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Resection for achalasia of the esophagus.
One hundred and twenty-two patients with advanced mega-esophagus managed by esophagectomy without thoracotomy and cervical gastroplasty were evaluated. Sixty-nine patients were followed up for periods of 6 months to 16 years. Clinical assessment included X-ray studies and endoscopy of the cervical esophagus and mobilized stomach. The most common postoperative complications were pleural effusion (22.1%) and cervical fistula (8.2%). Mortality was 4.18%. Regurgitation was the most frequent complaint in the late follow-up, followed by heartburn. Both symptoms were related to esophagitis and diffuse gastritis. Diarrhea and dumping also occurred due to vagotomy and pyloromyotomy performed at the same time as esophagectomy. The endoscopic study demonstrated esophagitis in 25.5% of the patients, and diffuse erosive gastritis in 12.7%. The symptoms and late complications were handled by clinical measures and careful endoscopic follow-up. Gastroplasty was considered a good procedure for replacing the esophagus, solving the serious problem of dysphagia and for providing nutritional improvement for the patient. |
362 So.2d 224 (1978)
G. Alvon DAMPIER, d/b/a Dampier-Harris & Associates
v.
James C. PEGUES et al.
77-241.
Supreme Court of Alabama.
September 15, 1978.
*225 William M. Acker, Jr., Susan E. Dominick, Birmingham, for appellant.
Gene R. Smitherman, The University of Alabama in Birmingham, for appellees.
ALMON, Justice.
Dampier filed an action seeking a writ of mandamus to require James C. Pegues, fiscal officer for the University of Alabama in Birmingham (UAB) and S. Richardson Hill, President of UAB, to pay him $14,325.66 allegedly due under a contract. The circuit court granted the defendant's motion to dismiss. We reverse.
The basic question is whether the allegations of the complaint are sufficient to bring the claim within the holding of Hardin v. Fullilove Excavating Co., Inc., 353 So.2d 779 (Ala.1977) and State Board of Administration, et al. v. Roquemore, 218 Ala. 120, 117 So. 757 (1928), so as to avoid UAB's constitutional immunity.
The essential allegations of the complaint are as follows:
"1. On or about November 20, 1972, the plaintiff and the Board of Trustees of the University of Alabama, by and through its lawfully authorized agent, Joseph F. Volker, then President of the University of Alabama in Birmingham, entered into a written contract for the performance by plaintiff of architectural and consulting services for the construction of the Holmes Addition to Spain Rehabilitation Center at the University of Alabama Hospitals and Clinics, Birmingham, Alabama. A copy of said contract is attached hereto as Exhibit "A" and incorporated herein by reference.
2. A sufficient sum was budgeted and available for the performance of this work according to a schedule of fees contained in said contract.
3. Plaintiff performed all of the work and services called for in the architectural contract, and said services were accepted, approved and used by the University of Alabama in Birmingham."
The motion to dismiss contains several grounds. The first ground is that the suit is barred by sovereign immunity. Article I, Section 14, Alabama Constitution 1901. The second ground asserts that mandamus is inappropriate because an adequate remedy exists through the State Board of Adjustment. The third and fourth grounds assert res judicata and collateral estoppel because of a prior suit by Dampier against the University of Alabama Board of Trustees.
Basically, Hardin holds that state officials may not arbitrarily withhold payment where the other party has undisputedly performed in accordance with the contract. In the defendant's answer to the petition they asserted as a defense that Dampier breached the contract and consequently is not entitled to the final payment. Whether the contract was breached or not is a fact question.
The complaint is not to be construed strictly against the pleader when challenged by a motion to dismiss. Butts v. Weiss, 346 So. 422 (Ala.1977). It is clear that if the plaintiff proved each allegation in his complaint, his claim would fall squarely within the holdings of Hardin and Roquemore. In these two cases, this Court affirmed the issuance of a writ of mandamus to compel state officials to pay sums of money due under executed contracts.
Therefore, considering the complaint in a light most favorable to the plaintiff, we hold that his petition is sufficient to withstand a motion to dismiss.
Motions to dismiss should only be granted when it appears beyond doubt that the plaintiff can prove no set of facts in support of his claim which would entitle him to relief. Jeannie's Grocery v. Baldwin County Elec. Mem. Corp., 331 So.2d 665 (Ala.1976).
The other grounds in the motion to dismiss are also without merit. Possible redress through the State Board of Adjustment will not prevent mandamus. Dunn Construction Co. v. State Board of Adjustment, 234 Ala. 372, 175 So. 383 (1937).
*226 The prior suit by Dampier against the Board of Trustees of the University of Alabama will not prevent the maintenance of this suit. The parties in the two suits were not the same. The sole issue decided in Dampier's prior suit was the immunity of UAB under § 14. See Ellison v. Abbott, 337 So.2d 756 (Ala.1976); Hutchinson v. Board of Trustees of University of Alabama, 288 Ala. 20, 256 So.2d 281 (1971). However, State immunity is subject to certain exceptions, one of which is recognized in Hardin. This cause is not barred by any principles of res judicata or collateral estoppel. Webster v. Gunter, 336 So.2d 170 (Ala. 1976).
The judgment of the circuit court is reversed and the cause is remanded.
REVERSED AND REMANDED.
TORBERT, C. J., and BLOODWORTH, FAULKNER and EMBRY, JJ., concur.
|
Fox News host Neil Cavuto took aim at President Trump on Monday, criticizing him over recent tweets and calling for him to act like the president.
Cavuto during his show brought up Trump's tweets lashing out at LaVar Ball for not giving the president enough credit in the release of UCLA basketball players, including Ball's son, detained in China.
He also referenced Trump's attacks on Sen. Jeff Flake Jeffrey (Jeff) Lane FlakeJeff Flake: Republicans 'should hold the same position' on SCOTUS vacancy as 2016 Republican former Michigan governor says he's voting for Biden Maybe they just don't like cowboys: The president is successful, some just don't like his style MORE (R-Ariz.). Trump called Flake's political career "toast" after Flake, who is not running for reelection, was caught on a hot mic criticizing Trump and Alabama GOP Senate candidate Roy Moore.
ADVERTISEMENT
"Forget about either comment not being presidential. At what point does the president see such remarks don’t even border on being human?” Cavuto said.
Cavuto said Trump is "using a bazooka to respond to a pea shooter."
"Is it really necessary?" he asked.
"Keep tweeting stuff like that, Mr. President, and those tax cuts just could be toast."
Cavuto said Trump has the White House and all the advantages of "being the most powerful human being on the planet."
"Let your achievements speak for themselves. Let your clear influence to get those players out of China speak for you. Not a kid's dad who just might be embarrassed, but that kid and his buddies who already told the nation and you they were grateful," Cavuto said.
ADVERTISEMENT
He called for Trump to pick his fights.
"Because neither of these seems worth the fuss," Cavuto said.
"Last time I checked, you are the president of the United States. Why don't you act like it?" |
Wardija Ridge
Wardija Ridge is a very scenic plateau, one of a group in the north of Malta. The ridge is sparsely populated making it an ideal walking location. There is lots of flora and fauna and Wardija Ridge has recently been protected.
Wardija ridge, Il-Ballut tal-Wardija and Il-Wied ta' San Martin run in a Southwestern - Northeastern direction from the Għajn Tuffieħa area in Mġarr to the Xemxija Bay area in St. Paul's Bay. The Northern ridge edge is interrupted by a series of valley systems which discharge into il-Wied tal-Pwales.
The landscape plateaus, that have been assigned a Level 2 degree of protection as Areas of Ecological Importance and Sites of Scientific Importance, support important garigue communities characterised by rare species.
Other levels of protection have been given to the three valley systems in the scheduled areas, the upper reaches of the watercourse of Il-Wied ta' San Martin and the Southern part of the promontory at the area known as L-Argentier.
In 1915, the British built Wardija Battery on the east end of the ridge. The battery became obsolete in 1938, but its gun emplacements still exist.
References
Category:Landforms of Malta
Category:St. Paul's Bay
Category:Plateaus of Europe |
Q:
Dask Does between_time exist?
In pandas there is the between_time method however my dataset is too large for pandas, however doing a quick control f in the Dask api shows 4 mentions for between_time but no actual use of it.
I just get a lovely: AttributeError: 'Series' object has no attribute 'between_time'. Does it actually exist? Is there a workaround?
A:
No. As of 2018-09-02 no between_time method exists for Dask Dataframes. It is not listed as a function in the Dask dataframe API
|
Ants vs Robots
Ants vs Robots
Hear the scuttling, scampering ants swarm to take on the computerised silicon wizardry of the clever robots. This is a fast paced, frenzy of technological circuitry against nature’s colonial wizards. Well, something like that anyway. It’s a fun, short, energetic piece ideally suited for videogames or children’s programming.
Use in one end product, free or commercial. Most web uses. 10,000 copy limit for a downloaded or physical end product. Plus up to 1 million broadcast audience. The total price includes the item price and a buyer fee.
Use in one end product, free or commercial. Most web uses. Up to 1 million broadcast audience. Plus unlimited copies of a downloaded or physical end product. The total price includes the item price and a buyer fee.
Use in one end product, free or commercial. Most web uses. Unlimited copies of a downloaded or physical end product. Plus a broadcast audience of up to 10 million. The total price includes the item price and a buyer fee.
Use in one end product, free or commercial. Most web uses. Unlimited copies of a downloaded or physical end product. Plus an unlimited broadcast audience, or a theatrically released film. The total price includes the item price and a buyer fee. |
# 包名
函数列表
- xxx1
- xxx2 |
Q:
Convert ternary relationship to binary in E/R model
When I am studying the database lecture on E/R model, it illustrates how to convert ternary relationship to binary. One way is using weak entity relationship as follows (each relationship is M:N cardinality):
ternary relationship:
convert the upper relationship with weak relationship
However, in another example:
it states in the lecture slide: "if each technician can be working on several projects and uses the same notebooks on each project, then we can decompose 3-ary relationship into binary relationships"as follows:
which I could not understand. I still kinda confused about when we should use weak entity approach and when we could just simply convert it to binary relationships as the latter one. Thanks!
A:
Your second image illustrates a confusion between conceptual and physical data models, or a confusion between the ER and network data models. The physical implementation of the models in the first two images are the same, what differs is the interpretation of entities and relationships. The entity-relationship model supports ternary relationships, but doesn't support multiple identifying relationships for a single weak entity set. I would advise you to disregard the second image completely.
The third and fourth images illustrate a fourth normal form decomposition using ER notation. This isn't something you can do with any ternary relationship, but rather something you do when 2 or 3 independent relations have been incorrectly combined into one. For more information, I suggest you read up on Fourth Normal Form.
|
Q:
How to query all entries from the database by id?
I was following this tutorial on Udemy about building login form. Everything went well.
Now I'm trying to reuse the code. The problem I'm having is when I'm trying to query all entries from a database in order to get a total number of websites. Could you please advise?
Here is my count function (this is where I'm having issues in obtaining a total number of entries to the database for pagination):
function get_all_websites(){
$all = array();
$db = DB::getInstance();
$all = $db->query("SELECT * FROM website");
if(!$all->count()){
echo 'No Websites available. Please check back later.';
} else {
foreach($all->results() as $all){
$all->id = $all;
}
}
return $all;
}
function get_websites_count(){
return count(get_all_websites());
}
if I use this I get all ID's listed.
function get_all_websites(){
$all = array();
$db = DB::getInstance();
$all = $db->query("SELECT * FROM website");
if(!$all->count()){
echo 'No Websites available. Please check back later.';
} else {
foreach($all->results() as $all){
echo $all->id;
}
}
}
Database class.
class DB{
private static $_instance = null;
private $_pdo,
$_query,
$_error = false,
$_results,
$_count = 0;
private function __construct(){
try {
$this->_pdo = new PDO('mysql:host=' .
Config::get('mysql/host') . ';dbname=' .
Config::get('mysql/db'),
Config::get('mysql/username'),
Config::get('mysql/password'));
} catch(PDOException $e){
die($e -> getMessage());
}
}
public static function getInstance(){
if(!isset(self::$_instance)) {
self::$_instance = new DB();
}
return self::$_instance;
}
public function query($sql, $params = array()){
$this->_error = false;
// Check if query has been prepared properly
if($this->_query = $this->_pdo->prepare($sql)){
$x = 1;
if(count($params)){
foreach($params as $param){
$this->_query->bindValue($x, $param);
$x++;
}
}
// If the query has been prepared successfuly, store the result
if($this->_query->execute()){
$this->_results = $this->_query->fetchAll(PDO::FETCH_OBJ);
$this->_count = $this->_query->rowCount();
} else {
$this->_error = true;
}
}
return $this;
}
public function action($action, $table, $where = array()){
if(count($where) === 3){
$operators = array('=', '>', '<', '>=', '<=');
$field = $where[0];
$operator = $where[1];
$value = $where[2];
if(in_array($operator, $operators)){
$sql = "{$action} FROM {$table} WHERE {$field} {$operator} ?";
if(!$this->query($sql, array($value))->error()){
return $this;
}
}
}
return false;
}
// QUERYING DATA FROM DATABASE
public function get($table, $where){
return $this->action('SELECT *', $table, $where);
}
// DELETING DATA FROM DATABASE
public function delete($table, $where){
return $this->action('DELETE', $table, $where);
}
// INSERTING DATA INTO DATABASE
public function insert($table, $fields = array()){
$keys = array_keys($fields);
$values = '';
$x = 1;
foreach($fields as $field){
$values .= "?";
if($x < count($fields)){
$values .= ', ';
}
$x++;
}
$sql = "INSERT INTO {$table} (`" . implode('`, `', $keys) . "`) VALUES({$values})";
if(!$this->query($sql, $fields)->error()){
return true;
}
return false;
}
public function results(){
return $this->_results;
}
public function update($table, $id, $fields){
$set = '';
$x = 1;
foreach($fields as $name => $value){
$set .= "{$name} = ?";
if($x < count($fields)){
$set .= ', ';
}
$x++;
}
$sql = "UPDATE {$table} SET {$set} WHERE id = {$id}";
if(!$this->query($sql, $fields)->error()){
return true;
}
return false;
}
public function first(){
return $this->results()[0];
}
public function error(){
return $this->_error;
}
public function count(){
return $this->_count;
}
}
A:
This part doesn't seem right:
} else {
foreach($all->results() as $all){
$all->id = $all;
}
}
return $all;
but I'm not exactly sure what you want.
We could append each id to an array called $allWebsiteIds:
} else {
foreach($all->results() as $all){
$allWebsiteIds[] = $all->id;
}
}
return $allWebsiteIds;
That might give you what you want.
|
/**
* Appcelerator Titanium Mobile
* Copyright (c) 2009-2012 by Appcelerator, Inc. All Rights Reserved.
* Licensed under the terms of the Apache Public License
* Please see the LICENSE included with this distribution for details.
*/
package ti.modules.titanium.ui.widget.picker;
import java.util.ArrayList;
import org.appcelerator.titanium.proxy.TiViewProxy;
import org.appcelerator.titanium.view.TiUIView;
import ti.modules.titanium.ui.PickerProxy;
public abstract class TiUIPicker extends TiUIView
{
protected boolean suppressChangeEvent = false;
public boolean batchModelChange =
false; // Set by proxy to indicate that several model changes are occurring and therefore view can wait to refresh
public TiUIPicker(TiViewProxy proxy)
{
super(proxy);
}
public abstract void selectRow(int columnIndex, int rowIndex, boolean animated);
public abstract int getSelectedRowIndex(int columnIndex);
protected abstract void refreshNativeView();
public void openPicker() {}
// When the whole set of columns has been changed out.
public void onModelReplaced()
{
if (!batchModelChange) {
refreshNativeView();
}
}
// When a column has been added.
public void onColumnAdded(int columnIndex)
{
}
// When a column has been removed.
public void onColumnRemoved(int oldColumnIndex)
{
}
// When a row has been added to / removed from a column
public void onColumnModelChanged(int columnIndex)
{
}
// When a row value has been changed.
public void onRowChanged(int columnIndex, int rowIndex)
{
}
protected PickerProxy getPickerProxy()
{
return (PickerProxy) proxy;
}
public void selectRows(ArrayList<Integer> selectionIndexes)
{
if (selectionIndexes == null || selectionIndexes.size() == 0) {
return;
}
for (int colnum = 0; colnum < selectionIndexes.size(); colnum++) {
int rownum = selectionIndexes.get(colnum).intValue();
selectRow(colnum, rownum, false);
}
}
}
|
[Level, distribution, and source identification of polychlorinated naphthalenes in surface agricultural soils from an electronic waste recycling area].
The concentration of 46 polychlorinated naphthalenes (PCNs) in the agricultural soils from Luqiao was analyzed by GC-NCI-MS. The objectives of this study were to investigate the contents, spatial distribution and sources of PCNs. The total concentrations of PCNs (sigma PCNs) in soil samples were in the range of 0.062 to 2.92 ng x g(-1), with a mean of 0.630 ng x g(-1). Tetra-CNs and penta-CNs were the predominant homologues in most of the samples, accounting for 18.4% - 88.8% and 8.40% - 53.1%, with average values of 46.7% and 30.7%, respectively, followed by tri-CNs, accounting for 0 - 47.3%, with a mean of 10.6%. Cluster analysis and combustion marker analysis showed that the sampling sites were mainly polluted by Halowax 1014 and Halowax 1013, also possibly polluted by PCBs mixtures and e-waste combustion process. Compared to other studies, the PCNs concentration in this study was at a medium level. |
The boyfriend of a teenager who had her leg amputated after a rollercoaster crash has recalled the horror after their carriage collided with another.
The couple's visit to Alton Towers was their first trip out of their hometown of Barnsley
Joe Pugh said his girlfriend Leah Washington screamed in pain and he described a bloody scene as they sat trapped in the Smiler.
The pair were among five seriously injured people caught up in the crash earlier this month which resulted in the Staffordshire theme park closing for six days.
Eighteen-year-old Mr Pugh, whose knees were shattered in the crash, said: "I remember a sickening bang with metal grinding against metal and the safety bar being rammed against my knees.
"I looked at my hands and there was blood everywhere. At first there was no pain, just numbness."
He added: "We didn't dare look at our own legs, so we took it in turns to inspect each other's injuries. It wasn't good."
But he vowed that nothing will change between him and his girlfriend - who were on their first trip out of Barnsley together at Alton Towers.
Joe said: "I want her for who she is and it would be very superficial if that changed over something such as her injury.
"We've both got a long road head of us as we come through this. We were together at the beginning, we're together now and we'll be together as we travel down that road – and we'll be leaning on each other every step of the way."
Leah, 17, from Barnsley in South Yorkshire, suffered the most serious injuries, having her left leg amputated above the knee and being treated for a fractured hand.
IG
Leah Washington had her left leg amputated after the crash
Mr Pugh said it was his fourth time on the Smiler ride, which was the first rollercoaster he had been on, but he vowed never to venture on to a ride again after what had happened.
He said he and the others on the ride had been forced to get off twice before it began so staff could carry out safety checks, and then described a nervous 20-minute delay as the rollercoaster paused on one of the loops.
Mr Pugh added that the excited screams of his fellow passengers turned to horrified ones moments after the crash as they were stuck in the carriage at an angle of 45 degrees.
"We couldn't believe what we were seeing," he said.
Immediately after the crash Mr Pugh said he became angry when he saw a girl taking a video of the trapped thrillseekers.
SWNS
Passengers could be seen with bloodied faces following the horror crash
Daniel Thorpe, a 27-year-old old hotel assistant manager from Buxton in Derbyshire, 20-year-old Vicky Balch from Leyland in Lancashire, and Chandaben Chauhan, 49, of Wednesbury, West Midlands, also suffered injuries.
Alton Towers has said it is in contact with the victims of the crash and their families.
A spokesman said: "We have made contact with all the families and have assured them that we will provide full support to all of those involved, now and throughout their recovery and rehabilitation." |
/*
Copyright 1993 by Davor Matic
Permission to use, copy, modify, distribute, and sell this software
and its documentation for any purpose is hereby granted without fee,
provided that the above copyright notice appear in all copies and that
both that copyright notice and this permission notice appear in
supporting documentation. Davor Matic makes no representations about
the suitability of this software for any purpose. It is provided "as
is" without express or implied warranty.
*/
#ifdef HAVE_XNEST_CONFIG_H
#include <xnest-config.h>
#endif
#include <X11/X.h>
#include <X11/Xproto.h>
#include "scrnintstr.h"
#include "window.h"
#include "windowstr.h"
#include "colormapst.h"
#include "resource.h"
#include "Xnest.h"
#include "Display.h"
#include "Screen.h"
#include "Color.h"
#include "Visual.h"
#include "XNWindow.h"
#include "Args.h"
DevPrivateKeyRec xnestColormapPrivateKeyRec;
static DevPrivateKeyRec cmapScrPrivateKeyRec;
#define cmapScrPrivateKey (&cmapScrPrivateKeyRec)
#define GetInstalledColormap(s) ((ColormapPtr) dixLookupPrivate(&(s)->devPrivates, cmapScrPrivateKey))
#define SetInstalledColormap(s,c) (dixSetPrivate(&(s)->devPrivates, cmapScrPrivateKey, c))
Bool
xnestCreateColormap(ColormapPtr pCmap)
{
VisualPtr pVisual;
XColor *colors;
int i, ncolors;
Pixel red, green, blue;
Pixel redInc, greenInc, blueInc;
pVisual = pCmap->pVisual;
ncolors = pVisual->ColormapEntries;
xnestColormapPriv(pCmap)->colormap =
XCreateColormap(xnestDisplay,
xnestDefaultWindows[pCmap->pScreen->myNum],
xnestVisual(pVisual),
(pVisual->class & DynamicClass) ? AllocAll : AllocNone);
switch (pVisual->class) {
case StaticGray: /* read only */
colors = xallocarray(ncolors, sizeof(XColor));
for (i = 0; i < ncolors; i++)
colors[i].pixel = i;
XQueryColors(xnestDisplay, xnestColormap(pCmap), colors, ncolors);
for (i = 0; i < ncolors; i++) {
pCmap->red[i].co.local.red = colors[i].red;
pCmap->red[i].co.local.green = colors[i].red;
pCmap->red[i].co.local.blue = colors[i].red;
}
free(colors);
break;
case StaticColor: /* read only */
colors = xallocarray(ncolors, sizeof(XColor));
for (i = 0; i < ncolors; i++)
colors[i].pixel = i;
XQueryColors(xnestDisplay, xnestColormap(pCmap), colors, ncolors);
for (i = 0; i < ncolors; i++) {
pCmap->red[i].co.local.red = colors[i].red;
pCmap->red[i].co.local.green = colors[i].green;
pCmap->red[i].co.local.blue = colors[i].blue;
}
free(colors);
break;
case TrueColor: /* read only */
colors = xallocarray(ncolors, sizeof(XColor));
red = green = blue = 0L;
redInc = lowbit(pVisual->redMask);
greenInc = lowbit(pVisual->greenMask);
blueInc = lowbit(pVisual->blueMask);
for (i = 0; i < ncolors; i++) {
colors[i].pixel = red | green | blue;
red += redInc;
if (red > pVisual->redMask)
red = 0L;
green += greenInc;
if (green > pVisual->greenMask)
green = 0L;
blue += blueInc;
if (blue > pVisual->blueMask)
blue = 0L;
}
XQueryColors(xnestDisplay, xnestColormap(pCmap), colors, ncolors);
for (i = 0; i < ncolors; i++) {
pCmap->red[i].co.local.red = colors[i].red;
pCmap->green[i].co.local.green = colors[i].green;
pCmap->blue[i].co.local.blue = colors[i].blue;
}
free(colors);
break;
case GrayScale: /* read and write */
break;
case PseudoColor: /* read and write */
break;
case DirectColor: /* read and write */
break;
}
return True;
}
void
xnestDestroyColormap(ColormapPtr pCmap)
{
XFreeColormap(xnestDisplay, xnestColormap(pCmap));
}
#define SEARCH_PREDICATE \
(xnestWindow(pWin) != None && wColormap(pWin) == icws->cmapIDs[i])
static int
xnestCountInstalledColormapWindows(WindowPtr pWin, void *ptr)
{
xnestInstalledColormapWindows *icws = (xnestInstalledColormapWindows *) ptr;
int i;
for (i = 0; i < icws->numCmapIDs; i++)
if (SEARCH_PREDICATE) {
icws->numWindows++;
return WT_DONTWALKCHILDREN;
}
return WT_WALKCHILDREN;
}
static int
xnestGetInstalledColormapWindows(WindowPtr pWin, void *ptr)
{
xnestInstalledColormapWindows *icws = (xnestInstalledColormapWindows *) ptr;
int i;
for (i = 0; i < icws->numCmapIDs; i++)
if (SEARCH_PREDICATE) {
icws->windows[icws->index++] = xnestWindow(pWin);
return WT_DONTWALKCHILDREN;
}
return WT_WALKCHILDREN;
}
static Window *xnestOldInstalledColormapWindows = NULL;
static int xnestNumOldInstalledColormapWindows = 0;
static Bool
xnestSameInstalledColormapWindows(Window *windows, int numWindows)
{
if (xnestNumOldInstalledColormapWindows != numWindows)
return False;
if (xnestOldInstalledColormapWindows == windows)
return True;
if (xnestOldInstalledColormapWindows == NULL || windows == NULL)
return False;
if (memcmp(xnestOldInstalledColormapWindows, windows,
numWindows * sizeof(Window)))
return False;
return True;
}
void
xnestSetInstalledColormapWindows(ScreenPtr pScreen)
{
xnestInstalledColormapWindows icws;
int numWindows;
icws.cmapIDs = xallocarray(pScreen->maxInstalledCmaps, sizeof(Colormap));
icws.numCmapIDs = xnestListInstalledColormaps(pScreen, icws.cmapIDs);
icws.numWindows = 0;
WalkTree(pScreen, xnestCountInstalledColormapWindows, (void *) &icws);
if (icws.numWindows) {
icws.windows = xallocarray(icws.numWindows + 1, sizeof(Window));
icws.index = 0;
WalkTree(pScreen, xnestGetInstalledColormapWindows, (void *) &icws);
icws.windows[icws.numWindows] = xnestDefaultWindows[pScreen->myNum];
numWindows = icws.numWindows + 1;
}
else {
icws.windows = NULL;
numWindows = 0;
}
free(icws.cmapIDs);
if (!xnestSameInstalledColormapWindows(icws.windows, icws.numWindows)) {
free(xnestOldInstalledColormapWindows);
#ifdef _XSERVER64
{
int i;
Window64 *windows = xallocarray(numWindows, sizeof(Window64));
for (i = 0; i < numWindows; ++i)
windows[i] = icws.windows[i];
XSetWMColormapWindows(xnestDisplay,
xnestDefaultWindows[pScreen->myNum], windows,
numWindows);
free(windows);
}
#else
XSetWMColormapWindows(xnestDisplay, xnestDefaultWindows[pScreen->myNum],
icws.windows, numWindows);
#endif
xnestOldInstalledColormapWindows = icws.windows;
xnestNumOldInstalledColormapWindows = icws.numWindows;
#ifdef DUMB_WINDOW_MANAGERS
/*
This code is for dumb window managers.
This will only work with default local visual colormaps.
*/
if (icws.numWindows) {
WindowPtr pWin;
Visual *visual;
ColormapPtr pCmap;
pWin = xnestWindowPtr(icws.windows[0]);
visual = xnestVisualFromID(pScreen, wVisual(pWin));
if (visual == xnestDefaultVisual(pScreen))
dixLookupResourceByType((void **) &pCmap, wColormap(pWin),
RT_COLORMAP, serverClient,
DixUseAccess);
else
dixLookupResourceByType((void **) &pCmap,
pScreen->defColormap, RT_COLORMAP,
serverClient, DixUseAccess);
XSetWindowColormap(xnestDisplay,
xnestDefaultWindows[pScreen->myNum],
xnestColormap(pCmap));
}
#endif /* DUMB_WINDOW_MANAGERS */
}
else
free(icws.windows);
}
void
xnestSetScreenSaverColormapWindow(ScreenPtr pScreen)
{
free(xnestOldInstalledColormapWindows);
#ifdef _XSERVER64
{
Window64 window;
window = xnestScreenSaverWindows[pScreen->myNum];
XSetWMColormapWindows(xnestDisplay, xnestDefaultWindows[pScreen->myNum],
&window, 1);
xnestScreenSaverWindows[pScreen->myNum] = window;
}
#else
XSetWMColormapWindows(xnestDisplay, xnestDefaultWindows[pScreen->myNum],
&xnestScreenSaverWindows[pScreen->myNum], 1);
#endif /* _XSERVER64 */
xnestOldInstalledColormapWindows = NULL;
xnestNumOldInstalledColormapWindows = 0;
xnestDirectUninstallColormaps(pScreen);
}
void
xnestDirectInstallColormaps(ScreenPtr pScreen)
{
int i, n;
Colormap pCmapIDs[MAXCMAPS];
if (!xnestDoDirectColormaps)
return;
n = (*pScreen->ListInstalledColormaps) (pScreen, pCmapIDs);
for (i = 0; i < n; i++) {
ColormapPtr pCmap;
dixLookupResourceByType((void **) &pCmap, pCmapIDs[i], RT_COLORMAP,
serverClient, DixInstallAccess);
if (pCmap)
XInstallColormap(xnestDisplay, xnestColormap(pCmap));
}
}
void
xnestDirectUninstallColormaps(ScreenPtr pScreen)
{
int i, n;
Colormap pCmapIDs[MAXCMAPS];
if (!xnestDoDirectColormaps)
return;
n = (*pScreen->ListInstalledColormaps) (pScreen, pCmapIDs);
for (i = 0; i < n; i++) {
ColormapPtr pCmap;
dixLookupResourceByType((void **) &pCmap, pCmapIDs[i], RT_COLORMAP,
serverClient, DixUninstallAccess);
if (pCmap)
XUninstallColormap(xnestDisplay, xnestColormap(pCmap));
}
}
void
xnestInstallColormap(ColormapPtr pCmap)
{
ColormapPtr pOldCmap = GetInstalledColormap(pCmap->pScreen);
if (pCmap != pOldCmap) {
xnestDirectUninstallColormaps(pCmap->pScreen);
/* Uninstall pInstalledMap. Notify all interested parties. */
if (pOldCmap != (ColormapPtr) None)
WalkTree(pCmap->pScreen, TellLostMap, (void *) &pOldCmap->mid);
SetInstalledColormap(pCmap->pScreen, pCmap);
WalkTree(pCmap->pScreen, TellGainedMap, (void *) &pCmap->mid);
xnestSetInstalledColormapWindows(pCmap->pScreen);
xnestDirectInstallColormaps(pCmap->pScreen);
}
}
void
xnestUninstallColormap(ColormapPtr pCmap)
{
ColormapPtr pCurCmap = GetInstalledColormap(pCmap->pScreen);
if (pCmap == pCurCmap) {
if (pCmap->mid != pCmap->pScreen->defColormap) {
dixLookupResourceByType((void **) &pCurCmap,
pCmap->pScreen->defColormap,
RT_COLORMAP,
serverClient, DixInstallAccess);
(*pCmap->pScreen->InstallColormap) (pCurCmap);
}
}
}
static Bool xnestInstalledDefaultColormap = False;
int
xnestListInstalledColormaps(ScreenPtr pScreen, Colormap * pCmapIDs)
{
if (xnestInstalledDefaultColormap) {
*pCmapIDs = GetInstalledColormap(pScreen)->mid;
return 1;
}
else
return 0;
}
void
xnestStoreColors(ColormapPtr pCmap, int nColors, xColorItem * pColors)
{
if (pCmap->pVisual->class & DynamicClass)
#ifdef _XSERVER64
{
int i;
XColor *pColors64 = xallocarray(nColors, sizeof(XColor));
for (i = 0; i < nColors; ++i) {
pColors64[i].pixel = pColors[i].pixel;
pColors64[i].red = pColors[i].red;
pColors64[i].green = pColors[i].green;
pColors64[i].blue = pColors[i].blue;
pColors64[i].flags = pColors[i].flags;
}
XStoreColors(xnestDisplay, xnestColormap(pCmap), pColors64, nColors);
free(pColors64);
}
#else
XStoreColors(xnestDisplay, xnestColormap(pCmap),
(XColor *) pColors, nColors);
#endif
}
void
xnestResolveColor(unsigned short *pRed, unsigned short *pGreen,
unsigned short *pBlue, VisualPtr pVisual)
{
int shift;
unsigned int lim;
shift = 16 - pVisual->bitsPerRGBValue;
lim = (1 << pVisual->bitsPerRGBValue) - 1;
if ((pVisual->class == PseudoColor) || (pVisual->class == DirectColor)) {
/* rescale to rgb bits */
*pRed = ((*pRed >> shift) * 65535) / lim;
*pGreen = ((*pGreen >> shift) * 65535) / lim;
*pBlue = ((*pBlue >> shift) * 65535) / lim;
}
else if (pVisual->class == GrayScale) {
/* rescale to gray then rgb bits */
*pRed = (30L * *pRed + 59L * *pGreen + 11L * *pBlue) / 100;
*pBlue = *pGreen = *pRed = ((*pRed >> shift) * 65535) / lim;
}
else if (pVisual->class == StaticGray) {
unsigned int limg;
limg = pVisual->ColormapEntries - 1;
/* rescale to gray then [0..limg] then [0..65535] then rgb bits */
*pRed = (30L * *pRed + 59L * *pGreen + 11L * *pBlue) / 100;
*pRed = ((((*pRed * (limg + 1))) >> 16) * 65535) / limg;
*pBlue = *pGreen = *pRed = ((*pRed >> shift) * 65535) / lim;
}
else {
unsigned limr, limg, limb;
limr = pVisual->redMask >> pVisual->offsetRed;
limg = pVisual->greenMask >> pVisual->offsetGreen;
limb = pVisual->blueMask >> pVisual->offsetBlue;
/* rescale to [0..limN] then [0..65535] then rgb bits */
*pRed = ((((((*pRed * (limr + 1)) >> 16) *
65535) / limr) >> shift) * 65535) / lim;
*pGreen = ((((((*pGreen * (limg + 1)) >> 16) *
65535) / limg) >> shift) * 65535) / lim;
*pBlue = ((((((*pBlue * (limb + 1)) >> 16) *
65535) / limb) >> shift) * 65535) / lim;
}
}
Bool
xnestCreateDefaultColormap(ScreenPtr pScreen)
{
VisualPtr pVisual;
ColormapPtr pCmap;
unsigned short zero = 0, ones = 0xFFFF;
Pixel wp, bp;
if (!dixRegisterPrivateKey(&cmapScrPrivateKeyRec, PRIVATE_SCREEN, 0))
return FALSE;
for (pVisual = pScreen->visuals;
pVisual->vid != pScreen->rootVisual; pVisual++);
if (CreateColormap(pScreen->defColormap, pScreen, pVisual, &pCmap,
(pVisual->class & DynamicClass) ? AllocNone : AllocAll,
0)
!= Success)
return False;
wp = pScreen->whitePixel;
bp = pScreen->blackPixel;
if ((AllocColor(pCmap, &ones, &ones, &ones, &wp, 0) !=
Success) ||
(AllocColor(pCmap, &zero, &zero, &zero, &bp, 0) != Success))
return FALSE;
pScreen->whitePixel = wp;
pScreen->blackPixel = bp;
(*pScreen->InstallColormap) (pCmap);
xnestInstalledDefaultColormap = True;
return True;
}
|
Pope Francis Pope FrancisNuns criticize Catholic group for giving Barr award for 'Christlike behavior' Pope seeks to prevent Mafia from using Virgin Mary imagery Pope: No one should seek to profit from pandemic MORE on Saturday warned against the dangers of disinformation and underscored what he called "an urgent need for reliable information" from the news media.
"There is an urgent need for reliable information, with verified data and news, which does not aim to amaze and excite, but rather to make readers develop a healthy critical sense, enabling them to ask themselves appropriate questions and reach justified conclusions," the pope said to a gathering of journalists at the Vatican.
"There is an urgent need for news communicated with serenity, precision and completeness, with a calm language, so as to favor a fruitful reflection; carefully weighted and clear words, which reject the inflation of allusive, strident and ambiguous speech," he said.
ADVERTISEMENT
In his speech to the Italian Periodical Press Union and the Italian Federation of Catholic Weeklies, Francis also warned journalists not to fall into what he called the "sins of communication."
"We must not fall prey to the 'sins of communication': disinformation — that is, giving just one side of the argument — slander, which is sensationalistic, or defamation, looking for outdated and old things, and bringing them to light today," he said. "They are very grave sins, which damage the heart of the journalist and harm people." |
NFL Playoffs 2019
NFL Playoffs 2019 : The team that started 11-2, boasts the NFL’s most dynamic young quarterback and, for so long, seemed like the obvious favorite to go to the Super Bowl? Or the team that started 1-5, only to win 10 of their last 11 games with an MVP candidate of their own,
That’s the big question surrounding the first Divisional Round game on tap for this weekend, when the Kansas City Chiefs will host the Indianapolis Colts for a chance to advance to the AFC Championship.
More NFL
A year after finishing 4-12 and playing without quarterback Andrew Luck, the Colts took a monumental step forward in 2018 under first-year head coach Frank Reich. Not only did Luck find his footing, throw the second most touchdowns of his career and return to the Pro Bowl, but Indy unearthed a Rookie of the Year candidate in Darius Leonard and shored up its defense, going 9-1 to finish the regular season and then stomping all over the rival Houston Texans in a 21-7 Wild-Card win. Judging purely off momentum, there isn’t a better team in the NFL outside of maybe the No. 1-seed New Orleans Saints than Reich’s Colts. NFL Playoffs Live Stream Free 2019. TV Schedule Game Watch Online
But then there’s the Chiefs, who had the rest of the league in awe for the majority of the season. Coach Andy Reid has had Kansas City competitive since he arrived back in 2013, but never has Arrowhead Stadium played host to video-game production like that of the 2018 Chiefs offense, which saw Patrick Mahomes toss 50 touchdowns in his first full year as a starter and the team score at least 30 points in 12 of its 16 regular-season games. The only downside in K.C. might be on defense, which has been an issue all season, and the fact that the Chiefs enter the playoffs on a 1-2 stretch and with a murky history of home postseason success under Reid |
Changes in incidence of in situ and invasive breast cancer by histology type following mammography screening.
To investigate secular trends and correlates of incidence of breast cancer by histology type following the introduction of population-based mammography screening. Analysis of age-standardised incidence rates for 1,423 in situ and 16,157 invasive carcinomas recorded on the South Australian population-based cancer registry for the 1985-2004 diagnostic period. Multiple logistic regression was undertaken to compare socio-demographic characteristics by histology. Progression from in situ disease was investigated using the Kaplan-Meier method. The incidence of in situ lesions increased approximately seven-fold over the 20-year period, compared with an increase of about 40% for invasive cancers. The increase for in situ lesions was due to increases for ductal carcinomas, with little change for lobular lesions. By comparison, the percentage increase in incidence for invasive cancer was greater for lobular than ductal cancers. Both for in situ and invasive cancers, percentage increases were greatest for the screening target age range of 50-69 years. One in 14 in situ cases was found to progress to invasive cancer within seven years of diagnosis, but insufficient detail was available to determine whether the invasive cancers were a progression of the in situ lesions or whether they originated separately. These invasive cancers were smaller than generally applying for other invasive cancers of the female breast. The larger secular increases in incidence for in situ than invasive cancers would reflect the dominant role of mammography in the detection of ductal carcinoma in situ. The lack of an increase for lobular in situ lesions may have resulted from their poorer radiological visibility. The greater percentage increase for lobular than ductal invasive lesions may have been due to an increase in imaging sensitivity for these lesions, plus real increases in incidence. The smaller sizes of invasive cancers found in women with a prior in situ diagnosis may have resulted from more intensive medical surveillance, although the possibility of biological differences cannot be discounted. |
//
// ChildViewController.swift
// ResponderChainDemo
//
// Created by Nicholas Outram on 17/01/2016.
// Copyright © 2016 Plymouth University. All rights reserved.
//
import UIKit
class ChildViewController: UIViewController {
@IBOutlet weak var passUpSwitch: UISwitch!
override func viewDidLoad() {
super.viewDidLoad()
// Do any additional setup after loading the view.
}
override func didReceiveMemoryWarning() {
super.didReceiveMemoryWarning()
// Dispose of any resources that can be recreated.
}
/*
// MARK: - Navigation
// In a storyboard-based application, you will often want to do a little preparation before navigation
override func prepareForSegue(segue: UIStoryboardSegue, sender: AnyObject?) {
// Get the new view controller using segue.destinationViewController.
// Pass the selected object to the new view controller.
}
*/
override func touchesBegan(touches: Set<UITouch>, withEvent event: UIEvent?) {
self.printNextRepsonderAsString()
if passUpSwitch.on {
//Pass up the responder chain
super.touchesBegan(touches, withEvent: event)
}
}
}
|
June 2011
June 1st, 2011: Glendale Recharge Ponds-A Tern Show
Hi everyone,
After Gary reported the Caspian Terns earlier today, I decided to give it a quick afternoon chase.
I headed straight to the northeast pond, where I immediately saw the two CASPIAN TERNS. What a pleasure they were to see, really the first time I have had good looks at this bird in Arizona. They were rather aggressive to all the other birds around, both vocal and taking dives at the herons and egrets closeby to them, as well as the Ring-billed Gull that Gary also reported. They did spend the majority of the time on the island too, as well as fly around the other basins throughout my visit. I saw one of the Caspians catch a fish in the much deeper northmiddle basin.
As I was watching the Caspians and was about to leave, I was very pleased to see two beautiful breeding plumaged BLACK TERNS fly in. They flew over the ponds for about five minutes before I lost track of them. I wasn't expecting them, but they were certainly a great surprise!
Other birds out there tonight included many GREAT and SNOWY EGRETS, GREAT BLUE HERONS, WHITE-FACED IBIS (good numbers), a single LONG-BILLED DOWITCHER, the RING-BILLED GULL, and HORNED LARKS. The two terns gave me 261 species so far in Maricopa County for 2011, which I didn't reach 261 until August 9th of last year in 2010. Caspian was also a new Glendale Recharge Pond bird for me, the birding location closest to me that I love to see new birds at. Thank you Gary!
Good Birding,
Tommy DeBardeleben (Glendale, Arizona)
June 3rd, 2011: Hassayampa River Preserve
Hi everyone,
I birded the Hassayampa River Preserve for three hours this morning, 8 to 11 A.M.
It wasn't as birdy as last time, but there are still some nice migrants moving through. My highlight of the day was a beautiful male TOWNSEND'S WARBLER that was singing away. I heard the bird first and knew it was one of the black-throated warblers, and was able to get on it with good views. It was the first time I've ever heard a Townsend's Warbler in song, it was a cool thing. The runner up to the Townsend's Warbler were at least two SWAINSON'S THRUSHES, which is the most recent addition to my Maricopa County list, so I was still very thrilled to see them for the second time in the county. This timeframe is sure productive throughout Arizona for this species, I'm shocked with how many there's been. WILSON'S WARBLERS were very numerous, as I counted at least 14 birds throughout the hike. Other migrants included two WESTERN WOOD-PEWEES, "WESTERN" FLYCATCHER, a WARBLING VIREO, and three BLACK-HEADED GROSBEAKS.
The regulars are always worth the trip. The GRAY HAWKS were calling throughout the morning at the Preserve, as I also caught a glimpse of a retreating bird along the Mesquite Meander. I found one TROPICAL KINGBIRD on the same trail also, where they are usually found. Once again, I located it by hearing it's voice, the easiest giveaway. And of course there are other greats to enjoy such as VERMILION and BROWN-CRESTED FLYCATCHERS, CANYON WREN, YELLOW WARBLERS, YELLOW-BREASTED CHATS, SUMMER TANAGERS, BLUE GROSBEAKS, and BULLOCK'S and HOODED ORIOLES.
Good birding,
Tommy DeBardeleben (Glendale, Arizona)
June 6th, 2011: Southwest Maricopa County
Hi everyone,
I attempted to re-find the Scissor-tailed Flycatcher that was seen on Saturday without success. I spent four hours in the area, driving back and fourth in the general area it was seen. I did this many times in between looking for other birds at some of the other locations. If in the area, do keep an eye out for this bird. I looked up Scissor-tailed Flycatcher records in the historical database on AZFO and saw some of them hung around in areas for a good amount of time. It could still be around somewhere, as the Roseate Spoonbill in this area has shown us time and time again!
One of the other locations I visited was the Arlington Wildlife Area. Highlights were a few CLAPPER RAILS calling, a LEAST BITTERN, BARN SWALLOW, YELLOW-HEADED BLACKBIRDS, and a singing WESTERN MEADOWLARK in the surrounding fields. The surrounding forest of tamarisk had tons of singing WHITE-WINGED DOVES, which was pretty cool to hear.
In the areas where I searched for the flycatcher and also along the Old Us 80, I had numerous BURROWING OWLS, RED-TAILED HAWKS, WESTERN KINGBIRDS, a flock of sixty or so WHITE-FACED IBIS, a GREATER ROADRUNNER, and a COMMON GROUND-DOVE. I also saw a total of four LESSER NIGHTHAWKS with one near 7 A.M., the second at 8 A.M., and then two flying together at 10:30 A.M. They are feeding like that over on this side of Arizona as well! GAMBEL'S QUAIL chicks are starting to fly and were very viewable throughout this area during the morning.
On my way home I stopped at the Glendale Recharge Ponds. The highlight there and of my day was a/or continuing BLACK TERN in breeding plumage. It was a beautiful bird, and fun to watch, as I spent about thirty minutes watching fly back and fourth at close range. The RING-BILLED GULL was still present, as well as the single LONG-BILLED DOWITCHER. Also present was a lingering drake AMERICAN WIGEON.
Good birding,
Tommy DeBardeleben (Glendale, Arizona)
June 17th, 2011: Sunflower and Mount Ord, Maricopa County, Arizona
Hi everyone,
I spent the first half of my day yesterday on June 17th, 2011, birding in the areas of Sunflower and Mount Ord.
My first stop was at Sunflower, which was a three hour stop starting at 5:45 A.M. It was a birdy stop, producing 51 species. I was hoping to get my first Yellow-billed Cuckoo of the year, but they remained unseen and quiet. Maybe next time. My favorite sighting here was a group of CASSIN'S KINGBIRDS mobbing a COMMON RAVEN. It was rather funny, and I got to observe at it close range. At first there were only a few Cassin's Kingbirds on the raven, and when the raven perched in a tree (close to me), the kingbirds kept on coming, as eventually seven of them crowded around the raven. As the raven took off, the kingbirds all seemed to take turns grabbing it's back, at times there were two of them at once striking the raven. I know I wouldn't want to be that raven, the kingbirds certainly don't have that aggressive of an approach when looking at one. Another highlight was a continously singing INDIGO BUNTING in the area of the Sunflower Work Station. I also birded the entrance road before I hiked back along the Old Beeline Highway. A few CLIFF SWALLOWS are nesting under a tunnel, and I found a good number of BRONZED COWBIRDS in an open field, which a male VERMILION FLYCATCHER was as well as numerous BLUE GROSBEAKS. Other highlights I had during the three hours included both COMMON BLACK and ZONE-TAILED HAWKS, a few COSTA'S HUMMINGBIRDS, two singing WESTERN WOOD-PEWEES, GRAY VIREOS singing from the hillsides, as well as great numbers of HOODED ORIOLES and a few BULLOCK'S ORIOLES. I also observed and heard a LESSER GOLDFINCH giving notes identical to the scream of a Gray Hawk, which was very odd. Obviously it wasn't that loud, but I've never heard them give that call out of all the other birds I've heard them immitate.
Next I went to Mount Ord. I had a strong feeling to come here when I woke up in the morning, despite the fact I was tired. It's a good thing I did as Mount Ord was closing yesterday as I left for temporary fire danger, who knows when it will be open again. I honestly can't complain however, I am glad Mount Ord will be reduced from fire chances. The birdlife was what Mount Ord basically brings on my usual outings for the most part. I observed PHAINOPEPLAS for the first time on Road 1688, where one will first meet the pine and oak forests. At the top of Ord, I found a family of CHIPPING SPARROWS with a young juvenille bird. They were up here last year too, and this is the only place I have found breeding Chipping Sparrows in Maricopa County. A HOUSE WREN was singing at the top, and I also encountered a male WESTERN BLUEBIRD. Other highlights throughout Ord included ACORN WOODPECKER, WESTERN WOOD-PEWEE, VIOLET-GREEN SWALLOWS, BRIDLED and JUNIPER TITMICE, BUSHTIT, WHITE-BREASTED NUTHATCH,(only two singing throughout the day, no other nuthatches heard or seen) singing HERMIT THRUSHES, OLIVE, VIRGINIA'S, GRACE's, and BLACK-THROATED GRAY WARBLERS; PAINTED REDSTARTS (including a recently fledged juv.), BLACK-CHINNED SPARROWS, five or more HEPATIC TANAGERS, one WESTERN TANAGER, two BLACK-HEADED GROSBEAKS, and a singing SCOTT'S ORIOLE.
I came down from Ord to find the forest service waiting for me, so they could officially lock the gate. Who knows how long it will be closed, but at least there is a better chance for Ord to be here longer, which is the most important thing! The habitat and birdlife is so great here and is one of my very favorite places on earth. However, Slate Creek was not on the closure map and I made a quick run over the FR 201 (Slate Creek's access road), and it was still open.
Also, I have recently published my own website. On that website, I've been working on a project called "Birding in Maricopa County". It's an online site to birding locations and hotspots around the county, and I've written information about many different sites and hotspots. Hotspots on the site are divided by area in the county. Each site has a description and information on birding at the site, what birds can be seen there, directions (with Google Maps), as well as many photographs of each site, and pictures of the birds taken at the sites that can be seen there. I have 42 sites done so far, many more to go. It's a work in progress and I'll never officially be done from adding to this. By the end of this year, I hope to have many more sites done. If anyone is interested, the area page to Highway 87 and it's hotspots on my website can be found at this link-http://www.birderfrommaricopa.com/area-a-highway-87-from-desert-to-the-high-country.htm. This includes Mesquite Wash, Sunflower, Mount Ord, and Slate Creek Divide. I hope you all enjoy this website!
Good birding,
Tommy DeBardeleben (Glendale, Arizona)
June 20th, 2011: Miller, Mesa and the Salt River-An expedition to see a male Broad-billed Hummingbird
Hi everyone,
I headed out to the Mesa/Salt River area today for a few hours of birding before I couldn't tollerate the heat anymore.
My first stop was at the Miller House of Mesa, where Jay Miller showed Jim Kopitzke and I his most recent addition to his fantastic yard, the male BROAD-BILLED HUMMINGBIRD. Broad-billed put on a show for us with great looks in the thirty or so minutes we observed. The three of us were talking and keeping an eye out for the bird, when it appeared just feet in front of my face. It was chasing other hummers away from the area it frequented. To bring back a good memory, Jay had a Magnificent Hummingbird in his yard for five days several years back, he certainly gets great birds. Other birds I enjoyed in the yard was a calling WESTERN KINGBIRD and an active VERDIN nest at very close views. Thanks again Jay for everything!
Jim and I then continued on to the Granite Reef Recreation Site at the Salt River. Marcus Watson has been seeing a Caspian Tern recently on a regular basis for about a week now here. The tern has been flying up and down the Salt River, so it has been at other locations besides Granite Reef. We ran into Marcus as we got there and he saw it there as recent as yesterday. So hopefully it will stick around. The birdlife here was regular. A juvenille BALD EAGLE perched on a dead snag across the river. Both ASH-THROATED and BROWN-CRESTED FLYCATCHERS were vocal. A few LESSER NIGHTHAWKS were flying over the river fifteen after seven. Singing BLUE GROSBEAKS and BULLOCK'S ORIOLES were also nice to see and hear in the area.
After Jim went home, I continued on to the area of the Foxtail Recreation Site. I read about this site in Mike Rupp's book, but it seems to be closed, but I have found an accessable area just a mile east of where the official Foxtail Site is, which is equally great in habitat. I've posted about it before. There is a nice wash here full of willows, which I found the Salt River to be flowing through the wash this time around. I had to bird at the sides of the wash, but it created good habitat for different birds. This is seasonal, sometimes the river goes through, sometimes it's dry. Jim said it's due to tubing, which makes perfect sense. The water flowing into the wash is a perfect pulloff for the tubers. Birdwise, my best highlight here was a pair of GREAT HORNED OWLS, who were calling back and fourth just after 9 A.M. I had great looks at both of them, which the first one I saw I spooked out of one of the willows. Many YELLOW WARBLERS were singing in the willows, as were a few BULLOCK'S ORIOLES. BROWN-CRESTED FLYCATCHERS were very vocal as well. With the river running through the wash, it created good habitat for COMMON YELLOWTHROATS, RED-WINGED BLACKBIRDS, SONG SPARROWS, and BLACK PHOEBES. I hoped for a Yellow-billed Cuckoo once more, no luck. Maybe nextime.
For anyone intersted in accessing these recreation sites or has never been to the Salt River and would like to bird the area, I'll include a link to the Salt River section on my website, which has directions and information about every recreation site at the Salt River, which is a thirteen mile stretch. Link- http://birderfrommaricopa.com/area-b-salt-river-area.htm
Another good but short day of birding. And I think the Ivory-billed Woodpecker still exists.
Good birding,
Tommy DeBardeleben (Glendale, Arizona)
June 31st, 2011: Phoenix Mountains Preserve-An expedition in pursuit of my first ever look at a Western Screech
Hey everyone,
I went with my friend, Norman Dong (who is avidly into herbs), this afternoon to explore a very shady area in the Phoenix Mountains close to a parking lot. Out of a quick hike I got my first ever look at a WESTERN SCREECH-OWL! I have heard Western Screech-Owls plenty of times, but this afternoon I finally tore that wall down with a perched owl right in front of me. We were able to get about six feet away from the bird without it spooking, which was perched up on rocks in a small and shady cave area. We watched the bird thirty minutes or so and walked away from the Screech-Owl still perched in the same spot. My first look at this bird was what I've always wanted, an open view without any branches in the way. I'd call the owl an "official lifer" now! I guess we all have overdue birds, I have my share.
Another standout highlight in the cave was another lifer for me, only from the reptile department, a TIGER RATTLESNAKE. It was coiled up peacefully just below the owl. We didn't want to spook the owl, so we enjoyed the snake at a further distance at ten feet away. Birding/herbing in a shady cave makes any summer day bearable! |
Venezuelans take to streets as uprising attempt sputters
International News
May 2, 2019
An anti-government protester winds up to throw a rock at security forces during clashes between the two, in Caracas, Venezuela, Wednesday, May 1, 2019. Opposition leader Juan Guaidó called for Venezuelans to fill streets around the country Wednesday to demand President Nicolás Maduro's ouster. Maduro is also calling for his supporters to rally. (AP Photo/Fernando Llano)
By SCOTT SMITH and CHRISTOPHER TORCHIA Associated Press
CARACAS, Venezuela (AP) — Venezuelans heeded opposition leader Juan Guaidó’s call to fill streets around the nation Wednesday but security forces showed no sign of answering his cry for a widespread military uprising, instead dispersing crowds with tear gas as the political crisis threatened to deepen.
Thousands cheered Guaidó in Caracas as he rolled up his sleeves and called on Venezuelans to remain out in force and prepare for a general strike, a day after his bold attempt to spark a mass military defection against President Nicolas Maduro failed to tilt the balance of power.“It’s totally clear now the usurper has lost,” Guaidó proclaimed, a declaration belied by events on the ground.
Across town at the Carlota air base near where Guaidó made his plea a day earlier for a revolt, intense clashes raged between protesters and troops loyal to Maduro, making clear the standoff would drag on. There and elsewhere, state security forces launched tear gas and fired rubber bullets while bands of mostly young men armed with makeshift shields threw rocks and set a motorcycle ablaze.“I don’t want to say it was a disaster, but it wasn’t a success,” said Marilina Carillo, who was standing in a crowd of anti-government protesters blowing horns and whistles.
Opposition leaders hoped Guaidó’s risky move would stir a string of high-ranking defections and shake Maduro’s grip on power. But only the chief of Venezuela’s feared intelligence agency broke ranks, while most others stood steadfast. Some analysts predicted that would make Maduro more emboldened.
The dramatic events could spell even more uncertainty for Venezuela, which has been rocked by three months of political upheaval since Guaidó re-energized a flagging opposition movement by declaring himself interim president, saying Maduro had usurped power.
Now the struggle has heightened geopolitical dimensions, with the United States and more than 50 other nations backing Guaidó as Venezuela’s legitimate president and Maduro allies like Russia lending the beleaguered president military and economic support.
U.S. National Security Adviser John Bolton said Wednesday that Maduro is surrounded by “scorpions in a bottle” and that key figures among his inner circle had been “outed” as dealing with the opposition.
The United States contends Maduro had been ready to flee Tuesday, an airplane already on the tarmac, but was talked out of it by Russian advisers.
Maria Zakharova, a spokeswoman for Russia’s Foreign Ministry, said such assertions were part of a “global information and psychological war against Venezuela and Caracas.”“There is no proof there was a Russian plane there,” she said. “The U.S. is big on Venezuela and wants to bring this to an end but that cannot do that.”
Protesters like Beatriz Pino, who took to the streets Wednesday waving flags and banging pots and pans, said they weren’t entirely surprised by the military’s response to Guaidó. She said the late President Hugo Chavez politicized Venezuela’s military as he installed a socialist system. Despite the setback, she said she remained committed to the opposition’s call for protest.“We can’t leave the streets,” she said. “We’ve been in this for years.”
As the standoff drags on, life is becoming even more difficult for Venezuelans, who are struggling with hyperinflation that has rendered salaries worthless as well as severe shortages of food and medicine that have driven about 3 million people to flee the country in recent years.“We need to get out of this tragedy,” said Ana Camarillo, a housewife.
David Smilde, a Venezuela expert, said the opposition’s thus far unsuccessful attempt to trigger an uprising should provoke a round of reflection.“Given the balance of power within Venezuela and the geopolitical struggle around it, they need to engage in real politics and real negotiations to move this conflict to a different place,” he said.
At a large pro-Maduro rally Wednesday, ruling party leader Diosdado Cabello said that “as a bloc” Venezuela’s military remained intact and united behind Maduro. He likened opposition leaders to “walking zombies.”
Luis Scott was among those wearing bright red shirts in solidarity with the socialist government and said he traveled seven hours on a bus to participate in the rally. He conceded Venezuela has deep economic troubles, but said the path set by Chavez and Maduro is firm.“We are fighting for our freedom,” the fisherman said.
While Maduro maintains a devout core of fervent supporters first inspired by Chavez, attendance at such shows of support is viewed as a requirement of their jobs.
At the Plaza Francia in Caracas’ Altamira neighborhood, protesters jammed the streets in one of the opposition’s biggest demonstrations yet. A few blocks away the scene quickly turned ugly. Protesters surrounded a suspected thief, beating him until he bled. A man with a megaphone appealed to the crowd to return to the fight against police.“The fight is down there!” he said, gesturing to the direction of a military base.
Mayor Gustavo Duque said the Salud Chacao medical center took in 27 patients by late afternoon Wednesday, one of whom was shot in the foot by a firearm. Those injuries are on top of more than 50 reported by the hospital’s director during clashes Tuesday.
Maduro appeared at the socialist party rally Wednesday afternoon, saying U.S. leaders had been fooled by the opposition into believing he was about to flee Venezuela. He said the Trump administration was part of a “pot of lies” and likened the ordeal to “fake news.”
He promised to put all conspirators behind bars.“Sooner or later they’ll go to jail and pay for their betrayal and their crimes,” he said.
Giancarlo Morelli of the British analysis group Economist Intelligence Unit said Maduro faces peril whatever path he takes with Guaidó over the uprising attempt.“Failing to arrest Mr. Guaidó would be perceived as an important sign for weakness from Mr. Maduro,” Morelli said. “But arresting Mr. Guaidó risks a strong counter-reaction from the U.S.,” which has been ratcheting up sanctions.
For many Venezuelans, the turmoil has become an almost normal state of affairs.
Johanns Davila walked his dog along a street in the capital littered with shotgun shells, tear gas canisters and a charred motorcycle, the remnants of skirmishes between the opposition and state security.“We need to get people out and recover the country,” Davila said.
___
Associated Press writers Christine Armario in Cucuta, Colombia, Fabiola Sanchez and Jorge Rueda in Caracas contributed to this report.
___
This story deletes extraneous ‘against’ in 4th paragraph. |
COLUMBUS, OHIO --- Columbus Blue Jackets captain Nick Foligno is expected to miss two to four weeks after suffering a lower body injury late in Saturday's game vs. St. Louis, club General Manager Jarmo Kekalainen announced today. The Blue Jackets have also recalled center Alex Broadhurst from the Cleveland Monsters, the club's American Hockey League affiliate.
Foligno, 30, has tallied 15 goals and 18 assists for 33 points with 50 penalty minutes and a +1 plus/minus rating in 72 games this season, including 8-11-19 and +7 over his past 30 outings. He registered the 400th point of his NHL career with an assist at the New York Rangers on March 20 and has registered 169-231-400 and 631 penalty minutes in 768 career games with the Blue Jackets and Ottawa Senators.
Broadhurst, 25, has recorded 19-22-41 and 20 penalty minutes in 66 games with the Monsters this season. He currently leads the club in goals, assists, points and games played and ranks second in shots on goal (133).
The Orland Park, Illinois native has posted 62-99-161 and 93 penalty minutes in 282 career AHL games with the Monsters and Rockford IceHogs. He helped Lake Erie capture the 2016 Calder Cup championship with 3-9-12 in 17 playoff games, finishing fifth-T in the AHL in postseason assists and points. Drafted by Chicago in the seventh round, 199th overall, in the 2011 NHL Draft, the 6-0, 178-pound center was acquired by the Blue Jackets from the Blackhawks as part of a trade on June 30, 2015. |
Simone Orton – once a runner, baseball player and figure skater – found herself curled up in a fetal position during a bout of agony so powerful, she wondered if she would ever live a pain-free and active life again
Her muscles were twitching from head to toe. The woman who prided herself on not letting anyone know the extent of her pain was vomiting repeatedly and moaning. Her terrified husband rushed her to the emergency room.
Orton had lived with chronic pain since an accident at age 12 that crushed two of her vertebrae. Many doctors told her that nothing serious was wrong with her in the years that followed. Some doctors suggested that the problem was all in her head. She persisted, until a rheumatologist finally diagnosed her with a kind of arthritis known as ankylosing spondylitis, which caused her spine to fuse together and to be completely solid.
Even after the diagnosis, however, Orton continued to suffer, often in silence. But this trip to the hospital altered everything. The peak of her pain was so high that she thought she might die. Doctors and nurses struggled to get her pain under control.
And on day six of her nine-day hospital stay, Orton, now 42, had a fateful meeting with Carmen R. Green, M.D., associate professor in the U-M Medical School's Department of Anesthesiology and pain specialist at the U-M Health System's Center for Interventional Pain Medicine.
Green “walked in and took over,” Orton recalls. “At that point she didn't feel I was adequately being taken care of, pain-wise.”
Green developed a pain-management plan for Orton that included regular fine-tunings of her pain medications, as well as ways to deal with the mental and social burdens associated with living with chronic pain.
Far too many people suffer from chronic pain without receiving adequate treatment, Green says. “Pain is a thief in the night; it steals people's livelihood,” she says. “Pain is under-treated. It really is a public health crisis. If we do not do something about the pain epidemic, it's going to significantly impact this society.”
Pain also is an issue that typically receives varying amounts of attention, depending on a patient's demographics, she says.
“The pain complaints of certain populations, including the elderly, minorities and women, do not receive the same attention as those of, in general, Caucasian men,” Green says. “Our research at U-M is focused on how age, race and gender influence the pain experience. We also look at how those factors influence health care providers' decision-making as it relates to pain.”
One-fifth to one-third of Americans live with pain, Green says. This number is increasing rapidly due to high rates of obesity and inactivity, improvements in medicine and technology that allow people to live longer, and other societal changes.
For instance, someone who would have died from a car accident or cancer 20 years ago now may be able to live a long life. “Despite the fact that we can save and prolong their lives, we now may end up treating them for chronic pain problems,” Green says.
But treatments are available, including medications – ranging from over-the-counter medicine to prescription-only opioid analgesics – as well as psychological counseling for the depression and anxiety that often accompany chronic pain, relaxation training, physical therapy to improve a person's function and mobility, and more. In addition, many types of nerve blocks are available to treat many painful conditions.
“We have a lot of things in our tool box,” Green says. “Nobody should have to suffer from pain when so many treatments are available.”
And how is Simone Orton doing? The woman who once needed a wheelchair when her pain and arthritis were at their most debilitating is back on her feet, continuing with an active lifestyle and in control of her pain.
“I'm doing so well that at my last visit with Dr. Green, she said she didn't need to see me for three months,” Orton says. “It felt like a graduation.”
In addition to his lab work, Adams is also the (non-paid) executive director of the non-profit Consumer Wellness Center (CWC), an organization that redirects 100% of its donations receipts to grant programs that teach children and women how to grow their own food or vastly improve their nutrition. Click here to see some of the CWC success stories.
With a background in science and software technology, Adams is the original founder of the email newsletter technology company known as Arial Software. Using his technical experience combined with his love for natural health, Adams developed and deployed the content management system currently driving NaturalNews.com. He also engineered the high-level statistical algorithms that power SCIENCE.naturalnews.com, a massive research resource now featuring over 10 million scientific studies.
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Q:
Java audio is not loading. toURI not working?
import java.io.File;
import javafx.application.Application;
import javafx.scene.media.Media;
import javafx.scene.media.MediaPlayer;
import javafx.stage.Stage;
public class SoundTest extends Application{
public static void main(String[] args) {
launch(args);
}
public static void sound() {
String path = "test.mp3";
Media media = new Media(new File(path).toURI().toString());
MediaPlayer mediaPlayer = new MediaPlayer(media);
mediaPlayer.play();
}
@Override
public void start(Stage arg0) throws Exception {
sound();
}
}
I have some issues. I googled and stumbled across a few helpful stackoverflow posts which provided explanations how sounds are loaded via media and media player.
What I am doing is, Im calling the sound function in the main() but my program fails to execute due to some failure in the second like of my sound function. The media object accepts an argument in the constructor which is the path to the audio file. Somehow it fails there as I get:
Exception in thread "Thread-0" java.lang.IllegalStateException: Toolkit not initialized
at com.sun.javafx.application.PlatformImpl.runLater(Unknown Source)
at com.sun.javafx.application.PlatformImpl.runLater(Unknown Source)
at javafx.application.Platform.runLater(Unknown Source)
at javafx.scene.media.Media$_MetadataListener.onMetadata(Unknown Source)
at com.sun.media.jfxmediaimpl.MetadataParserImpl.done(Unknown Source)
at com.sun.media.jfxmediaimpl.platform.java.ID3MetadataParser.parse(Unknown Source)
at com.sun.media.jfxmediaimpl.MetadataParserImpl.run(Unknown Source)
Exception in thread "main" java.lang.IllegalStateException: Toolkit not initialized
at com.sun.javafx.application.PlatformImpl.runLater(Unknown Source)
at com.sun.javafx.application.PlatformImpl.runLater(Unknown Source)
at javafx.application.Platform.runLater(Unknown Source)
at javafx.scene.media.MediaPlayer.init(Unknown Source)
at javafx.scene.media.MediaPlayer.<init>(Unknown Source)
at core.SoundTest.sound(SoundTest.java:43)
at core.SoundTest.main(SoundTest.java:13)
My sound file is located in the folder of my eclipse project where the class is in. It is a 3 minute long mp3 file located inside of the src and bin folders but not inside of the packages. (Im on windows).
How come this doesnt work? Why am I getting these errors.
A:
The problem here is that MediaPlayer is meant to be use in a JavaFX application only so you need to convert your application as a JavaFX application if you want to be able to use it.
To convert your class into a JavaFX application you need:
To make your class SoundTest extends javafx.application.Application
And modify your main method as next
public static void main(String[] args) {
Application.launch(args);
}
You can then call the method sound in your implementation of start
|
Q:
Simple bash function to find/replace string variable (no files)
I simply want a function (or just a 1-liner) to find/replace a string inside a variable, and not worry if the variables contain crazy characters.
Pseudo-code:
findReplace () {
#what goes here?
}
myLongVar="some long \crazy/ text my_placeholder bla"
replace="my_placeholder"
replaceWith="I like hamburgers/fries"
myFinalVar=$(findReplace $myLongVar $replace $replaceWith)
All similar questions seem complicated and use files
A:
You can define the function like this:
findReplace1() {
printf "%s" "${1/"$2"/$3}"
}
And then run it like this:
myFinalVar=$(findReplace "$myLongVar" "$replace" "$replaceWith")
Note the double-quotes -- they're very important, because without them bash will split the variables' values into separate words (e.g. "some long \crazy/ text..." -> "some" "long" "\crazy/" "text...") and also try to expand anything that looks like a wildcard into a list of matching filenames. It's ok to leave them off on the right side of an assignment (myFinalVar=...), but that's one of the few places where it's ok. Also, note that within the function I put double-quotes around $2 -- in that case again it's to keep it from being treated as a wildcard pattern, but here it'd a string-match wildcard rather than filenames. Oh, and I used printf "%s" instead of echo because some versions of echo do weird things with strings that contain backslashes and/or start with "-".
And, of course, you can just skip the function and do the replacement directly:
myFinalVar=${myLongVar/"$replace"/$replaceWith}
|
In the last installment in this series, we built oscillator and envelope generator (EG) components to add to our collection of needed components. In that discussion, we didn't really get into the type of devices that an EG might control or how EGs fit into the larger scheme of things. In this article, I'll discuss the digital equivalents of both a Voltage Controlled Amplifier (VCA) and a Voltage Controlled Filter (VCF), both of which will be controlled by EGs. We will continue to use the terms VCA and VCF to describe these devices even though voltage doesn't enter into the picture in the digital realm. Finally, we will touch on a MusicPlayer class for playing simple tunes within our synth environment. Before we get started, however, a little more history is in order.
Synthesizers
When synthesizers were first built, they were made up of modules connected together with patch cables. Early modular synths looked more like an old telephone switchboard than a musical instrument. Programming a synth to produce sound meant patching the modules together with a rats' nest of wires. Control signals were, in fact, voltages and these control signals (along with audio signals) were carried by the patch cables between modules. As flexible as this was for producing sound, it was very awkward from a performance perspective. To change sound in any significant way, musicians would have to tear down the current patch cord arrangement and patch in a new one not exactly a real-time operation. The MiniMoog was the first performance-oriented synthesizer that I am aware of. It did away with the patch cables, while offering players the ability to change sounds by manipulating pots and switches instead. The MiniMoog didn't have the flexibility of a modular synth, but it could perform in real time the functions most sought by musicians.
Here is an interesting quote from Wikipedia:
"A patch, in terms of music synthesizers, is a sound setting. Modular synthesizers used cables to patch the different sound modules together. Since these machines had no memory to save settings, musicians wrote down the locations of the patch cables and knob positions on a "patch sheet" (which usually showed a diagram of the synthesizer). After this, an overall sound setting for any type of synthesizer has been known as a patch."
For those interested in early synthesizer history there is a 2004 documentary called Moog by director Hans Fjellestad, which traces the roots of electronic music using interviews, photos, and archival footage. The story centers on inventor Robert Moog and his sometimes interesting views on creativity, interactivity, music, and machines. There are commentaries in the film by some of the early synthesizer adopters like Rick Wakeman of Yes and Keith Emerson of Emerson, Lake, and Palmer. This film is definitely worth seeking out for its historical perspective on electronic music.
Besides the mess of patch cords, early synthesizers were analog devices made up of resistors, capacitors, inductors, and transistors. One downside of these early analog synths was they were constantly going out of tune and needed to be periodically adjusted as the components aged and/or the temperature/humidity changed. Initially, these synths were used for avant garde music and sound effects so their ability to stay in tune wasn't much of an issue. However, as more people wanted to put them to use making traditional music, these stability problems needed to be addressed.
An EG in those days was a hardware device that would manipulate the charge on a capacitor for timing purposes. This voltage would be buffered and then fed out of the EG as a control voltage. This control signal would then be patched to other devices as a means of manipulating them dynamically. Besides using an EG to control amplitude dynamics with a VCA or cutoff frequency in a VCF it was possible to use the EG for many other purposes including: controlling the frequency of a low frequency oscillator (LFO), controlling filter resonance, controlling the amounts of modulation, and many other functions.
The MiniMoog had two EGs. One was coupled to a VCA for controlling amplitude dynamics and the other to a VCF for controlling filter cutoff frequency or resonance. My portable iPhone/iPod synthesizer, PSynth, has two EGs as well for the exact same purposes. In both the MiniMoog and PSynth, the EGs are triggered when a keyboard key is pressed and triggered again when the key is released.
Stability isn't an issue in the digital realm in which we are working. Aside from the occasional bug in the software, the electronic music modules we write will work consistently time after time and will never go out of tune. Of course, there are musicians who still prefer analog synths to digital ones, and that is why there is such a healthy market for these old analog devices. |
0010221201?
1122200010221202
In base 2, what is -101000111 - 100011000000010?
-100011101001001
In base 11, what is 5252 + 19a5?
7147
In base 11, what is -1300188 + -a?
-1300197
In base 4, what is 1311 - 2122021?
-2120110
In base 8, what is -74272237 + 2?
-74272235
In base 15, what is 87967 + -10a?
8785c
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11012
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-6162a4
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89762
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11121443314
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1132233313
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4143030144111
In base 5, what is 2030 + -23440201?
-23433121
In base 9, what is 751622 + -11?
751611
In base 14, what is 5 + 129662a1?
129662a6
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-31800
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-3343333312
In base 6, what is -10 - 51020053?
-51020103
In base 9, what is 27357340 - -121?
27357461
In base 3, what is -2122001011100 - 0?
-2122001011100
In base 10, what is 285958853 - 2?
285958851
In base 9, what is 121056 - 2035?
118021
In base 5, what is 20330 - -144402?
220232
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9106
In base 13, what is -1 - -67c0c56?
67c0c55
In base 13, what is 3 - -2314147?
231414a
In base 5, what is 110 - -1031232440?
1031233100
In base 16, what is dfca + 1219?
f1e3
In base 6, what is -1353333 + 201?
-1353132
In base 8, what is -12715 - -67676?
54761
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-100100100000000
In base 12, what is 1b61612 - -2?
1b61614
In base 8, what is -52 + -13412?
-13464
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188007
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111000101111011011000110
In base 14, what is -1c07 + 20c1?
2b8
In base 15, what is 3976 - 3c55?
-2ce
In base 2, what is 100 + 100110010000000000110101?
100110010000000000111001
In base 7, what is -1526434 - 31?
-1526465
In base 6, what is -5 - 132502400?
-132502405
In base 12, what is 2 - a431767?
-a431765
In base 12, what is 574a12 + -5?
574a09
In base 3, what is -10221020221112110 - 0?
-10221020221112110
In base 12, what is 100b + 3783?
4792
In base 7, what is 11 + -230522531?
-230522520
In base 15, what is -3be8 + -5e7?
-42e0
In base 4, what is -300 - -1030230300101?
1030230233201
In base 6, what is 4 - -10203345250?
10203345254
In base 10, what is -2 + 2181375?
2181373
In base 16, what is -5 + -19f66b?
-19f670
In base 16, what is -2f2c43 - -15?
-2f2c2e
In base 5, what is 3404103342 + 30?
3404103422
In base 4, what is 101312031030 + -1?
101312031023
In base 12, what is 176 + -81740b?
-817255
In base 13, what is 3c6 + -95a9?
-91b3
In base 11, what is 1143 + 3185?
4318
In base 14, what is -14114 + 2?
-14112
In base 13, what is 6 - -253665?
25366b
In base 7, what is 134352252 - -1?
134352253
In base 16, what is 446d - a82?
39eb
In base 14, what is -4c38a7 - 3?
-4c38aa
In base 7, what is 5 + -14360431?
-14360423
In base 3, what is -20102 - -1122110210?
1122020101
In base 14, what is -7 + 9c4978?
9c4971
In base 14, what is -1769 + -475b?
-60c6
In base 6, what is -1432 - -2502301?
2500425
In base 15, what is -5 + 95da78e?
95da789
In base 11, what is -1317 + -a14?
-2230
In base 9, what is -6103751 - -4?
-6103746
In base 4, what is -13321033333 - 1?
-13321100000
In base 9, what is -8235002 - -1?
-8235001
In base 7, what is -2414656 - -21?
-2414635
In base 5, what is -11413110 - 242?
-11413402
In base 11, what is -4 + 1421678?
1421674
In base 2, what is -10001001101011001 + -11?
-10001001101011100
In base 3, what is -200122220212 + 12?
-200122220200
In base 6, what is -1511 + -11334304?
-11340215
In base 2, what is -1110110011101110111 - 11?
-1110110011101111010
In base 3, what is 2210 + 22022020110?
22022100020
In base 9, what is 74232 - -160?
74402
In base 12, what is -4 + 47a1b94?
47a1b90
In base 16, what is -2 - -5d75efc?
5d75efa
In base 2, what is 11110101 + 11001000000111110?
11001000100110011
In base 16, what is 40638 - 58b?
400ad
In base 5, what is 11304030 + -114432?
11134043
In base 16, what is -30ce4 + -99?
-30d7d
In base 7, what is -334611330 - -4?
