if he could find an answer."
Fusako looked down in silence,
"All right, Fusako, now is your chance. Come out with everything. Say what you have to say You'll feel better, Kikuko is away."
"I was wrong, and I'm not going to complain. But I should think you could eat it even if it didn't come from Kikuko's hands." Fusako was weeping again. "Isn't it the truth? You sit there grimly forcing it down. I'm not happy myself."
"Fusako, You must have all sorts of things to say. When you went to the post office the other day—I imagine it was to mail a letter to Aihara?"
A tremor seemed to pass over Fusako, but she shook her head.
"I decided it had to be Aihara, because I couldn't think of anyone else you'd have any reason to write to." Yasuko's voice was not often so sharp. "Did you send money?"
So Yasuko had been giving Fusako money.
"Where is Aihara?" Shingo looked at Fusako, waiting for an answer. "He doesn't seem to be at home. I've been sending someone around from the office once a month or so to have a look at the place. No, not that so much, really, as to give a little money to his mother. If you were there you might be the one to take care of her."
Yasuko sat open-mouthed. "You send someone from the office?"
"Don't worry. He's someone you can depend on. Someone who doesn't give away secrets or ask questions. If Aihara were there I'd go and talk your problem over, but there's no point in talking to a lame old woman."
"What is Aihara doing?"
"Peddling drugs or something of the sort, it would seem. I imagine he was being used to peddle the stuff, and he moved from drink to drugs."
Yasuko gazed at him in fright. It seemed possible that she was less frightened by the matter of Aihara than by her own husband, who had kept his secret so long.
Shingo went on. "But now it seems that the old woman isn't there either. Someone else is in the place. In other words, Fusako no longer has a house."
"And what about Fusako's things?"
"My chests and trunks have been empty for a long time, Mother."
"I see." Yasuko sighed. "You're an easy target for him and come home with everything you own tied up in one kerchief."
Shingo wondered whether Fusako might know where Aihara was, and whether she might be in communication with him.
And as he looked out at the garden, moving into dusk, he wondered who it was that had been unable to keep Aihara from falling, Fusako or Shingo or Aihara himself. Or perhaps no one at all.
2
Shingo got to the office at about ten to find a note from Tanizaki Eiko.
She wanted to talk to him about the young mistress, and would come again later.
The young mistress could only be Kikuko.
Shingo questioned Iwamura Natsuko, who had replaced Eiko as his secretary.
"What time was Tanizaki here?"
"I had just come in and was dusting the desks. I suppose it would have been a little after eight."
"Did she wait?"
"Yes, for a while."
Shingo disliked the dull, heavy way in which Natsuko said "Yes." Perhaps it had to do with her native dialect.
"Did she see Shuichi?"
"I believe she went away without seeing him."
"Oh?" Shingo was talking to himself. "If it was a little after eight, then. . . ."
Eiko had probably come on her way to work. She would probably come again at noon.
After rereading the note, tiny at the edge of a large sheet of paper, he looked out of the window.
He looked out at the clear sky of the most May-like of May days.
He had seen it from the train. All the passengers looking out had their windows open.
The birds skimming low over the shining stream that marked the limits of Tokyo shone silver themselves. It seemed more than accidental that a red-banded bus should be crossing the bridge to the north.
"In the heavens, a high wind." For no particular reason. he repeated the motto on his counterfeit Ryokan.
"Well!" The Ikegami grove came into view, and he leaned forward as if he meant to jump out. "Maybe the pines aren't in the ikegami grove at ail."
This morning the two pines that stood out above the grove seemed nearer.
Had it been that, in the rains and the spring mists, the perspective had been blurred?
He gazed on, trying to make sure.
He gazed at them every morning, and he thought he would like to go and inspect the site itself.
But though he saw the grove every morning, he had only recently discovered the two pines. He had looked a it absently over the years, knowing that it was the grove of the Ikegami Hommonji Temple.
Today, in the clear May sky, he had discovered that the pine trees did not seem to be in the Ikegami grove at all.
And so he had twice discovered the two pines that leaned toward each other as if about to embrace.
When, last night after dinner, he had told of seeking out Aihara's house and giving modest help to his old mother, the agitated Fusako had fallen silent.
He had felt Sorry for her. He thought that he had discovered something in her, but what he had discovered was by no means as clear as the discovery in the lkegami grove.
Some days earlier, looking out at this same grove, he had questioned Shuichi, and drawn from him the news of Kikuko's abortion.
The pines were no longer just pines. They were entangled with the abortion. Perhaps he would always be reminded of it when he passed them to and from work.
This morning, of course, it had been so again.
On the morning of Shuichi's admission, the pines had melted back into the grove, dim in the wind and rain. This morning, standing apart, associated in his mind with Kikuko's abortion, they somehow looked dirty. Perhaps the weather was too good.
"Even when natural weather is good, human weather is bad," he muttered to himself, somewhat inanely. Turning away from the clear sky framed in the Office window, he set about the day's work.
Shortly after noon there was a telephone call from Eiko. Busy with summer clothes, she would riot be able to come by today.
You're so good at it that you're kept busy?"
"Yes." Eiko fell silent.
"You're at the shop?"
Yes. But Kinu isn't here." The name of Shuichi's woman came out smoothly. "I waited for her to leave."
"Oh?"
"Hello? I'll stop by tomorrow morning."
"Tomorrow morning? At eight again?"
"No. I'll wait for you."
"It's all that pressing?"
"Yes. Well, it is and it isn't. To me it seems pressing. I want to talk to you as soon as I can. I'm rather worked up about it."
"Worked up? About Shuichi?"
"I'll tell you when I see you."
He did not attach much importance to her being "worked up," but he was uneasy that she should so want to talk as to come two days in a row.
The uneasiness increased. At about three he called Kikuko's family house.
The Sagawa maid answered. Music came over the telephone as he waited for Kikuko.
He had not. talked to Shuichi of Kikuko since she had gone home to her family. Shuichi seemed anxious to avoid the subject.
And Shingo had avoided going to inquire after Kikuko, because to do so would only have been to give the matter unnecessary emphasis.
Shingo thought that, being what she was, Kikuko would have spoken to her family neither of Kinu nor of the abortion. But he could not be sure.
Kikuko's voice came up from the symphony over the telephone. "Father?" There was affection in it. "I've kept you waiting."
"Hello." A surge of relief swept over him. "And how are you?"
"Oh, I'm fine again. I'm pampering myself."
"Not at all." He found it hard to go on.
"Father," said Kikuko happily. "I want to see you. May I come now?"
"Now? Is it all right?"
"Yes. The quicker I see you the easier it will be for me to go home again."
"I'll be waiting here for you." The music went on. "Hello, hello," Shingo did not want to let her hang up. "That's very good music."
"I forgot to turn it down, didn't I? It's ballet music. Les Sylphides, by Chopin. I'll steal it from them and bring it home with me."
"You're coming right away?"
"Yes, But let me think a minute. I don't really want to go to the office."
She suggested that they meet at the Shinjuku Garden.
Shingo laughed, somewhat disconcerted at this proposed rendezvous.
Kikuko seemed to think that she had bit upon a remarkably good idea. "The green will bring you to life."
"The Shinjuku Garden? I've been there exactly once. For some reason or other I went to a dog show there."
"Come and let me show you myself instead." And after her laughter, Les Sylphides played on.
3
He went in through the main gate of the Shinjuku Garden.
A notice beside the gate announced that perambulators were available for thirty yen per hour, and straw mats for twenty yen per day and up.
There was an American couple ahead of him. The husband had a little girl in his arms, and the wife was leading a German pointer. There were other people too, all young couples. Only the Americans were walking at an easy pace.
Shingo fell in after them.
To the left: of the path, what at first seemed to be a stand of deciduous pines proved to be deodars. When he had come to the dog show, a benefit given by an organization for the prevention of cruelty to animals, he had seen a remarkable stand of deodars; but he could not remember where.
To the right were signs identifying trees and shrubs as Oriental arborvitae and utsukushimatsu* and the like.
He walked on at his leisure, thinking he would be ahead of Kikuko; but he found her on a bench under a gingko, by the pond to which the path shortly led.
Turning toward him and half rising to her feet, Kikuko bowed.
"You're early. It's still fifteen minutes to half-past four." He looked at his watch.
"I was so pleased when you called that I ran right out of the house." She spoke rapidly, "I can't tell you how pleased I was."
"So you've been waiting? Shouldn't you have on something heavier?"
"I've had this sweater since I was in school." A note of shyness came into her voice. "I don't have any clothes at home anymore. I couldn't very well borrow a kimono from my sister."
Kikuko was the youngest of eight children, and all her sisters were married. It was probably to a sister-in-law that she had reference.
The dark green sweater had short sleeves. It seemed to Shingo that he was seeing her bare arms for the first time this year.
She apologized in a somewhat more formal manner for having gone home to her family.
"Can you come back to Kamakura yet?" he asked softly, not knowing what sort of reply was called for.
"Yes" She nodded simply and quickly. "I've been wanting to come back." The beautiful shoulders moved as she gazed at Shingo. His eye had not caught the exact motion, but a gentle scent came from her to surprise him.
"Did Shuichi go to see you?"
"Yes. But if you hadn't called. . ."
It would have been difficult for her to go back?
The remark unfinished, Kikuko stepped out of the shade.
The green of the giant trees, so rich as to be almost heavy, seemed to fall upon the slender neck of the retreating figure.
The lake was Japanese, after a fashion. On the little island, his foot on a stone lantern, a foreign soldier was joking with a prostitute. There were other couples on the benches around the lake.
Shingo followed Kikuko out through the trees to the right of the lake. "Enormous," he said, surprised at the vastness of the expanse before them.
"It has brought you to life, Father," she said, openly pleased with herself. "I told you it would."
Shingo stopped before a loquat beside the path. He did not immediately go out upon the broad lawn.
"A splendid loquat. It has nothing to get in its way, and it spreads out just as it wants to, all the way to the bottom."
Shingo was deeply moved by the form the tree had taken in free and natural growth.
"Beautiful. Yes—when I came to the dog show there was a row of deodars growing just as they wanted to, spreading as far as they could spread, all the way down to the bottom. I felt like growing with them. I wonder where it was."
"Over in the direction of Shinjuku."
"Yes. I came in from Shinjuku."
"You said on the telephone that you came to look at dogs?"
"There weren't so many of them, but it was a benefit given by the Society for the Prevention of Cruelty to Animals. There were more foreigners than Japanese. Diplomats and people from the Occupation, I imagine. It was summer. The Indian girls were the most beautiful, all done up in red and blue silk gauzes. There were Indian and American stalls. We didn't have many such affairs in those days."
It had been two or three years before, but Shingo could not remember exactly when.
As he spoke, he moved away from the loquat tree.
"Let's get rid of the yatsude at the foot of the cherry. Remind me when you get home."
"I will."
"We've never cut back the cherry, I like it as it is."
"It has all those tiny branches loaded with flowers. We listened to the temple bell when it was in full bloom. Remember? Last month during the festival."
"Such a small thing—it's good of you to remember."
"I'll never forget. And there was the kite."
She came close to him. They walked from the shadow of a great keyaki* out over the broad lawn.
The vast green expanse set Shingo free.
"You can stretch out. It's like getting out of Japan—I wouldn't have dreamed that there was a place like this right in the middle of Tokyo." He gazed at the distant expanse of green toward Shinjuku.
"They paid a great deal of attention to the vista. It looks even farther off than it is."
"What's a vista?" Kikuko had used the Italian word.
"A line of vision, you might say. See how all the paths and the borders are in gentle curves."
Kikuko had come on a school outing, and her teacher had told them all about the garden. The wide lawn, with trees scattered over it, was in the English fashion, it seemed.
There were few people other than young couples, lying down, sitting up, strolling casually about. There were also children, and schoolgirls in groups of five and six. Shingo was surprised, and somehow thought it inappropriate, that the park should be an Eden for assignations.
Did the scene tell one that the youth of the land had been liberated, just as the imperial garden had been?
No one paid the slightest attention to the two of them as they made their way over the lawn, weaving in and out among the couples. Shingo stayed as far from them as he could.
And what would Kikuko be thinking? He was an old man who had brought, his young daughter-in-law to the garden, but there was something about the situation that did not rest well with him.
He had not given much thought to the matter when Kikuko had suggested over the telephone that they meet in the Shinjuku Garden, but now that they had come it all seemed very odd.
Shingo was drawn to one particularly high tree out on the lawn.
As he approached, looking up at it, the dignity and the mass of the towering green came grandly down to him, to wash away his and Kikuko's gloom. She had been right to think that the garden would bring him to life.
The tree was what is called in Japan a "lily tree," Coming nearer, he saw that it was in fact three trees. The sign explained that, since the flowers resemble both the lily and the tulip, it is also known as a tulip tree. A fast grower, it came originally from North America. These specimens were about fifty years old.
"Fifty years old? They're younger than I am." Shingo looked up in surprise.
The broad-leafed branches spread out as if to enfold and hide the two of them.
Shingo sat down on a bench, but he felt restless.
Kikuko looked at him, puzzled, as he stood up again.
"Let's go have a look at the flowers over there," he said.
There was a bed of white flowers, fresh in the distance beyond the lawn, about as high as the dipping branches of the tulip tree.
"They had a victory reception here for the generals in the Russo-Japanese War. I was in my teens, still out in the country."
There were trees in grand rows on either side of the flower bed. Shingo chose a bench set among them.
Kikuko stood before him. "I'll come home tomorrow morning. Tell Mother, and see that she doesn't scold me." She sat down beside him.
"Do you have anything you want to say to me first?"
"Say to you? All sorts of things, but. . . ."
4
Shingo waited hopefully the following morning, but Kikuko had not yet come back when he left for the office.
"She said I was to see that you didn't scold her."
"Scold her?" Yasuko's face was bright and happy. "We ought to apologize."
He had said only that he had telephoned Kikuko.
"You have a remarkably strong influence on her," Yasuko saw him to the door. "But that's all right."
Eiko came shortly after he arrived at the office.