-334611323
In base 16, what is -1902a384 - 4?
-1902a388
In base 6, what is 1452132 - -5?
1452141
In base 2, what is -10001011 - 1111001001111010?
-1111001100000101
In base 15, what is 116da + 62d?
11d18
In base 13, what is -a91cc091 + 3?
-a91cc08b
In base 13, what is -7ba + -271c?
-3209
In base 14, what is -1c + 30886?
30868
In base 11, what is -415 - -1983?
1569
In base 13, what is 10443020 + -b?
10443012
In base 14, what is -4b + 39607?
3959a
In base 15, what is a00c9 - -61?
a013a
In base 3, what is 11 + 12001220201001?
12001220201012
In base 2, what is -1100 - -101000010100000011010110?
101000010100000011001010
In base 10, what is 8 + 12571545?
12571553
In base 13, what is 687b08 - -c?
687b17
In base 2, what is -111111110011101000101000011 - 0?
-111111110011101000101000011
In base 10, what is -7 + 3459657?
3459650
In base 7, what is -14325203 - -2320?
-14322553
In base 11, what is 4965678 - -5?
4965682
In base 14, what is 109 - -82db01?
82dc0a
In base 10, what is 12513 - -5320?
17833
In base 4, what is -1233221 + -31331312?
-33231133
In base 16, what is -2 - 281cfa?
-281cfc
In base 9, what is -25264684 + -3?
-25264687
In base 16, what is 25 - -72cf26?
72cf4b
In base 11, what is -402a758 - -14?
-402a744
In base 16, what is -126 - 7bce9?
-7be0f
In base 2, what is -110110000101110101010011 - 0?
-110110000101110101010011
In base 4, what is 10 - -110213002203?
110213002213
In base 13, what is -b810 - 62b?
-c13b
In base 9, what is 1622 - -108028?
110651
In base 9, what is -3874 + 17675?
13701
In base 11, what is 27463975 + -3?
27463972
In base 2, what is 1001000010011 - -11010?
1001000101101
In base 8, what is 1 + 102662331?
102662332
In base 11, what is 299a608 - 7?
299a601
In base 11, what is 15913 - 1704?
1420a
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12a133a00
In base 6, what is -3 - 3052015553?
-3052020000
In base 13, what is 10b7 + -96c?
448
In base 7, what is -53413 + -16546?
-103262
In base 8, what is 167 - -140726?
141115
In base 3, what is -1020100 - -10021211?
2001111
In base 10, what is 15789022 + 5?
15789027
In base 16, what is 4fc6115 - 0?
4fc6115
In base 11, what is -2156 + -a8265?
-aa410
In base 12, what is -12407 - 1577?
-13982
In base 16, what is 12203 - d?
121f6
In base 6, what is 12 + -4041325?
-4041313
In base 10, what is -52148 - 80?
-52228
In base 16, what is 1344 - -7871?
8bb5
In base 14, what is 238 - -13685?
138bd
In base 10, what is -5727401 + 3?
-5727398
In base 4, what is -112333122021 + 11?
-112333122010
In base 7, what is -321 + 205354?
205033
In base 5, what is 22141431 + 10024?
22202010
In base 13, what is -285956 + -1c?
-285975
In base 6, what is 225444511 + -10?
225444501
In base 6, what is 2054311235 - -4?
2054311243
In base 14, what is 1c131c2b - 3?
1c131c28
In base 6, what is -1323542 + -10255?
-1334241
In base 13, what is -a2b - -a8a87?
a8059
In base 13, what is 5 + 14579?
14581
In base 6, what is -1455533405 + 3?
-1455533402
In base 15, what is -21b3 - -c5e?
-1454
In base 9, what is -164 - 145513?
-145677
In base 14, what is 17012b8 + -3?
17012b5
In base 6, what is -2300545 - -12?
-2300533
In base 6, what is -1044 - 53132?
-54220
In base 4, what is -1131 + 11322313303?
11322312112
In base 10, what is 4 - -3549435?
3549439
In base 2, what is 111110110100001110110 + -110?
111110110100001110000
In base 6, what is 25553310032 - 0?
25553310032
In base 3, what is 20112212021012 - -12?
20112212021101
In base 2, what is 10001 - 10000110001111110110101?
-10000110001111110100100
In base 6, what is -134315542511 + -4?
-134315542515
In base 4, what is -2033 + -301002023?
-301010122
In base 8, what is 3027 + -410442?
-405413
In base 7, what is 454653 + -4?
454646
In base 2, what is -100 - -1011010000101101000110?
1011010000101101000010
In base 2, what is 100100110 - 100011100100101110?
-100011100000001000
In base 9, what is -56740 - -230?
-56510
In base 9, what is 215 - 47444?
-47228
In base 5, what is -2014022143 - 3210?
-2014030403
In base 9, what is 366504708 - -5?
366504714
In base 15, what is 4b138 - 9?
4b12e
In base 13, what is 46a9 - 182?
4527
In base 13, what is 0 + 122ca521?
122ca521
In base 10, what is 7335 + 1938?
9273
In base 4, what is 3022011 - -133312?
3221323
In base 13, wh |
Star Wars: The Force Awakens is crushing it at the box office. It just passed Avatar for the top movie of all time when it comes to domestic sales. However, the movie had some major plot issues. There are two that really stood out to me. The first was the ever so controversial fight sequence between Rey and Finn and Kylo Ren. The second was how Rey and Finn escaped Jakku. Both of these sequences have parallels to the Star Wars Rebels “Siege of Lothal” television movie which made things much more believable while maintaining the core of the characters.
The Siege of Lothal television movie was written by co-executive producer Henry Gilroy and directed by Bosco Ng and Brad Rau. It shows the Rebels shifting to their larger role within the Rebellion as they attempt to rescue an Imperial officer in exchange for a list of Rebel sympathizers. However, Darth Vader has taken a personal interest in the Rebels and lays a trap that will make their rescue mission extremely difficult.
With this basic plot in mind, Darth Vader eventually confronts the Rebels leading to a lightsaber duel between Vader, Kanan Jarrus and Ezra. It is a very similar set-up to The Force Awakens. A much more experienced dark side user is combating two lesser-trained light side users. To give you a complete picture, Kanan never completed his Jedi training. He was forced to flee across the galaxy as a Padawan after the execution of his master during Order 66. Ezra Bridger is a young Force-sensitive boy who Kanan has taken under his wing and has begun training him.
If you thought the confrontation was similar, the actual combat is even more similar and something I would say is a running theme throughout Star Wars now, starting with the encounter with Darth Maul and later Count Dooku in Revenge of the Sith. Vader easily tosses Kanan aside with a Force push knocking him out of the battle early. The same thing happens when Rey is knocked aside by Kylo Ren. This allows both Vader and Ren to focus on their weaker opponents. While Ren is able to defeat his opponent, Ezra is saved by a recovering Kanan. Everything with both fight sequences is going great. It is all very believable and stays in tune with what we know about the characters. However, this is where the stories diverge. All of a sudden, Rey uses The Force to acquire Luke’s old lightsaber instead of the much stronger (at least from what we saw on-screen) Kylo Ren. The battle maintains Ren’s power and strength over Rey, pushing her to the brink. However, all of a sudden and completely out of nowhere, Ren makes a statement about training Rey. This is literally out of left field and, even after watching the movie twice, I am still baffled by this line of dialogue. This leads to Rey somehow becoming super-powered by The Force (think DBZ Super-Saiyan modes) and completely thrashing Ren like he was a youngling. Fortunately, the planet is exploding and saves Ren from complete destruction.
In contrast to the poor dialogue and unexplained Force empowerment of The Force Awakens, the sequence between Vader, Kanan, and Ezra in the Siege of Lothal is resolved through strategic thinking. Sabine expertly lands a couple of explosives onto the legs of an AT-AT and detonates them while Vader is standing underneath it. At the same time, Ezra and Kanan combine their Force abilities to push Vader further underneath the AT-AT. Not only is Vader caught off guard by the explosion, the battlefield awareness of Kanan allows them to gain the upper hand. However, this is Vader and a mere AT-AT is not going to stop him. He uses The Force to protect himself. This gives the Rebels enough time to escape the clutches of Vader. The characters remain true to themselves; they don’t have bizarre moments of epiphany, whether it is with The Force or randomly wanting to train people in the heat of battle.
After escaping Vader, the Rebels are still left without a plan to escape the besieged system. Luckily, off-planet communications haven’t been completely blocked off and the Rebels are able to reach out to Lando Calrissian who supplies them with equipment they use to devise a plan. It is a pretty brilliant plan too, relying on the fact that the Empire can’t be in all places at once. They are able to create a number of devices that mimic the signal of their ship. This will make the same signal appear in a number of different places across the planet. The Empire is forced to deploy a number of TIE fighters to obtain visuals on the various locations. Unfortunately, by the time they catch on to the strategy, the Rebels are able to calculate the jump to lightspeed and escape the blockade. It is a tight plot that expertly explains how they are able to escape the planet.
Contrast this with how Finn and Rey escape Jakku. They hijack the Millennium Falcon and then lead a pair of TIE fighters on a merry chase through and around an old Imperial Star Destroyer. After doing away with the TIE fighters, they just fly out into space on their merry way. Apparently, the TIE fighters weren’t communicating with the First Order Star Destroyer, The Finalizer, and providing updates on the whereabouts of the droid. They should have been, because Rey didn’t jam their transmissions. The actual troops stationed on The Finalizer must also be completely inept because they weren’t tracking the movements of ships exiting the system. However, Han Solo is able to find the Millennium Falcon as soon as it experiences a technical malfunction. It is just mind blowing how loosely explained these scenes are.
These are just two ways that Star Wars Rebels “Siege of Lothal” outclasses The Force Awakens when it comes to plot, believability, and staying true to the characters’ cores. Here is to hoping that Rian John will learn from the plot mistakes of The Force Awakens and take a cue from Star Wars Rebels for Star Wars Episode VIII, because we all want the best Star Wars experience possible!
(Visited 813 times, 1 visits today) |
Digestive disease management in Japan: a report on the 6th diagnostic pathology summer fest in 2012.
The 6th Diagnostic Pathology Summer Fest, held in Tokyo on August 25-26, 2012, opened its gates for everyone in the medical profession. Basic pathology training can contribute to the improvement of algorithms for diagnosis and treatment. The 6th Summer Fest with the theme 'Pathology and Clinical Treatment of Gastrointestinal Diseases' was held at the Ito International Research Center, The University of Tokyo. On August 25, 'Treatment of Early Gastrointestinal Cancer and New Guidelines' was discussed in the first session, followed by 'Biopsy Diagnosis of Digestive Tract: Key Points of Pathological Diagnosis for Inflammation and Their Clinical Significance' in the second session. On August 26, cases were discussed in the third session, and issues on pathological diagnosis and classification of neuroendorcrine tumor in the fourth session. The summaries of speeches and discussions are introduced along with the statements of each speaker. This meeting was not a formal evidence-based consensus conference, and 20 experts gave talks on their areas of specialty. Discussion was focused on how the management strategy should be standardized on the algorithm of patient care. |
using System.Reflection;
using System.Runtime.InteropServices;
[assembly: AssemblyTitle("Library")]
[assembly: AssemblyDescription("")]
[assembly: AssemblyConfiguration("")]
[assembly: AssemblyCompany("")]
[assembly: AssemblyProduct("")]
[assembly: AssemblyCopyright("")]
[assembly: AssemblyTrademark("")]
[assembly: AssemblyCulture("")]
[assembly: ComVisible(false)]
[assembly: AssemblyVersion("2.0.0.0")]
[assembly: AssemblyFileVersion("2.0.0.0")]
|
94 Ill. App.3d 295 (1981)
418 N.E.2d 891
THE PEOPLE OF THE STATE OF ILLINOIS, Plaintiff-Appellee,
v.
RICHARD CLARK, Defendant-Appellant.
No. 80-434.
Illinois Appellate Court First District (5th Division).
Opinion filed March 13, 1981.
Ralph Ruebner and Bradley S. Bridge, both of State Appellate Defender's Office, of Chicago, for appellant.
Richard M. Daley, State's Attorney, of Chicago (Marcia B. Orr and Pamela L. Gray, Assistant State's Attorneys, of counsel), for the People.
Judgment affirmed.
Mr. JUSTICE LORENZ delivered the opinion of the court:
On October 16, 1979, defendant entered a negotiated plea of guilty to an information charging him with three counts of armed robbery. (Ill. Rev. Stat. 1977, ch. 38, par. 18-2.) He was sentenced to concurrent terms of 9 years in the penitentiary. On February 8, 1980, the trial court denied defendant's motion to vacate the plea and defendant has appealed, contending the guilty plea was involuntary because the trial court denied his pro se motions for a bar association attorney and a continuance and failed, sua sponte, to admonish him about his constitutional right to proceed pro se as his own attorney.
When defendant's case was called, the following colloquy took place:
"Mr. BELL [assistant public defender]: * * * Mr. Clark presented to me this morning in the lockup of this courtroom a written motion. I indicated to Mr. Clark, at that time, I would not file that motion on his behalf. And he indicated that he wanted to file it in any event.
THE COURT: You are not making this motion, is that correct?
Mr. BELL: No.
*296 * * *
THE COURT: Motion for subpoena will be denied.
THE DEFENDANT: I also have this one.
THE COURT: Motion for bar association attorney; that motion is denied. Anything else?
THE DEFENDANT: I would like, on the record, the Public Defender and I have a communication problem, and I don't think I can get a fair trial.
THE COURT: Motion for bar association attorney is denied. All right.
You say you have a communication problem. What else do you have to say?
THE DEFENDANT: That is all.
THE COURT: Motion is denied. Anything further, now, before we start Jury selection?"
Mr. Bell then informed the court that defendant also had a list of the witnesses he wanted to call in his behalf, but defendant had left the list at the county jail; therefore defendant requested a continuance. The court denied this motion. When defendant began to interpose an objection, the court advised him to talk to his lawyer first.
Following a discussion off the record, the court was informed that there had been a conference with the State's Attorney regarding a negotiated disposition, and the court was requested to participate in a pretrial conference. After the court confirmed that it was defendant's wish that there be such a conference, the court admonished defendant that he had an absolute right to have a trial and told defendant that the court did not want defendant to feel coerced. Defendant responded, "I understand what you are telling me." After further admonitions defendant confirmed several times that he understood the proceedings and wished the conference to go forward.
Following the conference, defense counsel stated that there had been an agreement that in exchange for the plea of guilty the State would nolle prosequi two other cases, and the court dismissed those charges. The court, in compliance with Supreme Court Rule 402 (73 Ill.2d R. 402) proceeded to admonish defendant concerning his rights, elicited a factual basis for the plea, accepted the plea and in accordance with the plea agreement previously reached sentenced defendant to concurrent terms of 9 years in the penitentiary.
Defendant first contends that the trial court was required, under the circumstances presented, to advise defendant that he had a right to represent himself under Faretta v. California (1975), 422 U.S. 806, 45 L.Ed.2d 562, 95 S.Ct. 2525. Defendant points out that he had previously been represented by the public defender, who had withdrawn. He was then represented by two attorneys from a private law firm, who also *297 withdrew. From this, defendant argues he had difficulties getting along with his lawyers. Given the additional circumstances that the court denied defendant's request for a continuance and his pro se motion for a bar association attorney, defendant concludes that the plea of guilty was not voluntary.
1 We find no abuse in the trial court's denial of a continuance under these circumstances. (People v. Hahn (1980), 82 Ill. App.3d 173, 402 N.E.2d 895.) With respect to defendant's request for a bar association lawyer, we think this court could have concluded that like the request for continuance this was a dilatory tactic by defendant to delay his trial. (Cf. People v. Rivers (1978), 61 Ill. App.3d 376, 377 N.E.2d 1245.) In this regard we note that defendant's case had been pending for about 18 months before entry of the plea.
2, 3 We also conclude that, in the absence of any indication by defendant that he wished to discharge counsel and represent himself, the court was not required to admonish him concerning his rights to proceed pro se and to act as his own counsel. It is true that when a defendant expresses the desire to discharge counsel, the court must admonish him of his right to counsel as required by Supreme Court Rule 401(a) to insure that defendant's waiver of counsel is made with knowledge of the consequences. (Ill. Rev. Stat. 1979, ch. 110A, par. 401(a); People v. Johnson (1979), 75 Ill.2d 180, 387 N.E.2d 688.) However, we do not agree that the converse is true, that the court must advise a defendant of his constitutional right to proceed pro se as his own attorney where defendant has not requested such a right. Snead v. State (1979), 286 Md. 122, 406 A.2d 98, cited by defendant, is distinguishable, since in that case, the accused specifically requested to proceed without counsel. In State v. Garcia (1979), 92 Wash.2d 647, 600 P.2d 1010, cited by the State, the court found Faretta does not require that the trial court admonish defendant concerning his right to proceed pro se without counsel unless the defendant actually requests to be allowed to proceed pro se, and the court was under no obligation to advise him of his rights in this regard. In this case, defendant did not request to represent himself. (Cf. People v. Bell (1977), 49 Ill. App.3d 140, 145, 363 N.E.2d 1202.) We find that the court's failure to admonish defendant of his right to proceed pro se was not error. (People v. Slaughter (1980), 84 Ill. App.3d 88, 93, 404 N.E.2d 1058.) Furthermore, we have examined the record which affirmatively shows defendant's plea was voluntarily and intelligently made. People v. Wills (1975), 61 Ill.2d 105, 110-11, 330 N.E.2d 505.
Therefore, the judgment of the circuit court of Cook County is affirmed.
Judgment affirmed.
SULLIVAN, P.J., and MEJDA, J., concur.
|
A chapter of the Zeta Beta Tau fraternity has been placed on probation at Cornell University following an investigation into an alleged sex game where new members competed to sleep with the heaviest women they could for ‘points.’
After a slew of allegations throughout 2017, the university’s Fraternity and Sorority Review Board launched an investigation into the ZBT members’ behavior and practices at the Ivy league school located in upstate New York. Pledges were allegedly instructed to keep the contest a secret and, in the event of a tie, the new member who had sex with the heaviest partner won.
“The behavior that Zeta Beta Tau (ZBT) fraternity was recently found responsible for is abhorrent to me and antithetical to our values as a community. Behavior that degrades and dehumanizes women contributes to a climate and culture of tolerance for sexual violence,” Vice President for Student and Campus Life Ryan Lombardi told RT.com in a statement.
“While sanctions have been levied against this fraternity by the Fraternity and Sorority Review Board, the campus community will be watching to see whether the members of ZBT – as individuals and as a group – live up to their public pledge to demonstrate through our actions that this inexcusable behavior will not be tolerated,” Lombardi added.
“It is incumbent that all members of the Greek system and the campus as a whole challenge any form of sexual misconduct and the behaviors that foster it.”
Michigan student charged over 'peanut butter hazing' incident https://t.co/tbmRMQyZ1h — RT (@RT_com) April 11, 2017
The organization, which claims to be the world’s first Jewish fraternity, according to its website, was advised of the investigation in December. Zeta Beta Tau has reportedly been placed on "probationary recognition" for a period of two years following the investigation and has vowed to conduct a membership review and participate in a variety of educational programs about sexual violence.
“The IFC was appalled and disgusted by the activity described in the reports,” Interfraternity Council President Paul Russell said, as cited by The Cornell Sun. “The decision about the specific sanctions placed on ZBT was made jointly by administrators and IFC leadership in a review board hearing earlier this year after a hearing and a review of the allegations.”
The “allegations described are contrary to the values that Zeta Beta Tau Fraternity espouses and works in direct conflict with the beliefs and missions of the Kappa Chapter,” ZBT said in a statement as cited by The Cornell Sun, adding that, “this inexcusable behavior will not be tolerated.”
RT.com has contacted both Cornell University authorities and the Zeta Beta Tau fraternity for comment regarding the allegations and subsequent investigation.
The fraternity previously came under fire in January after its University of Michigan chapter was found to be in violation of rules regarding hazing, reports Detroit News. Zeta Beta Tau had already faced similar allegations at UM in 2005. In November, the fraternity's social activities were suspended amid multiple allegations of hazing, drugging and sexual misconduct by its members.
The so-called Greek Life system of fraternities and sororities at third-level educational institutions in the US has come under increased scrutiny in recent years following a number of high-profile cases of hazing, sexual misconduct. |
636 F.2d 1219
Simmonsv.E. W. Bliss Co., Inc.
79-1068
UNITED STATES COURT OF APPEALS Sixth Circuit
11/12/80
1
M.D.Tenn.
AFFIRMED
|
Male dominance, female mate choice, and intersexual conflict in the rose bitterling (Rhodeus ocellatus).
An intersexual conflict arises when males and females differ in their reproductive interests. Although experimental studies have shown that females often mate with dominant males, it may not always be in the interest of a female to do so. Here we investigated the impact of male dominance on female mate choice and offspring growth and survival in the rose bitterling (Rhodeus ocellatus), a freshwater fish with a resource-based mating system. Three experimental mating trials were conducted using males of known dominance rank, but with different levels of constraint on male behavior. Thus, females were able to choose among; (1) males that were isolated from each other; (2) males that could see and smell each other, but could not directly interact; (3) males that could interact fully. Using a combination of behavioral observation and parentage analyses it was shown that female preferences did not correspond with male dominance and that male aggression and dominance constrained female mate choice, resulting in a potential intersexual conflict. The survival of offspring to independence was significantly correlated with female mate preferences, but not with male dominance. A lack of strong congruence in female preference for males suggested a role for parental haplotype compatibility in mate choice. |
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Energy
Update: The Energy ON TO 2050 strategy paper is now available, and your feedback is welcome. Send your thoughts, ideas, or questions to [email protected].
Energy systems have significant influence over the economic and environmental health of the Chicago region. Development of the ON TO 2050 plan provides a good opportunity to examine the future of energy in a forward-looking but practical way. The previous long-range plan, GO TO 2040, emphasized the importance of increased energy efficiency and renewable energy generation. While both of these directions are expected to continue, a comprehensive examination of energy is warranted to clarify CMAP's role in forwarding energy issues, help stakeholders understand their role in a sustainable energy future, help meet local and regional greenhouse gas emissions targets, and contribute to other quality-of-life goals.
Understanding energy in the Chicago Region
The energy sector is complex. The organizational and regulatory framework that governs the generation, delivery, and use of energy is hindered by structural inefficiencies within the energy sector. Emerging trends, which pose potential opportunities for the region, also may experience conflict with this framework. Such trends include a steady shift away from fossil fuels to renewable energy sources and from centralized to decentralized energy systems, and significant technological advancements in building efficiency, "smart grid" applications, distributed generation and energy storage, and transportation energy. It is essential to remove barriers to innovative energy solutions to ensure a clean, smart, equitable, and resilient energy system for the region.
Energy Strategy for ON TO 2050
The Energy strategy paper will seek to identify barriers in the energy sector and prioritize potential near-term actions by both CMAP and entities with decision making authority, including local governments and utilities. As a comprehensive regional planning agency, CMAP also has the unique ability to identify and highlight areas where energy intersects with other topics; for the purposes of ON TO 2050, these topics include transportation technology, reinvestment and infill, housing, economic resilience, climate resilience, and water.
Public Engagement
CMAP is partnering with the Foresight Design Initiative to create the Energy strategy paper. Foresight will lead and convene a Resource Group, comprised of topical experts, in addition to several broader focus groups to prepare the Energy strategy paper, which is scheduled for release in Spring 2017. CMAP will also work with its Environment and Natural Resources Working Committee and other stakeholders as needed to shape its approach to energy in ON TO 2050. |
Fast fashion is everywhere. One second you’re browsing an online store for some new shoes; a minute later, those same exact shoes are staring you in the face in an ad on your Facebook.
Kind of rude. Pretty creepy. Basically inescapable.
But do you really know - or bother to check - where your cheap threads are coming from? Or do you just *add to cart* without really thinking?
Tips for shopping ethically Buy second hand / vintage. Cheap, unique and has no new impact on the environment or workers.
Buy less, but buy better: invest in fewer, good quality, and timeless pieces that you’ll wear for years.
Organise a clothes swap among friends
Take care of your clothes so they last longer
Mend or repurpose clothes before throwing them out
More fashion retailers in Australia are embracing transparency than ever before, but some companies are still failing to disclose the location of their factories - according to Oxfam’s annual “naughty and nice” list of retailers.
Cotton On, Big W and GAP were given kudos for recently revealing the factories where their clothes are made; Topshop, Uniqlo and The Just Group have remained on the “naughty” list.
Why transparency is important
In 2013, a building collapse that killed over 1100 people in Bangladesh served as a “wake up call” to the global fashion world - revealing an industry in tatters.
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Whatsapp Rescue workers continue their operations at a collapsed factory in Savar, 30 km outside Dhaka, Bangladesh.
It sparked awareness among consumers, and saw hundreds of companies sign the Accord on Fire and Building Safety in Bangladesh - a five year legally binding agreement to build a safe garment industry in Bangladesh.
Oxfam says that wake up call has been effective in shaking up the industry, but thinks there’s still plenty of retailers out there who are slacking off.
“Unless a company publishes the locations of its factories, there is still no way of checking if their clothing is being made under safe and fair conditions.
“And, workers can’t easily raise problems and get them fixed. There’s no other way out of the Naughty List.”
Why don’t brands say where their clothes are made?
Brands often have excuses for not publishing the location of their factories, Oxfam Australia Chief Executive Dr Helen Szoke told Hack.
“The reasons that they give us - are often along the lines of, ‘if we publish where our factories are, then people can go and steal our styles and ideas’.
Frankly, the whole of the fashion industry is based on copying so that's an empty threat.
“If everyone publishes where their factories are, it puts everyone on a level playing field. So that excuse can’t be given. We think that reason is rather thin.”
But Dr Szoke says you shouldn’t just boycott brands on the “naughty” list as protest. Rather, consumers should be more proactive and write to their favourite brands.
“We don’t want consumers boycotting brands on the “naughty” list. We want all of these retailers to improve these practices, so workers in Bangladesh have their rights protected.
“Instead, [consumers should] ask their favourite brand to publish where their factories are.There is nothing more powerful than consumer sentiment.”
Some of the retailers on the “naughty” list come with caveats - ASOS is on there for now, but “promises to publish” their factory locations soon, for example.
You can check out Oxfam’s full report card on each retailer in the list over here.
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Growing awareness
Oxfam’s list isn’t the only one shining a light on the ethics of the fashion industry.
There’s a bunch of resources out there with the same goal - like Good On You, an Australian app which assesses brands’ environmental impact, labour conditions, and animal treatment.
Baptist World Aid’s Behind The Barcode also has initiatives to end worker exploitation in the industry; they also have guides into ethical fashion and ethical electronics in Australia.
Meanwhile, Fashion Revolution’s social media campaigns and events reveal the faces behind the garment industry around the world, encouraging people to ask, #WhoMadeMyClothes?
Editor's note: a previous version of this story incorrectly published the number of casualties at the Rana Plaza factory complex in 2013, and also referred to the incident as a factory fire, rather than a building collapse. This story has been updated to reflect this. |
Q:
Does LeetCode ignore some semantic error in c++?
I just finished a question on LeetCode, check out this line: int visited[nums.size()] = {0};, apparently it's wrong because you can't statically initialize an array without specifying its size, but somehow it worked on LeetCode and I even submitted it. Can anyone explain what's going on here?
class Solution {
public:
vector<vector<int>> permuteUnique(vector<int>& nums) {
vector<vector<int>> result;
result.clear();
vector<int> v;
v.clear();
sort(nums.begin(), nums.end());
int visited[nums.size()] = {0};
helper(result, v, visited, nums);
return result;
}
void helper(vector<vector<int>> &result, vector<int> &v, int visited[], vector<int>& nums) {
if (v.size() == nums.size()) {
result.push_back(v);
return;
}
for (int i = 0; i < nums.size(); i++) {
if (visited[i] == 1) {
continue;
}
if (i > 0 && nums[i] == nums[i - 1] && visited[i - 1] == 0) {
continue;
}
visited[i] = 1;
v.push_back(nums[i]);
helper(result, v, visited, nums);
v.pop_back();
visited[i] = 0;
}
}
};
A:
leetcode uses g++ 5.4.0 compiler for C++ compilation. It supports variable length array definitions. After ISO C99 specification, arrays with variable length declarations are allowed. [Examples and Related Info]
|
/*
* ModeShape (http://www.modeshape.org)
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.modeshape.jcr;
import java.math.BigDecimal;
import java.util.Calendar;
import java.util.Date;
import java.util.Iterator;
import java.util.Set;
import javax.jcr.AccessDeniedException;
import javax.jcr.InvalidItemStateException;
import javax.jcr.Item;
import javax.jcr.ItemNotFoundException;
import javax.jcr.ItemVisitor;
import javax.jcr.Node;
import javax.jcr.NodeIterator;
import javax.jcr.PathNotFoundException;
import javax.jcr.Property;
import javax.jcr.RepositoryException;
import javax.jcr.ValueFormatException;
import javax.jcr.lock.Lock;
import javax.jcr.lock.LockException;
import javax.jcr.nodetype.ConstraintViolationException;
import javax.jcr.version.OnParentVersionAction;
import javax.jcr.version.VersionException;
import org.modeshape.common.annotation.Immutable;
import org.modeshape.common.annotation.NotThreadSafe;
import org.modeshape.common.util.CheckArg;
import org.modeshape.jcr.api.value.DateTime;
import org.modeshape.jcr.cache.CachedNode;
import org.modeshape.jcr.cache.MutableCachedNode;
import org.modeshape.jcr.cache.NodeKey;
import org.modeshape.jcr.cache.SessionCache;
import org.modeshape.jcr.value.BinaryValue;
import org.modeshape.jcr.value.Name;
import org.modeshape.jcr.value.Path;
import org.modeshape.jcr.value.PropertyFactory;
import org.modeshape.jcr.value.Reference;
import org.modeshape.jcr.value.ValueFactories;
import org.modeshape.jcr.value.ValueFactory;
import org.modeshape.jcr.value.basic.NodeKeyReference;
/**
* An abstract {@link Property JCR Property} implementation.
*/
@NotThreadSafe
abstract class AbstractJcrProperty extends AbstractJcrItem implements org.modeshape.jcr.api.Property, Comparable<org.modeshape.jcr.api.Property> {
@Immutable
private final static class CachedDefinition {
protected final PropertyDefinitionId propDefnId;
protected final int nodeTypesVersion;
protected CachedDefinition( PropertyDefinitionId propDefnId,
int nodeTypesVersion ) {
this.propDefnId = propDefnId;
this.nodeTypesVersion = nodeTypesVersion;
}
}
private final AbstractJcrNode node;
private final Name name;
private int propertyType;
private volatile CachedDefinition cachedDefn;
AbstractJcrProperty( AbstractJcrNode node,
Name name,
int propertyType ) {
super(node.session());
assert node != null;
assert name != null;
this.node = node;
this.name = name;
this.propertyType = propertyType;
}
final void setPropertyDefinitionId( PropertyDefinitionId propDefnId,
int nodeTypesVersion ) {
this.cachedDefn = new CachedDefinition(propDefnId, nodeTypesVersion);
}
final void releasePropertyDefinitionId() {
this.cachedDefn = null;
}
/**
* Get the property definition ID.
*
* @return the cached property definition ID; never null
* @throws ItemNotFoundException if the node that contains this property doesn't exist anymore
* @throws ConstraintViolationException if no valid property definition could be found
* @throws InvalidItemStateException if the node has been removed in this session's transient state
*/
final PropertyDefinitionId propertyDefinitionId()
throws ItemNotFoundException, ConstraintViolationException, InvalidItemStateException {
CachedDefinition defn = cachedDefn;
NodeTypes nodeTypes = session.nodeTypes();
if (defn == null || nodeTypes.getVersion() > defn.nodeTypesVersion) {
Name primaryType = node.getPrimaryTypeName();
Set<Name> mixinTypes = node.getMixinTypeNames();
PropertyDefinitionId id = node.propertyDefinitionFor(property(), primaryType, mixinTypes, nodeTypes).getId();
setPropertyDefinitionId(id, nodeTypes.getVersion());
return id;
}
return defn.propDefnId;
}
/**
* Get the definition for this property.
*
* @return the cached property definition ID; never null
* @throws ItemNotFoundException if the node that contains this property doesn't exist anymore
* @throws ConstraintViolationException if no valid property definition could be found
* @throws InvalidItemStateException if the node has been removed in this session's transient state
*/
final JcrPropertyDefinition propertyDefinition()
throws ItemNotFoundException, ConstraintViolationException, InvalidItemStateException {
CachedDefinition defn = cachedDefn;
NodeTypes nodeTypes = session.nodeTypes();
if (defn == null || nodeTypes.getVersion() > defn.nodeTypesVersion) {
Name primaryType = node.getPrimaryTypeName();
Set<Name> mixinTypes = node.getMixinTypeNames();
JcrPropertyDefinition propDefn = node.propertyDefinitionFor(property(), primaryType, mixinTypes, nodeTypes);
PropertyDefinitionId id = propDefn.getId();
setPropertyDefinitionId(id, nodeTypes.getVersion());
return propDefn;
}
return nodeTypes.getPropertyDefinition(defn.propDefnId);
}
final CachedNode cachedNode() throws ItemNotFoundException, InvalidItemStateException {
return node.node();
}
final MutableCachedNode mutable() {
return node.mutable();
}
final SessionCache sessionCache() {
return node.sessionCache();
}
final PropertyFactory propertyFactory() {
return node.session().propertyFactory();
}
final org.modeshape.jcr.value.Property property() throws ItemNotFoundException, InvalidItemStateException {
return cachedNode().getProperty(name, sessionCache());
}
final JcrValue createValue( Object value ) {
return new JcrValue(session().context().getValueFactories(), this.propertyType, value);
}
final JcrValue createValue( Object value,
int propertyType ) throws ValueFormatException {
try {
return new JcrValue(session().context().getValueFactories(), propertyType, value);
} catch (org.modeshape.jcr.value.ValueFormatException e) {
throw new ValueFormatException(e);
}
}
@Override
public JcrSession getSession() {
return node.getSession();
}
/**
* Checks that this property's parent node is not already locked by another session. If the parent node is not locked or the
* parent node is locked but the lock is owned by this {@code Session}, this method completes silently. If the parent node is
* locked (either directly or as part of a deep lock from an ancestor), this method throws a {@code LockException}.
*
* @throws LockException if the parent node of this property is locked (that is, if {@code getParent().isLocked() == true &&
* getParent().getLock().getLockToken() == null}.
* @throws RepositoryException if any other error occurs
* @see Node#isLocked()
* @see Lock#getLockToken()
*/
protected final void checkForLock() throws LockException, RepositoryException {
if (getParent().isLockedByAnotherSession()) {
Lock parentLock = this.getParent().getLock();
if (parentLock != null && parentLock.getLockToken() == null) {
throw new LockException(JcrI18n.lockTokenNotHeld.text(node.location()));
}
}
}
/**
* Verifies that this node is either not versionable or that it is versionable but checked out.
*
* @throws VersionException if the node is versionable but is checked in and cannot be modified
* @throws RepositoryException if there is an error accessing the repository
*/
protected final void checkForCheckedOut() throws VersionException, RepositoryException {
if (!node.isCheckedOut()) {
// Node is not checked out, so changing property is only allowed if OPV of property is 'ignore' ...
JcrPropertyDefinition defn = getDefinition();
if (defn.getOnParentVersion() != OnParentVersionAction.IGNORE) {
// Can't change this property ...
String path = getParent().getPath();
throw new VersionException(JcrI18n.nodeIsCheckedIn.text(path));
}
}
}
protected final void checkModifyPermission() throws AccessDeniedException {
session.checkPermission(node, ModeShapePermissions.SET_PROPERTY);
}
@Override
public final void accept( ItemVisitor visitor ) throws RepositoryException {
CheckArg.isNotNull(visitor, "visitor");
checkSession();
visitor.visit(this);
}
final Name name() {
return name;
}
@Override
Path path() throws RepositoryException {
return session().pathFactory().create(node.path(), name);
}
@Override
public int getType() throws RepositoryException {
checkSession();
return propertyType;
}
@Override
public final JcrPropertyDefinition getDefinition() throws RepositoryException {
checkSession();
return propertyDefinition();
}
@Override
public final String getName() {
return name.getString(namespaces());
}
@Override
public final AbstractJcrNode getParent() {
return node;
}
@Override
public final String getPath() throws RepositoryException {
return path().getString(namespaces());
}
@Override
public String getLocalName() {
return name.getLocalName();
}
@Override
public String getNamespaceURI() {
return name.getNamespaceUri();
}
@Override
public final boolean isModified() {
try {
checkSession();
CachedNode node = cachedNode();
return node instanceof MutableCachedNode && ((MutableCachedNode)node).isPropertyModified(sessionCache(), name);
} catch (RepositoryException re) {
throw new IllegalStateException(re);
}
}
@Override
public final boolean isNew() {
try {
checkSession();
CachedNode node = cachedNode();
return node instanceof MutableCachedNode && ((MutableCachedNode)node).isPropertyNew(sessionCache(), name);
} catch (RepositoryException re) {
throw new IllegalStateException(re);
}
}
@Override
public final boolean isNode() {
return false;
}
@Override
public final boolean isSame( Item otherItem ) throws RepositoryException {
checkSession();
if (otherItem instanceof Property) {
Property otherProperty = (Property)otherItem;
// The nodes that own the properties must be the same ...
if (!getParent().isSame(otherProperty.getParent())) return false;
// The properties must have the same name ...
return getName().equals(otherProperty.getName());
}
return false;
}
@Override
public void refresh( boolean keepChanges ) {
throw new UnsupportedOperationException();
}
@Override
public void remove() throws VersionException, LockException, ConstraintViolationException, RepositoryException {
checkSession();
checkNotProtected();
checkForLock();
checkForCheckedOut();
session.checkPermission(this, ModeShapePermissions.REMOVE);
AbstractJcrNode parentNode = getParent();
if (parentNode.isLockedByAnotherSession()) {
Lock parentLock = parentNode.getLock();
if (parentLock != null && !parentLock.isLockOwningSession()) {
throw new LockException(JcrI18n.lockTokenNotHeld.text(getPath()));
}
}
if (!parentNode.isCheckedOut()) {
throw new VersionException(JcrI18n.nodeIsCheckedIn.text(getPath()));
}
node.removeProperty(this);
}
private void checkNotProtected() throws RepositoryException {
if (this.getDefinition().isProtected()) {
throw new ConstraintViolationException(JcrI18n.propertyIsProtected.text(getPath()));
}
}
@Override
public abstract JcrValue[] getValues() throws ValueFormatException, RepositoryException;
@Override
public abstract JcrValue getValue() throws ValueFormatException, RepositoryException;
@SuppressWarnings( "deprecation" )
@Override
public void save() throws RepositoryException {
checkSession();
// This is not a correct implementation, but it's good enough to work around some TCK requirements for version tests
// Plus, Item.save() has been removed from the JCR 2.0 spec (and deprecated in JCR 2.0's Java API).
getParent().save();
}
@Override
public int compareTo( org.modeshape.jcr.api.Property that ) {
if (that == this) return 0;
try {
return this.getName().compareTo(that.getName());
} catch (RepositoryException e) {
throw new RuntimeException(e);
}
}
protected Node valueToNode( Object value ) throws RepositoryException {
ValueFactories factories = context().getValueFactories();
try {
if (value instanceof Reference) {
NodeKey key = null;
if (value instanceof NodeKeyReference) {
// REFERENCE and WEAKREFERENCE values are node keys ...
key = ((NodeKeyReference)value).getNodeKey();
} else {
throw new IllegalArgumentException("Unknown reference type: " + value.getClass().getSimpleName());
}
return session().node(key, null);
} else if (value instanceof String ) {
String valueString = (String) value;
// see MODE-2609 - some reference properties may have been incorrectly stored as strings, so try to resolve them
if (NodeKey.isValidFormat(valueString)) {
return session().node(new NodeKey(valueString), null);
}
}
// STRING, PATH and NAME values will be convertable to a Path object ...
Path path = factories.getPathFactory().create(value);
return path.isAbsolute() ? session().node(path) : session().node(getParent().node(), path);
} catch (org.modeshape.jcr.value.ValueFormatException e) {
throw new ValueFormatException(e.getMessage(), e);
}
catch (PathNotFoundException pathNotFound) {
//expected by the TCK
throw new ItemNotFoundException(pathNotFound.getMessage(), pathNotFound);
}
}
@Override
public String toString() {
ValueFactory<String> stringFactory = session().context().getValueFactories().getStringFactory();
StringBuilder sb = new StringBuilder();
try {
org.modeshape.jcr.value.Property property = cachedNode().getProperty(name, sessionCache());
sb.append(getName()).append('=');
if (isMultiple()) {
sb.append('[');
Iterator<?> iter = property.iterator();
while (iter.hasNext()) {
Object value = iter.next();
appendValueToString(stringFactory, sb, value);
if (iter.hasNext()) sb.append(',');
}
sb.append(']');
} else {
Object value = property.getFirstValue();
appendValueToString(stringFactory, sb, value);
}
} catch (RepositoryException e) {
// The node likely does not exist ...
sb.append(" on deleted node ").append(node.key());
}
return sb.toString();
}
private void appendValueToString( ValueFactory<String> stringFactory,
StringBuilder sb,
Object value ) {
if (value instanceof javax.jcr.Binary) {
sb.append("**binary-value-not-shown**");
} else {
sb.append(stringFactory.create(value));
}
}
@Override
public <T> T getAs( Class<T> type,
int index ) throws IndexOutOfBoundsException, ValueFormatException, RepositoryException {
checkSession();
Object value = property().getValue(index);
Object convertedValue = null;
try {
if (String.class.equals(type)) {
convertedValue = context().getValueFactories().getStringFactory().create(value);
} else if (Long.class.equals(type)) {
convertedValue = context().getValueFactories().getLongFactory().create(value);
} else if (Boolean.class.equals(type)) {
convertedValue = context().getValueFactories().getBooleanFactory().create(value);
} else if (Date.class.equals(type)) {
Calendar calendar = context().getValueFactories().getDateFactory().create(value).toCalendar();
convertedValue = calendar.getTime();
} else if (Calendar.class.equals(type)) {
convertedValue = context().getValueFactories().getDateFactory().create(value).toCalendar();
} else if (DateTime.class.equals(type)) {
convertedValue = context().getValueFactories().getDateFactory().create(value);
} else if (Double.class.equals(type)) {
convertedValue = context().getValueFactories().getDoubleFactory().create(value);
} else if (BigDecimal.class.equals(type)) {
convertedValue = context().getValueFactories().getDecimalFactory().create(value);
} else if (java.io.InputStream.class.equals(type)) {
BinaryValue binary = context().getValueFactories().getBinaryFactory().create(value);
convertedValue = binary.getStream();
} else if (javax.jcr.Binary.class.isAssignableFrom(type)) {
convertedValue = context().getValueFactories().getBinaryFactory().create(value);
} else if (Node.class.equals(type)) {
convertedValue = valueToNode(value);
} else if (NodeIterator.class.equals(type)) {
convertedValue = new JcrSingleNodeIterator((AbstractJcrNode)getNode());
} else {
throw new ValueFormatException(JcrI18n.unableToConvertPropertyValueAtIndexToType.text(getPath(), index, type));
}
} catch (org.modeshape.jcr.value.ValueFormatException e) {
throw new ValueFormatException(e);
}
return type.cast(convertedValue);
}
}
|
Does Doppler-detected fetal movement decrease the incidence of nonreactive nonstress tests?
To determine whether a decreased incidence of nonreactive nonstress tests (NSTs) in antepartum testing was attributable to the addition of fetal movement detection to the standard NST. Monitors with standard fetal heart rate recording capabilities were used, as were new monitors producing a Doppler-detected recording of fetal movement (NST-fetal movement). Cross-sectional retrospective analysis of NST results was carried out by chi 2. Comparison of the 10-month period before fetal movement detection to the 10 months including NST-fetal movement monitoring showed a significant decrease in nonreactive NSTs from 5.7% to 3.3% (chi 2 = 61.7, 95% confidence interval [CI] 0.97-0.98). This reduction in nonreactive tests disappeared (3.3% to 5.1%) when the NST-fetal movement-capable monitors were no longer available (chi 2 = 24.2, 95% CI 1.01-1.03). Nonreactive NSTs decreased by 58% with the introduction of fetal movement monitoring in our antepartum testing center and increased when the NST-fetal movement-capable monitors were removed. A reduced incidence of nonreactive NSTs associated with NST-Doppler-detected fetal movements should effect a savings in both time and resources. |
1. Field of the Invention
The present invention relates to mask control devices for controlling mask functions of LSI (Large Scaled Integrated circuit) testers, which can carry out a function test of LSIs designed according to an LSSD (Level Sensitive Scan Designing) rule.
2. Background Art
When testing logic functions of LSIs designed according to LSSD rules, a function test including a random pattern test and a serial pattern test is carried out on the LSIs by an LSI tester.
In the random pattern test, random pattern input waveform data are supplied to input terminals of a DUT (Device Under Test). Random pattern output waveform data are then obtained from output terminals of the DUT and the waveform data thus obtained are then compared with random pattern expected waveform data which are previously stored in a memory of the LSI tester.
In the serial pattern test, input waveform data are supplied to the DUT so that a shift register is formed by flip-flops which are internal circuit elements of the DUT, and data shift operation is carried out by the shift register. Serial pattern output waveform data are then obtained from one or more specified output terminals of the DUT as a result of the data shift operation. The serial pattern output waveform data thus obtained are compared with serial pattern expected waveform data which are previously stored in a memory of the LSI tester.
In the above tests, if the logical function of the DUT has no defect, the output waveform data obtained from the DUT are in accord with the expected waveform data. Therefore, the correctness of the logical function can be Judged by comparing the output waveform data with the expected waveform data.
However, there are cases in which the expected waveform data cannot be determined or are not necessary with respect to a part of the output waveform data. In such a case, it is necessary to control the LSI tester so that the comparison result regarding such output waveform data are not used for judging the correctness of the logic function of the DUT. In order to carry out this control, a mask control device is employed in LSI testers.
FIG. 5 is a block diagram showing the configuration of a conventional mask control device.
As shown in FIG. 5, an address generator 11 can selectively generate either one of a random pattern address or a serial pattern address. Furthermore, the address generator 11 can generate a pattern mode signal indicating which test is to be carried out, a random pattern test or a serial pattern test.
The random pattern address (referred to as "random address" hereinbelow) is supplied to n expected waveform random pattern memories 12.sub.1 -12.sub.n and n mask waveform random pattern memories 13.sub.1 -13.sub.n.
The serial pattern address (referred to as "serial address" hereinbelow) is supplied to m expected waveform serial pattern memories 14.sub.1 -14.sub.m and m mask waveform serial pattern memories 15.sub.1 -15.sub.n.
The expected waveform random pattern memories 12.sub.1 -12.sub.n store n random pattern expected waveform data to be used for the random pattern test. The LSI tester has n tester pins, some of which are connected to terminals of the DUT. The n random pattern waveform data stored in the memories 12.sub.1 -12.sub.n respectively correspond to the tester pins 1-n.
The mask waveform random pattern memories 13.sub.1 -13.sub.n store n random pattern mask waveform data which respectively correspond to the random pattern expected waveform data stored in the memories 12.sub.1 -12.sub.n and are to be used for the random pattern test.
In the random pattern test, the output waveform data obtained from the output terminals of the DUT are inputted to some tester pins of the LSI tester and the output waveform data thus inputted are compared with the random pattern expected waveform data. The random pattern mask waveform data are used for a mask control when judging the correctness of the logical function of the DUT based on the results of the comparison.
More specifically, if random pattern mask waveform data, at a random address corresponding to a pin k, designates "mask", the comparison result corresponding to the pin obtained at the random address is not used for the judgement. If random pattern mask waveform data, at a random address corresponding to a pin k, designates "no mask", the comparison result corresponding to the pin at the random address is used for the judgement.
The expected waveform serial pattern memories 14.sub.1 -14.sub.m store m serial pattern expected waveform data which are to be used for the serial pattern test.
The mask waveform serial pattern memories 15.sub.1 -15.sub.m store m serial pattern mask waveform data corresponding to the m serial pattern expected waveform data.
In the serial pattern test, the serial pattern output waveform data obtained from the specified output terminals of the DUT are inputted to one or more tester pins of the LSI tester and the serial pattern output waveform data thus inputted are compared with the serial pattern expected waveform data. The serial pattern mask waveform data are used for a mask control when Judging the correctness of the logical function of the DUT based on the results of the comparison.
Serial pattern expected waveform data SO.sub.1 -SO.sub.m read out from the expected waveform serial pattern memories 14.sub.1 -14.sub.m are supplied to a pin selector 16. The pin selector 16 has n output terminals for outputting n bits serial pattern expected waveform data corresponding to the pins 1-n of the LSI tester and the pin selector stores select data designating the pins to which the m bits of the serial pattern expected waveform data are to be assigned. The select data are previously programmed by an external CPU (not shown in the figure). In this pin selector 16, a pin assigning operation is carried out on the m bits of the serial pattern expected waveform data based on the select data and the results are outputted as n bits serial pattern expected waveform data PSO.sub.1 -PSO.sub.n corresponding to the pins 1-n.
Serial pattern mask waveform data SM.sub.1 -SM.sub.m read out from the mask waveform serial pattern memories 15.sub.1 -15.sub.m are supplied to a pin selector 17. The pin selector 17 also has n output terminals for outputting n bits serial pattern mask waveform data corresponding to the pins 1-n and the pin selector stores select data designating the pins to which the m bits of the serial pattern mask waveform data are respectively to be assigned. In this pin selector 17, a pin assigning operation which is the same as that of the pin selector 16 is carried out on the m bits serial pattern mask waveform data based on the select data and the results are outputted as n bits serial pattern mask waveform data PSM.sub.1 -PSM.sub.n corresponding to the pins 1-n.
The pin selectors 16 and 17 receive the pattern mode signal. The pin assigning operation described above is carried out when a serial pattern test is designated by the pattern mode signal.
Furthermore, the pin selector 16 has a function for outputting serial mode designating signals SP.sub.1 -SP.sub.n corresponding to the pins 1-n during the serial pattern mode.
FIG. 6 shows an example of the pin selector 16. This pin selector 16 has register groups R.sub.1 -R.sub.n which respectively correspond to the pins 1-n. Each one of the register groups R.sub.i (i=1-n) stores m bits select data which is programmed by the external CPU. The m bits select data stored in the register group R.sub.i are respectively supplied to AND gates AND.sub.i1 -AND.sub.im and an OR gate OR.sub.i. The output data of the OR gate OR.sub.i is supplied to an AND gate AND.sub.iO corresponding to the pin i.
The serial pattern expected waveform data SO.sub.1 -SO.sub.m are respectively supplied to the first input terminals of the AND gates AND.sub.i1 -AND.sub.im corresponding to the pin i. On the other hand, the m select data stored in the register group R.sub.i are respectively supplied to the second input terminals of the AND gates AND.sub.i1 -AND.sub.im. The AND gates AND.sub.i1 -AND.sub.im select the serial pattern expected waveform data SO.sub.1 -SO.sub.m according to the m select data supplied thereto and the selected result as the serial pattern expected waveform data PSO.sub.i.
The AND gates AND.sub.iO (i=1-n) receive the pattern mode signal. Each AND gate AND.sub.i outputs a serial mode designating signal SP.sub.i when the pattern mode signal indicating the serial pattern test is received and the data "H" is outputted by the OR gate OR.sub.i.
The configuration of the pin selector 17 is basically the same as that of the pin selector 16 but the pin selector 17 does not have the line for transmitting the pattern mode signal as shown in FIG. 6. Because the serial pattern mask waveform data SM.sub.1 -SM.sub.m respectively correspond to the serial pattern expected waveform data SO.sub.1 -SO.sub.m. If the pin selector 17 has a function for outputting the serial mode designating signals like the pin selector 16, the same serial mode designating signals may be outputted from the pin selectors 16 and 17 in most cases. Therefore, the function for outputting the serial mode designating signals is omitted in the pin selector 17.
In FIG. 5, the random pattern expected waveform data PO.sub.1 -PO.sub.n which are read out from the expected waveform random pattern memories 12.sub.1 -12.sub.n are supplied to an input port A of a selector 18. On the other hand, the serial pattern expected waveform data PS.sub.1 -PS.sub.n corresponding to the pins 1-n are supplied from the pin selector 16 to an input port B of the selector 18. The input data of the input port A or the input data of the input port B are selected by the selector based on the serial mode designating signals SP.sub.1 -SP.sub.n, and the selected data are outputted from the output terminal Q. The output data of the selector 18 are supplied to an input port A of a comparator 20 and thereby compared with the output data of the DUT which are inputted via an input port B.
The random pattern mask waveform data PM.sub.1 -PM.sub.n read out from the mask waveform random pattern memories 13.sub.1 -13.sub.n are supplied to an input port A of a selector 19. On the other hand, the serial pattern mask waveform data PSM.sub.1 -PSM.sub.n outputted by the pin selector 17 are supplied to an input port B of the selector 19. The input data of the input port A or the input data of the input port B are selected by the selector based on the serial mode setting signals SP.sub.1 -SP.sub.n and the selected data are outputted from the output terminal Q.
The output data of the selector 19 are supplied to a mask control circuit 21. The mask control circuit 21 masks the results of the comparison obtained from the comparator 20 based on the mask waveform data outputted from the selector 19.
FIGS. 7A and 7B shows an example of a waveform data and a command used for a function test of a DUT which is designed according to a LSSD rule. FIG. 8 shows an operation in which the waveform data read out from the random pattern memories and the serial pattern memories in the function test. FIG. 9 is a time chart showing the operation of the mask control device executing the function test. The operation of the mask control device will be described with reference to these drawings.
FIG. 7A shows the random pattern expected waveform data, and the random pattern mask waveform data, and the address generation control command at the random addresses (0)-(4) which are programmed in the random pattern memories 12.sub.1 and 13.sub.1.
In FIG. 7A, the random pattern expected waveform data at the random address (0) is "H" and the random pattern mask waveform data at the random address (0) is "no mask" which means that the comparison result regarding the waveform data is not to be masked. The address generation control command at the random address (0) is "no control" which means that no address control is required.
At the random address (1), the expected waveform data is "L", and the mask waveform data is "no mask", and the address generation control command is "no control".
At the random address (2), the expected waveform data is "H", and the mask waveform data is "mask" which means that the comparison result regarding the expected waveform data is to be masked, and the address generation control command is "no control".
At the random address (3), the expected waveform data is "L", and the mask waveform data is "no mask", and the address generation control command designates that the LSSD operation is to be carried out and the loop start address is to be set to (0)-(3).
At the random address (4), the expected waveform data is "H", and the mask waveform data is "no mask", and the address generation control command designates the end of the test.
On the other hand, FIG. 7B shows the serial pattern expected waveform data and the serial pattern mask waveform data corresponding to the serial addresses (0)-(4) which are programmed in the serial pattern memories 14.sub.1 and 15.sub.1.
In this example, at the serial address (0), the expected waveform data is "H", the mask waveform data is "no mask". At the serial address (1), the expected waveform data is "L", and the mask waveform is "mask". At the serial address (2), the expected waveform data is "L", and the mask waveform data is "mask". At the serial address (3), the expected waveform data is "H", and the mask waveform data is "mask".
FIG. 8 shows the random pattern memories 12.sub.1 -12.sub.n and 13.sub.1 -13.sub.n, and the serial pattern memories 14.sub.1 -14.sub.m and 15.sub.1 -15.sub.m, in some of which the waveform data described above are stored.
In this example, the select data are set in the pin selectors 16 and 17 so that the serial pattern waveform data SO.sub.1 and SM.sub.1 shown in FIG. 7B are assigned to the pin 1, for example.
The reason the pin selectors 16 and 17 are provided at the output stage of the serial pattern waveform data is as follows:
The number of the bits of the serial pattern waveform data SO.sub.1 -SO.sub.m and SM.sub.1 -SM.sub.m does not always correspond to that of the random pattern waveform data PO.sub.1 -PO.sub.n and PM.sub.1 -PM.sub.n. In most cases, the relationship of m>n or m<n exists. Therefore, the pin selectors 16 and 17 are provided to control the bit positions of the n bits output data at which the m bits serial data are outputted.
When the function test of a DUT starts, the random address, and the serial address, and the pattern mode signal are generated by the address generator 11 in real time, for example, as shown in FIG. 9.
In the example shown in FIG. 9, the pattern mode signal designating a random pattern mode is outputted by the address generator 11 while the random addresses (0)-(2) are sequentially outputted.
In this period, the random addresses (0)-(2) are sequentially supplied to the random pattern memories 12.sub.1 -12.sub.n and 13.sub.1 -13.sub.n which respectively store the random pattern expected waveform data and the random pattern mask waveform data as shown in FIG. 8. As a result, the random pattern expected and mask waveform data corresponding to the random address (0)-(2) are sequentially read out from the random pattern memories, and the data thus read out are supplied to the pin selectors 18 and
Since the pattern mode signal designates the random pattern mode, no serial mode designating signal is outputted by the pin selector 16. Therefore, the random pattern expected and mask waveform data from the random pattern memories are supplied to the comparator 20 and the mask control circuit 21.
Next, the pattern mode signal is changed so as to designate the serial pattern mode. While the pattern mode signal is designating the serial pattern mode, the serial addresses (0)-(3) are sequentially supplied from the address generator 11 to the expected waveform serial pattern memories 14.sub.1 -14.sub.m and the mask waveform serial pattern memories 15.sub.1 -15.sub.m which respectively store the serial pattern expected waveform data and the serial pattern mask waveform data as shown in FIG. 8. On the other hand, the random address supplied to the random pattern memories are fixed to (3) during the serial pattern mode.
During the serial pattern mode, the serial pattern expected and mask waveform data corresponding to the serial addresses (0)-(3) are sequentially read out from the serial pattern memories and the data thus read out are supplied to the pin selectors 16 and 17.
In each pin selector, each bit of the waveform data is assigned to a pin designated by the select data and the result of this pin assigning operation is outputted as the n bits serial waveform data PSO.sub.1 -PSO.sub.n or PSM.sub.1 -PSM.sub.n which corresponds to the assigned pin. The serial waveform data PSO.sub.1 -PSO.sub.n or PSM.sub.1 -PSM.sub.n are supplied to the selectors 18 and 19.
In this example, the select data are set in the pin selectors 16 and 17 so that the serial pattern waveform data SO.sub.1 and SM.sub.1 shown in FIG. 7B are assigned to the pin 1 as described above. Therefore, the serial pattern waveform data are outputted as the serial pattern expected and mask waveform data PSO.sub.1 and PSM.sub.1.
In selectors 18 and 19, the random pattern expected and mask waveform data or the serial pattern expected and mask waveform data are selected based on the serial mode designating signals SP.sub.1 -SP.sub.n corresponding to the pins 1-n.
More specifically, when a serial mode designating signal SP.sub.k is "H", the serial pattern expected and mask waveform data are selected by the selectors for the pin k. When a serial mode designating signal SP.sub.k is "L", the random pattern expected and mask waveform data are selected by the selectors for the pin k.
In this example, since the serial pattern expected and mask waveform data corresponding to the pin 1 are outputted from the pin selectors 16 and 17, the serial mode designating signal SP.sub.1 corresponding to the pin 1 and having the level "H" is supplied from the pin selector 16 to the selectors 18 and 19. Therefore, the serial pattern expected and mask waveform data are selected by the selectors 18 and 19 for the pin 1.
The n bits expected waveform data selected by the selector 18 may include the serial pattern expected waveform data corresponding to the specified pins (in this case, the pin 1) and the random pattern expected waveform data corresponding to the other pins. The expected waveform data thus selected are supplied to the comparator 20.
The n bits mask waveform data selected by the selector 19 may include the serial pattern mask waveform data corresponding to the specified pins (in this case, the pin 1) and the random pattern mask waveform data corresponding to the other pins. The mask waveform data thus selected are supplied to the mask control circuit 21.
The mask control is then carried out on the comparison results of the comparator 20 by the mask control circuit 21 based on the mask waveform data supplied from the selector 19.
Meanwhile, the conventional mask control device requires three signal interfaces for transmitting the serial pattern expected waveform data, and the serial pattern mask waveform data, and the serial mode designating signals in order to supply the serial pattern expected and mask waveform data to the comparator and the mask control circuit. Thus, the conventional mask control device has problems in that the scale of the device is large and the configuration of the device is complex. The shrinkage and simplification are required with respect to the mask control device. |
Can You Catch Kanto Starter Pokemons?
Can Be Caught in the Wild as Rare Pokemons
The three starter Pokemon can be caught in the wild in specific locations. Be warned, these Kanto starters are extremely rare, and will need lengthy hunting!
Given to You By Certain NPCs as Gift Pokemon
Each Kanto starters can be given to you as a gift Pokemon via event sequence with NPCs. In order to receive your gift Pokemon, you will need to have certain number of Pokemons registered in your Pokedex.
What Are Secret Techniques?
Special Moves to Aid You in Your Adventure
Secret Techniques are special move taught to your Partner Pokemon; allowing you to perform special actions on the field. These moves can only be used outside of battle.
Facility in Fuchsia City Used to Transfer from Pokemon Go
Easy Way to Catch Rare & Powerful Pokemon
By connecting with Pokemon Go, you will be able to acquire rare and powerful Pokemon with ease. You will be able to retry the capture sequence as much as you like.
Where Can Catch Legendary Pokemons?
3 Legendary Bird Pokemon Available at End-Game
3 Legendary Bird Pokemon: "Articuno", "Zapdos" and "Moltres" will be available to capture near the end-game of the Pokemon Let's Go's storyline. Each Pokemon represents Ice, Thunder, Fire and are extremely powerful.
Capture Mewtwo After Beating Game
An Unknown Cave in Cerulean City will be unlocked after you become the Pokemon Champion. The mighty legendary Pokemon, Mewtwo awaits your challenge there. Be sure to prepare yourself thoroughly before challenging!
(C)2018 Pokémon.(C)1995-2018 Nintendo/Creatures Inc./GAME FREAK inc. All Rights Reserved.All trademarks, character and/or image used in this article are the copyrighted property of their respective owners. |
Urinary ascorbic acid--HPLC determination and application as a noninvasive biomarker of hepatic response.
A high-performance liquid chromatograph (HPLC) procedure has been developed for the determination of rat urinary ascorbic acid, a major metabolite of the hepatic glucuronic acid pathway. The presence of EDTA and HCl effectively inhibited degradation of ascorbic acid during the collection of urine specimens. The reliability of the procedure was demonstrated by its high recovery (90%), specificity (characteristic absorption maximum and discrimination from isoascorbic acid), and reproducibility (2-3% coefficient of variation). The usefulness of this assay as an indicator of hepatic response was demonstrated in preliminary experiments where increases in urinary ascorbic acid excretion were detected in male rats treated with PCB 126 (3,3',4,4',5-pentachlorobiphenyl) or PCB 105 (2,3,3',4,4'-pentachlorobiphenyl). The HPLC measurement also showed that the two PCB congeners differed markedly in their potency in stimulating urinary ascorbic acid excretion. For example, 10 micrograms/kg bw/day of PCB 126 was sufficient to cause a fourfold increase in urinary ascorbic excretion while 5000 micrograms/kg bw/day of PCB 105 was required for a sevenfold increase. In response to the administration of PCB 105 or PCB 126, urinary ascorbic acid appeared to increase to the same extent as increases in hepatic ethoxyresorufin O-deethylase (EROD) and UDP-glucuronosyltransferase (UGT) activities, and to a much higher extent than changes in liver weight and hematological and serum clinical chemical parameters. The sensitivity and specificity, the ease in obtaining timed specimens, and the noninvasive nature make this assay a useful biomarker of hepatic response in dose-finding and various acute and chronic studies. |
Today, Mozilla proudly celebrates the 10th anniversary of the Mozilla Developer Network, one of the richest and also one of the few multilingual resources on the Web for documentation. It started in February 2005, when a small team dedicated to the open Web took DevEdge (Netscape’s developer materials) and set out to create an open, free, community-built online resource for all Web developers. Just a couple of months later, on 23 July, 2005 the original MDN wiki site launched and has evolved steadily ever since for the convenience and the benefit of its users.
Today, ten years later, not only has the amount of documentation grown – 34,500 documents and climbing – but also MDN’s global volunteer community is bigger than ever. Currently, MDN has more than 4 million users and over 1000 volunteer editors per month creating and translating documentation, sample code, tutorials and other learning resources for all open Web technologies, including CSS, HTML, JavaScript and everything that makes the open Web as rich and versatile as it is.
For a wide range of Web developers, from learners to hobbyists to full-time professionals, MDN provides useful explanations for coding practice. It aims to inspire ideas, encourage collaboration, innovation and ultimately, foster the growth of the open Web. Moreover, as the digital industry flourishes and the demand for coding skills at young age rises, the importance of well-organized resources like MDN grows exponentially. That is why in 2014 MDN started to feed and expand all its learning pages into a “Learn the Web” area for beginning web developers, including a web terminology glossary, which MDN’s technical writers and volunteers will continue to develop over the next years.
All these efforts, which would not be possible without the active MDN volunteer base, are being greatly acknowledged by developers from all over the world who would not be doing what they do without MDN – or at least not as good.
Let’s hear it for MDN!
For more information:
Web: https://developer.mozilla.org/
MDN at 10: https://developer.mozilla.org/en-US/docs/MDN_at_ten
Twitter: https://twitter.com/MozDevNet
All graphics are also available in French, German, Italien, Spanish and Polish. |
Ballistic Advantage 12.3" 5.56 Hanson Carbine Barrel w/ Lo Pro, Performance Series .750"
This 5.56 chambered 12.3 inch Performance Series Barrel is machined from 4150 Chrome Moly Vanadium steel with a QPQ Corrosion Resistant Finish. Our Performance Series Barrels feature a FailZero Nickel Boron Coated Extended M4 Feed Ramp Extension. Comes with a pinned low pro gas block.
Length - 12.3"
Material - 4150 Chrome Moly Vanadium
Profile - Hanson
Finish - QPQ Corrosion Resistant
Gas System Length - Carbine-Length
Gas Block Journal - 1.03" Gas Block Seat for .750" Low Profile Gas Blocks Only
Twist Rate - 1:7 inches
Muzzle - 1/2x28 Threaded
Weight - 22oz
Other Info - HP and MPI Tested
"BA Hanson Series Barrels provide a lightweight feel without lightweight limitations. Every barrel in the BA Hanson Series was designed specifically to its length and caliber to yield the best results with every aspect taken into consideration. It is truly ideal for how the AR was meant to perform. With the BA Hanson Profile, we found a more efficient way to harness a bullets energy while limiting felt recoil considerably. The balanced feel and overall performance of a Hanson barrel will create a more organic relationship between you and your rifle. It will simply be an extension of you!" - Clint Hanson |
July 30, 2014
Kids more afraid at school these days, despite being much safer
Most people today feel a lot safer walking around their neighborhood compared to 20 or 25 years ago, when the rates of violent and property crime were at their peak. Likewise, people felt less safe in the early '90s than they did 20 years earlier, when crime was rising but not yet at its peak. People's fear of their surroundings moves along with the crime rate, with a delay of about two years. See this old post, which looks at data on fear from the General Social Survey, alongside homicide statistics.
Something different is going on with teenagers, though, and probably children as well. The Youth Risk Behavior Survey (site) is a national probability survey like the GSS, only given to high school students — a group that is usually passed over in surveys because there are stronger regulations on studying minors. They have asked questions about your experiences with violence at school, and whether you've skipped school because you feared for your safety there. High schoolers, like adults, live in a much safer world today, but unlike adults, they have only grown more afraid.
The question on school violence asks whether you have been in a physical fight on school property at least once in the past year. The one on fearfulness asks whether you've skipped school at least once in the past month because you feared for your safety at school, or going to or coming from school. These make a great comparison since the potentially dangerous location is the same for both, namely the school.
Both questions were first asked in 1993, and most recently in 2013. I've calculated the changes in both violence levels and fear levels between those two years (only, not mapping the micro-changes in between), separated by sex and race (white, black, and Hispanic/Latino).
I've expressed the change in standard deviations rather than how many percentage points they've gone up or down.* This treats proneness to violence and fearfulness as "bell curve"-shaped traits, as most psychological and behavioral traits are. Standard deviations give us a better sense of how far the average teenager has moved in one direction or the other. It's like knowing how many inches taller or shorter they have become, not merely whether they are some percent taller or shorter.
The data are shown in the graph below (click to enlarge).
Change in violence is on the horizontal, and all values are negative, reflecting the overall decline in violence since the peak in the early 1990s. The change of about 0.5 SD for males and about 0.3 for females is huge, given how quickly they have occurred. If safety were a kind of "height," then males today are about an inch and a half "taller," and females nearly one inch taller, than their counterparts 20 years ago.
Change in fear is on the vertical, and five of the six groups are more fearful than they used to be (i.e., lying within the positive upper half). Bear in mind that the question about fear is not simply about feeling afraid, but actually skipping school because you were afraid. Most groups are around 0.1 SD more fearful. If we thought of feeling secure as a kind of "height," it's as though kids today feel about a third of an inch shorter than they used to — despite being taller!
The males (squares) cluster off to the left, where violence levels have plummeted further and kids are only a bit more fearful than before. The females (triangles) cluster to the right, where violence hasn't plummeted by as much (since it was so low to begin with), and kids are more fearful than before — particularly for white females. There's no overall pattern for where the different races lie within the clusters. (Remember these are changes over time, not absolute levels, which do show race differences.)
Is there at least a positive correlation, despite the general confusion that is shown by most points revealing declines in violence and increases in fear? If groups who have seen larger declines in violence showed larger declines in fearfulness, then the points should slope up and to the right.
When you look within race and compare males to females, the lines connecting points of the same color slope up in two cases and is flat in the other. This is the only sign of kids being in touch with reality here: males have seen greater declines in violence than females, and show greater declines in fearfulness, controlling for race.
However, when you look within each sex cluster, the line running through the three races slopes down! Controlling for sex, the racial groups that have seen greater declines in violence show greater rises in fearfulness.
Just about every way you look at it, there is a profound disconnect between school violence and fear of school violence, among the very teenagers who spend their days there. What gives?
It's not the rise in school shootings, since kids don't purposefully skip school because they're afraid that today is going to be the day when that one kid snaps and shoots up the school. The question is phrased to ask about fear of garden variety fights, bullies, and so on (e.g., the part about being afraid going to or coming from school, where spree shootings would not apply).
It's not cocooning and social isolation either, as though they were afraid that no one would have their back in case a fight broke out. That would apply to adults as well — yet adults are cocooning and less fearful of their neighborhoods, as crime has fallen.
My only good hunch is that it's the coddling from helicopter parents and their agents in the school system, especially all this propaganda about the omnipresent menace of bullying. Grown-ups bombard kids with a constant stream of dire appeals to watch out for bullies, report the slightest hint of bullying — IF YOU STAY SILENT, MILLIONS COULD DIE — etc. This makes it sound like school is one great big death trap, and if every grown-up sends you the same message, you'll tend to trust their expert consensus.
Now, if kids were more in touch with their surroundings, perhaps they wouldn't be so naive about what the grown-ups were warning them about. Over-protectiveness prevents the kid from having an awareness of the school's true danger level (or for that matter, how dangerous his own neighborhood is), which would lead him to dismiss the paranoia of his parents.
Adults may be cocooners these days, but they do get out some of the time, albeit with their guard always up. They are keenly aware of how safe every place has become over time, in a way that their sheltered kids are not, for want of exposure to the outside world.
I think this pattern can be found everywhere in young people's lives today — safer environments, but greater anxiety. And for the same reasons — over-protective sheltering has disconnected the kid's inner view of the world, from the actual state of the world.
Your kid is decently less likely to come home from school with a black eye than he was 20-25 years ago, but they're way more stressed-out and fearful than teenagers were back then. "No problem," the parents assure us — just pump them full of mind-altering drugs to calm them down.
Two abominations do not cancel each other out, it just looks even more fucked-up. Some kid who lives in such a safe world, lumbering around like a lobotomy victim because without the drugs he'd feel overwhelmed by the stress of living in such a safe world.
But y'know, it's really no big deal if the kid is warped into a confused, stressed-out, zombie-like freak — at least the parents feel relaxed that their pet child is all under control. Helicopter parents continue their glib, self-righteous boasting about their adherence to "good clean family values," while producing an entire generation of abominations against nature.
Pity the children, and punish the parents.
* This is La Griffe du Lion's "method of thresholds." Convert the fraction who meet some criterion into a Z-score, for each group, and subtract the Z-scores to see how far apart the means are.
3 comments:
Everybody seems to believe that the world has gotten more dangerous I still remember some middle-aged in a comment thread going on about how young college girls were being drugged by fraternity members.
There's more of a fear about stuff like that, where the threat is not in the person's own environment. But when it comes to their own lives, like walking around their neighborhood, they do report feeling a lot safer.
Here's an old post on how coverage of rape has been skyrocketing while actual rape rates have been plummeting. It's as of 2008, and would probably look worse today (although the NYT fucked up their search engine awhile ago, making it impossible to collect these data online anymore).
I bet the rising (or seemingly rising) number of school shootings is adding to this fear. They are heavily publicized, thus becoming a part of the collective consciousness of our present era. Meanwhile run-of-the-mill violence is declining. |
Motivation
Smartwatches are gaining in popularity as they are more affordable and offer good and easy APIs for developers. Moreover, they are easier to use in everyday actions. I published some time ago Open Sesame project that aims to enable the control of the doors using a Smartphones. In this project, I propose an application for pebble watches to control this system.
Now, when I back home from the grocery store with bags in my hands, I just use the watch while keeping my mobile device in my pocket :D
Description
Let's start with a video as it worst a 1000 words!
I will skip the details regarding the hardware setup and the Raspberry Pi configuration as they are explained in this project. The security aspect of this project is discussed in this article.
As illustrated in Figure 1, the application deployed in Pebble watch uses Bluetooth connection to the smartphone in order to access to the network. Nevertheless, this complication is mitigated by the Pebble SDK.
Figure 1
Software setup
The CLOUDPEBBLE IDE is a cool platform to develop Pebble applications. First you need to create a project and configure it. As illustrated in Figure 2, you need to specify the target platform. I used both aplite and basalt in this project.
Figure 2
One you project setting is complete, copy/paste the source from the github repository into your application sources.
The application is composed from a menu with two options. The first one is to open the door. the second one is to close it. this menu is available when you click on the "UP" button. When you click on the "Select" button, the application shows the status on the lock (Open or Close).
The application uses a WebSocket connection to access to the server deployed in the Raspberry Pi. The code below illustrate the WebSocket connection to the Raspberry Pi server.
... menu.on('select', function(e) { console.log('Selected item #' + e.itemIndex + ' of section #' + e.sectionIndex); console.log('The item is titled "' + e.item.title + '"'); var card = new UI.Card({ title:'Door Action', subtitle: e.item.title + ' operation in progress ...' }); card.show(); ws = new WebSocket('ws://YOUR-RASPBERRY-PI-IP:1337'); ws.onopen = function () { card.body( 'You are connected to Open Sesame System Server.'); if(e.itemIndex===0){ ws.send('22'); card.subtitle( "Door Opened"); status = "Open"; }else{ ws.send('2'); card.subtitle( "Door Closed"); status = "Close"; } }; ws.onerror = function (error) { card.body( 'Sorry, but there\'s some problem with your connection or the server is down.') ; }; }); menu.show(); }); ...
Conclusion
It is nice to have play with multiple technology to automate our homes. This project shows that the wearable devices may be used in a simple way to control stuffs in our modern smart homes. |
Apple Creek, Wisconsin
Apple Creek is an unincorporated community located in the towns of Grand Chute and Freedom in Outagamie County, Wisconsin, United States. It is in the Appleton, Wisconsin Metropolitan Statistical Area and the Appleton-Oshkosh-Neenah, Wisconsin Combined Statistical Area.
Geography
Apple Creek is located at (44.325833, -88.375). Its elevation is at 804 feet (245m).
Transportation
References
Category:Unincorporated communities in Wisconsin
Category:Unincorporated communities in Outagamie County, Wisconsin
Category:Appleton–Fox Cities metropolitan area |
In a move that will prevent foreign e-commerce players such as Amazon from expanding into India, the Government on Friday clarified that companies with foreign direct investment cannot sell their products online in India.
“E-commerce activities refer to the activity of buying and selling by a company through the e-commerce platform. Such companies would engage only in business-to-business (B2B) e-commerce and not in retail trading, inter-alia implying that existing restrictions on FDI in domestic trading would be applicable to e-commerce as well,” the Department of Industrial Policy and Promotion said.
As such, extant FDI policy does not permit FDI in B2C (business-to-consumers) e-commerce. The provisions are part of the notification by the industry department on the FDI policies for retail announced on Friday.
The new guidelines will also have a direct bearing on the plans of local e-commerce companies seeking strategic investment from foreign companies through the FDI route. The provision also applies to investments by foreign venture capital and private equity (PE) funds in such ventures.
The Department, while clarifying, said that e-commerce is a capital-intensive business. In many instances, foreign capital has come to the back end. But if the FDI in retail e-commerce is not allowed, it shuts the door for foreign capital in the future as PE players will not have any exit option.
“Retail trading, in any form, by means of e-commerce, would not be permissible, for companies with FDI, engaged in the activity of multi-brand retail trading,” it added.