"You're prettier," he said affably. "And you brought flowers."
"I can't get away once I'm at the shop, and so I walked around killing time. The florist's was beautiful."
But the expression on her face was solemn as she approached his desk, "Get rid of her," she wrote with her finger on the desk.
"What?" He was startled. "Would you mind leaving us alone for a minute?" he said to Natsuko.
Waiting for Natsuko to go, Eiko found a vase and put three roses in it. She was wearing a slip-on dress that gave her the look of one who worked for a modiste. She had put on a little weight, he thought.
"I'm sorry about yesterday." Her manner was strangely tense. "I—coming two days in a row, and all that."
"Have a seat."
"Thank you." She sat with bowed head.
"I'm making you late for work."
"It doesn't matter." Looking up at him, she drew in her breath sharply, as if she were about to weep. "Is it all right to talk to you? I'm boiling over, and I may be a little hysterical."
"Oh?"
"It's about the young mistress." She choked over the words. "I believe she had an abortion."
Shingo did not answer.
How could she have known? Shuichi would hardly have spoken to her of it. But Eiko worked with Shuichi's woman. He braced himself for unpleasantness.
"It's all right for her to have an abortion." Eiko hesitated again.
"Who told you?"
"Shuichi got the hospital money from Kinu."
Shingo felt a tightening in his chest.
"I thought it was outrageous. Really too insulting, too unfeeling. I felt so sorry for the young mistress that I wanted to cry. He gives Kinu money, and so I suppose you can think of it as his money, but it wasn't the right thing to do. He comes from a different class than the rest of us, and he could put together that amount of money any way he pleased. Does being on a different level make it all right for him to do things like that?" She fought to keep her slender shoulders from trembling. "And then there was Kinu, letting him have the money. I couldn't understand her. I was boiling over. I wanted to talk to you even if it meant that I couldn't work with her any more. I know I'm telling you more than I ought to, of course."
"Thank you."
"You were good to me here. I only met the young mistress once, but I liked her." Tears glistened in her eyes. "Have them separate."
"Yes."
She meant Shuichi and Kinu, of course—and yet the remark could also be interpreted as referring to Shuichi and Kikuko.
Into such depths Shuichi had been pushed.
Shingo was astonished at his son's spiritual paralysis and decay, but it seemed to him that he was caught in the same filthy slough. Dark terror swept over him.
Having had her say, Eiko prepared to leave.
"Don't rush off." He sought to detain her, but without enthusiasm.
"I'll come again. Today I'd weep for you and make a fool of myself."
He felt benevolence and a sense of responsibility in her.
He had thought it remarkably indelicate of her to go to work in the same shop as Kinu; but how much worse were Shuichi and Shingo himself.
He gazed absently at the crimson roses Eiko had brought.
Shuichi had said that squeamishness had kept Kikuko from bearing a child "with things as they are now." Was she not being trampled on for her squeamishness?
Unknowing, Kikuko would now be back in Kamakura. He closed his eyes.
## Footnotes
* A singlet worn under a kimono.
* A pine, to judge from the name. Not identified in botanical dictionaries.
* ZELKOVA SERRATA, related to the elms.
#
The Scar
On Sunday morning, Shingo sawed down the yatsude at the foot of the cherry.
He knew that to be quite rid of it he would have to dig up the roots; but he told himself that he could cut the shoots as they came up.
He had sawed it down before, and the effect had been to make it spread. Once again, however, digging up the roots seemed too much trouble. Perhaps he did not have the strength.
Though they put up little resistance to the saw, there were large numbers of stalks. His forehead was bathed in sweat.
"Shall I help you?" Shuichi had come up behind him.
'No, I can manage," he answered somewhat curtly.
Shuichi pulled up short.
"Kikuko called me. She said that you were cutting down the yatsude, and I should go help."
"Oh? But there's only a little more."
Sitting down on the yatsude he had cut away, Shingo looked toward the house. Kikuko, in a bright red obi, was leaning against a glass door at the veranda.
Shuichi took up the saw on Shingo's knee. "You're cutting it all, I suppose."
"Yes." He watched the youthful motions as the remaining four or five stalks were cut down.
"Shall I cut these too?" Shuichi turned toward Shingo.
"Just a minute." Shingo got up. "I'll have a look."
There were two or three young cherry trees; or possibly they were not independent trees but branches. They seemed to come up from the roots of the parent tree.
At the thick base of the trunk, as if grafted on, there were little branches with leaves.
Shingo backed off some paces. "I think it would look better if you cut the ones coming from the ground."
"Oh?" But Shuichi was in no hurry to set about cutting them down. He did not seem to think Shingo's idea a very good one.
Kikuko too came down into the garden.
Shuichi pointed the saw at the young trees. "Father is in process of deliberating whether to cut them or not." He laughed lightly.
"Yes, do cut them." Kikuko's solution came readily, "I don't know whether they're branches or not," said Shingo to Kikuko.
"Branches don't come from the ground."
"What do you call a branch coming from the roots?" Shingo laughed with the others.
In silence, Shuichi cut the shoots.
"I want to leave all the branches and let it grow and spread as it wants to. The yatsude was in the way. Leave the little branches there at the base."
"Tiny little branches, like chopsticks or toothpicks." Kikuko looked at Shingo. "They were very sweet when they were in bloom."
"Oh? They had blossoms, did they? I didn't notice."
"Oh, yes. One little cluster, and two and three. And I believe the ones like toothpicks had single blossoms."
"Oh?"
"But I wonder if they'll really grow. By the time they're like the bottom branches of the loquat and the wild cherry in the Shinjuku Garden, I'll be an old woman."
"Oh, no. Cherries are quick growers." He looked into Kikuko's eyes.
He had told neither his wife nor Shuichi of the visit to the Shinjuku Garden.
And had Kikuko revealed the secret to her husband immediately upon her return to Kamakura? Since it was not really a secret, she had probably spoken of it as a matter of no moment at all.
"I understand you met Kikuko at the Shinjuku Garden," Shuichi might have said; but if it was hard for him to broach the subject, then possibly Shingo should speak first. Both were silent, and there was a certain strain between them. Perhaps, having heard of the visit from Kikuko, Shuichi was feigning ignorance.
But there was no sign of embarrassment on Kikuko's face.
Shingo gazed at the tiny branches at the base of the tree. He painted in his mind a picture of them, now feeble, mere sprouts in an improbable place, growing and spreading like the under-branches in the Shinjuku Garden.
They would make a splendid sight, dipping to the ground and heavy with flowers; but he could not remember having seen such a cherry tree. He could not remember having seen a great cherry tree with branches sweeping from its base.
"What shall I do with the yatsude?" asked Shuichi.
"Throw it away in a corner somewhere."
Gathering the yatsude under his arm, Shuichi dragged it off. Kikuko followed with several branches he had left behind.
"Don't bother," he said. "You still have to take care of yourself."
Kikuko nodded and stood where she had dropped the branches.
Shingo went into the house.
"What was Kikuko doing in the garden?" asked Yasuko, taking off her glasses. She was trimming an old mosquito net to use for the baby's naps. "The two of them out in the garden together on a Sunday. Very unusual—they seem to be getting along better since she went home."
"She's lonely," muttered Shingo.
"Not necessarily." Yasuko spoke with emphasis. "She has a nice laugh, and it's been a long time since I last heard her laughing so. She's a little thinner, and when I see her laughing. . . ."
Shingo did not answer.
"He comes back early from the office, and he's at home on Sunday. Storms make trees take deeper root, they say."
Shingo still did not answer.
Shuichi and Kikuko came in together.
"Father, Satoko tore off your much-prized branches." Shuichi held the little branches between his fingers. "She was having a great time dragging away yatsude, and then she ripped off your branches."
"Oh? The sort of branches a child would be likely to rip off."
Kikuko was half hidden behind Shuichi.
2
When Kikuko came back from Tokyo, she brought Shingo an electric razor of Japanese make. Yasuko received an obi binder, and Fusako dresses for the two children.
"Did she bring Shuichi anything?" Shingo asked Yasuko.
"A collapsible umbrella. And she seems to have brought an American comb with a mirror on the case, I've always been told that you don't give people combs, because that means breaking off relations or something of the sort. I imagine Kikuko doesn't know."
"I don't suppose they'd say so in America."
"She brought a comb for herself, too, A little smaller, and a different color. Fusako admired it, and got it. It probably meant a lot to Kikuko to come back with a comb like Shuichi's; and Fusako reached in and grabbed it. Just a silly little comb."
Yasuko seemed to find her daughter hard to excuse. "The children's dresses are good silk, real party dresses. It's true that Fusako herself didn't get anything, but the children's dresses were really presents for her. Kikuko must have felt guilty about Fusako when she gave away the comb. I don't see how any of us can expect presents from her."
Shingo agreed, but had causes for gloom unknown to Yasuko.
Kikuko had no doubt borrowed money from her family. Since Shuichi had gone to Kinu for the medical expenses, it did not seem that either he or Kikuko had money for presents. Under the impression that the medical expenses had been paid by Shuichi, Kikuko had probably importuned her parents.
Shingo was sorry that for some time now he been giving Kikuko nothing resembling an allowance. He had, to be sure, had good intentions; but as Shuichi and Kikuko had drifted apart and he had drawn closer to Kikuko, it had become more difficult for him to give her money as if in secret. But perhaps, in his failure to put himself in her place, he had resembled Fusako as she took possession of the comb.
And since it was because of Shuichi's philandering that she was short of money, Kikuko could hardly come crying to her father-in-law for an allowance. Yet if Shingo had shown more sympathy, she would not have had to submit to the indignity of having the money for her abortion come from her husband's mistress.
"I would have felt better if she hadn't brought anything," said Yasuko meditatively. "How much do you suppose it all came to? A great deal, I'd imagine."
"I wonder." He made a mental reckoning. "I have no idea how much an electric razor costs. I've never noticed."
"Nor have I." Yasuko emphasized this admission with a nod. "If you think of it as a lottery, you got the top prize. That's the way Kikuko would want things to be. It makes noise and moves."
"The blades don't move."
"They must. How else would they cut?"
"No. I've stared and stared, and they don't move."
"Oh?" Yasuko was smiling broadly. "The top prize, absolutely, if only from the way it makes you look like a child with a new toy. You buzz and grind away every morning, absolutely delighted, and you feel your pretty, smooth skin all through breakfast. It embarrasses Kikuko a little. Not that she's not pleased, too, of course.
"I'll let you use it." He smiled, but Yasuko shook her head emphatically.
Shingo and Shuichi had come home together on the night of Kikuko's return; and the electric razor had been the object of much breakfast-room attention.
The electric razor, it might have been said, did the honors in place of the awkward greetings that would otherwise have been exchanged between Kikuko, absent without leave, and the family of Shuichi, by whom she had been driven to an abortion.
Fusako too smiled happily, getting the children into their new dresses and praising the good taste of the embroidery at the necklines. Having mastered the instruction booklet, Shingo gave the razor a trial.
The inquiring eyes of the whole family were upon him.
He moved his chin over the razor, the instruction booklet in his other hand. "It says here that it does well too with the downy hair at the nape of a lady's neck." His eyes met Kikuko's.
The hairline at her forehead was very beautiful. It seemed to him that he had not really seen it before. It drew a delicately graceful curve.
The division between the fine skin and the even, rich hair was sharp and clean.
For some reason the cheeks of the otherwise wan face were slightly flushed. Her eyes were shining happily.
"Father has a nice new toy," said Yasuko.
"It's not a toy," said Shingo. "It's a finely tooled product of modern civilization. A precision instrument. It has a number, and it's initialed by technicians for the trial and the adjustment and the final inspection."
In fine spirits, Shingo tried shaving with and against the grain.
"You won't cut yourself or give yourself a rash, I'm told," said Kikuko, "and you don't need soap and water."
"An old man is always getting his razor caught in wrinkles. It will do nicely for you too." He offered the razor to Yasuko.
But Yasuko pulled back as if in fright. "If you think I have whiskers, you're quite mistaken," she said.
He looked at the blades, and put on his glasses and looked again. "They don't move, I wonder how it cuts. The motor revolves, but the blades don't move."
"Let me see." Shuichi reached for the razor, but passed it on immediately to Yasuko.
"It's true. The blades don't seem to move. Maybe it's like a vacuum cleaner. You know how a vacuum cleaner sucks in dirt."
"Can you tell where the whiskers go?" asked Shingo. Kikuko looked down and smiled.
"Suppose we give a vacuum cleaner in return for the electric razor. Or a washing machine—that would do too. It would be a help to Kikuko."
Shingo agreed with his old wife.
"We don't have a single finely tooled product of modern civilization in this house. Every year you say you'll buy a refrigerator, and it's time for one again this year. And toasters. There are toasters that turn off automatically and send the bread flying when it's done."
"An old wife's views on domestic electrification?"
"You are very fond of Kikuko, and a lot of good it does her."
Shingo unplugged the electric razor. There were two brushes in the case. One was like a small toothbrush, the other like a small bottle brush. He gave them a try. Cleaning the hole behind the blades with the bottle brush, he looked down and saw that very short white hairs were falling on his knee. He could see only white hairs.
He slapped them from his knee.
3
Shingo at once bought a vacuum cleaner.
It struck him as amusing that, before breakfast. electric razor and Kikuko's vacuum cleaner should buzzing along together.
Perhaps he was hearing the sound of renewal in the house.
Satoko trailed after Kikuko, fascinated with the cleaner.
It may have been because of the electric razor that Shingo had a dream of chin-whiskers.
He was not a participant but a spectator. In a dream, however, the division between the two is not clear. It took place in America, where Shingo had never been. Shingo suspected that he had dreamed of America because the combs Kikuko had brought back were American.
In his dream, there were states in which the English were most numerous, and states in which the Spanish prevailed. Accordingly, each state had its own characteristic whiskers. He could not clearly remember, after he awoke, how the color and shape of the beards had differed, but in his dream he had clearly recognized differences in color, which is to say in racial origins, from state to state. In one state, the name of which he could not remember, there appeared a man who had gathered in his one person the special characteristics of all the states and origins. It was not that all the various whiskers were mixed in together on his chin. It was rather that the French variety would be set off from an Indian beard, each in its proper place. Varied tufts of whiskers, each for a different state and racial origin, hung in sprays from his chin.