Sectoral FII limit
On the sectoral FII limit for retail, the Department said the FDI limits for firms engaged in the activity of single-brand retail trading/ multi-brand retail trading. FII limits are governed by the relevant regulations on FII investments.
> [email protected] |
Here Comes 1969 Again
Back when the strongest substance in any major league clubhouse was brine (so as to toughen Nolan Ryan's finger against blisters), the New York Mets became champions of the baseball world. Presumably because it's the 40th anniversary of the 1969 triumph of triumphs, SNY is bringing back from Mets Classics[1] mothballs World Series Games Two through Five every night this week at 7:30, starting tonight. They make for fairly fascinating viewing[2] simply for television's sake. Throw in the Mets becoming champions of the baseball world, and ya think there's something better on?
Also this week, the Mets will win the 2006 National League East (Tuesday, 2 PM); beat the Giants on two balks and a blast (Wednesday, 1 PM); and ride two Robin Ventura grand slams to a doubleheader sweep over the Brewers (Thursday, 1:30 PM). Not that Mets Classics trend toward the predictable or anything.
Article printed from
Faith and Fear in Flushing:
http://www.faithandfearinflushing.com |
Since 1976,[^1^](#FN1){ref-type="fn"} the Supreme Court has consistently ruled that the death penalty is not inherently cruel and unusual and therefore does not categorically violate the Eighth Amendment's prohibition on the infliction of cruel and unusual punishments.[^2^](#FN2){ref-type="fn"} Yet litigation surrounding capital punishment continues at a high volume. The vast bulk of this litigation, over the past four decades, has tended to focus on what Justice Blackmun referred to as "tinkering with the machinery of death."[^3^](#FN3){ref-type="fn"} The consequence of this generation of tinkering has been the refinement of the process of selecting those eligible for the death penalty from the larger universe of those who commit homicide and the refinement of the process for sentencing convicted murderers to death. At this point in the history of the death penalty, therefore, the ostensible constitutionality of capital punishment, as a general proposition, is a given; nevertheless, specific modes of execution can in fact violate the Eighth Amendment's Cruel and Unusual punishments clause. Accordingly, our focus here is on modalities.
Any contemporary attention to a specific mode of execution will perforce focus on one form or another of lethal injection,[^4^](#FN4){ref-type="fn"} because nearly all executions in the United States occur by lethal injection. Thus, over the past decade, from 2009 through July 31, 2019, 364 inmates were put to death in the United States.[^5^](#FN5){ref-type="fn"} Of that total, 358 (or \> 98%) were executed by lethal injection. (Of the remaining six, one (in Utah) was killed by firing squad, and five (three in Virginia and two in North Carolina) were electrocuted.[^6^](#FN6){ref-type="fn"})
Driven in large part by the Supreme Court's unwillingness to revisit the general constitutional question involving the permissibility of capital punishment per se,[^7^](#FN7){ref-type="fn"} lawyers representing death row inmates have often shifted their focus to safeguarding their clients' interests by seeking to insure that the manner of inflicting death is humane; and because the dominant manner of inflicting death is lethal injection, the drug (or drugs) employed in the process -- including the method of obtaining them, the manufacturing process, their age and shelf life, and so forth -- as well as the mechanism by which the drugs are introduced into the condemned, have been litigated extensively.[^8^](#FN8){ref-type="fn"} Over the last two decades, hundreds of challenges to the lethal protocol,[^9^](#FN9){ref-type="fn"} pursued in the Supreme Court, the lower federal courts, and some state courts, have given rise to a rich body of case-law. This article examines this body of doctrine and comments on two defects in the decisional law pertaining to these lethal injection challenges, with a particular focus on one of these defects that has not received an appropriate degree of attention in either the case-law or the academic literature.
1. The Two Key Principles of Execution Protocol Litigation {#s1}
==========================================================
While litigation surrounding lethal injection is comparatively recent, this general category of constitutional attack (*i.e.*, challenges to a particular execution modality) has relatively deep roots, and these roots are evident in contemporary doctrine. Thus, challenges to a specific execution protocol have reached the Supreme Court of the United States since the 19th century. In the first such case, *Wilkerson v Utah*,[^10^](#FN10){ref-type="fn"} the Court held that death by firing squad did not constitute cruel and unusual punishment. (Indeed, the Court has never deemed a particular method of execution to be cruel and unusual.) *Wilkerson* appears to be the first instance where the Court attempted to draw the constitutional line between an execution method consistent with the Eighth Amendment and one that runs afoul:
*Difficulty would attend the effort to define with exactness the extent of the constitutional provision which provides that cruel and unusual punishments shall not be inflicted; but it is safe to affirm that punishments of torture, such as those mentioned by the commentator referred to, and all others in the same line of unnecessary cruelty, are forbidden by that emendment \[sic\] to the Constitution.*[^11^](#FN11){ref-type="fn"}
Out of this passage emerged the distinction between permissible modes of execution, on the one hand, and torture, on the other.[^12^](#FN12){ref-type="fn"} Put somewhat differently, unless the infliction of death involves torture, it is permissible, as a method. It follows that carrying out a lawfully authorized death sentence does not amount to cruel and unusual punishment unless it is accompanied by torture, where torture is defined as the infliction of gratuitous pain, *i.e*., any pain beyond that which is necessary to accomplish the death of the execution victim.[^13^](#FN13){ref-type="fn"} Put yet another way, although it may be the case that any method of putting someone to death involves some potential degree of pain, that possibility does not render the challenged method inherently cruel and unusual; rather, the method is constitutional unless death could be accomplished without inflicting at least the degree of pain associated with the challenged method. (While this manner of stating the principle seems to suggest an inherent degree of comparison between two or more methods, we suggest below the principle can be applied in many cases by focusing solely on the procedure being employed, rather than comparing it to other alternatives.)
Following the reinstatement of the death penalty in 1976, the first state to carry out an execution was Utah, which executed Gary Gilmore by firing squad in 1977. In 1982, Texas became the first state to put an inmate to death by lethal injection, but until the early 1990s, electrocution was as common as lethal injection as an execution method (with some states, including Mississippi, California, Arizona, and North Carolina continuing to use the gas chamber). By the mid to late 1990s, however, lethal injection was the dominant mode for carrying out death sentences in the US.[^14^](#FN14){ref-type="fn"}
For more than twenty years, death penalty states with a lethal injection protocol used a three-drug cocktail to carry out the execution.[^15^](#FN15){ref-type="fn"} The first drug, sodium thiopental, a barbiturate, rendered the inmate unconscious; the second, pancuronium bromide, acted as a paralytic agent; and the third, potassium chloride, induced cardiac arrest. Based on two ideas -- that the sodium thiopental could wear off before the execution was complete, and that the paralytic agent played no essential role in causing death and, to the extent it did contribute to an inmate's death, it caused excessive pain (hence, torture) in doing so -- challenges to this three-drug protocol began to reach the federal courts around 2005 and 2006. In 2008, the Supreme Court decided *Baze v Rees*[^16^](#FN16){ref-type="fn"} and affirmed the constitutionality of Kentucky's use of this three-drug cocktail for carrying out lethal injections.
Beginning in *Baze*, and continuing through its decision in *Glossip v Gross*,[^17^](#FN17){ref-type="fn"} which rejected a challenge to Oklahoma's lethal injection protocol, and *Bucklew v Precythe*,[^18^](#FN18){ref-type="fn"} which rejected a challenge to Missouri's protocol, the Court reaffirmed the central idea dating to *Wilkerson* -- that a particular method of execution violates the Eighth Amendment's cruel and unusual punishments clause only if it inflicts gratuitous pain -- and it also developed another.
This second idea, which had its roots in *Rees* but did not mature until later, is the requirement that any inmate challenging a particular method of execution as likely to inflict unnecessary pain must identify a "feasible, readily implemented" alternative method that would "significantly reduce a substantial risk of severe pain."[^19^](#FN19){ref-type="fn"} Put more directly, an inmate challenging a method of execution as likely to cause gratuitous pain cannot prevail on the Eighth Amendment claim without articulating a better (*i.e.*, less painful) mode. Inmates, in short, must prescribe their own methods of death before they can prevail on a claim that a state's chosen method amounts to cruel and unusual punishment.[^20^](#FN20){ref-type="fn"}
2. Legal and Scientific Shortcomings of the Court's Method-of-Execution Jurisprudence {#s2}
=====================================================================================
There is no comprehensive definition of what makes a legal rule defective, but one certain defect is when an ostensible rule applies to only a single and narrow category of disputes.[^21^](#FN21){ref-type="fn"} The Supreme Court's peculiar notion that a punishment cannot be cruel and unusual unless an inmate challenging that punishment can himself articulate an analogous punishment that would result in less pain is subject to criticism for this very reason: because the doctrine rests on no generalizable principle.[^22^](#FN22){ref-type="fn"}
In other constitutional contexts (*i.e.*, contexts other than capital punishment), there is no such burden placed on someone asserting her constitutional rights. A prisoner, for example, who argues that guards used excessive force to subdue her is not obligated to demonstrate precisely how she could have been subdued using lesser force. Indeed, it is noteworthy that the Court's excessive force doctrine, which is also rooted in the Eighth Amendment's Cruel and Unusual Punishments clause, is similar to its death penalty jurisprudence in terms of its prohibition on the infliction of unnecessary pain, yet prisoners raising a claim of excessive force are not estopped from prevailing even where they are unable to identify available alternative uses of force.[^23^](#FN23){ref-type="fn"}
Similarly, someone who sues a police force or officer for brutality following an arrest or detention does not have to show how the officer could have done the job without using brutality; the burden is merely to show that the force was excessive, *i.e.*, more than necessary.[^24^](#FN24){ref-type="fn"} If, for example, a fleeing suspect was immobilized by a taser, then kicking that suspect following immobilization was excessive, because it was unnecessary to the objective. By way of analogy, an inmate who challenges a three-drug protocol by arguing the intermediate drug (the paralytic agent) is unnecessary is making a similar claim. Yet the additional burden -- that the inmate provide evidence comparing the risk of pain under one protocol with the risk of pain under another -- is a distinctive and unique burden. Measured by this standard, the burden on inmates facing execution to delineate an execution protocol that would not create a similar risk of infliction of gratuitous pain is defective because it is a *sui generis* burden for a narrow class of inmates raising an otherwise generalizable Eighth Amendment claim.
That defect, however, is not the worst problem with this dubious rule. In two different respects, the rule apparently became part of the fabric of Eighth Amendment jurisprudence inadvertently, or at least surreptitiously, and the manner in which it seeped into current doctrine illuminates its larger weakness. We discuss these two inadvertencies in turn.
First, prior to the Supreme Court's decision in *Baze v Rees*,[^25^](#FN25){ref-type="fn"} the principal factors in determining whether a punishment was constitutionally cruel and unusual were whether the punishment involved gratuitous pain, and whether state officials acted intentionally in inflicting it. For example, in the 1940s, the State of Louisiana attempted to carry out an execution by electrocution,[^26^](#FN26){ref-type="fn"} but the inmate did not die. The inmate sought to prevent the state from trying again, on the grounds that a second attempt would be cruel and unusual. By a vote of five-to-four, the Supreme Court permitted the execution to go forward, ruling that the botched attempt was not intentional, and that a second attempt would therefore not be gratuitous or a purposeful effort to impose cruelty.[^27^](#FN27){ref-type="fn"}
In *Rees*, this erstwhile focus subtly shifted. Writing for a plurality, Chief Justice Roberts concluded an inmate cannot demonstrate that an execution modality is cruel and unusual unless he can identify an alternative procedure that "significantly reduce\[s\] a substantial risk of severe pain."[^28^](#FN28){ref-type="fn"} The question, therefore, was no longer the inherent cruelty of a given procedure or even the intentionality of the state's actions; the issue had started to become comparative. (But, despite this early shift, it was not yet precisely a burden of proof imposed on the inmate.)
The case cited in closest proximity to this new proposition by Chief Justice Roberts was a decision called *Farmer v Brennan*;[^29^](#FN29){ref-type="fn"} *Farmer*, however, involved an entirely distinct issue: namely, whether an Eighth Amendment violation existed when state officials had no knowledge that their actions carried with them a "significant risk of harm."[^30^](#FN30){ref-type="fn"} In other words, *Farmer* was a case about intentionality. But by an act of judicial alchemy, the Court in *Rees* took the question of whether state officials were *aware* of a "significant risk" of harm and transformed it into the question of whether a procedure other than the one the state intend to employ posed a significantly lower risk of harm.
Second, notwithstanding *Rees*, until the Supreme Court's decision in *Glossip*, the focus in challenges to a state's method of execution protocol lay simply in the question of whether that protocol presented a substantial risk of serious harm. Whether something presents a substantial risk is not an inherently comparative inquiry. For example, whether jumping out of an airplane at 30,000 feet without a parachute presents a substantial risk of serious harm can be answered without comparison to the risk of serious harm presented by jumping out of an airplane from 3 feet.[^31^](#FN31){ref-type="fn"} The focus on inherent risk shifted in *Glossip*, however, when the Court ruled that the burden on prisoners of identifying "a known and available alternative method of execution that entails a lesser risk of pain" is "*a requirement* of all Eighth Amendment method-of-execution claims."[^32^](#FN32){ref-type="fn"}
In support of this critical proposition -- that a *requirement* of successful method of execution challenges is to identify a better method -- the *Glossip* Court cited two sentences from Chief Justice Roberts' plurality opinion in *Rees*, where he wrote:
*A stay of execution may not be granted on grounds such as those asserted here unless the condemned prisoner establishes that the State's lethal injection protocol creates a demonstrated risk of severe pain. He must show that the risk is substantial when compared to the known and available alternatives.*[^33^](#FN33){ref-type="fn"}
What is important to understand about these two sentences, however, is that, in *Rees*, the issue of alternatives was introduced by the *inmates*. Concerned that Kentucky's three-drug protocol might result in the infliction of gratuitous pain, and that state officials were aware of this possibility, the inmates proposed that a single-drug protocol be used in place of the more familiar three-drug regimen. The *inmates* elected to identify an alternative because the evidence was substantial that the use of the paralytic agent as part of the lethal injection protocol presented a risk of inflicting unnecessary pain, and because it was unnecessary for carrying out the execution -- it was akin to beating a suspect who has already been subdued.
In rejecting their challenge, the Court in *Rees* focused primarily on the risks inherent in the three-drug protocol; it did not accept the inmate's invitation to conduct a side-by-side comparison of one method with another. In point of fact, Chief Justice Roberts' opinion expressly *rejected* this approach. The federal courts, he explained, are not "boards of inquiry charged with determining 'best practices' for executions."[^34^](#FN34){ref-type="fn"} Paradoxically, however, the Court in *Glossip*, by taking a line from *Rees* out of context, inverted the Chief Justice's warning and reached exactly the conclusion Chief Justice Roberts had eschewed. As a result, from *Glossip* until now, the burden on inmates raising an Eighth Amendment challenge to an execution modality is to persuade the Court that there is a superior method -- a better (if not best) practice for carrying out an execution.
Whereas *Rees*, properly read, rejected transforming the federal courts into boards of inquiry, *Glossip* now requires precisely that role. Further, this evidentiary burden is one uniquely borne by inmates facing execution. Hence, insofar as one feature of a jurisprudentially infirm rule is contextual uniqueness, the core of the Court's method of execution jurisprudence is fatally infirm.
But there is a second defect with this line of jurisprudence. This second problem is largely unremarked upon, yet substantially more significant. For even if it were appropriate to carve out a *sui generis* legal requirement applicable only to method of execution challenges, the specific requirement imposed by the Court in *Glossip* is scientifically unsound.
Ironically, Chief Justice Roberts foresaw the very problem we will address. In rejecting the inmates' invitation that the federal courts serve as so-called boards of inquiry, Roberts noted that, aside from finding "no support in our cases," asking courts to compare one execution protocol to another "would embroil the courts in ongoing scientific controversies beyond their expertise."[^35^](#FN35){ref-type="fn"} Yet the very inquiry Roberts accurately characterized in *Rees* as beyond the competence of federal courts became entrenched by *Glossip* at the core of what federal courts must examine in challenges to a state's execution protocol. Consequently, in contemporary method of execution litigation, federal courts are grappling with issues Chief Justice Roberts recognized as lying beyond their competence; worse still, those very questions have no clinically-supported answers.
It may be useful to tease apart the two related yet distinct empirical questions that are embedded in *Glossip*'s requirement that inmates compare the risks of one protocol with the risks associated with available alternatives. Because, in this context, *risk* refers to a risk of severe pain, the first question is how to define *severe pain*. The second distinct issue is to calculate the probability a given inmate will experience that degree of pain in any specified procedure.
*Severity of Pain* -- There is no medically established and accepted definition of *severe* pain, and there are no clinical data identifying factors that might result in that degree of pain (nor, *a fortiori*, are there any data concerning the probability those factors would inhere in any particular execution protocol).[^36^](#FN36){ref-type="fn"} For example, there are at least seven accepted subjective pain scales. They do not use a uniform set of criteria for characterizing a certain level of pain, and while some scales identify severe pain as the most intense, others have various degrees of severity.[^37^](#FN37){ref-type="fn"} *Glossip*'s requirement, therefore, that an inmate facing execution demonstrate some probability he will experience severe pain as a result of the state's execution method is a requirement that is conceptually impossible to meet, at least as a medical or scientific matter, because there is no rigorous definition of that standard. We return to this issue, and propose a potential solution, below in Part 3. For now, it would suffice to improve the Eighth Amendment's method of execution jurisprudence, and make it coherent, by returning the rule of *Wilkerson* and replacing the adjective "severe" with "unnecessary" or "gratuitous."
*Probability* -- Most litigation challenging existing state execution protocols has focused on the likelihood an inmate will experience some degree of pain during the procedure. The weight of authority certainly suggests that, at least with respect to the three-drug protocol, or any protocol using midazolam as the anaesthetizing agent, there is a nontrivial risk the inmate will feel the effects of additional drugs, including the paralytic agent, thereby experiencing the sensation of suffocation, and may even experience pain associated with cardiac arrest, when potassium chloride is introduced. (We refer to some of this authority in the notes.)[^38^](#FN38){ref-type="fn"} It must be said, however, that regardless of the soundness of these predictions, and regardless of a small sample of anecdotal evidence consistent with these predictions, there are no reliable published clinical data that would permit a court to assess precisely the probability an inmate would experience that level of pain.[^39^](#FN39){ref-type="fn"}
(To our knowledge, there has been only a single attempt to determine the risk that an inmate facing execution by the three-drug protocol would not be in a state of deep unconsciousness -- and therefore could experience severe pain[^40^](#FN40){ref-type="fn"} -- when the second and third drugs of the protocol are administered. In two related studies, Dr. Leonidas Koniaris and his team examined post-mortem serum levels of sodium thiopental in cadavers of inmates executed by lethal injection in Arizona, Georgia, North Carolina, and South Carolina and reported their findings.[^41^](#FN41){ref-type="fn"} Although these findings have been subject to some criticism,[^42^](#FN42){ref-type="fn"} it is not our intention to examine either those criticisms or the responses of Drs. Koniaris and Zimmers; instead, we simply stress that the data-set Leonidas examined -- execution victims -- was small, that the time between execution and post-mortem blood sampling varied, and that other characteristics varied from one inmate to another (including, for example, the time between injection of the various drugs comprising the cocktail). Regardless, therefore, of the correctness of their analysis -- *i.e.*, that the inmates subjected to this three-drug protocol were not in states of deep unconsciousness when the second and/or third drugs of the cocktail were administered -- it is quite clear that the conditions for drawing rigorous conclusions based on carefully controlled clinical environments were not present.)
3. Acquiring the Data and Repairing Doctrine {#s3}
============================================
Lethal injection litigation turns on the answers to two empirical questions and one analytic issue: first, whether a particular inmate is enduring significant pain during his execution is a factual question, the answer to which can be ascertained by contemporaneous monitoring; second, whether there is a reasonable probability any given inmate will endure significant pain during an execution is also an empirical question, the answer to which can be illuminated by acquiring a sufficiently robust data-set; finally, whether any given protocol inherently includes the potential of gratuitous pain is an analytical question that can be answered outside the context of a particular execution. We address each of these issues in reverse order.
3.1. Potential for Gratuitous Pain {#s3a}
----------------------------------
In the commonly used three-drug execution protocol, the intermediate drug, a paralytic agent, typically pancuronium bromide, plays no essential role in the achievement of the execution.[^43^](#FN43){ref-type="fn"} In ordinary application, a paralytic is administered to facilitate surgery.[^44^](#FN44){ref-type="fn"} A substantial medical-legal literature (much of which has been at issue in execution protocol litigation in various states) has demonstrated that the potential for any barbiturate or anesthetizing agent to wear off or be improperly administered creates the potential for an inmate to experience pain during the execution from the paralytic agent, and that same agent would prevent observers for recognizing the pain being endured by the inmate. Under the Supreme Court's original and coherent understanding of the Cruel and Unusual Punishments clause, because the intermediate drug plays no role in the carrying out of the execution, it is therefore, by definition, gratuitous, and any pain suffered as a result of the introduction of that drug would therefore also, by definition, be gratuitous. A return to and faithful application of the *Wilkerson* standard would result in the elimination of the paralytic agent from the lethal injection protocol.
3.2. Probability of Pain Given a Defined Protocol {#s3b}
-------------------------------------------------
As we have observed, there is no body of robust data examining either the efficacy of or potential complications from any particular execution protocol. However, it would be possible to acquire data germane to answering the question of probability of pain in either a controlled experimental environment involving animals, or by compiling anecdotal reports from witnesses of individual executions, or by acquiring both contemporaneous as well as post-mortem data from executions.
Using laboratory animals to help ascertain whether human victims might experience significant pain during an execution is contrary to ethical guidelines for the use of animal research,[^45^](#FN45){ref-type="fn"} and we therefore do not consider it further.
Modern litigation challenging lethal injection protocols has employed both anecdotal reports of individual executions as well as data gathered from post-mortem analyses of blood serum and other tissue.[^46^](#FN46){ref-type="fn"} For example, the challenge to the Arkansas lethal injection protocol (a three-drug cocktail employing midazolam as the sedating agent), which concluded in May 2019, relied heavily on testimony from eyewitnesses present for the executions of four inmates put to death in Arkansas in 2017 using the same three-drug protocol.[^47^](#FN47){ref-type="fn"} In addition, in Arkansas and in similar litigation challenging the use of midazolam in Ohio, pharmacologists or anesthesiologists testified about the risk that midazolam would (or would not) result in deep unconsciousness,[^48^](#FN48){ref-type="fn"} which is significant testimony because an inmate in the deepest level of unconsciousness[^49^](#FN49){ref-type="fn"} would not feel pain, but an inmate is a shallower state might, unless the sedating agent also included an analgesic, which neither Arkansas nor Ohio protocols employ.
Insofar as eyewitness accounts of so-called botched executions credibly suggest the inmates were reacting to painful stimuli, that testimony is evidence that the sedative effect of the drug they received produced only minimal or moderate sedation. However, the combination of this anecdotal reporting, coupled with expert testimony that midazolam, for example, provides neither deep unconsciousness nor analgesia[^50^](#FN50){ref-type="fn"} has not been adequate to demonstrate an Eighth Amendment violation, in part, it appears, because this evidence is incapable of providing the comparative analysis contemporary doctrine requires. One obvious remedy to contemporary doctrine, therefore, which would be consistent with *Rees* but would require a modification of *Glossip* and *Bucklew*, would be simply to require that a party challenging the execution protocol demonstrate only a substantial likelihood of pain. Moreover, given that state officials can be charged with knowledge of these anecdotal reports as well as expert testimony concerning the inefficacy of midazolam as a sedating agent during the execution protocol, this approach would perhaps be consistent with Justice Thomas's position that an Eighth Amendment violation occurs only when state officials act purposefully.[^51^](#FN51){ref-type="fn"} Under this understanding of the Eighth Amendment, the evidence presented in Arkansas, Ohio, and elsewhere would establish an unacceptable probability that a three-drug protocol, or any protocol relying on midazolam as the anesthetizing agent, presents an unconstitutionally high risk of the infliction of gratuitous pain.
3.3. Experience of Pain During a Specified Execution {#s3c}
----------------------------------------------------
Our discussion in each of the two preceding subsections assumes a modification (albeit minor) of existing legal doctrine. However, it is also possible, from entirely within the confines of the *Glossip*-*Bucklew* framework, to obtain contemporaneous data during an execution that would reveal whether the execution victim is experiencing pain or distress. This data could in turn be used both to address the issue during the ongoing execution and to modify the procedure for future executions to avoid that pain or distress level in other execution victims.
As we have indicated, an inmate might experience pain for a variety of reasons: because of inherent defects in the execution protocol; because drugs are adulterated; because IV lines are improperly set; because the drugs are administered too quickly, or too slowly, or with imprecise spacing; etc. Contemporaneous monitoring could remediate problems resulting from any of these occurrences.
Most death penalty states do not provide for continuous technical monitoring of inmates during the execution;[^52^](#FN52){ref-type="fn"} such monitoring, however, could reduce the potential for the experience of pain to a level approaching zero, if performed by trained monitors. Lawyers who challenged the execution protocol in Ohio specifically requested that inmates undergoing execution have their potential pain assessed objectively and contemporaneously using various physiological markers.[^53^](#FN53){ref-type="fn"} The federal district court and, on appeal, the United States Court of Appeals for the Sixth Circuit dismissed this request for monitoring, citing, among other reasons, the fact nonmedical personnel might not understand the data, might not have the knowledge to reconcile potentially conflicting parameters, and would lack training to identify or implement any required remedial measures dictated by the data.[^54^](#FN54){ref-type="fn"}
There are, however, common monitoring mechanisms widely available and reliably useful to avoid pain during an execution. An inmate requesting such monitoring would not be challenging a state's method of execution -- and would therefore not be required by *Glossip*-*Bucklew* to identify another readily implementable method for carrying out the death sentence. Instead, the inmate would simply request that he have his level of consciousness continually monitored so as to assure he was in a state such that he would not experience gratuitous pain from either a paralytic agent or a drug intended to induce cardiac arrest. For example, the so-called BIS monitor[^55^](#FN55){ref-type="fn"} is widely used in general surgery and reveals in real time the depth of (un)consciousness occupied by someone receiving a general anesthesia. There are various other similar monitoring devices available. As far as we can determine, however, no currently adopted execution protocols provide for such monitoring to occur.
In addition, it would also be possible, although somewhat less useful than EEG monitoring, to use (as the Ohio lawyers requested) an EKG to measure both heart rate and blood pressure during the execution procedure. An execution victim experiencing pain or distress would show an elevated heart rate and blood pressure. Again, however, as far as we can determine, no currently adopted execution protocols provide for such contemporaneous cardiac or blood pressure monitoring.
Crafting the precise dimensions of the legal argument in support of contemporaneous monitoring is beyond our present scope; the salient point is that the argument would not require overruling or alteration of the Supreme Court's decisions in either *Glossip* or *Bucklew*, nor would the argument preclude the states from continuing to execute inmates.
4. Conclusion {#s4}
=============
In the medical anaesthesia literature relating to wakefulness during surgery, there are, as Dr. J. Bruhn and his colleagues report, an alarming high number of such incidents which, according to Dr. Bruhn, make for grim reading.[^56^](#FN56){ref-type="fn"} We should not be surprised, therefore, to learn there could well be consciousness of pain during an alarmingly high number of executions, especially given that, according to the published protocols of the most active death penalty states, medical professionals, if present at all, are involved in neither the introduction of the lethal drugs nor state-of-the-art monitoring that could reveal in real time distress or severe pain during the execution. The fact that inmates challenging these protocols, including the accompanying lack of contemporaneous monitoring, repeatedly lose their legal challenges is not a reflection that inmates undergoing execution are not experiencing pain; it is simply a reflection that legal doctrine requires they prove a proposition for which the data necessary to establish such proof have not been collected.
Requiring contemporaneous monitoring by personnel trained to identify indications of pain or distress and take steps to remediate that distress would not require states to abandon the death penalty or even alter their execution protocols. It would merely require them to take easily achievable steps to reduce or eliminate the possibility inmates will suffer unnecessary pain during their executions. In contrast, not adopting these easily implementable measures may help preserve the fiction that executions are painless and simple, but the refusal also demonstrates an indifference to the possibility that existing execution protocols inflict gratuitous pain and therefore involve torture.
We gratefully acknowledge the assistance of Dr. Neville Leibman, M.D., and Dr. Benjamin Musher, M.D; and Maurie Levin, Esq. We also profited from comments of three anonymous reviewers. We are grateful to Dean Leonard Baynes and Associate Dean Greg Vetter, and to the University of Houston Law Foundation for its generous financial support. Views expressed are our own.
1 In 1972, a narrowly divided Court had invalidated the death penalty. See Furman v Georgia, 408 U.S. 238 (1972). But capital punishment was reinstated a mere four years later, and there has been no serious categorical challenge to it since that time. See Gregg v Georgia, 428 U.S. 153 (1976); and its companions, including Jurek v Texas, 428 U.S. 262 (1976); and Proffitt v Florida, 428 U.S. 242 (1976).
2 In full, the Eighth Amendment to the U.S. Constitution provides: "Excessive bail shall not be required, nor excessive fines imposed, nor cruel and unusual punishments inflicted." The Amendment has been held to apply to the states (via incorporation through the Fourteenth Amendment's Due Process clause) in a somewhat piecemeal fashion. The "excessive fines" limitation was finally incorporated by the Supreme Court's recent decision in Timbs v Indiana, 139 S.Ct. 682 (2019).
3 Callins v Collins, 510 U.S. 1141 (1994) (Blackmun, J., dissenting).
4 Historically, as discussed below, the widely-used form was a three-drug cocktail. Some states now employ a single drug. For a useful summary of the various drugs used in different lethal injection protocols, as well as their therapeutic class and unique pharmacological characteristics, see Sean Riley, Navigating the New Era of Assisted Suicide and Execution Drugs, 4 J. Law and the Biosciences 424, 426 (2017).
5 These executions take place in a cluster of states, mostly in the south. For although thirty states (as well as the federal government and the U.S. military) allow for the death penalty, a much smaller number of states actually put inmates to death. The federal government has not carried out an execution since 2003 (when three inmates were executed), and of the 30 states with death penalty statutes, half (i.e., 15) have not carried out an execution for at least eight years. Thus, 15 states have carried out all executions in the U.S. since 2009. Of the 364 executions from 2009 through July 31, 2019, Texas has accounted for 158 of this number, followed by Florida (32), Georgia (31), Alabama (28), Oklahoma (24), Missouri (22), and Virginia (11).
6 The best source of information on specific execution protocols in each of the death penalty states is the regularly updated database maintained by the Death Penalty Information Center. https://deathpenaltyinfo.org/states-and-without-death-penalty
7 On occasion, Justice Breyer has intimated an interest in revisiting the general question of whether capital punishment necessarily amounts to cruel and unusual punishment, but his interest is rather desultory. Thus, he questioned the death penalty's reliability, and raised questions of its arbitrariness, in his dissenting opinion in Glossip v Gross, 576 U.S. \_\_\_ , 135 S.Ct. 2726, 2755 (2015) (Breyer, J., dissenting), and in his dissent from the denial of certiorari in Jordan v Mississippi, 585 U.S. \_\_\_, 138 S. Ct. 2567, 2568 (2018) (Breyer, J., dissenting from denial of certiorari). Apart from these occasional Hamlet-like indications of an interest in addressing the broad constitutional question of the death penalty's permissibility, however, Justice Breyer routinely votes in favor of allowing individual executions to proceed.
8 Traditional Eighth Amendment challenges pursued in federal court will be discussed at greater length below. Not all lethal injection litigation has followed the traditional trajectory, however. For example, Cook v Food and Drug Administration, 733 F.3d 1 (D.C. Cir. 2013), was brought by three death row inmates in three different states to challenge the importation of foreign manufactured sodium thiopental. The inmates alleged the importation of this drug violated the Food, Drug, and Cosmetics Act, and they further alleged a violation of the Administrative Procedures Act. See Beaty v FDA, 853 F.Supp.2d 30 (D.D.C. 2012), aff'd in part and rev'd in part by *Cook, supra*. A useful discussion of this litigation is contained in Nicholas Meyers, Cook v FDA and the Importation and release of Lethal Injection Drugs, 1 J. Law & Biosciences 209 (2014). (However, the effect of this litigation could soon be undermined due to a recently released opinion from the Department of Justice's Office of Legal Counsel, which concludes the FDA lacks jurisdiction to regulate the drugs used in executions. Whether the Food and Drug Administration Has Jurisdiction over Articles Intended for Use in Lawful Executions, 43 Op. O.L.C. (May 3, 2019).)
9 As of May 1, 2019, the Westlaw federal courts data base identifies 391 cases when search with the following search terms: "death /p lethal /1 injection /s constitutionality constitutional unconstitutional & da(after 2000)."
10 99 U.S. 130 (1879). The other notable 19th century case dealing with mode of execution is, In re Kimmler, 136 U.S. 436 (1890). *Kimmler* involved a constitutional challenge to electrocution, which was used for the first time in the U.S. in 1890. As Professor Debbie Denno has shown, *Kimmler* does not actually rest on the Eighth Amendment's Cruel and Unusual Punishments clause, even though it is frequently cited, including by the Supreme Court, for that very proposition. See Deborah W. Denno, Adieu to Electrocution, 26 Ohio Northern L. Rev 665 (2000).
11 *Wilkerson*, 99 U.S. at 135-36.
12 While beyond the immediate scope of our analysis here, it is perhaps worth observing that the Supreme Court has never explicitly addressed the incorporation of the Cruel and Unusual Punishments clause. *Cf. Timbs*, supra note 2 (noting incorporation of excessive fines clause). In O'Neil v State of Vermont, the Court noted the Eighth Amendment did not apply to the states. 144 U.S. 323, 331-32 (1892). In contrast, when the Court struck down the death penalty in *Furman*, see *supra* note 1, the Court did not provide any incorporation analysis or formally overrule *O'Neil*, but, as Judge (and Professor) Stephen McAllister notes, following *Furman*, the Court seems to have assumed that *Furman* deemed the Cruel and Unusual Punishments Clause to have been incorporated. See Stephen R. McAllister, The Problem of Implementing a Constitutional System of Capital Punishment, 43 U. Kan. L. Rev 1039, 1039-40 & nn. 6, 63 (1995). On the other hand, former Chief Justice Rehnquist did think the Court had deemed the Cruel and Unusual Punishments Clause to be incorporated, and therefore applied to the states, by the Fourteenth Amendment, although his statement to that effect does not include a specific citation. See Hutto v Finney, 437 U.S. 678, 717-18 (1978)(Rehnquist, J., dissenting). He may, however, have had in mind either State of La. ex rel. Francis v Resweber, 329 U.S. 459, 463 (1947), where the Court assumed without deciding the Constitution did not prohibit the state from attempting a second execution after the first attempt failed ("When an accident, with no suggestion of malevolence, prevents the consummation of a sentence, the state's subsequent course in the administration of its criminal law is not affected on that account by any requirement of due process under the Fourteenth Amendment."), or the subsequent opinion in Robinson v California, 370 U.S. 660 (1962), where the Court, citing *Resweber*, held: "\[I\]n the light of contemporary human knowledge, a law which made a criminal offense of such a disease would doubtless be universally thought to be an infliction of cruel and unusual punishment in violation of the Eighth and Fourteenth Amendments." Id. at 666; see also *id*. at 668-69 (Douglas, J., concurring). In any case, as a practical matter, it is clear the Cruel and Unusual Punishments clause applies to the states, despite the arguable absence of an express holding so stating.
13 See, e.g., In re Kimmler, 136 U.S. 436, 443-44 (1890) (noting electrocution presumably causes death "instantaneous\[ly\], and consequently . . . painless\[ly\]"); *id*. at 447 ("Punishments are cruel when they involve torture or a lingering death."). We refer at times in the following discussion to "gratuitous" pain; in this context, gratuitous is coterminous with torture, i.e., it refers to any pain in excess of the quantum required to carry out the execution of the prisoner.
14 No legal academician has written more perceptively about lethal injection litigation and the broad range of constitutional issues associated with method of execution challenges, including lethal injection, than Professor Debbie Denno. See, e.g., Deborah W. Denno, Courting Abolition, 130 Harv L. Rev 1827, 1864-66 (2017)(reviewing Steiker & Steiker, Courting Death (2016)); Deborah W. Denno, Lethal Injection Chaos Post-*Baze*, 102 Geo. L.J. 1331 (2014); Deborah W. Denno, The Lethal Injection Debate: Law and Science, 35 Fordham Urban L.J. 701 (2008).
15 The displacement of other modes of execution by lethal injection has been authoritatively addressed by Debbie Denno. On the history of the lethal injection protocol, see, e.g., Deborah W. Denno, When Legislatures Delegate Death, 63 Ohio St. L.J. 63, 90-91 (2002).
16 553 U.S. 35 (2008).
17 135 S.Ct. 2726 (2015).
18 139 S.Ct. 1112 (2019). *Bucklew* was more precisely an as-applied challenge that grew out of Bucklew's specific medical condition. But that distinction is not relevant to our discussion.
19 553 U.S. at 52.
20 The development of this requirement that inmates identify a superior mode of execution before they could prevail on their challenge to a state's chosen mode is succinctly covered in Megan McCracken, Legally Indefensible: Requiring Death Row Prisoners to Prove Available Execution Alternatives, 41 FEB Champion 46, 47-48 (2017). As one anonymous reviewer stressed, the Court has steadily, and almost surreptitiously, imposed an increasingly heightened burden on inmates in the lethal injection cases. We address this stealthy augmentation of the burden facing inmates below in Part 2. The upshot is that inmates who challenge the method the state intends to use to carry out their executions cannot prevail unless they identify a better modality of death. (In this context, "better" is defined as a method involving less gratuitous pain, where "gratuitous" is defined as any pain beyond that which is inherently required to put someone to death. We return to how one might measure these alternatives below.) See, e.g., *Bucklew*, 139 S.Ct. at 1123-25. However, as the same reviewer reminded us, Justice Gorsuch fleetingly suggested in *Bucklew* that even if there are better methods, the Eighth Amendment may not compel the state to use them. See 139 S.Ct. at 1125 ("Nor do *Baze* and *Glossip* suggest that traditionally accepted methods of execution---such as hanging, the firing squad, electrocution, and lethal injection---are necessarily rendered unconstitutional as soon as an arguably more humane method like lethal injection becomes available.").
Finally, the Court in *Bucklew*, citing *Glossip*, stressed that any alternative mode of execution identified by the inmate must also be capable of ready implementation. See 139 S.Ct. at 1129. The protocol modifications we suggest below in Part 3 meet this requirement of being readily implementable.
21 In a sense, when a doctrine applies to a single, narrow factual context, it does not resemble what is known as the rule of law. For a thoughtful (and non-tendentious) discussion, see, e.g., Charles Fried, Saying What the Law Is, esp. 70-77 (2004). Not surprisingly, much of the criticism of the Supreme Court's decision in Bush v Gore, 531 U.S. 98 (2000), focused on the sense in which the Court's ruling did not resemble a decision based on a generally applicable rule of law. See, e.g., Robin West, Reconstructing the Rule of Law, 90 Georgetown L.J. 215 (2001).
22 See, e.g., McCracken, supra note 20.
23 See Hudson v McMillian, 503 U.S. 1, 8-11 (1992).
24 E.g., Sallenger v Oakes, 473 F.3d 731, 740-42 (7th Cir. 2007).
25 553 U.S. 35 (2008).
26 State of La. ex rel. Francis v Resweber, 329 U.S. 459 (1947).
27 For excellent analysis, see Brent E. Newton, The Slow Wheels of *Furman*'s Machinery of Death, 13 J. App. L. & Proc. 42 (2012). Among the Court's current members, Justice Thomas has been most committed to the idea that a punishment is not cruel and unusual unless state officials act intentionally in inflicting gratuitous pain. See, e.g., *Rees*, 553 U.S. at 94 (Thomas, J., concurring)
28 *Rees*, 553 U.S. at 52 (Roberts, C.J., joined by Kennedy and Alito, JJ.).
29 511 U.S. 825 (1994).
30 *Id*. at 837-38.
31 But for a cautionary tale on experimental design, see R.W. Yeh, *Parachute use to prevent death and major trauma when jumping from aircraft: randomized controlled trial*, *BMJ* 363:k5094 (2018), available at https://www.bmj.com/content/363/bmj.k5094.
32 *Glossip*, 135 S. Ct at 2731 (emphasis added). For the evolution of the burden of proof placed on inmates challenging the execution protocol, see supra note 20 and text accompanying notes 25-29.
33 *Rees*, 553 U.S. at 61.
34 *Rees*, 553 U.S. at 51.
35 *Rees*, 553 U.S. at 51.
36 As Drs. Zimmers and Koniaris correctly observed, "\[l\]ethal injection was designed and carried out without any research at all." Teresa A. Zimmers and Leonidas G. Koniaris, Peer-Reviewed Studies Identifying Problems in the Design and Implementation of Lethal Injection for Execution, 35 Fordham Urban L.J. 919 (June 2008). In contrast, data on animal euthanasia are widely reported, and veterinarians regularly update and modify the protocol for euthanizing animals. See Eric Berger, Lethal Injection and the Problem of Constitutional Remedies, 27 Yale Law & Policy Rev 259, 267 & n. 31 (2009); Ty Alper, Anesthetizing the Public Conscience, 35 Fordham Urban L. J. 817 (2008) (noting function of paralytic is to protect witnesses, who would otherwise see a thrashing inmate, rather than to accomplish the death of the inmate).
37 See "Pain Assessment Scales," Pain Assessment and Management Initiative, University of Florida College of Medicine - Jacksonville, available at http://pami.emergency.med.jax.ufl.edu/resources/pain-assessment-scales/ As one reviewer has pointed out, the data on pain severity are thin, but not non-existent. For one meta-analysis of non-communicative ICU patients, see Isabela Freire Azevedo-Santos and Josimari Melo DeSantana, Pain Measurement Techniques: Spotlight on mechanically ventilated patients, J.Pain Res. 11:2969-80 (2018), available at https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6255280/
38 Lethal injection litigation has been pursued in a number of jurisdictions. In Ohio, the most recent litigation has focused on potential problems with the use of midazolam as the first drug in the cocktail. See In re: Ohio Execution Protocol Litigation, No. 2:11-cv-01016 (S.D. Ohio, Eastern Div 2017), ECF 1295-1 testimony of Dr. Ashish Sinha, practicing anesthesiologist, and professor of anesthesiology in the Department of Anesthesiology at the Lewis Katz School of Medicine, Temple University, in Philadelphia, Pennsylvania, at para 12 (noting, inter alia, that "midazolam has no analgesic properties" and that, on the contrary, midazolam "*increases*sensitivity to pain, and creates the condition hyperalgesia" (citing Michael A. Frölich, Kui Zhang & Timothy J. Ness, *Effect of Sedation on Pain Perception*, 118 Anesthesiology 611, 611 (2013)). See also para 18 of Sinha affidavit (noting district court's misunderstanding of relevant science). See generally Michael A. Frölich, Kui Zhang & Timothy J. Ness, *Effect of Sedation on Pain Perception*, 118 Anesthesiology 611, 611 (2013); Miller's Anesthesia (8th ed., 2015); Cowen et al., *Assessing pain objectively: the use of physiological markers*, Anaesthesia 70: 828-847 (2015).
39 See Zimmers and Koniaris, supra note 36.
40 We repeat here that there is no medically accepted definition of what constitutes *severe*pain. Commonly used pain scales are widely recognized as entirely subjective. See, e.g., James Giordano, Kim Abramson, and Mark V Boswell, Pain Assessment: Subjectivity, Objectivity, and the Use of Neurotechnology - Part One: Practical and Ethical Issues, *Pain Physician* 13:305-315 (2010); see also Tandon, M., Singh, A., Saluja, V, Dhankhar, M., Pandey, C.K., and Jain, P., Validation of a New "Objective Pain Score" Vs. "Numeric Rating Scale" For the Evaluation of Acute Pain: A Comparative Study, *Anesth Pain Med*. 2016:6(1), available at https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4834447/. We return to the issue of measuring pain objectively below in Part 3.
41 Koniaris, et al., Inadequate Anaesthesia in Lethal Injection for Execution, 365 Lancet 1412 (2005); Zimmers, et al., Lethal Injection for Execution: Chemical Asphyxiation?, 4 PLoS Med. 0646 (2007). The Lancet study in particular proved controversial. See, e.g., Jonathan I. Groner, Response, 366 Lancet 1073 (2005); Mark Heath, et al., Response, 366 Lancet 1073 (2005) (both suggesting limitations of post-mortem extrapolation of serum thiopental levels).
42 Both Koniaris and Zimmers stand by their conclusions. See supra note 37, at 925-29.
43 See supra note 37. One anonymous reviewer correctly points out that the paralytic agent, by preventing the inmate from thrashing or otherwise exhibiting distress during the execution, helps preserve the fiction that the procedure is carefully clinically controlled and painless. This view finds support in the legal literature. See, e.g., Seema Shah, How Lethal Injection Reform Constitutes Impermissible Research on Prisoners, 45 Am. Crim. L. Rev 1101, 1136 & n. 234 (2008) (noting that, in *Rees*, the State of Kentucky acknowledged the role played by the paralytic was to "maintain\[\] an appearance of dignity").
44 See, e.g., Charles J. Cote, et. al. (eds), A Practice of General Anesthesia for Infants and Children (6th ed. 2019) (noting role pancuronium bromide plays in cardiac surgery and other high-risk procedures).
45 Using animals in this context is obviously simply immoral, period. It is also the case, however, that any experimental design that might provide robust data relevant to the Eighth Amendment inquiry would be inconsistent with accepted scientific ethical guidelines pertaining to animal experimentation. See, e.g., Bernard E. Rollin, The Ethics of Animal Research, in Lida Kalof, ed., The Oxford Handbook of Animal Studies (2017); Curzer, H.J., Perry, G., Wallace, M.C. et al., The Three Rs of Animal Research, Sci Eng Ethics 22:549 (2016), available at https://doi.org/10.1007/s11948-015-9659-8.
In addition, as an anonymous reviewer has pointed out, it is by no means clear that results from animal studies would be admissible to establish a risk that human subjects would face a significant possibility of experiencing gratuitous pain during an execution, because many courts have excluded the results of animal studies on *Daubert* grounds. (The so-called *Daubert* standard, named after Daubert v Merrell Dow Pharmaceuticals, 509 U.S. 579 (1993), establishes the criteria for the admission at trial of expert or scientific evidence.) See, e.g., Erica Beecher-Monas, The Heuristics of Intellectual Due Process: A Primer for Triers of Science, 75 N.Y.U. L. Rev 1563,1609-13 (2000) (identifying cases). Professor Beecher-Monas is rightly critical of many of these holdings. See also Amanda Hungerford, Note, Back to Basics: Courts' Treatment of Agency Animal Studies After *Daubert*, 110 Colum. L. Rev 70 (2010) (addressing cases involving possible carcinogens).
Finally, while animal studies in which laboratory specimens are put to death simply in order to acquire data regarding execution protocols would be unethical, there are in fact valuable animal studies regarding monitoring of EEG oscillations to obtain contemporaneous information regarding the experience of pain. Recently, analgesic effect in rats was observed by monitoring theta wave oscillations. See Suguru Koyama, et al., An Electroencephalography Bioassay for Preclinical Testing of Analgesic Efficacy, Scientific Reports 8:16402 (November 2018). We are not aware of any similar study in humans. These studies would, however, face the problems imposed by some courts' reading of *Daubert* identified in the preceding paragraph.
46 Regarding post-mortem blood serum analysis, see Koniaris, supra notes 36, 41. For a sampling of contemporaneous witness accounts reporting apparent distress exhibited by execution victims during procedures in Florida, Oklahoma, and Arizona, see Lillian Segura, Our Most Cruel Experiment Yet, in The Intercept, August 5, 2018, available at https://theintercept.com/2018/08/05/death-penalty-lethal-injection-trial-tennessee/
47 A summary of the testimony, with hyperlinks to local news coverage, can be found at https://deathpenaltyinfo.org/node/7388.
48 For the Arkansas case, see *id*.; for the Ohio litigation, see supra note 38.
49 There are various stages of anesthesia depths, ranging from minimal, where the subject remains conscious, to the stage where there is no response to painful stimuli. We are particularly grateful to Dr. Neville Leibman for discussing these stages with us, and how to determine during the administration of anesthesia the depth at which the recipient has reached. See generally Musizza, B. and Ribaric, S., Monitoring the Depth of Anaesthesia, *Sensors (Basel)* 10(12):10896--10935. doi:10.3390/s101210896, available at https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3231065/
50 Here again we are grateful to Dr. Neville Leibman. See testimony of Dr. Sinha, supra note 38.
51 *Rees*, 553 U.S. at 94 (Thomas, J., concurring) (noting Eighth Amendment is violated if and only if the protocol is designed to inflict pain).
52 We are grateful to the following death penalty lawyers for providing us with copies of their respective states's execution protocols: John Palombi and Spencer Hahn (Alabama uses an EKG only at the completion of the execution, to confirm death); Kim Stout (Arizona monitors inmate level of consciousness with an EKG, which is marked at the outset and conclusion of the execution, but protocol provides nothing in the way of adjustments if the EKG reveals distress); Kelson Bohnet (California, Kentucky, and Nevada use heart monitors, but only for purposes of confirming death by flat-line); Jonah Horwitz (Idaho monitors inmate level of consciousness with an EKG, which is marked at the outset and conclusion of the execution, but protocol provides nothing in the way of adjustments if the EKG reveals distress); Adam Rusonak (Ohio protocol provides for no monitoring); Kelley Henry (Tennessee protocol provides for no monitoring); Maurie Levin (Texas protocol provides for no monitoring). All death penalty states do provide for visual monitoring during the execution procedure.
One anonymous reviewer has communicated to us an important caveat: There may be some states that use medical professionals to monitor executions without publicly acknowledging this fact. (The anonymous reviewer informed us that discovery in a particular case revealed a certain state's use of medical professionals for this very purpose; it so happens, however, that this state no longer carries out executions.) Further, despite the shibboleth that physicians may not participate in executions because of prohibitory AMA Guidelines, the AMA Guidelines are precisely that: guidelines; and it is impossible to gainsay that physicians have in fact participated in numerous executions. See generally Ty Alper, The Truth About Physician Participation in Lethal Injection Executions, 88 N.C. L. Rev 11 (2009); see also Jonathan I. Groner, M.D., The Hippocratic Paradox: The Role of the Medical Professional in Capital Punishment, 35 Fordham Urban L.J. 883, 889-90 (2008) (noting that, with the exception of Georgia, where physicians have played what Dr. Groner characterizes as an "active role" in executions, the involvement of physicians elsewhere is to pronounce the time of death).
Nevertheless, it remains correct to state that the published execution protocols in the states carrying out most of the nation's executions do not provide for contemporaneous monitoring (by physicians or otherwise) that could be used to identify the infliction of gratuitous pain and to take measures to ameliorate such pain should it be identified.
53 See supra note 38. The Ohio legal team identified, inter alia, blood pressure, heart rate, and ECG.
54 See Campbell v Kasich, 881 F.3d 447, 454 (6th Cir. 2018). Considering that the lack of expertise by those who implement the execution protocol is one principal reason the inmate might experience severe pain during the procedure, the Sixth Circuit's identification of lack of expertise as a reason not to take steps that might mitigate the risk of pain is notable.
55 The bispectral index monitor, which uses an EEG, measures consciousness from a scale of 0 to 100, with 100 being wakefulness and numbers closer to 0 indicating deeper levels of unconciousness. General surgery is typically performed at a BIS level of 40-60. Thanks again to Dr. Neville Leibman for discussing this monitoring technology with us. See, e.g., Rani, D. and Harsoor, S., Depth of General Anaesthesia Monitors. *Indian J Anaesth*. 56(5):437--441 doi:10.4103/0019-5049.103956 (2012), available at https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3530997/ . We do not address the cost of necessary electrodes here. For a discussion, see, e.g., Anderson, R.E., et al., Use of Conventional ECG Electrodes for Depth of Anaesthesia Monitoring Using the Cerebral State Index, British Journal of Anaesthesia 98:5, 645-648 (2007), available at https://bjanaesthesia.org/article/S0007-0912(17)34850-X/fulltext
56 See Bruhn, J., Myles, P.S., Sneyd, R., and Struys, M.M.R.F., Depth of Anaesthesia Monitoring, British Journal of Anaesthesia , 97:1, 85-94 (2006), available at https://bjanaesthesia.org/article/S0007-0912(17)35187-5/fulltext
|
Off-pump coronary artery bypass grafting: a case-matched comparison of hemodynamic outcome.
The objective of this study was to assess improved myocardial protection by performing coronary artery bypass grafting (CABG) on the beating heart. A case-matched study was conducted among patients who underwent CABG either on-pump (group 1), or off-pump (group 2). Forty-five pairs of patients, having a similar clinical profile, were selected on the basis of five variables: age, gender, body surface area, ejection fraction, extent of coronary disease. Operative risk predicted by the The Society of Thoracic Surgeons national database was 1.80+/-0.35% in group 1, and 1.89+/-0.37% in group 2 (NS). Cold blood cardioplegia and 28 degrees C cardiopulmonary bypass were used in group 1. In group 2, beating heart coronary grafting was achieved with the Octopus 1 and 2 stabilizers. The average number of distal anastomoses was 2.8+/-0.1 in group 1 and 2.3+/-0.1 in group 2 (P=0.015). There was no significant difference among the groups regarding the trend in cardiac index, left and right ventricular stroke work indexes, and systemic and pulmonary vascular resistance indexes. However, heart rate trend was slower in group 2 (P=0.05). Pharmacological support was required in 65% of the patients in group 1, and in 33% in group 2 (P<0.001). The total amount of Dobutamine and/or Dopamine administered during the first 48 h was 3914+/-1306 gamma/kg in group 1 and 1645+/-697 gamma/kg in group 2 (P=0.049). Release of creatine kinase MB mass isoenzyme (CK-MB mass) was markedly reduced in group 2 (P<10(-4)). Hemodynamic outcome following off-pump CABG is similar to on-pump CABG but the need for inotropic support is significantly reduced and CPK-MB mass release is markedly lower. |
---
abstract: |
We study the enumerative geometry of rational curves on the Hilbert schemes of points of a K3 surface.
Let $S$ be a K3 surface and let $\operatorname{\mathsf{Hilb}}^d(S)$ be the Hilbert scheme of $d$ points of $S$. In case of elliptically fibered K3 surfaces $S \to {\mathbb{P}}^1$, we calculate genus $0$ Gromov-Witten invariants of $\operatorname{\mathsf{Hilb}}^d(S)$, which count rational curves incident to two generic fibers of the induced Lagrangian fibration $\operatorname{\mathsf{Hilb}}^d(S) \to {\mathbb{P}}^d$. The generating series of these invariants is the Fourier expansion of a power of the Jacobi theta function times a modular form, hence of a Jacobi form.
We also prove results for genus $0$ Gromov-Witten invariants of $\operatorname{\mathsf{Hilb}}^d(S)$ for several other natural incidence conditions. In each case, the generating series is again a Jacobi form. For the proof we evaluate Gromov-Witten invariants of the Hilbert scheme of $2$ points of ${\mathbb{P}}^1 \times E$, where $E$ is an elliptic curve.
Inspired by our results, we conjecture a formula for the quantum multiplication with divisor classes on $\operatorname{\mathsf{Hilb}}^d(S)$ with respect to primitive curve classes. The conjecture is presented in terms of natural operators acting on the Fock space of $S$. We prove the conjecture in the first non-trivial case $\operatorname{\mathsf{Hilb}}^2(S)$. As a corollary, we find that the full genus $0$ Gromov-Witten theory of $\operatorname{\mathsf{Hilb}}^2(S)$ in primitive classes is governed by Jacobi forms.
We present two applications. A conjecture relating genus $1$ invariants of $\operatorname{\mathsf{Hilb}}^d(S)$ to the Igusa cusp form was proposed in joint work with R. Pandharipande in [@K3xE]. Our results prove the conjecture in case $d=2$. Finally, we present a conjectural formula for the number of hyperelliptic curves on a K3 surface passing through $2$ general points.
author:
- Georg Oberdieck
date: October 2015
title: |
Gromov-Witten invariants of\
the Hilbert schemes of points of a K3 surface
---
Introduction {#Intro}
============
The Yau-Zaslow formula
----------------------
Let $S$ be a smooth projective K3 surface, and let $\beta_h \in H_2(S,{{\mathbb{Z}}})$ be a primitive effective curve class of self-intersection $\beta_h^2 = 2 h - 2$. The Yau-Zaslow formula [@YZ] predicts the number $\mathsf{N}_h$ of rational curves in class $\beta_h$ in the form of the generating series $$\sum_{h \geq 0} \mathsf{N}_h\, q^{h-1} = \frac{1}{q} \prod_{m \geq 1} \frac{1}{(1-q^m)^{24}} \,. \label{YZ}$$ The right hand side is the reciprocal of the Fourier expansion of a classical modular form of weight 12, the modular discriminant $$\Delta(\tau) = q \prod_{m \geq 1} (1-q^m)^{24} \label{Delta}$$ where $q = \exp(2 \pi i \tau)$ and $\tau \in {{\mathbb{H}}}$. The prediction was proven by Beauville [@Beauville2] and Chen [@ChenK3] using the compactified Jacobian, and by Bryan and Leung [@BL] using Gromov-Witten theory. It is the starting point of further research into the enumerative geometry of algebraic curves on K3 surfaces, see for example [@MPT; @KKV; @KKP] and the references therein.
The Hilbert scheme of $d$ points on $S$, denoted $$\operatorname{\mathsf{Hilb}}^d(S),$$ is the moduli space of zero-dimensional subschemes of $S$ of length $d$, see [@Lehn; @Nakajima] for an introduction. It is a non-singular projective variety of dimension $2d$, which is simply-connected and carries a holomorphic symplectic form. In case $d = 1$ we recover the original surface, $$\operatorname{\mathsf{Hilb}}^1(S) = S,$$ while for $d \geq 2$ the Hilbert schemes $\operatorname{\mathsf{Hilb}}^d(S)$ may be thought of as analogues of K3 surfaces in higher dimensions.
In this paper we study the enumerative geometry of rational curves on the Hilbert scheme of points $\operatorname{\mathsf{Hilb}}^d(S)$ for all $d \geq 1$. In particular, we obtain a generalization of the Yau-Zaslow formula .
Gromov-Witten invariants {#Section_Definition_of_Gromov_Witten}
------------------------
For all $\alpha \in H^{\ast}(S ; {{\mathbb{Q}}})$ and $i > 0$ let $${{\mathfrak{p}}}_{-i}(\alpha) : H^{\ast}(\operatorname{\mathsf{Hilb}}^d(S);{{\mathbb{Q}}}) \to H^{\ast}(\operatorname{\mathsf{Hilb}}^{d+i}(S);{{\mathbb{Q}}}), \ \gamma \mapsto {{\mathfrak{p}}}_{-i}(\alpha) \gamma$$ be the Nakajima creation operator obtained by adding length $i$ punctual subschemes incident to a cycle Poincare dual to $\alpha$. By a result of Grojnowski [@GrojH] and Nakajima [@Nakajima] the cohomology of $\operatorname{\mathsf{Hilb}}^d(S)$ is completely described by the cohomology of $S$ via the action of the operators ${{\mathfrak{p}}}_{-i}(\alpha)$ on the vacuum vector $$1_S \in H^{\ast}(\operatorname{\mathsf{Hilb}}^0(S);{{\mathbb{Q}}}) = {{\mathbb{Q}}}.$$
Let ${{\omega}}$ be the class of a point on $S$. For $\beta \in H_2(S;{{\mathbb{Z}}})$, define the class $$C(\beta) = {{\mathfrak{p}}}_{-1}(\beta) {{\mathfrak{p}}}_{-1}({{\omega}}_S)^{d-1} 1_S \, \, \in H_2(\operatorname{\mathsf{Hilb}}^d(S);{{\mathbb{Z}}}) \,.$$ If $\beta = [C]$ for a curve $C \subset S$, then $C(\beta)$ is the class of the curve obtained by fixing $d-1$ distinct points away from $C$ and letting a single point move on $C$. For brevity, we often write $\beta$ for $C(\beta)$. For $d \geq 2$ let $$A = {{\mathfrak{p}}}_{-2}({{\omega}}_S) {{\mathfrak{p}}}_{-1}({{\omega}}_S)^{d-2} 1_S$$ be the class of an exceptional curve – the locus of 2-fat points centered at a point $P \in S$ plus $d-2$ distinct points away from $P$. For $d \geq 2$ we have $$H_2(\operatorname{\mathsf{Hilb}}^d(S);{{\mathbb{Z}}}) = \big\{ \beta + kA\ \big|\ \beta \in H_2(S;{{\mathbb{Z}}}), k \in {{\mathbb{Z}}}\big\}.$$\
Let $\beta + k A \in H_2(\operatorname{\mathsf{Hilb}}^d(S))$ be a non-zero effective curve class and consider the moduli space $${{\overline M}}_{g,m}( \operatorname{\mathsf{Hilb}}^d(S), \beta + kA) \label{rigrge22r}$$ of $m$-marked stable maps[[^1]]{} $f : C \to \operatorname{\mathsf{Hilb}}^d(S)$ of genus $g$ and class $\beta + k A$. Since $\operatorname{\mathsf{Hilb}}^d(S)$ carries a holomorphic symplectic $2$-form, the virtual class on defined by ordinary Gromov-Witten theory vanishes [@KL]. A modified reduced theory was defined in [@GWNL] and gives rise to a non-zero *reduced* virtual class $$[ {{\overline M}}_{g,m}( \operatorname{\mathsf{Hilb}}^d(S), \beta + kA) ]^{\text{red}}$$ of dimension $(1-g) (2d - 3) + 1$, see also [@STV; @Pridham]. Let $${\mathop{\rm ev}\nolimits}_i : {{\overline M}}_{g,m}( \operatorname{\mathsf{Hilb}}^d(S), \beta + kA) \to \operatorname{\mathsf{Hilb}}^d(S), \quad i = 1, \dots, n$$ be the evaluation maps. The *reduced Gromov-Witten invariant* of $\operatorname{\mathsf{Hilb}}^d(S)$ in genus $g$ and class $\beta + kA$ with primary insertions $$\gamma_1, \dots, \gamma_m \in H^{\ast}(\operatorname{\mathsf{Hilb}}^d(S))$$ is defined by $$\big\langle\, \gamma_1, \dots, \gamma_m\, \big\rangle^{\operatorname{\mathsf{Hilb}}^d(S)}_{g,\beta + k A} =
\int_{[\overline{M}_{g,m}(\operatorname{\mathsf{Hilb}}^d(S), \beta + kA)]^{\text{red}}}
{\mathop{\rm ev}\nolimits}_1^{\ast}(\gamma_1) \cup \dots \cup {\mathop{\rm ev}\nolimits}_m^{\ast}(\gamma_m) \,. \label{bbbm}$$ In case $d = 1$ and $k \neq 0$, the moduli space $\overline{M}_{g,m}(\operatorname{\mathsf{Hilb}}^d(S), \beta + kA)$ is empty by convention and the invariant is defined to vanish.
The Yau-Zaslow formula in higher dimensions {#YZ_section_statement_of_results}
-------------------------------------------
Let $\pi : S \to {\mathbb{P}}^1$ be an elliptically fibered K3 surface and let $$\pi^{[d]} : \operatorname{\mathsf{Hilb}}^d(S) {{\ \longrightarrow\ }}\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1) = {\mathbb{P}}^d \,,$$ be the induced *Lagrangian* fibration with generic fiber a smooth Lagrangian torus. Let $$L_z \subset \operatorname{\mathsf{Hilb}}^d(S)$$ denote the fiber of $\pi^{[d]}$ over a point $z \in {\mathbb{P}}^d$.
Let $F \in H_2(S;{{\mathbb{Z}}})$ be the class of a fiber of $\pi$, and let $\beta_h$ be a primitive effective curve class on $S$ with $$F \cdot \beta_h = 1 \quad \text{ and } \quad \beta_h^2 = 2h - 2 \,.$$ For points $z_1, z_2 \in {\mathbb{P}}^d$ and for all $d \geq 1$ and $k \in {{\mathbb{Z}}}$, define the Gromov-Witten invariant $$\begin{aligned}
\mathsf{N}_{d,h,k} &= \big\langle L_{z_1}, L_{z_2} \big\rangle_{\beta_h + kA}^{\operatorname{\mathsf{Hilb}}^d(S)} \\
&=
\int_{[{{\overline M}}_{0,2}(\operatorname{\mathsf{Hilb}}^d(S), \beta_h + kA)]^{\text{red}}} {\mathop{\rm ev}\nolimits}_1^{\ast}(L_{z_1}) \cup {\mathop{\rm ev}\nolimits}_2^{\ast}(L_{z_2})\end{aligned}$$ which (virtually) counts the number of rational curves in class $\beta_h + kA$ incident to the Lagrangians $L_{z_1}$ and $L_{z_2}$. The first result of this paper is a complete evaluation of the invariants $\mathsf{N}_{d,h,k}$.
Define the Jacobi theta function $$\label{FFFdef}
F(z,\tau) = \frac{\vartheta_1(z,\tau)}{\eta^3(\tau)}
= (y^{1/2} + y^{-1/2}) \prod_{m \geq 1} \frac{ (1 + yq^m) (1 + y^{-1}q^m)}{ (1-q^m)^2 }
$$ considered as a formal power series in the variables $$y = -e^{2 \pi i z} \quad \text{ and } \quad q = e^{2 \pi i \tau}$$ where $|q|<1$.
\[MThm0\] For all $d \geq 1$, we have $$\sum_{h \geq 0}\sum_{k \in {{\mathbb{Z}}}} \mathsf{N}_{d,h,k} y^k q^{h-1}\ =\ F(z,\tau)^{2d-2} \cdot \frac{1}{\Delta(\tau)} \label{005}$$ under the variable change $y = - e^{2\pi i z}$ and $q = e^{2 \pi i \tau}$.
The right hand side of is the Fourier expansion of a Jacobi form[^2] of index $d-1$. For $d = 1$ the class $A$ vanishes on $S$ and by convention only the term $k = 0$ is taken in the sum on the left side of . Then, specializes to the Yau-Zaslow formula .
Further Gromov-Witten invariants {#Section_More_evaluations_Introduction}
--------------------------------
Let $S$ be a smooth projective K3 surface, let $\beta_h \in H_2(S,{{\mathbb{Z}}})$ be a primitive curve class of square $$\beta_h^2 = 2h-2,$$ and let $\gamma \in H^2(S, {{\mathbb{Z}}})$ be a cohomology class with $\gamma \cdot \beta_h = 1$ and $\gamma^2 = 0$. We define three sets of invariants.
For $d \geq 2$, consider the classes $$\begin{aligned}
C(\gamma) & = {{\mathfrak{p}}}_{-1}(\gamma) {{\mathfrak{p}}}_{-1}({{\omega}})^{d-1} 1_S \\
A & = {{\mathfrak{p}}}_{-2}({{\omega}}) {{\mathfrak{p}}}_{-1}({{\omega}})^{d-2} 1_S\end{aligned}$$ which were defined in Section \[Section\_Definition\_of\_Gromov\_Witten\]. Define the first two invariants $$\mathsf{N}_{d,h,k}^{(1)} = {\Big\langle}\, C(\gamma)\, {\Big\rangle}_{\beta_h + kA}^{\operatorname{\mathsf{Hilb}}^d(S)}, \quad \quad \quad
\mathsf{N}_{d,h,k}^{(2)} = {\Big\langle}\, A\, {\Big\rangle}_{\beta_h + kA}^{\operatorname{\mathsf{Hilb}}^d(S)}$$ counting rational curves incident to a cycle of class $C(\gamma)$ and $A$ respectively.
For a point $P \in S$ consider the incidence scheme of $P$, $$I(P) = \{\, \xi \in \operatorname{\mathsf{Hilb}}^d(S)\ |\ P \in \xi\, \} \,.$$ For generic points $P_1, \dots, P_{2d-2} \in S$ define the third invariant $$\mathsf{N}_{d,h,k}^{(3)} = {\Big\langle}\, I(P_1),\, \dots,\, I(P_{2d-2})\, {\Big\rangle}^{\operatorname{\mathsf{Hilb}}^d(S)}_{\beta_h + kA}. \label{GW33invs}$$
By geometric recursions, the invariants $\mathsf{N}_{d,h,k}^{(i)}, i=1,2,3$ and $\mathsf{N}_{d,h,k}$ determine the full Gromov-Witten theory of $\operatorname{\mathsf{Hilb}}^2(S)$ in genus $0$. In case $d = 2$ the invariants are also related to counting hyperelliptic curves on a K3 surface passing through $2$ generic points, see Section \[Section\_hyperelliptic\_curves\] below. The following theorem provides a full evaluation of the invariants $\mathsf{N}_{d,h,k}^{(i)}$. Consider the formal variables $$y = - e^{2 \pi i z} \quad \text{ and } \quad q = e^{2 \pi i \tau}$$ expanded in the region $|y|<1$ and $|q| < 1$, and the function $$\label{G_Function_def}
\begin{aligned}
G(z, \tau) & = F(z,\tau)^2 \left( y \frac{d}{dy} \right)^2 \log( F(z,\tau) ) \\
& = F(z,\tau)^2 \cdot \bigg\{ \frac{y}{(1+y)^2} - \sum_{d \geq 1} \sum_{m | d} m \big( (-y)^{-m} + (-y)^m \big) q^d \bigg\} \\
& = 1 + (y^{-2} + 4 y^{-1} + 6 + 4 y^{1} + y^2) q \\
& \quad \quad \quad \quad \quad \quad + (6y^{-2} + 24y^{-1}+ 36 + 24y + 6y^2)q^2 + \, \dots \ .
\end{aligned}$$
\[MThm\] For all $d \geq 2$, we have $$\begin{aligned}
\sum_{h \geq 0} \sum_{k \in {{\mathbb{Z}}}} \mathsf{N}^{(1)}_{d,h,k} y^k q^{h-1} & = G(z,\tau)^{d-1}\frac{1}{\Delta(\tau)} \\[4pt]
\sum_{h \geq 0} \sum_{k \in {{\mathbb{Z}}}} \mathsf{N}^{(2)}_{d,h,k} y^k q^{h-1} & = \frac{1}{2-2d} \Big( y \frac{d}{dy} \big( G(z,\tau)^{d-1} \big) \Big) \frac{1}{\Delta(\tau)} \\[4pt]
\sum_{h \geq 0} \sum_{k \in {{\mathbb{Z}}}} \mathsf{N}^{(3)}_{d,h,k} y^k q^{h-1} & = \frac{1}{d} \binom{2d-2}{d-1} \Big( q \frac{d}{dq} F(z,\tau)\Big)^{2d-2} \frac{1}{\Delta(\tau)}\end{aligned}$$ under the variable change $y = - e^{2 \pi i z}$ and $q = e^{2 \pi i \tau}$.
Quantum cohomology
------------------
### Reduced quantum cohomology
Let ${\hbar}$ be a formal parameter with ${\hbar}^2 = 0$. The reduced quantum cohomology of $\operatorname{\mathsf{Hilb}}^d(S)$ is a formal deformation of the ordinary cup-product multiplication in $H^{\ast}(\operatorname{\mathsf{Hilb}}^d(S))$ defined by $$\langle a \ast b, c \rangle = \langle a \cup b, c \rangle + {\hbar}\sum_{\beta > 0} \langle a,b,c \rangle_{0,\beta}^{\operatorname{\mathsf{Hilb}}^d(S)} q^\beta \,, \label{rgrgrg}$$ where $\langle a,b \rangle = \int_{\operatorname{\mathsf{Hilb}}^d(S)} a \cup b$ is the standard intersection form, $\beta$ runs over all non-zero elements of the cone ${\rm Eff}_{\operatorname{\mathsf{Hilb}}^d(S)}$ of effective curve classes in $\operatorname{\mathsf{Hilb}}^d(S)$, the symbol $q^{\beta}$ denotes the corresponding element in the semi-group algebra, and $\langle a,b,c \rangle_{0,\beta}^{\operatorname{\mathsf{Hilb}}^d(S)}$ denote the *reduced* genus $0$ Gromov-Witten invariants of $\operatorname{\mathsf{Hilb}}^d(S)$ in class $\beta$; see [@QGQC] for the related case of equivariant quantum cohomology.
By the WDVV equation for reduced virtual classes (see Appendix \[section\_reducedWDVV\]), equality defines a commutative and associative product on $$H^{\ast}(\operatorname{\mathsf{Hilb}}^d(S), {{\mathbb{Q}}}) \otimes {{\mathbb{Q}}}[[ {\rm Eff}_{\operatorname{\mathsf{Hilb}}^d(S)} ]] \otimes {{\mathbb{Q}}}[{\hbar}]/{\hbar}^2,$$ which we call the reduced quantum cohomology ring $$Q H^{\ast}( \operatorname{\mathsf{Hilb}}^d(S) ) \,. \label{15fsdfsf}$$
The ordinary cohomology ring structure on $H^{\ast}( \operatorname{\mathsf{Hilb}}^d(S) , {{\mathbb{Q}}})$ has been explicitly determined by Lehn and Sorger in [@LS_K3]. In this paper, we put forth several conjectures and results about its quantum deformation . Our results will concern only the quantum multiplication with a divisor class on $\operatorname{\mathsf{Hilb}}^d(S)$. In other cases [@Lehn2; @LQW; @MO2; @OP; @QGQC], this has been the first step towards a more complete understanding. We will also restrict to *primitive* classes $\beta$ below.
### Elliptic K3 surfaces
Let $\pi : S \to {\mathbb{P}}^1$ be an elliptic K3 surface with a section, and let $B$ and $F$ denote the class of a section and fiber respectively. For every $h \geq 0$, we set $$\beta_h = B + hF \,.$$ For $d \geq 1$ and cohomology classes $\gamma_1, \dots, \gamma_m \in H^{\ast}(\operatorname{\mathsf{Hilb}}^d(S);{{\mathbb{Q}}})$, define the quantum bracket $$\big\langle \gamma_1, \dots, \gamma_m \big\rangle_q^{\operatorname{\mathsf{Hilb}}^d(S)}
= \sum_{h \geq 0} \sum_{k \in {{\mathbb{Z}}}} y^k q^{h-1} \big\langle \gamma_1, \dots, \gamma_m \big\rangle^{\operatorname{\mathsf{Hilb}}^d(S)}_{\beta_h + k A} \,. $$ Define the *primitive* quantum multiplication $\ast_{\text{prim}}$ by $$\big\langle a , b \ast_{\text{prim}} c \big\rangle\
=\ \big\langle a , b \cup c \big\rangle
+ {\hbar}\cdot \big\langle a, b, c \big\rangle_q \,. \label{vsdfdsjf}$$ for all $a,b,c$. Since ${\hbar}^2 = 0$, different curve classes $\beta$ dont interact, and $\ast_{\text{prim}}$ defines a commutative and associative product on $$H^{\ast}(\operatorname{\mathsf{Hilb}}^d(S),{{\mathbb{Q}}}) \otimes {{\mathbb{Q}}}((y))((q)) \otimes {{\mathbb{Q}}}[{\hbar}]/{\hbar}^2 \,. \label{jfnvief}$$ The main result of Section \[Section\_Quantum\_Cohomology\] is a conjecture for the primitive quantum multiplication with divisor classes. By the divisor axiom and by deformation invariance the conjecture explicitly predicts the full $2$-point genus $0$ Gromov-Witten theory of all Hilbert schemes of points of a K3 surface in primitive classes. By direct calculations using the WDVV equation and the evaluations of Section \[Section\_More\_Evaluations\], we prove the conjecture in case $\operatorname{\mathsf{Hilb}}^2(S)$.
### Quasi-Jacobi forms {#Section_Quantum_Cohomology_Jacobi_Forms}
Let $(z,\tau) \in {{\mathbb{C}}}\times {{\mathbb{H}}}$. The ring ${\mathop{\rm QJac}\nolimits}$ of quasi-Jacobi forms is defined as the linear subspace $${\mathop{\rm QJac}\nolimits}\subset {{\mathbb{Q}}}[ F(z,\tau), E_2(\tau), E_4(\tau), \wp(z,\tau), \wp^{\bullet}(z,\tau), J_1(z,\tau)]$$ of functions which are holomorphic at $z=0$ for generic $\tau$; here $F(z,\tau)$ is the Jacobi theta function , $E_{2k}$ are the classical Eisenstein series, $\wp$ is the Weierstrass elliptic function, $\wp^{\bullet}$ is its derivative with respect to $z$, and $J_1$ is the logarithmic derivative of $F$ with respect to $z$, see Appendix \[Appendix\_Quasi\_Jacobi\_Forms\].
We will identify a quasi Jacobi form $\psi \in {\mathop{\rm QJac}\nolimits}$ with its power series expansions in the variables $$q = e^{2 \pi i \tau} \quad \text{ and } \quad y = - e^{2 \pi i z}.$$ The space ${\mathop{\rm QJac}\nolimits}$ is naturally graded by index $m$ and weight $k$: $${\mathop{\rm QJac}\nolimits}= \bigoplus_{m \geq 0} \bigoplus_{k \geq -2m} {\mathop{\rm QJac}\nolimits}_{k,m}$$ with finite-dimensional summands ${\mathop{\rm QJac}\nolimits}_{k,m}$.
Based on the proven case of $\operatorname{\mathsf{Hilb}}^2(S)$ and effective calculations for $\operatorname{\mathsf{Hilb}}^d(S)$ for any $d$, we have the following results that link curve counting on $\operatorname{\mathsf{Hilb}}^d(S)$ to quasi-Jacobi forms.
\[jac\_thm\] For all $\mu, \nu \in H^{\ast}(\operatorname{\mathsf{Hilb}}^2(S))$, we have $$\langle \mu, \nu \rangle^{\operatorname{\mathsf{Hilb}}^2(S)}_q = \frac{\psi(z,\tau)}{\Delta(\tau)}$$ for a quasi-Jacobi form $\psi(z,\tau)$ of index $1$ and weight $\leq 6$.
Since ${{\overline M}}_{0}(\operatorname{\mathsf{Hilb}}^2(S), \gamma)$ has virtual dimension $2$ for all $\gamma$, Theorem \[jac\_thm\] implies that the full genus $0$ Gromov-Witten theory of $\operatorname{\mathsf{Hilb}}^2(S)$ in primitive classes is governed by quasi-Jacobi forms.
[**Conjecture J.**]{} [*For $d \geq 1$ and for all $\mu, \nu \in H^{\ast}(\operatorname{\mathsf{Hilb}}^d(S))$, we have $$\langle \mu, \nu \rangle^{\operatorname{\mathsf{Hilb}}^d(S)}_q = \frac{\psi(z,\tau)}{\Delta(\tau)}$$ for a quasi-Jacobi form $\psi(z,\tau)$ of index $d-1$ and weight $\leq 2 + 2d$.*]{}
A sharper formulation of Theorem \[jac\_thm\] and Conjecture J specifying the weight appears in Lemma \[index\_weight\_lemma\_jacforms\].
Application 1: Genus 1 invariants of $\operatorname{\mathsf{Hilb}}^d(S)$
------------------------------------------------------------------------
Let $S$ be a K3 surface and let $\beta_h \in H^2(S, {{\mathbb{Z}}})$ be a primitive curve class of square $\beta_h^2 = 2h-2$. Let $(E,0)$ be a nonsingular elliptic curve with origin $0\in E$, and let $$\overline{M}_{(E,0)}(\operatorname{\mathsf{Hilb}}^d(S), \beta_h + kA) \label{ofovfvfvgd}$$ be the fiber of the forgetful map $${{\overline M}}_{1,1}(\operatorname{\mathsf{Hilb}}^d(S), \beta_h + k A) \to {{\overline M}}_{1,1} \,.$$ over the moduli point $(E,0) \in {{\overline M}}_{1,1}$. Hence, is the moduli space parametrizing stable maps to $\operatorname{\mathsf{Hilb}}^d(S)$ with 1-pointed domain with complex structure [*fixed*]{} after stabilization to be $(E,0)$. The moduli space carries a reduced virtual class of dimension $1$.
For $d>0$ consider the reduced Gromov-Witten potential $$\label{wssw}
\mathsf{H}_d(y,q) =
\sum_{k \in {{\mathbb{Z}}}} \sum_{h\geq 0}y^k q^{h-1}
\int_{[ \overline{M}_{(E,0)}(\operatorname{\mathsf{Hilb}}^d(S), \beta_h + kA) ]^{\text{red}}} \text{ev}_0^*(\beta_{h,k}^\vee)\,,$$ where the divisor class $\beta_{h,k}^\vee \in H^2(\operatorname{\mathsf{Hilb}}^d(S),\mathbb{Q})$ is chosen to satisfy $$\label{ffqq}
\int_{\beta_h + kA} \beta_{h,k}^\vee = 1 \, .$$ The invariants virtually count the number of maps from the elliptic curve $E$ to the Hilbert scheme $\operatorname{\mathsf{Hilb}}^d(S)$ in the classes $\beta_h + kA$. By degenerating $E$ to a nodal curve, resolving and using the divisor equation, the series $\mathsf{H}_d(y,q)$ is seen to not depend on the choice of $\beta_{h,k}^{\vee}$.
The case $d=1$ of the series $\mathsf{H}_d(y,q)$ is determined by the Katz-Klemm-Vafa formula [@MPT]. In case $d=2$ we have the following result.
\[rgergegssa\]Under the variable change $y = - e^{2 \pi i z}$ and $q = e^{2 \pi i \tau}$, $$\mathsf{H}_2(y,q) =
F(z,\tau)^2 \cdot \left( 54 \cdot \wp(z,\tau) \cdot E_2(\tau) - \frac{9}{4} E_2(\tau)^2 + \frac{3}{4} E_4(\tau) \right) \frac{1}{\Delta(\tau)}$$
In joint work with Rahul Pandharipande a correspondence between curve counting on $\operatorname{\mathsf{Hilb}}^d(S)$ and the enumerative geometry of the product Calabi-Yau $S \times E$ was proposed in [@K3xE]. This in turn lead to a explicit conjecture for $\mathsf{H}_d(y,q)$ for all $d$ in terms of the reciprocal of the Igusa cusp form $\chi_{10}$. Proposition verifies this conjecture in case $d=2$.
Application 2: Hyperelliptic curves {#Section_hyperelliptic_curves}
-----------------------------------
A projective nonsingular curve $C$ of genus $g\geq 2$ is [*hyperelliptic*]{} if $C$ admits a degree 2 map to ${\mathbb{P}}^1$, $$C \rightarrow {\mathbb{P}}^1\,.$$ The locus of hyperelliptic curves in the moduli space $M_g$ of non-singular curves of genus $g$ is a closed substack of codimension $g-2$. Let $$\mathcal{H}_g \in H^{2(g-2)}({{\overline M}}_g, {{\mathbb{Q}}})$$ be the stack fundamental class of the closure of nonsingular hyperelliptic curves inside ${{\overline M}}_g$. By results of Faber and Pandharipande [@FPM], $\mathcal{H}_g$ is a tautological class [@FP13] of codimension $g-2$. Let $S$ be a K3 surface, let $\beta_h \in H^2(S)$ be a primitive curve class of square $\beta_h^2 = 2h-2$, and let $$\pi : {{\overline M}}_g(S,\beta_h) \to {{\overline M}}_g$$ be the forgetful map from the moduli space of genus $g$ stable maps to $S$ in class $\beta_h$. A virtual count of genus $g \geq 2$ hyperelliptic curves on $S$ in class $\beta_h$ passing through $2$ general points is defined by the integral $$\mathsf{H}_{g, h} = \int_{ [ {{\overline M}}_{g,2}(S, \beta_h) ]^{\text{red}} } \pi^{\ast}(\mathcal{H}_g) {\mathop{\rm ev}\nolimits}_1^{\ast}( \omega ) {\mathop{\rm ev}\nolimits}_2^{\ast}( \omega ),$$ where $\omega \in H^4(S, {{\mathbb{Z}}})$ is the class of a point. In [@Grab] T. Graber used the genus $0$ Gromov-Witten theory of $\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^2)$ to obtain enumerative results on hyperelliptic curves in ${\mathbb{P}}^2$. A similar strategy has been applied for ${\mathbb{P}}^1 \times {\mathbb{P}}^1$ [@Pon07] and for abelian surfaces [@Ros14; @BOPY] (modulo a transversality result). By arguments parallel to the abelian case [@BOPY], Theorem \[MThm\] above leads to the following prediction for $\mathsf{H}_{g, h}$. Let $\Delta(\tau) = q \prod_{m \geq 1} (1-q^m)^{24}$ be the modular discriminant and let $$\begin{aligned}
F(z,\tau) & = u \, \exp \left( {\sum_{k \geq 1}} \frac{(-1)^k B_{2k}}{2k (2k)!} E_{2k}(\tau) u^{2k} \right)\end{aligned}$$ be the Jacobi theta function which appeared already in expanded in the variables $$q = e^{2 \pi i \tau} \quad \text{ and } \quad u = 2 \pi z \,. \label{wwrewrew}$$ [**Conjecture H.**]{} Under the variable change , $$\sum_{h \geq 0} \sum_{g \geq 2} u^{2g+2} q^{h-1} \mathsf{H}_{g, h} =
\left( q \frac{d}{dq} F(z,\tau) \right)^2 \cdot \frac{1}{\Delta(\tau)}$$ By a direct verification using results of [@BL; @MPT] and an explicit expression [@HM] for $$\mathcal{H}_3 \in H^2({{\overline M}}_3, {{\mathbb{Q}}})\, ,$$ Conjecture H holds in the first non-trivial case $g = 3$. Similar conjectures relating the Gromov-Witten count of $r$-gonal curves on the K3 surface $S$ to the genus $0$ Gromov-Witten invariants of $\operatorname{\mathsf{Hilb}}^r(S)$ can be made.
The virtual counts $\mathsf{H}_{g, h}$ have contributions from the boundary of the moduli space, and do *not* correspond to the actual, enumerative count of hyperelliptic curves on $S$. For example, $\mathsf{H}_{3,1} = - \frac{1}{4}$ is both rational and negative. For $h \geq 0$ BPS numbers $\mathsf{h}_{g, h}$ of genus $g$ hyperelliptic curves on $S$ in class $\beta_h$ are defined by the expansion $$\sum_{g \geq 2} \mathsf{h}_{g, h}
\left( 2 \sin( u/2) \right)^{2g+2}
= \sum_{g \geq 2}\, \mathsf{H}_{g,h} \, u^{2g+2} \,.
\label{BPS_expansion}$$ The invariants $\mathsf{h}_{g, h}$ are expected to yield the enumerative count of genus $g$ hyperelliptic curves in class $\beta_h$ on a generic K3 surface $S$ carrying a curve class $\beta_h$, compare [@BOPY Section 0.2.4].
The invariants $\mathsf{h}_{g,h}$ vanish for $h=0,1$. The first non-vanishing values of $\mathsf{h}_{g, h}$ are presented in Table \[hyptable\] below. The distribution of the non-zero values in Table \[hyptable\] matches precisely the results of Ciliberto and Knutsen in [@CK14 Theorem 0.1]: there exist curves on a generic K3 surface in class $\beta_h$ with normalization a hyperelliptic curve of genus $g$ if and only if $$h\ \geq\ g + \Big\lfloor \frac{g}{2} \Big\rfloor \Big( g - 1 - \Big\lfloor \frac{g}{2} \Big\rfloor \Big) \,.$$
------ ---------------- ---------------- --------------- ------------- --------------------------- -- -- -- -- --
$2$ $3$ $4$ $5$ $6$
$2$ $1$ $0$ $0$ $0$ $\phantom{0}0\phantom{0}$
$3$ $36$ $0$ $0$ $0$ $0$
$4$ $672$ $6$ $0$ $0$ $0$
$5$ $8728$ $204$ $0$ $0$ $0$
$6$ $88830$ $3690$ $9$ $0$ $0$
$7$ $754992$ $47160$ $300$ $0$ $0$
$8$ $5573456$ $476700$ $5460$ $0$ $0$
$9$ $36693360$ $4048200$ $70848$ $36$ $0$
$10$ $219548277$ $29979846$ $730107$ $1134$ $0$
$11$ $1210781880$ $198559080$ $6333204$ $19640$ $0$
$12$ $6221679552$ $1197526770$ $47948472$ $244656$ $36$
$13$ $30045827616$ $6666313920$ $324736392$ $2438736$ $1176$
$14$ $137312404502$ $34612452966$ $2002600623$ $20589506$ $20895$
$15$ $597261371616$ $169017136848$ $11396062440$ $152487720$ $265860$
------ ---------------- ---------------- --------------- ------------- --------------------------- -- -- -- -- --
: The first values for the counts $\mathsf{h}_{g,h}$ of hyperelliptic curves of genus $g$ and class $\beta_h$ on a generic K3 surface $S$ passing through $2$ general points, as predicted by Conjecture H and the BPS expansion .[]{data-label="hyptable"}
Plan of the paper
-----------------
In Section 1, we introduce the bare notational necessities and prove a few general lemmas. In Section 2 we prove Theorem \[MThm0\] by reducing to an elliptic K3 surface with 24 rational nodal fibers and by comparision with rational curve counts on a Kummer K3. In Section 3 and 4 we prove Theorem \[MThm\] by reducing the statement to a calculation of Gromov-Witten invariants of $\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1 \times E)$. This approach is mainly independent from the Kummer K3 geometry used in Section 2, and yields a second proof of Theorem \[MThm0\]. In Section 5, we present the conjectures and results on the quantum cohomology ring of $\operatorname{\mathsf{Hilb}}^d(K3)$. Here we also prove Theorem \[jac\_thm\] and Proposition \[rgergegssa\]. In Appendix A, we present the precise form of the WDVV equations for reduced invariants. In Appendix B, we introduce the notion of a quasi-Jacobi form, and list numerical results related to the conjectures of Section 5.
Acknowledgements
----------------
I would like to thank the following people. First and foremost, my advisor Rahul Pandharipande for suggesting the topic and for all his support over the years. Aaron Pixton for guessing a key series needed in the proof. Qizheng Yin for pointing out the connection between $\operatorname{\mathsf{Hilb}}^2(K3)$ and the Kummer geometry. And Ben Bakker, , Lothar Göttsche, Simon Häberli, Jonas Jermann, Alina Marian, , Davesh Maulik, Andrew Morrison, Martin Raum, Emanuel Scheidegger, Timo Schürg, Qizheng Yin and Paul Ziegler for various discussions and comments related to the topic.
The author was supported by the Swiss National Science Foundation grant SNF 200021\_143274.
Preliminaries
=============
Let $S$ be a smooth projective surface and let $\operatorname{\mathsf{Hilb}}^d(S)$ be the Hilbert scheme of $d$ points of $S$. By definition, $\operatorname{\mathsf{Hilb}}^0(S)$ is a point parametrizing the empty subscheme.
Notation
--------
We always work over ${{\mathbb{C}}}$. All cohomology coefficients are in ${{\mathbb{Q}}}$ unless denoted otherwise. We let $[V]$ denote the homology class of an algebraic cycle $V$.
On a connected smooth projective variety $X$, we will freely identify cohomology and homology classes by Poincare duality. We write $$\begin{aligned}
{{\omega}}= {{\omega}}_X & \in H^{2 \dim(X)}(X;{{\mathbb{Z}}}), \\
e = e_X & \in H^0(X;{{\mathbb{Z}}})\end{aligned}$$ for the class of a point and the fundamental class of $X$ respectively. Using the degree map we identify the top cohomology class with the underlying ring: $$H^{2 \dim(X)}(X, {{\mathbb{Q}}}) \equiv {{\mathbb{Q}}}.$$ The tangent bundle of $X$ is denoted by $T_X$.
A homology class $\beta \in H_2(X, {{\mathbb{Z}}})$ is an *effective curve class* if $X$ admits an algebraic curve $C$ of class $[C] = \beta$. The class $\beta$ is *primitive* if it is indivisible in $H_2(X,{{\mathbb{Z}}})$.
Cohomology of $\operatorname{\mathsf{Hilb}}^d(S)$ {#cohhil}
-------------------------------------------------
### The Nakajima basis
Let $(\mu_1, \dots, \mu_l)$ with $\mu_1 \geq \ldots \geq \lambda_l \geq 1$ be a partition and let $$\alpha_1, \dots, \alpha_l \in H^{\ast}(S; {{\mathbb{Q}}})$$ be cohomology classes on $S$. We call the tuple $$\mu = \big( (\mu_1, \alpha_1), \dots, (\mu_l, \alpha_l) \big) \label{iejfisfsg}$$ a cohomology-weighted partition of size $|\mu| = \sum \mu_i$.
If the set $\{ \alpha_1, \dots, \alpha_l \}$ is ordered, we call ordered if for all $i \leq j$ $$\mu_i \geq \mu_j \quad \text{ or } \quad ( \mu_i = \mu_j\ \text{ and }\ \alpha_i \geq \alpha_j ) \,.$$ For $i > 0$ and $\alpha \in H^{\ast}(S ; {{\mathbb{Q}}})$, let $${{\mathfrak{p}}}_{-i}(\alpha) : H^{\ast}( \operatorname{\mathsf{Hilb}}^d(S) , {{\mathbb{Q}}}) \to H^{\ast}( \operatorname{\mathsf{Hilb}}^{d+i}(S) , {{\mathbb{Q}}})$$ be the Nakajima creation operator [@N2], and let $$1_S \in H^{\ast}( \operatorname{\mathsf{Hilb}}^0(S), {{\mathbb{Q}}}) = {{\mathbb{Q}}}$$ be the vacuum vector. A cohomology weighted partition defines the cohomology class $${{\mathfrak{p}}}_{-\mu_1}(\alpha_1) \dots {{\mathfrak{p}}}_{-\mu_l}(\alpha_l) \, 1_S \, \in\, H^{\ast}( \operatorname{\mathsf{Hilb}}^{|\mu|}(S) ) \,.$$
Let $\alpha_1, \dots, \alpha_p$ be a homogeneous ordered basis of $H^{\ast}(S ; {{\mathbb{Q}}})$. By a theorem of Grojnowski [@GrojH] and Nakajima [@N2], the cohomology classes associated to all ordered cohomology weighted partitions of size $d$ with cohomology weighting by the $\alpha_i$ not repeating factors $(\alpha_j,k)$ with $\alpha_j$ odd, form a basis of the cohomology $H^{\ast}(\operatorname{\mathsf{Hilb}}^d(S) ; {{\mathbb{Q}}})$.
### Special cycles {#special_cycles}
We will require several natural cycles and their cohomology classes. In the definitions below, we set ${{\mathfrak{p}}}_{-m}(\alpha)^{k} = 0$ whenever $k < 0$.
**(i) The diagonal**\
The diagonal divisor $$\Delta_{\operatorname{\mathsf{Hilb}}^d(S)} \subset \operatorname{\mathsf{Hilb}}^d(S)$$ is the reduced locus of subschemes $\xi \in \operatorname{\mathsf{Hilb}}^d(S)$ such that ${\rm len}({{\mathcal O}}_{\xi,x}) \geq 2$ for some $x \in S$. It has cohomology class $$[ \Delta_{\operatorname{\mathsf{Hilb}}^d(S)} ]\, =\, \frac{1}{(d-2)!} \, {{\mathfrak{p}}}_{-2}(e) {{\mathfrak{p}}}_{-1}(e)^{d-2} 1_S \, =\, -2 \cdot c_1({{\mathcal O}}_{S}^{[d]}),$$ where we let $E^{[d]}$ denote the tautological bundle on $\operatorname{\mathsf{Hilb}}^d(S)$ associated to a vector bundle $E$ on $S$, see [@Lehn2; @Lehn].
**(ii) The exceptional curve**\
Let $\operatorname{Sym}^d(S)$ be the $d$-th symmetric product of $S$ and let $$\rho : \operatorname{\mathsf{Hilb}}^{d}(S) \to \operatorname{Sym}^d(S),\ \xi \mapsto \small{\sum}_{x \in S} {\mathop{\rm len}\nolimits}({{\mathcal O}}_{\xi,x}) x$$ be the Hilbert-Chow morphism.
For distinct points $x, y_1, \dots, y_{d-2} \in S$ where $d \geq 2$, the fiber of $\rho$ over $$2x + \textstyle{\sum}_i y_i \in \operatorname{Sym}^d(S)$$ is isomorphic to ${\mathbb{P}}^1$ and called an *exceptional curve*. For all $d$ define the cohomology class $$A \, = \, {{\mathfrak{p}}}_{-2}({{\omega}}) {{\mathfrak{p}}}_{-1}({{\omega}})^{d-2} 1_S \,,$$ where ${{\omega}}\in H^4(S, {{\mathbb{Z}}})$ is the class of a point on $S$. If $d \geq 2$ every exceptional curve has class $A$.
**(iii) The incidence subschemes**\
Let $z \subset S$ be a zero-dimensional subscheme. The incidence scheme of $z$ is the locus $$I(z) = \{\ \xi \in \operatorname{\mathsf{Hilb}}^d(S) \ |\ z \subset \xi\ \}$$ endowed with the natural subscheme structure.
**(iv) Curve classes**\
For $\beta \in H_2(S)$ and $a,b \in H_1(S)$, define $$\label{xiudemjck}
\begin{alignedat}{2}
C(\beta) & = {{\mathfrak{p}}}_{-1}(\beta) {{\mathfrak{p}}}_{-1}({{\omega}})^{d-1} 1_S\ & & \in H_{2}(\operatorname{\mathsf{Hilb}}^d(S)), \\
C(a, b) & = {{\mathfrak{p}}}_{-1}(a) {{\mathfrak{p}}}_{-1}(b) {{\mathfrak{p}}}_{-1}({{\omega}})^{d-2} 1_S \ & & \in H_{2}(\operatorname{\mathsf{Hilb}}^d(S)) \,.
\end{alignedat}$$ In unambiguous cases, we write $\beta$ for $C(\beta)$. By Nakajima’s theorem, the assignment induces for $d \geq 2$ the isomorphism $$\begin{aligned}
H_2(S, {{\mathbb{Q}}}) \oplus \wedge^2 H_1(S, {{\mathbb{Q}}}) \oplus {{\mathbb{Q}}}& \to H_2( \operatorname{\mathsf{Hilb}}^d(S) ; {{\mathbb{Q}}}) \\
(\beta, a \wedge b, k) & \mapsto \beta + C(a,b) + k A \,.\end{aligned}$$ If $d \leq 1$ and we write $$\beta + \sum_i C(a_i,b_i) + k A \, \in H_2(\operatorname{\mathsf{Hilb}}^d(S), {{\mathbb{Q}}})$$ for some $\beta, a_i, b_i, k$, we *always assume* $a_i = b_i = 0$ and $k = 0$. If $d = 0$, we also assume $\beta = 0$. This will allow us to treat $\operatorname{\mathsf{Hilb}}^d(S)$ simultaneously for all $d$ at once, see for example Section \[Section\_Curves\_in\_Hilbd\].
**(v) Partition cycles**\
Let $V \subset S$ be a subscheme, let $k \geq 1$ and consider the diagonal embedding $$\iota_k : S \to \operatorname{Sym}^k(S)$$ and the Hilbert-Chow morphism $$\rho : \operatorname{\mathsf{Hilb}}^k(S) \to \operatorname{Sym}^k(S).$$ The $k$-*fattening* of $V$ is the subscheme $$V[k] = \rho^{-1}( i_k(V) ) \subset \operatorname{\mathsf{Hilb}}^k(S) \,.$$
Let $d= d_1 + \dots + d_r$ be a partition of $d$ into integers $d_i \geq 1$, and let $$V_1, \dots, V_r \subset S$$ be pairwise disjoint subschemes on $S$. Consider the open subscheme $$U = \big\{ (\xi_1, \dots, \xi_r) \in \operatorname{\mathsf{Hilb}}^{d_1}(S) \times \cdots \times \operatorname{\mathsf{Hilb}}^{d_r}(S) \ |\ \xi_i \cap \xi_j = \varnothing \text{ for all } i \neq j \big\} \label{111}$$ and the natural map $\sigma : U \to \operatorname{\mathsf{Hilb}}^d(S)$, which sends a tuple of subschemes $(\xi_1, \dots, \xi_r)$ defined by ideal shaves $I_{\xi_i}$ to the subscheme $\xi \in \operatorname{\mathsf{Hilb}}^d(S)$ defined by the ideal sheaf . We often use the shorthand notation[^3] $$\sigma(\xi_1, \dots, \xi_r)\, =\, \xi_1 + \dots + \xi_r. \label{116}$$ We define the *partition cycle* as $$V_1[d_1]\, \cdots \, V_r[d_r]\, =\, \sigma(\, V_1[d_1] \times \dots \times V_r[d_r]\, ) \subset \operatorname{\mathsf{Hilb}}^d(S). \label{505}$$ By [@Nakajima Thm 9.10], the subscheme has cohomology class $${{\mathfrak{p}}}_{-d_1}(\alpha_1) \cdots {{\mathfrak{p}}}_{-d_r}(\alpha_r) 1_S \in H^{\ast}(\operatorname{\mathsf{Hilb}}^d(S)),$$ where $\alpha_i = [V_i]$ for all $i$.
Curves in $\operatorname{\mathsf{Hilb}}^d(S)$ {#Section_Curves_in_Hilbd}
---------------------------------------------
### Cohomology classes
Let $C$ be a projective curve and let $f : C \to \operatorname{\mathsf{Hilb}}^d(S)$ be a map. Let $p : {{\mathcal Z}}_d \to \operatorname{\mathsf{Hilb}}^d(S)$ be the universal subscheme and let $q : {{\mathcal Z}}_d \to S$ be the universal inclusion. Consider the fiber diagram $$\label{pullback_diagram}
\begin{tikzcd}
\widetilde{C} \ar{r}{\widetilde{f}} \ar{d}{\widetilde{p}} & {{\mathcal Z}}_d \ar{d}{p} \ar{r}{q} & S \\
C \ar{r}{f} & \operatorname{\mathsf{Hilb}}^d(S)
\end{tikzcd}
$$ and let $f' = q \circ \widetilde{f}$. The embedded curve $\widetilde{C} \subset C \times S$ is flat of degree $d$ over $C$. By the universal property of $\operatorname{\mathsf{Hilb}}^d(S)$, we can recover $f$ from $\widetilde{C}$. Here, even when $C$ is a smooth connected curve, $\widetilde{C}$ could be disconnected, singular and possibly non-reduced.
\[pullback\_lemma\] Let $C$ be a reduced projective curve and let $f : C \to \operatorname{\mathsf{Hilb}}^d(S)$ be a map with $$f_{\ast} [C] = \beta + \sum_{j} C(\gamma_j, \gamma_j') + k A \label{xxvv}$$ for some $\beta \in H_2(S), \gamma_j, \gamma_j' \in H_1(S)$ and $k \in {{\mathbb{Z}}}$. Then, $$( q \circ \widetilde{f} )_{\ast} [\widetilde{C}] = \beta \,.$$
We may assume $d \geq 2$ and $C$ irreducible. Since $\widetilde{p}$ is flat, $$f'_{\ast} [\widetilde{C}] = f'_{\ast} \widetilde{p}^{\ast} [C] = q_{\ast} p^{\ast} f_{\ast} [C].$$ Therefore, the claim of Lemma \[pullback\_lemma\] follows from and $$q_{\ast} p^{\ast} A = 0, \quad q_{\ast} p^{\ast} C(\beta) = \beta, \quad q_{\ast} p^{\ast} C(a,b) = 0$$ for all $\beta \in H_2(S)$ and $a,b \in H_1(S)$. By considering an exceptional curve of class $A$, one finds $q_{\ast} p^{\ast} A = 0$. We will verify $q_{\ast} p^{\ast} C(\beta) = \beta$; the equation $q_{\ast} p^{\ast} C(a,b) = 0$ is similar.
Let $U \subset S^d$ be the open set defined in and let $\sigma : U \to \operatorname{\mathsf{Hilb}}^d(S)$ be the sum map. We have $C(\beta) = \sigma_{\ast}( {{\omega}}^{d-1} \times \beta )$. Consider the fiber square $$\begin{tikzcd}
\widetilde{U} \ar{r} \ar{d}{p'} & {{\mathcal Z}}_d \ar{d}{p} \ar{r}{q} & S \\
U \ar{r} & \operatorname{\mathsf{Hilb}}^d(S) \,.
\end{tikzcd}$$ Let $\Delta_{i,d+1} \subset S^d \times S$ be the $(i,d+1)$ diagonal. Then $\widetilde{U} \subset S^d \times S$ is the disjoint union $\bigcup_{i=1, \dots,d} \Delta_{i,d+1} \cap (U \times S)$. Therefore $$\begin{aligned}
q_{\ast} p^{\ast} C(\beta)
& = q_{\ast} p^{\ast} \sigma_{\ast} ({{\omega}}^{d-1} \times \beta) \\
& = {\mathop{\rm pr}\nolimits}_{d+1 \ast} p^{\prime \ast} ({{\omega}}^{d-1} \times \beta) \\
& = \sum_{i = 1}^{d} {\mathop{\rm pr}\nolimits}_{d+1 \ast} ([\Delta_{i,d+1}] \cdot ({{\omega}}^{d-1} \times \beta \times e_S)) \\
& = \beta \,. \qedhere\end{aligned}$$
\[eulcharlemma\] Let $C$ be a smooth, projective, connected curve of genus $g$ and let $f: C \to \operatorname{\mathsf{Hilb}}^d(S)$ be a map of class . Then $$k = \chi({{\mathcal O}}_{\widetilde{C}}) - d (1-g)$$
The intersection of $f_{\ast} [C]$ with the diagonal class $\Delta = -2 c_1( {{\mathcal O}}_S^{[d]} )$ is $-2k$. Therefore $$k = \deg(c_1({{\mathcal O}}_{S}^{[d]}) \cap f_{\ast}[C]) = \deg( f^{\ast} {{\mathcal O}}_{S}^{[d]} ) = \chi( f^{\ast} {{\mathcal O}}_{S}^{[d]} ) - d (1-g),$$ where we used Riemann-Roch in the last step. Since we have $$f^{\ast} {{\mathcal O}}_{S}^{[d]} = f^{\ast} p_{\ast} q^{\ast} {{\mathcal O}}_S = \widetilde{p}_{\ast} \widetilde{f}^{\ast} q^{\ast} {{\mathcal O}}_S = \widetilde{p}_{\ast} {{\mathcal O}}_{\widetilde{C}}$$ and $\widetilde{p}$ is finite, we obtain $\chi( f^{\ast} {{\mathcal O}}_{S}^{[d]} ) = \chi( \widetilde{p}_{\ast} {{\mathcal O}}_{\widetilde{C}} ) = \chi( {{\mathcal O}}_{\widetilde{C}} )$.
\[bounded\] Let $\gamma \in H_2(\operatorname{\mathsf{Hilb}}^d(S), {{\mathbb{Z}}})$ and let ${{\overline M}}_0(\operatorname{\mathsf{Hilb}}^d(S), \gamma)$ be the moduli space of stable maps of genus $0$ in class $\gamma$. Then for $m \ll 0$, $${{\overline M}}_0(\operatorname{\mathsf{Hilb}}^d(S), \gamma + m A) = \varnothing$$
Let $f : {\mathbb{P}}^1 \to \operatorname{\mathsf{Hilb}}^d(S)$ be a map in class $\gamma + mA$. The cohomology class of the corresponding curve $\widetilde{C} = f^{\ast} {{\mathcal Z}}_d \subset {\mathbb{P}}^1 \times S$ is independent of $m$. Hence, the holomorphic Euler characteristic $\chi({{\mathcal O}}_{\widetilde{C}})$ is bounded from below by a constant independent of $m$. Therefore, by Lemma \[eulcharlemma\], we find $m$ to be bounded from below when the domain curve is ${\mathbb{P}}^1$. Since an effective class $\gamma + m A$ decomposes in at most finitely many ways in a sum of effective classes, the claim is proven.
### Irreducible Components {#irreducible_components}
\[Section\_irreducible\_components\] Let $f:C \to \operatorname{\mathsf{Hilb}}^d(S)$ be a map and consider the fiber diagram $$\begin{tikzcd}
\llap{$\widetilde{C} ={}$} f^{\ast} {{\mathcal Z}}_d \ar{r} \ar{d}{\widetilde{p}} & {{\mathcal Z}}_d \ar{d}{p} \\
C \ar{r}{f} & \operatorname{\mathsf{Hilb}}^d(S) \,,
\end{tikzcd}$$ where $p : {{\mathcal Z}}_d \to \operatorname{\mathsf{Hilb}}^d(S)$ is the universal family.
The map $f$ is *irreducible*, if $f^{\ast} {{\mathcal Z}}_d$ is irreducible.
Let $d \geq 1$ and let $f : C \to \operatorname{\mathsf{Hilb}}^d(S)$ be a map from a connected non-singular projective curve $C$. Consider the (reduced) irreducible components $$G_1, \dots, G_r$$ of the curve $\widetilde{C} = f^{\ast} {{\mathcal Z}}_d$, and let $$\xi = \cup_{i \neq j}\, \widetilde{p}(G_i \cap G_j)\, \subset\, C$$ be the image of their intersection points under $\widetilde{p}$. Every *connected* component $D$ of $\widetilde{C} \setminus \widetilde{p}^{-1}(\xi)$ is an irreducible curve and flat over $C \setminus \xi$. Since $C$ is a non-singular curve, also the closure $\overline{D}$ is flat over $C$, and by the universal property of $\operatorname{\mathsf{Hilb}}^{d'}(S)$ yields an associated irreducible map $$C \to \operatorname{\mathsf{Hilb}}^{d'}(S)$$ for some $d' \leq d$. Let $\phi_1, \dots, \phi_r$ be the irreducible maps associated to all connected components of $\widetilde{C} \setminus \widetilde{p}^{-1}(\xi)$. We say $f$ *decomposes into the irreducible components* $\phi_1, \dots, \phi_r$.
Conversely, let $\phi_i : C \to \operatorname{\mathsf{Hilb}}^{d_i}(S), i=1,\dots,n$ be irreducible maps with
- $\sum_i d_i = d$,
- $\phi_i^{\ast} {{\mathcal Z}}_{d_i} \cap \phi_{j}^{\ast} {{\mathcal Z}}_{d_j}$ is of dimension $0$ for all $i \neq j$.
Let $U$ be the open subset defined in . The map $$(\phi_1, \dots, \phi_n) : C {{\ \longrightarrow\ }}\operatorname{\mathsf{Hilb}}^{d_1}(S) \times \dots \times \operatorname{\mathsf{Hilb}}^{d_n}(S)$$ meets the complement of $U$ in a finite number of points $x_1, \dots, x_m \in C$. By smoothness of $C$, the composition $$\sigma \circ (\phi_1, \dots, \phi_n) : C \setminus \{ x_1, \dots, x_m \} {{\ \longrightarrow\ }}\operatorname{\mathsf{Hilb}}^d(S)$$ extends uniquely to a map $f : C \to \operatorname{\mathsf{Hilb}}^d(S)$.
A direct verification shows that the two operations above are inverse to each other. We write $$f = \phi_1 + \dots + \phi_r$$ for the decomposition of $f$ into the irreducible components $\phi_1, \dots, \phi_r$.
Let $\beta, \beta_i \in H_2(S)$, $\gamma_{j}, \gamma'_{j}, \gamma_{i,j}, \gamma'_{i,j} \in H_1(S)$ and $k, k_i \in {{\mathbb{Z}}}$ such that $$\begin{aligned}
f_{\ast}[C] & = C(\beta) + \sum_j C(\gamma_j, \gamma'_j) + k A\ \in H_{2}(\operatorname{\mathsf{Hilb}}^d(S))\\
\phi_{i \ast}[C] & = C(\beta_i) + \sum_j C(\gamma_{i,j}, \gamma'_{i,j}) + k_i A\ \in H_2(\operatorname{\mathsf{Hilb}}^{d_i}(S)) \,.\end{aligned}$$
\[hom\_add\_up\] We have
- $\sum_i \beta_i = \beta \, \in H_2(S;{{\mathbb{Z}}})$
- $\sum_{i,j} \gamma_{i,j} \wedge \gamma'_{i,j} = \sum_j \gamma_j \wedge \gamma'_j\, \in \bigwedge^2 H_1(S)$ .
This follows directly from [@Nakajima Theorem 9.10].
The Yau-Zaslow formula in higher dimensions {#YZ_formula_in_higher_dim}
===========================================
\[Section\_higher\_dimensional\_Yau\_Zaslow\]
Overview
--------
In the remainder of section \[YZ\_formula\_in\_higher\_dim\] we give a proof of Theorem \[MThm0\]. The proof proceeds in the following steps.
In section \[sec\_ellK3case\] we use the deformation theory of K3 surfaces to reduce Theorem \[MThm0\] to an evaluation on a specific elliptic K3 surface $S$. Here, we also analyse rational curves on $\operatorname{\mathsf{Hilb}}^d(S)$ and prove a few Lemmas. This discussion will be used also later on.
In section \[basic\_case\], we study the structure of the moduli space of stable maps which are incident to the Lagrangians $L_{z_1}$ and $L_{z_2}$. The main result is a splitting statement (Proposition \[W0splitprop\]), which reduces the computation of Gromov-Witten invariants to integrals associated to fixed elliptic fibers.
In section \[section\_Kummer\_evaluation\], we evaluate these remaining integrals using the geometry of the Kummer K3 surfaces, the Yau-Zaslow formula and a theta function associated to the $\mathsf{D}_4$ lattice.
The Bryan-Leung K3 {#sec_ellK3case}
------------------
### Definition {#Section_BLK3_Defn}
Let $\pi : S \to {\mathbb{P}}^1$ be an elliptic K3 surface with a unique section $s : {\mathbb{P}}^1 \to S$ and 24 rational nodal fibers. We call $S$ a *Bryan-Leung K3 surface*.
Let $x_1, \dots, x_{24} \in {\mathbb{P}}^1$ be the basepoints of the nodal fibers of $\pi$, let $B_0$ be the image of the section $s$, and let $$F_x \subset S$$ denote the fiber of $\pi$ over a point $x \in {\mathbb{P}}^1$. The Picard group $${\mathop{\rm Pic}\nolimits}(S) = H^{1,1}(S;{{\mathbb{Z}}}) = H^{2}(S;{{\mathbb{Z}}}) \cap H^{1,1}(S;{{\mathbb{C}}})$$ is of rank $2$ and generated by the section class $B$ and the fiber class $F$. We have the intersection numbers $B^2 = -2$, $B \cdot F = 1$ and $F^2 = 0$. Hence for all $h \geq 0$ the class $$\beta_h = B + hF \in H_2(S;{{\mathbb{Z}}}) \label{BL_betah}$$ is a primitive and effective curve class of square $\beta_h^2 = 2h - 2$.
The projection $\pi$ and the section $s$ induce maps of Hilbert schemes $$\pi^{[d]} : \operatorname{\mathsf{Hilb}}^d(S) \to \operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1) = {\mathbb{P}}^d, \quad \quad s^{[d]} : {\mathbb{P}}^d \to \operatorname{\mathsf{Hilb}}^d(S),$$ such that $\pi^{[d]} \circ s^{[d]} = \operatorname{id}_{{\mathbb{P}}^d}$. The map $s^{[d]}$ is an isomorphism from $\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)$ to the locus of subschemes of $S$, which are contained in $B_0$. This gives natural identifications $${\mathbb{P}}^d = \operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1) = \operatorname{\mathsf{Hilb}}^d( B_0 ) \,,$$ that we will use sometimes. In unambiguous cases we also write $\pi$ and $s$ for $\pi^{[d]}$ and $s^{[d]}$ respectively.
### Main statement revisited {#K3statements}
For $d \geq 1$ and cohomology classes $\gamma_1, \dots, \gamma_m \in H^{\ast}(\operatorname{\mathsf{Hilb}}^d(S);{{\mathbb{Q}}})$ define the quantum bracket $$\big\langle \gamma_1, \dots, \gamma_m \big\rangle_q^{\operatorname{\mathsf{Hilb}}^d(S)}
= \sum_{h \geq 0} \sum_{k \in {{\mathbb{Z}}}} y^k q^{h-1} {\Big\langle}\gamma_1, \dots, \gamma_m {\Big\rangle}^{\operatorname{\mathsf{Hilb}}^d(S)}_{\beta_h + k A} \,,$$ where the bracket on the right hand side was defined in .
\[ellthm\] For all $d \geq 1$, $${\Big\langle}{{\mathfrak{p}}}_{-1}(F)^d 1_S \ , \ {{\mathfrak{p}}}_{-1}(F)^d 1_S {\Big\rangle}^{\operatorname{\mathsf{Hilb}}^d(S)}_q \ =\ \frac{F(z,\tau)^{2d-2}}{\Delta(\tau)},$$ where $q = e^{2 \pi i \tau}$ and $y = -e^{2 \pi i z}$.
We begin the proof of Theorem \[ellthm\] in Section \[basic\_case\].
Let $\pi' : S' \to {\mathbb{P}}^1$ be any elliptic K3 surface, and let $F'$ be the class of a fiber of $\pi'$. A fiber of the induced Lagrangian fibration $$\pi^{\prime [d]} : \operatorname{\mathsf{Hilb}}^d(S') \to {\mathbb{P}}^d$$ has class ${{\mathfrak{p}}}_{-1}(F')^d 1_S$. Hence, Theorem \[MThm0\] implies Theorem \[ellthm\]. The following Lemma shows that conversely Theorem \[ellthm\] also implies Theorem \[MThm0\], and hence the claims in both Theorems are equivalent.
\[fijerfmimv\] Let $S$ be the fixed Bryan-Leung K3 surface defined in Section \[Section\_BLK3\_Defn\], and let $\beta_h = B + hF$ be the curve class .
Let $S'$ be a K3 surface with a primitive curve class $\beta$ of square $2h-2$, and let $\gamma \in H^2(S', {{\mathbb{Z}}})$ be any class with $\beta \cdot \gamma = 1$ and $\gamma^2 = 0$. Then $${\Big\langle}{{\mathfrak{p}}}_{-1}(\gamma)^d 1_{S'}\, ,\, {{\mathfrak{p}}}_{-1}(\gamma)^d 1_{S'} {\Big\rangle}_{\beta + kA}^{\operatorname{\mathsf{Hilb}}^d(S')}
=
{\Big\langle}{{\mathfrak{p}}}_{-1}(F)^d 1_S \, , \, {{\mathfrak{p}}}_{-1}(F)^d 1_S {\Big\rangle}^{\operatorname{\mathsf{Hilb}}^d(S)}_{\beta_h + k A} \,.$$
We will construct an algebraic deformation from $S'$ to the fixed K3 surface $S$ such that $\beta$ deforms to $\beta_h$ through classes of Hodge type $(1,1)$, and $\gamma$ deforms to $F$. By the deformation invariance of reduced Gromov-Witten invariants the claim of Lemma \[fijerfmimv\] follows.
Let $E_8(-1)$ be the negative $E_8$ lattice, let $U$ be the hyperbolic lattice and consider the K3 lattice $$\Lambda = E_8(-1)^{\oplus 2} \oplus U^{\oplus 3} \,.$$ Let $e,f$ be a hyperbolic basis for one of the $U$ summands of $\Lambda$ and let $$\phi: \Lambda \overset{\cong}{\longrightarrow} H^{2}(S;{{\mathbb{Z}}})$$ be a fixed marking with $\phi(e) = B + F$ and $\phi(f) = F$. We let $$b_h = e + (h-1) f$$ denote the class corresponding to $\beta_h = B + hF$ under $\phi$.
The orthogonal group of $\Lambda$ is transitive on primitive vectors of the same square, see [@GH Lemma 7.8] for references. Hence there exists a marking $$\phi' : \Lambda \overset{\cong}{\longrightarrow} H^2(S';{{\mathbb{Z}}})$$ such that $\phi'(b_h) = \beta$. Let $g = \phi^{\prime -1}(\gamma) \in \Lambda$ be the vector that corresponds to the class $\gamma$ under $\phi'$. The span $$\Lambda_0 = \langle g, b_h \rangle \subset \Lambda$$ defines a hyperbolic sublattice of $\Lambda$ which, by unimodularity, yields the direct sum decomposition $$\Lambda = \Lambda_0 \oplus \Lambda_0^{\perp}.$$ Because the irreducible unimodular factors of a unimodular lattice are unique up to order, we find $$\Lambda_0^{\perp} \cong E_8(-1)^{\oplus 2} \oplus U^{\oplus 2} \,.$$ Hence there exists a lattice isomorphism $\sigma : \Lambda \to \Lambda$ with $\sigma(b_h) = b_h$ and $\sigma(g) = f$. Replacing $\phi'$ by $\phi' \circ \sigma^{-1}$, we may therefore assume $\phi'(b_h) = \beta$ and $\phi'(f) = \gamma$.
Since the period domain $\Omega$ associated to $b_h$ is connected, there exists a curve inside $\Omega$ connecting the period point of $S'$ to the period point of $S$. Restricting the universal family over $\Omega$ to this curve, we obtain a deformation with the desired properties.
### Rational curves in $\operatorname{\mathsf{Hilb}}^d(S)$ {#Section_Rational_curves}
Let $h \geq 0$ and let $k$ be an integer. We consider rational curves on $\operatorname{\mathsf{Hilb}}^d(S)$ in the classes $\beta_h + kA$ and $h F + kA$.
**Vertical maps**\
Let $u_1, \dots, u_d \in {\mathbb{P}}^1$ be points such that
- $u_i$ is not the basepoint of a nodal fiber of $\pi : S \to {\mathbb{P}}^1$ for all $i$,
- the points $u_1, \dots, u_n$ are pairwise distinct.
Then, the fiber of $\pi^{[d]}$ over $u_1 + \dots + u_d \in \operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)$ is isomorphic to the product of smooth elliptic curves $$F_{u_1} \times \ldots \times F_{u_d} \,.$$ The subset of points in $\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)$ whose preimage under $\pi^{[d]}$ is not of this form is the divisor $${{{{\mathcal W}}}}= I(x_1) \cup \dots I(x_{24}) \cup \Delta_{\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)} \subset \operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1), \label{501}$$ where $x_1, \dots, x_{24}$ are the basepoints of the nodal fibers of $\pi$, $I(x_i)$ is the incidence subscheme, and $\Delta_{\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)}$ is the diagonal, see Section \[special\_cycles\]. Since a fiber of $\pi^{[d]}$ over a point $z \in {\mathbb{P}}^d$ is non-singular if and only if $z \notin {{{{\mathcal W}}}}$, we call ${{{{\mathcal W}}}}$ the *discriminant* of $\pi^{[d]}$.
Consider a stable map $f : C \to \operatorname{\mathsf{Hilb}}^d(S)$ of genus $0$ and class $h F + kA$. Since the composition $$\pi^{[d]} \circ f : C \to \operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)$$ is mapped to a point, and since non-singular elliptic curves do not admit non-constant rational maps, we have the following Lemma.
\[200\] Let $f : C \to \operatorname{\mathsf{Hilb}}^d(S)$ be a non-constant genus $0$ stable map in class $h F + kA$. Then the image of $\pi^{[d]} \circ f$ lies in the discriminant ${{{{\mathcal W}}}}$.
**Non-vertical maps**\
Let $f : C \to \operatorname{\mathsf{Hilb}}^d(S)$ be a stable genus $0$ map in class $f_{\ast} [C] = \beta_h + kA$. The composition $$\pi^{[d]} \circ f : C {{\ \longrightarrow\ }}{\mathbb{P}}^d$$ has degree $1$ with image a line $$L \subset {\mathbb{P}}^d.$$ Let $C_0$ be the unique irreducible component of $C$ on which $\pi \circ f$ is non-constant. We call $C_0 \subset C$ the *distinguished component* of $C$.
Since $C_0 \cong {\mathbb{P}}^1$, we have a decomposition $$f|_{C_0} = \phi_0 + \dots + \phi_r \label{115}$$ of $f|_{C_0}$ into irreducible maps $\phi_i : C_0 \to \operatorname{\mathsf{Hilb}}^{d_i}(S)$ where $d_i$ are positive integers such that $d = d_0 + \dots + d_r$, see Section \[irreducible\_components\]. By Lemma \[hom\_add\_up\], exactly one of the maps $\pi^{[d_i]} \circ \phi_i$ is non-constant; we assume this map is $\phi_0$.
\[117\] Let ${{{{\mathcal W}}}}$ be the discriminant of $\pi^{[d]}$. If $L \nsubseteq {{{{\mathcal W}}}}$, then
- $d_i = 1$ for all $i \in \{1,\dots, r\}$,
- $\phi_i : C_0 \to S$ is constant for all $i \in \{ 1,\dots,r \}$,
- $\phi_0 : C_0 \to \operatorname{\mathsf{Hilb}}^{d_0}(S)$ is an isomorphism onto a line in $\operatorname{\mathsf{Hilb}}^{d_0}(B_0)$.
Assume $L \nsubseteq {{{{\mathcal W}}}}$.
1. If $d_i \geq 2$, then $\pi^{[d_i]} \circ \phi_i$ maps $C_0$ into $\Delta_{\operatorname{\mathsf{Hilb}}^{d_i}({\mathbb{P}}^1)}$. Hence $$\pi^{[d]} \circ f = \textstyle{\sum}_i \pi^{[d_i]} \circ \phi_i$$ maps $C_0$ into $\Delta_{\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)} \subset {{\mathcal W}}$. Since $L = \pi^{[d]} \circ f(C_0)$, we find $L \subset {{\mathcal W}}$, which is a contradiction.
2. If $\phi_i : C_0 \to S$ is non-constant, then $\pi \circ \phi_i$ maps $C_0$ to a basepoint of a nodal fiber of $\pi : S \to {\mathbb{P}}^1$. By an argument identical to (i) this implies $L \subset {{\mathcal W}}$, which is a contradiction. Hence, $\phi_i$ is constant.
3. The universal family of curves on the elliptic K3 surface $\pi : S \to {\mathbb{P}}^1$ in class $\beta_{h} = B + hF$ is the $h$-dimensional linear system $$|\beta_h| = \operatorname{\mathsf{Hilb}}^h({\mathbb{P}}^1) = {\mathbb{P}}^h \,.$$ Explicitly, an element $z \in \operatorname{\mathsf{Hilb}}^h({\mathbb{P}}^1)$ corresponds to the comb curve $$B_0 + \pi^{-1}(z) \subset S \,, \label{307}$$ where $\pi^{-1}(z)$ denotes the fiber of $\pi$ over the subscheme $z \subset {\mathbb{P}}^1$.
Let ${{\mathcal Z}}_d \to \operatorname{\mathsf{Hilb}}^d(S)$ be the universal family and consider the fiber diagram $$\begin{tikzcd}
\widetilde{C_0} \ar{r}{\widetilde{f}} \ar{d}{\widetilde{p}} & {{\mathcal Z}}_d \ar{d}{p} \ar{r}{q} & S \\
C_0 \ar{r}{f} & \operatorname{\mathsf{Hilb}}^d(S) \,.
\end{tikzcd}$$
By Lemma \[pullback\_lemma\], the map $f' = q \circ \widetilde{f} : \widetilde{C_0} \to S$ is a curve in the linear system $|\beta_{h'}|$ for some $h' \leq h$. Its image is therefore a comb of the form .
Let $G_0$ be the irreducible component of $\widetilde{C_0}$ such that $\pi \circ f'|_{G_0}$ is non-constant. The restriction $$\widetilde{p}|_{G_0} : G_0 \to C_0 \label{fvmskfmvdf}$$ is flat. Since $\pi \circ f' : \widetilde{C}_0 \to {\mathbb{P}}^1$ has degree $1$, the curve $\widetilde{C}_0$ has multiplicity $1$ at $G_0$, and the map to the Hilbert scheme of $S$ associated to is equal to $\phi_0$.
Since $G_0$ is reduced and $f'|_{G_0} : G_0 \to S$ maps to $B_0$, the map $\phi_0$ maps with degree $1$ to $\operatorname{\mathsf{Hilb}}^{d_0}(B_0)$. The proof of (iii) is complete.
**The normal bundle of a line**\
Let $s^{[d]} : \operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1) \hookrightarrow \operatorname{\mathsf{Hilb}}^d(S)$ be the section, and consider the normal bundle $$N = s^{[d] \ast} T_{\operatorname{\mathsf{Hilb}}^d(S)} \big/ T_{\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)}.$$
\[normaltoL\] For every line $L \subset \operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)$, $$T_{\operatorname{\mathsf{Hilb}}^d(S)}\big|_{L} = T_{\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)}\big|_{L} \oplus N\big|_{L}$$ with $N\big|_{L} = T^{\vee}_{\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)}\big|_{L} = {{\mathcal O}}_L(-2) \oplus {{\mathcal O}}_L(-1)^{\oplus (d-1)}$.
Because the embedding $s^{[d]} : \operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1) \hookrightarrow \operatorname{\mathsf{Hilb}}^d(S)$ has the right inverse $\pi^{[d]}$, the restriction $$T_{\operatorname{\mathsf{Hilb}}^d(S)} \big|_{\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1) }$$ splits as a direct sum of the tangent and normal bundle of $\operatorname{\mathsf{Hilb}}^d(B_0)$.
The vanishing $H^0( {\mathbb{P}}^d, \Omega_{{\mathbb{P}}^d}^2 ) = 0$ implies that the holomorphic symplectic form on $\operatorname{\mathsf{Hilb}}^d(S)$ restricts to $0$ on $\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)$ and hence, by non-degeneracy, induces an isomorphism $$T_{\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)} \to N^{\vee} \,.$$ Since $T_{\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)}\big|_{L} = {{\mathcal O}}_L(1)^{\oplus (d-1)} \oplus {{\mathcal O}}_L(2)$, the proof is complete.
Analysis of the moduli space {#basic_case}
----------------------------
### Overview {#Section_basic_case_overview}
Let $S$ be the fixed elliptic Bryan-Leung K3 surface, let $z_1, z_2 \in \operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)$ be generic points, and for $i \in \{1,2\}$ let $$Z_{i} = \pi^{[d] -1}(z_i) \subset \operatorname{\mathsf{Hilb}}^d(S)$$ be the fiber of $\pi^{[d]}$ over $z_i$. The subscheme $Z_i$ has class $[Z_i] = {{\mathfrak{p}}}_{-1}(F)^d 1_S$. Let $${\mathop{\rm ev}\nolimits}: {{\overline M}}_{0,2}( \operatorname{\mathsf{Hilb}}^d(S), \beta_h + kA) {{\ \longrightarrow\ }}\operatorname{\mathsf{Hilb}}^d(S) \times \operatorname{\mathsf{Hilb}}^d(S)$$ be the evaluation map from the moduli space of genus $0$ stable maps in class $\beta_h = B + hF$, and define the moduli space $$M_Z = M_Z(h,k) = {\mathop{\rm ev}\nolimits}^{-1}(Z_1 \times Z_2) \label{cvcvdfvf}$$ parametrizing maps which are incident to $Z_1$ and $Z_2$.
In Section \[basic\_case\], we begin the proof of Theorem \[ellthm\] by studying the moduli space $M_Z$ and its virtual class. First, we prove that $M_Z$ is naturally isomorphic to a product of moduli spaces associated to specific fibers of the elliptic fibration $\pi \colon S \to {\mathbb{P}}^1$. Second, we show that the virtual class splits as a product of virtual classes on each factor. Both results are summarized in Proposition \[W0splitprop\]. As a consequence, Theorem \[ellthm\] is reduced to the evaluation of a series $F^{\textup{GW}}(y,q)$ encoding integrals associated to specific fibers of $\pi$.
### The set-theoretic product {#basic_case_settheoretic_splitting}
\[Section\_settheoretic\_splitting\] Consider a stable map $$[f : C \to \operatorname{\mathsf{Hilb}}^d(S), p_1, p_2]\, \in\, M_Z$$ with markings $p_1,p_2 \in C$. By definition of $M_Z$, we have $$\pi^{[d]}(f(p_1)) = z_1, \quad \quad \pi^{[d]}(f(p_2)) = z_2 .$$ Hence, the image of $C$ under $\pi^{[d]} \circ f$ is the unique line $$L \subset {\mathbb{P}}^d$$ incident to the points $z_1, z_2 \in {\mathbb{P}}^d$. Because $z_1, z_2 \in {\mathbb{P}}^d$ are generic, also $L$ is generic. In particular, since $z_1 \cap z_2 = \varnothing$, we have $$L \nsubseteq I(x) \quad \text{ for all } x \in {\mathbb{P}}^1 . \label{Lnon_deg}$$
Let $C_0$ be the distinguished irreducible component of $C$ on which $\pi \circ f$ is non-constant. By , the restriction $f|_{C_0}$ is irreducible, and by Lemma \[117\] (iii), the map $f|_{C_0}$ is an isomorphism onto the embedded line $$L \subset \operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1) \overset{s}{\subset} \operatorname{\mathsf{Hilb}}^d(S).$$ We will identify $C_0$ with $L$ via this isomorphism.
Let $x_1, \dots, x_{24} \in {\mathbb{P}}^1$ be the basepoints of the nodal fibers of $\pi$, and let $$y_1, \dots, y_{2d-2} \in {\mathbb{P}}^1$$ be the points such that $2 y_i \subset z$ for some $z \in L$. For $x \in {\mathbb{P}}^1$, let $$\widetilde{x} = I(x) \cap L \ \in \ \operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)$$ denote the unique point on $L$ which is incident to $x$. Then, the points $$\label{L_with_W_int_points} \widetilde{x}_1, \dots, \widetilde{x}_{24}, \widetilde{y}_1, \dots, \widetilde{y}_{2d-2}$$ are the intersection points of $L$ with the discriminant of $\pi^{[d]}$ defined in . Hence, by Lemma \[200\], components of $C$ can be attached to $C_0$ only at the points . Consider the decomposition $$C = C_0 \cup A_1 \cup \dots \cup A_{24} \cup B_1 \cup \dots \cup B_{2d-2}, \label{ogorgdfgfdg}$$ where $A_i$ and $B_j$ are the components of $C$ attached to the points $\widetilde{x}_i$ and $\widetilde{y}_j$ respectively. We consider the restriction of $f$ to $A_i$ and $B_j$ respectively.
1. Let $\widetilde{x_i} = x_i + w_1 + \dots + w_{d-1}$ for some points $w_\ell \in {\mathbb{P}}^1$. By genericity of $L$, the $w_\ell$ are basepoints of smooth elliptic fibers. Hence, $f|_{A_i}$ decomposes as $$f|_{A_i} = \phi + w_1 + \dots + w_{d-1}, \label{201}$$ where $w_\ell \in {\mathbb{P}}^1 \subset S$ for all $\ell$ denote constant maps, and $\phi : A_i \to F_{x_i}$ is a map to $i$-th nodal fiber which sends $\tilde{x}_i$ to the point $s(x_i) \in S$.
2. Let $\widetilde{y_j} = 2 y_j + w_1 + \dots + w_{d-2}$ for some points $w_\ell \in {\mathbb{P}}^1$. Then, $f|_{B_j}$ decomposes as $$f|_{B_j} = \phi + w_1 + \dots + w_{d-2}, \label{212}$$ where $\phi : B_j \to \operatorname{\mathsf{Hilb}}^2(S)$ maps to the fiber $(\pi^{[2]})^{-1}(2y)$ and sends the point $\widetilde{y}_j \in L \equiv C_0$ to $s(2 y_j)$.
Since $L$ is independent of $f$, we conclude that the moduli space $M_Z$ is *set-theoretically*[^4] a product of moduli spaces of maps of the form $f|_{A_i}$ and $f|_{B_j}$. The next step is to prove the splitting is *scheme-theoretic*.
### Deformation theory {#defsplii}
Let $[f : C \to \operatorname{\mathsf{Hilb}}^d(S),p_1, p_2] \in M_Z$ be a point and let $$\xymatrix{
\widehat{C} \ar@<+6pt>[d]^p \ar[r]^{\widehat{f}} & \operatorname{\mathsf{Hilb}}^d(S) \\
{\mathop{\rm Spec}\nolimits}( {{\mathbb{C}}}[\epsilon]/\epsilon^2 ) \ar@<+0pt>[u] \ar@<+6pt>[u]^{\widehat{p_1}, \widehat{p}_2}
}
\label{221}$$ be a first order deformation of $f$ inside $M_Z$. In particular, $p$ is a flat map, $\widehat{p}_1, \widehat{p}_2$ are sections of $p$, and $\widehat{f}$ restricts to $f$ at the closed point.
Consider the decomposition and let $\widetilde{x}_i$ for $i = 1,\dots, 24$ and $\widetilde{y}_j$ for $j = 1,\dots, 2d-2$ be the node points $A_i \cap C_0$ and $B_j \cap C_0$ respectively.
\[doesnotsmooth\] The deformation does not resolve the nodal points $\widetilde{x}_1, \dots, \widetilde{x}_{24}$ and $\widetilde{y}_1, \dots, \widetilde{y}_{2d-2}$.
Assume $\widehat{f}$ smoothes the node $\widetilde{x}_i$ for some $i$. Let ${{\mathcal Z}}_d \to \operatorname{\mathsf{Hilb}}^d(S)$ be the universal family and consider the pullback diagram $$\begin{tikzcd}
\mathllap{f^{\ast} {{\mathcal Z}}_d = }\ \widetilde{C} \ar{r} \ar{d} & {{\mathcal Z}}_d \ar{d} \ar{r} & S \\
C \ar{r}{f} & \operatorname{\mathsf{Hilb}}^d(S)
\end{tikzcd}$$ Let $E$ be the connected component of $f|_{A_i}^{\ast} {{\mathcal Z}}_d$, which defines the non-constant map $\phi$ in the decomposition , and let $G_0 = f|_{C_0}^{\ast} {{\mathcal Z}}_d$. Then, the projection $\widetilde{C} \to C$ is étale at the intersection point $q = G_0 \cap E$,
The deformation $\widehat{f} : \widehat{C} \to \operatorname{\mathsf{Hilb}}^d(S)$ induces the deformation $$K = \widehat{f}^{\ast} {{\mathcal Z}}_d {{\ \longrightarrow\ }}{\mathop{\rm Spec}\nolimits}\bigl( {{\mathbb{C}}}[\epsilon]/\epsilon^2 \bigr)$$ of the curve $\widetilde{C}$. Since $\widehat{C}$ smoothes $\widetilde{x}_i$ and $\widetilde{C} \to C$ is étale near $q$, the deformation $K$ resolves $q$. Then, the natural map $K \to S$ defines a deformation of the curve $\widetilde{C} \to S$ which resolves $q$. Since $\widetilde{C} \to S$ has class $\beta_h$, such a deformation can not exist by the geometry of the linear system $|\beta_h|$. Hence, $\widehat{f}$ does not smooth the node $\widetilde{x}_i$.
Assume $\widehat{f}$ smoothes the node $\widetilde{y}_j$ for some $j$. We follow closely the argument of T. Graber in [@Grab page 19]. Let $F_{y_j}$ be the fiber of $\pi : S \to {\mathbb{P}}^1$ over $y_j$, let $$D(F_{y_j}) = \{ \xi \in \operatorname{\mathsf{Hilb}}^d(S)\ |\ \xi \cap F_{y_j} \neq 0 \}$$ be the divisor of subschemes with non-zero intersection with $F_{y_j}$, and consider the divisor $$D = \Delta_{\operatorname{\mathsf{Hilb}}^d(S)} + D(F_{y_j}) \,.$$ Let $C_1$ be the irreducible component of $C$ that attaches to $C_0$ at $q = \widetilde{y}_j$, and let $C_2$ be the union of all irreducible components of $B_j$ except $C_1$. The curves $C_2$ and $C_1$ intersect in a finite number of nodes $\{ q_i \}$. The deformation $\widehat{f}$ resolves the node $q$ and may also resolve some of the $q_i$.
The first order neighborhood $\widetilde{C_1}$ of $C_1$ in the total space of the deformation $\widehat{C}$ can be identified with the first order neighborhood of ${\mathbb{P}}^1$ in the total space of the bundle ${{\mathcal O}}(- \ell)$, where $\ell \geq 1$ is the number of nodes on $C_1$ which are smoothed by $\widehat{f}$. Let $$f' : \widetilde{C}_1 \to \operatorname{\mathsf{Hilb}}^d(S)$$ be the induced map on $\widetilde{C}_1$. We consider the case, where $f'|_{C_1}$ is a degree $k \geq 1$ map to the exceptional curve at $\widetilde{y}_j$. The general case is similar.
Let $N$ be the pullback of ${{\mathcal O}}(D)$ by $f' : \widetilde{C}_1 \to \operatorname{\mathsf{Hilb}}^d(S)$, and let $s \in H^0(\widetilde{C}, N)$ be the pullback of the section of ${{\mathcal O}}(D)$ defined by $D$. The bundle $N$ restricts to ${{\mathcal O}}(-2k)$ on $C_1$. By [@Grab page 20], giving $N$ and $s$ is equivalent to an element of the vector space $$\operatorname{Hom}_{{{\mathcal O}}_{C_1}}({{\mathcal O}}(-\ell), f|_{C_1}^{\ast} {{\mathcal O}}(D)),$$ of dimension $\ell-2k+1 \leq \ell-1$.
The neighborhood $\widetilde{C_1}$ intersects $C_0$ in a double point. Since $C_0$ intersects the divisor $D$ transversely, $s$ is non-zero on $\widetilde{C}_1$. Let $q_1, \dots, q_{\ell-1}$ be the other nodes on $C_1$ which get resolved by $\widehat{f}$. Since $C_2 \subset D$, the section $s$ vanishes at $q_1, \dots, q_{\ell-1}$. By dimension reasons, we find $s = 0$. This contradicts the non-vanishing of $s$. Hence, $\widehat{f}$ does not smooth the node $\widetilde{y}_j$.
By Lemma \[doesnotsmooth\], any first order (and hence any infinitesimal) deformation of $[f : C \to \operatorname{\mathsf{Hilb}}^d(S), p_1, p_2] \in M_Z$ inside $M$ preserves the decomposition $$C = C_0 \cup_i A_i \cup_j B_j$$ and therefore induces a deformation of the restriction $$f|_{C_0} : C_0 \overset{\cong}{{{\ \longrightarrow\ }}} L \subset \operatorname{\mathsf{Hilb}}^d(S). \label{frmgerg}$$ By Lemma \[normaltoL\], every deformation of $L \subset \operatorname{\mathsf{Hilb}}^d(S)$ moves the line $L$ in the projective space $\operatorname{\mathsf{Hilb}}^{d}(B_0)$. Since any deformations of $f$ inside $M_Z$ must stay incident to $Z_1, Z_2 \subset \operatorname{\mathsf{Hilb}}^d(S)$, we conclude that such deformations induce the constant deformation of . The image line $f(C_0)$ stays completely fixed.
### The product decomposition {#splitting}
\[Section\_splitting\] For $h > 0$ and for $x \in {\mathbb{P}}^1$ a basepoint of a nodal fiber of $\pi : S \to {\mathbb{P}}^1$, let $$M^{\textup{(N)}}_{x}(h)$$ be the moduli space of $1$-marked genus $0$ stable maps to $S$ in class $h F$ which map the marked point to $x$. Hence, $M^{(N)}_{x}(h)$ parametrizes degree $h$ covers of the nodal fiber $F_{x}$. By convention, $M^{(N)}_{x}(0)$ is taken to be a point. For $h \geq 0,\, k \in {{\mathbb{Z}}}$ and for $y \in {\mathbb{P}}^1$ a basepoint of a smooth fiber of $\pi$, let $$M^{\textup{(F)}}_{y}(h,k) \label{234_1}$$ be the moduli space of $1$-marked genus $0$ stable maps to $\operatorname{\mathsf{Hilb}}^2(S)$ in class $h F + k A$ which map the marked point to $s^{[2]}(2 y)$. By convention, $M^{(F)}_{y}(0,0)$ is taken to be a point.
Let $T$ be a connected scheme and consider a family $$\begin{tikzcd}
C \ar{r}{F} \ar{d} & \operatorname{\mathsf{Hilb}}^d(S) \\ T
\end{tikzcd}
\label{fam_dekfodkfo}$$ of stable maps in $M_Z$. By Lemma \[doesnotsmooth\], the curve $C \to T$ allows a decomposition $$C = C_0 \cup A_1 \cup \dots \cup A_{24} \cup B_1 \cup \dots \cup B_{2d-2},$$ where $C_0$ is the distinguished component of $C$ and the components $A_i$ and $B_j$ are attached to $C_0$ at the points $\tilde{x}_i$ and $\tilde{y}_j$ respectively.
The restriction of the family to the components $A_i$ (resp. $B_j$) defines a family in the moduli space $M^{\textup{(N)}}_{x_i}(h_{x_i})$ (resp. $M^{\textup{(F)}}_{y_j}(h_{y_j}, k_{y_j})$) for some $h_{x_i}$ (resp. $h_{y_j}, k_{y_j}$). Since, by Section \[Section\_Curves\_in\_Hilbd\], the line $f(C_0) = L$ has class $$[L] = B - (d-1) A \ \in H_2( \operatorname{\mathsf{Hilb}}^d(S), {{\mathbb{Z}}}),$$ and by the additivity of cohomology classes under decomposing (Lemma \[hom\_add\_up\]), we must have $\sum_i h_{x_i} + \sum_j h_{y_j} = h$ and $\sum_j k_{y_j} = k + (d-1)$. Let $$\Psi: M_{Z}
{{\ \longrightarrow\ }}\bigsqcup_{\textbf{h}, \textbf{k}} \bigg( \prod_{i=1}^{24} M^{\textup{(N)}}_{x_i}(h_{x_i})
\times \prod_{j=1}^{2d-2} M^{\textup{(F)}}_{y_j}(h_{y_j}, k_{y_j}) \bigg)
\label{spl1}.$$ be the induced map on moduli spaces, where the disjoint union runs over all $$\label{spl1_indexing}
\begin{aligned}
\textbf{h} & = (h_{x_1}, \dots, h_{x_{24}}, h_{y_1}, \dots, h_{y_{2d-2}} ) \in ({{\mathbb{N}}}^{\geq 0})^{ \{ x_i, y_j \}} \\
\textbf{k} & = (k_{y_1}, \dots, k_{y_{2d-2}}) \in {{\mathbb{Z}}}^{2d-2}
\end{aligned}$$ such that $$\sum_i h_{x_i} + \sum_j h_{y_j} = h \quad \text{ and } \quad \sum_j k_{y_j} = k + (d-1) \,. \label{spl2_indexing}$$ Since $L \subset \operatorname{\mathsf{Hilb}}^d(S)$ is fixed under deformations, we can glue elements of the right hand side of to $C_0$ and obtain a map in $M_{Z}$. By a direct verification, the induced morphism on moduli spaces is the inverse to $\Psi$. Hence, $\Psi$ is an isomorphism.
### The virtual class {#anvirclass}
\[Section\_analysis\_of\_virtual\_class\] Let $Z_1, Z_2$ be the Lagrangian fibers of $\pi^{[d]}$ defined in Section \[Section\_basic\_case\_overview\], and let $Z = Z_1 \times Z_2$. We consider the fiber square $$\label{iweufruf}
\begin{tikzcd}
M_Z \ar{r}{j} \ar{d}{p} & M \ar{d}{{\mathop{\rm ev}\nolimits}} \\
Z \ar{r}{i} & ( \operatorname{\mathsf{Hilb}}^d(S) )^2,
\end{tikzcd}$$ where $M = {{\overline M}}_{0,2}( \operatorname{\mathsf{Hilb}}^d(S), \beta_h + kA)$. The map $i$ is the inclusion of a smooth subscheme of codimension $2d$. Hence, the restricted virtual class $$[M_Z]^{\text{vir}} = i^{!} [M]^{\text{red}} \label{204}$$ is of dimension $0$. By the push-pull formula we have $$\int_{[M_Z]^{\text{vir}}} 1\ =\ \big\langle {{\mathfrak{p}}}_{-1}(F)^d 1_S , {{\mathfrak{p}}}_{-1}(F)^d 1_S \big\rangle^{\operatorname{\mathsf{Hilb}}^d(S)}_{\beta_h+kA} \,.
\label{virt_pushpull_mddfds}$$ Let $\Psi$ be the splitting morphism . We will show that $\Psi_{\ast} [ M_Z ]^{\text{vir}}$ splits naturally as a product of virtual cycles.
Let ${{\mathbb{L}}}_X$ denote the cotangent complex on a space $X$. Let $E^{\bullet} \to {{\mathbb{L}}}_{M}$ be the reduced perfect obstruction theory on $M$, and let $F^{\bullet}$ be the cone of the map $$p^{\ast} i^{\ast} \Omega_{(\operatorname{\mathsf{Hilb}}^d(S))^2} {{\ \longrightarrow\ }}j^{\ast} E^{\bullet} \oplus p^{\ast} \Omega_Z$$ induced by the diagram . The cone $F^{\bullet}$ maps to ${{\mathbb{L}}}_{M_Z}$ and defines a perfect obstruction theory on $M_Z$. By [@BF Proposition 5.10], the associated virtual class is $[M_Z]^{\text{vir}}$.
Let $[f : C \to \operatorname{\mathsf{Hilb}}^d(S), p_1, p_2] \in M_Z$ be a point. For simplicity, we consider all complexes on the level of tangent spaces at the moduli point $[f]$. Let $E_{\bullet}$ and $F_{\bullet}$ denote the derived duals of $E^{\bullet}$ and $F^{\bullet}$ respectively.
We recall the construction of $E_{\bullet}$, see [@GWNL; @STV]. Consider the semi-regularity map $$b : R \Gamma( C, f^{\ast} T_{\operatorname{\mathsf{Hilb}}^d(S)} ) \to V[-1] \label{semiregu_map}$$ where $V = H^0(\operatorname{\mathsf{Hilb}}^d(S), \Omega^2_{\operatorname{\mathsf{Hilb}}^d(S)})^{\vee}$, and recall the ordinary (non-reduced) perfect obstruction theory of $M$ at the point $[f]$, $$E_{\bullet}^{\text{vir}} = {\mathop{\rm Cone}\nolimits}\Big( R \Gamma( C, {{\mathbb{T}}}_{C} (-p_1 - p_2) ) \to R \Gamma( C, f^{\ast} T_{\operatorname{\mathsf{Hilb}}^d(S)} ) \Big),$$ where ${{\mathbb{T}}}_C = {{\mathbb{L}}}_{C}^{\vee}$ is the tangent complex on $C$. Then, by the vanishing of the composition $$R \Gamma( C, {{\mathbb{T}}}_{C} (-p_1 - p_2) ) \to R \Gamma( C, f^{\ast} T_{\operatorname{\mathsf{Hilb}}^d(S)} ) \xrightarrow{b} V[-1], \label{virclass_vanishing}$$ the map induces a morphism $\overline{b} : E_{\bullet}^{\text{vir}} \to V[-1]$ with co-cone $E_{\bullet}$.
By a diagram chase, $F_{\bullet}$ is the co-cone of $$(\overline{b}, d {\mathop{\rm ev}\nolimits}) \colon E_{\bullet}^{\text{vir}} \to V[-1] \oplus N_{Z, (z_1, z_2)}$$ where $z_1,z_2$ are the basepoints of the Lagrangian fiber $Z_1,Z_2$ respectively, $N_{Z,(z_1,z_2)}$ is the normal bundle of $Z$ in $\operatorname{\mathsf{Hilb}}^d(S)^2$ at $(z_1, z_2)$, and $d {\mathop{\rm ev}\nolimits}$ is the differential of the evaluation map. Since taking the cone and co-cone commutes, the complex $F_{\bullet}$ is therefore the cone of $$\gamma: R \Gamma(C, {{\mathbb{T}}}_C(-p_1-p_2)) \to K, \label{210}$$ where $$K = {\mathop{\rm Cocone}\nolimits}\Big[ (b, d {\mathop{\rm ev}\nolimits}) : R \Gamma(C, f^{\ast} T_{\operatorname{\mathsf{Hilb}}^d(S)}) \to V[-1] \oplus N_{Z, (z_1, z_2)} \Big] \,. \label{206}$$
Consider the decomposition $$C = C_0 \cup A_1 \cup \dots \cup A_{24} \cup B_1 \cup \dots \cup B_{2d-2}, \label{rfofdjf}$$ where the components $A_i$ and $B_j$ are attached to $C_0$ at the points $\tilde{x}_i$ and $\tilde{y}_j$ respectively. Tensoring $R \Gamma( C, {{\mathbb{T}}}_{C} (-p_1 - p_2) )$ and $K$ against the partial renormalization sequence associated to decomposition , we will show that the dependence on $L$ cancels in the cone of .
The map $(b, d\!{\mathop{\rm ev}\nolimits})$ fits into the diagram $$\label{207}
\xymatrix{
R \Gamma(C, f^{\ast} T_{\operatorname{\mathsf{Hilb}}^d(S)}) \ar[r]^u \ar[d]^{(b, d {\mathop{\rm ev}\nolimits})} & R \Gamma(L, f^{\ast} T_{\operatorname{\mathsf{Hilb}}^d(S)}) \ar[d]^{v = (b, d {\mathop{\rm ev}\nolimits})} \\
V[-1] \oplus N_{Z, (z_1, z_2)} \ar[r]^{(\sigma, \operatorname{id})} & V[-1] \oplus N_{Z, (z_1, z_2)},
}$$ where $u$ is the restriction map and $\sigma$ is the induced map[^5]. By Lemma \[normaltoL\], the co-cone of $v$ is $R \Gamma( {{\mathbb{T}}}_L(-p_1 - p_2) )$.