The American government designated the beard a national monument; and so he could not of his own free Will cut or dress it.
That was the whole of the dream. Looking at the wondrous assortment of colors in the beard, Shingo half felt that it was his own. Somehow he felt the man's pride and confusion as his own.
The dream had had scarcely any plot. He had just seen a bearded man.
The beard was of course a long one. Perhaps it was because he shaved his own face clean every morning that he had dreamed of that unfettered beard. He liked the idea of its becoming a national monument.
A naïve, uncomplicated dream, and he looked forward to telling it in the morning. He woke to the sound of rain, however, and, shortly going back to sleep, woke again, this time from an unpleasant dream.
His hands were against drooping, vaguely pointed breasts. They remained soft, refusing to rise. The woman was refusing to respond. All very stupid.
Even though he was touching her breasts, he did not know who the woman was. It was not so much that he did not know as that he did not seek to find out. She had no face and no body; just two breasts floating in space. Asking for the first time who she was, he saw that she had become the younger sister of a friend of Shuichi's; but the recognition brought neither excitement nor feelings of guilt. The impression that it was the sister was a fleeting one. She remained a dim figure. Her breasts were those of a woman who had not had children, but Shingo did not think she was a virgin. He was startled to find traces of her purity on his finger. He felt disconcerted, but not especially guilty.
"We can say that she was an athlete," he muttered.
Startled at the remark, he awoke.
"All very stupid"—he recognized Mori Ogai's* dying words. It seemed he had once read them in a newspaper.
But it had probably been an evasion on his part, waking from an unpleasant dream, to think first of Mori Ogai's dying words and then to tie them to the dream.
The Shingo of the dream had felt neither delight nor affection, nor even wantonness. All very stupid indeed. And a dreary way to wake up.
He had not sought to assault the girl. Perhaps he had been about to. Had he assaulted her, trembling with love or terror, the dream would have had more life after he waked.
He thought of wanton dreams he had had in recent years. They had generally been of women he would have to call coarse and vulgar. So it had been tonight. Was it that even in a dream he feared adultery?
He remembered the friend's sister as having full breasts. Before Shuichi married there had been some not-very-serious talk of arranging a marriage with her, and the two had kept company.
A bolt flashed across his mind.
Had not the girl in the dream been an incarnation of Kikuko, a substitute for her? Had not moral considerations after all had their way even in his dream, had he not borrowed the figure of the girl as a substitute for Kikuko? And, to coat over the unpleasantness, to obscure the guilt, had he not made her a less attractive girl than she was?
And might it not be that, if his desires were given free rein, if he could remake his life as he wished, he would want to love the virgin Kikuko, before she was married to Shuichi?
Suppressed and twisted, the subconscious wish had taken an unlovable form in his dream. Even in the dream, had he sought to hide it, to deceive himself?
That he had transferred it to the girl who had been talked of for Shuichi, that he had given her an elusive, uncertain form—was it not because he feared in the extreme having the woman be Kikuko?
And the fact that, upon awakening, he had difficulty remembering it, that his companion in the dream, and the plot as well, was blurred, and the fact that there had been no pleasure in the hand against the breast—might these be because, at the moment of awakening, a certain cunning went adroitly to work at erasing the dream?
"A dream. And the national monument was a dream too. Don't put faith in what dreams decide for you." He wiped his face with the palm of his hand.
The dream had had a chilling effect, but when he woke Shingo was bathed in a disagreeable sweat.
The rain which after the dream of whiskers had been only enough to tell him that it was rain was now driven by a wind, and beating against the house. The dampness seemed to come up through the floor mats. It had the sound of a rain, however, that would have its brief rampage and go.
He remembered an ink wash by Watanabe Kazan* that he had seen at a friend's house a few days before.
It had been of a single crow at the tip of a leafless tree, and had born the legend: "A stubborn crow in the dawn: the rains of June. Kazan."
Shingo thought he understood Kazan's feelings, and the intent of the picture. The crow, high in a naked tree, bearing up under strong wind and rain, was awaiting the dawn. The storm was shown in faint ink. He did not remember the tree very well, but he thought it had been broken off, leaving only a thick trunk. He remembered the crow vividly. Perhaps from sleep, perhaps from the wind—most likely both—its feathers were somewhat ruffled. It had a heavy bill. The upper bill, blackly stained where the ink had run, was thicker and heavier than the lower. The eyes were sleepy, as if it had not yet fully awakened. Yet they were strong, and somehow angry. It was a large figure for the size of the picture.
Shingo knew of Kazan only that he had been impoverished and that he had committed suicide, but he could see that this "Crow in the Stormy Dawn" gave expression to Kazan's feelings at a certain point in his life.
No doubt the friend had put the painting up to match the season.
Shingo ventured an opinion: "A very strong-minded bird. Not at all likeable."
"Oh? I used to look at it during the war. Damned crow, I used to think. Damned crow it is. But it has a quietness about it. If Kazan had to kill himself for no better reasons than he had, then you and I probably ought to kill ourselves time after time. It's a question of the age you live in."
"We waited for the dawn, too."
The crow would be hanging in the friend's parlor this rainy night, thought Shingo.
He wondered where his own kite and crow would be.
4
Unable to sleep after waking from the second dream, Shingo lay waiting for the dawn. He did not wait with the stubborn resistance of the Kazan crow, however.
Whether the woman in the dream had been Kikuko or the friend's sister, he thought it altogether too dreary that no flicker of lust had come over him.
The dream had been uglier than any waking adultery. The ugliness of old age, might it be?
Women had left his life during the war, and had been absent since. He was not very old, but that was how it was with him. What had been killed by the war had not come to life again. It seemed too that his way of thinking was as the war had left it, pushed into a narrow kind of common sense.
He wanted to inquire among his friends whether many old men his age felt as he did. But perhaps he would but be laughed at and called weak and feckless.
What was wrong with loving Kikuko in a dream? What was there to fear, to be ashamed of, in a dream? And indeed what would be wrong with secretly loving her in his waking hours? He tried this new way of thinking.
But a haiku by Buson came into his mind: "I try to forget this senile love; a chilly autumn shower." The gloom only grew denser.
Shuichi's marital relations had ripened since he had taken a mistress. Since Kikuko had had her abortion, they had softened, warmed. On the night of that wild storm, Kikuko had been much more coquettish toward Shuichi than usual: on the night he had come home drunk, she had forgiven him more gently than usual.
Was she sad, or silly?
And was she aware of these facts herself? Perhaps, not alive to them, she was but giving herself in all innocence to the wonders of creation, riding the wave of life.
She had protested by not having the baby and by going back to her family, and so given expression to an unbearable loneliness; and then, returning a few days later, she had drawn closer to Shuichi, as if apologizing for some misdeed, or treating a wound.
Shingo could, if he chose, think that this too was "all very stupid." But probably it was to the good.
He was even able to think that he might as well wait for the Kinu affair to settle itself.
Shuichi was his son; but were they so ideal a couple, were they so fated for each other, that Kikuko must put up with such treatment? Once he began doubting, the doubts were endless.
Not wanting to arouse Yasuko, he could not turn on the light to look at his watch; but dawn seemed to be breaking, and it would soon be time for the temple bell.
He remembered the bell at the Shinjuku Garden.
It had signaled closing time, but he had said to Kikuko: "It sounds like a church bell."
He had felt as if he were making his way through some wooded park on his way to a Western church, and as if the cluster of people at the gate were also going to church.
He got up without having had enough sleep.
He left early with Shuichi. He did not want to have to face Kikuko.
Suddenly he asked: "Did you kill anyone during the war?"
"I wonder. If anyone got in the way of a bullet from my machine gun, he probably died. But you might say I wasn't shooting the machine gun."
Shuichi looked away in displeasure.
The rain stopped during the day and began again in the evening. Tokyo was wrapped in a heavy fog.
When he left the restaurant after a business dinner, he found himself in the predicament of having to see the geisha home in the last automobile.
Two elderly geisha and Shingo were side by side, and three young ones sat on their knees.
"Please." Shingo put his hand to the front of the girl's obi.
"If you'll excuse me, then," Reassured, she settled in his lap. She was four or five years younger than Kikuko.
He meant to write her name down in his memorandum book once he was on the train. It was only a passing thought, however, which he seemed likely to forget.
## Footnotes
* Writer, 1862-7922.
* Scholar and painter, 1793-1841.
#
In the Rain
Kikuko was the first to read the newspaper that morning.
Rain had apparently blown into the mailbox. She dried the paper over the gas as she was cooking breakfast.
Sometimes, when he was awake early, Shingo went for the newspaper and took it back to bed with him; but now going for it seemed to have become Kikuko's work.
Usually he saw the newspaper only after Shuichi had left for the office.
"Father, Father," Kikuko called softly through the door.
"What is it?"
"If you're awake, would you come out for a minute?'
"Is something wrong?"
Alarmed by the tone of her voice, he got up immediately.
She was standing on the veranda with the newspaper in her hand.
"What's happened?"
"Mr. Aihara is in the paper."
"Has Aihara been taken in by the police?"
"No." Retreating a step, she handed him the paper.
"It's still wet."
He reached for it, reluctantly. It sagged limply from his hand. Kikuko held it up for him.
"I can't see. What happened to Aihara?"
"It was suicide with a woman."
"Is he dead?"
"He will probably be saved, it says."
"Wait a minute." He started off, leaving the newspaper with Kikuko. "I suppose Fusako is here?"
"Yes."
It was scarcely likely that Fusako, who had gone to bed here with her two children late the night before, had committed suicide with Aihara, or that she would be in the paper.
Looking at the wind-driven rain outside the toilet window, Shingo sought to calm himself. The drops fell in rapid succession from the long leaves of pampas grass at the foot of the mountain.
"It's a real downpour. Not the usual thing for June."
In the breakfast room, he took up the newspaper, but before he could begin reading his glasses slipped down over his nose. Snorting in annoyance, he took them off and rubbed impatiently at the bridge of his nose. It was damply unpleasant.
His glasses slipped down again as he was reading the short article.
The incident had occurred at Rendaiji Spa on the Izu Peninsula. The woman was dead. She was twenty-five or -six and had the look of a maid or waitress, but had not been identified. The man seemed to be a drug addict. The probability was that he would be saved. Because of his addiction and because there was no suicide note, there was a suspicion that he had himself been playing a game and had lured the woman on.
Shingo clutched at his glasses, which had slipped to the tip of his nose, as if to give them a cuffing. He did not know whether he was angry that Aihara had tried suicide or angry that his glasses slipped.
Rubbing at his face, he went off to the washstand.
The newspaper said that Aihara had given the inn a Yokohama address. Fusako was not mentioned.
The article was thus unrelated to Shingo's family.
Perhaps the registration was false, and Aihara was in fact a vagrant. And perhaps Fusako was no longer his wife.
He washed his face before he brushed his teeth.
Was it only sentimentality that left him troubled and confused at the thought that Fusako might still be Ai-hara's wife?
"Is this what they call letting time take care of things?" he muttered to himself.
Had time finally brought the solution he had so put off seeking?
But might it not be that Shingo had had no recourse other than to hope for desperate action on Aihara's part?
He did not know whether Fusako had pushed Aihara to destruction, or whether Aihara had led her into misery. There were no doubt those whose nature it was to push their partners into misery and destruction, and those whose it was to be led into misery and destruction.
"Kikuko," he said, going back into the breakfast room and sipping at hot tea. "You knew, didn't you, that Aihara mailed us a divorce notice five or six days ago?"
"Yes. You were furious."
"That I was. And Fusako said there was a limit to the insults a person could take. But maybe he was getting ready for suicide. He wasn't pretending, he meant to kill himself. I imagine he just took the woman along for company."
Kikuko wrinkled her beautiful eyebrows and did not answer. She had on a striped silk kimono.
"Would you get Shuichi up, please?"
The retreating figure seemed taller than usual, perhaps because of the broad vertical stripes.
"So Aihara did it?" Shuichi took up the newspaper. "Has Fusako sent in the notice?"
"Not yet."
"Not yet?" Shuichi looked up. "Why not? Send it in this morning. We won't want to be sending in a divorce notice from a corpse."
"But what about the children? Aihara said nothing about them, and they're too young to decide for themselves which family they want to be in."
The divorce notice, with Fusako's seal on it, had been going to and from the office in Shingo's briefcase.
He had occasionally sent money to Aihara's mother. He had thought to have the same messenger take the divorce notice to the ward office, but he had delayed from day to day.
"They're here, and you can't do a damned thing about it. I imagine the police will be coming."
"What for?"
"Looking for someone to hand Aihara over to."
"I don't think so. I think that must have been exactly why he sent the notice."
Banging the door open, Fusako came in, still in her night kimono.
She ripped the newspaper to pieces and flung it away after only glancing at it. Though she had put more than enough strength into the ripping, it did not rebound when she threw it. Falling to the floor on her knees, she pushed violently at the fragments.
"Close the door, please, Fusako," said Shingo.
He could see the sleeping children through the open door.
Her hands trembling, Fusako tore the newspaper into smaller pieces.
Shuichi and Kikuko were silent.
"Fusako. Do you feel like going for Aihara?"
"No!" Raising herself on an elbow, she turned and glared at Shingo, her eyes rolled upwards. "How do you feel about your daughter, Father? You coward. Seeing your own daughter into this, and not upset, not the least little bit. Swallow your pride and go for him yourself. You be the one to do it. Who was it that married me to a man like that?"
Kikuko went off to the kitchen.
Shingo had only said what floated into his mind; but he continued to think that if Fusako went for Aihara in this extremity, the two might come together again, they might make a new start. Human beings were capable of such things.
2
There was nothing more in the newspaper to tell them whether Aihara was dead or alive.
Since the ward office had accepted the divorce notice, it would seem that he was not registered as dead.
Or if he was dead, had his identity not been established? It hardly seemed likely. There was his lame mother. Even if she had not seen the newspaper, someone among their acquaintances and relatives would surely have noticed. Shingo concluded that Aihara had been saved.
But, having taken in Aihara's two children, was it enough for him just to conclude? For Shuichi the answer was clear, but Shingo himself still had doubts.
The two children were now Shingo's responsibility. Shuichi apparently did not consider the fact that they might one day be his.
Quite aside from the worry of rearing and educating the children, it would seem that what chance Fusako and the children had had for happiness had been cut in half; and was that fact too a part of Shingo's responsibility?
As he sent off the divorce notice, Shingo thought of the woman with Aihara.
A woman had died, that much was certain. What were the fife and death of a woman?
"Come back and haunt us," he muttered to himself. Startled, he added: "And what a stupid life you had."
If Aihara and Fusako had gone on living together as an ordinary husband and wife, the woman need not have died; and so it was not impossible to call Shingo himself a murderer by remote control. Should there not then come into his mind pious thoughts about the dead woman?
But there was no way of conjuring up her image. Suddenly he saw Kikuko's baby. He could not of course see the face of a baby disposed of so early in pregnancy. Still he pictured the varieties of beauty in children.
The baby had not been born; and was he not then twice a distant murderer?
Unpleasantly wet days went on, when even his glasses seemed damp and clammy. He felt a heaviness in his right chest.
The sun blazed forth during lulls in the June rains.
"The house that had sunflowers last summer," said Shingo as he stepped into his trousers. "This year it has some white flower, I don't know the name of it. Something like a Western chrysanthemum. Four or five houses in a row have the same flower. They must have arranged it. Last year they all had sunflowers."
Kikuko stood in front of him, holding his coat.
"I imagine it's because the sunflowers were blown over in the storm."
"Probably so. Haven't you grown a little, Kikuko?"
"I've been getting taller since I came here, but lately I've begun to shoot up. Shuichi was very surprised."
"When?"
Flushing scarlet, Kikuko stepped behind him to help him into his coat.
"I thought you were taller, and it wasn't just the kimono. It's a good idea to keep growing for years after you're married."
"I've been too small. A late bloomer."
"Not at all. I think it's splendid." Shingo did feel something splendidly fresh in this new blossoming. Had Kikuko so grown that Shuichi noticed the difference when he held her in his arms?
It also seemed to Shingo, as he left the house, that the lost life of the child was growing in Kikuko herself.
Squatting at the edge of the street, Satoko was watching some little girls of the neighborhood play house.
Shingo too stopped to watch. He looked admiringly at the neatly clipped mounds of grass on the abalone shells and yatsude leaves they were using as dishes.
Dahlia and marguerite petals, also cut into fine bits, had been added for color.
They had spread a straw mat, over which marguerites cast a heavy shadow.
"Marguerites. That's what they are," said Shingo, remembering.
Marguerites had been planted before the several houses that had last year had sunflowers.
It seemed that Satoko was too young to be admitted to the company.
"Grandfather." She followed after him.
He led her by the hand to the corner of the main street. There was summer in the figure running back toward home.
Natsuko, her white arms bare, was polishing the office windows.
"Did you see the newspaper this morning?" he asked lightly.
"Yes." The word was, as usual, dull and heavy.
"The newspaper. Which paper was it, I wonder?"
"Which newspaper?"
"I don't remember which newspaper it was, but some sociologists at Harvard University and Boston University sent out a questionnaire to a thousand secretaries, asking what it was that gave them the greatest pleasure. Every last one of them said that it was being praised when there was someone around to hear it. Every last one of them. Are girls the same in the East and in the West? How is it with you?"
"But wouldn't it be embarrassing?"
"Embarrassing things and pleasant things often go together. Isn't it that way when a man makes a pass at you?"
Natsuko looked down and did not answer. Not the sort of girl one often comes upon these days, thought Shingo.
"I imagine that's how it was with Tanizaki. I should have praised her more often when there were people around."
"Miss Tanizaki was here," said Natsuko awkwardly. "At about eight-thirty."
"And?"
"She said she'd come again at noon."
Shingo sensed the approach of unhappiness.
He did not go out for lunch.
Eiko stood in the door. She was breathing heavily and seemed on the edge of tears.
"No flowers today?" Shingo hid his uneasiness.
She approached him solemnly, as if reprimanding him for his own want of solemnity.
"You want me to get rid of her again?" But Natsuko had gone to lunch and he was alone.
He was offered the startling news that Shuichi's woman was pregnant.
"I told her she must not have the baby." Eiko's thin lips were trembling. "I got hold of her yesterday on the way home from work, and told her so."
"I see."
"But isn't that right? It's too awful."
Shingo had no answer. He was frowning.
Eiko was thinking of Kikuko.
Kikuko, Shuichi's wife, and Kinu, his mistress, had become pregnant the one after the other. The sequence was of course not impossible, but it had not occurred to Shingo that his own son could be the agent. And Kikuko had had an abortion.
3
"Would you see whether Shuichi is here, please. If he is, ask him to come in for a minute."
"Yes, sir." Eiko took out a small mirror. "I'd be ashamed to have him see me this way," she added, somewhat hesitantly. "And then Kinu will find out that I've been bringing stories."
"I see."
"Not that I'd mind having to leave the shop."
"Don't do that."
Shingo inquired by telephone. He did not, at this moment, want to have to face Shuichi in front of other employees. Shuichi was out.
Inviting Eiko to a foreign restaurant nearby, Shingo left the office.
Eiko, who was small, walked close to him and looked up into his face. "Do you remember?" she asked nonchalantly. "You took me dancing just once when I was in your office."
"Yes. You had a white ribbon in your hair."
"No." She shook her head. "I had the white ribbon the day after the typhoon. I remember because I was very upset. It was the day you first asked about Kinu."
"Was that it?"
It had been that day, he remembered. Eiko had told him that Kinu's husky voice was erotic.
"Last September. I really asked too much of you." Shingo had come without a hat. The sun was hot on his bare head.
"I was no help at all."
"Because we didn't give you anything to work with. A family to be ashamed of."
"I admire you. Even more since I left the office." Her voice was strained and unnatural. After a moment she went on: "When I told her she must not have the child, she hit back at me as if I were a child myself that needed a spanking. I knew nothing about it, she said, and I couldn't understand. I'd do better to mind my own business. And finally she said she had it there inside her."
"Oh?"
"Who had asked me to give her stupid advice? If it had to do with being separated from Shuichi, there was nothing she could do but be separated when he left her, But the child was hers and no one else's. No one could do anything about it. If I could I should ask a baby inside my own self whether it was wrong to have it. I'm young and she was making fun of me. She said I wasn't to make fun of her. She may intend to go on and have it. I remembered afterwards that she and her husband had no children. He was killed in the war."
Walking beside her, Shingo nodded.
"Maybe she just said it because I irritated her. Maybe she doesn't mean to have it."
"How far along is it?"
"Four months. I didn't notice, but some of the others in the shop did. They say the owner heard and told her not to have it. She's very talented, and I imagine it would be a loss to the shop." She raised a hand to her face. "I didn't know what to do to. I thought if I told you you might speak to Shuichi."
"Yes."
"I think you should see her as soon as possible."
Shingo had been thinking the same thing. "The lady who came to the office with you that day—are they still living together?"
"Mrs. Ikeda."
"Yes. Which is older?"
"I believe Kinu is two or three years younger."
Eiko saw him back as far as his building. Her smile seemed on the edge of tears.
"Thank you."
"Thank you. Are you going back to the shop?"
"Yes, Kinu generally leaves early these days. The shop is open till six-thirty."
"You don't mean I'm to go there!"
It had been as if Eiko were urging him to see Kinu even today; but the thought was more than he could tolerate. And it would not be easy to face Kikuko when he got back to Kamakura.
Evidently, from her squeamishness, her irritation at being pregnant while Shuichi had another woman, Kikuko had refused to bear her child. Doubtless she had not even dreamed that the other woman was pregnant.
Kikuko had come back from a few days with her family after Shingo had heard of the abortion, and had since seemed closer to Shuichi. Home early every day, he was considerate as he had not been before. What did it all mean?
The more favorable interpretation was that Shuichi, deeply troubled by Kinu and her resolution to have the baby, was pulling away from her, apologizing to Kikuko.
But a scent of ugly decay and want of principle filled Shingo's nostrils.
Wherever it came from, the embryonic life itself seemed evil.
"And if it's born it will be my grandchild," Shingo muttered.
#
The Cluster of Mosquitoes
Shingo walked up the main Hongo street on the side that skirted the Tokyo University campus.
He had left the cab on the side lined by shops, and would of course turn from that side into Kinu's lane. He had purposely crossed the car tracks to the other side.
He was most reluctant to visit the house of his so mistress. He would be meeting her for the first time, and she was already pregnant. Would he be able to ask her not to have the child?
"So there is to be another murder," he said to himself. "Can't it be accomplished without adding to the crimes of an old man? But all solutions are cruel, I suppose."
The solution in this case should have been up to the son. It was not the father's place to interfere. Shingo was going off to see Kinu without telling Shuichi; and he was thus no doubt providing evidence that he had lost faith in his son.
When, he asked himself, startled, had this gap come between them? Might it be that this visit to Kinu was less out of a wish to find a solution for Shuichi than out of pity and anger at what had been done to Kikuko?
The strong evening sunlight touched only the tips of the branches. The sidewalk was in shade. On the university lawns, men students in shirt sleeves were talking to girl students. It was a scene that told of a break in the early summer rains.
Shingo touched a hand to his cheek. The effects of the sake had left him.
Knowing when Kinu would be finishing work, he had invited a friend from another company to a Western restaurant. He had not seen the friend in rather a long time and had forgotten what a drinker he was. They had had a short drink downstairs before going up to dinner, and after dinner they had again sat for a time in the bar.
"You're not going already?" the friend had asked in surprise. Thinking that, at this first meeting in such a long time, they would want to have a talk, said the friend, he had called for reservations in the Tsukiji geisha district.
Shingo had replied he would come after paying an unavoidable visit of perhaps an hour or so. The friend had written the Tsukiji address and telephone number on a calling card. Shingo had had no intention of going.
He walked along the wall of the University, looking across the street for the mouth of the lane. He was relying on vague memories, but they did not prove wrong.
Inside the dark doorway, which faced north, there was a shabby chest for footwear. On it was a potted occidental plant of some description from which hung a woman's umbrella.
A woman in an apron came from the kitchen.
Her face went tense as she started to take off the apron. She had on a navy-blue skirt, and her feet were bare.
"Mrs. Ikeda, I believe. You once honored us at the office with a visit."
"Yes. It was rude of me, but Eiko dragged me along."
Her apron wadded in one hand, she looked at him inquiringly. There were freckles even around her eyes, all the more conspicuous because she did not seem to be wearing powder. She had a delicate, well-shaped nose, and one saw a certain elegance in the narrow eyes and the fair skin.
No doubt the new blouse had been made by Kinu.
"I was hoping to see Miss Kinu."
He spoke as if requesting a favor.
"She should be home soon. Would you like to wait?"
A smell of grilling fish came from the kitchen.
Shingo thought it might be better to come later, when Kinu had had her dinner. On the urging of the Ikeda woman, however, he went inside.
Fashion magazines were piled in the alcove of the medium-sized parlor, among them considerable numbers of what seemed to be foreign magazines. Beside them were two French dolls, their frills quite out of harmony with the shabby old walls. From the sewing machine hung a length of silk. The bright, flowery pattern made the dirty floor matting look all the dirtier.
To the left of the machine was a little desk on which were numerous primary-school textbooks and a photograph of a small boy.
Between the machine and the desk was a dressing table, and in front of the closet to the rear a full-length mirror, the most conspicuous piece of furniture in the room. Perhaps Kinu used it to try on clothes she had made, perhaps she gave fittings to customers for whom she did extra work. There was a large ironing board beside it.
The Ikeda woman brought orange juice from the kitchen.
"It's my son," she said immediately. Shingo was looking at the picture.
"Is he in school?"
"I don't have him here. I left him with my husband's family. The books—i don't have regular work like Kinu, and so I do tutoring. There are six or seven houses I go to."
"I see. I thought there were too many for one child."
"They're all ages and grades. The schools these days are a great deal different from before the war, and I'm afraid I don't really do very well. But when I'm teaching I feel as if he were with me."
Shingo nodded. There was nothing he could say to the war widow.
The other, Kinu, was working.
"How did you find the place? Did Shuichi tell you?"
"No. I came once before, but I couldn't make myself come inside. It must have been last autumn."
"Really?" She looked up at him, and looked down again. "Shuichi hasn't been coming around lately," she said abruptly, after a time.
Shingo thought it might be better to tell her why he had come. "I understand that Kinu is going to have a child," he said.
The woman shrugged her shoulders very slightly and turned to the photograph of her son.
"Does she mean to go ahead and have it?"
She continued to look at the photograph. "I think you'd better ask her."
"I agree. But won't it be a great misfortune for both mother and child?"
"I think you can call Kinu unfortunate whether she has the child or not."
"But I'd imagine that you yourself might have been advising her to break with Shuichi."
"That's what I think she should do. But Kinu is much stronger than I, and it hasn't amounted to advice. We're two very different people, but somehow we get along well. She's been a great help to me since we started living together. We met at the war widows club, you know. Both of us have left our husbands' families an not gone back to our own—we're free agents, you say. We want our minds to be free too, and so we've put our husbands' pictures away. I do have the boy's out, course. Kinu reads all sorts of American then she can get the gist of French too with a dictionary, she says. After all, it's about sewing and there aren't many words. She wants to have a shop of her own some day. We both say that when the chance comes we'll remarry. And so I don't understand why she had to be all tangled up with Shuichi."
The front door opened. She got up somewhat hastily and went out to the hall.
"Mr. Ogata's father is here," Shingo heard her say.
"Do I have to see him?" replied a husky voice.