The partial normalization sequence of $C$ with respect to $\widetilde{x}_i$ and $\widetilde{y}_j$ is $$0 {{\ \longrightarrow\ }}{{\mathcal O}}_C {{\ \longrightarrow\ }}{{\mathcal O}}_L \oplus_{D \in \{ A_i, B_j \}} {{\mathcal O}}_{D} {{\ \longrightarrow\ }}\oplus_{s \in \{ \widetilde{x}_i, \widetilde{y}_j \}} {{\mathcal O}}_{C,s} {{\ \longrightarrow\ }}0 \label{normalization} \,.$$ Tensoring with $f^{\ast} T_{\operatorname{\mathsf{Hilb}}^d(S)}$, applying $R \Gamma( \cdot )$ and factoring with , we obtain the exact triangle $$K {{\ \longrightarrow\ }}R \Gamma(L, {{\mathbb{T}}}_{L}(-p_1 - p_2)) \oplus_D R \Gamma(D, f_{|D}^{\ast} T_{\operatorname{\mathsf{Hilb}}^d(S)}) {{\ \longrightarrow\ }}\oplus_s T_{\operatorname{\mathsf{Hilb}}^d(S),s} {{\ \longrightarrow\ }}K[1]. \label{209}$$
For each node $t \in C$, let $N_t$ (resp. $T_{t}$) be the tensor product (resp. the direct sum) of the tangent spaces to the branches of $C$ at $t$. Tensoring with ${{\mathbb{T}}}_C(-p_1 - p_2)$ and applying $R \Gamma( \cdot )$, we obtain the exact triangle $$R \Gamma {{\mathbb{T}}}_C(-p_1 - p_2) \to R \Gamma({{\mathbb{T}}}_L(-p_1-p_2)) \oplus_{D} R \Gamma({{\mathbb{T}}}_{D}) \oplus_t N_{t}[-1] \to \oplus_t T_{t} \to \ldots \,. \label{208}$$ By the vanishing of (applied to $C = L$), the sequence maps naturally to . Consider the restriction of this map to the summand $R \Gamma( {{\mathbb{T}}}_L(-p_1-p_2) )$ which appears in the second term of , $$\varphi : R \Gamma( {{\mathbb{T}}}_L(-p_1-p_2) ) \to R \Gamma(L, {{\mathbb{T}}}_{L}(-p_1 - p_2)) \oplus_D R \Gamma(D, f_{|D}^{\ast} T_{\operatorname{\mathsf{Hilb}}^d(S)}) \,.$$ Then, the composition of $\varphi$ with the projection to $R \Gamma(L, {{\mathbb{T}}}_{L}(-p_1 - p_2))$ is the identity. Hence, $F_{\bullet} = {\mathop{\rm Cone}\nolimits}(\gamma)$ admits the exact sequence $$F_{\bullet} {{\ \longrightarrow\ }}\oplus_D G_D \overset{\psi}{{{\ \longrightarrow\ }}} \oplus_D H_D {{\ \longrightarrow\ }}F_{\bullet}[1], \label{211}$$ where $D$ runs over all $A_i$ and $B_j$, and $$\begin{aligned}
G_D & = {\mathop{\rm Cone}\nolimits}\Big[ R \Gamma({{\mathbb{T}}}_{D}) \oplus_t N_{t}[-1] {{\ \longrightarrow\ }}R \Gamma(D, f_{|D}^{\ast} T_{\operatorname{\mathsf{Hilb}}^d(S)}) \Big]\\
H_D & = {\mathop{\rm Cone}\nolimits}\Big[ \oplus_t T_{t} {{\ \longrightarrow\ }}\oplus_t T_{\operatorname{\mathsf{Hilb}}^d(S),t} \Big].\end{aligned}$$ Here $t = t(D) = D \cap C_0$ is the attachment point of the component $D$.
The map $\psi$ in maps the factor $G_D$ to $H_D$ for all $D$. For $D = A_i$ consider the decomposition $$f|_{A_i} = \phi + w_1 + \dots + w_{d-1}.$$ The trivial factors which arise in $G_D$ and $H_D$ from the tangent space of $\operatorname{\mathsf{Hilb}}^d(S)$ at the points $w_1, \dots, w_{d-1}$ cancel each other in ${\mathop{\rm Cone}\nolimits}(G_D \to H_D)$. Hence ${\mathop{\rm Cone}\nolimits}(G_D \to H_D)$ only depends on $\phi : C \to S$, and therefore only on the image of $[f]$ in the factor $M^{\textup{(N)}}_{x_i}(h_{x_i})$, where $M^{\textup{(N)}}_{x_i}(h_{x_i})$ is the moduli space defined in Section \[splitting\]. The case $D = B_j$ is similar.
Hence, $F_{\bullet}$ splits into a sum of complexes pulled back from each factor of the product splitting . Since $F_{\bullet}$ is a perfect obstruction theory on $M$, the complexes on each factor are perfect obstruction theories. Let $$[M^{\textup{(N)}}_{x_i}(h_{x_i})]^{\text{vir}} \quad \text{ and } \quad [ M^{\textup{(F)}}_{y_j}(h_{y_j}, k_{y_j}) ]^{\text{vir}}$$ be their virtual classes respectively. We have proved the following.
\[W0splitprop\] Let $\Psi$ be the splitting morphism . Then, $\Psi$ is an isomorphism and we have $$\Psi_{\ast} [M_{Z}]^{\text{vir}}
= \sum_{\textbf{h}, \textbf{k}} \left(
\prod_{i=1}^{24} [M^{\textup{(N)}}_{x_i}(h_{x_i})]^{\text{vir}} \times \prod_{j=1}^{2d-2} [ M^{\textup{(F)}}_{y_j}(h_{y_j}, k_{y_j}) ]^{\text{vir}} \right)$$ where the sum is over the set satisfying .
### The series $F^{\textup{GW}}$ {#concl_basic_case}
We consider the left hand side of Theorem \[ellthm\]. By , we have $$\label{ellthm_eval_44231}
{\Big\langle}{{\mathfrak{p}}}_{-1}(F)^d 1_S \ , \ {{\mathfrak{p}}}_{-1}(F)^d 1_S {\Big\rangle}^{\operatorname{\mathsf{Hilb}}^d(S)}_q
=\ \sum_{h \geq 0} \sum_{k \in {{\mathbb{Z}}}} y^k q^{h-1} \int_{[M_{Z}(h,k)]^{\text{vir}}} 1 \,.$$ By Proposition \[W0splitprop\], this equals $$\begin{aligned}
& \sum_{\substack{h \geq 0 \\ k \in {{\mathbb{Z}}}}} y^k q^{h-1}
\sum_{\substack{ (\textbf{h}, \textbf{k}) \\ \sum_i h_{x_i} + \sum_j h_{y_j} = h \\ \sum_{j} k_{y_j} = k + (d-1) }}
\bigg( \prod_{i=1}^{24} \int_{[M^{\textup{(N)}}_{x_i}(h_{x_i})]^{\text{vir}}}1 \bigg)
\cdot
\bigg( \prod_{j=1}^{2d-2} \int_{[ M^{\textup{(F)}}_{y_j}(h_{y_j}, k_{y_j}) ]^{\text{vir}}} 1 \bigg) \\
=\ & y^{-(d-1)}q^{-1}
\bigg( \prod_{i=1}^{24} \sum_{h_{x_i} \geq 0} q^{h_{x_i}} \int_{[M^{\textup{(N)}}_{x_i}(h_{x_i})]^{\text{vir}}}1 \bigg) \\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\times \bigg( \prod_{j=1}^{2d-2} \sum_{\substack{h_{y_j} \geq 0 \\ k_{y_j} \in {{\mathbb{Z}}}}} y^{k_{y_j}} q^{h_{y_j}} \int_{[ M^{\textup{(F)}}_{y_j}(h_{y_j}, k_{y_j}) ]^{\text{vir}}} 1 \bigg) \\
=\ & \Big( \prod_{i = 1}^{24} \sum_{h \geq 0} q^{h - \frac{1}{24}} \int_{[M^{\textup{(N)}}_{x_i}(h)]^{\text{vir}}}1 \Big) \cdot
\Big( \prod_{i = 1}^{2d-2} \sum_{\substack{ h \geq 0 \\ k \in {{\mathbb{Z}}}}} q^{h} y^{k - \frac{1}{2}} \int_{[ M^{\textup{(F)}}_{y_j}(h, k) ]^{\text{vir}}} 1 \Big) \,.\end{aligned}$$ The integrals in the first factor were calculated by Bryan and Leung in their proof of the Yau-Zaslow conjecture [@BL]. The result is $$\sum_{h \geq 0} q^h \int_{[M^{\textup{(N)}}_{x_i}(h)]^{\text{vir}}}1 \ =\ \prod_{m \geq 0} \frac{1}{1-q^m} \,. \label{nodal_fiber_contribution}$$ By deformation invariance, the integrals $$\int_{[ M^{\textup{(F)}}_{y_j}(h, k) ]^{\text{vir}}} 1$$ only depend on $h$ and $k$. Define the generating series $$F^{\textup{GW}}(y,q) = \sum_{h \geq 0} \sum_{k \in {{\mathbb{Z}}}} q^{h} y^{k - \frac{1}{2}} \int_{[ M^{\textup{(F)}}_{y_j}(h, k) ]^{\text{vir}}} 1 \,. \label{FGW}$$ By our convention on $M^{\textup{(F)}}_{y_j}(0,0)$, the $y^{-1/2} q^0$-coefficient of $F^{\textup{GW}}$ is $1$.
Let $\Delta(q) = q \prod_{m \geq 1} (1-q^m)^{24}$ be the modular discriminant $\Delta(\tau)$ considered as a formal expansion in the variable $q = e^{2 \pi i \tau}$. We conclude $${\Big\langle}{{\mathfrak{p}}}_{-1}(F)^d 1_S \ , \ {{\mathfrak{p}}}_{-1}(F)^d 1_S {\Big\rangle}^{\operatorname{\mathsf{Hilb}}^d(S)}_q = \frac{F^{\textup{GW}}(y,q)^{2d-2}}{\Delta(q)} \,.$$ The proof of Theorem \[ellthm\] now follows directly from Theorem \[thm\_F\_evaluation\] below.
Evaluation of $F^{\textup{GW}}$ and the Kummer K3 {#section_Kummer_evaluation}
-------------------------------------------------
Let $F$ be the theta function which already appeared in Section \[YZ\_section\_statement\_of\_results\], $$F(z,\tau) = \frac{\vartheta_1(z,\tau)}{\eta^3(\tau)}
= (y^{1/2} + y^{-1/2}) \prod_{m \geq 1} \frac{ (1 + yq^m) (1-y^{-1}q^m)}{ (1-q^m)^2 } \,,$$ where $q = e^{2 \pi i \tau}$ and $y = - e^{2 \pi i z}$.
\[thm\_F\_evaluation\] Under the variable change $q = e^{2 \pi i \tau}$ and $y = -e^{2 \pi i z}$, $$F^{\textup{GW}}(y,q) = F(z,\tau).$$
In Section \[section\_Kummer\_evaluation\] we present a proof of Theorem \[thm\_F\_evaluation\] using the Kummer K3 surface and the Yau-Zaslow formula. An independent proof is given in Section \[Section\_Hilb2P1xE\] through the geometry of $\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1 \times E)$, where $E$ is an elliptic curve.
The Yau-Zaslow formula was used in the geometry of Kummer K3 surfaces before by S. Rose [@Ros14] to obtain virtual counts of hyperelliptic curves on abelian surfaces. While the geometry used in [@Ros14] is similar to our setting, the closed formula of Theorem \[thm\_F\_evaluation\] in terms of the Jacobi theta function $F$ is new. For example, Theorem \[thm\_F\_evaluation\] yields a new, closed formula for hyperelliptic curve counts on an abelian surface, see [@BOPY].
### The Kummer K3
Let ${{\mathsf{A}}}$ be an abelian surface. The *Kummer* of ${{\mathsf{A}}}$ is the blowup $$\rho : {\operatorname{Km}}({{\mathsf{A}}}) \to {{\mathsf{A}}}/ \pm 1 \label{bldn_kummer}$$ of ${{\mathsf{A}}}/ \pm 1$ along its 16 singular points. It is a smooth projective K3 surface. Alternatively, consider the composition $$s : \operatorname{\mathsf{Hilb}}^2({{\mathsf{A}}}) {{\ \longrightarrow\ }}\operatorname{Sym}^2({{\mathsf{A}}}) {{\ \longrightarrow\ }}{{\mathsf{A}}}$$ of the Hilbert-Chow morphism with the addition map. Then, ${\operatorname{Km}}({{\mathsf{A}}})$ is the fiber of $s$ over the identity element $0_{{\mathsf{A}}}\in {{\mathsf{A}}}$, $${\operatorname{Km}}({{\mathsf{A}}}) = s^{-1}(0_{{\mathsf{A}}}) \,. \label{Kummer_cnstruction_23142}$$
Let $E$ and $E'$ be generic elliptic curves and let $${{\mathsf{A}}}= E \times E' .$$ Let $t_1, \dots, t_4$ and $t'_1, \dots, t'_4$ denote the 2-torsion points of $E$ and $E'$ respectively. The *exceptional curves* of ${\operatorname{Km}}({{\mathsf{A}}})$ are the divisors $$A_{ij} = \rho^{-1}(\, (t_i, t'_j) \,), \ \ \ i,j = 1, \dots, 4 \,.$$
The projection of ${{\mathsf{A}}}$ to the factor $E$ induces the elliptic fibration $$p : {\operatorname{Km}}({{\mathsf{A}}}) {{\ \longrightarrow\ }}{{\mathsf{A}}}/ \pm 1 {{\ \longrightarrow\ }}E / \pm 1 = {\mathbb{P}}^1.$$ Hence, ${\operatorname{Km}}({{\mathsf{A}}})$ is an elliptically fibered K3 surface. Similarly, we let $p' : {\operatorname{Km}}({{\mathsf{A}}}) \to {\mathbb{P}}^1$ denote the fibration induced by the projection ${{\mathsf{A}}}\to E'$. Since $E$ and $E'$ are generic, the fibration $p$ has exactly 4 sections $$s_1, \dots, s_4 : {\mathbb{P}}^1 \to {\operatorname{Km}}({{\mathsf{A}}})$$ corresponding to the torsion points $t'_1, \dots, t_4'$ of $E'$. We write $B_i \subset {\operatorname{Km}}({{\mathsf{A}}})$ for the image of $s_i$, and we let $F_x$ denote the fiber of $p$ over $x \in {\mathbb{P}}^1$
Let $y_1, \dots, y_4 \in {\mathbb{P}}^1$ be the image of the $2$-torsion points $t_1, \dots, t_4 \in E$ under $E \to E/\pm 1={\mathbb{P}}^1$. The restriction $$p: {\operatorname{Km}}({{\mathsf{A}}}) \setminus \{ F_{y_1}, \dots, F_{y_4} \} {{\ \longrightarrow\ }}{\mathbb{P}}^1 \setminus \{ y_1, \dots, y_4 \}$$ is an isotrivial fibration with fiber $E'$. For $i \in \{1,\dots, 4\}$, the fiber $F_{y_i}$ of $p$ over the points $y_i$ is singular with divisor class $$F_{y_i} = 2 T_i + A_{i1} + \dots + A_{i4},$$ where $T_i$ denotes the image of the section of $p' : {\operatorname{Km}}({{\mathsf{A}}}) \to {\mathbb{P}}^1$ corresponding to the $2$-torsion points $t_i$. We summarize the notation in figure \[Kummer\_diagram\].
(1.0,0) node\[below, align=center\][$T_1$]{} – (1.0,1.0) – (1.0,2.5) node\[above right, align=center\][ $\vdots$]{} – (1.0,4.0) node\[above right, align=left\][$A_{12}$]{} – (1.0,5.5) node\[above right, align=left\][$A_{11}$]{} – (1.0,6.5); (2.5,0) node\[below, align=center\][$T_2$]{} – (2.5,5.5) node\[above right, align=left\][$A_{21}$]{} – (2.5,6.5); (4.0,0) node\[below, align=center\][$T_3$]{} – (4.0,5.5) node\[above right, align=left\][$\cdots$]{} – (4.0,6.5); (5.5,0) node\[below, align=center\][$T_4$]{} – (5.5,6.5); (0,1.0) node\[left, align=right\][$B_4$]{} – (6.5,1.0); (0,2.5) node\[left, align=right\][$B_3$]{} – (6.5,2.5); (0,4.0) node\[left, align=right\][$B_2$]{} – (6.5,4.0); (0,5.5) node\[left, align=right\][$B_1$]{} – (6.5,5.5); (0, -2.0) node\[left, align=right\][${\mathbb{P}}^1 = E/\pm 1$]{} – (6.5, -2.0); (1.0,5.5) circle \[radius=0.1\]; (1.0,4.0) circle \[radius=0.1\]; (2.5,5.5) circle \[radius=0.1\]; (1.0,-2.0) circle \[radius=0.06\]; at (1,-2) [$y_1$]{}; (2.5,-2.0) circle \[radius=0.06\]; at (2.5,-2) [$y_2$]{}; (4.0,-2.0) circle \[radius=0.06\]; at (4.0,-2) [$y_3$]{}; (5.5,-2.0) circle \[radius=0.06\]; at (5.5,-2) [$y_4$]{}; (3.25, -1.0) – (3.25, -1.5); at (3.35, -1.25) [$p$]{}; (8.0, 0.0) – (8.0, 6.5) node\[above, align=center\][$E'/\pm 1$]{}; (6.75, 3.25) – (7.5, 3.25); at (7.125, 3.35) [$p'$]{};
Let $F$ and $F'$ be the class of a fiber of $p$ and $p'$ respectively. We have the intersections $$F^2 = 0, \quad F \cdot F' = 2, \quad F'^2 = 0$$ and $$F \cdot A_{ij} = F' \cdot A_{ij} = 0, \quad A_{ij} \cdot A_{k \ell} = -2\, \delta_{ik} \delta_{j \ell}
\ \ \ \ \ \text{ for all } i,j,k, \ell \in \{1, \dots, 4 \} \,.$$ By the relation $$\label{241_3}
\begin{aligned}
F & = 2 T_i + A_{i1} + A_{i2} + A_{i3} + A_{i4} \\
F' & = 2 B_i + A_{1i} + A_{2i} + A_{3i} + A_{4i}
\end{aligned}$$ for $i \in \{1, \dots, 4 \}$ this determines the intersection numbers of all the divisors above.
### Rational curves and $F^{\textup{GW}}$
Let $\beta \in H_2( {\operatorname{Km}}({{\mathsf{A}}}), {{\mathbb{Z}}})$ be an effective curve class and let $$\big\langle\, 1 \, \big\rangle_{0, \beta}^{{\operatorname{Km}}({{\mathsf{A}}})} =
\int_{[ {{\overline M}}_{0}( {\operatorname{Km}}({{\mathsf{A}}}), \beta) ]^{\text{red}} } 1$$ denote the genus $0$ Gromov-Witten invariants of ${\operatorname{Km}}({{\mathsf{A}}})$. For an integer $n \geq 0$ and a tuple $\textbf{k} = (k_{ij})_{i,j = 1, \dots, 4}$ of half-integers $k_{ij} \in \frac{1}{2} {{\mathbb{Z}}}$, define the class $$\beta_{n, \textbf{k}} = \frac{1}{2} F' + \frac{n}{2} F + \sum_{i,j = 1}^{4} k_{ij} A_{ij}\ \in H_2( {\operatorname{Km}}({{\mathsf{A}}}) , {{\mathbb{Q}}}).$$ We write $\beta_{n, \textbf{k}} > 0$, if $\beta_{n, \textbf{k}}$ is effective.
\[Kummer\_prop\_1\] We have $$\sum_{\substack{n, \textbf{k} \\ \beta_{n,\textbf{k}} > 0}} \big\langle\, 1 \, \big\rangle_{0, \beta_{n,\textbf{k}}}^{{\operatorname{Km}}({{\mathsf{A}}})} q^n y^{\sum_{i,j} k_{ij}}
\ =\ 4 \cdot F^{\textup{GW}}(y,q)^{4},$$ where the sum runs over all $n \geq 0$ and $\textbf{k} = (k_{ij})_{i,j} \in (\frac{1}{2} {{\mathbb{Z}}})^{4 \times 4}$ for which $\beta_{n,\textbf{k}}$ is an effective curve class.
Let $f : C \to {\operatorname{Km}}({{\mathsf{A}}})$ be a genus $0$ stable map in class $\beta_{n,\textbf{k}}$. By genericity of $E$ and $E'$ the fibration $p$ has only the sections $B_1, \dots, B_4$. Since $p \circ f$ has degree $1$, the image divisor of $f$ is then of the form $${\mathop{\rm Im}\nolimits}(f) = B_{\ell} + D'$$ for some $1 \leq \ell \leq 4$ and a divisor $D'$, which is contracted by $p$. Since the fibration $p$ has fibers isomorphic to $E'$ away from the points $y_1, \dots, y_4 \in {\mathbb{P}}^1$, the divisor $D'$ is supported on the singular fibers $F_{y_i}$. Hence, there exist non-negative integers $$a_i, \ \ i=1,\dots,4 \quad \text{ and } \quad b_{ij}, \ \ i,j = 1, \dots, 4$$ such that $${\mathop{\rm Im}\nolimits}(f) = B_{\ell} + \sum_{i=1}^4 a_i T_i + \sum_{i,j = 1}^{4} b_{ij} A_{ij} .$$ Let $C_0$ be the component of $C$ which gets mapped by $f$ isomorphically to $B_{\ell}$, and let $D_i$ be the component of $C$, that maps into the fiber $F_{y_i}$. Then, $$C = C_0 \cup D_1 \cup \dots \cup D_4, \label{2345}$$ with pairwise disjoint $D_i$. Under $f$ the intersection points $C_0 \cap D_j$ gets mapped to $s_{\ell}(y_j)$, where $s_{\ell} : {\mathbb{P}}^1 \to {\operatorname{Km}}({{\mathsf{A}}})$ denotes the $\ell$-th section of $p$.
By arguments similar to the proof of Lemma \[doesnotsmooth\] or by the geometry of the linear system $|\beta_{n,\textbf{k}}|$, the nodal points $C_0 \cap D_j$ do not smooth under infinitesimal deformations of $f$. The decomposition is therefore preserved under infinitesimal deformations. This implies that the moduli spaces ${{\overline M}}_{0}( {\operatorname{Km}}({{\mathsf{A}}}), \beta_{n, \textbf{k}})$ admits the decomposition $${{\overline M}}_{0}( {\operatorname{Km}}({{\mathsf{A}}}), \beta_{n, \textbf{k}})\ =\
\bigsqcup_{\ell = 1}^{4}\ \bigsqcup_{n = n_1 + \dots + n_4}\ \prod_{i=1}^{4}\ M_{y_i}^{(\ell)}(n_i, (k_{ij} + \frac{1}{2} \delta_{j \ell})_j ),
\label{242_1}$$ where $M_{y_i}^{(\ell)}(n_i, (k_{ij})_j )$ is the moduli space of stable $1$-pointed genus $0$ maps to ${\operatorname{Km}}({{\mathsf{A}}})$ in class $$\frac{n_i}{2} F + \sum_{j = 1}^4 k_{ij} A_{ij}$$ and with marked point mapped to $s_{\ell}(y_i)$. The term $\frac{1}{2} \delta_{j \ell}$ appears in since $$B_{\ell} = \frac{1}{2} ( F' - A_{1 \ell} - A_{2 \ell} - A_{3 \ell} - A_{4 \ell}).$$
For $n_i \geq 0$ and $k_i \in {{\mathbb{Z}}}/ 2$, let $$M^{(\ell)}_{y_i}(n_i,k_i) = \bigsqcup_{\substack{k_{i1}, \dots, k_{i4} \in {{\mathbb{Z}}}/2 \\ k_i = k_{i1} + \dots + k_{i4}}} M_{y_i}^{(\ell)}(n_i, (k_{ij})_j ) \,.
\label{moduli_space_Kummer_case}$$
be the moduli space parametrizing stable $1$-pointed genus $0$ maps to ${\operatorname{Km}}({{\mathsf{A}}})$ in class $\frac{n_i}{2} F + \sum_{j} k_{ij} A_{ij}$ for some $k_{ij}$ with $\sum_{j} k_{ij} = k_i$ and such that the marked points maps to $s^{\ell}(y_i)$.
Let $n \geq 0$ and $k \in {{\mathbb{Z}}}/2$ be fixed. Taking the union of over all $\textbf{k}$ such that $k = \sum_{i,j} k_{ij}$, interchanging sum and product and reindexing, we get $$\bigsqcup_{\substack{ \textbf{k} \colon \sum_{i,j} k_{ij} = k}} {{\overline M}}_{0}( {\operatorname{Km}}({{\mathsf{A}}}), \beta_{n, \textbf{k}})
=
\bigsqcup_{\ell = 1}^{4}\ \bigsqcup_{\substack{n = n_1 + \dots + n_4 \\ k + 2 = k_1 + \dots + k_4}} \prod_{i=1}^{4} M^{(\ell)}_{y_i}(n_i,k_i)
\label{242_2}$$ By arguments essentially identical to those in Section \[anvirclass\] the moduli space $M^{(\ell)}_{y_i}(n_i,k_i)$ carries a natural virtual class $$[ M^{(\ell)}_{y_i}(n_i,k_i) ]^{\text{vir}} \label{virtual_class_XYZ}$$ of dimension $0$ such that the splitting holds also for virtual classes: $$\bigsqcup_{\substack{ \textbf{k} \colon \sum_{i,j} k_{ij} = k}} [ {{\overline M}}_{0}( {\operatorname{Km}}({{\mathsf{A}}}), \beta_{n, \textbf{k}}) ]^{\text{red}}
=
\bigsqcup_{\ell = 1}^{4}\ \bigsqcup_{\substack{n = n_1 + \dots + n_4 \\ k + 2 = k_1 + \dots + k_4}} \prod_{i=1}^{4}\ [ M^{(\ell)}_{y_i}(n_i,k_i) ]^{\text{vir}} \,.
\label{242_3}$$
Consider the Bryan-Leung K3 surface $\pi_S : S \to {\mathbb{P}}^1$. Let[^6] $$L \subset \operatorname{\mathsf{Hilb}}^2(B)$$ be a fixed generic line and let $y \in {\mathbb{P}}^1$ be a point with $2y \in L$. Let $$M^{\textup{(F)}}_{S,y}(n,k)$$ be the moduli space parametrizing $1$-marked genus $0$ stable maps to $\operatorname{\mathsf{Hilb}}^2(S)$ in class $n F + k A$, which map the marked point to $s^{[2]}(2y)$, see . The subscript $S$ is added to avoid confusion. By Section \[Section\_analysis\_of\_virtual\_class\], the moduli space $M^{\textup{(F)}}_{S,y}(n,k)$ carries a natural virtual class.
\[242\_lemma\] We have $$\int_{ [ M^{(\ell)}_{y_i}(n,k) ]^{\text{vir}} } 1 \ =\ \int_{ [ M^{\textup{(F)}}_{S,y}(n,k) ]^{\text{vir}} } 1 \,. \label{242_lemma_vkmdsv}$$
The Lemma is proven below. We finish the proof of Proposition \[Kummer\_prop\_1\]. By the decomposition , $$\begin{gathered}
\quad \quad \quad \sum_{n \geq 0} \sum_{\substack{ \textbf{k} \\ \beta_{n,\textbf{k}} > 0}} \big\langle\, 1 \, \big\rangle_{0, \beta_{n,\textbf{k}}}^{{\operatorname{Km}}({{\mathsf{A}}})} q^n y^{\sum_{i,j} k_{ij}} \\
= \
\sum_{\substack{n \geq 0 \\ k \in {{\mathbb{Z}}}}} \sum_{\ell = 1}^{4}
\sum_{\substack{n = n_1 + \dots + n_4 \\ k + 2 = k_1 + \dots + k_4}}
\prod_{i=1}^{4} q^{n_i} y^{k_i - \frac{1}{2}} \int_{ [ M^{(\ell)}_{y_i}(n_i,k_i) ]^{\text{vir}} } 1 \quad \quad \quad\end{gathered}$$ An application of Lemma \[242\_lemma\] then yields $$\sum_{\ell = 1}^4 \prod_{i=1}^{4} \Big( \sum_{ \substack{ n_i \geq 0 \\ k_i \in {{\mathbb{Z}}}}} q^{n_i} y^{k_i - \frac{1}{2}} \int_{ [ M^{(\ell)}_{y_i}(n_i,k_i) ]^{\text{vir}} } 1 \Big)
= \ 4 \cdot ( F^{\textup{GW}}(y,q) )^4 \,.$$ This completes the proof of Proposition \[Kummer\_prop\_1\].
Let $F_y = \pi_S^{-1}(y)$ denote the fiber of $\pi_S$ over $y \in {\mathbb{P}}^1$. Consider the deformation of $S$ to the normal cone of $F_y$, $${{\mathcal S}}= {\mathop{\rm Bl}\nolimits}_{F_{y} \times 0}(S \times {{\mathbb{A}}}^1) \to {{\mathbb{A}}}^1,$$ and let ${{\mathcal S}}^{\circ} \subset {{\mathcal S}}$ be the complement of the proper transform of $S \times 0$. The *relative* Hilbert scheme $$\operatorname{\mathsf{Hilb}}^2( {{\mathcal S}}^{\circ} / {{\mathbb{A}}}^1 ) \to {{\mathbb{A}}}^1 \label{sofjosfosdf}$$ parametrizes length $2$ subschemes on the fibers of ${{\mathcal S}}^{\circ} \to {{\mathbb{A}}}^1$. Let $$p : M' \to {{\mathbb{A}}}^1$$ be the moduli space of $1$-pointed genus $0$ stable maps to $\operatorname{\mathsf{Hilb}}^2( {{\mathcal S}}^{\circ} / {{\mathbb{A}}}^1 )$ in class $n F + kA$, with the marked point mapping to the proper transform of $s^{[2]}(2 y) \times {{\mathbb{A}}}^1$. The fiber of $p$ over $t \neq 0$ is $$p^{-1}(t) = M^{\textup{(F)}}_{S,y}(n,k).$$ The fiber over $t = 0$ parametrizes maps to $\operatorname{\mathsf{Hilb}}^2({{\mathbb{C}}}\times F_y)$. Since the domain curve has genus $0$, these map to a fixed fiber of the natural map $$\operatorname{\mathsf{Hilb}}^2( {{\mathbb{C}}}\times F_y ) \xrightarrow{\ \rho\ } \operatorname{Sym}^2 ( {{\mathbb{C}}}\times F_y ) \xrightarrow{\ + \ } F_y \,.$$ We find, that $p^{-1}(0)$ parametrizes $1$-pointed genus $0$ stable maps into a singular $\mathsf{D}_4$ fiber of a trivial elliptic fibration, with given conditions on the class and the marking. Comparing with the construction of ${\operatorname{Km}}({{\mathsf{A}}})$ via and the definition of $M^{(\ell)}_{y_i}(n_i,k_i)$, one finds $$p^{-1}(0) \cong M^{(\ell)}_{y_i}(n_i,k_i).$$
The moduli space $M'$ carries the perfect obstruction theory obtained by the construction of section \[basic\_case\] in the relative context. On the fibers over $t \neq 0$ and $t=0$ the perfect obstruction theory of $M'$ restricts to the perfect obstruction theories of $M^{\textup{(F)}}_{S,y}(n,k)$ and $M^{(\ell)}_{y_i}(n_i,k_i)$ respectively. Hence, the associated virtual class $[M']^{\text{vir}}$ restricts on the fibers to the earlier defined virtual classes: $$\begin{aligned}
\quad \quad \quad t^{!} [M']^{\text{vir}} & = [M^{\textup{(F)}}_{S,y}(n,k)]^{\text{vir}}
\quad \quad (t \neq 0), \\
0^{!} [M']^{\text{vir}} & = [ M^{(\ell)}_{y_i}(n_i,k_i) ]^{\text{vir}}.\end{aligned}$$ Since $M' \to {{\mathbb{A}}}^1$ is proper, the proof of Lemma \[242\_lemma\] follows now from the principle of conversation of numbers, see [@Fulton Section 10.2].
### Effective classes
By Proposition \[Kummer\_prop\_1\], the evaluation of $F^{\textup{GW}}(y,q)$ is reduced to the evaluation of the series $$\sum_{\substack{ n, \textbf{k} \\ \beta_{n,\textbf{k}} > 0}} \big\langle\, 1 \, \big\rangle_{0, \beta_{n,\textbf{k}}}^{{\operatorname{Km}}({{\mathsf{A}}})} q^n y^{\sum_{i,j} k_{ij}} .
\label{Kummer_Series}$$ Since ${\operatorname{Km}}({{\mathsf{A}}})$ is a K3 surface, the Yau-Zaslow formula applies to the invariants $\langle 1 \rangle^{{\operatorname{Km}}({{\mathsf{A}}})}_{\beta}$, when $\beta$ is effective[^7] The remaining difficulty is to identify precisely the set of effective classes of the form $\beta_{n, \textbf{k}}$.
Let $n \geq 0$ and $\textbf{k} \in ({{\mathbb{Z}}}/2)^{4 \times 4}$. If $\beta_{n,\textbf{k}}$ is effective, then there exists a unique $\ell = \ell(n, \textbf{k}) \in \{ 1, \dots, 4 \}$ such that $$\beta_{n,\textbf{k}} = B_{\ell} + \sum_{i=1}^4 a_i T_i + \sum_{i,j = 1}^{4} b_{ij} A_{ij} \,.$$ for some integers $a_i \geq 0$ and $b_{ij} \geq 0$.
If $\beta_{n,\textbf{k}}$ is effective, then by the argument in the proof of Proposition \[Kummer\_prop\_1\], there exist non-negative integers $$a_i, \ \ i=1,\dots,4 \quad \text{ and } \quad b_{ij}, \ \ i,j = 1, \dots, 4$$ such that $$\beta_{n,\textbf{k}} = B_{\ell} + \sum_{i=1}^4 a_i T_i + \sum_{i,j = 1}^{4} b_{ij} A_{ij}$$ for *some* $\ell \in \{ 1, \dots, 4 \}$. We need to show, that $\ell$ is unique. By , we have $$\beta_{n,\textbf{k}} = \frac{F'}{2} + \frac{\sum_{i=1}^{4} a_i}{2} F + \sum_{i,j=1}^{4} \Big( b_{ij} - \frac{a_{i}}{2} - \frac{1}{2} \delta_{j \ell} \Big) A_{ij},$$ hence $k_{ij} = b_{ij} - \frac{a_{i1}}{2} - \frac{1}{2} \delta_{j \ell}$. We find, that $\ell$ is the unique integer such that for every $i$ one of the following holds:
- $k_{ij} \in {{\mathbb{Z}}}$ for all $j \neq \ell$ and $k_{i \ell} \notin {{\mathbb{Z}}}$,
- $k_{ij} \notin {{\mathbb{Z}}}$ for all $j \neq \ell$ and $k_{i \ell} \in {{\mathbb{Z}}}$.
In particular, $\ell$ is uniquely determined by $\textbf{k}$.
By the proof of proposition \[Kummer\_prop\_1\], the contribution from all classes $\beta_{n, \textbf{k}}$ with a given $\ell$ to the sum is independent of $\ell$. Hence, equals $$4 \cdot \sum_{ n, \textbf{k}}
\big\langle\, 1 \, \big\rangle_{0, \beta_{n,\textbf{k}}}^{{\operatorname{Km}}({{\mathsf{A}}})}
q^n y^{\sum_{i,j} k_{ij}}\,, \label{Kummer_Series2}$$ where the sum runs over all $(n, \textbf{k})$ such that $\beta_{n,\textbf{k}}$ is effective *and* $\ell(n, \textbf{k}) = 1$. Hence, we may assume $\ell = 1$ from now on.
It will be useful to rewrite the classes $\beta_{n, \textbf{k}}$ in the basis $$B_1,\ F \quad \text{ and } \quad T_i,\ A_{i2},\ A_{i3},\ A_{i4}, \ \ \ i = 1, \dots, 4. \label{243_1}$$ Consider the class $$\begin{aligned}
\beta_{n, \textbf{k}}
& = \frac{1}{2} F' + \frac{n}{2} F + \sum_{i,j = 1}^{4} k_{ij} A_{ij}\ \in H_2( {\operatorname{Km}}({{\mathsf{A}}}) , {{\mathbb{Q}}}) \\
& = B_1 + \tilde{n} F + \sum_{i=1}^{4} \Big( a_i T_i + \sum_{j=2}^{4} b_{ij} A_{ij} \Big) \,,\end{aligned}$$ where $(n,\textbf{k})$ and $(\tilde{n}, a_i, b_{ij})$ are related by $$n = 2 \tilde{n} + \textstyle{\sum}_i a_i, \quad
k_{i1} = -\frac{1}{2} (a_i + 1), \quad
k_{ij} = b_{ij} - \frac{a_i}{2} \ \ (j > 2).
\label{eff_trans_eqn}$$
\[eff\_lemma\_1\] If $\beta_{n, \textbf{k}}$ is *effective*, then $\tilde{n}, a_i, b_{ij}$ are integers for all $i,j$.
If $\beta_{n, \textbf{k}}$ is effective with $\ell(n, \textbf{k}) = 1$, there exist non-negative integers $$\tilde{a}_i, \ \ i=1,\dots,4 \quad \text{ and } \quad \tilde{b}_{ij}, \ \ i,j = 1, \dots, 4$$ such that $$\beta_{n,\textbf{k}} = B_{1} + \sum_{i=1}^4 a_i T_i + \sum_{i,j = 1}^{4} b_{ij} A_{ij} .$$ In the basis we obtain $$\beta_{n,\textbf{k}}
= B_1 + \Big( \sum_{i=1}^{4} \tilde{b}_{i1} \Big) F
+ \sum_{i=1}^{4} \Big( (\tilde{a}_i - 2 \tilde{b}_{i1} ) T_i + \sum_{j=2}^{4} (\tilde{b}_{ij} - \tilde{b}_{i1}) A_i \Big).$$ The claim follows.
\[eff\_lemma\_2\] If $\tilde{n}, a_i, b_{ij}$ are integers and $\beta_{n, \textbf{k}}^2 \geq -2$, then $\beta_{n, \textbf{k}}$ is effective.
If $\tilde{n}, a_i, b_{ij}$ are integers, then $\beta_{n, \textbf{k}}$ is the class of a divisor $D$. By Riemann-Roch we have $$\frac{\chi({{\mathcal O}}(D)) + \chi({{\mathcal O}}(-D))}{2} = \frac{D^2}{2} + 2,$$ and by Serre duality we have $$h^0(D) + h^0(-D)\ \geq\ \frac{\chi({{\mathcal O}}(D)) + \chi({{\mathcal O}}(-D))}{2} \,.$$ Hence, if $\beta_{n,\textbf{k}}^2 = D^2 \geq -2$, then $h^0(D) + h^0(-D) \geq 1$. Since $F \cdot \beta_{n, \textbf{k}} = 1$, we have $h^0(-D) = 0$, and therefore $h^0(D) \geq 1$ and $D$ effective.
We are ready to evaluate the series .
By Lemma \[eff\_lemma\_1\] we may replace the sum in by a sum over all integers $\tilde{n} \in {{\mathbb{Z}}}$ and all elements $$x_i = a_i T_i + \sum_{j=2}^{4} b_{ij} A_{ij}, \ \ \ \ i=1,\dots 4$$ such that
- $a_i, b_{i2}, b_{i3}, b_{i4}$ are integers for $i \in \{1,\dots, 4\}$,
- $B_1 + \tilde{n} F + \sum_{i} x_i$ is effective.
Hence, using the series equals $$4 \cdot \sum_{\tilde{n}} \sum_{x_1, \dots, x_4} q^{2 \tilde{n} + \sum_i a_i} y^{-2 + \sum_i \langle x_i, T_i \rangle}
\big\langle\, 1 \, \big\rangle_{0, B_1 + \tilde{n} F + \sum_i x_i}^{{\operatorname{Km}}({{\mathsf{A}}})}, \label{KumS_3}$$ where the sum runs over all $(\tilde{n}, x_1, \dots, x_4)$ satisfying (i) and (ii) above. By the Yau-Zaslow formula , we have $$\big\langle\, 1 \, \big\rangle_{0, B_1 + \tilde{n} F + \sum_i x_i}^{{\operatorname{Km}}({{\mathsf{A}}})}
=
\left[ \frac{1}{\Delta(\tau)} \right]_{q^{\tilde{n}-1 + \sum_i \langle x_i, x_i \rangle/2}},
\label{eff_YZ_eval}$$ whenever $B_1 + \tilde{n} F + \sum_i x_i$ is effective; here $[\ \cdot\ ]_{q^m}$ denotes the coefficient of $q^m$. The term vanishes, unless $$\tilde{n}-1 + \frac{1}{2} \sum_i \langle x_i, x_i \rangle = \frac{1}{2} \left(B_1 + \tilde{n} F + \sum_i x_i\right)^2 \geq -1 \,.$$ When evaluating , we may therefore restrict to tuples $(\tilde{n}, x_1, \dots, x_4)$, that also satisfy
- $\big(B_1 + \tilde{n} F + \sum_i x_i \big)^2 \geq -2$.
By Lemma \[eff\_lemma\_2\], condition (i) and (iii) together imply condition (ii). In we may therefore sum over tuples $(\tilde{n}, x_1, \dots, x_4)$ satisfying (i) and (iii) alone. Rewriting (iii) as $$\tilde{n} \geq - \sum_i \langle x_i, x_i \rangle/2$$ and always assuming (i) in the following sums, equals $$\begin{aligned}
& 4 \cdot \sum_{x_1, \dots, x_4}
\sum_{\tilde{n} \geq \sum_i \frac{\langle x_i, x_i \rangle}{-2}}
q^{2 \tilde{n} + \sum_i a_i} y^{-2 + \sum_i \langle x_i, T_i \rangle}
\left[ \frac{1}{\Delta(\tau)} \right]_{q^{\tilde{n}-1 + \sum_i \langle x_i, x_i \rangle/2}} \\
=\ & 4 \cdot \sum_{x_1, \dots, x_4} y^{-2 + \sum_i \langle x_i, T_i \rangle} q^{2 + \sum_i (a_i - \langle x_i, x_i \rangle)} \\
& \quad \quad \quad \quad \quad \quad \quad \ \ \ \times \sum_{\tilde{n} \geq \sum_i \frac{\langle x_i, x_i \rangle}{-2}} q^{2 \tilde{n} - 2 + \sum_i \langle x_i, x_i \rangle}
\left[ \frac{1}{\Delta(\tau)} \right]_{q^{\tilde{n}-1 + \sum_i \frac{\langle x_i, x_i \rangle}{2}}} \\
=\ & \frac{4}{\Delta(2 \tau)} \cdot \sum_{x_1, \dots, x_4} y^{-2 + \sum_i \langle x_i, T_i \rangle} q^{2 + \sum_i (a_i - \langle x_i, x_i \rangle)} \\
=\ & \frac{4}{\Delta(2 \tau)} \cdot \prod_{i=1}^{4} \Big( \sum_{x_i} y^{- \frac{1}{2} + \langle x_i, T_i \rangle} q^{\frac{1}{2} + a_i - \langle x_i, x_i \rangle} \Big)\,.
$$ Consider the $\mathsf{D}_4$ lattice, defined as ${{\mathbb{Z}}}^4$ together with the bilinear form $${{\mathbb{Z}}}^4 \times {{\mathbb{Z}}}^4 \ni (x,y) \mapsto \langle x,y \rangle := x^T M y,$$ where $$M = \left[ \begin{array}{rrrr} 2 & -1 & -1 & -1 \\ -1 & 2 & 0 & 0 \\ -1 & 0 & 2 & 0 \\ -1 & 0 & 0 & 2 \end{array} \right].$$ Let $(e_1, \dots, e_4)$ denote the standard basis of ${{\mathbb{Z}}}^4$ and let $$\alpha = 2 e_1 + e_2 + e_3 + e_4.$$ Consider the function $$\Theta(z,\tau)
= \sum_{x \in {{\mathbb{Z}}}^4} \exp\Big( - 2 \pi i \Big\langle x + \frac{\alpha}{2} , z e_1 + \frac{e_1}{2} \Big\rangle \Big) \cdot
q^{\left\langle x + \frac{\alpha}{2}, x + \frac{\alpha}{2} \right\rangle}$$ where $z \in {{\mathbb{C}}}, \tau \in {{\mathbb{H}}}$ and $q = e^{2 \pi i \tau}$. The function $\Theta(z,\tau)$ is a theta function with characteristics associated to the lattice $\mathsf{D}_4$. In particular $\Theta(z,\tau)$ is a Jacobi form of index $1/2$ and weight $2$, see [@EZ Section 7].[^8]
\[theta\_lemma\_xzz\] For every $i \in \{ 1, \dots, 4 \}$, $$\sum_{x_i} y^{- \frac{1}{2} + \langle x_i, T_i \rangle} q^{\frac{1}{2} + a_i - \langle x_i, x_i \rangle}
\ = \ \Theta(z,\tau)$$ under $q = e^{2 \pi i \tau}$ and $y = -e^{2 \pi i z}$.
Let $\mathsf{D}_4(-1)$ denote the lattice ${{\mathbb{Z}}}^4$ with intersection form $$(x,y) \mapsto {-}x^{T}M y.$$ The ${{\mathbb{Z}}}$-homomorphism defined by $$e_1 \mapsto T_i,\ \ e_2 \mapsto A_{i2},\ \ e_3 \mapsto A_{i3}, \ \ e_4 \mapsto A_{i4}$$ is an isomorphism from $\mathsf{D}_4(-1)$ to $$\Big( {{\mathbb{Z}}}T_i \oplus {{\mathbb{Z}}}A_{i2} \oplus {{\mathbb{Z}}}A_{i3} \oplus {{\mathbb{Z}}}A_{i4}, \langle \cdot , \cdot \rangle \Big),$$ where $\langle \ , \, \rangle$ denotes the intersection product on ${\operatorname{Km}}({{\mathsf{A}}})$. Hence, $$\sum_{x_i} y^{- \frac{1}{2} + \langle x_i, T_i \rangle} q^{\frac{1}{2} + a_i - \langle x_i, x_i \rangle}
=
\sum_{x \in {{\mathbb{Z}}}^4} y^{- \frac{1}{2} - \langle x, e_1 \rangle} q^{\frac{1}{2} + \langle x, \alpha \rangle + \langle x, x \rangle}$$ Using the substitution $y = \exp( 2 \pi i z + \pi i )$, we obtain $$\sum_{x \in {{\mathbb{Z}}}^4} \exp \Big( - 2 \pi i \cdot \big\langle x + \frac{\alpha}{2}, z e_1 + \frac{e_1}{2} \big\rangle \Big)
\cdot q^{ \langle x + \frac{\alpha}{2}, x + \frac{\alpha}{2} \rangle} = \Theta(z,\tau) \,. \qedhere \\$$
By Lemma \[theta\_lemma\_xzz\], we conclude $$\sum_{\substack{ n, \textbf{k} \\ \beta_{n,\textbf{k}} > 0}} \big\langle\, 1 \, \big\rangle_{0, \beta_{n,\textbf{k}}}^{{\operatorname{Km}}({{\mathsf{A}}})} q^n y^{\sum_{i,j} k_{ij}}
=
\frac{4}{\Delta(2 \tau)} \cdot \Theta(z,\tau)^4
\label{YZ_Kummer_jdhnr}$$
### The theta function of the $\mathsf{D}_4$ lattice
Consider the Dedekind eta function $$\eta(\tau) = q^{1/24} \prod_{m \geq 1} (1 - q^m)$$ and the first Jacobi theta function $$\vartheta_1(z,\tau) = -i q^{1/8} (p^{1/2} - p^{-1/2}) \prod_{m \geq 1} (1 - q^m) (1 - p q^m) (1 - p^{-1} q^m),$$ where $q = e^{2 \pi i \tau}$ and $p = e^{2 \pi i z}$.
\[YZ\_Theta\_identity\] We have $$\Theta(z,\tau) = \frac{ -\vartheta_1(z,\tau) \cdot \eta(2 \tau)^6 }{ \eta(\tau)^3 } \label{YZ_Theta_identity_eqn}$$
The proof of Proposition \[YZ\_Theta\_identity\] is given below. We complete the proof of Theorem \[thm\_F\_evaluation\].
By Proposition \[Kummer\_prop\_1\], we have $$4 \cdot F^{\textup{GW}}(y,q)^{4}
= \sum_{\substack{n, \textbf{k} \\ \beta_{b,\textbf{k}} > 0}} \big\langle\, 1 \, \big\rangle_{0, \beta_{n,\textbf{k}}}^{{\operatorname{Km}}({{\mathsf{A}}})} q^n y^{\sum_{i,j} k_{ij}} \,.$$ The evaluation and Proposition \[YZ\_Theta\_identity\] yields $$F^{\textup{GW}}(y,q)^{4} = \frac{1}{\Delta(2 \tau)} \left( \frac{ \vartheta_1(z,\tau) \cdot \eta(2 \tau)^{6} }{ \eta(\tau)^3 } \right)^4 \,.$$ Since $\Delta(\tau) = \eta(\tau)^{24}$, we conclude $$F^{\textup{GW}}(y,q) = \pm \frac{ \vartheta_1(z,\tau) }{\eta^3(\tau)}.$$ By the definition of $F^{\textup{GW}}$ in Section \[concl\_basic\_case\], the coefficient of $y^{-1/2} q^0$ is $1$. Hence $$F^{\textup{GW}}(y,q) = \frac{ \vartheta_1(z,\tau) }{\eta^3(\tau)} = F(z, \tau). \qedhere$$
Both sides of are Jacobi forms of weight $2$ and index $1/2$ for a certain congruence subgroup of the Jacobi group. The statement would therefore follow by the theory of Jacobi forms [@EZ] after comparing enough coefficients of both sides. For simplicity, we will instead prove the statement directly.
We will work with the variables $q = e^{2 \pi i \tau}$ and $p = e^{2 \pi i z}$. Consider $$F(z,\tau) = \frac{\vartheta_1(z,\tau)}{\eta^3(\tau)} = -i(p^{1/2} - p^{-1/2}) \prod_{m \geq 1} \frac{ (1-pq^m) (1-p^{-1}q^m)}{ (1-q^m)^2 } \label{ffffpqdf}$$ By direct calculation one finds $$\begin{aligned}
\label{15343r4g}
F( z+ \lambda \tau + \mu, \tau) & = (-1)^{\lambda + \mu} q^{-\lambda/2} p^{-\lambda} K(z,\tau) \\
\Theta(z + \lambda \tau + \mu, \tau) & = (-1)^{\lambda + \mu} q^{-\lambda/2} p^{-\lambda} \Theta(z,\tau) \,.
\end{aligned}$$ We have $$\begin{aligned}
\Theta(0, \tau)
& = \sum_{x \in {{\mathbb{Z}}}^4} \exp\Big( - 2 \pi i \Big\langle x + \frac{\alpha}{2} , \frac{e_1}{2} \Big\rangle \Big) q^{\left\langle x + \frac{\alpha}{2}, x + \frac{\alpha}{2} \right\rangle} \\
& = \sum_{x' \in {{\mathbb{Z}}}^4 + \frac{\alpha}{2}} \exp\big( - \pi i \langle x', e_1 \rangle \big) q^{\langle x', x' \rangle}\end{aligned}$$ Since for every $x' = m + \frac{\alpha}{2}$ with $m \in {{\mathbb{Z}}}^4$ one has $$\begin{aligned}
\exp\big( - \pi i \langle x', e_1 \rangle \big) + \exp\big( - \pi i \langle - x', e_1 \rangle \big)
& = -i (-1)^{\langle m, e_1\rangle} + i (-1)^{- \langle m, e_1 \rangle} \\ & = 0,\end{aligned}$$ we find $\Theta(0,\tau) = 0$. By , we also have $F(0, \tau) = 0$.
Since $\Theta$ and $F$ are Jacobi forms of index $1/2$ (see [@EZ Theorem 1.2]), the point $z = 0$ is the only zero of $\Theta$ resp. $F$ in the standard fundamental region. Therefore, the quotient $$\frac{\Theta(z,\tau)}{F(z,\tau)}$$ is a double periodic entire function, and hence a constant in $\tau$. Using the evaluations $$F\left( \frac{1}{2}, \tau \right) = 2 \prod_{m \geq 1} \frac{(1+q^m)^2}{(1-q^m)^2} = 2 \frac{\eta(2 \tau)^2}{\eta(\tau)^4}$$ and $$\Theta\left( \frac{1}{2} , \tau \right)
= \sum_{x \in {{\mathbb{Z}}}^4} (-1) q^{\langle x + \frac{\alpha}{2}, x + \frac{\alpha}{2} \rangle},$$ the statement therefore follows directly from Lemma \[xmcisforg\] below.
\[xmcisforg\] We have $$\sum_{x \in {{\mathbb{Z}}}^4} q^{\langle x + \frac{\alpha}{2}, x + \frac{\alpha}{2} \rangle} = 2 \, \frac{ \eta(2 \tau)^8 }{ \eta(\tau)^4 } \,.$$
As a special case of the Jacobi triple product [@Chandra], we have $$2 \frac{ \eta(2 \tau)^2 }{ \eta(\tau) } = 2 q^{1/8} \prod_{m \geq 1} \frac{(1-q^{2m})^{2}}{(1-q^m)} = \sum_{m \in {{\mathbb{Z}}}} q^{(m + \frac{1}{2})^{2} / 2}$$ For $m = (m_1, \dots, m_4) \in {{\mathbb{Z}}}^4$, let $$x_m =
\left( m_1 + \frac{1}{2} \right) \frac{\alpha}{2}
+ \left( m_2 + \frac{1}{2} \right) \frac{e_2}{2}
+ \dots + \left(m_4 + \frac{1}{2}\right) \frac{e_4}{2}$$ Using that $\alpha, e_2, e_3, e_4$ are orthogonal, we find $$16 \, \frac{ \eta(2 \tau)^8 }{ \eta(\tau)^4 }
= \left( \sum_{m \in {{\mathbb{Z}}}} q^{(m + \frac{1}{2})^{2} / 2} \right)^{4}
= \sum_{m \in {{\mathbb{Z}}}^4} q^{\langle x_m, x_m \rangle}$$ We split the sum over $m = (m_1, \dots, m_4) \in {{\mathbb{Z}}}^4$ depending upon whether $m_1 + m_i$ is odd or even for $i=2,3,4$, $$\sum_{m \in {{\mathbb{Z}}}^4} q^{\langle x_m, x_m \rangle}
=
\sum_{s_2, s_3, s_4 \in \{0,1\}}
\sum_{\substack{(m_1, \dots,m_r) \in {{\mathbb{Z}}}^4 \\ m_1 + m_i \equiv s_i\, (2) }} q^{\langle x_m, x_m \rangle}
\label{dmsodvfsdg}$$ For every choice of $s_2, s_3, s_4 \in \{0,1\}$, we have $$\sum_{\substack{(m_1, \dots,m_r) \in {{\mathbb{Z}}}^4 \\ m_1 + m_i \equiv s_i (2) }} q^{\langle x_m, x_m \rangle}
= \sum_{x \in {{\mathbb{Z}}}^4} q^{\left\langle x + \frac{\beta}{2} , x + \frac{\beta}{2} \right\rangle},$$ where $\beta \in {{\mathbb{Z}}}^4$ is a root of the $\mathsf{D}_4$-lattice (i.e. $\langle \beta, \beta \rangle = 2$). Since the isometry group of $\mathsf{D}_4$ acts transitively on roots, $$\sum_{x \in {{\mathbb{Z}}}^4} q^{\left\langle x + \frac{\beta}{2} , x + \frac{\beta}{2} \right\rangle} =
\sum_{x \in {{\mathbb{Z}}}^4} q^{\langle x + \frac{\alpha}{2}, x + \frac{\alpha}{2} \rangle}.$$ Inserting this into and dividing by $8$, the proof is complete.
Evaluation of further Gromov-Witten invariants {#Chapter_More_Evaluations}
==============================================
\[Section\_More\_Evaluations\]
Overview {#overview-1}
--------
In Section \[Chapter\_More\_Evaluations\] and Section \[Section\_Hilb2P1xE\] we prove Theorem \[MThm\]
In Section \[BL\_reformulation\] we reduce the calculation to a Bryan-Leung K3. We also state one extra evaluation on the Hilbert scheme of $2$ points of a K3 surface, which is required in Section \[Section\_Quantum\_Cohomology\]. Next, for each case separately, we analyse the moduli space of maps which are incident to the given conditions. In each case, the main result is a splitting statement similar to Proposition \[W0splitprop\].
As a result, the proof of Theorem \[MThm\] is reduced to the calculation of certain universal contributions associated to single elliptic fibers. These contributions will be determined in Section \[Section\_Hilb2P1xE\] using the geometry of $\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1 \times E)$, where $E$ is an elliptic curve. The strategy is parallel but more difficult to the evaluation considered in Section \[basic\_case\].
Reduction to the Bryan-Leung K3 {#BL_reformulation}
-------------------------------
Let $\pi : S \to {\mathbb{P}}^1$ be an elliptic K3 surface with a unique section and $24$ nodal fibers. Let $B$ and $F$ be the section and fiber class respectively, and let $$\beta_h = B + h F$$ for $h \geq 0$. The quantum bracket $\langle \ \ldots \ \rangle_q$ on $\operatorname{\mathsf{Hilb}}^d(S), d \geq 1$ is defined by $$\big\langle \gamma_1, \dots, \gamma_m \big\rangle_q^{\operatorname{\mathsf{Hilb}}^d(S)}
= \sum_{h \geq 0} \sum_{k \in {{\mathbb{Z}}}} y^k q^{h-1} \langle \gamma_1, \dots, \gamma_m \rangle^{\operatorname{\mathsf{Hilb}}^d(S)}_{\beta_h + k A} \,,$$ where $\gamma_1, \dots, \gamma_m \in H^{\ast}(\operatorname{\mathsf{Hilb}}^d(S))$ are cohomology classes. By arguments parallel to Section \[K3statements\], Theorem \[MThm\] is equivalent to the following Theorem.
\[ellthm2\] Let $P_1, \dots, P_{2d-2} \in S$ be generic points. For $d \geq 2$, $$\begin{aligned}
{\Big\langle}C(F) {\Big\rangle}^{\operatorname{\mathsf{Hilb}}^d(S)}_q
& = \frac{G(z,\tau)^{d-1}}{\Delta(\tau)} \\[3pt]
{\Big\langle}A\, {\Big\rangle}^{\operatorname{\mathsf{Hilb}}^d(S)}_q
& = - \frac{1}{2} \Big( y \frac{d}{dy} G(z, \tau) \Big) \frac{G(z,\tau)^{d-2}}{\Delta(\tau)} \\[3pt]
{\Big\langle}I(P_1), \ldots, I(P_{2d-2}) {\Big\rangle}^{\operatorname{\mathsf{Hilb}}^d(S)}_q
& = \frac{1}{d} \binom{2d-2}{d-1} \Big( q \frac{d}{dq} F(z,\tau)\Big)^{2d-2} \frac{1}{\Delta(\tau)}\end{aligned}$$ under the variable change $q = e^{2 \pi i \tau}$ and $y = - e^{2 \pi i z}$.
Later we will require one additional evaluation on $\operatorname{\mathsf{Hilb}}^2(S)$. Let $P \in S$ be a generic point and let $${{\mathfrak{p}}}_{-1}(F)^2 1_S$$ be the class of a generic fiber of $\pi^{[2]} : \operatorname{\mathsf{Hilb}}^2(S) \to {\mathbb{P}}^2$.
\[extra\_eval\] Under the variable change $q = e^{2 \pi i \tau}$ and $y = - e^{2 \pi i z}$, $${\Big\langle}{{\mathfrak{p}}}_{-1}(F)^2 1_S ,\, I(P) {\Big\rangle}^{\operatorname{\mathsf{Hilb}}^2(S)}_q = \frac{F(z,\tau) \cdot q \frac{d}{dq} F(z,\tau) }{ \Delta(\tau) }$$
Case $\langle C(F) \rangle_q$ {#CASE1}
-----------------------------
We consider the evaluation of $\big\langle C(F) \big\rangle_q^{\operatorname{\mathsf{Hilb}}^d(S)}$. Let $P_1, \dots, P_{d-1} \in S$ be generic points, let $F_0$ be a generic fiber of the elliptic fibration $\pi : S \to {\mathbb{P}}^1$, and let $$Z = F_0[1] P_1[1] \cdots P_{d-1}[1] \subset \operatorname{\mathsf{Hilb}}^d(S)$$ be the induced subscheme of class $[Z] = C(F)$, where we used the notation of Section \[special\_cycles\] (v). Consider the evaluation map $${\mathop{\rm ev}\nolimits}: {{\overline M}}_{0,1}( \operatorname{\mathsf{Hilb}}^d(S), \beta_h + kA) \to S, \label{evalmap_lkvkffgk}$$ the moduli space parametrizing maps incident to the subscheme $Z$ $$M_Z = {\mathop{\rm ev}\nolimits}^{-1}(Z) \,, \label{301}$$ and an element $$[f : C \to \operatorname{\mathsf{Hilb}}^d(S), p] \in M_Z \,.$$ By Lemma \[200\], there does not exist a non-constant genus $0$ stable map to $\operatorname{\mathsf{Hilb}}^d(S)$ of class $h' F + k'A$ which is incident to $Z$. Hence, the marking $p \in C$ must lie on the distinguished irreducible component $$C_0 \subset C$$ on which $\pi^{[d]} \circ f$ is non-constant. By Lemma \[117\], the restriction $f|_{C_0}$ is therefore an isomorphism $$\begin{aligned}
f|_{C_0}\colon C_0 & \to B_0[1] P_1[1] \cdots P_{d-1}[1] \\
& \quad \quad \quad \quad = I(B_0) \cap I(P_1) \cap \ldots \cap I(P_{d-1}) \subset \operatorname{\mathsf{Hilb}}^d(S) \,,
\label{v5gkfkf}
\end{aligned}$$ where $B_0$ is the section of $S \to {\mathbb{P}}^1$. In particular, $f(p) = (F_0 \cap B_0) + \sum_j P_j$. We identify $C_0$ with its image in $\operatorname{\mathsf{Hilb}}^d(S)$.
Let $x_1, \dots, x_{24}$ be the basepoints of the rational nodal fibers of $\pi$ and let $u_i = \pi(P_i)$ for all $i$. The image line $L = \pi^{[d]} \circ f(C)$ meets the discriminant locus of $\pi^{[d]}$ in the points $$x_i + \sum_{j=1}^{d-1} u_j\ \ (i = 1,\dots, 24) \quad \text{ and } \quad 2 u_i + \sum_{j \neq i} u_j \ \ (i =1,\dots, d-1)$$ By Lemma \[200\], the curve $C$ is therefore of the form $$C = C_0 \cup A_1 \cup \ldots \cup A_{24} \cup B_1 \cup \ldots \cup B_{d-1}$$ where the components $A_i$ and $B_j$ are attached to the points $$x_i + P_1 + \dots + P_{d-1} \quad \text{ and } \quad u_j + P_1 + \ldots + P_{d-1} \label{fowjroirgfrg}$$ respectively. Hence, the moduli space $M_Z$ is set-theoretically a product of spaces parametrizing maps of the form $f|_{A_i}$ and $f|_{B_j}$ respectively. We show that the set-theoretic product is scheme-theoretic and the virtual class splits. The argument is similar to Section \[basic\_case\].
First, the attachment points do not smooth under infinitesimal deformations: this follows since the projection $$f^{\ast} {{\mathcal Z}}_d = \widetilde{C} \to C$$ is étale over the points , see the proof of Lemma \[doesnotsmooth\]; here ${{\mathcal Z}}_d \to \operatorname{\mathsf{Hilb}}^d(S)$ is the universal family. Therefore, any infinitesimal deformation of $f$ inside $M_Z$ induces a deformation of the image $f(C_0)$. This deformation corresponds to moving the points $P_1, \dots, P_{d-1}$ in , which is impossible since $f$ continues to be incident to $Z$. Hence, $f(C_0)$ is fixed under infinitesimal deformations.[^9]
By a construction parallel to Section \[splitting\], we have a splitting map $$\Psi : M_Z \to \bigsqcup_{(\textbf{h}, \textbf{k})}
\bigg( \prod_{i=1}^{24} M^{\textup{(N)}}_{x_i}(h_{x_i})
\times \prod_{j=1}^{d-1} M^{\textup{(G)}}_{u_j}(h_{y_j}, k_{y_j}) \bigg),
\label{vfivofvg}$$ where $M^{\textup{(N)}}_{x_i}(h_{x_i})$ was defined in Section \[splitting\], and for an appropriately defined moduli space $M^{\textup{(G)}}_{u_j}(h_{y_j}, k_{y_j})$; since $f(C_0)$ has class $B$, the disjoint union in runs over all $$\begin{aligned}
\textbf{h} & = (h_{x_1}, \dots, h_{x_{24}}, h_{u_1}, \dots, h_{u_{d-1}} ) \in ({{\mathbb{N}}}^{\geq 0})^{ \{ x_i, u_j \}} \\
\textbf{k} & = (k_{u_1}, \dots, k_{u_{d-1}}) \in {{\mathbb{Z}}}^{d-1}
\end{aligned}$$ such that $\sum_i h_{x_i} + \sum_j h_{u_j} = h$ and $\sum_j k_{u_j} = k$. Since $f(C_0)$ is fixed under infinitesimal deformations, the map $\Psi$ is an isomorphism.
Let $[ M_Z ]^{\text{vir}}$ be the natural virtual class on $M_Z$. By arguments parallel to Section \[anvirclass\], the pushforward $\Psi_{\ast} [M_Z]^{\text{vir}}$ is a product of virtual classes defined on each factor. Hence, by a calculation identical to Section \[concl\_basic\_case\], $\langle C(F) \rangle_q$ is the product of series corresponding to the points $x_i$ and $u_j$ respectively.
For the points $x_1, \dots, x_{24}$, the contributing factor agrees with the contribution from the nodal fibers in the case of Section \[Section\_higher\_dimensional\_Yau\_Zaslow\]. It is the series . For $u_1, \dots, u_{d-1}$ define the formal series $$G^{\textup{GW}}(y,q) = \sum_{h \geq 0} \sum_{k \in {{\mathbb{Z}}}} y^k q^{h} \int_{[M^{\textup{(G)}}_{u_j}(h, k)]^{\text{vir}} } 1, \label{GGGdef}$$ where we let $[ M^{\textup{(G)}}_{u_j}(h, k) ]^{\text{vir}}$ denote the induced virtual class on $M^{\textup{(G)}}_{u_j}(h, k)$. We conclude $${\Big\langle}C(F) {\Big\rangle}_q^{\operatorname{\mathsf{Hilb}}^d(S)} = \frac{G^{\textup{GW}}(y,q)^{d-1}}{\Delta(q)} \,. \label{401}$$
Case $\langle A \rangle_q$ {#ABCD}
--------------------------
We consider the evaluation of $\langle A\, \rangle_q^{\operatorname{\mathsf{Hilb}}^d(S)}$. Let $P_0, \ldots , P_{d-2} \in S$ be generic points, let $$Z = P_0[2] P_1[1] \cdots P_{d-2}[1] \subset \operatorname{\mathsf{Hilb}}^d(S)$$ be the exceptional curve (of class $A$) centered at $2 P_0 + P_1, \dots + P_{d-2}$, and let $$M_Z = {\mathop{\rm ev}\nolimits}^{-1}(Z),$$ where ${\mathop{\rm ev}\nolimits}$ is the evaluation map . We consider an element $$[f : C \to \operatorname{\mathsf{Hilb}}^d(S), p] \in M_Z.$$ Let $C_0 \subset C$ be the distinguished component of $C$ on which $\pi^{[d]} \circ f$ is non-constant, and let $C'$ be the union of all irreducible components of $C$ which map into the fiber $$(\pi^{[d]})^{-1}( 2 u_0 + u_1 + \ldots + u_{d-2} ),$$ where $u_i = \pi(P_i)$. Since $f(C_0)$ cannot meet the exceptional curve $Z$, the component $C'$ contains the marked point $p$, $$p \in C'.$$ The restriction $f|_{C'}$ decomposes into the components $$f|_{C'} = \phi + P_1 + \dots + P_{d-2},$$ where $\phi : C' \to \operatorname{\mathsf{Hilb}}^2(S)$ maps into the fiber $\pi^{[2] -1}(2 u_0)$ and the $P_i$ denote constant maps.
Consider the Hilbert-Chow morphism $$\rho : \operatorname{\mathsf{Hilb}}^2(S) \to \operatorname{Sym}^2(S)$$ and the Abel-Jacobi map $$\mathsf{aj} : \operatorname{Sym}^2(F_{u_0}) \to F_{u_0} \,.$$ Since $\rho(\phi(p)) = 2 P_0$, the image of $\phi$ lies inside the fiber $V$ of $$\rho^{-1}(\operatorname{Sym}^2(F_{u_0}) ) \xrightarrow{\rho} \operatorname{Sym}^2(F_{u_0}) \xrightarrow{\mathsf{aj}} F_{u_0}$$ over the point $\mathsf{aj}(2 P_0)$. Hence, $f|_{C'}$ maps into the subscheme $$\widetilde{V} = V + P_1 + \ldots + P_{d-2} \subset \operatorname{\mathsf{Hilb}}^d(S)\,.$$
The intersection of $\widetilde{V}$ with the divisor $D(B_0) \subset \operatorname{\mathsf{Hilb}}^d(S)$ is supported in the reduced point $$s(u_0) + Q + P_1 + \dots + P_{d-2} \in \operatorname{\mathsf{Hilb}}^d(S), \label{3ofidofg}$$ where $s : {\mathbb{P}}^1 \to S$ is the section and $Q \in F_{u_0}$ is defined by $$\mathsf{aj}(s(u_0) + Q) = \mathsf{aj}( 2 P_0 ) \,.$$ Since the distinguished component $C_0 \subset C$ must map into $D(B_0)$, the point $f(C_0 \cap C')$ therefore equals . Hence, the restriction $f|_{C_0}$ yields an isomorphism $$f|_{C_0} : C_0 \xrightarrow{\ \cong\ } B_0[1] Q[1] P_1[1] \cdots P_{d-2}[1]\,,$$ and we will identify $C_0$ with its image.
Following the lines of Section \[CASE1\], we find that the domain $C$ is of the form $$C = C_0 \cup C' \cup A_1 \cup \ldots \cup A_{24} \cup B_1 \cup \ldots \cup B_{d-2},$$ where the components $A_i$ and $B_j$ are attached to the points $$x_i + Q + P_1 + \dots + P_{d-2}, \quad \quad \quad u_j + Q + P_1 + \dots + P_{d-2}$$ respectively. Hence, $M_Z$ is set-theoretically a product of spaces corresponding to the points $$u_0, u_1,\dots, u_{d-2}, x_1, \dots, x_{24}. \label{vkkmfvmf}$$ By arguments parallel to Section \[CASE1\], the moduli scheme $M_Z$ splits scheme-theoretic as a product, and also the virtual class splits. Hence, $\langle A \, \rangle_q$ is a product of series corresponding to the points respectively.
For $x_1, \dots, x_{24}$ the contributing factor is the same as in Section \[concl\_basic\_case\], and for $u_1, \dots, u_{d-2}$ it is the same as in Section \[CASE1\]. Let $$\widetilde{G}^{\textup{GW}}(y,q) \in {{\mathbb{Q}}}((y))[[q]] \label{gggt}$$ denote the contributing factor from the point $u_0$. Then we have $${\Big\langle}A \, {\Big\rangle}_q^{\operatorname{\mathsf{Hilb}}^d(S)} = \frac{ G^{\textup{GW}}(y,q)^{d-2} \widetilde{G}^{\textup{GW}}(y,q) }{\Delta(q) }. \label{402}$$
Case $\langle I(P_1), \dots, I(P_{2d-2}) \rangle_q$
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Let $P_1, \dots, P_{2d-2} \in S$ be generic points. In this section, we consider the evaluation of $${\Big\langle}I(P_1), \dots, I(P_{2d-2}) {\Big\rangle}_q^{\operatorname{\mathsf{Hilb}}^d(S)} \label{IP_eval_xcxczcxcxc}$$ In Section \[Section\_intermediate\_lemma\], we discuss the geometry of lines in $\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)$. In Section \[special\_case\_3\], we analyse the moduli space of stable maps incident to $I(P_1), \dots, I(P_{2d-2})$.
### The Grassmannian {#Section_intermediate_lemma}
Let ${{\mathcal Z}}_d \to \operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)$ be the universal family, and let $$L \hookrightarrow \operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)$$ be the inclusion of a line such that $L \nsubseteq I(x)$ for all $x \in {\mathbb{P}}^1$. Consider the fiber diagram $$\begin{tikzcd}
\widetilde{L} \ar{d} \ar{r} & {{\mathcal Z}}_d \ar{d} \ar{r} & {\mathbb{P}}^1 \\
L \ar{r} & \operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1).
\end{tikzcd}$$ The curve $\widetilde{L} \subset L \times {\mathbb{P}}^1$ has bidegree $(d,1)$, and is the graph of the morphism $$I_L \colon {\mathbb{P}}^1 \to L,\ x \mapsto I(x) \cap L \,. \label{MapI_Lasd}$$ By definition, the subscheme corresponding to a point $y \in L$ is $I_L^{-1}(y)$. Hence, the ramification index of $I_L$ at a point $x \in {\mathbb{P}}^1$ is the length of $I_L(x)$ (considered as a subscheme of ${\mathbb{P}}^1$) at $x$. In particular, for $y \in L$, we have $y \in \Delta_{\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)}$ if and only if $I_L(x) = y$ for a branchpoint $x$ of $I_L$.
Let $R(L) \subset {\mathbb{P}}^1$ be the ramification divisor of $I_L$. Since $I_L$ has $2d-2$ branch points counted with multiplicity (or equivalently, $L$ meets $\Delta_{\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)}$ with multiplicity $2d-2$), $$R(L) \in \operatorname{\mathsf{Hilb}}^{2d-2}({\mathbb{P}}^1).$$
Let $G = G(2,d+1)$ be the Grassmannian of lines in $\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)$. By the construction above relative to $G$, we obtain a rational map $$\phi : G \dashrightarrow \operatorname{\mathsf{Hilb}}^{2d-2}({\mathbb{P}}^1),\ L \mapsto R(L) \label{303}$$ defined on the open subset of lines $L \in G$ with $L \nsubseteq I(x)$ for all $x \in {\mathbb{P}}^1$.
The map $\phi$ will be used in the proof of the following result. For $u \in {\mathbb{P}}^1$, consider the incidence subscheme $$I(2u) = \{ z \in \operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)\ |\ 2u \subset z \}$$ Under the identification $\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1) \equiv {\mathbb{P}}^d$, the inclusion $I(2u) \subset \operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)$ is a linear subspace of codimension $2$. Let $$\label{Grassmann_diagram}
\xymatrix{
{{\mathcal Z}}\ar[r]^{q} \ar[d]^{p} & \, {\mathbb{P}}^d\, \mathrlap{= \operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)} \\
G
}$$ be the universal family of $G$, and let $$S_u = p(q^{-1}(I(2u))) = \{ L \in G | L \cap I(2u) \neq \varnothing \} \subset G$$ be the divisor of lines incident to $I(2u)$.
\[intermediate\_lemma\] Let $u_1, \dots, u_{2d-2} \in {\mathbb{P}}^1$ be generic points. Then, $$S_{u_1} \cap \ldots \cap S_{u_{2d-2}} \label{Su_intersection}$$ is a collection of $\frac{1}{d} \binom{2d-2}{d-1}$ reduced points.
The class of $S_u$ is the Schubert cycle $\sigma_1$. By Schubert calculus the expected number of intersection points is $$\int_G \sigma_1^{2d-2} = \frac{1}{d} \binom{2d-2}{d-1}.$$ It remains to prove that the intersection is transverse.
Given a line $L \subset I(x) \subset \operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)$ for some $x \in {\mathbb{P}}^1$, there exist at most $2d-1$ different points $v \in {\mathbb{P}}^1$ with $2 v \subset z$ for some $z \in L$. Hence, for every $L$ in we have $L \nsubseteq I(x)$ for all $x \in {\mathbb{P}}^1$. Therefore, $S_{u_1} \cap \ldots \cap S_{u_{2d-2}}$ lies in the domain of $\phi$. Then, by construction of $\phi$, the intersection is the fiber of $\phi$ over the point $$u_1 + \dots + u_{2d-2} \in \operatorname{\mathsf{Hilb}}^{2d-2}({\mathbb{P}}^1).$$ We will show that $\phi$ is generically finite. Since $u_1, \dots, u_{2d-2}$ are generic, the fiber over $u_1 + \dots + u_{2d-2}$ is then a set of finitely many reduced points.
We determine an explicit expression for the map $\phi$. Let $L \in G$ be a line with $L \nsubseteq I(x)$ for all $x \in {\mathbb{P}}^1$, let $f,g \in L$ be two distinct points and let $x_0, x_1$ be coordinates on ${\mathbb{P}}^1$. We write $$\begin{aligned}
f & = a_n x_0^n + a_{n-1} x_0^{n-1} x_1 + \dots + a_0 x_1^n \\
g & = b_n x_0^n + b_{n-1} x_0^{n-1} x_1 + \dots + b_0 x_1^n\end{aligned}$$ for coefficients $a_i, b_i \in {{\mathbb{C}}}$. The condition $L \nsubseteq I(x)$ for all $x$ is equivalent to $f$ and $g$ having no common zeros. Consider the rational function $$h(x) = h(x_0/x_1) = f/g = \frac{ a_n x^n + \dots + a_0 }{ b_n x^n + \dots + b_0 },$$ where $x = x_0/x_1$. The ramification divisor $R(L)$ is generically the zero locus of the nominator of $h' = (f/g)' = (f' g - f g')/g^2$; in coordinates we have $$f' g - f g' = \sum_{m = 0}^{2d-2} \Big( \sum_{ \substack{ i + j = m+1 \\ i < j }} (i - j) (a_i b_j - a_j b_i) \Big) x^m.$$ Let $M_{ij} = a_i b_j - a_j b_i$ be the Plücker coordinates on $G$. Then we conclude $$\phi(L) = \sum_{m = 0}^{2d-2} \Big( \sum_{ \substack{ i + j = m+1 \\ i < j }} (i - j) M_{ij} \Big) x^m \ \in \operatorname{\mathsf{Hilb}}^{2d-2}({\mathbb{P}}^1) \,. $$ By a direct verification, the differential of $\phi$ at the point with coordinates $$(a_0, \dots, a_n) = (1, 0, \dots, 0 , 1), \quad \quad(b_0, \dots, b_n) = (0,1,0,\dots,0,1)$$ is an isomorphism. Hence, $\phi$ is generically finite.