2
Kinu went to the kitchen and seemed to be having a glass of water.
"You come in too," she said, looking back toward Mrs. Ikeda as she came into the room.
She had on a very bright suit. Perhaps because she was so large, it was not apparent to Shingo that she was pregnant. He found it hard to believe that the hoarse voice could have come from the small, puckered mouth.
The mirrors were in the parlor, and it seemed that she had retouched her face from a compact.
Shingo's first impression was not unfavorable. The face, round yet hollow, did not suggest the strength of will which the Ikeda woman had described. There was a gentle roundness about the hands too.
"My name is Ogata."
Kinu did not answer.
"You've kept us waiting," said Mrs. Ikeda, seating herself before the mirror stand. Still Kinu said nothing.
Perhaps because surprise and hostility did not show themselves well on the essentially cheerful face, she seemed about to weep. Shingo remembered that in this house Shuichi had gotten drunk and had made her weep by insisting that the Ikeda woman sing for him.
Kinu had hurried home through muggy streets. Her face was flushed, and her rich breasts rose and fell.
"It must seem strange that I should be calling on you," said Shingo, unable to approach his subject with complete directness, "but I imagine that you will have guessed what brings me."
Kinu still did not answer.
"Shuichi, of course."
"If it's about Shuichi, then I have nothing to say." Suddenly she pounced. "Are you asking that I apologize?
"No. I think the apologies should come from me."
"We've separated, and I will be no more trouble to you." She looked at Mrs. Ikeda. "Shouldn't that take care of things?"
Shingo had difficulty replying, but at length he found words: "There is still the question of the child, you know."
"I don't know what you're talking about." Kinu blanched, but all her strength seemed to go into the words. As her voice fell it was even huskier.
"You must forgive me for asking, but I believe you are to have a child?"
"Do I have to answer that sort of question? If a woman wants to have a child, are outsiders to step in and prevent it? Do you think a man would understand that sort of thing?" She spoke rapidly and there were tears in her voice.
"Outsiders, you say—but I am Shuichi's father. I imagine your child will have a father too?"
"It will not. A war widow has decided to have a bastard, that's all. I have nothing to ask of you except that you leave me alone to have it. Just ignore it, as an act of charity, if you will. The child is inside me, and it is mine."
"That is true. And when you get married you will have other children. I see no need at this point in having unnatural children."
"And what is unnatural about it?"
"I didn't mean that."
"There is no guarantee that I will marry again, or that I will have children. Are you willing to play God and give us an oracle? I had no children last time."
"Relations between the child and its father are the main point. The child will suffer and so will you."
"A great many children were left behind by men who died in the war, and a great many mothers were left to suffer. Think of it as if he had gone off into the islands and left behind a half-breed. Women bring up children that men have forgotten long ago."
"The matter has to do with Shuichi's child."
"I can't see that it makes any difference as long as I don't mean to bother you. I won't come crying to you, I swear I won't. And Shuichi and I have separated."
"The child will live for a long time. The bond with its father will last after you think you've cut it."
"The child is not Shuichi's."
"You must know that Shuichi's wife did not have her child."
"She can have as many as she wants, and if she has none the regrets are hers. Do you think a pampered wife can understand how I feel?"
"And you do not know how Kikuko feels."
In spite of himself, Shingo spoke the name.
"Did Shuichi send you around?" She set upon him like an inquisitor. "He told me I was not to have the child, and beat me and stamped on me and kicked me and dragged me downstairs to try to get me to a doctor, It was a fine show, and I think we have acquitted ourselves of our duty to his wife."
Shingo smiled bitterly.
"It really was quite a display, wasn't it?" she said to the Ikeda woman, who nodded.
"Kinu is already collecting scraps that she thinks might do as diapers."
"I went to the doctor afterwards because I thought the kicking might have injured the child. I told Shuichi it was not his. It most definitely is not yours, I said. And with that we separated. He hasn't been here since."
"Another man's, then?"
"Take it so, and that will be that."
Kinu looked up. She had been weeping for some time, and there were new tears on her face.
Even now, at the end of his resources, Shingo thought the woman beautiful. On close examination her features were not perfect; but the first impression was of beauty all the same.
Despite the apparent softness, she was not a woman to let Shingo come near.
3
His head bowed, Shingo left Kinu's house.
Kinu had accepted the check he had offered her.
"If you're leaving Shuichi it might be better to take it." Mrs. Ikeda had been very direct, and Kinu had nodded.
"So you're buying me off. That's the sort of thing I've come to. Shall I give you a receipt?"
As he got into a cab, Shingo wondered whether it might not be better to effect a reconciliation between Shuichi and the woman. An abortion might still be possible. Or should the separation be considered final?
Kinu had been antagonized by Shuichi and now by Shingo's visit. Her longing for a child seemed unshakable.
It would be dangerous to push Shuichi toward the woman again; and yet as matters stood the child would be born.
Kinu had said that it belonged to another man. Not even Shuichi could be sure. If Kinu made the assertion out of pride and Shuichi was prepared to believe her, then the world might be described as in order. There need be no further complications. Yet the child would be a fact. Shingo would die, and he would have a grandson on whom he had never laid eyes.
"And so?" he muttered.
In some haste, they had submitted the divorce notice after Aihara's attempt at suicide. In effect, Shingo had taken in his daughter and two grandchildren. If Shuichi and his woman were to part, another child would remain, out in the world somewhere. Were they not but a clouding-over of the moment, these two solutions that Were no solutions?
He had contributed to no one's happiness.
On a different level, he did not like to think of the ineptness with which he had faced Kinu.
He had intended to take a train home from Tokyo Central Station, but, coming upon the friend's card, he took a cab instead to the Tsukiji geisha district.
He hoped to ask advice of the friend. The latter was getting drunk with two geisha, however, and there was no opportunity.
Shingo thought of a young geisha who had once sat on his lap. It had been after a party, and they had been in an automobile. He called her again tonight. When she arrived the friend made a number of not very interesting remarks: that Shingo was not to be underestimated, that he had a good eye, and the like. It was rather an achievement for Shingo, who could not remember the girl's face, to have remembered her name. She proved to be winsome and elegant.
Shingo went into a small room with her, but did nothing out of the ordinary.
Soon he found her face pressed gently against his chest. He thought she was being coquettish, but she seemed in fact to have gone to sleep.
He looked inquiringly down at her. She was too near for him to see her face.
He smiled. There was warm comfort in having a young girl peacefully asleep in one's arms. She was still in her teens, four or five years younger than Kikuko.
Perhaps there was in his feelings a touch of pity at the plight of the prostitute. In any case, he felt himself bathed in a soft repose, the repose of sleeping with a young girl.
Happiness, he thought, might be just such a matter of the fleeting instant.
He considered vaguely the fact that in sex too there were riches and poverty, good luck and bad. Slipping away, he caught the last train home.
Yasuko and Kikuko were waiting up in the breakfast room. It was past one.
"Shuichi?" asked Shingo, avoiding Kikuko's eyes.
"He's already in bed."
"Oh? And Fusako?"
"She's in bed too." Kikuko was putting away his suit for him. "The good weather managed to hold out, but it seems to have clouded over again."
"Oh? I hadn't noticed."
As she stood up, she lost her hold on the suit. She straightened the trousers again.
Shingo noticed that her hair was shorter. She seemed to have been to the beauty parlor.
With Yasuko breathing heavily beside him, he slept fitfully. Soon he had a dream.
He was a young army officer in uniform. He had a sword at his hip, and three pistols. The sword seemed to be the family heirloom that Shuichi had taken off to the war.
Shingo was walking a mountain path. He had a woodcutter with him.
"The roads are dangerous at night. I seldom go out," said the woodcutter. "You would do well to walk on the right."
Shingo felt uneasy as he moved to the right. He turned on a flashlight. Diamonds glittered around the edge, making it brighter than most flashlights. A dark form loomed up in the darkness—two or three cedars, one against another. But he looked more carefully and saw instead a great cluster of mosquitoes in the shape of a tree trunk. What to do, he wondered. Cut his way through. He took out his sword and hacked away at the mosquitoes.
Looking back, he saw that the woodcutter was in headlong flight. Here and there flames were shooting from Shingo's uniform. The strange thing was that there were two Shingos. Another Shingo was watching the Shingo along whose uniform the flames were creeping. The flames licked the sleeves and the shoulder seam and the hem of the tunic, and disappeared again. It was less that they blazed up than that they came and went like wisps from a charcoal fire, giving forth tiny noises.
Shingo was finally at home. It seemed to be his childhood home, in Shinshu. Yasuko's beautiful sister was there. Though exhausted, Shingo felt no itching from the mosquitoes.
The woodcutter who had fled in such haste also made his way to Shingo's old home. He fell unconscious as he stepped through the door.
From his body they took a great bucketful of mosquitoes.
Shingo did not know by what process this was accomplished, but he could see the piling up of mosquitoes in the bucket as he awakened.
"A mosquito in the net?" He listened carefully, but his head was heavy.
It was raining.
#
The Snake's Egg
As autumn came on and the full Weariness of summer overtook him, Shingo would sometimes go to sleep on his way home from work.
During rush hours there were trains on the Yokosuka Line every fifteen minutes. The second-class car was not crowded.
In his mind, as he dozed off lightly, was a row of acacia trees in bloom. Not long before, he had passed under the trees that now came to him, and he had marveled, as he looked up, that even in Tokyo rows of acacias came into bloom. It had been on the street leading from the foot of Kudan Hilt toward the Palace moat. It had been a damp, drizzly day in mid-August. A single acacia in the row had scattered its flowers on the sidewalk. Why should that be, he had asked, looking back from the cab. The picture was still in his mind. The flowers had been delicate ones, pale yellow tinged with green. Even had there not been the single tree shedding its flowers, the fact of the row of flowering trees would no doubt have left its impression. He had been on his way from a hospital, where he. had visited a friend dying of liver cancer.
Although they had been college classmates, the man was not one whom Shingo saw regularly. He was in an advanced state of emaciation and had with him only a nurse.
Shingo did not know whether or not his wife was still living.
"Do you ever see Miyamoto?" asked the friend. "Even if you don't have a chance to see him, would you mind telephoning and asking about it?"
"About what?"
"You remember. What we talked about at the class reunion. At New Year's."
Shingo remembered. It had been about potassium cyanide. The friend apparently knew that he had cancer.
At a gathering of men in their sixties, talk of senile disabilities and mortal ailments tended to loom Iarge in the conversation. Knowing that Miyamoto's factory made use of potassium cyanide, someone had said that, should he perhaps fall victim to an inoperable cancer, he would hope to be given a dose of the poison. To prolong the hideous ailment would only bring meaningless suffering. And, when a person knew he was doomed to die, he would at least wish to choose his own time.
Shingo had trouble finding an answer. "But we were in our cups, after all," he said.
"I won't use it. I won't use it. I just want to have the freedom of choice we were talking about. I think I'll be able to stand the pain if only I know I have a way of being rid of it. You understand, don't you? It's all I have—call it my last liberty, my only way of resisting. But I promise you that I won't use it."
A certain fire came into the man's eyes as he spoke. The nurse, who was knitting a white woolen sweater, said nothing.
Unable to make the request, of Miyamoto, Shingo had dropped the matter; but he did not like to think that a man who would soon die might still be depending on him.
At a certain remove from the hospital, the acacia trees, Shingo found, somehow brought relief. And now, as he dozed off on the train, the same row of trees appeared before him. The sick man had not left his mind.
He went to sleep, and when he opened his eyes the train had stopped.
It was not in a station.
The roar as a Tokyo-bound train passed had been more startling with Shingo's train stopped. Probably it had awakened him.
Shingo's train would move forward a little and stop, move forward a little and stop.
A group of children were running down a narrow road toward the train.
Several passengers were leaning out the windows and looking ahead.
Outside the left window was the concrete wall of a factory, a dirty, stagnant ditch between it and the train. The stench flooded in through the window.
To the right was the road along which the children were running. A dog stood motionless by the road, its nose in the green grass.
At the point where the road met the tracks there were two or three little huts, the cracks nailed over with old boards. From a window that was no more than a square hole a girl who seemed to be feeble-minded was beckoning to the train. Her motions were weak and languid.
"The train that left just before us seems to have had an accident in Tsurumi Station," said the conductor. "It is stopped there. We must apologize for keeping you waiting."
The foreigner opposite Shingo shook the Japanese boy sleeping beside him and asked in English what the conductor had said.
Holding the large arm of the foreigner in his hands, the boy had been sleeping with his head on the other's shoulder. In the same position after opening his eyes, he looked up coquettishly. His eyes were somewhat inflamed, and ringed with dark circles. His hair was dyed red, but had grown out black at the roots, to make it a dirty brown. Only the tips were that strange reddish color. Shingo suspected that the boy was a male prostitute who specialized in foreigners.
The boy turned the hand on the foreigner's knee palm up, and pressed it gently with his own, for all the world like a satisfied woman.
The foreigner's arms, below the short sleeves, made one think of a shaggy red bear. Though the boy was not particularly small, he looked like a child beside the giant foreigner. The latter's arms were heavy, his neck thick. Perhaps because he found it too much trouble to turn his head, he appeared quite unaware of the boy clinging to him. He had a fierce countenance, and his florid robustness made the muddy quality of the boy's weary face stand out more.
The ages of foreigners are not easy to guess. The large bald head, the wrinkles at the throat, and the blotches on the bare arms, however, made Shingo suspect that the man's age was not too far from his own. That such a man should come to a foreign country and appropriate a boy for himself—Shingo suddenly felt as if he were faced with a monster. The boy had on a maroon shirt, open at the throat to reveal a bony chest.
He would soon be dead, thought Shingo, averting his eyes.
The foul ditch was lined with green weeds. Still the train did not move.
2
Shingo found mosquito nets heavy and oppressive. He was no longer using one.
Yasuko complained of the deprivation every night and Would make a great ceremony of swatting mosquitoes.
"Kikuko and Shuichi still have one."
"Suppose you go sleep with them, then," said Shingo, gazing up at the ceiling now liberated from the net.