Let $u_1, \dots, u_{2d-2} \in {\mathbb{P}}^1$ be generic points. Consider a line $$L\ \in\ S_{u_1} \cap \ldots \cap S_{u_{2d-2}} = \phi^{-1}(u_1 + \dots + u_{2d-2})$$ and let $U_L$ be the formal neighborhood of $L$ in $G$. By the proof of Lemma \[intermediate\_lemma\], the map $$\phi : G \dashrightarrow \operatorname{\mathsf{Hilb}}^{2d-2}({\mathbb{P}}^1)$$ is étale near $L$. Hence, $\phi$ induces an isomorphism from $U_L$ to $${\mathop{\rm Spec}\nolimits}\big( \widehat{{{\mathcal O}}}_{\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1),u_1+\dots+u_{2d-2}} \big) \equiv \prod_{i=1}^{2d-2} {\mathop{\rm Spec}\nolimits}( \widehat{{{\mathcal O}}}_{{\mathbb{P}}^1,u_i} ) \,, \label{sos34mkgmrg}$$ the formal neighborhood of $\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)$ at $u_1+\dots+u_{2d-2}$. Composing $\phi$ with the projection to the $i$-th factor of , we obtain maps $$\kappa_{i} : U_L \xrightarrow{\ \phi\ }{\mathop{\rm Spec}\nolimits}\big( \widehat{{{\mathcal O}}}_{\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1),u_1+\dots+u_{2d-2}} \big) \to {\mathop{\rm Spec}\nolimits}( \widehat{{{\mathcal O}}}_{{\mathbb{P}}^1,u_i} ) \subset {\mathbb{P}}^1 \,,
\label{kappa_map_def}$$ which parametrize the deformation of the branch points of $I_L$ (defined in ).
In the notation of the diagram , consider the map $$q^{-1}(\Delta_{\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)})\ \to\ G \label{408}$$ whose fiber over a point $L' \in G$ are the intersection points of $L'$ with the diagonal $\Delta_{\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)}$. Since $L$ is in the fiber of a generically finite map over a generic point, we have $$L \cap \Delta_{\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)} = \{ \xi_1, \dots, \xi_{2d-2} \}$$ for pairwise disjoint subschemes $\xi_i \in \operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)$ of type $(21^{d-2})$ with $2 u_i \subset \xi_i$. The restriction of to $U_L$ is a $(2d-2)$-sheeted trivial fibration, and hence admits sections $$v_1, \dots, v_{2d-2} : U_L \to q^{-1}(\Delta_{\operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)})|_{U_L}, \label{ifimsoidf}$$ such that for every $i$ the composition $q \circ v_i$ restricts to $\xi_i$ over the closed point. Moreover, since $q \circ v_i$ is incident to the diagonal and must contain twice the branchpoint $\kappa_i$ defined in , we have the decomposition $$q \circ v_i = 2 \kappa_i + h_1 + \dots + h_{d-2} \label{fatt_kfmvodmfo}$$ for maps $h_1, \dots, h_{d-2} : U_L \to {\mathbb{P}}^1$.
### The moduli space {#special_case_3}
Let $P_1, \dots, P_{2d-2} \in S$ be generic points and let $u_i = \pi(P_i)$ for all $i$. Let $${\mathop{\rm ev}\nolimits}: {{\overline M}}_{0,2d-2}(\operatorname{\mathsf{Hilb}}^d(S), \beta_h + kA) \to \big(\operatorname{\mathsf{Hilb}}^d(S)\big)^{2d-2}$$ be the evaluation map and let $$M_Z = {\mathop{\rm ev}\nolimits}^{-1}\bigl(I(P_1) \times \dots \times I(P_{2d-2})\bigr)$$ be the moduli space of stables maps incident to $I(P_1), \dots, I(P_{2d-2})$. We consider an element $$[f : C \to \operatorname{\mathsf{Hilb}}^d(S), p_1, \dots, p_{2d-2}] \in M_Z \,.$$
Since $P_i \in f(p_i)$ and $P_i$ is generic, the line $L = \pi(f(C)) \subset \operatorname{\mathsf{Hilb}}^d({\mathbb{P}}^1)$ is incident to $I(2u_i)$ for all $i$, and therefore lies in the finite set $$S_{u_1} \cap \dots \cap S_{u_{2d-2}} \subset G(2,d+1)
\label{Su_intersection_new}$$ defined in Section \[Section\_intermediate\_lemma\]; here $G(2,d+1)$ is the Grassmannian of lines in ${\mathbb{P}}^d$.
Because the points $u_1, \dots, u_{2d-2}$ are generic, by the proof of Lemma \[intermediate\_lemma\] also $L$ is generic. By arguments identical to the case of Section \[basic\_case\_settheoretic\_splitting\], the map $f|_{C_0} : C_0 \to L$ is an isomorphism. We identify $C_0$ with the image $L$.
For $x \in {\mathbb{P}}^1$, let $\widetilde{x} = I(x) \cap L$ be the unique point on $L$ incident to $x$. The points $$\label{L_with_W_int_points_2} \widetilde{x}_1, \dots, \widetilde{x}_{24}, \widetilde{u}_1, \dots, \widetilde{u}_{2d-2}$$ are the intersection points of $L$ with the discriminant of $\pi^{[d]}$ defined in . Hence, by Lemma \[200\], the curve $C$ admits the decomposition $$C = C_0 \cup A_1 \cup \dots \cup A_{24} \cup B_1 \cup \dots \cup B_{2d-2}, \label{ogorgdfgfdg2}$$ where $A_i$ and $B_j$ are the components of $C$ attached to the points $\widetilde{x}_i$ and $\widetilde{u}_j$ respectively; see also Section \[basic\_case\_settheoretic\_splitting\].
By Lemma \[doesnotsmooth\], the node points $C_0 \cap A_i$ and $C_0 \cap B_j$ do not smooth under deformations of $f$ inside $M_Z$. Hence, by the construction of Section \[splitting\], we have a splitting morphism $$\Psi: M_{Z}
{{\ \longrightarrow\ }}\bigsqcup_{L} \bigsqcup_{\textbf{h}, \textbf{k}} \bigg( \prod_{i=1}^{24} M^{\textup{(N)}}_{x_i}(h_{x_i})
\times \prod_{j=1}^{2d-2} M^{\textup{(H)}}_{u_j}(h_{u_j}, k_{u_j}) \bigg),
\label{405}$$ where $\textbf{h}, \textbf{k}$ runs over the set (with $y_j$ replaced by $u_j$) satisfying , and $L$ runs over the set of lines , and where $M^{\textup{(H)}}_{u_j}(h', k')$ is the moduli space defined as follows:
Consider the evaluation map $${\mathop{\rm ev}\nolimits}: {{\overline M}}_{0,2}( \operatorname{\mathsf{Hilb}}^2(S) , h' F + k' A ) {{\ \longrightarrow\ }}(\operatorname{\mathsf{Hilb}}^2(S))^2$$ and let $${\mathop{\rm ev}\nolimits}^{-1}\big( I(P_j) \times \operatorname{\mathsf{Hilb}}^2(B_0) \big) \label{erewrewg}$$ be the subscheme of maps incident to $I(P_j)$ and $\operatorname{\mathsf{Hilb}}^d(B_0)$ at the marked points. We define $M^{\textup{(H)}}_{u_j}(h', k')$ to be the open and closed component of whose ${{\mathbb{C}}}$-points parametrize maps into the fiber $\pi^{[2] -1}(2 u_j)$. Using this definition, the map $\Psi$ is well-defined (for example, the intersection point $C_0 \cap B_j$ maps to the second marked point in ).
In the case considered in Section \[basic\_case\], the image line $L = f(C_0)$ was fixed under infinitesimal deformations. Here, this does not seem to be the case; the line $L$ may move infinitesimal. Nonetheless, the following Proposition shows that these deformations are all captured by the image of $\Psi$.
\[352\_splitting\_lemma\] The splitting map is an isomorphism.
We will require the following Lemma, which will be proven later.
\[410\_lemma\] Let $\phi : C \to \operatorname{\mathsf{Hilb}}^2(S)$ be a family in $M^{\textup{(H)}}_{u_j}(h', k')$ over a connected scheme $Y$, $$\label{fam_diag}
\begin{tikzcd}
C \ar{r}{\phi} \ar{d} & \operatorname{\mathsf{Hilb}}^2(S) \,. \\ Y
\end{tikzcd}$$ Then $\pi^{[2]} \circ \phi$ maps to $\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1) \cap I(u_j)$.
We define an inverse to $\Psi$. Let $$\Big((\phi_{i}' : A_i \to S,q_{x_i})_{i = 1, \dots,24} , (\phi_{j} : B_j \to \operatorname{\mathsf{Hilb}}^2(S), p_j, q_j)_{j=1,\dots, 2d-2}\Big) \label{410}$$ be a family of maps in the right hand side of over a connected scheme $Y$. By Lemma \[410\_lemma\], $\pi^{[2]} \circ \phi_{j} : B_j \to \operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)$ maps into $I(u_j) \cap \Delta_{\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)}$. Since the intersection of the line $I(u_j)$ and the diagonal $\Delta_{\operatorname{\mathsf{Hilb}}^2(S)}$ is infinitesimal, we have the inclusion $$I(u_j) \cap \Delta_{\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)} \hookrightarrow {\mathop{\rm Spec}\nolimits}( \widehat{{{\mathcal O}}}_{\Delta_{\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)},2 u_j} ) = {\mathop{\rm Spec}\nolimits}( \widehat{{{\mathcal O}}}_{{\mathbb{P}}^1, u_j} ),$$ and therefore the induced map $\iota_j \colon Y \to {\mathop{\rm Spec}\nolimits}( \widehat{{{\mathcal O}}}_{{\mathbb{P}}^1, u_i} )$ making the diagram $$\begin{tikzcd}
B_j \ar{d} \ar{dr}{\pi^{[2]} \circ \phi_j} & \\
Y \ar{r}{\iota_j} &
{\mathop{\rm Spec}\nolimits}( \widehat{{{\mathcal O}}}_{\Delta_{\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)},2 u_j} )\mathrlap{\, = {\mathop{\rm Spec}\nolimits}( \widehat{{{\mathcal O}}}_{{\mathbb{P}}^1, u_j} )}
\end{tikzcd}$$ commutative. Let $\ell = (\iota_j)_{j} \colon Y \to U_L$, where $$U_L = \textstyle{\prod}_{j = 1}^{2d-2} {\mathop{\rm Spec}\nolimits}( \widehat{{{\mathcal O}}}_{{\mathbb{P}}^1, u_j} ) \equiv {\mathop{\rm Spec}\nolimits}( \widehat{{{\mathcal O}}}_{\operatorname{\mathsf{Hilb}}^{2d-2}({\mathbb{P}}^1), \sum_{i} u_i} ).$$
Under the generically finite rational map $$G(2, d+1) \dashrightarrow \operatorname{\mathsf{Hilb}}^{2d-2}({\mathbb{P}}^1),$$ defined in , the formal scheme $U_L$ is isomorphic to the formal neighborhood of $G(2,d+1)$ at the point $[L]$. We identify these neighborhoods under this isomorphism.
Let ${{\mathcal Z}}_L \to U_L$ be the restriction of the universal family ${{\mathcal Z}}\to G(2,d+1)$ to $U_L$. By pullback via $\ell$, we obtain a family of lines in ${\mathbb{P}}^d$ over the scheme $Y$, $$\ell^{\ast} {{\mathcal Z}}_L \to Y, \label{dwofmoofdg}$$ together with an induced map $$\psi : \ell^{\ast} {{\mathcal Z}}_L \xrightarrow{\ell} {{\mathcal Z}}_L \to {\mathbb{P}}^d \equiv \operatorname{\mathsf{Hilb}}^d(B_0) \xrightarrow{s^{[d]}} \operatorname{\mathsf{Hilb}}^d(S).$$
We will require sections of $\ell^{\ast} {{\mathcal Z}}_L \to Y$, which allow us to glue the domains of the maps $\phi_{i}'$ and $\phi_{j}$ to $\ell^{\ast} {{\mathcal Z}}_L$. Consider the sections $$v_1, \dots, v_{2d-2} : Y \to \ell^{\ast} {{\mathcal Z}}_L$$ which are the pullback under $\ell$ of the sections $v_i : U_L \to {{\mathcal Z}}_L$ defined in . By construction, the section $v_i : Y \to \ell^{\ast} {{\mathcal Z}}$ parametrizes the points of $\ell^{\ast} {{\mathcal Z}}_L$ which map to the diagonal $\Delta_{\operatorname{\mathsf{Hilb}}^d(S)}$ under $\psi$ (in particular, over closed points of $Y$ they map to $I(u_j) \cap L$).
For $j=1,\dots,2d-2$, consider the family of maps $\phi_j : B_j \to \operatorname{\mathsf{Hilb}}^2(S)$, $$\xymatrix{
B_j \ar@<+6pt>[d]^{\pi_j} \ar[r]^{\phi_{j}} & \mathrlap{\operatorname{\mathsf{Hilb}}^2(S)} \\
Y \ar@<+6pt>[u]^{p_j, q_j} \ar@<+0pt>[u],
}
\label{411}$$ where $p_j$ is the marked point mapping to $I(P_j)$, and $q_j$ is the marked point mapping to $\operatorname{\mathsf{Hilb}}^2(B_0)$. Let $C'$ be the curve over $Y$ which is obtained by glueing the component $B_j$ to the line $\ell^{\ast} {{\mathbb{Z}}}_L$ along the points $q_j, v_j$ for all $j$: $$C' = \Big( \ell^{\ast} {{\mathbb{Z}}}_L \sqcup B_1 \sqcup \ldots \sqcup B_{2d-2} \Big)\Big/ q_1 \sim v_1, \dots, q_{2d-2} \sim v_{2d-2} \,.$$ We will define a map $f' : C' \to \operatorname{\mathsf{Hilb}}^d(S)$.
For all $j$, let $\kappa_j : U_L \to {\mathop{\rm Spec}\nolimits}( \widehat{{{\mathcal O}}}_{{\mathbb{P}}^1,u_j} ) \subset {\mathbb{P}}^1$ be the map defined in . By construction, we have $\kappa_{j} \circ \ell = \iota_j$. Hence, by , there exist maps $h_1, \dots, h_{d-2} : Y \to S$ with $$\psi \circ v_j = \phi_{j} \circ q_j + h_1 + \dots + h_{d-2} \,.
\label{fomvodmvfd}$$ Let $\pi_j : B_j \to Y$ be the map of the family $B_j/Y$, and define $$\widetilde{\phi}_{u_j} = \Big( \phi_{j} + \sum_{i=1}^{n-2} h_i \circ \pi_j\Big) : B_j {{\ \longrightarrow\ }}\operatorname{\mathsf{Hilb}}^d(S).$$ Define the map $$f' : C' \to \operatorname{\mathsf{Hilb}}^d(S)$$ by $f'|_{C_0} = \psi$ and by $f'|_{B_j} = \widetilde{\phi}_{j}$ for every $j$. By , the map $\widetilde{\phi}_{u_j}$ restricted to $q_j$ agrees with $\psi : C' \to \operatorname{\mathsf{Hilb}}^d(S)$ restricted to $v_j$. Hence $f'$ is well-defined.
By a parallel construction, we obtain a canonical glueing of the components $A_i$ to $C'$ together with a glueing of the maps $f'$ and $\phi_i' : A_i \to S$. We obtain a family of maps $$f : C \to \operatorname{\mathsf{Hilb}}^d(S)$$ over $Y$, which lies in $M_Z$ and such that $\Psi(f)$ equals . By a direct verification, the induced morphism on the moduli spaces is the desired inverse to $\Psi$. Hence, $\Psi$ is an isomorphism.
The remaining steps in the evaluation of are similar to Section \[basic\_case\]. Using the identification $$H^0(C_0, f^{\ast} T_{\operatorname{\mathsf{Hilb}}^d(S)}) = H^0(C_0, T_{C_0}) \oplus \bigoplus_{j=1}^{2d-2} T_{\Delta_{\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)}, \phi_j(q_j)},$$ where $q_j = C_0 \cap B_j$ are the nodes and $\phi_j$ is as in the proof of Proposition \[352\_splitting\_lemma\], one verifies that the virtual class splits according to the product . Hence, the invariant is a product of series associated to the points $x_i$ and $u_j$ respectively. Let $$H^{\textup{GW}}(y,q) = \sum_{h \geq 0} \sum_{k \in {{\mathbb{Z}}}} y^{k - \frac{1}{2}} q^{h} \int_{[M^{\textup{(H)}}_{u_j}(h, k)]^{\text{vir}}} 1 \in {{\mathbb{Q}}}((y^{1/2}))[[q]],
\label{hhh}$$ be the contribution from the point $u_j$. By Lemma \[intermediate\_lemma\], there are $\frac{1}{d} \binom{2d-2}{d-1}$ lines in the set . Hence, $${\Big\langle}I(P_1), \dots, I(P_{2d-2}) {\Big\rangle}_q^{\operatorname{\mathsf{Hilb}}^d(S)}
=
\frac{1}{d} \binom{2d-2}{d-1}
\frac{ H^{\textup{GW}}(y,q)^{2d-2} }{\Delta(q)}.
\label{403}$$
Since $\phi$ is incident to $I(P_j)$, the composition $\pi^{[2]} \circ \phi$ maps to $I(u_j)$. Therefore, we only need to show that $\pi^{[2]} \circ \phi$ maps to $\Delta_{\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)}$.
It is enough to consider the case $Y = {\mathop{\rm Spec}\nolimits}( {{\mathbb{C}}}[\epsilon]/\epsilon^2 )$. Let $f_0 : C_0 \to \operatorname{\mathsf{Hilb}}^2(S)$ be the restriction of $f$ over the closed point of $Y$, and consider the diagram $$\begin{tikzcd}
C \ar{d}{\pi_C} \ar{r}{\phi} & \operatorname{\mathsf{Hilb}}^2(S) \ar{d}{\pi^{[2]}} \\
{\mathop{\rm Spec}\nolimits}( {{\mathbb{C}}}[\epsilon]/\epsilon^2 ) \ar{r}{a} & \operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1) \,.
\end{tikzcd}$$ where $\pi_C$ is the given map of the family and $a$ is the induced map. Let $s$ be the section of ${{\mathcal O}}(\Delta_{{\mathbb{P}}^1})$ with zero locus $\Delta_{{\mathbb{P}}^1}$, and assume the pullback $\phi^{\ast} s$ is non-zero.
Let $\Omega_{\pi_2}$ be the sheaf of relative differentials of $\pi_2 := \pi^{[2]}$. The composition $$\phi^{\ast} \pi_2^{\ast} \pi_{2 \ast} \Omega_{\pi_2} \to \phi^{\ast} \Omega_{\pi_2} \overset{d}{\to} \Omega_{\pi_C} \label{304}$$ factors as $$\phi^{\ast} \pi_2^{\ast} \pi_{2 \ast} \Omega_{\pi_2} \to \pi_C^{\ast} \pi_{C \ast} \Omega_{\pi_C} \to \Omega_{\pi_C}. \label{304_11}$$ Since the second term in is zero, the map is zero. Hence, $d$ factors as $$\phi^{\ast} \Omega_{\pi_2} \to \phi^{\ast}( \Omega_{\pi_2} / \pi_2^{\ast} \pi_{2 \ast} \Omega_{\pi_2} ) \to \Omega_{\pi_C}. \label{214}$$ By Lemma \[213\] below, $\Omega_{\pi_2} / \pi_2^{\ast} \pi_{2 \ast} \Omega_{\pi_2}$ is the pushforward of a sheaf supported on $\pi_2^{-1}(\Delta_{{\mathbb{P}}^1})$. After trivializing ${{\mathcal O}}(\Delta_{{\mathbb{P}}^1})$ near $2 u_j$, write $\phi^{\ast} s = \lambda \epsilon$ for some $\lambda \in {{\mathbb{C}}}\setminus \{ 0\}$. Then, by , $$0 = d( s \cdot \Omega_{\pi_2} ) = \lambda \epsilon \cdot d( \Omega_{\pi_2} ) \subset \Omega_{\pi_{C}}.$$ In particular, $d b = 0$ for every $b \in \Omega_{\pi_2}$, which does not vanish on $\pi_2^{-1}(\Delta_{{\mathbb{P}}^1})$. Since $\phi|_{C}$ is non-zero, this is a contradiction.
\[213\] Let $x \in {\mathbb{P}}^1$ be the basepoint of a smooth fiber of $\pi : S \to {\mathbb{P}}^1$. Then, there exists a Zariski-open $2x \in U \subset \operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)$ and a map $$u: {{\mathcal O}}_U^{\oplus 2} \to \pi_{2 \ast} \Omega_{\pi_2}|_{U} \label{215}$$ with cokernel equal to $j_{\ast} {{\mathcal F}}$ for a sheaf ${{\mathcal F}}$ on $\Delta_{{\mathbb{P}}^1} \cap U$.
Let $U$ be an open subset of $x \in {\mathbb{P}}^1$ such that $\pi_{\ast} \Omega_{\pi}|_U$ is trivialized by a section $$\alpha \in \pi_{\ast} \Omega_{\pi}(U) = \Omega_{\pi}( S_U ),$$ where $S_U = \pi^{-1}(U)$. Consider the open neighborhood $\widetilde{U} = \operatorname{\mathsf{Hilb}}^2(U)$ of the point $2x \in \operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)$.
Let $D_U \subset S_U \times S_U$ be the diagonal and consider the ${{\mathbb{Z}}}_2$ quotient $${\mathop{\rm Bl}\nolimits}_{D_U} ( S_U \times S_U ) \overset{ /{{\mathbb{Z}}}_2 }{{{\ \longrightarrow\ }}} \operatorname{\mathsf{Hilb}}^2(S_U) = \pi_2^{-1}( \operatorname{\mathsf{Hilb}}^2(U) ).$$ For $i \in \{ 1,2 \}$, let $$q_i : {\mathop{\rm Bl}\nolimits}_{D_U} ( S_U \times S_U ) \to S_U$$ be the composition of the blowdown map with the $i$-th projection. Let $t$ be a coordinate on $U$ and let $$t_i = q_{i}^{\ast} t, \quad \alpha_i = q_{i}^{\ast} \alpha$$ for $i = 1,2$ be the induced global functions resp. 1-forms on ${\mathop{\rm Bl}\nolimits}_{D_U} ( S_U \times S_U )$. The two 1-forms $$\alpha_1 + \alpha_2 \quad \text{ and } \quad (t_1 - t_2) (\alpha_1 - \alpha_2)$$ are ${{\mathbb{Z}}}_2$ invariant and descend to global sections of $\pi_{2 \ast} \Omega_{\pi_2}|U$. Consider the induced map $$u: {{\mathcal O}}_U^{\oplus 2} {{\ \longrightarrow\ }}\pi_{2 \ast} \Omega_{\pi_2}|_{U}$$
The map $u$ is an isomorphism away from the diagonal $$\Delta_{\pi} \cap \widetilde{U} = V( (t_1 - t_2)^2 ) \subset \widetilde{U} \,. \label{216}$$ Hence, it is left to check the statement of the lemma in an infinitesimal neighborhood of . Let $U'$ be a small analytic neighborhood of $v \in U$ such that the restriction $\pi_{U'}: S_{U'} \to U'$ is analytically isomorphic to the quotient $$(U' \times {{\mathbb{C}}}) \mathclose{}/\mathopen{} \sim {{\ \longrightarrow\ }}U',$$ where $\sim$ is the equivalence relation $$(t, z) \sim (t', z') \quad \Longleftrightarrow \quad t = t' \text{ and } z - z' \in \Lambda_{t},$$ with an analytically varying lattice $\Lambda_t : {{\mathbb{Z}}}^2 \to {{\mathbb{C}}}$. Now, a direct and explicit verification yields the statement of the lemma.
Case $\langle {{\mathfrak{p}}}_{-1}(F)^2 1_S, I(P) \rangle_q$ {#Section_CaseI_P}
-------------------------------------------------------------
Let $F^{\text{GW}}(y,q)$ and $H^{\text{GW}}(y,q)$ be the power series defined in and respectively, let $P \in S$ be a point and let $F$ be the class of a fiber of $\pi :S \to {\mathbb{P}}^1$.
\[Lemma\_in\_special\_case\_4\] We have $$\big\langle \, {{\mathfrak{p}}}_{-1}(F)^2 1_S\, ,\, I(P)\, \big\rangle_q^{\operatorname{\mathsf{Hilb}}^2(S)} = \frac{F^{\text{GW}}(y,q) \cdot H^{\text{GW}}(y,q)}{\Delta(q)}$$
Let $F_1, F_2$ be fibers of $\pi \colon S \to {\mathbb{P}}^1$ over generic points $x_1, x_2 \in {\mathbb{P}}^1$ respectively, and let $P \in S$ be a generic point. Define the subschemes $$Z_1 = F_1[1] F_2[1] \quad \text{ and } \quad Z_2 = I(P) \,.$$ Consider the evaluation map $${\mathop{\rm ev}\nolimits}: {{\overline M}}_{0,1}(\operatorname{\mathsf{Hilb}}^2(S), \beta_h + kA ) \to \operatorname{\mathsf{Hilb}}^2(S)$$ from the moduli space of stable maps with *one* marked point, let $$M_{Z_2} = {\mathop{\rm ev}\nolimits}^{-1}(Z_2),$$ and let $$M_Z \subset M_{Z_2}$$ be the closed substack of $M_{Z_2}$ of maps which are incident to both $Z_1$ and $Z_2$.
Let $[f : C \to \operatorname{\mathsf{Hilb}}^2(S), p_1] \in M_Z$ be an element, let $C_0$ be the distinguished component of $C$ on which $\pi^{[2]} \circ f$ is non-zero, and let $L = \pi^{[2]}(f(C_0))$ be the image line. Since $P \in S$ is generic, we have $2v \in L$ where $v = \pi(P)$. Hence, $L$ is the line through $2v$ and $u_1 + u_2$, and has the diagonal points $$L \cap \Delta_{\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)} = \{ 2u, 2v \} \label{dmvofdvfv}$$ for some fixed $u \in {\mathbb{P}}^1 \setminus \{ v \}$. By Lemma \[117\], the restriction $f|_{C_0}$ is therefore an isomorphism onto the embedded line $L \subset \operatorname{\mathsf{Hilb}}^2(B_0)$. Using arguments parallel to Section \[Section\_settheoretic\_splitting\], the moduli space $M_Z$ is *set-theoretically* a product of the moduli space of maps to the nodal fibers, the moduli space $M^{\textup{(F)}}_{u}(h',k')$ parametrizing maps over $2u$, and the moduli space $M^{\textup{(H)}}_{v}(h'',k'')$ parametrizing maps over $2v$. Under infinitesimal deformations of $[f : C \to \operatorname{\mathsf{Hilb}}^2(S)]$ inside $M_Z$, the line $L$ remains incident to $x_1 + x_2$, but may move to first order at the point $2 v$ (see Section \[special\_case\_3\]); hence, it may move also at $2 u$ to first order. In particular, the moduli space is scheme-theoretically *not* a product of the above moduli spaces. Nevertheless, by degeneration, we will reduce to the case of a scheme-theoretic product. For simplicity, we work on the component of $M_Z$ which parametrizes maps with no component mapping to the nodal fibers of $\pi$; the general case follows by completely analog arguments with an extra $1/\Delta(q)$ factor appearing as contribution from the nodal fibers.
Let $N \subset M_{Z_2}$ be the *open* locus of maps $f : C \to \operatorname{\mathsf{Hilb}}^2(S)$ in $M_{Z_2}$ with $$\pi^{[2]}(f(C)) \cap \Delta_{\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)} =
\{ 2 t, 2 v \}$$ for some point $t \in {\mathbb{P}}\setminus \{ x_1, \dots, x_{24}, v \}$. Under deformations of an element $[f] \in N$, the intersection point $2t$ may move freely and independently of $v$. Hence, we have a splitting *isomorphism* $$\Psi : N {{\ \longrightarrow\ }}\bigsqcup_{\substack{h = h_1 + h_2 \\ k=k_1 + k_2 - 1}} M^{\textup{(F)}}(h_1, k_1) \times M^{\textup{(H)}}_{v}(h_2, k_2), \label{412}$$ where
- $M^{\textup{(F)}}(h, k)$ is the moduli space of $1$-pointed stable maps to $\operatorname{\mathsf{Hilb}}^2(S)$ of genus $0$ and class $hF + kA$ such that the marked point is mapped to $s^{[2]}(2t)$ for some $t \in {\mathbb{P}}\setminus \{ x_1, \dots, x_{24}, v \}$,
- $M^{\textup{(H)}}_{v}(h,k)$ is the moduli space defined in Section \[special\_case\_3\].
For every decomposition $h=h_1 + h_2$ and $k = k_1 + k_2 - 1$ separately, let $$M^{\textup{(F)}}(h_1, k_1) \times M^{\textup{(H)}}_{v}(h_2, k_2) {{\ \longrightarrow\ }}\Delta_{\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)} \times {\mathop{\rm Spec}\nolimits}\bigl( \widehat{{{\mathcal O}}}_{\Delta_{\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)}, 2 v} \bigr)$$ be the product of the compositions of the first evaluation map with $\pi^{[2]}$ on each factor, let $$\iota \colon V \hookrightarrow
\Delta_{\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)} \times {\mathop{\rm Spec}\nolimits}\big( \widehat{{{\mathcal O}}}_{\Delta_{\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)}, 2 v} \big)
\label{REGMMM}$$ be the subscheme parametrizing the intersection points $L \cap \Delta_{\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)}$ of lines $L$ which are incident to $x_1 + x_2$, and consider the fiber product $$\label{diagram_fkmvkfmv}
\begin{tikzcd}
M_{Z,(h_1,h_2,k_1,k_2)} \ar{r} \ar{d} & M^{\textup{(F)}}(h_1, k_1) \times M^{\textup{(H)}}_{v}(h_2, k_2) \ar{d} \\
V \ar{r}{\iota} & \Delta_{\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)} \times {\mathop{\rm Spec}\nolimits}\big( \widehat{{{\mathcal O}}}_{\Delta_{\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)}, 2 v} \big)
\end{tikzcd}$$ Then, by definition, the splitting isomorphism restricts to an isomorphism $$\Psi : M_Z \to \bigsqcup_{\substack{h = h_1 + h_2 \\ k=k_1 + k_2 - 1}}M_{Z,(h_1,h_2,k_1,k_2)}.$$
Restricting the natural virtual class on $M_{Z_2}$ to the open locus, we obtain a virtual class $[N]^{\text{vir}}$ of dimension $1$. By the arguments of Section \[Section\_analysis\_of\_virtual\_class\], $$\Psi_{\ast}[N]^{\text{vir}}
= \sum_{\substack{h = h_1 + h_2 \\ k=k_1 + k_2 - 1}} [ M^{\textup{(F)}}(h_1, k_1) ]^{\text{vir}} \times [M_v^{\textup{(H)}}(h_2,k_2)]^{\text{vir}}, \label{ABBAC}$$ where $[ M^{\textup{(F)}}(h_1, k_1) ]^{\text{vir}}$ is a $\Delta_{\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)}$-relative version of the virtual class considered in Section \[Section\_analysis\_of\_virtual\_class\], and $[M^{\textup{(H)}}_{v}(h_2, k_2)]^{\text{vir}}$ is the virtual class constructed in Section \[special\_case\_3\]. The composition of $\iota$ with the projection to the second factor is an isomorphism. Hence $\iota$ is a regular embedding and we obtain $$\begin{gathered}
\label{qqqq}
{\Big\langle}{{\mathfrak{p}}}_{-1}(F)^2, I(P) {\Big\rangle}^{\operatorname{\mathsf{Hilb}}^2(S)}_{\beta_h+kA}
= \deg( \Psi_{\ast} [M_Z]^{\text{vir}} ) \\
= \sum_{\substack{h = h_1 + h_2 \\ k=k_1 + k_2 - 1}} \deg\, \iota^{!} \Big([ M^{\textup{(F)}}(h_1, k_1) ]^{\text{vir}} \times [M_v^{\textup{(H)}}(h_2,k_2)]^{\text{vir}} \Big) .\end{gathered}$$
We proceed by degenerating the first factor in the product $$M^{\textup{(F)}}(h_1, k_1) \times M^{\textup{(H)}}_{v}(h_2, k_2),$$ while keeping the second factor fixed. Let $${{\mathcal S}}= {\mathop{\rm Bl}\nolimits}_{F_{u} \times 0}(S \times {{\mathbb{A}}}^1) \to {{\mathbb{A}}}^1,$$ be a deformation of $S$ to the normal cone of $F_u$, where $u$ was defined in . Let ${{\mathcal S}}^{\circ} \subset {{\mathcal S}}$ be the complement of the proper transform of $S \times 0$ and consider the *relative* Hilbert scheme $\operatorname{\mathsf{Hilb}}^2( {{\mathcal S}}^{\circ} / {{\mathbb{A}}}^1 ) \to {{\mathbb{A}}}^1$, which appeared already in . Let $$p \colon \widetilde{M}^{\textup{(F)}}(h_1, k_1) \to {{\mathbb{A}}}^1 \label{eigreigrg}$$ be the moduli space of $1$-pointed stable maps to $\operatorname{\mathsf{Hilb}}^2({{\mathcal S}}^{\circ}/{{\mathbb{A}}}^1)$ of genus $0$ and class $h_1 F + k_1 A$, which map the marked point to the closure of $$(\Delta_{\operatorname{\mathsf{Hilb}}^2(B_0)} \setminus \{ x_1, \dots, x_{24}, v\}) \times ({{\mathbb{A}}}^1 \setminus \{ 0 \}).$$ Over $t \neq 0$, restricts to $M^{\textup{(F)}}(h_1, k_1)$, while the fiber over $0$, denoted $$M_0^{\textup{(F)}}(h_1, k_1) = p^{-1}(0),$$ parametrizes maps into the trivial elliptic fibration $\operatorname{\mathsf{Hilb}}^2({{\mathbb{C}}}\times E)$ incident to the diagonal $\Delta_{\operatorname{\mathsf{Hilb}}^2({{\mathbb{C}}}\times e)}$ for a fixed $e \in E$. Since addition by ${{\mathbb{C}}}$ acts on $M_0^{\textup{(F)}}(h_1, k_1)$ we have the product decomposition $$M_0^{\textup{(F)}}(h_1, k_1) = M^{\textup{(F)}}_{0,\text{fix}}(h_1, k_1) \times \Delta_{\operatorname{\mathsf{Hilb}}^2({{\mathbb{C}}}\times e)}, \label{dmvfmvvbcbv}$$ where $M^{\textup{(F)}}_{0,\text{fix}}(h_1, k_1)$ is a fixed fiber of $$M_0^{\textup{(F)}}(h_1, k_1) \to \Delta_{\operatorname{\mathsf{Hilb}}^2({{\mathbb{C}}}\times e)}.$$
Consider a deformation of the diagram to $0 \in {{\mathbb{A}}}^1$, $$\begin{tikzcd}
M'_{Z,(h_1,h_2,k_1,k_2)} \ar{r} \ar{d} & M_0^{\textup{(F)}}(h_1, k_1) \times M^{\textup{(H)}}_{v}(h_2, k_2) \ar{d} \\
V' \ar{r}{\iota'} & \Delta_{\operatorname{\mathsf{Hilb}}^2({{\mathbb{C}}}\times E)} \times {\mathop{\rm Spec}\nolimits}\bigl( \widehat{{{\mathcal O}}}_{\Delta_{\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)}, 2 v} \bigr),
\end{tikzcd}
$$ where $(V',\iota')$ is the fiber over $0$ of a deformation of $(V,\iota)$ such that the composition with the projection to ${\mathop{\rm Spec}\nolimits}( \widehat{{{\mathcal O}}}_{\Delta_{\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)}, 2 v} )$ remains an isomorphism. By construction, the total space of the deformation $$M_{Z,(h_1,h_2,k_1,k_2)} \rightsquigarrow M'_{Z,(h_1,h_2,k_1,k_2)}$$ is proper over ${{\mathbb{A}}}^1$. Using the product decomposition , we find $$M'_{Z,(h_1,h_2,k_1,k_2)} \cong M^{\textup{(F)}}_{0,\text{fix}}(h_1, k_1) \times M^{\textup{(H)}}_{v}(h_2, k_2)$$ Hence, after degeneration, we are reduced to a scheme-theoretic product. It remains to consider the virtual class.
By the relative construction of Section \[Section\_analysis\_of\_virtual\_class\] the moduli space $\widetilde{M}^{\textup{(F)}}(h_1, k_1) $ carries a virtual class $$[ \widetilde{M}^{\textup{(F)}}(h_1, k_1) ]^{\text{vir}} \label{SOSOSO123}$$ which restricts to $[ M^{\textup{(F)}}(h_1, k_1) ]^{\text{vir}}$ over $t \neq 0$, while over $t=0$ we have $$0^{!}[ \widetilde{M}^{\textup{(F)}}(h_1, k_1) ]^{\text{vir}} = {\rm pr}_1^{\ast} \left( [ M^{\text{fix}}_1(h_1,k_1) ]^{\text{vir}} \right)\,.
\label{kgfjreufr}$$ where ${\rm pr}_1$ is the projection to the first factor in and $[ M^{\text{fix}}_1(h_1,k_1) ]^{\text{vir}}$ is the virtual class obtained by the construction of Section \[Section\_analysis\_of\_virtual\_class\]. We conclude, $$\begin{gathered}
\label{rrrr}
\deg\, \iota^{!} \Big([ M^{\textup{(F)}}(h_1, k_1) ]^{\text{vir}} \times [M_v^{\textup{(H)}}(h_2,k_2)]^{\text{vir}} \Big) \\
=
\deg \, (\iota')^{!} \Big({\rm pr}_1^{\ast} \big( [ M^{\text{fix}}_1(h_1,k_1) ]^{\text{vir}} \big) \times [M_v^{\textup{(H)}}(h_2,k_2)]^{\text{vir}} \Big) \\
=
\deg \big( [ M^{\text{fix}}_1(h_1,k_1) ]^{\text{vir}} \big) \cdot \deg \big( [M_v^{\textup{(H)}}(h_2,k_2)]^{\text{vir}} \big).\end{gathered}$$
By definition (see ), $$\deg [M_v^{\textup{(H)}}(h_2,k_2)]^{\text{vir}} = \left[ H^{\textup{GW}}(y,q) \right]_{q^{h_2} y^{k_2-1/2}},$$ where $[\ \cdot\ ]_{q^{a} y^b}$ denotes the $q^a y^b$ coefficient. The moduli space $M^{\text{fix}}_1(h_1,k_1)$ is isomorphic to the space $M^{(\ell)}_{y_i}(h_1,k_1)$ defined in . Since the construction of the virtual class on both sides agree, the virtual class is the same under this isomorphism. Hence, by Lemma \[242\_lemma\], $$\deg [ M^{\text{fix}}_1(h_1,k_1) ]^{\text{vir}} = \left[ F^{\textup{GW}}(y,q) \right]_{q^{h_1} y^{k_1-1/2}}.$$
Inserting into yields $$\begin{gathered}
\deg\, \iota^{!} \Big([ M^{\textup{(F)}}(h_1, k_1) ]^{\text{vir}} \times [M_v^{\textup{(H)}}(h_2,k_2)]^{\text{vir}} \Big) \\
= \left[ H^{\textup{GW}}(y,q) \right]_{q^{h_2} y^{k_2-1/2}} \cdot \left[ F^{\textup{GW}}(y,q) \right]_{q^{h_1} y^{k_1-1/2}},\end{gathered}$$ which completes the proof by equation .
The Hilbert scheme of $2$ points of ${\mathbb{P}}^1 \times E$ {#Section_Hilb2P1xE}
=============================================================
Overview {#overview-2}
--------
In previous sections we expressed genus $0$ Gromov-Witten invariants of the Hilbert scheme of points of an elliptic K3 surface $S$ in terms of universal series which depend only on specific fibers of the fibration $S \to {\mathbb{P}}^1$. The contributions from nodal fibers have been determined before by Bryan and Leung in their proof [@BL] of the Yau-Zaslow formula . The yet undetermined contributions from smooth fibers, denoted $$\label{undetermined_series} F^{\textup{GW}}(y,q),\ G^{\textup{GW}}(y,q),\ \widetilde{G}^{\textup{GW}}(y,q),\ H^{\textup{GW}}(y,q)$$ in equations , , , respectively, depend only on infinitesimal data near the smooth fibers, and not on the global geometry of the K3 surface. Hence, one may hope to find similar contributions in the Gromov-Witten theory of the Hilbert scheme of points of other elliptic fibrations.
Let $E$ be an elliptic curve with origin $0_E \in E$, and let $$X = {\mathbb{P}}^1 \times E$$ be the trivial elliptic fibration. Here, we study the genus $0$ Gromov-Witten theory of the Hilbert scheme $$\operatorname{\mathsf{Hilb}}^2(X) \,.$$ and use our results to determine the series .
From the view of Gromov-Witten theory, the variety $\operatorname{\mathsf{Hilb}}^2(X)$ has two advantages over the Hilbert scheme of $2$ points of an elliptic K3 surface. First, $\operatorname{\mathsf{Hilb}}^2(X)$ is not holomorphic symplectic. Therefore, we may use ordinary Gromov-Witten invariants and in particular the main computation method which exists in genus $0$ Gromov-Witten theory – the WDVV equation. Second, we have an additional map $$\operatorname{\mathsf{Hilb}}^2(X) \to \operatorname{\mathsf{Hilb}}^2(E)$$ induced by the projection of $X$ to the second factor which is useful in calculations. Our study of the Gromov-Witten theory of $\operatorname{\mathsf{Hilb}}^2(X)$ will proceed in two independent directions. First, we directly analyse the moduli space of stable maps to $\operatorname{\mathsf{Hilb}}^2(X)$ which are incident to certain geometric cycles. Similar to the K3 case, this leads to an explicit expression of generating series of Gromov-Witten invariants of $\operatorname{\mathsf{Hilb}}^2(X)$ in terms of the series . This is parallel to the study of the Gromov-Witten theory of the Hilbert scheme of points of a K3 surface in Sections \[Section\_higher\_dimensional\_Yau\_Zaslow\] and \[Section\_More\_Evaluations\].
In a second independent step, we will calculate the Gromov-Witten invariants of $\operatorname{\mathsf{Hilb}}^2(X)$ using the WDVV equations and a few explicit calculations of initial data. Then, combining both directions, we are able to solve for the functions . This leads to the following result.
Let $F(z,\tau)$ be the Jacobi theta function and, with $y = -e^{2\pi i z}$, let $$G(z,\tau) = F(z,\tau)^2 \left( y \frac{d}{dy} \right)^2 \log(F(z,\tau))$$ be the function which appeared already in Section \[Section\_More\_evaluations\_Introduction\].
\[Hilb2P1e\_complete\_evaluation\_theorem\] Under the variable change $y=-e^{2\pi iz}$ and $q = e^{2 \pi i \tau}$, $$\begin{aligned}
F^{\textup{GW}}(y,q) & = F(z,\tau) \\
G^{\textup{GW}}(y,q) & = G(z,\tau) \\
\widetilde{G}^{\textup{GW}}(y,q) & = - \frac{1}{2} \left( y \frac{d}{dq} \right) G(z,\tau) \\
H^{\textup{GW}}(y,q) & = \left( q \frac{d}{dq} \right) F(y,q)\end{aligned}$$
The proof of Theorem \[Hilb2P1e\_complete\_evaluation\_theorem\] via the geometry of $\operatorname{\mathsf{Hilb}}^2(X)$ is independent from the Kummer K3 geometry studied in Section \[section\_Kummer\_evaluation\]. In particular, our approach here yields a second proof of Theorem \[thm\_F\_evaluation\].
The fiber of $\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1 \times E) \to E$
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### Definition {#definition}
The projections of $X = {\mathbb{P}}^1 \times E$ to the first and second factor induce the maps $$\pi : \operatorname{\mathsf{Hilb}}^2(X) \to \operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1) = {\mathbb{P}}^2 \quad \text{ and } \quad \tau : \operatorname{\mathsf{Hilb}}^2(X) \to \operatorname{\mathsf{Hilb}}^2(E) \label{induced_morphisms}$$ respectively. Consider the composition $$\sigma : \operatorname{\mathsf{Hilb}}^2(X) \xrightarrow{\, \tau \, } \operatorname{\mathsf{Hilb}}^2(E) \xrightarrow{\, + \, } E$$ of $\tau$ with the addition map $+ \colon \operatorname{\mathsf{Hilb}}^2(E) \to E$. Since $\sigma$ is equivariant with respect to the natural action of $E$ on $\operatorname{\mathsf{Hilb}}^2(X)$ by translation, it is an isotrivial fibration with smooth fibers. We let $$Y = \sigma^{-1}(0_E)$$ be the fiber of $\sigma$ over the origin $0_E \in E$.
Let $\gamma \in H_2(\operatorname{\mathsf{Hilb}}^2(X))$ be an effective curve class and let $${{\overline M}}_{0,m}(\operatorname{\mathsf{Hilb}}^2(X), \gamma)$$ be the moduli space of $m$-pointed stable maps to $\operatorname{\mathsf{Hilb}}^2(X)$ of genus $0$ and class $\gamma$. The map $\sigma$ induces an isotrivial fibration $$\sigma : {{\overline M}}_{0,m}(\operatorname{\mathsf{Hilb}}^2(X), \gamma) \to E$$ with fiber over $0_E$ equal to $$\bigsqcup_{\gamma'}\, {{\overline M}}_{0,m}(Y, \gamma'),$$ where the disjoint union runs over all effective curve classes $\gamma' \in H_2(Y;{{\mathbb{Z}}})$ with $\iota_{\ast} \gamma' = \gamma$; here $\iota : Y \to \operatorname{\mathsf{Hilb}}^2(X)$ is the inclusion.
For cohomology classes $\gamma_1, \dots, \gamma_m \in H^{\ast}(\operatorname{\mathsf{Hilb}}^2(X))$, we have $$\begin{gathered}
\int_{[ {{\overline M}}_{0,m}(\operatorname{\mathsf{Hilb}}^2(X), \gamma) ]^{\text{vir}}} {\mathop{\rm ev}\nolimits}_1^{\ast} ( \gamma_1 \cup [Y] )\, \cdots \, {\mathop{\rm ev}\nolimits}_{m}^{\ast}( \gamma_m ) \\
= \sum_{\substack{ \gamma' \in H_2(Y) \\ \iota_{\ast} \gamma' = \gamma} }
\int_{ [ {{\overline M}}_{0,m}(Y, \gamma') ]^{\text{vir}}} (\iota \circ {\mathop{\rm ev}\nolimits}_1)^{\ast}( \gamma_1 ) \cdots (\iota \circ {\mathop{\rm ev}\nolimits}_{m})^{\ast}( \gamma_m ),\end{gathered}$$ where we let $[\ \cdot \ ]^{\text{vir}}$ denote the virtual class defined by ordinary Gromov-Witten theory. Hence, for calculations related to the Gromov-Witten theory of $\operatorname{\mathsf{Hilb}}^2(X)$ we may restrict to the threefold $Y$.
### Cohomology {#cohomology_of_Y}
\[Section\_cohomology\_of\_Y\] Let $D_{X} \subset X \times X$ be the diagonal and let $${\mathop{\rm Bl}\nolimits}_{D_{X}}(X \times X) \to \operatorname{\mathsf{Hilb}}^2(X) \label{fmdefg}$$ be the ${{\mathbb{Z}}}_2$-quotient map which interchanges the factors. Let $$W = {\mathbb{P}}^1 \times {\mathbb{P}}^1 \times E\ \hookrightarrow\ X \times X, \quad (x_1, x_2,e) \mapsto (x_1, e, x_2,-e)$$ be the fiber of $0_E$ under $X \times X \to E \times E \overset{+}{\to} E$ and consider the blowup $$\rho : \widetilde{W} = {\mathop{\rm Bl}\nolimits}_{D_{X} \cap W} W \to W. \label{blowup_map}$$ Then, the restriction of to $\widetilde{W}$ yields the ${{\mathbb{Z}}}_2$-quotient map $$g : \widetilde{W} \to \widetilde{W} / {{\mathbb{Z}}}_2 = Y. \label{gquotientmap}$$
Let $D_{X,1}, \dots, D_{X,4}$ be the components of the intersection $$D_{X} \cap W = \{ (x_1, x_2, f) \in {\mathbb{P}}^1 \times {\mathbb{P}}^1 \times E\ | \ x_1 = x_2 \text{ and } f = -f \}$$ corresponding to the four $2$-torsion points of $E$, and let $$E_1, \dots, E_4$$ be the corresponding exceptional divisors of the blowup $\rho : \widetilde{W} \to W$. For every $i$, the restriction of $g$ to $E_i$ is an isomorphism onto its image. Define the cohomology classes $$\Delta_i = g_{\ast} [ E_i ], \quad \quad A_i = g_{\ast} [ \rho^{-1}(y_i) ]$$ for some $y_i \in D_{X,i}$. We also set $$\Delta = \Delta_1 + \dots + \Delta_4 \,, \quad \quad A = \frac{1}{4} ( A_1 + \dots + A_4 ).$$ Let $x_1, x_2 \in {\mathbb{P}}^1$ and $f \in E$ be points, and define $$B_1 = g_{\ast} \big[ \rho^{-1}( {\mathbb{P}}^1 \times x_2 \times f ) \big],
\quad \quad
B_2 = \frac{1}{2} \cdot g_{\ast} \big[ \rho^{-1}(x_1 \times x_2 \times E) \big] \,.$$ Identify the fiber of $\operatorname{\mathsf{Hilb}}^2(E) \to E$ over $0_E$ with ${\mathbb{P}}^1$, and consider the diagram $$\label{some_mmm_diagram}
\begin{tikzcd}
Y \ar{r}{\tau} \ar{d}{\pi} & {\mathbb{P}}^1 \\ {\mathbb{P}}^2
\end{tikzcd}$$ induced by the morphisms . Let $h \in H^2({\mathbb{P}}^2)$ be the class of a line and let $x \in {\mathbb{P}}^1$ be a point. Define the divisor classes $$D_1 = [ \tau^{-1}(x) ], \quad \quad D_2 = \pi^{\ast} h \,.$$
The cohomology classes $$D_1, D_2, \Delta_1, \dots, \Delta_4 \quad \quad (\text{resp. } B_1, B_2, A_1, \dots, A_4 \big) \label{305}$$ form a basis of $H^2(Y;{{\mathbb{Q}}})$ (resp. of $H^4(Y;{{\mathbb{Q}}}))$.
Since the map $g$ is the quotient map by the finite group ${{\mathbb{Z}}}_2$, we have the isomorphism $$g^{\ast} \colon H^{\ast}(Y;{{\mathbb{Q}}}) \to H^{\ast}(\widetilde{W};{{\mathbb{Q}}})^{{{\mathbb{Z}}}_2},$$ where the right hand side denotes the ${{\mathbb{Z}}}_2$ invariant part of the cohomology of $\widetilde{W}$. The Lemma now follows from a direct verification.
By straight-forward calculation, we find the following intersections between the basis elements .
[lcl]{}
------------ ------- ------- ------------------
$\cdot$ $B_1$ $B_2$ $A_i$
$D_1$ $0$ $1$ $0$
$D_2$ $1$ $0$ $0$
$\Delta_j$ $0$ $0$ $-2 \delta_{ij}$
------------ ------- ------- ------------------
& $\quad \quad$ &
------------ --------- --------- -----------------------------
$\cdot$ $D_1$ $D_2$ $\Delta_i$
$D_1$ $0$ $2 B_1$ $0$
$D_2$ $2 B_1$ $2 B_2$ $2 A_i$
$\Delta_j$ $0$ $2 A_j$ $4 (A_i - B_1) \delta_{ij}$
\[5pt\]
------------ --------- --------- -----------------------------
Finally, using intersection against test curves, the canonical class of $Y$ is $$K_Y = -2 D_2 \,.$$
### Gromov-Witten invariants {#gromov-witten-invariants}
Let $r,d \geq 0$ be integers and let $\textbf{k} = (k_1, \dots, k_4)$ be a tuple of half-integers $k_i \in \frac{1}{2} {{\mathbb{Z}}}$. Define the class $$\beta_{r,d,\textbf{k}} = r B_1 + d B_2 + k_1 A_1 + k_2 A_2 + k_3 A_3 + k_4 A_4.$$ Every algebraic curve in $Y$ has a class of this form.
For cohomology classes $\gamma_1, \dots, \gamma_m \in H^{\ast}(Y;{{\mathbb{Q}}})$ define the genus $0$ potential $$\big\langle \gamma_1, \dots, \gamma_l \big\rangle^Y = \sum_{r,d \geq 0} \sum_{\textbf{k} \in (\frac{1}{2} {{\mathbb{Z}}})^4 } \zeta^r q^d y^{\sum_i k_i}
\int_{[ {{\overline M}}_{0,m}(Y, \beta_{r,d,\textbf{k}}) ]^{\text{vir}} } {\mathop{\rm ev}\nolimits}_{1}^{\ast}(\gamma_1) \cdots {\mathop{\rm ev}\nolimits}_m^{\ast}(\gamma_m),
\label{306}$$ where $\zeta, y, q$ are formal variables and the integral on the right hand side is defined to be $0$ whenever $\beta_{r,d, \textbf{k}}$ is not effective.
The virtual class of ${{\overline M}}_{0,m}(Y,\beta_{r,d,\textbf{k}})$ has dimension $2r+m$. Hence, for homogeneous classes $\gamma_1, \dots, \gamma_m$ of complex degree $d_1, \dots, d_m$ respectively satisfying $\sum_i d_i = 2r + m$, only terms with $\zeta^r$ contribute to the sum . In this case, we often set $\zeta = 1$.
### WDVV equations
Let $\iota \colon Y \to \operatorname{\mathsf{Hilb}}^2(X)$ denote the inclusion and consider the subspace $$i^{\ast} H^{\ast}(\operatorname{\mathsf{Hilb}}^2(X) ;{{\mathbb{Q}}}) \subset H^{\ast}(Y;{{\mathbb{Q}}}). \label{subspace_abc}$$ of classes pulled back from $\operatorname{\mathsf{Hilb}}^2(X)$. The tuple of classes $$b = (T_i)_{i=1}^{8} = (e_Y, D_1, D_2, \Delta, B_1, B_2, A, {{\omega}}_Y),$$ forms a basis of ; here $e_Y = [Y]$ is the fundamental class and ${{\omega}}_Y$ is the class of point of $Y$. Let $(g_{ef})_{e,f}$ with $$g_{ef} = \langle T_e, T_f \rangle = \int_Y T_e \cup T_f$$ be the intersection matrix of $b$, and let $( g^{ef} )_{e,f}$ be its inverse.
Let $\gamma_1, \dots, \gamma_4 \in i^{\ast} H^{\ast}(\operatorname{\mathsf{Hilb}}^2(X) ;{{\mathbb{Q}}})$ be homogeneous classes of complex degree $d_1, \dots, d_4$ respectively such that $\sum_i d_i = 5$. Then, $$\label{313}
\sum_{e,f=1}^{8} \big\langle \gamma_1, \gamma_2, T_e \big\rangle^Y g^{ef} \big\langle \gamma_3, \gamma_4, T_f \big\rangle^Y
=
\sum_{e,f=1}^{8} \big\langle \gamma_1, \gamma_4, T_e \big\rangle^Y g^{ef} \big\langle \gamma_2, \gamma_3, T_f \big\rangle^Y.$$
The claim follows directly from the classical WDVV equation [@FP] and direct formal manipulations.
We reformulate equation into the form we will use. Let $$\gamma \in i^{\ast} H^2(\operatorname{\mathsf{Hilb}}^2(X);{{\mathbb{Q}}})$$ be a divisor class and let $$Q(\zeta, y, q) = \sum_{i,d,k} a_{ikd} \zeta^i y^k q^d$$ be a formal power series. Define the differential operator $\partial_{\gamma}$ by $$\partial_{\gamma} Q(\zeta, y, q) = \sum_{i,d,k} \Big( \int_{i B_1 + d B_2 + k A} \gamma \Big) a_{ikd} \zeta^i y^k q^d.$$ Explicitly, we have $$\partial_{D_1} = q \frac{d}{d q}, \quad \quad \partial_{D_2} = \zeta \frac{d}{d \zeta}, \quad \quad \partial_{\Delta} = -2 y \frac{d}{d y}.$$ Then, for homogeneous classes $\gamma_1, \dots, \gamma_4 \in i^{\ast} H^{\ast}(\operatorname{\mathsf{Hilb}}^2(X) ;{{\mathbb{Q}}})$ of complex degree $2,1,1,1$ respectively, the left hand side of equals $$\begin{gathered}
\partial_{\gamma_2} \big\langle \gamma_1, \gamma_3 \cup \gamma_4 \big\rangle^Y
+ \partial_{\gamma_4} \partial_{\gamma_3} \big\langle \gamma_1 \cup \gamma_2 \big\rangle^Y \\
+
\sum_{\substack{ T_e \in \{ B_1, B_2, A \} \\ T_f \in \{ D_1, D_2, \Delta \} }}
\partial_{\gamma_2}\left( \big\langle \gamma_1, T_e \big\rangle^Y \right) g^{ef}
\partial_{\gamma_3} \partial_{\gamma_4} \partial_{T_f} \big\langle 1 \big\rangle^Y, \label{315}\end{gathered}$$ where we let $\big\langle 1 \big\rangle$ denote the Gromov-Witten potential with no insertions. The expression for the right hand side of is similar.
### Relation to the Gromov-Witten theory of $\operatorname{\mathsf{Hilb}}^2(K3)$
Recall the power series , $$F^{\textup{GW}}(y,q),\ G^{\textup{GW}}(y,q),\ \widetilde{G}^{\textup{GW}}(y,q),\ H^{\textup{GW}}(y,q).$$
\[comparision\_proposition\] There exist a power series $$\widetilde{H}^{\textup{GW}}(y,q) \in {{\mathbb{Q}}}((y^{1/2}))[[q]]$$ such that $$\begin{aligned}
\big\langle B_2, B_2 \big\rangle^Y & = (F^{\textup{GW}})^2 \tag{i} \\
\big\langle {{\omega}}_{Y} \big\rangle^Y & = 2 G^{\textup{GW}} \tag{ii} \\
\big\langle B_1, B_2 \big\rangle^Y & = 2 F^{\textup{GW}} \cdot H^{\textup{GW}} + G^{\textup{GW}} \tag{iii} \\
\big\langle A, B_1 \big\rangle^Y & = \widetilde{G}^{\textup{GW}} + \widetilde{H}^{\textup{GW}} \cdot H^{\textup{GW}} \tag{iv} \\
\big\langle A, B_2 \big\rangle^Y & = \frac{1}{2} \widetilde{H}^{\textup{GW}} \cdot F^{\textup{GW}} \,. \tag{v}\end{aligned}$$
Let $d \geq 0$ be an integer, let $\textbf{k} = (k_1, \dots, k_4) \in (\frac{1}{2} {{\mathbb{Z}}})^4$ be a tuple of half-integers, and let $$\beta_{d, \textbf{k}} = B_1 + d B_2 + k_1 A_1 + \dots + k_4 A_4.$$ Consider a stable map $f : C \to Y$ of genus $0$ and class $\beta_{d, \textbf{k}}$. The composition $\pi \circ f : C \to {\mathbb{P}}^2$ has degree $1$ with image a line $L$. Let $C_0$ be the component of $C$ on which $\pi \circ f$ is non-constant.
Let $g : \widetilde{W} \to Y$ be the quotient map , and consider the fiber diagram $$\begin{tikzcd}
\widetilde{C} \ar{r}{\widetilde{f}} \ar{d} & \widetilde{W} \ar{d}{g} \ar{r}{p} & {\mathbb{P}}^1 \times E \\
C \ar{r}{f} & Y,
\end{tikzcd}$$ where $p = {\mathop{\rm pr}\nolimits}_{23} \circ \rho$ is the composition of the blowdown map with the projection to the $(2,3)$-factor of ${\mathbb{P}}^1 \times {\mathbb{P}}^1 \times E$. Then, parallel to the case of elliptic K3 surfaces, the image of $\widetilde{C}$ under $p \circ \widetilde{f}$ is a comb curve $$B_e + {\mathop{\rm pr}\nolimits}_1^{-1}(z),$$ where $B_e$ is the fiber of the projection $X \to E$ over some point $e \in E$, the map ${\mathop{\rm pr}\nolimits}_1 : {\mathbb{P}}^1 \times E \to {\mathbb{P}}^1$ is the projection to the first factor, and $z \subset {\mathbb{P}}^1$ is a zero-dimensional subscheme of length $d$.
Let $G_0 \subset \widetilde{C}$ be the irreducible component which maps with degree $1$ to $B_e$ under $p \circ \widetilde{f}$. The projection $\widetilde{C} \to C$ induces a flat map $$G_0 \to C_0 \,. \label{flll_map}$$
If has degree $2$, then similar to the arguments of Lemma \[117\], the restriction $f|_{C_0}$ is an isomorphism onto an embedded line $$L \subset \operatorname{\mathsf{Hilb}}^2(S_e) \subset Y,$$ where $e = -e \in E$ is a $2$-torsion point of $E$. Since $f|_{C_0}$ is irreducible, we have $L \nsubseteq I(x)$ for all $x \in {\mathbb{P}}^1$. The tangent line to $\Delta_{\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)}$ at $2x$ is $I(x)$ for every $x \in {\mathbb{P}}^1$. Hence, $L$ intersects the diagonal $\Delta_{\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)}$ in two distinct points.
If has degree $1$, the map $f|_{C_0}$ is the sum of two maps $C_0 \to X$. The first of these must map $C_0$ to a section of $X \to {\mathbb{P}}^1$, the second must be constant since there are no non-constant maps to the fiber of $X \to {\mathbb{P}}^1$. Hence, the restriction $f|_{C_0}$ is an isomorphism onto the embedded line[^10] $$B_e + (x',-e) = g\left( \rho^{-1}\left( x' \times B_e \times -e \right) \right), \label{308}$$ for some $x' \in {\mathbb{P}}^1$ and $e \in E$; here we used the notation .
Every irreducible component of $C$ other then $C_0$ maps into the fiber of $$\pi : Y \to \operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1) = {\mathbb{P}}^2$$ over a diagonal point $2x \in \Delta_{\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)}$.
Summarizing, the map $f : C \to Y$ therefore satisfies one of the following.
1. The restriction $f|_{C_0}$ is an isomorphism onto a line $$L \subset \operatorname{\mathsf{Hilb}}^2(B_e) \subset Y \label{312}$$ where $e \in E$ is a $2$-torsion point. The line $L$ intersects the diagonal in the distinct points $2 x_1$ and $2 x_2$. The curve $C$ has a decomposition $$C = C_0 \cup C_1 \cup C_2 \label{Csplitting_eqn2}$$ such that for $i=1,2$ the restriction $f|_{C_i}$ maps in the fiber $\pi^{-1}(2 x_i)$.
2. The restriction $f|_{C_0}$ is an isomorphism onto the line for some $x' \in {\mathbb{P}}^1$ and $e \in E$. The image $f(C_0)$ meets the fiber $\pi^{-1}(\Delta_{\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)})$ only in the point $(x',e) + (x',-e)$. Hence, the curve $C$ admits the decomposition $$C = C_0 \cup C_1 \label{Csplitting_eqn}$$ where $f|_{C_1}$ maps to the fiber $\pi^{-1}(2 x')$.
According to the above cases, we say that $f : C \to Y$ is of type (A) or (B).
We consider the different cases of Proposition \[comparision\_proposition\].
**Case (i).** Let $Z_1, Z_2$ be generic fibers of the natural map $$\pi : Y \to \operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1).$$ The fibers $Z_1, Z_2$ have class $2 B_2$. Let $f : C \to Y$ be a stable map of class $\beta_{d, \textbf{k}}$ incident to $Z_1$ and $Z_2$. Then $f$ must be of type (A) above, with the line $L$ in uniquely determined by $Z_1, Z_2$ up the choice of the 2-torsion point $e \in E$. After specifying a $2$-torsion point, we are in a case completely parallel to Section \[Section\_higher\_dimensional\_Yau\_Zaslow\], except for the existence of the nodal fibers in the K3 case. Following the argument there, we find the contribution from each fixed $2$-torsion point to be $(F^{\textup{GW}})^2$. Hence, $$\big\langle 2 B_2, 2 B_2 \big\rangle^Y = \left| \left\{ e \in E\ |\ 2e = 0 \right\} \right| \cdot (F^{\textup{GW}})^2 = 4 \cdot (F^{\textup{GW}})^2.$$
**Case (ii).** Let $x_1, x_2 \in {\mathbb{P}}^1$ and $e \in E$ be generic, and consider the point $$y = (x_1,e) + (x_2,-e) \in Y\,.$$ A stable map $f : C \to Y$ of class $\beta_{d,\textbf{k}}$ incident to $y$ must be of type (B) above, with $x' = x_1$ or $x_2$ in . In each case, the calculation proceeds completely analogous to Section \[CASE1\] and yields the contribution $G^{\textup{GW}}$. Summing up both cases, we therefore find $\langle y \rangle^Y = 2 G^{\textup{GW}}$.
**Case (iii).** Let $x', x_1, x_2 \in {\mathbb{P}}^1$ and $e \in E$ be generic points. Let $$\label{owrogr}
Z_1 = g(\rho^{-1}( {\mathbb{P}}^1 \times x' \times e )) = ({\mathbb{P}}^1 \times e) + (x', -e)$$ and let $Z_2$ be the fiber of $\pi$ over the point $x_1 + x_2$. The cycles $Z_1, Z_2$ have the cohomology classes $[Z_1] = B_1$ and $[Z_2] = 2 B_2$ respectively. Let $$f \colon C \to Y$$ be a $2$-marked stable map of genus $0$ and class $\beta_{d, \textbf{k}}$ with markings $p_1, p_2 \in C$ incident to $Z_1, Z_2$ respectively. Since $f(p_1) \in Z_1$, we have $$f(p_1) = (x'', e) + (x',-e)$$ for some $x'' \in {\mathbb{P}}^1$. Since also $f(p_2) \in Z_2$ and $e$ is generic, $x'' \in \{ x', x_1, x_2 \}$.
Assume $x'' = x_1$. Then, $f$ is of type (B) and the restriction $f|_{C_0}$ is an isomorphism onto the line $\ell = B_e + (x_1,-e)$. The line $\ell$ meets the cycle $Z_2$ in the point $(x_2,e) + (x_1,-e)$ and no marked point of $C$ lies on the component $C_1$ in the splitting . Parallel to (ii), the contribution of this case is $G^{\textup{GW}}$. The case $x'' = x_2$ is identical.
Assume $x'' = x'$. Then, $\pi(f(p_1)) = 2x'$. Since $\pi(f(p_2)) = x_1 + x_2$, we have $\pi(f(p_1)) \cap \pi(f(p_2)) = \varnothing$. Hence, $f$ is of type (A) and we have the decomposition $$C = C_0 \cup C_1 \cup C_2,$$ where $f|_{C_0}$ maps to a line $L \subset \operatorname{\mathsf{Hilb}}^2(B_{e'})$ for a $2$-torsion point $e' \in E$, the restriction $f|_{C_1}$ maps to $\pi^{-1}(2 x')$, and $f|_{C_2}$ maps to the fiber of $\pi$ over the diagonal point of $L$ which is not $2x'$. We have $p_1 \in C_1$ with $f(p_1) \in Z_1$, and $p_2 \in C_0$ with $f(p_2) = (x_1,e') + (x_2, -e')$. The contribution from maps to the fiber over $2x'$ matches the contribution $H^{\textup{GW}}$ considered in Section \[Section\_CaseI\_P\]. Since there is no marking on $C_2$, the contribution from maps $f|_{C_2}$ is $F^{\textup{GW}}$. For each fixed $2$-torsion point $e' \in E$, we therefore find the contribution $F^{\textup{GW}} \cdot H^{\textup{GW}}$.
In total, we obtain $$\big\langle B_1, 2 B_2 \big\rangle = 2 \cdot G^{\textup{GW}} + 4 \cdot F^{\textup{GW}} \cdot H^{\textup{GW}} \,.$$
**Case (iv).** Let $x, x' \in {\mathbb{P}}^1$ and $e' \in E$ be generic points, and let $e \in E$ be the $i$-th $2$-torsion point. Consider the exceptional curve at $(x,e)$, $$Z_1 = g(\rho^{-1} (x,x,e) )$$ and the cycle which appeared in above, $$Z_2 = g(\rho^{-1}( {\mathbb{P}}^1 \times x' \times e' )) = ({\mathbb{P}}^1 \times e') + (x', -e').$$ We have $[Z_1] = A_i$ and $[Z_2] = B_1$. Consider a $2$-marked stable map $f : C \to Y$ of class $\beta_{d, \textbf{k}}$ with markings $p_1, p_2 \in C$ incident to $Z_1, Z_2$ respectively.
If $f$ is of type (B), we must have $\pi(f(p_1)) \cap \pi(f(p_2)) \neq \varnothing$. Hence, $f(p_2) = (x, e') + (x', -e')$ and the restriction $f|_{C_0}$ is an isomorphism onto $$\ell = ( \rho^{-1} ( x \times {\mathbb{P}}^1 \times e') ) = B_{(-e')} + (x,e')$$ In the splitting , the component $C_1$ is attached to the component $C_0 \equiv \ell$ at $(x, -e') + (x, e')$. Then, the contribution here matches precisely the contribution of the point $u_0$ in the K3 case of Section \[ABCD\]; it is $\widetilde{G}^{\textup{GW}}$.
Assume $f$ is of type (A). The line $L$ in lies inside $\operatorname{\mathsf{Hilb}}^2(B_{e''})$ for some $2$-torsion point $e'' \in E$. Since $e'$ is generic, $\pi(L)$ is the line through $2x$ and $2x'$. Consider the splitting with $C_1$ and $C_2$ mapping to the fibers of $\pi$ over $2x$ and $2x'$ respectively. The contribution from maps $f|_{C_2}$ over $2x'$ is parallel to Section \[special\_case\_3\]; it is $H^{\textup{GW}}$. Let $\widetilde{H}_0$ (resp. $\widetilde{H}_1$) be the contribution from maps $f|_{C_1}$ over $2x$ if $e'' = e$ (resp. if $e'' \neq e)$. Then, summing up over all $2$-torsion points, the total contribution is $\widetilde{H}^{\textup{GW}} \cdot H^{\textup{GW}}$, where $\widetilde{H}^{\textup{GW}} = \widetilde{H}_0 + 3 \widetilde{H}_1$.
Adding up both cases, we obtain $\langle A_i, B_1 \rangle^Y = \widetilde{G}^{\textup{GW}} + \widetilde{H}^{\textup{GW}} \cdot H^{\textup{GW}}$.
**Case (v).** This is identical to the second case of (iv) above, with the difference that the second marked point does lie on $C_0$, not $C_2$.
Calculations
------------
### Initial Conditions {#Section_Initial_conditions}
Define the formal power series $$\begin{aligned}
H & = \sum_{d \geq 0} \sum_{k \in {{\mathbb{Z}}}} H_{d,k} y^k q^d = \big\langle B_2, B_2 \big\rangle^Y \\
I & = \sum_{d \geq 0} \sum_{k \in {{\mathbb{Z}}}} I_{d,k} y^k q^d = \big\langle {{\omega}}_Y \big\rangle^Y \\
T & = \sum_{d \geq 0} \sum_{k \in {{\mathbb{Z}}}} T_{d,k} y^k q^d = \big\langle 1 \big\rangle^Y,\end{aligned}$$ where $\big\langle 1 \big\rangle^Y$ is the Gromov-Witten potential with no insertion, and we have set $\zeta = 1$ in . We have the following initial conditions.
\[320\] We have
1. for all $k \geq 1$
2. for all $d \geq 1$
3. 4. if $(d = 0, k \leq -2)$ or $( d > 0, k < -2d )$
5. if $k < -2d$
6. if $k < -2d$.
**Case (i).** The moduli space ${{\overline M}}_{0}(Y,\sum_i k_i A_i)$ is non-empty only if there exists a $j \in \{1, \dots, 4\}$ with $k_i = \delta_{ij} k$ for all $i$. Hence, $$T_{0,k}
= \sum_{k_1 + \dots + k_4 = k} \int_{[{{\overline M}}_{0}(Y,\sum_{i} k_i A_i)]^{\text{vir}}} 1
\ =\ \sum_{i = 1}^{4} \int_{[{{\overline M}}_{0}(Y,k A_i)]^{\text{vir}}} 1 \,.$$ Since the term in the last sum is independent of $i$, $$\label{ssssss} T_{0,k} = 4 \int_{[{{\overline M}}_{0}(Y,k A_1)]^{\text{vir}}} 1 \,.$$
Let $e \in E$ be the first $2$-torsion point, let $$D_{X,1} = \{\, (x,x,e)\, |\, x \in {\mathbb{P}}^1\, \} \subset {\mathbb{P}}^1 \times {\mathbb{P}}^1 \times E$$ and consider the subscheme $$\Delta_1 = g( \rho^{-1}( D_{X,1} ) )$$ which already appeared in Section \[Section\_cohomology\_of\_Y\]. The divisor $\Delta_1$ is isomorphic to the exceptional divisor $E_1$ of the blowup $\rho : \widetilde{W} \to W$, see . Hence $\Delta_1 = {\mathbb{P}}(V)$, where $$V = {{\mathcal O}}_{{\mathbb{P}}^1}(2) \oplus {{\mathcal O}}_{{\mathbb{P}}^1} \to {\mathbb{P}}^1.$$ Under the isomorphism $\Delta_1 = {\mathbb{P}}(V)$, the map $$\pi : \Delta_1 \to \Delta_{\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)} \equiv {\mathbb{P}}^1. \label{ffibrt_map}$$ is identified with the natural ${\mathbb{P}}(V) \to {\mathbb{P}}^1$.
The normal bundle of the exceptional divisor $E_1 \subset \widetilde{W}$ is ${{\mathcal O}}_{{\mathbb{P}}(V)}(-1)$. Hence, taking the ${{\mathbb{Z}}}_2$ quotient of $\widetilde{W}$, the normal bundle of $\Delta_1 \subset Y$ is $$N = N_{\Delta_1/Y} = {{\mathcal O}}_{{\mathbb{P}}(V)}(-2).$$
For $k \geq 1$, the moduli space $$M = {{\overline M}}_0(Y, kA_1)$$ parametrizes maps to the fibers of the fibration . Since the normal bundle $N$ of $\Delta_{1}$ has degree $-2$ on each fiber, there is no infinitesimal deformations of maps out of $\Delta_1$. Hence, $M$ is isomorphic to ${{\overline M}}_0( {\mathbb{P}}(V), d {{\mathfrak{f}}})$, where ${{\mathfrak{f}}}$ is class of a fiber of ${\mathbb{P}}(V)$. In particular, $M$ is smooth of dimension $2k-1$.
By smoothness of $M$ and convexity of ${\mathbb{P}}(V)$ in class $k {{\mathfrak{f}}}$, the virtual class of $M$ is the Euler class of the obstruction bundle ${\rm Ob}$ with fiber $${\rm Ob}_f = H^1(C, f^{\ast} T_Y)$$ over the moduli point $[f : C \to Y] \in M$. The restriction of the tangent bundle $T_Y$ to a fixed fiber $A_0$ of is $$T_{Y}|_{A_0} \cong {{\mathcal O}}_{A_0}(2) \oplus {{\mathcal O}}_{A_0} \oplus {{\mathcal O}}_{A_0}(-2),$$ Hence, $${\rm Ob}_f = H^1(C, f^{\ast} T_Y) = H^1(C, f^{\ast} N).$$
Consider the relative Euler sequence of $p : {\mathbb{P}}(V) \to {\mathbb{P}}^1$, $$0 \to \Omega_{p} \to p^{\ast} V \otimes {{\mathcal O}}_{{\mathbb{P}}(V)}(-1) \to {{\mathcal O}}_{{\mathbb{P}}(V)} \to 0. \label{euler}$$ By direct calculation, $\Omega_{p} = {{\mathcal O}}_{{\mathbb{P}}(V)}(-2) \otimes p^{\ast} {{\mathcal O}}_{{\mathbb{P}}^1}(-2)$. Hence, twisting by $p^{\ast} {{\mathcal O}}_{{\mathbb{P}}^1}(2)$, we obtain the sequence $$0 \to N \to p^{\ast} V(2) \otimes {{\mathcal O}}_{{\mathbb{P}}(V)}(-1) \to p^{\ast} {{\mathcal O}}_{{\mathbb{P}}^1}(2) \to 0. \label{euler2}$$ Let $q : {{\mathcal C}}\to M$ be the universal curve and let $f : {{\mathcal C}}\to \Delta_1 \subset Y$ be the universal map. Pulling back by $f$, pushing forward by $q$ and taking cohomology we obtain the exact sequence $$0 \to R^0q_{\ast} f^{\ast} p^{\ast} {{\mathcal O}}_{{\mathbb{P}}^1}(2) \to R^1q_{\ast} f^{\ast} N \to R^1q_{\ast} f^{\ast} p^{\ast} V(2) \otimes {{\mathcal O}}_{{\mathbb{P}}(V)}(-1) \to 0.$$ The bundle $R^1q_{\ast} f^{\ast} N$ is the obstruction bundle ${\rm Ob}$, and $$R^0q_{\ast} f^{\ast} p^{\ast} {{\mathcal O}}_{{\mathbb{P}}^1}(2) = q_{\ast} q^{\ast} p^{\prime \ast} {{\mathcal O}}_{{\mathbb{P}}^1}(2) = p^{\prime \ast} {{\mathcal O}}_{{\mathbb{P}}^1}(2),$$ where $p' : M \to {\mathbb{P}}^1$ is the map induced by $p : {\mathbb{P}}(V) \to {\mathbb{P}}^1$. We find $$c_1(p^{\prime \ast} {{\mathcal O}}_{{\mathbb{P}}^1}(2)) = 2 p^{\prime \ast} {{\omega}}_{{\mathbb{P}}^1},$$ where ${{\omega}}_{{\mathbb{P}}^1}$ is the class of a point on ${\mathbb{P}}^1$. Taking everything together, we have $$\begin{aligned}
\int_{[{{\overline M}}_{0}(Y,k A_1)]^{\text{vir}}} 1
& = \int_M e(R^1q_\ast f^{\ast} N) \notag \\
& = \int_M c_1(p^{\prime \ast} {{\mathcal O}}_{{\mathbb{P}}^1}(2)) c_{2k-2}( R^1q_{\ast} f^{\ast} p^{\ast} V(2) \otimes {{\mathcal O}}_{{\mathbb{P}}(V)}(-1) ) \notag \\
& = 2 \int_{M_x} c_{2k-2}( R^1q_{\ast} f^{\ast} p^{\ast} V(2) \otimes {{\mathcal O}}_{{\mathbb{P}}(V)}(-1) )|_{M_x}, \label{rree}\end{aligned}$$ where $M_x = {{\overline M}}_{0}({\mathbb{P}}^1,k)$ is the fiber of $p' : M \to {\mathbb{P}}^1$ over some $x \in {\mathbb{P}}^1$. Since $$p^{\ast} V(2) \otimes {{\mathcal O}}_{{\mathbb{P}}(V)}(-1)|_{p^{-1}(x)} = {{\mathcal O}}_{{\mathbb{P}}(V)_x}(-1) \oplus {{\mathcal O}}_{{\mathbb{P}}(V)_x}(-1),$$ the term equals $2 \int_{{{\overline M}}_{0,0}({\mathbb{P}}^1, k)} c_{2k-2}({{\mathcal E}})$, where ${{\mathcal E}}$ is the bundle with fiber $$H^1(C, f^{\ast} {{\mathcal O}}_{{\mathbb{P}}^1}(-1)) \oplus H^1(C, f^{\ast} {{\mathcal O}}_{{\mathbb{P}}^1}(-1)).$$ over a moduli point $[f \colon C \to {\mathbb{P}}^1] \in M_x$. Hence, using the Aspinwall-Morrison formula [@MirSym Section 27.5] the term is $$\int_{[{{\overline M}}_{0}(Y,k A_1)]^{\text{vir}}} 1 = 2 \cdot \int_{{{\overline M}}_{0,0}({\mathbb{P}}^1, k)} c_{2k-2}({{\mathcal E}}) = \frac{2}{k^3}.$$ Combining with , the proof of case (i) is complete.
**Case (ii) and (v).** Let $f \colon C \to Y$ be a stable map of genus $0$ and class $d B_2 + \sum k_i A_j$. Then, $f$ maps into the fiber of $$\pi : Y \to \operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)$$ over some diagonal point $2x \in \Delta_{\operatorname{\mathsf{Hilb}}^2({\mathbb{P}}^1)}$. The reduced locus of such a fiber is the union $$\Sigma_{x} \cup A_{x,e_1} \cup \ldots \cup A_{x,e_4} \label{310}$$ where $e_1, \dots, e_4 \in E$ are the $2$-torsion points of $E$, $$A_{x,e} = g(\rho^{-1}( x \times x \times e ))$$ is the exceptional curve of $\operatorname{\mathsf{Hilb}}^2(X)$ at $(x,e) \in X$, and $\Sigma_{x}$ is the fiber of the addition map $\operatorname{\mathsf{Hilb}}^2(F_x) \to F_x = E$ over the origin $0_E$. Hence, $$f_{\ast} [ C ] = a [\Sigma_x] + \sum_i b_i [ A_{x,e_i} ]$$ for some $a, b_1, \dots, b_4 \geq 0$. Since $[ A_{x,e_i} ] = A_i$ and $$[ \Sigma_x ] = B_2 - \frac{1}{2} ( A_1 + A_2 + A_3 + A_4 )$$ we must have $d = a$ and therefore $$f_{\ast} [ C ] = d B_2 + \sum_i (b_i - d/2) A_i.$$ Since $b_i \geq 0$ for all $i$, we find $\sum_i k_i \geq -2d$ with equality if and only if $k_i = -d/2$ for all $i$. This proves (v) and shows $$T_{d, -2d} = \int_{[{{\overline M}}_{0}(Y, d B_2 - \sum_{i} (d/2) A_i )]^{\text{vir}}} 1 \,. \label{xxvvb}$$ Moreover, if $f : C \to Y$ has class $d B_2 - \sum_i (d/2) A_i$, it is a degree $d$ cover of the curve $\Sigma_x$ for some $x$.
We evaluate the integral . Let $Z'$ be the proper transform of $${\mathbb{P}}^1 \times E \hookrightarrow W, \ (x,e) \mapsto (x,x,e)$$ under the blowup map $\rho \colon \widetilde{W} \to W$, and let $$Z = g(Z') = Z' / {{\mathbb{Z}}}_2 \subset Y$$ be its image under $g : \widetilde{W} \to Y$. The projection map ${\mathop{\rm pr}\nolimits}_{1,3} \circ \rho : Z' \to {\mathbb{P}}^1 \times E$ descends by ${{\mathbb{Z}}}_2$ quotient to the isomorphism $$(\tau_{|Z}, \pi_{|Z}) : Z \to {\mathbb{P}}^1 \times {\mathbb{P}}^1, \label{gfrmgfg}$$ where $\tau : Y \to {\mathbb{P}}^1$ is the morphism defined in . Under the isomorphism , the curve $\Sigma_x$ equals ${\mathbb{P}}^1 \times x$. Since moreover the normal bundle of $Z \subset Y$ has degree $-2$ on $\Sigma_x$, we find $${{\overline M}}_{0}(Y, d B_2 - 2 d A) \cong {{\overline M}}_0({\mathbb{P}}^1, d) \times {\mathbb{P}}^1.$$ The normal bundle $Z \subset Y$ is the direct sum $$N = N_{Z/Y} = {\mathop{\rm pr}\nolimits}_{1}^{\ast} {{\mathcal O}}_{{\mathbb{P}}^1}(a) \otimes {\mathop{\rm pr}\nolimits}_{2}^{\ast} {{\mathcal O}}_{{\mathbb{P}}^1}(-2).$$ for some $a$. We determine $a$. Under the isomorphism , the curve $$R = x \times {\mathbb{P}}^1 \subset {\mathbb{P}}^1 \times {\mathbb{P}}^1$$ corresponds to the diagonal in a generic fiber of $\tau : Y \to {\mathbb{P}}^1$. The generic fiber of $\tau$ is isomorphic to ${\mathbb{P}}^1 \times {\mathbb{P}}^1$, hence $c_1(N) \cdot R = 2$ and $a=2$. The result now follows by an argument parallel to (i).
**Case (iii).** This follows directly from the proof of Proposition \[comparision\_proposition\] Case (i) since the line in has class $B_1 - A_i$ for some $i$.
**Case (iv).** Let $f : C \to Y$ be a stable map of genus $0$ and class $\beta_{d, \textbf{k}}$ incident to the cycles $Z_1, Z_2$ of the proof of Proposition \[comparision\_proposition\] Case (i). Then, there exists an irreducible component $C_0 \subset C$ which maps isomorphically to the line $L$ considered in . We have $[L] = B_1 - A_i$ for some $i$
Since all irreducible components of $C$ except for $C_0$ gets mapped under $f$ to curves of the form $\Sigma_x$ or $A_{x,i}$, we have $$\begin{aligned}
f_{\ast} [ C ] = \beta_{d, \textbf{k}}
& = [L] + d [\Sigma_x ] + \sum_j b_j A_j \\
& = B_1 + d B_2 + \sum_j (-d/2 - \delta_{ij} + b_j) \end{aligned}$$ for some $b_1, \dots, b_4 \geq 0$. If $d = 0$ we find $k = \sum_{i} k_i \geq -1$. If $d > 0$, then $f$ maps to at least one curve of the form $\Sigma_x$ with non-zero degree. Since $L$ and $\Sigma_x$ are disjoint, we must have $b_j > 0$ for some $j$. This shows $k = \sum_j k_j \geq -2d$.
**Case (vi).** This case follows by an argument parallel to (iv).
### The system of equations {#system_of_equations}
Let $\frac{d}{d q}$ and $\frac{d}{d y}$ be the formal differentiation operators with respect to $q$ and $y$ respectively. We will use the notation $$\partial_{\tau} = q \frac{d}{d q} \quad \text{ and } \quad \partial_z = y \frac{d}{d y}.$$
The WDVV equation , applied to the cohomology insertions $$\xi = (\gamma_1, \dots, \gamma_4)$$ specified below yield the following relations: $$\label{relations_first_batch}
\begin{alignedat}{2}
& \xi = (B_2, D_2, D_2, \Delta) && : \quad \quad \big\langle B_2, A \big\rangle = - \frac{1}{2} \partial_z(H) \\
& \xi = (B_2, D_2, D_2, D_1) && : \quad \quad \big\langle B_1, B_2 \big\rangle = \partial_{\tau} H + \frac{1}{2} I \\
& \xi = (A, D_2, D_2, \Delta) && : \quad \quad \big\langle A, A \big\rangle = \frac{1}{4} \partial_z^2 H - \frac{1}{4} I \\
& \xi = (A, D_2, D_2, D_1) && : \quad \quad \big\langle B_1 , A \big\rangle = - \frac{1}{2} \partial_z \partial_{\tau} H \\
& \xi = (B_1, D_2, D_2, \Delta) && : \quad \quad \big\langle B_1 , A \big\rangle - \frac{1}{4} \partial_z I = - \frac{1}{2} \partial_z \big\langle B_1, B_2 \big\rangle \\
& \xi = (B_1, D_2, D_2, D_1) && : \quad \quad 2 \big\langle B_1, B_1 \big\rangle + \partial_{\tau} I = 2 \partial_{\tau} \big\langle B_1, B_2 \big\rangle \\
& && \quad \quad \quad \Leftrightarrow \big\langle B_1, B_1 \big\rangle = \partial_{\tau}^2 H
\end{alignedat}$$ Using and the WDVV equations with insertions $\xi$ further yields:
**W1.** $\xi = (B_2, D_1, D_1, D_2)$: $$0 = 2 \partial_{\tau}^2 H + 2 \partial_{\tau} I - H \cdot \partial_{\tau}^3 T + \frac{1}{2} \partial_{z} H \cdot \partial_{z} \partial_{\tau}^2 T$$
**W2.** $\xi = (B_2, \Delta, \Delta, D_2)$: $$0 = 2 \partial_{z}^2 H + 4 \partial_{\tau} H + 2 I - H \cdot \partial_{z}^2 \partial_{\tau} T + \frac{1}{2} \partial_{z} H \cdot ( 4 + \partial_{z}^3 T )$$
**W3.** $\xi = (B_2, \Delta, \Delta, D_1)$: $$\begin{gathered}
0 = 4 \partial_{\tau}^2 H + 2 \partial_{\tau} I - \partial_{z}^2 I + \frac{1}{2} \partial_{z} \partial_{\tau} H \cdot (4 + \partial_{z}^3 T) - \partial_{\tau} H \cdot \partial_{z}^2 \partial_{\tau} T \notag \\
- \frac{1}{2} \partial_{z}^2 H \cdot \partial_{z}^2 \partial_{\tau} T + \partial_{z} H \cdot \partial_{z} \partial_{\tau}^2 T \end{gathered}$$
**W4.** $\xi = (A, \Delta, \Delta, D_2)$: $$0 = -8 \partial_{z} \partial_{\tau} H - 4 \partial_{z}^3 H + 8 \partial_{z} I + 2 \partial_{z} H \cdot \partial_{z}^2 \partial_{\tau} T - \partial_{z}^2 H \cdot (4 + \partial_{z}^3 T) + I \cdot (4 + \partial_{z}^3 T)$$
**W5.** $\xi = (A, \Delta, D_1, D_1)$: $$\begin{gathered}
0 = -2 \partial_{\tau}^2 I + \frac{1}{2} \partial_{z}^2 \partial_{\tau} H \cdot \partial_{z}^2 \partial_{\tau} T - \partial_{z} \partial_{\tau} H \cdot \partial_{z} \partial_{\tau}^2 T - \frac{1}{2} \partial_{z}^3 H \cdot \partial_{z} \partial_{\tau}^2 T \notag \\
+ \partial_{z}^2 H \cdot \partial_{\tau}^3 T - \frac{1}{2} \partial_{\tau} I \cdot \partial_{z}^2 \partial_{\tau} T + \frac{1}{2} \partial_{z} I \cdot \partial_{z} \partial_{\tau}^2 T \end{gathered}$$
**W6.** $\xi = (B_1, D_1, D_1, D_2)$: $$0 = 2 \partial_{\tau}^3 H - \partial_{\tau}^2 I - \partial_{\tau} H \cdot \partial_{\tau}^3 T - \frac{1}{2} I \cdot \partial_{\tau}^3 T + \frac{1}{2} \partial_{z} \partial_{\tau} H \cdot \partial_{z} \partial_{\tau}^2 T$$
### Non-degeneracy of the equations
\[321\] The initial conditions of Proposition \[320\] and the equations **W1** - **W6** together determine $H_{d,k}, I_{d,k}, T_{d,k}$ for all $d$ and $k$.
For all $d,k$, taking the coefficient of $q^d y^k$ in equations **W1** - **W6** yields $$\begin{gathered}
2 d^2 H_{d,k} + 2 d I_{d,k} = \sum_{j,l} (d-l)^2 \Big( (d-l) - \frac{1}{2} j (k-j) \Big) H_{l,j} T_{d-l,k-j} \tag{W1}\end{gathered}$$ $$\begin{gathered}
(2 k (k+1) + 4 d) H_{d,k} + 2 I_{d,k} = \sum_{j,l} (k-j)^2 \Big( (d-l) - \frac{1}{2} j (k-j) \Big) H_{l,j} T_{d-l,k-j} \tag{W2}
$$ $$\begin{gathered}
2d (2d + k) H_{d,k} + (2d - k^2) I_{d,k} = \tag{W3} \\
- \sum_{j,l} (k-j) \Big( j (d-l) - l (k-j) \Big) \Big( (d-l) - \frac{1}{2} j (k-j) \Big) H_{l,j} T_{d-l,k-j} \notag\end{gathered}$$ $$\begin{gathered}
(2 k + 1) I_{d,k} - k( k^2 + k + 2d )H_{d,k} = \tag{W4} \\
- \frac{1}{2} \sum_{j,l} (k-j)^2 \Big( (j (d-l) - \frac{1}{2} (k-j)) H_{l,j} + \frac{1}{2} (k-j) I_{l,j} \Big) T_{d-l,k-j} \notag\end{gathered}$$ $$\begin{gathered}
2 d^2 I_{d,k} =
\sum_{j,l} (d-l) \Big( j (d-l) - l (k-j) \Big) \cdot \tag{W5} \\
\Big( j (d-l) H_{l,j} - \frac{1}{2} j^2 (k-j) H_{l,j} + \frac{1}{2} (k-j) I_{l,j} \Big) T_{d-l,k-j}\end{gathered}$$ $$\begin{gathered}
2 d^3 H_{d,k} - d^2 I_{d,k} =
\sum_{j,l} (d-l)^2 \cdot \tag{W6} \\
\Big( (d-l) ( l H_{l,j} + \frac{1}{2} I_{l,j} ) - \frac{1}{2} j l (k-j) H_{l,j} \Big) T_{d-l,k-j}.\end{gathered}$$
*Claim 1.* The initial conditions and **W1** - **W6** determine $H_{0,k}, I_{0,k}, T_{0,k}$ for all $k$, except for $H_{0,0}$ *Proof of Claim 1.* The values $T_{0,k}$ are determined by the initial conditions. Consider the equation **W2** for $(d,k) = (0,0)$. Plugging in $(d,k) = (0,0)$ and using $H_{0,-1} = 1, T_{0,1} = 8$, we find $I_{0,0} = 2$.
Let $d = 0$ and $k > 0$, and assume we know the values $H_{0,j}, I_{0,j}$ for all $j < k$ except for $H_{0,0}$. Then, equations **W3** and **W4** read $$\begin{aligned}
-4 k^2 I_{0,k} + \text{ (known terms) } & = 0 \\
b -4 k^2 (k + 1) H_{0,k} + \text{ (known terms) } & = 0.\end{aligned}$$ Hence, also $I_{0,k}$ and $H_{0,k}$ are uniquely determined. By induction, the proof of Claim 1 is complete.
Let $d >0$. We argue by induction. Calculating the first values of $H_{0,k}, I_{0,k}$ and $T_{0,k}$, and plugging them into equations **W1** - **W6** for $(d,k) = (1,-2)$ and $(d,k) = (1,-1)$, we find by direct calculation that the values $$H_{0,0},\ H_{1,-2},\ H_{1,-1},\ I_{1,-2},\ I_{1,-1},\ T_{1,-1},\ T_{1,0}$$ are determined.
Let now $(d = 1, k \geq 0)$ or $(d > 1, k \geq -2d)$, and assume we know the values $H_{l,j}, I_{l,j}, T_{l,j}$ for all $l < d, j \leq k + 2 (d-l)$ and for all $l = d, j < k$. Also assume, that we know $T_{d,k}$. The proof of Proposition \[321\] follows now from the following claim.
*Claim 2:* The values $H_{d,k}, I_{d,k}, T_{d,k + 1}$ are determined.
*Proof of Claim 2.* Solving for the terms $H_{d,k}, I_{d,k}, T_{d,k+1}$ in the equations **W1**, **W6** and **W5**, we obtain: $$\begin{aligned}
2 d^2 H_{d,k} + 2d I_{d,k} - d^2 \left(d + \frac{1}{2} (k+1)\right) T_{d,k+1} & = \text{ (known terms) } \tag{W1} \\
2 d^3 H_{d,k} - d^2 I_{d,k} & = \text{ (known terms) } \tag{W6} \\
-2 I_{d,k} + \left(d + \frac{1}{2} (k + 1)\right) T_{d,k+1} & = \text{ (known terms) }, \tag{W5}\end{aligned}$$ where in the last line we divided by $d^2$. These equations in matrix form read $$\begin{pmatrix}
2d & 2 & -d (d + \frac{1}{2} (k+1)) \\ 2d & -1 & 0 \\ 0 & -2 & d + \frac{1}{2} (k+1)
\end{pmatrix} \cdot
\begin{pmatrix}
H_{d,k} \\ I_{d,k} \\ T_{d,k+1}
\end{pmatrix} = \text{ (known terms) }$$ The matrix on the left hand side has determinant $(2d - 3) (k + 2d + 1) d$. It vanishes if $d = \frac{3}{2}$ or $k = -2d - 1$ or $d = 0$. By assumption, each of these cases were excluded. Hence the values $H_{d,k}, I_{d,k}, T_{d,k + 1}$ are uniquely determined.
**Remark.** We have selected very particular WDVV equations for $Y$ above. Using additional equations, one may show that the values $$H_{0,-1} = 1, \quad T_{0,0} = 0, \quad T_{0,1} = 8, \quad T_{1,-2} = 2$$ together with the vanishings of Proposition \[320\] (iv) - (vi) suffice to determine the series $H, I, T$.
### Solution of the equations
Let $z \in {{\mathbb{C}}}$ and $\tau \in {{\mathbb{H}}}$ and consider the actual variables $$y = -e^{2 \pi i z} \quad \text{ and } \quad q = e^{2 \pi i \tau} \,. \label{var_change_1000}$$ Let $F(z,\tau)$ and $G(z,\tau)$ be the functions and respectively.
\[fIT\] \[HIT\] We have $$\begin{aligned}
H & = F(z,\tau)^2 \\
I & = 2\, G(z,\tau) \\
T & = 8 \sum_{k \geq 1} \frac{1}{k^3} y^k + 12 \sum_{k,n \geq 1} \frac{1}{k^3} q^{kn} \\
& \quad + 8 \sum_{k,n \geq 1} \frac{1}{k^3} (y^k + y^{-k}) q^{kn} + 2 \sum_{k,n \geq 1} \frac{1}{k^3} (y^{2k} + y^{-2k}) q^{(2n-1) k}.\end{aligned}$$ under the variable change $y = -e^{2 \pi i z}$ and $q=e^{2 \pi i \tau}$.
By Proposition \[321\], it suffices to show that the functions defined in the statement of Theorem \[fIT\] satisfy the initial conditions of Proposition \[320\] and the WDVV equations **W1** - **W6**. By a direct check, the initial conditions are satisfied. We consider the WDVV equations.
For the scope of this proof, define $H = F(z,\tau)^2$ and $I = 2\, G(z,\tau)$ and $$\begin{gathered}
T = 8 \sum_{k \geq 1} \frac{1}{k^3} y^k + 12 \sum_{k,n \geq 1} \frac{1}{k^3} q^{kn} \\
+ 8 \sum_{k,n \geq 1} \frac{1}{k^3} (y^k + y^{-k}) q^{kn} + 2 \sum_{k,n \geq 1} \frac{1}{k^3} (y^{2k} + y^{-2k}) q^{(2n-1) k}.\end{gathered}$$ considered as a function in $z$ and $\tau$ under the variable change . We show these functions satisfy the equations **W1** - **W6**.
For a function $A(z,\tau)$, we write $$A^{\bullet} = \partial_z A := \frac{1}{2 \pi i} \frac{\partial A}{\partial z} = y \frac{d}{d y} A, \quad \quad
A' = \partial_\tau A := \frac{1}{2 \pi i} \frac{\partial A}{\partial \tau} = q \frac{d}{d q} A$$ for the differential of $A$ with respect to $z$ and $\tau$ respectively.
For $n \geq 1$, define the deformed Eisenstein series [@O] $$\begin{aligned}
J_{2,n}(z,\tau) & = \delta_{n,1} \frac{y}{y - 1} + B_n - n \sum_{k, r \geq 1} r^{n - 1} (y^k + (-1)^n y^{-k}) q^{k r} \\
J_{3,n}(z, \tau) & = -B_n \Big( 1 - \frac{1}{2^{n - 1}} \Big) - n \sum_{k , r \geq 1} \left( r - 1/2 \right)^{n - 1} (y^k + (-1)^n y^{-k}) q^{k (r - \frac{1}{2})},\end{aligned}$$ where $B_{n}$ are the Bernoulli numbers (with $B_1 = -\frac{1}{2}$) and we used the variable change . We also let $$\begin{aligned}
G_n(z, \tau) & = J_{4,n}(2z, 2 \tau) \\
& = -B_n \Big( 1 - \frac{1}{2^{n - 1}} \Big) - n \sum_{k , r \geq 1} (r - 1/2)^{n - 1} (y^{2k} + (-1)^n y^{-2k}) q^{k (2 r - 1)}. \end{aligned}$$ Then we have $$\begin{aligned}
\partial_z^3 T & = -4 - 8 J_{2,1} - 16 G_1 \\
\partial_z^2 \partial_\tau T & = -4 J_{2,2} - 8 G_2 \\
\partial_z \partial_\tau^2 T & = - \frac{8}{3} J_{2,3} - \frac{16}{3} G_3 \\
\partial_\tau^3 T & = -2 J_{2,4} - 4 G_4 + \frac{1}{20} E_4.
\end{aligned} \label{Trelations}$$ Since $T(z,\tau)$ appears only as a third derivative in the equations **W1** - **W6**, we may trade it for deformed Eisenstein series using equations .
The first theta function $\vartheta_1(z,\tau)$ satisfies the heat equation $$\partial_z^2 \vartheta_1 = 2 \partial_{\tau} \vartheta_1,$$ which implies that $F = F(z,\tau) = \vartheta_1(z,\tau) / \eta^3(\tau)$ satisfies $$\partial_{\tau}F = \frac{1}{2} \partial_z^2 F - \frac{1}{8} E_2(\tau) F, \label{yyyy}$$ where $E_2(\tau) = 1 - 24 \sum_{d \geq 1} \sum_{k|d} k q^d$ is the second Eisenstein series. With a small calculation, we obtain the relation $$I = 4 \partial_{\tau}(H) - \partial_z^2(H) + E_2 \cdot H. \label{ItoH}$$
Hence, using equations and , we may replace in the equations **W1** - **W6** the function $T$ with deformed Eisenstein Series and $I$ with terms involving only $H$ and $E_2$. Hence, we are left with a system of partial differential equations between the square of the Jacobi theta function $F$, deformed Eisenstein series and classical modular forms.
These new equations may now be checked directly by methods of complex analysis as follows. Divide each equation by $H$; derive how the quotients $$\frac{H^{k \bullet}}{H} \quad \text{ and } \quad \frac{H^{k \prime}}{H}$$ (with $H^{k \bullet}$ and $H^{k \prime}$ the $k$-th derivative of $H$ with respect to $z$ resp. $\tau$ respectively) transform under the variable change $$(z,\tau) \mapsto (z + \lambda \tau + \mu, \tau) \quad \quad (\lambda, \mu \in {{\mathbb{Z}}}) \,;$$ using the periodicity properties of the deformed Eisenstein series proven in [@O], show that each equation is is double periodic in $z$; calculate all appearing poles using the expansions of the deformed Eisenstein series in [@O]; prove all appearing poles cancel; finally prove that the constant term is $0$ by evaluating at $z = 1/2$. Using this procedure, the proof reduces to a long, but standard calculation.
### Proof of Theorem \[Hilb2P1e\_complete\_evaluation\_theorem\]
We will identify functions in $(z,\tau)$ with their expansion in $y,q$ under the variable change . By Proposition \[comparision\_proposition\], the definition of $H$ in , and Theorem \[HIT\], we have $$(F^{\textup{GW}})^2 = \langle B_2, B_2 \rangle^Y = H = F(z,\tau)^2$$ which implies $$F^{\textup{GW}}(y,q) = \pm F(z,\tau) \,. \label{12355}$$ By definition , the $y^{-1/2} q^0$-coefficient of $F^{\textup{GW}}(y,q)$ is $1$. Hence, there is a positive sign in , and we have equality. This proves the first equation of Theorem \[Hilb2P1e\_complete\_evaluation\_theorem\]. The case $G^{\textup{GW}} = G$ is parallel.
Finally, the two remaining cases follow directly from Proposition \[comparision\_proposition\], the relations and Theorem \[HIT\]. This completes the proof of Theorem \[Hilb2P1e\_complete\_evaluation\_theorem\].
Quantum Cohomology {#Section_Quantum_Cohomology}
==================
Overview {#overview-3}
--------
Let $S$ be a K3 surface. In section \[FockSpace\_section\] we recall basic facts about the Fock space $${{\mathcal F}}(S) = \bigoplus_{d \geq 0} H^{\ast}( \operatorname{\mathsf{Hilb}}^d(S) ; {{\mathbb{Q}}}) \,.$$ In Section \[Section\_Main\_Conjecture\_WDVV\] we define a $2$-point quantum operator ${{\mathcal E}}^{\operatorname{\mathsf{Hilb}}}$, which encodes the quantum multiplication with a divisor class. In section \[Main\_conjecture\_section\] we introduce natural operators ${{\mathcal E}}^{(r)}$ acting on ${{\mathcal F}}(S)$. In Section \[Main\_conjecture\_section2\], we state a series of conjectures which link ${{\mathcal E}}^{(r)}$ to the operator ${{\mathcal E}}^{\operatorname{\mathsf{Hilb}}}$. In section \[Main\_conjecture\_examples\_section\] we present several example calculations and prove our conjectures in the case of $\operatorname{\mathsf{Hilb}}^2(S)$. Here, we also discuss the relationship of the K3 surface case to the case of ${{\mathcal A}}_1$-resolution studied by Maulik and Oblomkov in [@MO2].
The Fock space {#FockSpace_section}
--------------
The Fock space of the K3 surface $S$, $${{\mathcal F}}(S) = \bigoplus_{d \geq 0} {{\mathcal F}}_d(S) = \bigoplus_{d \geq 0} H^{\ast}(\operatorname{\mathsf{Hilb}}^d(S),{{\mathbb{Q}}}) \label{P1},$$ is naturally bigraded with the $(d,k)$-th summand given by $${{\mathcal F}}_{d}^k(S) = H^{2 (k + d)}(\operatorname{\mathsf{Hilb}}^d(S),{{\mathbb{Q}}})$$ For a bihomogeneous element $\mu \in {{\mathcal F}}_{d}^k(S)$, we let $$| \mu | = d, \quad \quad k(\mu) = k.$$ The Fock space ${{\mathcal F}}(S)$ carries a natural scalar product $\big\langle \cdot\, \big|\, \cdot \big\rangle$ defined by declaring the direct sum orthogonal and setting $$\big\langle \mu\, \big|\, \nu \big\rangle = \int_{\operatorname{\mathsf{Hilb}}^d(S)} \mu \cup \nu$$ for $\mu, \nu \in H^{\ast}(\operatorname{\mathsf{Hilb}}^d(S),{{\mathbb{Q}}})$. If $\alpha, \alpha' \in H^{\ast}(S,{{\mathbb{Q}}})$ we also write $$\langle \alpha, \alpha' \rangle = \int_S \alpha \cup \alpha'.$$ If $\mu, \nu$ are bihomogeneous, then $\langle \mu | \nu \rangle$ is nonvanishing only if $|\mu| = |\nu|$ and $k(\mu) + k(\nu) = 0$. For all $\alpha \in H^{\ast}(S,{{\mathbb{Q}}})$ and $m \neq 0$, the Nakajima operators ${{\mathfrak{p}}}_m(\alpha)$ act on ${{\mathcal F}}(S)$ bihomogeneously of bidegree $(-m, k(\alpha))$, $${{\mathfrak{p}}}_{m}(\alpha) : {{\mathcal F}}_d^k \to {{\mathcal F}}_{d-m}^{k+ k(\alpha)} \,.$$ The commutation relations $$[ {{\mathfrak{p}}}_{m}(\alpha), {{\mathfrak{p}}}_{n}(\beta) ] = -m \delta_{m + n,0} \langle \alpha, \beta \rangle\, {\rm id}_{{{\mathcal F}}(S)}, \label{N1}$$ are satisfied for all $\alpha, \beta \in H^{\ast}(S)$ and all $m, n \in {{\mathbb{Z}}}\setminus \{ 0 \}$.
The inclusion of the diagonal $S \subset S^m$ induces a map $$\tau_{\ast m} : H^{\ast}(S,{{\mathbb{Q}}}) \to H^{\ast}(S^m,{{\mathbb{Q}}}) \stackrel{\sim}{=} H^{\ast}(S,{{\mathbb{Q}}})^{\otimes m} \, .$$ For $\tau_{\ast} = \tau_{\ast 2}$, we have $$\tau_{\ast}(\alpha) = \sum_{i,j} g^{ij} \, (\alpha \cup \gamma_i) \otimes \gamma_j,$$ where $\{ \gamma_i \}_i$ is a basis of $H^{\ast}(S)$ and $g^{ij}$ is the inverse of the intersection matrix $g_{ij} = \langle \gamma_i, \gamma_j \rangle$.
For $\gamma \in H^{\ast}(S,{{\mathbb{Q}}})$ and $n \in {{\mathbb{Z}}}$ define the Virasoro operator $$L_n(\gamma) = - \frac{1}{2} \sum_{k \in {{\mathbb{Z}}}} : {{\mathfrak{p}}}_k {{\mathfrak{p}}}_{n-k} : \tau_{\ast}(\gamma),$$ where $: -- :$ is the normal ordered product [@Lehn] and we used $${{\mathfrak{p}}}_{k}{{\mathfrak{p}}}_{l} \cdot \alpha \otimes \beta = {{\mathfrak{p}}}_{k}(\alpha) {{\mathfrak{p}}}_{l}(\beta).$$ We are particularly interested in the degree $0$ Virasoro operator $$\begin{aligned}
L_0(\gamma)
& = - \frac{1}{2} \sum_{k \in {{\mathbb{Z}}}\setminus 0} : {{\mathfrak{p}}}_k {{\mathfrak{p}}}_{-k} : \tau_{\ast}(\gamma) \\
& = - \sum_{k \geq 1} \sum_{i,j} g^{ij} {{\mathfrak{p}}}_{-k}(\gamma_i \cup \gamma) {{\mathfrak{p}}}_k(\gamma_j) \,,\end{aligned}$$ The operator $L_0(\gamma)$ is characterized by the commutator relations $$\big[ {{\mathfrak{p}}}_{k}(\alpha), L_0(\gamma) \big] = k \, {{\mathfrak{p}}}_{k}(\alpha \cup \gamma).$$ Let $e \in H^{\ast}(S)$ denote the unit. The restriction of $L_0(\gamma)$ to ${{\mathcal F}}_d(S)$, $$L_0(\gamma)\big|_{{{\mathcal F}}_d(S)} : H^{\ast}(\operatorname{\mathsf{Hilb}}^d(S),{{\mathbb{Q}}}) \to H^{\ast}(\operatorname{\mathsf{Hilb}}^d(S),{{\mathbb{Q}}})$$ is the cup product by the class $$D(\gamma) = \frac{1}{(d-1)!} {{\mathfrak{p}}}_{-1}(\gamma) {{\mathfrak{p}}}_{-1}(e)^{d-1} \in H^{\ast}(\operatorname{\mathsf{Hilb}}^d(S),{{\mathbb{Q}}}) \label{divisor_class_hilbd}$$ of subschemes incident to $\gamma$, see [@Lehn2]. In the special case $\gamma = e$, the operator $L_0 = L_0(e)$ is the *energy operator*, $$L_0\big|_{{{\mathcal F}}_d(S)} = d \cdot {\rm id}_{{{\mathcal F}}_d(S)} \,. \label{energy_operator}$$
Finally, define Lehn’s diagonal operator [@Lehn2] $$\partial = - \frac{1}{2} \sum_{i,j \geq 1} ( {{\mathfrak{p}}}_{-i} {{\mathfrak{p}}}_{-j} {{\mathfrak{p}}}_{i+j} + {{\mathfrak{p}}}_i {{\mathfrak{p}}}_j {{\mathfrak{p}}}_{-(i+j)} ) \tau_{3 \ast}( [S] ) \,.$$ For $d \geq 2$, the operator $\partial$ acts on ${{\mathcal F}}_d(S)$ by cup product with $-\frac{1}{2} \Delta_{\operatorname{\mathsf{Hilb}}^d(S)}$, where $$\Delta_{\operatorname{\mathsf{Hilb}}^d(S)} = \frac{1}{(d-2)!} {{\mathfrak{p}}}_{-2}(e) {{\mathfrak{p}}}_{-1}(e)^{d-2}$$ is the class of the diagonal in $\operatorname{\mathsf{Hilb}}^d(S)$.
The WDVV equation {#Section_Main_Conjecture_WDVV}
-----------------
Let $S$ be an elliptic K3 surface with a section. Let $B$ and $F$ be the section and fiber class respectively, and let $$\beta_h = B + hF \,.$$ For $d \geq 1$ and cohomology classes $\gamma_1, \dots, \gamma_m \in H^{\ast}(\operatorname{\mathsf{Hilb}}^d(S);{{\mathbb{Q}}})$ define the quantum bracket $$\big\langle \gamma_1, \dots, \gamma_m \big\rangle_q^{\operatorname{\mathsf{Hilb}}^d(S)}
= \sum_{h \geq 0} \sum_{k \in {{\mathbb{Z}}}} y^k q^{h-1} {\Big\langle}\gamma_1, \dots, \gamma_m {\Big\rangle}^{\operatorname{\mathsf{Hilb}}^d(S)}_{\beta_h + k A} \,,$$ where the bracket on the right hand side was defined in .
Define the 2-point quantum operator $${{\mathcal E}}^{\operatorname{\mathsf{Hilb}}} : {{\mathcal F}}(S) \otimes {{\mathbb{Q}}}((y))((q)) {{\ \longrightarrow\ }}{{\mathcal F}}(S) \otimes {{\mathbb{Q}}}((y))((q))$$ by the following two conditions.
- for all homogeneous $a,b \in {{\mathcal F}}(S)$, $$\big\langle a\ |\ {{\mathcal E}}^{\operatorname{\mathsf{Hilb}}} b \big\rangle =
\begin{cases}
\big\langle a , b \big\rangle_q & \text{ if } |a| = |b| \\
0 & \text{ else, }
\end{cases}$$
- ${{\mathcal E}}^{\operatorname{\mathsf{Hilb}}}$ is linear over ${{\mathbb{Q}}}((y))((q))$.
Since ${{\overline M}}_{0,2}(\operatorname{\mathsf{Hilb}}^d(S),\alpha)$ has reduced virtual dimension $2d$, the operator ${{\mathcal E}}^{\operatorname{\mathsf{Hilb}}}$ is self-adjoint of bidegree $(0,0)$.
For $d \geq 0$, consider a divisor class $$D \in H^2(\operatorname{\mathsf{Hilb}}^d(S)),$$ and the operator of primitive quantum multiplication[^11] with $D$, $$\mathsf{M}_D \colon a \mapsto D \ast a$$ for all $a \in {{\mathcal F}}_d(S) \otimes {{\mathbb{Q}}}((y))((q)) \otimes {{\mathbb{Q}}}[{\hbar}]/{\hbar}^2$. If $$D = D(\gamma) \text{ for some } \gamma \in H^2(S)\quad \text{ or } \quad D = -\frac{1}{2} \Delta_{\operatorname{\mathsf{Hilb}}^d(S)}, \label{fomofddf}$$ by the divisor axiom we have $$\begin{aligned}
\mathsf{M}_{D(\gamma)} \big|_{{{\mathcal F}}_d(S)}
& = \Big( L_0(\gamma) + {\hbar}\, {{\mathfrak{p}}}_{0}(\gamma) {{\mathcal E}}^{\operatorname{\mathsf{Hilb}}} \Big)\Big|_{{{\mathcal F}}_d(S)} \\
-\frac{1}{2} \mathsf{M}_{\Delta_{\operatorname{\mathsf{Hilb}}^d(S)}}\big|_{{{\mathcal F}}_d(S)}
& = \Big( \partial + {\hbar}\, y \frac{d}{d y} {{\mathcal E}}^{\operatorname{\mathsf{Hilb}}} \Big)\Big|_{{{\mathcal F}}_d(S)} \,,\end{aligned}$$ where $\frac{d}{dy}$ is formal differentiation with respect to the variable $y$, and ${{\mathfrak{p}}}_0(\gamma)$ for $\gamma \in H^{\ast}(S)$ is the degree $0$ Nakajima operator defined by the following conditions:[^12] $$\label{dowofgefg}
\begin{aligned}
\bullet\ \, & [ {{\mathfrak{p}}}_0(\gamma), {{\mathfrak{p}}}_{m}(\gamma') ] = 0 && \text{ for all } \gamma' \in H^{\ast}(S),\ m \in {{\mathbb{Z}}}, \\
\bullet\ \, & {{\mathfrak{p}}}_0(\gamma)\, q^{h-1} y^k \, 1_S = \big\langle \gamma, \beta_h \big\rangle q^{h-1} y^k\, 1_S && \text{ for all } h,k.
\end{aligned}$$ Since the classes $D(\gamma)$ and $\Delta_{\operatorname{\mathsf{Hilb}}^d(S)}$ span $H^2(\operatorname{\mathsf{Hilb}}^d(S)$, the operator ${{\mathcal E}}^{\operatorname{\mathsf{Hilb}}}$ therefore determines quantum multiplication $\mathsf{M}_D$ for every divisor class $D$.
Let $D_1, D_2 \in H^2(\operatorname{\mathsf{Hilb}}^d(S),{{\mathbb{Q}}})$ be divisor classes. By associativity and commutativity of quantum multiplication, we have $$D_1 \ast ( D_2 \ast a ) = D_2 \ast ( D_1 \ast a ) \label{12333}$$ for all $a \in {{\mathcal F}}_d(S)$. After specializing $D_1$ and $D_2$, we obtain the main commutator relations for the operator ${{\mathcal E}}^{\operatorname{\mathsf{Hilb}}}$:
For all $\gamma, \gamma' \in H^2(S,{{\mathbb{Q}}})$, after restriction to ${{\mathcal F}}(S)$, we have $$\label{P4}
\begin{aligned}
{{\mathfrak{p}}}_0(\gamma)\, \big[ {{\mathcal E}}^{\operatorname{\mathsf{Hilb}}}, L_0(\gamma') \big] & = {{\mathfrak{p}}}_0(\gamma')\, \big[ {{\mathcal E}}^{\operatorname{\mathsf{Hilb}}}, L_0(\gamma) \big] \\
{{\mathfrak{p}}}_0(\gamma)\, \big[ {{\mathcal E}}^{\operatorname{\mathsf{Hilb}}}, \partial \big] & = y \frac{d}{dy}\, \big[ {{\mathcal E}}^{\operatorname{\mathsf{Hilb}}}, L_0(\gamma) \big] \,.
\end{aligned}$$
The equalities hold only after restricting to ${{\mathcal F}}(S)$. In both cases, the extension of these equations to ${{\mathcal F}}(S) \otimes {{\mathbb{Q}}}((y))((q))$ does *not* hold, since ${{\mathfrak{p}}}_0(\gamma)$ is not $q$-linear, and $y \frac{d}{dy}$ is not $y$-linear.
Equations show that the commutator of ${{\mathcal E}}^{\operatorname{\mathsf{Hilb}}}$ with a divisor intersection operator is essentially independent of the divisor.
The operators ${{\mathcal E}}^{(r)}$ {#Main_conjecture_section}
------------------------------------
For all $(m,\ell) \in {{\mathbb{Z}}}^2 \setminus \{ 0 \}$ consider fixed formal power series $$\varphi_{m,\ell}(y,q)\, \in {{\mathbb{C}}}((y^{1/2}))[[q]] \label{006}$$ which satisfy the symmetries $$\begin{aligned} \label{symmetries_phi}
\varphi_{m,\ell} & = - \varphi_{-m, -\ell} \\
\ell \varphi_{m,\ell} & = m \varphi_{\ell, m} \,.
\end{aligned}$$ Let $\Delta(q) = q \prod_{m \geq 1} (1-q^m)^{24}$ be the modular discriminant and let $$F(y,q) = (y^{1/2} + y^{-1/2}) \prod_{m \geq 1} \frac{ (1 + yq^m) (1 + y^{-1}q^m)}{ (1-q^m)^2 }$$ be the Jacobi theta function which appeared in Section , considered as formal power series in $q$ and $y$ in the region $|q| < 1$. Depending on the functions , define for all $r \in {{\mathbb{Z}}}$ operators $${{\mathcal E}}^{(r)} : {{\mathcal F}}(S) \otimes {{\mathbb{C}}}((y^{1/2}))((q)) {{\ \longrightarrow\ }}{{\mathcal F}}(S) \otimes {{\mathbb{C}}}((y^{1/2}))((q)) \label{Def_CEr_operators}$$ by the following recursion relations:
[**Relation 1.**]{} For all $r \geq 0$, $${{\mathcal E}}^{(r)} \Big|_{{{\mathcal F}}_0(S) \otimes {{\mathbb{C}}}((y^{1/2}))((q))} = \frac{\delta_{0r}}{F(y,q)^2 \Delta(q)} \cdot {\rm id}_{{{\mathcal F}}_0(S) \otimes {{\mathbb{C}}}((y^{1/2}))((q))},$$
[**Relation 2.**]{} For all $m \neq 0$, $r \in {{\mathbb{Z}}}$ and homogeneous $\gamma \in H^{\ast}(S)$, $$[ {{\mathfrak{p}}}_{m}(\gamma), {{\mathcal E}}^{(r)} ] = \sum_{\ell \in {{\mathbb{Z}}}} \frac{\ell^{k(\gamma)}}{m^{k(\gamma)}} : {{\mathfrak{p}}}_\ell(\gamma) {{\mathcal E}}^{(r+m-\ell)} : \, \varphi_{m,\ell}(y,q),$$ where $k(\gamma)$ denotes the shifted complex cohomological degree of $\gamma$, $$\gamma \in H^{2(k(\gamma) + 1)}(S;{{\mathbb{Q}}}) \,,$$ and $: -- :$ is a variant of the normal ordered product defined by $$: {{\mathfrak{p}}}_\ell(\gamma) {{\mathcal E}}^{(k)} : = \begin{cases}
{{\mathfrak{p}}}_{\ell}(\gamma) {{\mathcal E}}^{(k)} & \text{ if } \ell \leq 0 \\
{{\mathcal E}}^{(k)} {{\mathfrak{p}}}_{\ell}(\gamma) & \text{ if } \ell > 0 \,.
\end{cases}$$
By definition, the operator ${{\mathcal E}}^{(r)}$ is homogeneous of bidegree $(-r,0)$; it is $y$-linear, but *not* $q$ linear.
The operators ${{\mathcal E}}^{(r)}, r \in {{\mathbb{Z}}}$ are well-defined.
By induction, Relation 1 and 2 uniquely determine the operators ${{\mathcal E}}^{(r)}$. It remains to show that the Nakajima commutator relations are preserved by ${{\mathcal E}}^{(r)}$. Hence, we need to show $$\Big[ \big[ {{\mathfrak{p}}}_m(\alpha), {{\mathfrak{p}}}_n(\beta) \big] , {{\mathcal E}}^{(r)} \Big]
= \big[ -m \delta_{m + n,0} \langle \alpha, \beta \rangle\, {\rm id}_{{{\mathcal F}}(S)}, {{\mathcal E}}^{(r)} \big] = 0$$ for all homogeneous $\alpha, \beta \in H^{\ast}(S)$ and all $m, n \in {{\mathbb{Z}}}\setminus \{ 0 \}$. We have $$\Big[ \big[ {{\mathfrak{p}}}_m(\alpha), {{\mathfrak{p}}}_n(\beta) \big] , {{\mathcal E}}^{(r)} \Big]
= \Big[ {{\mathfrak{p}}}_m(\alpha), \big[ {{\mathfrak{p}}}_n(\beta), {{\mathcal E}}^{(r)} \big] \Big]
- \Big[ {{\mathfrak{p}}}_n(\beta) , \big[ {{\mathfrak{p}}}_m(\alpha) , {{\mathcal E}}^{(r)} \big] \Big].
\label{Muasfdagd}$$ Using Relation 2, we obtain $$\begin{aligned}
\notag & \Big[ {{\mathfrak{p}}}_m(\alpha), \big[ {{\mathfrak{p}}}_n(\beta), {{\mathcal E}}^{(r)} \big] \Big] \\
\notag =\ &
\Big[ {{\mathfrak{p}}}_m(\alpha),\ \sum_{\ell \in {{\mathbb{Z}}}} \frac{\ell^{k(\beta)}}{n^{k(\beta)}} : {{\mathfrak{p}}}_\ell(\beta) {{\mathcal E}}^{(r+m-\ell)} : \, \varphi_{m,\ell}(y,q) \Big] \\
\label{fovmdfd} =\ & \frac{(-m)^{k(\beta) + 1}}{n^{k(\beta)}} \langle \alpha, \beta \rangle {{\mathcal E}}^{(r+n+m)} \varphi_{n,-m} \\
\notag & \quad + \sum_{\ell, \ell' \in {{\mathbb{Z}}}} \frac{ \ell^{k(\beta)} (\ell')^{k(\alpha)} }{ n^{k(\beta)} m^{k(\alpha)} }
: {{\mathfrak{p}}}_\ell(\beta) \bigl( : {{\mathfrak{p}}}_{\ell'}(\alpha) {{\mathcal E}}^{(r+n+m-\ell- \ell')} : \bigr)\! :\varphi_{m,\ell'} \varphi_{n,\ell}.\end{aligned}$$ Similarly, we have $$\begin{gathered}
\label{equation2_epsr_check}
\Big[ {{\mathfrak{p}}}_n(\beta) , \big[ {{\mathfrak{p}}}_m(\alpha) , {{\mathcal E}}^{(r)} \big] \Big]
= \frac{(-n)^{k(\alpha) + 1}}{m^{k(\alpha)}} \langle \alpha, \beta \rangle {{\mathcal E}}^{(r+n+m)} \varphi_{m,-n} \\
+ \sum_{\ell, \ell' \in {{\mathbb{Z}}}} \frac{ \ell^{k(\beta)} (\ell')^{k(\alpha)} }{ n^{k(\beta)} m^{k(\alpha)} }
: {{\mathfrak{p}}}_{\ell'}(\alpha) \bigl( : {{\mathfrak{p}}}_{\ell}(\beta) {{\mathcal E}}^{(r+n+m-\ell- \ell')} : \bigr)\! :\varphi_{m,\ell'} \varphi_{n,\ell}.\end{gathered}$$ Since for all $\ell, \ell' \in {{\mathbb{Z}}}$ we have $$: {{\mathfrak{p}}}_\ell(\beta) \bigl( : {{\mathfrak{p}}}_{\ell'}(\alpha) {{\mathcal E}}^{(r+n+m-\ell- \ell')} : \bigr)\! :
\ \, =\ \,
: {{\mathfrak{p}}}_{\ell'}(\alpha) \bigl( : {{\mathfrak{p}}}_{\ell}(\beta) {{\mathcal E}}^{(r+n+m-\ell- \ell')} : \bigr)\! :$$ the second terms in and agree. Hence, equals $$\langle \alpha, \beta \rangle {{\mathcal E}}^{(r+m+n)}
\bigg\{ \frac{(-m)^{k(\beta) + 1}}{n^{k(\beta)}} \varphi_{n,-m}
- \frac{(-n)^{k(\alpha) + 1}}{m^{k(\alpha)}} \varphi_{m,-n} \bigg\}
\label{hovsmdovsv}$$ If $\langle \alpha, \beta \rangle = 0$ we are done, hence we may assume otherwise. Then, for degree reasons, $k(\alpha) = - k(\beta)$. Using the symmetries , we find $$\varphi_{m,-n} = - \frac{m}{n} \varphi_{-n,m} = \frac{m}{n} \varphi_{n,-m}$$ Inserting both equations into , this yields $$\langle \alpha, \beta \rangle {{\mathcal E}}^{(r+m+n)} \varphi_{n,-m} \bigg\{
- \frac{m^{-k(\alpha) + 1}}{n^{-k(\alpha)}}
\ +\ \frac{n^{k(\alpha) + 1}}{m^{k(\alpha)}} \cdot \frac{m}{n}
\bigg\} = 0. \qedhere$$
Conjectures {#Main_conjecture_section2}
-----------
Let $G(y,q)$ be the formal expansion in the variables $y,q$ of the function $G(z,\tau)$ which already appeared in Section \[Section\_More\_evaluations\_Introduction\], $$\begin{aligned}
G(y,q) & = F(y,q)^2 \left( y \frac{d}{dy} \right)^2 \log( F(y,q) ) \\
& = F(y,q)^2 \cdot \bigg\{ \frac{y}{(1+y)^2} - \sum_{d \geq 1} \sum_{m | d} m \big( (-y)^{-m} + (-y)^m \big) q^d \bigg\}.\end{aligned}$$
[**Conjecture A.**]{}
*There exist unique series $\varphi_{m,\ell}$ for $(m,\ell) \in {{\mathbb{Z}}}^2 \setminus \{ 0\}$ such that the following hold:*
1. the symmetries are satisfied,
2. the initial conditions $$\varphi_{1,1} = G(y,q) - 1, \quad \varphi_{1,0} = -i \cdot F(y,q), \quad \varphi_{1,-1} = - \frac{1}{2} \, q \frac{d}{dq}\big( F(y,q)^2 \big) \,,$$ hold, where $i=\sqrt{-1}$ is the imaginary number,
3. Let ${{\mathcal E}}^{(r)}, r \in {{\mathbb{Z}}}$ be the operators defined by the functions $\varphi_{m,\ell}$. Then, ${{\mathcal E}}^{(0)}$ satisfies after restriction to ${{\mathcal F}}(S)$ the WDVV equations $$\label{WDVV_eqn_for_CEr}
\begin{aligned}
{{\mathfrak{p}}}_0(\gamma)\, [ {{\mathcal E}}^{(0)}, L_0(\gamma') ] & = {{\mathfrak{p}}}_0(\gamma')\, [ {{\mathcal E}}^{(0)}, L_0(\gamma) ] \\
{{\mathfrak{p}}}_0(\gamma)\, [ {{\mathcal E}}^{(0)}, \partial ] & = y \frac{d}{dy}\, [ {{\mathcal E}}^{(0)}, L_0(\gamma) ]
\end{aligned}$$ for all $\gamma, \gamma' \in H^2(S,{{\mathbb{Q}}})$.
Conjecture A is a purely algebraic, non-degeneracy statement for the WDVV equations . It has been checked numerically on ${{\mathcal F}}_d(S)$ for all $d \leq 5$. The first values of the series $\varphi_{m,\ell}$ are given in Appendix \[Appendix\_Numerical\_Values\]. For the remainder of Section \[Section\_Quantum\_Cohomology\], we *assume* conjecture A to be true, and we let ${{\mathcal E}}^{(r)}$ denote the operators defined by the (hence unique) functions $\varphi_{m,\ell}$ satisfying (i)-(iii) above. Since Conjecture A has been shown to be true for ${{\mathcal F}}_d(S)$ for all $d \leq 5$, the restriction of ${{\mathcal E}}^{(0)}$ to the subspace $\oplus_{d \leq 5} {{\mathcal F}}_d(S)$ is well-defined unconditionally.
The following conjecture relates ${{\mathcal E}}^{(0)}$ to the quantum operator ${{\mathcal E}}^{\operatorname{\mathsf{Hilb}}}$. Let $L_0$ be the energy operator . Define the operator $$G(y,q)^{L_0} : {{\mathcal F}}(S) \otimes {{\mathbb{Q}}}((y))((q)) {{\ \longrightarrow\ }}{{\mathcal F}}(S) \otimes {{\mathbb{Q}}}((y))((q))$$ by the assignment $$G(y,q)^{L_0}( \mu ) = G(y,q)^{|\mu|} \cdot \mu$$ for any homogeneous $\mu \in {{\mathcal F}}(S)$.
[**Conjecture B.**]{} After restriction to ${{\mathcal F}}(S)$, $${{\mathcal E}}^{\operatorname{\mathsf{Hilb}}} \ = \ {{\mathcal E}}^{(0)} - \frac{1}{F(y,q)^2 \Delta(q)} G(y,q)^{L_0} \, . \label{MMMDCDVD}$$
Combining Conjectures A and B we obtain an algorithmic procedure to determine the 2-point quantum bracket $\langle \cdot , \cdot \rangle_q$. The equality of Conjecture B is conjectured to hold only after restriction to ${{\mathcal F}}(S)$. The extension of to ${{\mathcal F}}(S) \otimes {{\mathbb{Q}}}((y))((q))$ is clearly false: The operators ${{\mathcal E}}^{\operatorname{\mathsf{Hilb}}}$ and $G^{L_0}/(F^2 \Delta)$ are $q$-linear by definition, but ${{\mathcal E}}^{(0)}$ is not.
Let ${\mathop{\rm QJac}\nolimits}$ be the ring of holomorphic quasi-Jacoi forms defined in Appendix \[Appendix\_Quasi\_Jacobi\_Forms\], and let $${\mathop{\rm QJac}\nolimits}= \bigoplus_{m \geq 0} \bigoplus_{k \geq -2m} {\mathop{\rm QJac}\nolimits}_{k,m}$$ be the natural bigrading of ${\mathop{\rm QJac}\nolimits}$ by index $m$ and weight $k$, where $m$ runs over all non-negative half-integers $\frac{1}{2} {{\mathbb{Z}}}^{\geq 0}$.
[**Conjecture C.**]{} [*For every $(m,\ell) \in {{\mathbb{Z}}}^2 \setminus \{ 0 \}$, the series $$\varphi_{m,\ell} + {\rm sgn}(m) \delta_{m \ell}$$ is a quasi-Jacobi form of index $\frac{1}{2} (|m|+|\ell| )$ and weight $-\delta_{0\ell}$.*]{}
Define a new degree functions $\underline{\deg}(\gamma)$ on $H^{\ast}(S)$ by the assignment
- $\gamma \in {{\mathbb{Q}}}F \mapsto \underline{\deg}(\gamma) = -1$
- $\gamma \in {{\mathbb{Q}}}(B+F) \mapsto \underline{\deg}(\gamma) = 1$
- $\gamma \in \{ F,\, B+F \}^{\perp} \mapsto \underline{\deg}(\gamma) = 0$,
where the orthogonal complement $\{ F,\, B+F \}^{\perp}$ is defined with respect to the inner product $\langle \cdot , \cdot \rangle$.
\[oiejgojeogerg\] Assume Conjectures A and C hold. Let $\gamma_i, \widetilde{\gamma}_i \in H^{\ast}(S)$ be $\underline{{\rm deg}}$-homogeneous classes, and let $$\label{munu_coh_classes}
\mu = \prod_{i} {{\mathfrak{p}}}_{-m_i}(\gamma_i) 1_S, \quad \quad \nu = \prod_{j} {{\mathfrak{p}}}_{-n_j}(\widetilde{\gamma}_j) 1_S$$ be cohomology classes of $\operatorname{\mathsf{Hilb}}^m(S)$ and $\operatorname{\mathsf{Hilb}}^n(S)$ respectively. Then $$\Big\langle \mu\ \Big|\ {{\mathcal E}}^{(n-m)} \nu \Big\rangle = \frac{\Phi}{F(y,q)^2 \Delta(q)}$$ for a quasi-Jacobi form $\Phi \in {\mathop{\rm QJac}\nolimits}$ of index $\frac{1}{2} (|m|+|n|)$ and weight $$\sum_{i} \underline{\deg}(\gamma_i) + \sum_j \underline{\deg}(\gamma'_j).$$
By a straight-forward induction on $|\mu| + |\nu|$.
Let $\mu, \nu \in H^{\ast}(\operatorname{\mathsf{Hilb}}^d(S))$. By Lemma and Conjecture B, we have $$\langle \mu, \nu \rangle_q = \frac{\varphi(y,q)}{F(y,q)^2 \Delta(q)} \label{xxxzzz}$$ for a quasi-Jacobi form $\varphi(y,q)$. Since $F(y,q)$ has a simple zero at $z=0$, we expect the function to have a pole of order $2$ at $z=0$. Numerical experiments (Conjecture J) or deformation invariance[^13] suggest that the series $\langle \mu, \nu \rangle_q$ is nonetheless holomorphic at $z=0$. Combining everything, we obtain the following prediction.
\[index\_weight\_lemma\_jacforms\] Assume Conjectures A, B, C, J hold. Let $\mu, \nu \in H^{\ast}(\operatorname{\mathsf{Hilb}}^d(S))$ be cohomology classes of the form . Then, $$\big\langle \mu , \nu \big\rangle^{\operatorname{\mathsf{Hilb}}^d(S)}_q = \frac{\Phi(y,q)}{\Delta(q)}$$ for a quasi-Jacobi form $\Phi(y,q)$ of index $d-1$ and weight $$2 + \sum_{i} \underline{\deg}(\gamma_i) + \sum_j \underline{\deg}(\gamma'_j).$$
Examples {#Main_conjecture_examples_section}
--------
### The higher-dimensional Yau-Zaslow formula {#Examples_higher_dim_YZ}
**(i)** Let $F$ be the fiber of the elliptic fibration $\pi : S \to {\mathbb{P}}^1$. Then [$$\begin{aligned}
& {\Big\langle}{{\mathfrak{p}}}_{-1}(F)^d 1_S\ \Big|\ \Big( {{\mathcal E}}^{(0)} - \frac{1}{F^2 \Delta} G^{L_0} \Big) {{\mathfrak{p}}}_{-1}(F)^d 1_S {\Big\rangle}\\[4pt]
= & {\Big\langle}{{\mathfrak{p}}}_{-1}(F)^d 1_S\ \Big|\ {{\mathcal E}}^{(0)} {{\mathfrak{p}}}_{-1}(F)^d 1_S {\Big\rangle}\\
=\ & (-1)^d {\Big\langle}1_S\ \Big|\ {{\mathfrak{p}}}_{1}(F)^d {{\mathcal E}}^{(0)} {{\mathfrak{p}}}_{-1}(F)^d 1_S {\Big\rangle}\\
=\ & (-1)^d {\Big\langle}1_S\ \Big|\ {{\mathfrak{p}}}_0(F)^d {{\mathcal E}}^{(d)} \varphi_{1,0}^d {{\mathfrak{p}}}_{-1}(F)^d 1_S {\Big\rangle}\\
=\ & (-1)^d {\Big\langle}1_S\ \Big|\ {{\mathfrak{p}}}_0(F)^{2d} {{\mathcal E}}^{(0)} (-1)^d \varphi_{1,0}^{d} \varphi_{-1,0}^d 1_S {\Big\rangle}\\
=\ & \frac{\varphi_{1,0}^{d} \varphi_{-1,0}^d}{F(y,q)^2 \Delta(q)} \\
=\ & \frac{F(y,q)^{2d-2}}{\Delta(q)}\end{aligned}$$ shows Conjecture B to be in agreement with Theorem \[MThm0\]; here we have used ${{\mathfrak{p}}}_0(F) = 1$ above. ]{}
**(ii)** Let $B$ be the class of the section of $\pi : S \to {\mathbb{P}}^1$ and consider the class $$W = B + F.$$ We have $\langle W, W \rangle = 0$ and $\langle W, \beta_h \rangle = h-1$. Hence, ${{\mathfrak{p}}}_0(W)$ acts as $q \frac{d}{dq}$ on functions in $q$. We have [$$\begin{aligned}
& {\Big\langle}{{\mathfrak{p}}}_{-1}(W)^d 1_S\ \Big|\ \Bigl( {{\mathcal E}}^{(0)} - \frac{1}{F^2 \Delta} G^{L_0} \Bigr) {{\mathfrak{p}}}_{-1}(W)^d 1_S {\Big\rangle}\\[4pt]
=\ & {\Big\langle}{{\mathfrak{p}}}_{-1}(W)^d 1_S\ \Big|\ {{\mathcal E}}^{(0)} {{\mathfrak{p}}}_{-1}(W)^d 1_S {\Big\rangle}\\
=\ & (-1)^d {\Big\langle}1_S\ \Big|\ {{\mathfrak{p}}}_0(W)^d {{\mathcal E}}^{(d)} \varphi_{1,0}^d {{\mathfrak{p}}}_{-1}(W)^d 1_S {\Big\rangle}\\
=\ & {\Big\langle}1_S\ \Big|\ {{\mathfrak{p}}}_0(W)^{2d} {{\mathcal E}}^{(0)} \varphi_{1,0}^{d} \varphi_{-1,0}^d 1_S {\Big\rangle}\\
=\ & \left( q \frac{d}{dq} \right)^{2d} \left( \frac{\varphi_{1,0}^{d} \varphi_{-1,0}^d}{F(y,q)^2 \Delta(q)} \right)\\
=\ & \left( q \frac{d}{dq} \right)^{2d} \left( \frac{F(y,q)^{2d-2}}{\Delta(q)} \right).\end{aligned}$$ ]{}
### Further Gromov-Witten invariants {#Section_Examples_More_evaluations}
**(i)** Let ${{\omega}}\in H^4(S;{{\mathbb{Z}}})$ be the class of a point. For $d \geq 1$, let $$C(F) = {{\mathfrak{p}}}_{-1}(F) {{\mathfrak{p}}}_{-1}({{\omega}})^{d-1} 1_S \in H_2(\operatorname{\mathsf{Hilb}}^2(S),{{\mathbb{Z}}})$$ and $$D(F) = {{\mathfrak{p}}}_{-1}(F) {{\mathfrak{p}}}_{-1}(e)^{d-1} 1_S \in H^2(\operatorname{\mathsf{Hilb}}^2(S),{{\mathbb{Z}}}) \,.$$ Then, [$$\begin{aligned}
& {\Big\langle}C(F) \ \Big|\ \bigl( {{\mathcal E}}^{(0)} - \frac{1}{F^2 \Delta} G^{L_0} \bigr) D(F) {\Big\rangle}\\[4pt]
& = \frac{1}{(d-1)!} {\Big\langle}{{\mathfrak{p}}}_{-1}(F) {{\mathfrak{p}}}_{-1}({{\omega}})^{d-1} 1_S \ \Big|\ {{\mathcal E}}^{(0)} {{\mathfrak{p}}}_{-1}(F) {{\mathfrak{p}}}_{-1}(e)^{d-1} 1_S {\Big\rangle}\\
& = \frac{1}{(d-1)!} {\Big\langle}{{\mathfrak{p}}}_{-1}({{\omega}})^{d-1} 1_S \ \Big|\ {{\mathcal E}}^{(0)} \varphi_{1,0} \varphi_{-1,0} {{\mathfrak{p}}}_{-1}(e)^{d-1} 1_S {\Big\rangle}\\
& = \frac{(-1)^{d-1}}{(d-1)!} {\Big\langle}1_S \ \Big|\ {{\mathcal E}}^{(0)} \varphi_{1,0} \varphi_{-1,0} (\varphi_{1,1} + 1)^{d-1} {{\mathfrak{p}}}_{1}({{\omega}})^{d-1} {{\mathfrak{p}}}_{-1}(e)^{d-1} 1_S {\Big\rangle}\\
& = \frac{ \varphi_{1,0} \varphi_{-1,0} (\varphi_{1,1} + 1)^{d-1} }{F(y,q)^2 \Delta(q)} \\
& = \frac{ G(y,q)^{d-1} }{\Delta(q)}.\end{aligned}$$ By the divisor equation and $\langle D(F) , \beta_h + kA \rangle = 1$ for all $h,k$, Conjecture B is in full agreement with Theorem \[ellthm2\] equation 1. ]{}
**(ii)** Let $A = {{\mathfrak{p}}}_{-2}(\omega) {{\mathfrak{p}}}_{-1}(\omega)^{d-2} 1_S$ be the class of an exceptional curve. Then, $$\begin{aligned}
& {\Big\langle}A\ \Big|\ \bigl( {{\mathcal E}}^{(0)} - \frac{1}{F^2 \Delta} G^{L_0} \bigr) D(F) {\Big\rangle}\\[4pt]
& = \frac{(-1)^d}{(d-1)!} {\Big\langle}1_S \ \Big|\ {{\mathfrak{p}}}_{2}(\omega) {{\mathcal E}}^{(0)} {{\mathfrak{p}}}_1(\omega)^{d-2} {{\mathfrak{p}}}_{-1}(F) {{\mathfrak{p}}}_{-1}(e)^{d-1} (\varphi_{1,1} + 1)^{d-2} 1_S {\Big\rangle}\\
& = \frac{(-1)^d}{(d-1)!} {\Big\langle}1_S \ \Big|\ \frac{1}{2} {{\mathcal E}}^{(1)} {{\mathfrak{p}}}_1(\omega)^{d-1} {{\mathfrak{p}}}_{-1}(F) {{\mathfrak{p}}}_{-1}(e)^{d-1} \varphi_{2,1} (\varphi_{1,1} + 1)^{d-2} 1_S {\Big\rangle}\\
& = - \frac{1}{2} {\Big\langle}1_S \ \Big|\ {{\mathcal E}}^{(1)} {{\mathfrak{p}}}_{-1}(F) \varphi_{2,1} (\varphi_{1,1} + 1)^{d-2} {\Big\rangle}\\
& = - \frac{1}{2} \frac{ (-\varphi_{-1,0}) \varphi_{2,1} (\varphi_{1,1} + 1)^{d-2}}{F^2(y,q) \Delta} \\
& = - \frac{1}{2} \frac{ \Big( y \frac{d}{dy} G \Big) \cdot G^{d-2} }{\Delta}.\end{aligned}$$ Hence, again, Conjecture B is in full agreement with Theorem \[ellthm2\] equation 2.
**(iii)** For a point $P \in S$, the incidence subscheme $$I(P) = \{ \xi \in \operatorname{\mathsf{Hilb}}^2(S) \ |\ P \in \xi \}$$ has class $[I(P)] = {{\mathfrak{p}}}_{-1}(\omega) {{\mathfrak{p}}}_{-1}(e) 1_S$. We calculate $$\begin{aligned}
& {\Big\langle}I(P)\ \Big|\ \bigl( {{\mathcal E}}^{(0)} - \frac{1}{F^2 \Delta} G^{L_0} \bigr) I(P) {\Big\rangle}\\[4pt]
& = - {\Big\langle}{{\mathfrak{p}}}_{-1}(e) 1_S \ \Big|\ {{\mathfrak{p}}}_1(\omega) {{\mathcal E}}^{(0)} I(P) {\Big\rangle}- \frac{G^2}{F^2 \Delta} \\
& = - {\Big\langle}{{\mathfrak{p}}}_{-1}(e) 1_S \ \Big|\ \Big( {{\mathcal E}}^{(0)} {{\mathfrak{p}}}_{1}(\omega) (\varphi_{1,1} + 1) - {{\mathfrak{p}}}_{-1}(\omega) {{\mathcal E}}^{(2)} \varphi_{1,-1} \Big) I(P) {\Big\rangle}- \frac{G^2}{F^2 \Delta} \\
& = {\Big\langle}{{\mathfrak{p}}}_{-1}(e) 1_S\ \Big|\ {{\mathcal E}}^{(0)} {{\mathfrak{p}}}_{-1}(\omega) (\varphi_{1,1} + 1) 1_S {\Big\rangle}\\
& \quad \quad \quad + {\Big\langle}1_S\ \Big|\ {{\mathcal E}}^{(2)} {{\mathfrak{p}}}_{-1}(\omega) {{\mathfrak{p}}}_{-1}(e) \varphi_{1,-1} 1_S {\Big\rangle}- \frac{G^2}{F^2 \Delta} \\
& = \frac{ (\varphi_{1,1} + 1)^2 }{ F^2 \Delta } + \frac{- \varphi_{-1,1} \varphi_{1,-1} }{ F^2 \Delta} - \frac{G^2}{F^2 \Delta} \\
& = \frac{ \Big( q \frac{d}{dq} F \Big)^2 }{\Delta(q)} .
$$ Hence, Conjecture B agrees with Theorem \[ellthm2\] equation 3, case $d=2$.
**(iv)** For a point $P \in S$, we have $$\begin{aligned}
& {\Big\langle}{{\mathfrak{p}}}_{-1}(F)^2 1_S \ \Big|\ \bigl( {{\mathcal E}}^{(0)} - \frac{1}{F^2 \Delta} G^{L_0} \bigr) I(P) {\Big\rangle}\\[4pt]
& = - {\Big\langle}1_S \ \Big|\ {{\mathcal E}}^{(2)} \varphi_{1,0}^2 I(P) {\Big\rangle}\\
& = \frac{- \varphi_{1,0}^2 \varphi_{-1,1}}{F^2 \Delta} \\
& = \frac{F(y,q) \cdot q \frac{d}{dq} F(y,q) }{ \Delta(\tau) } .\end{aligned}$$ Hence, Conjecture B is in agreement with Theorem \[extra\_eval\].
### The Hilbert scheme of $2$ points {#Section_Conj_in_Hilb2}
We check conjectures A, B, C, J in the case $\operatorname{\mathsf{Hilb}}^2(S)$. Conjecture A is seen to hold for $\operatorname{\mathsf{Hilb}}^2(S)$ by direct calculation. The corresponding functions $\varphi_{m,\ell}$ are given in Appendix \[Appendix\_Numerical\_Values\]. This implies Conjecture C by inspection. Conjecture B and J hold by the following result.
\[ConjC\_for\_Hilb2\] \[Theorem\_for\_Hilb2\] For all $\mu, \nu \in H^{\ast}(\operatorname{\mathsf{Hilb}}^2(S))$, $$\langle \mu, \nu \rangle_q = \Big\langle\, \mu\ \Big|\ \Big( {{\mathcal E}}^{(0)} - \frac{G^{L_0}}{F^2 \Delta} \Big)\, \nu\, \Big\rangle.$$
\[ConjJ\_for\_Hilb2\] Let $\mu, \nu \in H^{\ast}(\operatorname{\mathsf{Hilb}}^2(S)$ be cohomology classes of the form . Then, $$\big\langle \mu , \nu \big\rangle_q = \frac{\Phi}{\Delta(q)}$$ for a quasi-Jacobi form $\Phi$ of index $1$ and weight $$2 + \sum_{i} \underline{\deg}(\gamma_i) + \sum_j \underline{\deg}(\gamma'_j).$$
By Sections \[Examples\_higher\_dim\_YZ\] and \[Section\_Examples\_More\_evaluations\] above, Theorem \[ConjC\_for\_Hilb2\] holds in the cases considered in Theorems \[ellthm\], \[ellthm2\] and \[extra\_eval\] respectively. Applying the WDVV equation successively to these base cases, one evaluates the bracket $\langle \mu, \nu \rangle_q$ for all $\mu, \nu \in H^{\ast}(\operatorname{\mathsf{Hilb}}^2(S))$ in finitely many steps. Since for $\operatorname{\mathsf{Hilb}}^2(S)$ the WDVV equation also holds for ${{\mathcal E}}^{(0)} - G^{L_0}/(F^2 \Delta)$, this implies Theorem \[ConjC\_for\_Hilb2\]. Finally, Theorem \[ConjJ\_for\_Hilb2\] follows now from direct inspection.
By degenerating $(E,0)$ to the nodal elliptic curve and using the divisor equation we may rewrite $\mathsf{H}_d(y,q)$ as $$\label{aaasd}
\mathsf{H}_d(y,q) =
\sum_{k \in {{\mathbb{Z}}}} \sum_{h\geq 0}
y^k q^{h-1}
\int_{[ \overline{M}_{0,2}(\operatorname{\mathsf{Hilb}}^d(S), \beta_h + kA) ]^{\text{red}}}
({\mathop{\rm ev}\nolimits}_1 \times {\mathop{\rm ev}\nolimits}_2)^{\ast} [\Delta^{[d]}]$$ where $[\Delta^{[d]}] \in H^{2d}( \operatorname{\mathsf{Hilb}}^d(S) \times \operatorname{\mathsf{Hilb}}^d(S))$ is the diagonal class. Proposition now follows from calculating the right hand side of using Theorem \[Theorem\_for\_Hilb2\].
### The ${{\mathcal A}}_1$ resolution.
Let $[q^{-1}]$ be the operator that extracts the $q^{-1}$ coefficient, and let $${{\mathcal E}}_B^{\operatorname{\mathsf{Hilb}}} = [q^{-1}] {{\mathcal E}}^{\operatorname{\mathsf{Hilb}}}$$ be the restriction of ${{\mathcal E}}^{\operatorname{\mathsf{Hilb}}}$ to the case of the section class $B$. The corresponding local case was considered before in [@MO; @MO2].
Define operators ${{\mathcal E}}_{B}^{(r)}$ by the relations
- $\big\langle 1_S\ \big|\ {{\mathcal E}}_{B}^{(r)} 1_S \big\rangle = \frac{y}{(1+y)^2} \delta_{0r}$
- $\big[ {{\mathfrak{p}}}_{m}(\gamma), {{\mathcal E}}_B^{(r)} \big] = \langle \gamma, B \rangle \left( (-y)^{-m/2} - (-y)^{m/2} \right) {{\mathcal E}}_B^{(r+m)}$
for all $m \neq 0$ and all $\gamma \in H^{\ast}(S)$, see [@MO2 Section 5.1]. Translating the results of [@MO; @MO2] to the $K3$ surface leads to the following evaluation.
\[8765\] After restriction to ${{\mathcal F}}(S)$, $${{\mathcal E}}_B^{\operatorname{\mathsf{Hilb}}} + \frac{y}{(1+y)^2} {\rm id_{{{\mathcal F}}(S)}} = {{\mathcal E}}_{B}^{(0)} \,.$$
By the numerical values of Appendix \[Appendix\_Numerical\_Values\], we expect the expansions $$\begin{aligned}
{2}
\varphi_{m,0} & = \big( (-y)^{-m/2} - (-y)^{m/2} \big) + O(q) \quad \quad && \text{ for all } m \neq 0 \\
\varphi_{m,\ell} & = O(q) && \text{ for all } m \in {{\mathbb{Z}}}, \ell \neq 0 \,.\end{aligned}$$ Because of $$[q^{-1}] \frac{G^{L_0}}{F^2 \Delta} = \frac{y}{(1+y)^2} \text{id}_{{{\mathcal F}}(S)}\, ,$$ we find conjectures A and B in agreement with Theorem \[8765\].