"I couldn't very well do that. But I do think I'll move in with Fusako tomorrow night."
"Do. Sleep with one of your grandchildren in your arms."
"Why do you suppose, with the baby there, Satoko has to go on clinging to her mother? Don't you think there's something abnormal about her? She gets the strangest look in her eyes."
Shingo did not answer.
"I wonder if not having a father does that to a child."
"It might help if you were to make yourself more approachable."
"And you might do the same thing. I prefer the baby myself."
"Not a word from Aihara to let us know whether he's dead or alive."
"You sent in the divorce notice. And so it makes no difference."
"And that is that?"
"I know what you mean. But even if he were alive we would have no way of knowing where he might be. We 11 have to resign ourselves to it—the marriage failed. But is that the way it should be? You produce two children and then separate? It doesn't give you a great deal of confidence in marriage."
"If a marriage has to break up, the echoes might be a little pleasanter. Fusako hasn't been all that good herself. He was a failure in life, and I don't imagine she gave him much sympathy. He must have suffered."
"There are things a woman can't do when a man is desperate. He won't let her come near. If Fusako and the children had just let themselves be thrown away, then I suppose there would have been nothing left for them but suicide. A man can always find another woman to commit suicide with him. And Shuichi," Yasuko went on after a pause. "He's all right now, but who can tell when he'll be up to something again. It wasn't good for Kikuko."
"You mean the baby?"
Shingo's word referred to two different matters: the fact that Kikuko had refused to have her child, and the fact that Kinu was determined to have hers. Yasuko did not know of the latter.
Kinu had said that the child was not Shuichi's and that she would take no interference from him. Shingo could not be sure of the truth, but he felt all the same that the woman was lying.
"Maybe I should go sleep with Shuichi and Kikuko after all. You can't tell what sort of discussions they might be having."
"And what do you mean by that?"
Yasuko, who had been lying on her back, turned toward him. She seemed about to take his hand, but he did not extend it to her.
She touched the edge of his pillow gently. Then, as if whispering a secret: "It's just possible that she's pregnant again."
"What!"
"I think it's a little too early, but Fusako has suspicions."
Nothing remained in Yasuko's manner from the days When she had announced her own pregnancy.
"Fusako said so?"
"It's a little early," said Yasuko again. "But they say another often follows along after that sort of thing."
"Did Kikuko or Shuichi speak to Fusako?"
"No. Fusako's own investigations."
"Investigations" was a strange word. It seemed that Fusako, who had left her own husband, was particularly inquisitive in matters having to do with her brother's wife.
"You should say something to her yourself," Yasuko went on. "Persuade her to have it this time."
Shingo felt a tightening at the throat. The news that Kikuko might be pregnant again made the fact of Kinu's pregnancy weigh on him the more oppressively.
It was not so very unusual, perhaps, that two women should simultaneously be pregnant by the same man. But if the man was one's son, then it brought with it a strange fear. It had a hellish aspect, as of retribution, or a curse.
One might look upon these various events as evidence of the healthiest physiological processes; but such magnanimity was at the moment rather beyond Shingo.
This would be Kikuko's second pregnancy. Kinu bad been pregnant at the time of the abortion. Before the latter had had her child, the former was pregnant again. Kikuko did not know of Kinu's condition. Kinu would already be attracting attention, and feeling the motions of the child within her.
"If she knows we know, then she won't be able to do quite as she pleases this time."
"I suppose not," said Shingo weakly. "You ought to have a talk with her."
Shingo could not sleep.
He found sinister thoughts coming to him. He asked himself irritably if violence of some description might not prevent Kinu from having her child.
She had said that the child was not Shuichi's; if he were to investigate her activities might he not come upon something to ease his mind?
There was a loud humming of insects in the garden outside. It was past two. The humming was not the clear and distinct sound of bell crickets or pine crickets. It was blurry and ill-defined, rather. It made Shingo think of sleep in dark, dank earth.
He had been much given to dreams lately, and toward dawn he had another long dream.
He did not know by what road he had come. When he awakened he could still see the two white eggs in the dream. He was on a sandy moor, there was sand as far as he could see. Two eggs lay side by side, one of them large, an ostrich's egg, and the other small, a snake's. The shell of the latter was cracked and an engaging little snake was waving its head back and forth. To Shingo it did seem engaging.
There could be no doubt that he had been thinking about Kikuko and Kinu. He did not know which child was the ostrich's, which the snake's.
It occurred to him to wonder whether snakes were oviparous or viviparous.
3
The next day was Sunday. Feeling quite drained of energy, Shingo stayed in bed until nine.
Now in the morning, both the ostrich egg and the little snake's head seemed vaguely sinister.
He brushed his teeth gloomily and went into the breckfast room.
Kikuko was tying up the accumulated newspapers, no doubt preparing to sell them to a junk dealer.
It was among her duties, for Yasuko convenience, to arrange the morning and evening newspapers in order.
She went to get tea for him.
"Did you see the news about the lotuses?" She put two newspapers on the table before him. "Two articles. I kent them out for you."
"It does seem to me that I read something of the sort."
He took the papers up all the same.
Lotus seeds some two thousand years old had been dug from a Yayoi tumulus. The "lotus doctor, a botanist who specialized in lotuses, had succeeded in making them sprout. News that they had bloomed had been in the papers earlier, and Shingo had taken it to Kikuko's room. She had been resting, having recently had her abortion.
Items about lotuses had appeared twice since. One described how the lotus doctor had divided the roots and transferred a part of them to Sanshiro's Lake, on the grounds of Tokyo University, from which he had graduated. The other had to do with America. A scientist at Tohoku University had found lotus seeds, apparently fossilized, in a marl stratum in Manchuria and sent them to America. The rock-like outer shell had been removed at the National Botanical Gardens and the seeds wrapped in permeated cotton wadding and put under glass. They had sent out delicate shoots the year before.
This year, set out in a lake, they had produced two buds, which had opened into pink flowers. The national park service announced that the seeds were from a thousand to fifty thousand years old.
"I thought so when I read it the first time," laughed Shingo. "A thousand to fifty thousand years old—a broadish sort of calculation." He came upon a Japanese scholar's opinion; that, to judge from the nature of the marl stratum, the seeds would be some tens of thousand of years old. Carbon radiation tests run on the shells in America, however, had shown them to be a thousand years old.
The two articles were reports from Washington correspondents.
"Are you finished?" asked Kikuko, picking up the newspapers. No doubt she meant to ask whether she had permission to sell them when next the junk dealer came by.
Shingo nodded. "A thousand years or fifty thousand, a lotus seed lives a long time. Almost an eternity, when you compare it with a human life." He looked at Kikuko. "It would be good to lie in the ground a thousand years or two without dying."
"Lie in the ground!" Kikuko half muttered the words.
"Not in a grave. And not dying. Just resting. If it were possible just to rest in the ground—you would wake up after fifty thousand years and find all your own problems settled and the problems of the world, and you would be in paradise."
"Kikuko, will you see to Father's breakfast, please?" called Fusako from the kitchen, where she seemed to be feeding the children.
Kikuko came back with the breakfast.
"You're all by yourself. The rest of us have eaten."
"Oh? What about Shuichi?"
"He's gone out to the fishing pond."
"And Yasuko?"
"Out in the garden."
"I think I'll do without eggs this morning," he said, handing back the saucer that contained eggs. He disliked the memory it brought of the snake's egg.
Fusako came in with a dried and roasted flounder. She put it down in silence and went back to the children.
Looking Kikuko in the eye as he took the bowl of rice she handed him, Shingo said in a low voice: "Are you going to have a baby?"
"No." She answered readily, and seemed only afterwards to be taken by surprise. "No. Nothing of the sort." She shook her head.
"So it wasn't true."
"No."
She looked at him curiously, and flushed.
"I hope you'll treat it better next time. I argued with Shuichi over the last one. I asked if he could guarantee that you would have another, and he said he could. As if it were all very simple. I told him he ought to be a little more God-fearing. I asked him whether anyone could guarantee that he would be alive the next day. The baby would be yours and Shuichi's, of course, but it would be our grandchild too. A child you would have would be too good to lose."
"I'm sorry," said Kikuko, looking down.
He was sure that she was telling the truth.
And why then had Fusako thought her pregnant? Fusako's investigations had evidently outdone themselves. She could scarcely be aware of a situation of which Kikuko herself was ignorant.
Shingo looked around, afraid that Fusako might have overheard the conversation. She seemed to be out in front with her children, however.
"Has Shuichi been to the pond before?"
"No. I think he musy have herad abour it from a friend."
To Shingo the unusual event seemed evidence that Shuichi had in fact left Kinu. He had on occasion used his Sundays to visit her.
"Would you like to go have a look at it yourself?"
"Yes."
Shingo stepped into the garden. Yasuko was looking up at the cherry tree.
"What's the trouble?"
"Nothing. But it's lost most of its leaves. I wonder if something might be eating it. The summer crickets are still singing, and here it has lost most of its leaves."
Even as they talked, yellowish leaves came down, one after another. In the still air, they fell straight to the ground.
"I hear Shuichi's gone fishing. I'm going to take Kikuko for a look."
"Fishing?" Yasuko looked around.
"I asked her about it, and she said it wasn't true. Fusako's investigations have misled her."
"You asked her about it?" There was something a little slow-witted about Yasuko. "What a shame."
"Why does Fusako have to be so energetic with those investigations of hers?"
"Why?"
"I'm the one who's asking."
Back in the house, Kikuko had put on a white sweater and was waiting for him. She had touched her cheeks with rouge, and seemed unusually bright and lively.
4
One day, without warning, there were red flowers outside the train window, equinox lilies all along the railway filling, so near that they seemed to quiver as the train passed.
Shingo gazed too at the lilies on the cherry-lined Totsuka embankment. Just coming into bloom, they were a fresh, clear red.
It was the sort of morning when flowers made one feel the quiet of the autumn meadows.
The pampas grass was beginning to send out plumes.
Taking off his shoe, Shingo raised his right foot to his knee and rubbed at the instep.
"Is there something the matter with it?" asked Shuichi.
"It seems so heavy. Sometimes climbing the stairs in the station my feet seem so heavy. This hasn't been a good year. The life is going out of me."
"Kikuko has been worried. She says you seem tired.
"I'd like to rest in the ground for fifty thousand years—that's the sort of thing I've said to her."
Shuichi looked at him curiously.
"There was something in the paper about old lotuses. Remember? Some ancient lotus seeds that sent out shoots and finally bloomed."
"Oh?" Shuichi lit a cigarette. "You asked her whether she was going to have a baby. She was very upset."
"Well, is she?"
"It's too soon, I think."
"And what about Kinu's? That's more important."
Though cornered, Shuichi took the offensive. "I understand you went to see her. To give her consolation money. There was no need for that."
"When did you hear about it?"
"Oh, I heard indirectly. We've separated, you know."
"Is the child yours?"
"Kinu says it isn't."
"The matter has to do with your own conscience." Shingo's voice was trembling. "What about that?"
"I don't think it's the sort of thing your conscience tells you much about."
"What do you mean by that?"
"Suppose I am suffering. Will that do anything to shake her? There is something demented about the woman and that determination of hers to have the baby."
"She's suffering more than you are. So is Kikuko."
"Now that we've separated, I can see that she's been going her own way all along."
"And that's enough for you? You really don't want to know whether or not it's your child? Or does your conscience tell you?"
Shuichi did not answer. His large eyes, almost too good-looking for a man, were blinking.
On Shingo's desk was a black-bordered postcard. The cancer patient had died somewhat more swiftly than the natural course of the illness would have led one to expect.
Had someone brought him poison? Perhaps Shingo had not been the only one of whom the request had been made. Or perhaps the man had found another way to commit suicide.
There was also a letter from Tanizaki Eiko. She had moved to another shop. Kinu had left the earlier shop shortly afterwards, the letter continued, and was in seclusion in Numazu. She meant to open a small business of her own, she had told Eiko. Tokyo would present too many complications.
Although Eiko had not touched upon the matter, it seemed likely that Kinu had retired to Numazu to have the baby.
Was it as Shuichi had said, that she went her own way quite without regard for others, for Shuichi or for Shingo himself?
He sat for a time looking absently into the clear sunlight.
What would the Ikeda woman, now left alone, be doing?
Shingo thought he would like to see either her or Eiko and make inquiries about Kinu.
In the afternoon he went to pay his condolences to the cancer victim's family. He learned for the first time that the wife had died seven years before. The man had apparently lived with his oldest son, and there were five children in the house. It did not seem to Shingo that either the son or the grandchildren resembled the dead man.
Shingo suspected suicide, but could not of course make inquiries. Giant chrysanthemums were conspicuous among the flowers by the coffin.
Going over the mail with his secretary, he had an unexpected telephone call from Kikuko. He feared that something untoward had happened.
"Where are you? In Tokyo?"
"Yes. Visiting my family." There was bright laughter in her voice. "Mother said she had something to talk over with me, and here I am, and it turns out to be nothing at all. She was just lonely and wanted to see my face."
"Oh?" Softness flooded into his chest, and the pleasingly girlish voice over the telephone was not the whole explanation.
"Will you be going home soon?" asked Kikuko.
"Yes. And is everyone well there?"
"Very well. I thought I'd like to go back with you."
"Take your time, now that you're here. I'll tell Shuichi."
"I'm ready to go."
"Suppose you come to the office, then."
"That will be all right? I thought I might wait at the station."
"No, come here. Shall I connect you with Shuichi? The three of us might have dinner together."
"The operator tells me he isn't at his desk."
"Oh?"
"I can start out right away."
Shingo felt warm to the eyelids, and the city beyond the window seemed lighter and clearer.
#
Fish in Autumn
It was an October morning. Shingo. tying his necktie, felt his hands go wrong.
"Wait a minute." He paused, and a troubled expression came over his face. "How does it go?"
He untied it and tried again, but was no more successful the second time.
Pulling the two ends up to his face, he gazed at them inquiringly.
"What seems to be the trouble?"
Behind him and a little to one side, Kikuko was holding his coat. She came around in front of him.
"I can't tie my tie. Very strange."
Slowly and awkwardly, he wound an end around a finger and tried to pull it through the loop, but the result was a strange lump. The word "strange" was most appropriate for describing the performance, but fear and despair were written on Shingo's face.
It was an expression that seemed to startle Kikuko. "Father!" she cried.
"What shall I do?"
Shingo stood as if without strength for trying to remember.
Unable to watch in silence, Kikuko came up to him, the coat over her arm.
"How do you do it?"
In some consternation, she took up the tie. Her hands were dim to Shingo's old eyes.
"That's what I've forgotten."
"But you tie it yourself every day."
"So I do."
Why should he suddenly this morning have forgotten a process he had repeated every morning through the forty years of his office career? His hands should have moved automatically. He should have been able to tie his tie without even thinking.
It seemed to Shingo that he faced a collapse, a loss of self.
"I've been watching you every morning," said Kikuko solemnly as she twisted the tie and then straightened it out to begin again.
Quite giving himself up to her, he was like a small, spoiled child that is feeling somehow neglected.
The scent of her hair came to him.
"I can't do it." Kikuko flushed.
"Haven't you ever tied Shuichi's?"
"No."
"Just untied it when he's come home drunk?"
She drew back a little and, her shoulders taut, gazed at the tie.
"Mother might know," she said, at length releasing her breath. "Mother," she called, "would you come here, please? Father says he can't tie his tie."
"And why in the world should that be?" Yasuko's face suggested that she had never before been witness to such nonsense. "Why can't he tie it for himself?"
"He says he's forgotten how."
"Something went wrong, and I forgot everything. Very strange."
"Very strange indeed."
Kikuko moved aside and Yasuko took her place.
"I don't seem to remember it all that well myself." She gave his chin a gentle shove upward as she took the tie in her hands. Shingo closed his eyes.
Yasuko did somehow seem to be producing a knot.
Perhaps because of the pressure at the base of his skull, he felt a little giddy, and a golden mist of snow flowed past his closed eyelids. A mist of snow from an avalanche, gold in the evening light. He thought he could hear the roar.
Startled, he opened his eyes. Might he be having a hemorrhage?
Kikuko was holding her breath, and her eyes were on Yasuko's hands.
It was an avalanche he had seen in the mountain home of his boyhood.
"Will this do?"
Yasuko was putting the last touches on the knot.
"Yes."
His fingers brushed against hers as he reached to feel it.
He remembered that when he had left college and first discarded his choke-collared student's uniform for an ordinary business suit, it had been Yasuko's beautiful sister who had tied his tie for him.
Shingo turned to the mirror on the wardrobe, avoiding the eyes of Kikuko and Yasuko.
"This should do nicely. Well, old age has finally caught up with me. It's not a very comfortable feeling when you find all of a sudden that you can't tie your own tie."
To judge from the facility with which she had tied it for him, Yasuko would appear to have performed the function in the early days of their marriage, but he could not remember when it might have been.
Or perhaps, when she had gone to help after the death of her sister, she had tied her handsome brother-in-law's tie.
Slipping into sandals, a worried Kikuko saw him to the gate.
"What are your plans for this evening?"
"Nothing scheduled. I'll be home early."
"Make it very early."
Gazing at Mount Fuji in the autumn blue as the train passed Ofuna, Shingo again felt his tie. He found that left and right were reversed. Facing him, Yasuko had made the left end the longer.
He untied it and retied it with no effort.
That he should earlier have forgotten the process seemed scarcely credible.
2
It was not uncommon now for Shingo and Shuichi to take the same train home.
Normally there were trains on the Yokosuka Line every half-hour, but during rush hours the number was increased to one every fifteen minutes. Sometimes rush-hour trains were emptier than normal ones.
At Tokyo Station a young girl occupied one of the seats opposite them.
"Would you save this for me, please?" she said to Shuichi, putting a red suede handbag on the seat.
"Both seats?"
She murmured an answer that was not entirely clear. As she turned and went out, however, there was no suggestion of embarrassment on her somewhat heavily powdered face. The narrow shoulders of her coat had a most winsome upthrust, and the coat flowed down over a gently elegant figure.
Shingo was puzzled. How had Shuichi guessed that the girl wanted both seats saved? He seemed to have an instinct for such things; but how had he known that the girl would be waiting for someone?
Now that his son had taken the lead, however, Shingo too thought it most evident that the girl had gone to look for her companion.
And why, since she had been sitting by the window, opposite Shingo, was it Shuichi to whom she had spoken? Probably because, as she had stood up, she had found herself facing him; and then again, perhaps Shuichi was for a woman the more approachable of the two.
Shingo looked at his son's profile.
Shuichi was reading the evening paper.
The girl got back onto the train. Clutching the frame of the open door, she looked up and down the platform. Apparently the person with whom she had an appointment had not come. Her light-colored coat, as she returned to her seat, flowed rhythmically from shoulder to hem. It was held together by a large button at the throat. The pockets were well down and forward. She swayed from side to side, one hand in a pocket, as she came down the aisle to her seat. The cut, though somewhat strange, was most becoming.
Sitting down opposite Shuichi this time, she looked repeatedly at the door. It would appear that she had chosen the aisle seat because it offered the better view.
Her handbag still lay on the seat opposite Shingo. It was a sort of flattened cylinder, and had a large clasp.
The diamond earrings were no doubt imitation, but they had a good luster. The wide nose stood out on the firm, regular face, and the mouth was small and well shaped. The thick eyebrows, with a tendency to sweep upwards, had been clipped short. The line of the wide eyes was equally graceful, but disappeared before it reached the corners. The jaw was firm and strong. These various features added up to a face that was in its way beautiful.
There was a certain weariness in the eyes, and Shingo had trouble guessing her age.
The doorway was suddenly crowded. Shingo's eyes and the girl's were on it. Five or six men, apparently on their way home from an excursion, came aboard with large maple branches in their arms.
The dark red of the leaves suggested cold mountain country.
Presently he learned, from the boisterous talk, that the men had been deep in the mountains of Echigo.
"The maples in Shinshu will be their best," he said to Shuichi.
He was thinking less, however, of the wild maples in the mountains of his old home than of the large potted maple, its leaves crimson, among the memorial tablets when Yasuko's sister had died.
Shuichi, of course, had not been born.
He gazed at the red leaves, speaking so vividly of the season.
He came to himself. The father of the girl was seated before him.
So she had been waiting for her father! The thought somehow brought relief to Shingo.
The father had the same wide nose, so similar indeed that the effect was almost comical. The hairlines were identical. The father wore dark-rimmed glasses.
Like strangers, father and daughter neither spoke to nor looked at each other. The father was asleep before they had left the outskirts of Tokyo. The daughter also closed her eyes; and even the eyelashes seemed identical.
Shuichi did not resemble Shingo as closely.
Although waiting for the two to exchange even a remark, Shingo felt somehow envious of this complete indifference.
Theirs was no doubt a peaceful family.
He was therefore startled when, in Yokohama, the girl got off by herself. They had in fact not been father and daughter but complete strangers!
He felt that he had been deceived.
The man opened his eyes slightly as they stopped in Yokohama, and went untidily back to sleep.
Now that the girl had gone the middle-aged man before him seemed untidy to Shingo.
3
Shingo nudged Shuichi with his elbow. "So they weren't father and daughter."
Shuichi did not give as much evidence of interest as Shingo had hoped for.
"You saw them, didn't you?"
Shuichi nodded perfunctorily.
"Very strange."
Shuichi did not seem to think the matter strange at all.
"They did look alike."
"Yes, I suppose they did."
The man was asleep, and the train would have drowned out Shingo's voice; but still it did not seem right to be loudly assessing the man right before one's eyes.
Shingo looked away, feeling guilty even at staring; and as he did so a sadness came over him.
It was at first sadness for the man, and then it came to be directed at Shingo himself.
The train was on the long run between Hodogaya and Totsuka. The autumn sky was darkening.
The man was younger than Shingo, but in his late fifties even so. And the girl—would she perhaps be the age of Kikuko? There had been in her nothing corresponding to the cleanness of Kikuko's eyes.
But how could it be, Shingo wondered, that she was not the man's child?
The more he thought about the problem the more his wonder grew.
There were in the world people so resembling each other that one could only take them for parent and child. There could hardly, however, be large numbers of such people. Probably in all the world there was only the one man to go with the girl, only the one girl to go with the man Only the one for either of them; and indeed perhaps in all the world there was only one such couple. They lived as strangers, with no suggestion of a bond between them. Perhaps they were even ignorant of each other's existence.
And quite by chance they were aboard the same train. They had come together for the first time, and probably would never meet again. Thirty minutes, in the length of a human life. They had parted without exchanging words. Sitting side by side, they had not looked at each other, and neither could have noticed the resemblance. And they had separated, participants in a miracle of which they had been unaware.
And the only one struck by the strangeness of it all was an outsider.
He wondered whether, accidental witness to it all, he too had partaken of the miracle.
What had it meant, creating a man and woman who looked like father and daughter, and putting them side by side for a half hour in their whole lives, and showing them to Shingo?
There she had sat, knee to knee with a man who could only be her father; and only because the person she had been waiting for had not come.
Was such the way, Shingo could only mutter to himself, with human life?
The man got up in some confusion as the train pulled into Totsuka. Taking his hat from the luggage rack, he dropped it at Shingo's feet. Shingo picked it up for him. "Thank you."
Without bothering to dust it, he put it on.
"Very odd." Shingo at length felt free to speak. "They were strangers."
"They looked alike, but they weren't gotten up alike."
"Gotten up?"
"The woman paid attention to herself, and the man was a shambles."
"But that's the way it is—girls done up in the best. fathers in rags."
"Their clothes were on two completely different levels."
Shingo had to nod his assent. "The girl got off in Yokohama. And the minute she left it seemed to me too that the man went to pieces."
"He was in pieces from the beginning."
"But it happened in such a hurry. It struck home, somehow. He was a good deal younger than I am."
"Well, there's no doubt about it." Shuichi threw the matter off with a joke. "An old man looks better when he's out with a young girl. How is it with you, Father?"
"You youngsters are envious."
"Nothing of the sort. There's something uncomfortable about a handsome man out with a pretty girl, and you feel sorry for an ugly man when the girl is beautiful. Let's leave the beauties to old people."
But the strangeness of the pair was still with Shingo.
"Maybe they really are father and daughter. Maybe she's a girl he fathered away from home somewhere and left behind. They've never introduced themselves to each other, and don't know they are father and child."
Shuichi looked away.
Shingo was a bit startled at his own remark.
Having made what seemed like an innuendo, however, he had to go ahead: "Twenty years from now the same thing may happen to you."
"That was what you were trying to say, was it? Well, I'm not that sort of sentimental fatalist myself. The bullets used to go whistling by my ears, and not a one of them touched me. I may have left behind a child or two in the islands or in China. It's nothing at all, meeting your own bastard and not recognizing it, when you've had bullets whistling by your ear. No threat to your life. And then there's no guarantee that Kinu will have a girl, and if she says it isn't mine that's enough for me."
"Wartime and peacetime are not the same thing."
"But maybe another war is on its way. And maybe the other one is still haunting people like me. Still somewhere inside us." Shuichi spoke with asperity. "There was something a little strange about her, and you were attracted to her, and so you go on with these imaginings of yours. Men always get caught when a woman is just a little different."
"And that's all right, is it? Because a woman is a little different, you get her pregnant and leave her to bring up the child?"
"I don't want it. It's the woman herself."
Shingo fell silent.
"The woman that got off in Yokohama—she's a free agent. Perfectly free."
"Free?"
"She's not married, and she'd come if you called. She may put on airs, but she doesn't have a decent living, and she's tired of the insecurity."
The words upset Shingo deeply. "So that's how far you've fallen," he said.
"Kikuko's free too." There was challenge in Shuichi's tone. "She's not a soldier and she's not a prisoner."
"What do you mean saying that about your own wife? Have you said so to her?"
"Suppose you say it to her yourself."
"You're telling me I should send her away?" Shingo fought to control his voice.
"Not at all." Shuichi too was carefully controlling his voice. "We were saying that the girl who got off in Yokohama was free. Don't you suppose you thought they were father and daughter because she was about Kikuko's age?"
Shingo was taken by surprise. "It was just that if they weren't father and daughter they looked enough alike to make it a miracle."
"It wasn't anything to be all that impressed with."
"It was to me." But now, having had it pointed out that Kikuko had been on his mind, he felt a tightening in the throat.
The men with the maple branches got off in Ofuna.
"Why don't we go to Shinshu to see the maples?" said Shingo, watching the branches move off down the platform. "With Yasuko and Kikuko too."
"I don't have much interest in maple leaves myself."
"I'd like to see the old mountains again. Yasuko says she has dreams that her house is going to pieces."
"It is in bad shape."
"We ought to repair it while there's time."
"The frame is strong, and it's not going to pieces exactly. But if you were to start repairing it—what would be the point?"
"We may want a place to retire. And then you may have to get out of the city again some day yourself."
"I'll stay behind this time and watch the house. Kikuko can go have a look at the old place. She's never seen it." "How is Kikuko these days?"
"Well, she seems a little bored, now that my affair is over."
Shingo smiled wryly.
4
Once again it was Sunday, and Shuichi seemed to have gone once more to the fish pond.
Lining up a row of cushions that had been airing in the hall, Shingo lay down in the warm autumn sun, his head on his arm.
Teru was sunning herself on the stone step below him.
In the breakfast room Yasuko was reading through the pile of newspapers on her knee, perhaps ten days' worth of them.
When she came on something interesting she would tell Shingo. It happened so often that Shingo's answers tended to be perfunctory.
"I wish you'd stop this business of reading all the newspapers on Sundays," he said, turning over sluggishly.
At the alcove in the parlor, Kikuko was putting together an arrangement of red crow-gourds.
"You found them on the mountain?"
"Yes. They seemed very pretty."
"Are there still some left?"
"Just a few. Five or six."
Three gourds hung from the vine in her hand.
Every morning from the washstand Shingo could see red gourds on the mountain, above the pampas grass. Here inside the parlor they were an even more dazzling red.
Kikuko also came into his range of vision.
There was an indescribable freshness about the line from her jaw to her throat. It was not the product of a single generation, thought Shingo, somehow saddened.
Perhaps because the style of her hair set off the neck and throat, her face seemed a little thin.
Shingo had of course been aware all along of the beauty of that line, and the long, slender throat. Was it that, given the considerable distance and the angle from which he was watching her, it stood out in more beauty than usual?
Perhaps the autumn radiance added something.
That line from jaw to throat spoke first of maidenly freshness. It was beginning to swell a little, however, and that maidenliness would soon disappear.
"Just one more," Yasuko called. "Here's a very interesting one."
"Oh?"
"It's about America. A place called Buffalo, New York. Buffalo. A man 'had his left ear cut off in an automobile accident, and went to a doctor. The doctor ran off to where the accident happened and found the ear, all dripping blood, and stuck it back on. And it's worked perfectly since."
"They say you can put a finger back on if you do it soon enough."
"Oh?" She read on for a time, and seemed to remember something. "I suppose that's true of husband and wife too. If you put them back together soon enough they'll stick. But it's been too long."
"What do you mean?" said Shingo, not really asking a question.
"Don't you suppose it's that way with Fusako.
"Aihara's disappeared," answered Shingo lightly, "and we don't know whether he's dead or alive.'
"Oh, we could find that out if we tried. But what's to happen?"
"So Granny still has her regrets. Give them up. We sent in the divorce notice long ago."
"I've been good at giving things up since I was a girl. It's just that I have her and the two children right here in front of me, and wonder what's to become of them."
Shingo did not answer.
"Fusako's not the prettiest girl in the world. And suppose she were to remarry—it would be really too much for Kikuko to have the two children left on her hands."
"Kikuko and Shuichi would have to live somewhere else. And it would be up to Granny to raise the children."
"I don't think anyone could call me lazy, but how old do you think I am?"
"Do your best and leave what's undone to the gods. Where's Fusako?"
"They've gone to see the Buddha. Children are very strange. Satoko almost got run over once on her way back, and she still loves the place. She's always crying to go there."
"I doubt if it's the Buddha itself that she likes."
"It does seem to be."
"Come, now."
"Don't you suppose Fusako could go back to the country? They might make her their heir."
"They don't need an heir," said Shingo curtly.
Yasuko read her newspapers in silence.
"Mother's ear story reminds me." This time it was Kikuko who spoke. "Do you remember how you once said you'd like to leave your head in a hospital and have it cleaned and restored?"
"We were looking at the sunflowers down the street. I think the need is more pressing now that I find myself forgetting how to tie my tie. Before long I'll be reading the newspaper upside down and not noticing."
"I often think about it, how it would be after you left your head in a hospital."
Shingo looked at her. "Well, it's as if you were leaving your head at a hospital every night for a sleep cure, I suppose. Maybe it's because I'm old, but I'm always having dreams. 'When I am in pain, I have dreams that continue reality.' I seem to remember reading that line in a poem somewhere. Not that my own dreams go on with reality."
Kikuko was surveying her completed arrangement.
Shingo too gazed at the gourds. "Kikuko. Why don't you and Shuichi go live somewhere else?"
Kikuko looked up in surprise, and came over to him. "I'd be afraid." It was a voice too low for Yasuko to overhear. "I'm afraid of him."
"Do you intend to leave him?"
"If I were to, I'd be able to look after you as I pleased," she said solemnly.
"Your misfortune."
"It's no misfortune when you're doing something you want to do."
Shingo was startled. The remark was like a first expression of ardor. He sensed in it a certain danger.
"You're very diligent in looking after me, but don't you have me confused with Shuichi? I should think it would only drive him farther away."
"There are things about him I don't understand." The white face seemed to be pleading with him. "Sometimes all of a sudden I'm so frightened I don't know what to do."
"I know. He changed after he went to war. Sometimes he seems to behave on purpose so that I myself can't tell what's on his mind. But then if you just stick to him like that ear, all dripping blood, maybe things will come out all right."
Kikuko was gazing at him.
"Has he told you that you are a free agent?"
"No." She looked at him in curiosity. "A free agent?"
"I asked him myself what he meant by saying that about his own wife. I suspect he may have meant partly that you should be freer. I should arrange to let you go free."
"You mean from you yourself?"
"Yes. He said I should tell you you're free."
That moment a sound came from the heavens. To Shingo it was really as if he had heard a sound from the heavens.
Five or six pigeons cut a low diagonal across the garden.
Kikuko also heard them. She went to the edge of the veranda.
"Am I free, then?" she said, tears in her voice, as she watched the pigeons fly off.
The dog Teru left the step to run off across the garden in pursuit of the wings.
5
All seven members of the family were present at dinner.
Fusako and her two children were now members of the family too, no doubt.
"There were only three trout left at the store," said Kikuko. "One of them is for Satoko." She set the three before Shingo, Shuichi, and Satoko.
"Trout are not for children." Fusako put out her hand. "Give it to Grandmother."
"No." Satoko clutched at the dish.
"What big trout," observed Yasuko calmly. "The last of the year, I imagine. I'll just pick away at Grandfather's here, and I don't need any of yours. Kikuko can have some of Shuichi's."
They formed three separate factions. Perhaps they should be in three separate houses.
Satoko's attention was concentrated on the trout.
"Is it good?" asked Fusako, frowning. "But what a messy way to eat." She scooped out the roe and gave it to Kuniko, the younger child. Satoko did not object.
"Roe," muttered Fusako, tearing off one end of the roe in Shingo's trout.
"Back in the old days in the country, Yasuko's sister got me interested in writing haiku. There are all sorts of expressions about trout—'autumn trout,' and 'descending trout,' and 'rusty trout.' That sort of thing." Shingo glanced at Yasuko and went on. "'Descending trout' and 'rusty trout' are trout that have laid their eggs. Worn out, completely exhausted, they are going down to sea."
"Just like me." Fusako's response was immediate. "Not that I was much to look at as a healthy trout."
Shingo pretended not to hear."'A trout in the autumn, abandoning itself to the water.' 'Trout swimming down the shallows, not knowing they must die,' That sort of old poem. I imagine they would apply to me."
"To me," said Yasuko, "Do they die when they've laid their eggs and gone down to sea?"
"I believe that's the way it went. Though of course there were occasionally trout that spent the winter in deep pools. They were called 'remaining trout.'"
"Maybe that's the kind of trout I am."
"I don't think I'll be able to stay on," said Fusako.
"But you've put on weight since you came home," said Yasuko, looking at her daughter, "and your color has improved."
"I don't want to put on weight."
"Being at home is like hiding in a deep pool," said Shuichi.
"I don't want to stay all that long. I'd rather go down to the sea, Satoko." Her voice rose. "You haven't anything there but bones. Stop worrying them."
"Your talk about the trout has spoiled the flavor of the trout," said Yasuko, a quizzical expression on her face.
Fusako looked down, and her mouth was working nervously. Then she gathered herself to bring out the words: "Father. Won't you open a little shop for me? A cosmetics shop, a stationery shop, anything. I don't care what part of town it's in. I don't mind if it's just a street stall. A drinking place."
"You think you'd be able to manage that kind of business?" asked Shuichi in surprise.
"I would. Customers don't come to drink a woman's face. They come to drink sake. Are you comparing me with your pretty wife?"
"That's not what I meant at all."
"Of course she can do it," put in Kikuko, to the surprise of the others. "And if she decides to have a try, I'll ask her to let me help her."
"A very fine plan indeed," said Shuichi.
The dinner table fell silent.
Kikuko alone among them flushed. She was crimson to the ears.
"How about next Sunday?" said Shingo. "I'd been thinking it would be good if we could all go to the country to see the maples."
Yasuko's eyes sparkled.
"Kikuko too. Kikuko hasn't seen our old home."
"I'd love to," said Kikuko.
Shuichi and Fusako sat in perverse silence.
"Who will watch the house?" asked Fusako at length.
"I will," said Shuichi.
"No, I will. But I'd like to have your answer, Father, before you leave."
"I'll let you know my decision," said Shingo. He was thinking of Kinu, said to have opened a small dressmaking shop in Numazu, the child still inside her.
The moment the meal was over Shuichi left the table.
Shingo too got up, rubbing at a cramp in the small of his back. He looked absently into the living room and turned on the light.
"Your gourds are sagging," he called to Kikuko. "They seem to be too heavy."
She apparently could not hear him over the sound of the dishes.
Other Great Names in the TUTTLE CLASSICS Scries
KOBO ABE
The Woman in the Dunes ISBN: 4-8053-0207-0
First published in 1962. this avant-garde work is esteemed as one of the finest Japanese novels of" the postwar period, and it was the first of Abe's novels to be translated into English. Niki Jumpei. an amateur entomologist on a weekend trip from the city, comes upon a bizarre village among lowering dunes where the residents are living deep within pits. Drawn to assist a widow, he is entrapped and must shovel along with her against the ever-encroaching dunes that threaten to bury the community. White Niki attempts to escape his prison of sand, his relationship with the woman evolves and he comes to accept a whole new identity . . . With striking similarities to the work of Franz Kafka, The Woman in the Dunes entices with its unusual plot, its vivid detail, and its existential examination of the human condition.
Secret Rendezvous ISBN: 4-8053-0472-3
In this surrealistic detective story, a man awakens one morning with the groggy recollection that an ambulance had taken his wife from their home in the middle of the night. In his laborious quest to locate her at the hospital, his senses are assaulted by the depraved environment he enters, a labyrinth filled with odd and insane sights, sounds and people.
YUSUNARI KAWABATA
Thousand Cranes ISBN: 4-8053-0667-X
This melancholy tale uses Japan's classical tea ceremony as a backdrop for the relationships between the young man Kikuji and two of his late father's mistresses. Whereas (he bitter and domineering Chikako has become as coarse and meddlesome as Kikuji's late father, the beguiling and charming Mrs Ota remains as young and as alluring as ever. Further complicating these relationships is Mrs Ota's 20-year-old daughter, Fumiko. Kikuji retraces his father's fickle loot-steps as he tries to escape the domineering Chikako while harboring a guilty passion for Mrs Ota, and Fumiko's efforts to intervene ultimately lead to the novel's stunning climax. Kawabata deftly employs the symbolism of the Japanese tea ceremony as well as the minimalist style of sumi-e. haiku, and Noh theater in this exploration of sex, love, guilt, and raw revenge.
YUKIO MISHIMA
After the Banquet ISBN 4-8053-0628-9
Kazu is a successful, independent woman determined never to fall in love again—that is, until a former government cabinet minister walks into her life. Attracted despite herself by Noguehi's aristocratic hack-ground and intellectualism, she marries him—even as the wide social and moral gulf between them signals catastrophe. In her own selfish attempt to resurrect Noguehi's political career, Kazu secretly funds his idealistic campaign, but Noguchi finds out and is morally offended. Now Kazu has to decide: should she comply with her husband's strict code of conduct, or return to die independent life she'd once cherished? The climax of this tale reveals Mishima's full range of power as a master storyteller and novelist.
Forbidden Colors ISBN: 4-8053-0630-0
Widely regarded as Mishima's finest work. Forbidden Colors explores the issue of sexual hedonism in postwar Japan, An aging, embittered novelist employs the beautiful and young Yuichi Minami to avenge himself on the women who have betrayed him. As his mentor forces him into a loveless marriage and a string of meaningless love affairs, Yuichi looks for physical and emotional solace in Japan's vast gay underworld. As he works his way through the gay world's secret spaces. Yuichi meets with blackmailers, pimps and male prostitutes, soon finds himself as vulnerable and defenseless as the women he preyed on.
SOSEKI NATSUME
And Then ISBN: 0-8048-1537-2
One of Natsume Soseki's most admired works, And Then is a novel of love and disillusionment that tells the story of a young man for whom idleness has become an expression of rebellion. Having become thoroughly alienated by the cultural upheaval in Japan, Daisuke has lost the values to guide his life. Yet his refined indifference to everything is completely disrupted when he falls in love with his best friend's wife. Now for the first time. Daisuke must choose his own fate.
Kokoro ISBN: 4-8053-0161-9
Set in the turbulent Meiji era, a chance encounter on a Kamakura beach irrevocably links a young student to a man he simply calls "Sensei." Intrigued by Sensei's aloofness and wanting to know more about him, the student calls upon Sensei with increasing frequency. Eventually. Sensei and his beautiful wife open up their home and their lives to him. The young man graduates from university and is called home to his dying father, but the Sensei draws him back to Tokyo with a letter of heartfelt confession ... Written in 1914. Kokoro provides a timeless psychological analysis of one man's alienation from society, and starkly but gently shows the depth of both friendship and love.
JUNICHIRO TANIZAKI
Some Prefer Nettles ISBN: 4-8053-0633-5
The conflict between traditional and modern Japanese culture is at the heart of this novel. Kaname is a smug, modem man living in a modern marriage. He gamely allows his wife to become the lover of another man, an act that does not cure the profound sadness at the heart of their relationship. So Kaname gradually retreats into the protection of traditional rituals, attitudes and tastes, eventually making love to O-Hisa. his father-in-law's old-fashioned mistress, as he abandons the modern world entirely. The novel's other characters, including Kaname's wife, his lover, his father-in-law, and even the cities in which they live, all symbolize the modern and ancient ways of life in Japan. Tanizaki's characteristic irony, eroticism, and psychological undertones make Some Prefer Nettles an exceptional and compelling read.
The Key ISBN: 4-8053-0632-7
In The Key, a middle-aged husband and his wife both seek passion outside of their loveless marriage, telling their stories in the form of parallel diary entries. The novel carefully explores the theme of sexual desire, and Tanizaki's restrained and delicate prose artfully conveys not only the consequences of the couple's pursuits, but also demonstrates the very best of Japanese literature.
#
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