The reduced WDVV equation {#section_reducedWDVV}
=========================
Let ${{\overline M}}_{0,4}$ be the moduli space of stable genus $0$ curves with $4$ marked points. The boundary of ${{\overline M}}_{0,4}$ is the union of the divisors $$D(12|34),\ D(14|23),\ D(13|24) \label{530}$$ corresponding to a broken curve with the respective prescribed splitting of the marked points. Since ${{\overline M}}_{0,4}$ is isomorphic to ${\mathbb{P}}^1$, any two of the divisors are rationally equivalent.
Let $Y$ be a smooth projective variety and let ${{\overline M}}_{0,n}(Y,\beta)$ be the moduli space of stable maps to $Y$ of genus $0$ and class $\beta$. Let $$\pi : {{\overline M}}_{0,n}(Y,\beta) \to {{\overline M}}_{0,4}$$ be the map that forgets all but the last four points. The pullback of the boundary divisors under $\pi$ defines rationally equivalent divisors on ${{\overline M}}_{0,n}(Y,\beta)$. The intersection of these divisors with curve classes obtained from the virtual class yields relations among Gromov-Witten invariants of $Y$, the WDVV equations [@FP]. We derive the precise form of these equations for reduced Gromov-Witten theory. For simplicity, we restrict to the case $n = 4$.
Let $Y$ be a holomorphic symplectic variety and let $${\Big\langle}\gamma_1, \ldots, \gamma_n {\Big\rangle}_\beta^{\text{red}} = \int_{[ {{\overline M}}_{0,n}(Y,\beta) ]^{\text{red}} } {\mathop{\rm ev}\nolimits}_1^{\ast}(\gamma_1) \cup \cdots \cup {\mathop{\rm ev}\nolimits}_n^{\ast}(\gamma_n)$$ denote the *reduced* Gromov-Witten invariants of $Y$ of genus $0$ and class $\beta \in H_2(Y;{{\mathbb{Z}}})$ with primary insertions $\gamma_1, \cdots, \gamma_m \in H^{\ast}(Y)$.
\[WDVVprop\] Let $\gamma_1, \dots, \gamma_4 \in H^{2 \ast}(Y;{{\mathbb{Q}}})$ be cohomology classes with $$\sum_i \deg(\gamma_i) = \mathop{\rm vdim}\nolimits {{\overline M}}_{0,4}(Y,\beta) - 1 = \dim Y + 1,$$ where $\deg(\gamma_i)$ denotes the complex degree of $\gamma_i$. Then, $${\Big\langle}\gamma_1, \gamma_2, \gamma_3 \cup \gamma_4 {\Big\rangle}^{\text{red}}_{\beta} + {\Big\langle}\gamma_1 \cup \gamma_2, \gamma_3, \gamma_4 {\Big\rangle}^{\text{red}}_{\beta}
=
{\Big\langle}\gamma_1, \gamma_4, \gamma_2 \cup \gamma_3 {\Big\rangle}^{\text{red}}_{\beta} + {\Big\langle}\gamma_1 \cup \gamma_4, \gamma_2, \gamma_3 {\Big\rangle}^{\text{red}}_{\beta}.$$
Consider the fiber of $\pi$ over $D(12|34)$, $$D = \pi^{-1}( D(12|34) ) \,.$$ The intersection of $D$ with the class $$\left( \prod_{i = 1}^{4} {\mathop{\rm ev}\nolimits}_i^{\ast}(\gamma_i) \right) \cap [{{\overline M}}_{0,4}(Y,\beta)]^{\text{red}}. \label{wdvv_kcmvknvcv}$$ splits into a sum of integrals over the product $$M' = {{\overline M}}_{0,3}(Y,\beta_1) \times {{\overline M}}_{0,3}(Y, \beta_2),$$ for all effective decompositions $\beta = \beta_1 + \beta_2$.
The reduced virtual class $[{{\overline M}}_{0,4}(Y,\beta)]^{\text{red}}$ restricts to $M'$ as the sum of $$({\mathop{\rm ev}\nolimits}_3 \times {\mathop{\rm ev}\nolimits}_3)^{\ast} \Delta_Y \cap [{{\overline M}}_{0,3}(Y, \beta_1)]^{\text{red}} \times [{{\overline M}}_{0,3}(Y,\beta_2)]^{\text{ord}}$$ with the same term, except for ’red’ and ’red’ interchanged; here $$\Delta_Y \in H^{2 \dim Y}(Y \times Y;{{\mathbb{Z}}})$$ is the class of the diagonal and $[\ \cdot \ ]^{\text{ord}}$ denotes the ordinary virtual class.
Since $[{{\overline M}}_{0,3}(Y,\beta)]^{\text{ord}} = 0$ unless $\beta = 0$, we find $$\begin{aligned}
\notag \int_{[ {{\overline M}}_{0,4}(Y,\beta) ]^{\text{red}} } D \cup \prod_{i} \gamma_i & =
\sum_{e,f} {\Big\langle}\gamma_1, \gamma_2, T_e {\Big\rangle}^{\text{red}}_{\beta} g^{ef} {\Big\langle}\gamma_3, \gamma_4, T_f {\Big\rangle}^{\text{ord}}_{0} + \\
\notag & \quad \quad + {\Big\langle}\gamma_1, \gamma_2, T_e {\Big\rangle}^{\text{ord}}_{0} g^{ef} {\Big\langle}\gamma_3, \gamma_4, T_f {\Big\rangle}^{\text{red}}_{\beta} \\
& = {\Big\langle}\gamma_1, \gamma_2, \gamma_3 \cup \gamma_4 {\Big\rangle}^{\text{red}}_{\beta} + {\Big\langle}\gamma_1 \cup \gamma_2, \gamma_3, \gamma_4 {\Big\rangle}^{\text{red}}_{\beta},
\label{ifijvofj}\end{aligned}$$ where $\{ T_e \}_e$ is a basis of $H^{\ast}(Y;{{\mathbb{Z}}})$ and $(g^{ef})_{e,f}$ is the inverse of the intersection matrix $g_{ef} =\int_Y T_e\cup T_f$.
After comparing with the integral of over the pullback of $D(14|23)$, the proof of Proposition \[WDVVprop\] is complete.
We may use the previous proposition to define reduced quantum cohomology. Let ${\hbar}$ be a formal parameter with ${\hbar}^2 = 0$. Let ${\rm Eff}_{Y}$ be the cone of effective curve class on $Y$, and for any $\beta \in {\rm Eff}_{Y}$ let $q^{\beta}$ be the corresponding element in the semi-group algebra ${{\mathbb{Q}}}[ {\rm Eff}_{Y} ]$. Define the *reduced* quantum product $\ast$ on $$H^{\ast}(Y;{{\mathbb{Q}}}) \otimes {{\mathbb{Q}}}[[ {\rm Eff}_{Y} ]] \otimes {{\mathbb{Q}}}[{\hbar}]/{\hbar}^2 \,.$$ by $$\big\langle \gamma_1 \ast \gamma_2, \gamma_3 \big\rangle
= \big\langle \gamma_1 \cup \gamma_2, \gamma_3 \big\rangle +
{\hbar}\sum_{\beta > 0} q^{\beta} {\Big\langle}\gamma_1, \gamma_2, \gamma_3 {\Big\rangle}_\beta^{\text{red}}$$ for all $a,b,c \in H^{\ast}(Y)$, where $\langle \gamma_1, \gamma_2 \rangle = \int_Y \gamma_1 \cup \gamma_2$ is the standard inner product on $H^{\ast}(Y;{{\mathbb{Q}}})$ and $\beta$ runs over all non-zero effective curve classes of $Y$. Then, Proposition \[WDVVprop\] implies that $\ast$ is associative.
Quasi-Jacobi forms {#Appendix_Quasi_Jacobi_Forms}
==================
### Definition {#definition-1}
Let $(z,\tau) \in {{\mathbb{C}}}\times {{\mathbb{H}}}$, and let $y = - p = - e^{2 \pi i z}$ and $q = e^{2 \pi i \tau}$. For all expansions below, we will work in the region $|y| < 1$.
Consider the Jacobi theta functions $$F(z,\tau) = \frac{\vartheta_1(z,\tau)}{\eta^3(\tau)} = (y^{1/2} + y^{-1/2}) \prod_{m \geq 1} \frac{ (1 + yq^m) (1 + y^{-1}q^m)}{ (1-q^m)^2 },$$ the logarithmic derivative $$J_1(z,\tau) = y \frac{d}{dy} \log( F(y,q) ) =
\frac{y}{1 + y} - \frac{1}{2} - \sum_{d \geq 1} \sum_{m|d} \big( (-y)^m - (-y)^{-m} \big) q^{d},$$ the Weierstrass elliptic function $$\wp(z,\tau) = \frac{1}{12} - \frac{y}{(1+y)^2} + \sum_{d \geq 1} \sum_{m|d} m ((-y)^m - 2 + (-y)^{-m}) q^{d},
\label{wp_function_def}$$ the derivative $$\wp^{\bullet}(z,\tau) = y \frac{d}{dy} \, \wp(z,\tau) = \frac{y (y-1)}{(1+y)^3}
+ \sum_{d \geq 1} \sum_{m|d} m^2 ((-y)^m - (-y)^{-m}) q^{d},$$ and for $k \geq 1$ the Eisenstein series $$E_{2k}(\tau) = 1 - \frac{4k}{B_{2k}} \sum_{d \geq 1} \Big( \sum_{m |d} m^{2k-1} \Big) q^d, \label{Eisenstein_function_def}$$ where $B_{2k}$ are the Bernoulli numbers. Define the free polynomial algebra $$\mathsf{V} = {{\mathbb{C}}}\big[ F(z,\tau), E_2(\tau),\, E_4(\tau),\, J_1(z,\tau),\, \wp(z,\tau),\, \wp^{\bullet}(z,\tau) \big] .$$ Define the weight and index of the generators of $\mathsf{V}$ by the following table. Here, we list also their pole order at $z = 0$ for later use.
$F(z,\tau)$ $E_{2k}(\tau)$ $J_1(z,\tau)$ $\wp(z,\tau)$ $\wp^{\bullet}(z,\tau)$
-- ------------- ---------------- --------------- --------------- -------------------------
$0$ $0$ $1$ $2$ $3$
$-1$ $2k$ $1$ $2$ $3$
$1/2$ $0$ $0$ $0$ $0$
: Weight and pole order at $z=0$[]{data-label="weight_pole_order_table"}
The grading on the generators induces a natural bigrading on $\mathsf{V}$ by weight $k$ and index $m$, $$\mathsf{V} = \bigoplus_{m \in (\frac{1}{2} {{\mathbb{Z}}})^{\geq 0}} \bigoplus_{k \in {{\mathbb{Z}}}} \mathsf{V}_{k,m},$$ where $m$ runs over all non-negative half-integers.
In the variable $z$, the functions $$E_{2k}(\tau),\ J_1(z,\tau),\ \wp(z,\tau),\ \wp^{\bullet}(z,\tau) \label{ckvmsvff}$$ can have a pole in the fundamental region $$\big\{ x + y \tau\ \big|\ 0 \leq x,y < 1 \big\} \label{fundamental_region}$$ only at $z = 0$. The function $F(z,\tau)$ has a simple zero at $z = 0$ and no other zeros (or poles) in the fundamental region .
Let $m$ be a non-negative half-integer and let $k \in {{\mathbb{Z}}}$. A function $$f(z,\tau) \in \mathsf{V}_{k,m}$$ which is holomorphic at $z=0$ for generic $\tau$ is called a *quasi-Jacobi form* of weight $k$ and index $m$.
The subring ${\mathop{\rm QJac}\nolimits}\subset \mathsf{V}$ of quasi-Jacobi forms is graded by index $m$ and weight $k$, $${\mathop{\rm QJac}\nolimits}= \bigoplus_{m \geq 0} \bigoplus_{k \geq -2m} {\mathop{\rm QJac}\nolimits}_{k,m}$$ with finite-dimensional summands ${\mathop{\rm QJac}\nolimits}_{k,m}$.
By the classical relation $$\big( \wp^{\bullet}(z) \big)^{2} = 4 \wp(z)^3 - \frac{1}{12} E_4(\tau) \wp(z) + \frac{1}{216} E_6(\tau).$$ we have $E_6(\tau) \in \mathsf{V}$ and therefore $E_6(\tau) \in {\mathop{\rm QJac}\nolimits}$. Hence, ${\mathop{\rm QJac}\nolimits}$ contains the ring of quasi-modular forms ${{\mathbb{C}}}[E_2, E_4, E_6]$. Since the functions $$\varphi_{-2,1} = - F(z,\tau)^2, \quad \varphi_{0,1} = -12 F(z,\tau)^2 \wp(z,\tau),$$ lie both in ${\mathop{\rm QJac}\nolimits}$, it follows from [@EZ Theorem 9.3] that ${\mathop{\rm QJac}\nolimits}$ also contains the ring of weak Jacobi forms.
\[closed\_under\_diff\] The ring ${\mathop{\rm QJac}\nolimits}$ is closed under differentiation by $z$ and $\tau$.
We write $$\partial_{\tau} = \frac{1}{2 \pi i} \frac{\partial}{\partial \tau} = q \frac{d}{dq}
\quad \text{ and } \quad
\partial_z = \frac{1}{2 \pi i} \frac{\partial}{\partial z} = y \frac{d}{dy}$$ for differentiation with respect to $\tau$ and $z$ respectively. The lemma now direct follows from the relations. $$\begin{aligned}
\partial_{\tau}(F) & = F \cdot \left( \frac{1}{2} J_1^{2} - \frac{1}{2} \wp - \frac{1}{12} E_2 \right),
& \partial_z(F) & = J_1 \cdot F, \\
\partial_{\tau}(J_1) & = J_1 \cdot \left( \frac{1}{12} E_2 - \wp \right) - \frac{1}{2} \wp^{\bullet},
& \partial_z(J_1) & = - \wp + \frac{1}{12} E_2, \\
\partial_{\tau}(\wp) & = 2 \wp^{2} + \frac{1}{6} \wp E_2 + J_1 \wp^{\bullet} - \frac{1}{36} E_4,
& \partial_z(\wp) & = \wp^{\bullet}, \\
\partial_{\tau}(\wp^{\bullet})
& = 6 J_1 \wp^{2} - \frac{1}{24} J_1 E_4 + 3 \wp \wp^{\bullet} + \frac{1}{4} E_2 \wp^{\bullet},
\quad \quad & \partial_z(\wp^{\bullet}) & = 6 \wp^2 - \frac{1}{24} E_4 \,. \qedhere
$$
### Numerical values {#Appendix_Numerical_Values}
We present the first values of the functions $\varphi_{m,\ell}$ satisfying the conditions of Conjecture A of Section \[Main\_conjecture\_section2\]. Let $K = iF$, where $i = \sqrt{-1}$. Then, [ ]{}
[HKK[[$^{+}$]{}]{}03]{}
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Departement Mathematik\
ETH Zürich\
[email protected]
[^1]: The domain of a [*stable map*]{} is always taken here to be connected.
[^2]: see [@EZ] for an introduction
[^3]: For functions $f_i : X \to \operatorname{\mathsf{Hilb}}^{d_i}(S),i=1,\dots, r$ with $(f_1, \dots,f_r) : X \to U$ we also use $f_1 + \ldots + f_r = \sigma \circ (f_1, \dots, f_r) : X \to \operatorname{\mathsf{Hilb}}^d(S)$.
[^4]: i.e. the set of ${{\mathbb{C}}}$-valued points of $M_Z$ is a product
[^5]: $\sigma$ is the inverse to the natural isomorphism in the other direction induced by the sequence of surjections $H^1(C,\Omega_C) {{\ \longrightarrow\ }}\oplus H^1(C_i, \Omega_{C_i}) {{\ \longrightarrow\ }}H^1(C,\omega_C) {{\ \longrightarrow\ }}0$.
[^6]: We may restrict here to the Hilbert scheme of $2$ points, since the evaluation of $F^{\textup{GW}}$ is independent of the number of points.
[^7]: In fact, the Yau-Zaslow formula applies to all classes $\beta \in H_2({\operatorname{Km}}({{\mathsf{A}}}), {{\mathbb{Z}}})$ which are of type $(1,1)$ and pair positively with an ample class.
[^8]: The general form of these theta functions is $$\Theta_\mathsf{v}\left[ \begin{array}{cc}A\\B\end{array} \right](z, \tau)
= \sum_{x \in {{\mathbb{Z}}}^4} q^{\frac{1}{2} \langle x+A, x+A \rangle} \exp\Big( 2\pi i \cdot \big\langle x+A, z \cdot \mathsf{v} + B \big\rangle \Big) \,.$$ for characteristics $A,B \in {{\mathbb{Q}}}^4$ and a direction vector $\mathsf{v} \in {{\mathbb{C}}}^4$. Here, $$\Theta(z,\tau) = \Theta_{(-e_1)} \left[ \begin{array}{cc} \alpha/2 \\ -e_1/2 \end{array} \right](z, 2 \tau).$$
[^9]: Although $f(C_0)$ is fixed under infinitesimal deformations, the point $u_j$ in the attachment point $f(C_0 \cap B_j) = u_j + P_1 + \dots + P_{d-1}$ may move to first order, compare Section \[Section\_CaseI\_P\].
[^10]: If $e$ is a $2$-torsion point of $E$, we take the proper transform instead of $\rho^{-1}$ in . This case will not appear below.
[^11]: defined in
[^12]: The definition precisely matches the action of the extended Heisenberg algebra $\big\langle\, {{\mathfrak{p}}}_k(\gamma),\ k \in {{\mathbb{Z}}}\, \big\rangle$ on the full Fock space ${{\mathcal F}}(S) \otimes {{\mathbb{Q}}}[ H^{\ast}(S,{{\mathbb{Q}}}) ]$ under the embedding $q^{h-1} \mapsto q^{B + hF}$, see [@KY section 6.1].
[^13]: See [@thesis] for a discussion of the monodromy action by deformations of $\operatorname{\mathsf{Hilb}}^d(S)$ in the moduli space of irreducible holomorphic-symplectic varieties.
|
DoD News
News Article
Army Works to Accelerate Leader Development
By Donna MilesAmerican Forces Press Service
WASHINGTON, Oct. 10, 2007 The Army is exploring new ways to accelerate the development of leaders prepared for the broad challenges they’ll face in what’s expected to be an era of persistent conflict, the Army’s chief of staff said here yesterday. (Video)
“We are committed to investing in our officer, warrant officer, noncommissioned officer and civilian leaders,” Army Gen. George W. Casey Jr. told attendees at the annual Association of the U.S. Army convention. “In this era of persistent conflict, it is absolutely essential that we develop leaders that can handle the challenges of full-spectrum operations.”
Full-spectrum operations include the broad range of missions soldiers can be called on to carry out: from supporting peacetime operations to conducting major combat operations, and everything in between.
“Our leaders in the 21st century must be competent in their core competencies, broad enough to operate across the full spectrum of conflict, able to operate in joint, interagency and combined environments, at home in other cultures and courageous enough to see and exploit opportunities in the complex environments they will be operating in,” he said.
Just as warfare has changed, so too has the way the Army develops leaders, Army Lt. Gen. William Caldwell IV, commander of the Army’s Combined Arms Center at Fort Leavenworth, Kan., said during a panel discussion about accelerating leader development.
“We don’t want to teach you what to think,” he said. “We want to teach you how to think.”
This effort extends throughout the Army’s education and professional development system, through a blend of formal education, operational experience and guided self-development, Caldwell said.
Officer candidates are getting more field and operational experience, Maj. Gen. W. Montague Winfield, commander of U.S. Army Cadet Command, told attendees. Officers are getting more educational opportunities and more access to joint, interagency, intergovernmental and multinational training, Brig. Gen. Mark O’Neill, deputy commandant of the Army’s Command and General Staff College, told the group.
Warrant officers are attending more officer training courses, said Col. Mark Jones, command of the Army Warrant Officer Career Center. And recognizing that its enlisted force is “taking on more responsibility earlier in their careers than ever before, the Army is adapting its training programs so they’re better prepared, said Col. Donald Gentry, commandant of the Army Sergeants Major Academy.
Meanwhile, the Army is tapping into best practices from the private and public sectors to accelerate training and development of its civilian work force that’s filling critical positions and maintaining continuity, said Volney “Jim” Warner, director of the Army’s Civilian Development Office.
How well the Army develops soldiers and leaders able to operate effectively and efficiently in an era of persistent conflict will have far-reaching impact on the force and its ability to succeed, the officials agreed.
“Soldiers are the strength of this Army, and they make this Army the strength of this nation,” Casey said. “It will be our soldiers who lead us to victory over the nation’s enemies, and it will be soldiers who preserve the peace for us and for our allies.” |
Pembroke House
Pembroke House, located on Whitehall, was the London residence of the earls of Pembroke.
History
It was built by the architect earl Henry Herbert in 1723–24 (under Colen Campbell and latterly his assistant Roger Morris), on ground leased by the earl in 1717 and 1729 amidst the ruins of the parts of Whitehall Palace that burned down in 1698 (and still covered in its rubble). Its design may have inspired the 9th earl's designs for Marble Hill House. The 9th earl died here in 1733, as did his great-grandson the 11th Earl, in 1827.
It was the subject of a major rebuild by the 10th Earl in 1756–59, and in 1762 Lady Hervey wrote that it was "taken for the Duc de Nivernois, the French Ambassador". Gardens were created in 1818 by demolishing the house's riding-house and stables, and the main floor-level terrace (including the portion over the water-gate) was retained. The lease was repeatedly renewed (passing to the Earl of Harrington) until in or around 1853, when the land and house became crown freehold (housing the Ministry of Transport c.1930, and later parts of what would become the Ministry of Defence). It was demolished to build the Ministry of Defence main building in 1938.
Notes
Bibliography
Steven Brindle, 'Pembroke House, Whitehall', in The Georgian Group Journal, vol. VIII, 1998, pp.88-113.
'Pembroke House', Survey of London: volume 13: St Margaret, Westminster, part II: Whitehall I (1930), pp. 167-179.
'Whitehall: Precinct and gardens', Old and New London: Volume 3 (1878), pp. 376-382.
See also
List of demolished buildings and structures in London
External links
Pembroke House in "The Opening of Westminster Bridge" by Constable
THE FIRST PEMBROKE HOUSE. ELEVATION AND PLAN
Category:Former houses in the City of Westminster
Category:National government buildings in London
Category:Georgian architecture in London |
Q:
Value of Internationalization in the iPhone App store?
I have several iOS/iPhone apps that have been continually selling in small amounts in over 2 dozen different countries, even though the app UIs and all the store descriptions are only in English. In a few countries where English is not the official or native language, a few apps are selling far better than is proportionate for those country's population size compared with the U.S.
So why Internationalize apps?
What kind of increase, if any, in sales might a typical app see if it is Internationalized into given local languages? Which major languages might be likely to see the greatest improvement in app sales or downloads due to a localized app description?
A:
It's going to depend on your market.
If your market is a largely technical or is quite niche then translating probably isn't an issue as your users are probably OK with it being in English and in some cases may actually prefer it being in English.
If your market is the general population then translation is probably a good idea as it will increase the reach of your application. In certain countries that protect their native languages (e.g. France) translation would be pretty much essential.
So you need to analyse your current market and where you want to go with your apps to see if internationalisation is actually worth it. Then tackle the languages in market order, but languages like French, Spanish and Portuguese that are either 1st or 2nd languages in a number of countries would seem to be the most obvious choices.
A:
I agree with ChrisF, it really depends on your target market.
In my opinion, if you want the see more revenue based on sales in non-English speaking markets, the best languages to tackle are:
Japanese
Chinese (Traditional)
Chinese (Simplified)
Especially China, as users in this market tend to shy away from products containing English text, (it also may become illegal soon - http://mcaf.ee/7a162).
|
T.C. Memo. 1996-406
UNITED STATES TAX COURT
THWAITES TERRACE HOUSE OWNERS CORP., Petitioner v.
COMMISSIONER OF INTERNAL REVENUE, Respondent
Docket No. 15777-94. Filed September 3, 1996.
Terrence J. Dwyer, for petitioner.1
Andrew J. Mandell, for respondent.
1
Amici curiae briefs were filed by Mark A. Levy and Mayer
Greenberg for Stewart Tenants Corp., and Joel E. Miller for the
National Association of Housing Cooperatives, the Council of New
York Cooperatives, and the Federation of New York Housing
Cooperatives.
2
MEMORANDUM OPINION
COLVIN, Judge: Respondent determined deficiencies in
petitioner's Federal income tax of $8,117 for 1989 and $6,606 for
1990.
The issues for decision are:
1. Whether petitioner, a housing cooperative under section
216, is subject to subchapter T (sections 1381-1388),
aspetitioner contends, or is a membership organization under
section 277, as respondent contends. We hold that petitioner is
subject to subchapter T and is not subject to section 277.
2. Whether interest income earned by petitioner is
patronage sourced income under section 1388(j)(1). We hold that
it is not.
Section references are to the Internal Revenue Code in
effect for the years at issue. Rule references are to the Tax
Court Rules of Practice and Procedure.
The facts have been fully stipulated and are so found. The
relevant facts are summarized below.
Background
A. Petitioner
Petitioner's principal place of business was in New York
City when it filed the petition in this case.
3
Petitioner was formed on September 2, 1983, under New York
business corporation law. Petitioner uses the accrual method of
accounting.
Petitioner is a cooperative housing corporation under
section 216(b)(1) and is not tax-exempt under section 501.
Petitioner's certificate of incorporation was filed on
August 30, 1983, and was amended on May 7, 1984. Petitioner's
certificate of incorporation states in part that it was formed to
provide homes for its stockholders by leasing apartments to them
under proprietary leases that entitle them to live in the
building.
Petitioner's certificate of incorporation authorizes
petitioner to issue 70,000 shares of one class of common stock at
a par value of $1 each. Petitioner may make distributions to its
shareholders only from its earnings and profits unless petitioner
is completely or partially liquidated.
Petitioner's bylaws did not authorize it to pay patronage
dividends to its members in the years at issue. Petitioner's
bylaws have no provisions relating to whether petitioner may
distribute net earnings to its tenant-shareholders. Petitioner
has no rules or regulations requiring it to distribute patronage
dividends to its tenant-shareholders.
Petitioner could use net earnings to reduce maintenance.
Petitioner has never paid or allocated "net margins" (the excess
of its operating revenues over its cost of operations) to its
4
patrons as patronage dividends. The record does not show if
petitioner has ever had net margins.
Petitioner's bylaws require petitioner to hold an annual
meeting of the shareholders to elect directors and to conduct
other business. The bylaws also provide for special meetings of
the shareholders. Petitioner must give written notice of all
shareholders' meetings to each shareholder. Under the bylaws,
each shareholder has one vote at each shareholder's meeting for
each share of stock in his or her name. The bylaws permit proxy
voting at shareholder's meetings. Petitioner's bylaws require
petitioner to have at least 3 but not more than 7 directors, the
majority of whom must live in petitioner's building. The board
of directors manages petitioner, oversees its operations,
oversees the management company, and holds meetings not less than
once every 8 weeks to discuss problems referred to the Board.
Directors serve without pay unless pay is approved by
shareholders owning two-thirds of the outstanding shares.
Petitioner generally maintains the building and its grounds,
fixtures, elevators, lighting and heating, and other common areas
by hiring a superintendent and janitors. Petitioner's management
agent collects rents from petitioner's shareholders, keeps
petitioner's books, pays petitioner's expenses, prepares
petitioner's annual operating budget to be approved by
petitioner's directors, and hires and supervises petitioner's
5
employees. Petitioner provides laundry facilities for its
tenant-shareholders.
Petitioner is not required to rebate to its members the
excess of its charges collected from them over its operating
costs, and its members have no right to receive those
distributions.
B. Petitioner's Proprietary Lease
Petitioner's tenant-shareholders, because of their ownership
of stock in the corporation, may have proprietary leases2 which
entitle them to live in an apartment of petitioner.
Each shareholder must sign a proprietary lease with
petitioner. The proprietary lease used by petitioner during the
years in issue referred to tenants as "lessees". It required
each lessee to pay rent (called "maintenance") in equal monthly
installments. A lessee could occupy only the apartment he or she
leased. Monthly rent equaled the lessee's pro rata share of "the
estimated amount in cash which the Directors * * * determine to
be necessary" to operate, maintain and improve the property, and
to create a reserve for contingencies, repairs, and replacements.
The lease provided that petitioner's Board of Directors "from
time to time in its judgment" shall determine the annual
obligation of each lessee. The lease authorizes the Board of
2
A proprietary lease allows a shareholder in a cooperative
to possess an apartment in the cooperative. Black's Law
Dictionary 890 (6th ed. 1990).
6
Directors to "modify its prior determination and increase or
diminish the amount previously determined as cash requirements"
of petitioner.
Petitioner charges a maintenance fee to its tenant-
shareholders that varies according to the number of shares each
shareholder owns. Petitioner collects monthly apartment
maintenance payments from each tenant and a monthly parking space
rental fee from some of the tenants.
C. Petitioner's Income from Savings and Other Accounts
Petitioner earned interest income of $52,468 in 1989 and
$44,041 in 1990 from various savings and money market accounts,
and certificates of deposit with terms ranging from 2 months to 2
years. The parties stipulated that petitioner was not required
by law to have savings or money market accounts or certificates
of deposit.D. Characterization of Petitioner's Income
Petitioner's books and records did not distinguish between
patronage and nonpatronage sourced income. Petitioner did not
prepare records for 1989 and 1990 characterizing its current
earnings, liabilities, and net operating losses as patronage or
nonpatronage sourced.
Petitioner did not pay patronage dividends in the years at
issue. The parties stipulated that, because petitioner did not
pay patronage dividends, petitioner was not required to and did
not file information returns under section 6044. Petitioner
7
issued no Forms 1099-PATR (Taxable Distributions Received From
Cooperatives) to its tenant-shareholders for the years at issue.
Petitioner serves its members at less than cost and realized
a loss from its activities in 1989 and 1990. It did not pay tax
on its investment income.
Respondent determined that petitioner's interest income of
$52,468 in 1989 and $44,041 in 1990 was taxable as nonmembership
income by reason of section 277.
Discussion
A. Background
The issue for decision is whether petitioner, a section 216
cooperative housing corporation, is a cooperative under
subchapter T (sections 13813-1388), as petitioner contends, or
3
Section 1381 provides in part:
SEC. 1381(a). In General.--This part shall apply to--
(1) any organization exempt from tax under section 521
(relating to exemption of farmers' cooperatives from tax),
and
(2) any corporation operating on a cooperative basis
other than an organization--
(A) which is exempt from tax under this chapter,
(B) which is subject to the provisions of--
(i) part II of subchapter H (relating to
mutual savings banks, etc.), or
(ii) subchapter L (relating to insurance
companies), or
(continued...)
8
whether it is a membership organization under section 2774, as
respondent contends. If petitioner is subject to subchapter T,
we must also decide whether interest income petitioner received
is patronage or nonpatronage sourced under section 1388(j)(1).
We consider several sections in our analysis of this case:
(1) section 216, which defines cooperative housing corporations;
(2) section 277, which applies to social clubs and other
membership organizations; and (3) sections 1381-1388 (subchapter
T), which apply to corporations that operate on a cooperative
3
(...continued)
(C) which is engaged in furnishing electric
energy, or providing telephone service, to persons in
rural areas.
4
Section 277 provides as follows:
SEC. 277(a). General Rule.--In the case of a social club or
other membership organization which is operated primarily to
furnish services or goods to members and which is not exempt
from taxation, deductions for the taxable year attributable
to furnishing services, insurance, goods, or other items of
value to members shall be allowed only to the extent of
income derived during such year from members or transactions
with members (including income derived during such year from
institutes and trade shows which are primarily for the
education of members). If for any taxable year such
deductions exceed such income, the excess shall be treated
as a deduction attributable to furnishing services,
insurance, goods, or other items of value to members paid or
incurred in the succeeding taxable year. The deductions
provided by sections 243, 244, and 245 (relating to
dividends received by corporations) shall not be allowed to
any organization to which this section applies for the
taxable year.
9
basis and to farmers' cooperatives that are exempt under section
521.
We first decide whether petitioner is subject to subchapter
T. If petitioner is subject to subchapter T, then we must also
decide whether petitioner's interest income was patronage or
nonpatronage sourced income.
B. Whether Petitioner Is Subject to Subchapter T
1. Background
Respondent argues that petitioner is not subject to
subchapter T because it does not operate on a cooperative basis.
Respondent argues in the alternative that, if subchapter T
applies, petitioner's interest income is nonpatronage sourced
income that cannot be offset with petitioner's patronage
expenses.
Petitioner argues that it operates on a cooperative basis
within the meaning of section 1381(a)(2), and that it is governed
by subchapter T.
Section 1381(a) specifies the organizations that are subject
to subchapter T. Petitioner is not a farmers' cooperative and is
not exempt under section 521. Thus, we consider whether
petitioner is a corporation "operating on a cooperative basis".
Sec. 1381(a)(2). If a corporation operates on a cooperative
basis under section 1381(a)(2), then it is subject to subchapter
T. Trump Village Section 3, Inc. v. Commissioner, T.C. Memo.
1995-281.
10
2. Whether a Section 216 Cooperative Housing Corporation Is
a Cooperative Under Subchapter T
In Park Place, Inc. v. Commissioner, 57 T.C. 767 (1972), we
held that the taxpayer was a cooperative under subchapter T
because it was a section 216 cooperative housing corporation. We
concluded that:
We disagree with the Commissioner's assertion that
subchapter T, section 1381, * * * does not apply. Part
I of that subchapter applies to the taxable year of any
corporation operating on a cooperative basis after
December 31, 1962, and that necessarily includes a
section 216 cooperative housing corporation. [Citation
omitted.]
Id. at 779.
The parties have stipulated that petitioner is a section 216
cooperative housing corporation. Thus, as we discuss further
below in par. B-3-a, we conclude that petitioner is subject to
the provisions of subchapter T. Id.
3. Subordination of Capital, Control by Members, and
Allocation of Profit to Members
In Puget Sound Plywood, Inc. v. Commissioner, 44 T.C. 305,
308 (1965), we identified three factors that we said form the
core of economic cooperative theory:
(1) Subordination of capital, both as regards control
over the cooperative undertaking, and as regards the
ownership of the pecuniary benefits arising therefrom; (2)
democratic control by the worker-members themselves; and (3)
the vesting in and the allocation among the worker-members
of all fruits and increases arising from their cooperative
endeavor (i.e., the excess of the operating revenues over
the costs incurred in generating those revenues), in
proportion to the worker-members' active participation in
the cooperative endeavor.
11
a. Respondent’s Contention That Park Place Does Not Control
Respondent contends that the fact that we applied the Puget
Sound factors in Trump Village Section 3, Inc. v. Commissioner,
supra, shows that Park Place does not establish that section 216
cooperative housing corporations are subject to subchapter T and
that we should apply the Puget Sound factors to decide whether a
section 216 cooperative housing corporation operates on a
cooperative basis for purposes of subchapter T. We disagree.
There is no indication that the parties in Trump Village
asked the Court to consider (or that the Court did consider)
whether Park Place establishes that a section 216 cooperative
housing corporation operates on a cooperative basis for purposes
of section 1381. Thus, Trump Village does not bar our reliance
on Park Place.
Sections 216 and 1381 both use the term "cooperative".
Congress' use of the same word in both sections supports the
inference that a “cooperative” housing corporation under section
216 is operated on a “cooperative” basis for purposes of section
1381. The legislative history of the Revenue Act of 1942, ch.
19, tit. I, sec. 128, 56 Stat. 798, 826, which added section
23(z) (the predecessor to section 216) to the Code, states:
The definitions of the terms "cooperative apartment
corporation" and "tenant-stockholder" prescribe certain
standards which are designed to safeguard the revenue by
assuring that the apartment corporations involved are bona
fide cooperative apartment corporations and that the
individuals entitled to deductions under section 23(z) are
bona fide tenant-stockholders of such corporations.
12
S. Rept. 1361, 77th Cong., 2d Sess. (1942), 1942-2 C.B. 504, 577.
We interpret this to mean that Congress expected that a
cooperative apartment corporation (the predecessor to a
cooperative housing corporation) would be operated as a
cooperative.
b. Respondent’s Contention That Petitioner Does Not Operate
as a Cooperative
Respondent argues that petitioner does not meet the Puget
Sound factors and thus does not operate on a cooperative basis.
We disagree. First, petitioner meets the subordination of
capital factor because its tenant-shareholders and patrons are
identical and petitioner operated for the benefit of its patrons.
Second, petitioner is democratically controlled by its tenant-
stockholders. The fact that petitioner's shareholders may vote
by proxy is akin to voting by absentee ballot. See Rev. Rul. 75-
97, 1975-1 C.B. 167 (a farmer's cooperative is not denied exempt
status by allowing proxy voting by shareholders). Also, the fact
that petitioner's shareholders have one vote for each share they
own (instead of one vote per shareholder) and that they own
shares based on the relative sizes of their various dwelling
units is not contrary to democratic principles. The ownership
percentage of shareholders of a housing cooperative is not only a
measure of their investment; it is also a measure of their
relative “patronage” of the housing cooperative. Third,
petitioner did not fail to allocate profits to its members; in
13
fact, it operated at a loss in the years at issue. We conclude
that petitioner is a cooperative under the three factors stated
in Puget Sound Plywood, Inc. v. Commissioner, supra, and that
petitioner operates on a cooperative basis under section
1381(a)(2).
Because petitioner is subject to subchapter T, it is not
subject to section 277. Buckeye Countrymark, Inc. v.
Commissioner, 103 T.C. 547, 581 (1994); Trump Village Section 3,
Inc. v. Commissioner, T.C. Memo 1995-281; see also Landmark,
Inc., v. United States, 25 Cl. Ct. 100 (1992). In Buckeye
Countrymark, Inc., v. Commissioner, supra at 581, we said that
“the provisions of section 277 conflict with the provisions of
subchapter T and that the application of section 277 to nonexempt
cooperatives would lead to absurd or futile results.”
C. Whether Petitioner's Interest Income Is Patronage-Sourced
Income
Even if petitioner is a cooperative subject to subchapter T,
petitioner must pay tax on its investment income if the interest
was not patronage sourced. Petitioner bears the burden of
proving that its interest income is patronage sourced under
subchapter T. Rule 142(a); Welch v. Helvering, 290 U.S. 111, 115
(1933).
Petitioner argues that its interest income is patronage
sourced because petitioner earned the interest on funds its
14
shareholders deposited with petitioner to pay its expenses. We
disagree.
Subchapter T prohibits cooperatives from using patronage
losses to offset nonpatronage income. Buckeye Countrymark, Inc.
v. Commissioner, supra at 559; Certified Grocers of Calif., Ltd.
v. Commissioner, 88 T.C. 238, 250 (1987). A cooperative earns
patronage income from business it does with or for its patrons.
Sec. 1388(a); Illinois Grain Corp. v. Commissioner, 87 T.C. 435,
450 (1986). Income is patronage sourced if it is derived from an
activity that is so closely intertwined with the main cooperative
effort that it may be characterized as directly related to, and
inseparable from, the cooperative's principal business activity,
and thus facilitates the accomplishment of the cooperative's
business purpose. Illinois Grain Corp. v. Commissioner, supra at
459-460. However, if the transaction or account which produces
the income merely enhances the overall profitability of the
cooperative, then the income is from nonpatronage sources. Id.
at 452-453. Investment income is not patronage sourced. Sec.
1.1382-3(c)(2), Income Tax Regs.
In Illinois Grain Corp. v. Commissioner, supra at 442, 459-
460, the taxpayer-cooperative had a specific business need for
large amounts of cash at short notice. As a result, it invested
its temporary surplus funds in short-term (e.g., overnight,
weekend, and 10-day or less deposits) debt instruments because it
did not know when it would need the temporary surplus funds in
15
its business. We held that the interest earned by the taxpayer
on its short-term instruments was income from patronage sources
and not investment income. Id. at 460. The taxpayer's money
management activities were "inseparably intertwined with the
overall conduct of its cooperative enterprise, and the interest
income which it earned was therefore patronage-sourced". Id.
Petitioner earned interest income from money market and
savings accounts and from certificates of deposits with terms
ranging from 2 months to 2 years. The record contains no
evidence linking the savings and money market accounts to
petitioner's cooperative activities. The 2-month to 2-year
certificates of deposit were investments that provided income to
petitioner and did not facilitate the accomplishment of
petitioner's cooperative business activities. See Washington-
Oregon Shippers Coop., Inc. v. Commissioner, T.C. Memo. 1987-32
(taxpayer's money management activities were not integrally
linked to the overall conduct of its cooperative enterprise and
did nothing more than add to its overall profitability).
Petitioner alleged in the petition5 that its interest income
was patronage-sourced income that can be offset by patronage
deductions. Respondent denied this allegation in the answer.
Thus, petitioner has been on notice that the character of its
5
In the petition, petitioner stated: “This is a co-
operative housing, not a membership, corporation. Further,
income from ancillary sources is used to maintain and/or reduce
maintenance. Deductions are allowed. IRC 1388(j)(1).”
16
interest income as either patronage or nonpatronage-sourced
income has been at issue since the pleadings were filed. A
taxpayer bears the burden of proving allegations it makes in the
petition when it submits a case fully stipulated. Rules 122,
217(c)(1)(A); Kitch v. Commissioner, 104 T.C. 1, 5 (1995). The
stipulation does not include facts that support a finding that
petitioner's interest income was patronage sourced. The record
includes nothing to suggest that petitioner’s interest income was
earned from required reserves or that petitioner maintained its
savings and money market accounts and certificates of deposit for
business, rather than investment, purposes. Petitioner failed to
carry its burden of proving that its interest income was
patronage sourced under section 1388.
We hold that petitioner's interest income is taxable to
petitioner as determined by respondent.
To reflect the foregoing,
Decision will be entered for
respondent.
|
/*******************************************************************************
* Copyright (c) 2011 GigaSpaces Technologies Ltd. All rights reserved
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
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* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
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*******************************************************************************/
service {
name "network"
type "APP_SERVER"
compute {
template "SMALL_UBUNTU"
}
lifecycle{
install 'printContext.groovy'
}
network {
template "My_Network"
accessRules {
incoming = ([
accessRule {
portRange "80"
type "PUBLIC"
},
accessRule {
portRange "81"
type "SERVICE"
}
])
}
}
}
|
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Sometimes we have interpretations of what it means to be alive. Because I am alive, I can do this, I can do that. The focus is not about all those things. The focus is your existence, you being on the face of this earth. This will never happen again. The house you live in, other people will live in, too. Right now, you say, “It’s mine.” One day, somebody else will say, “It’s mine.” |
---
abstract: 'Pure hadronic compact stars, above a threshold value of their gravitational mass (central pressure), are metastable to the conversion to quark stars (hybrid or strange stars). In this paper, we present a systematic study of the metastability of pure hadronic compact stars using different relativistic models for the equation of state (EoS). In particular, we compare results for the quark-meson coupling (QMC) model with those for the Glendenning–Moszkowski parametrization of the non-linear Walecka model (NLWM). For QMC model, we find large values ($M_{cr} = 1.6$ – $1.9 M_\odot$) for the critical mass of the hadronic star sequence and we find that the formation of a quark star is only possible with a soft quark matter EoS. For the Glendenning–Moszkowski parametrization of the NLWM, we explore the effect of different hyperon couplings on the critical mass and on the stellar conversion energy. We find that increasing the value of the hyperon coupling constants shifts the bulk transition point for quark deconfinement to higher densities, increases the stellar metastability threshold mass and the value of the critical mass, and thus makes the formation of quark stars less likely. For the largest values of the hyperon couplings we find a critical mass which may be as high as 1.9 - 2.1 $M_\odot$. These stellar configurations, which contain a large central hyperon fraction ($f_{Y,cr} \sim 30 \%$), would be able to describe highly-massive compact stars, such as the one associated to the millisecond pulsars PSR B1516+02B with a mass $M = 1.94^{+ 0.17}_{- 0.19} M_{\odot}$.'
author:
- Ignazio Bombaci
- 'Prafulla K. Panda'
- Constança Providência
- Isaac Vidaña
title: Metastability of hadronic compact stars
---
PACS number(s): [97.60.s, 97.60.Jd, 26.60.Dd, 26.60.Kp]{}
Introduction
============
The nucleation of quark matter in neutron stars has been studied by many authors, due to its potential connection with explosive astrophysical events such as supernovae and gamma ray burst. Some of the earlier studies on quark matter nucleation (see e.g., [@ho92; @ho94; @ol94] and references therein) dealt with thermal nucleation in hot and dense hadronic matter. In these studies, it was found that the prompt formation of a critical size drop of quark matter via thermal activation is possible above a temperature of about $2-3$ MeV. As a consequence, it was inferred that pure hadronic stars are converted to quark stars (hybrid stars (HyS) or strange stars (SS)) within the first seconds after their birth. However, neutrino trapping in the protoneutron star phase strongly precludes the formation of a quark phase [@prak97; @lu98; @be99; @vi05]. Then, it is possible that the compact star survives the early stages of its evolution as a pure hadronic star. In this case, the nucleation of quark matter would be triggered by quantum fluctuations in degenerate ($T=0$) neutrino free hadronic matter [@gr98; @iida97; @iida98; @be03; @bo04; @drago04; @lug05; @blv07].
Quantum fluctuations could form, in principle, a drop of $\beta$-stable quark matter (hereafter the $Q^{\beta}$ phase). However, this process is strongly suppressed with respect to the formation of a non $\beta$-stable drop by a factor $\sim G^{2N/3}_{Fermi}$, where $N\sim 100-1000$ is the number of quarks in a critical-size quark drop. This is so because the formation of a $\beta$-stable drop would involve the almost simultaneous conversion of $\sim N/3$ up ($u$) and down ($d$) quarks into strange ($s$) quarks. Alternatively, quantum fluctuations can form a non $\beta$-stable drop (hereafter the $Q^*$ phase), in which the flavor content of the quark phase is equal to that of the $\beta$-stable hadronic phase at the same pressure [@iida97; @iida98; @bo04]. Since no flavor conversion is involved, there are no suppressing Fermi factors, and a $Q^*$ drop can be nucleated much more easily. Once a critical-size $Q^*$ drop is formed, the weak interactions will have enough time to act, changing the quark flavor fraction of the deconfined droplet to lower its energy, and a drop of the $Q^\beta$ phase is formed. This first seed of quark matter will trigger the conversion [@oli87; @hbp91; @grb] of the pure hadronic star to a hybrid star or to a strange star (depending on the details of the equation of state for quark matter used to model the phase transition). The stellar conversion process liberates a total energy of the order of $10^{53}$ erg [@grb].
When finite-size effects at the interface between the quark and hadron phases are taken into account, it is necessary to have an overpressure $\Delta P = P - P_0> 0$ with respect to the bulk transition point $P_0$, to create a drop of deconfined quark matter. As a consequence, pure hadronic stars with values of the central pressure larger than $P_0$ are metastable to the decay (conversion) to hybrid stars or to strange stars [@be03; @bo04; @drago04; @lug05; @blv07]. The mean lifetime of the metastable stellar configuration is related to the time needed to nucleate the first drop of quark matter in the stellar center and depends dramatically on the value of the stellar central pressure [@be03; @bo04; @drago04; @lug05; @blv07].
The possibility of having in nature both metastable hadronic stars and stable quark stars, has led the authors of ref. [@bo04] to extend the concept of limiting mass of a “neutron star” with respect to the [*classical*]{} one introduced by Oppenheimer and Volkoff [@ov39]. Since metastable HS with a “short” [*mean-life time*]{} are very unlikely to be observed, the extended concept of limiting mass has been introduced in view of the comparison with the values of the mass of compact stars deduced from direct astrophysical observation (see sect. 3.1 of ref. [@bo04] for the definition of the [*limiting mass*]{}, $M_{lim}$, of compact stars in the case of metastable pure hadronic stars).
As it is well known, neutron star mass measurements give one of the most stringent test on the overall [*stiffness*]{} of dense matter EoS. Recent measurements of Post Keplerian orbital parameters in relativistic binary stellar systems (containing millisecond pulsars) give strong evidence for the existence of highly-massive “neutron stars”. For example, the compact star associated to the millisecond pulsar PSR B1516+02B in the Globular Cluster NGC 5904 (M5) has a mass $M = 1.94^{+ 0.17}_{- 0.19} M_{\odot}$ (1 $\sigma$) [@frei07a]. In the case of PSR J1748-2021B, a millisecond pulsar in the Globular Cluster NGC 6440, the measured mass is $M = 2.74^{+ 0.41}_{- 0.51} M_{\odot}$ (2 $\sigma$) [@frei07b]. These measurements challenge most of the existing models for dense matter EoS.
In this work, we carry out a systematic study of the properties of metastable hadronic compact stars obtained within different relativistic mean-field models for the equation of state (EoS) of hadronic matter. In particular, we compare the predictions of the Quark-Meson Coupling (QMC) model [@guichon; @ST] with those of the non-linear Walecka model (NLWM) [@qhd] parametrizations given by Glendenning–Moszkowski (GM) [@gm91].
For the quark phase we have adopted a phenomenological EOS [@farhi] which is based on the MIT bag model for hadrons. The parameters here are: the mass $m_s$ of the strange quark, the so-called pressure of the vacuum $B$ (bag constant) and the QCD structure constant $\alpha_s$. For all the quark matter model used in the present work, we take $m_u = m_d =0$, $m_s = 150$ MeV and $\alpha_s = 0$.
In the QMC model quark degrees of freedom are explicitly taken into account: baryons are described as a system of non-overlapping MIT bags which interact through the effective scalar and vector mean fields. The coupling constants are defined at the quark level. An attractive aspect of the model is that different phases of hadronic matter, from very low to very high baryon densities, can be described within the same underlying model, namely the MIT bag model: matter at low densities is a system of nucleons interacting through meson fields, with quarks and gluons confined within MIT bag; at very high density one expects that baryons and mesons dissolve and the entire system of quarks and gluons becomes confined within a single, big MIT bag.
In the case of the Glendenning–Moszkowski EoS [@gm91], we have paid special attention to the role played by the hyperon-meson couplings. In fact, all previous works on metastable hadronic stars [@be03; @bo04; @drago04; @lug05; @blv07] have uniquely considered the case of “low” values for these quantities ($x_\sigma = 0.6$ for the ratio between the hyperon–$\sigma$ meson to nucleon–$\sigma$ meson coupling). As it is well known, larger values of the hyperon-meson couplings (constrained by the empirical binding energy of the $\Lambda$ particle in nuclear matter) make the EoS stiffer and increase the value of the Oppenheimer–Volkoff mass for the hadronic stellar sequence [@gm91]. In addition, as we demonstrate in the present work, increasing the values of the hyperon-meson couplings shifts the bulk transition point for quark deconfinement to higher densities and increments the value of the [*critical mass*]{} $M_{cr}$ (see ref.[@be03; @bo04; @drago04] and Section \[sec:quantum\] for the explicit definition of this quantity) for the hadronic stellar sequence. Thus our study is relevant in connections with the recent measurements of highly-massive “neutron stars” mentioned above.
A brief review of the NLW and QMC models is given in Section \[sec:formalism\]. The quantum nucleation of a quark matter drop inside hadronic matter is briefly reviewed in Section \[sec:quantum\]. Our main results are presented in Section \[sec:results\], whereas the main conclusions are given in Section \[sec:conclusions\]
The formalism {#sec:formalism}
=============
In the present section we review the models used in this work, namely the GM parametrizations [@gm91] of the NLWM and the quark-meson coupling (QMC) model including hyperons.
The non-linear Walecka model
----------------------------
The Lagrangian density, including the baryonic octet, in terms of the scalar $\sigma$, the vector-isoscalar $\omega_\mu$ and the vector-isovector $\vec \rho_\mu$ meson fields reads (see [*e.g.*]{} [@prak97; @glen00; @mp03])
$${\cal L}={\cal L}_{hadrons}+{\cal L}_{leptons}$$
where the hadronic contribution is
$${\cal L}_{hadrons}={\cal L}_{baryons}+{\cal L}_{mesons}$$
with
$${\cal L}_{baryons}=\sum_{\mbox{baryons}} \bar \psi\left[\gamma^\mu D_\mu -M^*_B\right]\psi,$$
where $$D_\mu=i\partial_{\mu}
-g_{\omega B} \omega_{\mu}-{g_{\rho B}} \vec{t_B} \cdot \vec{\rho}_\mu,$$ and $M^*_B=M_B-g_{\sigma B} \sigma.$ The quantity $\vec{t_B}$ designates the isospin of baryon $B$. The mesonic contribution reads $${\cal L}_{mesons}={\cal L}_{\sigma}+{\cal L}_{\omega}+ {\cal L}_{\rho},$$ with $${\cal L}_\sigma=\frac{1}{2}(\partial_{\mu}\sigma\partial^{\mu}\sigma
-m_{\sigma}^2 \sigma^2)+ \frac{1}{3!} \kappa \sigma^3+ \frac{1}{4!} \lambda
\sigma^4,$$ $${\cal L}_{\omega}=-\frac{1}{4}\Omega_{\mu\nu}\Omega^{\mu\nu}+\frac{1}{2}
m_{\omega}^2 \omega_{\mu}\omega^{\mu}, \qquad \Omega_{\mu\nu}=\partial_{\mu}\omega_{\nu}-\partial_{\nu}\omega_{\mu},$$ $${\cal L}_{\rho}=
{ -\frac{1}{4}\vec B_{\mu\nu}\cdot\vec B^{\mu\nu}}+\frac{1}{2}
m_\rho^2 \vec \rho_{\mu}\cdot \vec \rho^{\mu}, \quad \vec B_{\mu\nu}=\partial_{\mu}\vec \rho_{\nu}-\partial_{\nu} \vec \rho_{\mu}
- g_\rho (\vec \rho_\mu \times \vec \rho_\nu)$$ For the lepton contribution we take $${\cal L}_{leptons}=\sum_{\mbox{leptons}} \bar \psi_l \left(i \gamma_\mu \partial^{\mu}-
m_l\right)\psi_l,$$ where the sum is over electrons and muons. In uniform matter, we get for the baryon Fermi energy $
\epsilon_{FB}=g_{\omega B} \omega_0+ g_{\rho B} t_{3B} \rho_{03} + \sqrt{k_{FB}^2+{M^*_B}^2},
$ with the baryon effective mass $M^*_B=M-g_{\sigma B}\sigma.$
We will use the GM1 and GM3 parametrizations of NLWM [@gm91] both fitted to the bulk properties of nuclear matter: for GM1 (GM3) the compressibility is 300 (240) MeV and the effective mass at saturation $M^{*} = 0.7\, M$ ($M^{*} = 0.78\, M$). The inclusion of hyperons involves new couplings, the hyperon-nucleon couplings: $g_{\sigma B}=x_{\sigma B}~ g_{\sigma},~~g_{\omega B}=x_{\omega B}~ g_{\omega},~~g_{\rho B}=x_{\rho B}~ g_{\rho}$. For nucleons we take $x_{\sigma B}$, $x_{\omega B}$, $x_{\rho B} = 1$ and for hyperons we will consider the couplings proposed by Glendenning and Moszkowski [@gm91]. They have considered the binding energy of the $\Lambda$ in nuclear matter, $B_\Lambda$, $$\left(\frac{B_\Lambda}{A}\right)=-28 \mbox{ MeV}= x_{\omega} \, g_{\omega}\, \omega_0-x_{\sigma}\, g_{\sigma} \sigma$$ to establish a relation between $x_{\sigma}$ and $x_{\omega}$. Moreover, known neutron star masses restrict $x_{\sigma}$ to the range $0.6-0.8$. We will take $x_{\rho}=x_{\sigma}$ and will consider $x_{\sigma}=0.6,\,0.7,\,0.8$.
The quark-meson coupling model for hadronic matter
--------------------------------------------------
In the QMC model, the nucleon in nuclear medium is assumed to be a static spherical MIT bag in which quarks interact with the scalar and vector fields, $\sigma$, $\omega$ and $\rho$ and these fields are treated as classical fields in the mean field approximation [@guichon; @ST]. The quark field, $\psi_q(x)$, inside the bag then satisfies the equation of motion: $$\left[i\,\rlap{/}\partial-(m_q^0-g_\sigma^q\, \sigma)
-g_\omega^q\, \omega\,\gamma^0 + \frac{1}{2} g^q_\rho \tau_z \rho_{03}\right]
\,\psi_q(x)=0\ , \quad q=u,d,s,
\label{eq-motion}$$ where $m_q^0$ is the current quark mass, and $g_\sigma^q$, $g_\omega^q$ and $g_\rho^q$ and denote the quark-meson coupling constants. The normalized ground state for a quark in the bag is given by $$\psi_q({\bf r}, t) = {\cal N}_q \exp \left(-i\epsilon_q t/R_B \right)
\left(
\begin{array}{c}
j_0\left(x_q r/R_B\right)\\
i\beta_q \vec{\sigma} \cdot \hat r j_1\left(x_q r/R_B\right)
\end{array}\right)
\frac{\chi_q}{\sqrt{4\pi}} ~,$$ where $$\epsilon_q=\Omega_q +R_B\left(g_\omega^q\, \omega+
\frac{1}{2} g^q_\rho \tau_z \rho_{03} \right) ~; ~~~
\beta_q=\sqrt{\frac{\Omega_q-R_B\, m_q^*}{\Omega_q\, +R_B\, m_q^* }}\ ,$$ with the normalization factor given by $${\cal N}_q^{-2} = 2R_B^3 j_0^2(x_q)\left[\Omega_q(\Omega_q-1)
+ R_B m_q^*/2 \right] \Big/ x_q^2 ~,$$ where $\Omega_q\equiv \sqrt{x_q^2+(R_B\, m_q^*)^2}$, $m_q^*=m_q^0-g_\sigma^q\, \sigma$, $R_B$ is the bag radius of the baryon, and $\chi_q$ is the quark spinor. The quantities $\psi_q,\, \epsilon_q,\,
\beta_q,\, {\cal N}_q,\, \Omega_q,\, m^*_q$ all depend on the baryon considered. The bag eigenvalue, $x_q$, is determined by the boundary condition at the bag surface $$j_0(x_q)=\beta_q\, j_1(x_q)\ .
\label{bun-con}$$ The energy of a static bag describing baryon $B$ consisting of three ground state quarks can be expressed as $$E^{\rm bag}_B=\sum_q n_q \, \frac{\Omega_q}{R_B}-\frac{Z_B}{R_B}
+\frac{4}{3}\, \pi \, R_B^3\, B_B\ ,
\label{ebag}$$ where $Z_B$ is a parameter which accounts for zero-point motion and $B_B$ is the bag constant. The effective mass of a nucleon bag at rest is taken to be $M_B^*=E_B^{\rm bag}.$ The equilibrium condition for the bag is obtained by minimizing the effective mass, $M_B^*$ with respect to the bag radius $$\frac{d\, M_B^*}{d\, R_B^*} = 0\ .
\label{balance}$$ For the QMC model, the equations of motion for the meson fields in uniform static matter are given by $$m_\sigma^2\sigma = \sum_B g_{\sigma B} C_B(\sigma) \frac{2J_B + 1}{2\pi^2}
\int_0^{k_B} \frac{M_B^*(\sigma)}
{\left[k^2 + M_B^{* 2}(\sigma)\right]^{1/2}} \: k^2 \ dk ~,
\label{field1}$$ $$m_\omega^2\omega_0 = \sum_B g_{\omega B} \left(2J_B + 1\right)
k_B^3 \big/ (6\pi^2) ~,
\label{field2}$$ $$m_\rho^2\rho_{03} = \sum_B g_{\rho B} I_{3B} \left(2J_B + 1\right)
k_B^3 \big/ (6\pi^2) ~.
\label{field3}$$ In the above equations $J_B$, $I_{3B}$ and $k_B$ are respectively the spin, isospin projection and the Fermi momentum of the baryon species $B$. For the hyperon couplings we take $x_{\omega}=0.78$ and $x_{\rho}=0.7$. The coupling $x_{\sigma}$ is an output of the model and is approximately equal to 0.7. Note that the $s$-quark is unaffected by the $\sigma$ and $\omega$ mesons i.e. $g_\sigma^s=g_\omega^s=0\ .$
In Eq. (\[field1\]) we have $$g_{\sigma B}C_B(\sigma) = - \frac{\partial M_B^*(\sigma)}{\partial \sigma}
= - \frac{\partial E^{\rm bag}_B}{\partial \sigma}
= \sum_{q=u,d} n_q g^q_\sigma S_B(\sigma)$$ where $$S_B(\sigma) = \int_{bag} d{\bf r} \ {\overline \psi}_q \psi_q
= \frac{\Omega_q/2 + R_Bm^*_q(\Omega_q - 1)}
{\Omega_q(\Omega_q -1) + R_Bm_q^*/2} ~; ~~~~ q \equiv (u,d) ~.$$
The total energy density and the pressure including the leptons can be obtained from the grand canonical potential and they read $$\begin{aligned}
\varepsilon &=& \frac{1}{2}m_\sigma^2 \sigma^2
+ \frac{1}{2}m_\omega^2 \omega^2_0
+ \frac{1}{2} m_\rho^2 \rho^2_{03} \nonumber\\
&+& \sum_B \frac{2J_B +1}{2\pi^2} \int_0^{k_B}k^2 dk
\left[k^2 + M_B^{* 2}(\sigma)\right]^{1/2}
+ \sum_l \frac{1}{\pi^2} \int_0^{k_l} k^2 dk\left[k^2 + m_l^2\right]^{1/2}~,
\end{aligned}$$ $$\begin{aligned}
P &=& - \frac{1}{2}m_\sigma^2 \sigma^2
+ \frac{1}{2}m_\omega^2 \omega^2_0
+ \frac{1}{2} m_\rho^2 \rho^2_{03} \nonumber\\
&+& \frac{1}{3} \sum_B \frac{2J_B +1}{2\pi^2} \int_0^{k_B}
\frac{k^4 \ dk}{\left[k^2 + M_B^{* 2}(\sigma)\right]^{1/2}}
+ \frac{1}{3} \sum_l \frac{1}{\pi^2} \int_0^{k_l} \frac{k^4 dk}
{\left[k^2 + m_l^2\right]^{1/2}} ~.
\end{aligned}$$
For the bag radius we take $R_N=0.6$ fm. The two unknowns $Z_N$ and $B_N$ for nucleons are obtained by fitting the nucleon mass $M=939$ MeV and enforcing the stability condition for the bag at free space. The values obtained are $Z_N=3.98699$ and $B_N^{1/4}=211.303$ MeV for $m_u=m_d=0$ MeV and $Z_N=4.00506$ and $B_N^{1/4}=210.854$ MeV for $m_u=m_d=5.5$ MeV. We take these bag values, $B_B$, for all baryons and the parameter $Z_B$ and $R_B$ of the other baryons are obtained by reproducing their physical masses in free space and again enforcing the stability condition for their bags. Note that for a fixed bag value, the equilibrium condition in free space results in an increase of the bag radius and a decrease of the parameters $Z_{B}$ for the heavier baryons. The set of parameters used in the present work is given in Ref. [@parameter].
Next we fit the quark-meson coupling constants $g_\sigma^q$, $g_\omega = 3g_\omega^q$ and $g_\rho = g_\rho^q$ for the nucleon to obtain the correct saturation properties of the nuclear matter, $E_B \equiv \epsilon/\rho - M = -15.7$ MeV at $\rho~=\rho_0=~0.15$ fm$^{-3}$, $a_{sym}=32.5$ MeV, $K=257$ MeV and $M^*=0.774 M$. We have $g_\sigma^q=5.957$, $g_{\omega N}=8.981$ and $g_{\rho N}=8.651$. We take the standard values for the meson masses, $m_\sigma=550$ MeV, $m_\omega=783$ MeV $m_\rho=770$ MeV.
Quantum nucleation of quark matter in hadron stars {#sec:quantum}
==================================================
Let us consider a pure hadronic star whose central pressure (density) is increasing due to spin-down or due to mass accretion (from a companion or from the interstellar medium). As the central pressure approaches the deconfinement threshold pressure $P_0$ (see Fig. \[gm1\]), a drop of non $\beta$-stable quark matter ($Q^*$), but with flavor content equal to that of the $\beta$-stable hadronic phase, can be formed in the central region of the star. The process of drop formation is regulated by its quantum fluctuations in the potential well created from the difference in the Gibbs free energies of the hadron and quark phases [@iida97; @iida98; @be03] $$U({\cal R})=\frac{4}{3}\pi n_{b,Q^*}(\mu_{Q^*}-\mu_H){\cal R}^3 + 4\pi \sigma {\cal R}^2
\label{eq:potential}$$ where ${\cal R}$ is the radius of the $Q^*$ droplet (supposed to be spherical), $n_{b,Q^*}$ is the quark baryon number density, $\mu_{Q^*}$ and $\mu_H$ are the quark and hadron chemical potentials at a fixed pressure $P$ and $\sigma$ is the surface tension for the surface separating the hadron from the $Q^*$ phase. Notice that $\mu$ is the same as the bulk Gibbs energy per baryon $g=(P+\epsilon)/n_B=(\sum_i\mu_in_i)/n_B$. Notice also that we have neglected the term associated with the curvature energy, and also the terms connected with the electrostatic energy, since they are known to introduce small corrections [@iida98; @bo04]. The value of the surface tension $\sigma$ for the interface separating the quark and hadron phase is poorly known, and typically values used in the literature range within $10-50$ MeV fm$^{-2}$ [@he93; @iida98].
The time needed to form the first drop (nucleation time) can be straightforwardly evaluated within a semi-classical approach [@iida97; @iida98]. First one computes, in the Wentzel–Kramers–Brillouin (WKB) approximation, the ground state energy $E_0$ and the oscillation frequency $\nu_0$ of the drop in the potential well $U({\cal R})$. Then, the probability of tunneling is given by $$p_0=exp\left[-\frac{A(E_0)}{\hbar}\right]
\label{eq:prob}$$ where $A$ is the action under the potential barrier which in a relativistic framework reads $$A(E)=\frac{2}{c}\int_{{\cal R}_-}^{{\cal R}_+}\sqrt{[2{\cal M}({\cal R})c^2 +E-U({\cal R})][U({\cal R})-E]} \ ,
\label{eq:action}$$ being ${\cal R}_\pm$ the classical turning points and $${\cal M}({\cal R}) = \frac{4\pi}{3} \rho_H\left(1-\frac{n_{b,Q^*}}{n_H}\right)^2 {\cal R}^3
\label{eq:mass}$$ the droplet effective mass, with $\rho_H$ and $n_H$ the hadron energy density and the hadron baryon number density, respectively. The nucleation time is then equal to $$\tau=(\nu_0 p_0 N_c)^{-1} \ ,
\label{eq:time}$$ where $N_c$ is the number of virtual centers of droplet formation in the star. A simple estimation gives $N_c \sim 10^{48}$ [@iida97; @iida98]. The uncertainty in the value of $N_c$ is expected to be within one or two orders of magnitude. In any case, all the qualitative features of our scenario will not be affected by this uncertainty. As a consequence of the surface effects it is necessary to have an overpressure $\Delta P= P-P_0 > 0$ with respect to the bulk transition point $P_0$ to create a drop of deconfinement quark matter in the hadronic environment. The higher the overpressure, the easier to nucleate the first drop of $Q^*$ matter. In other words, the higher the mass of the metastable pure hadronic star, the shorter the time to nucleate a quark matter drop at the center of the star.
In order to explore the astrophysical implications of quark matter nucleation, following ref. [@be03; @bo04], we introduce the concept of [*critical mass*]{} for the hadronic star sequence. The [*critical mass*]{} $M_{cr}$ is the value of the gravitational mass of a metastable hadronic star for which the nucleation time is equal to one year: $M_{cr}=M_{HS} (\tau=1 yr)$. Therefore, pure hadronic stars with $M_{HS} > M_{cr}$ are very unlikely to be observed, while pure hadronic stars with $M_{HS} < M_{cr}$ are safe with respect to a sudden transition to quark matter. Then $M_{cr}$ plays the role of an [*effective maximum mass for the hadronic branch of compact stars*]{} (see discussion in Ref. [@bo04]). While the Oppenheimer–Volkov maximum mass is determined by the overall stiffness of the equation of state for hadronic matter, the value of $M_{cr}$ will depend in addition on the properties of the intermediate non $\beta$-stable $Q^*$ phase.
Results and discussion {#sec:results}
======================
In this section we present and discuss our results for stellar configurations obtained using the equation of state (EoS) models described in section II. In particular, we determine the region of the pure hadronic star sequence where these compact stars are metastable, the value of the corresponding critical mass $M_{cr}$, and the final fate of this configuration after quark matter nucleation, [*i.e.*]{} whether it will evolve to a quark star or to a black hole.
In Fig \[eos\] the EoS for the models discussed are plotted for the range of densities of relevance for the discussion that follows. For GM1 and GM3 we have considered three different hyperon-meson coupling as discussed above. The QMC EoS corresponds approximately to $x_{\sigma}=0.7$. A higher value of the hyperon couplings $x_i$ corresponds to stiffer EoSs: at high densities we have vector dominance defined by the magnitude of $x_{\omega}, \,
x_{\rho}$. It is clear from Fig. \[eos\] that the onset of hyperons (represented by the change of slope in the EoS curves) occurs for the smaller $x_{\sigma}$ values at lower densities. The nucleonic EoS for QMC is very soft and therefore the onset of hyperons occurs at quite high densities, $\varepsilon=373.87$ MeV/fm$^3$. As a consequence although QMC is softer than GM1 EoS at lower densities, it becomes, at higher densities, stiffer than GM1($x_{\sigma}=0.6$) and very close to GM1($x_{\sigma}=0.7$).
-- --
-- --
In Figs. \[gm1\], we plot the Gibbs‘ free energy per baryon for the hadronic phase and for the corresponding $Q^*$ phase using the various EoS models (couple of continuous and dashed curves with the same color) considered in the present work. It is clearly seen that in the case of the GM1 or GM3 EoS models, the lower the value of the hyperon coupling $x_{\sigma}$, the softer the EoS (see also Fig \[eos\]) and the lower the pressure $P_0$ at the crossing between the hadronic and the $Q^*$ phase. This will give rise to lower critical masses for the smaller $x_{\sigma}$ values (see Tabl. I-III below). The $Q^*$ phase is very sensitive to the particle content and it is due to this fact that, although in Fig. \[gm1\] the EoS for QMC is softer than the EoS for GM1 with $x_{\sigma}=0.8$ and 0.7, its crossing with the $Q^*$ phase occurs at higher pressures. A similar observation occurs in the figure with the GM3 results. This behavior will reflect itself on values of the critical masses $M_{cr}$.
[c]{}\
In Fig. \[QMC\_MR\], we show the mass-radius (MR) curve for pure HS within the QMC model for the EoS of the hadronic phase, and that for hybrid stars or strange stars for different values of the bag constant $B$. The configuration marked with an asterisk on the hadronic MR curves represents the hadronic star for which the central pressure is equal to the threshold value $P_0$ and the quark matter nucleation time is $\tau = \infty$. The full circle on the hadronic star sequence represents the critical mass configuration, in the case $\sigma = 30$ MeV/fm$^2$. The full circle on the HyS (SS) mass-radius curve represents the hybrid (strange) star which is formed from the conversion of the hadronic star with $M_{HS} = M_{cr}$. We assume [@grb] that during the stellar conversion process the total number of baryons in the star (or in other words the stellar baryonic mass) is conserved. Thus the total energy liberated in the stellar conversion is given by the difference between the gravitational mass of the initial hadronic star ($M_{in} \equiv M_{cr}$) and that of the final hybrid or strange stellar configuration with the same baryonic mass ($M_{fin} \equiv M_{QS}(M^b_{cr}) \,$): $$E_{conv} = (M_{in} - M_{fin}) c^2 \, .
\label{eq:eq11}$$ As we can see from Fig. \[QMC\_MR\], for the case of the QMC model, the region of metastability of pure hadronic stars (the part of the MR curve between the asterisk and the full circle) is very narrow. For this hadronic EoS, the quark star sequence can be populated only in the case of “small” values of the bag constant ($B \leq 80$ MeV/fm$^3$, in this case the final star is a strange star). In all the other cases the critical mass hadronic star will form a black hole.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![Mass-radius relation for a pure HS described within the QMC model and that of the HyS or SS configurations for several values of the bag constant and $m_s=150$ MeV and $\alpha_s=0$. The configuration marked with an asterisk represents in all cases the HS for which the central pressure is equal to $P_0$. The conversion process of the HS, with a gravitational mass equal to $M_{cr}$, into a final HyS or SS is denoted by the full circles connected by an arrow. In all the panels $\sigma$ is taken equal to $30$ MeV/fm$^2$.[]{data-label="QMC_MR"}](fig3.eps "fig:"){height="9cm"}
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For comparison we plot the MR curve obtained with the GM1 parametrization for the same surface tension ($\sigma =30 $MeV/fm$^2$) and two values for bag constant ($B=75$ and 100 MeV/fm$^3$). We consider the two extreme values of the hyperon couplings studied in this work. The dots and stars have the same meaning as in Fig. \[QMC\_MR\]. We see that for the cases plotted the only configuration that does not end in a black hole has the smallest bag constant and hyperon coupling considered. In the present model, however, the configuration with the central pressure $P_0$ and the $M_{cr}$ configuration are quite separated, contrary to what was observed with QMC, Fig. \[QMC\_MR\].
The larger mass difference between the star with the central pressure $P_0$ and the one with the $M_{cr}$ occurs when these stars have small masses. A small change in the central energy density corresponds to a large change in the mass. If instead of plotting the MR graph we would have plotted the corresponding mass–central pressure (MP) graph a larger difference between these two configurations would be expected. This is seen in Figs. \[QMC\_MP\] and \[GM1\_MP\] where the mass-pressure curves for the family of stars obtained respectively within QMC and GM1 are plotted for two bag constants and two values of the surface tension ($\sigma = 10$ and 30 MeV/fm$^2$). We conclude that when the $M_{cr}$ star is almost on top of the $P_0$ star in the MR curves, these stars lie on or close to the plateau that contains the maximum mass configuration. A large separation between these two configurations corresponds to a phase transition which occurs during the rise of the MR curve before the plateau. Due to the softness of the QMC EOS, hyperons set on at quite large energy densities and the star with the central $P_0$ pressure only occurs at high densities. We also conclude that a smaller surface tension hastens the transition and the critical mass is closer to the $P_0$ mass.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![ Mass-radius relation for a pure HS described within the GM1 parametrization and that of the HyS or SS configurations for two values of the bag constant ($B=75$ and 100 MeV/fm$^3$) and two values of the hyperon-meson coupling ($x_\sigma=0.6,$ and 0.8) and $m_s=150$ MeV and $\alpha_s=0$. The configuration marked with an asterisk represents in all cases the HS for which the central pressure is equal to $P_0$. The conversion process of the HS, with a gravitational mass equal to $M_{cr}$, into a final HyS or SS is denoted by the full circles connected by an arrow. In all the panels $\sigma$ is taken equal to 30 MeV/fm$^2$.[]{data-label="GM1_MR"}](fig4.eps "fig:"){height="9cm"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![Mass-pressure relation for a pure HS described within the QMC model and that of the HyS or SS configurations two values of the Bag constant(75 and 100 MeV/fm$^3$) and $m_s=150$ MeV and $\alpha_s=0$. The configuration marked with an asterisk represents in all cases the HS for which the central pressure is equal to $P_0$. The conversion process of the HS, with a gravitational mass equal to $M_{cr}$, into a final HyS or SS is denoted by the full circles connected by an arrow. Two values of the surface energy $\sigma$ were considered $10$ MeV/fm$^2$ (left) and $30$ MeV/fm$^2$ (right).[]{data-label="QMC_MP"}](fig5.eps "fig:"){height="9cm"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![The same as Fig \[QMC\_MP\] for the GM1 parametrization with the $x_\sigma=0.6$ hyperon coupling. []{data-label="GM1_MP"}](fig6.eps "fig:"){height="9cm"}
------------------------------------------------------------------------------------------------------------------------------------------------------------- --
In Tables \[tabgm1\], \[tabgm3\], and \[tabqmc\] we give the gravitational ($M_{cr}$) and baryonic ($M_{cr}^b$) critical mass values for the hadronic star sequence, together with the central hyperon fraction ($f_{Y,cr} = n_Y/n_B$, [*i.e.*]{} the ratio between the total hyperon number density and the total baryon number density at the center of the critical mass star). We also report the value of the gravitational mass ($M_{fin}$) of the final quark star configuration and the total energy [@grb] $E_{conv} = (M_{cr} - M_{fin}) c^2$ released in the stellar conversion process, assuming baryon mass conservation ([*i.e.*]{} no matter ejection) [@grb]. The gravitational ($M_{QS,max}$) and the baryonic ($M_{QS,max}^b$) mass of the maximum mass configuration for the quark (hybrid or strange) star sequence are also included. The value of the latter quantity is relevant to establish whether the critical mass hadronic star will evolve to a quark star ($M_{cr}^b < M_{QS,max}^b$) or will form a black hole ($M_{cr}^b > M_{QS,max}^b$). The entries in Tables \[tabgm1\], \[tabgm3\] [[^1]]{}, and \[tabqmc\] are relative respectively to the GM1, GM3 and QMC equation of state for the hadronic phase. For the quark phase we consider four different values of bag constants, 75, 85, 100 and 150 MeV/fm$^3$, and two different values for quark-hadron surface tension, 10 and 30 MeV. Notice that for the quark matter parameter set adopted in the present work (see Section I), strange quark matter is absolutely stable [@bod71; @witt84] only for $B = 75$ MeV/fm$^3$.
Some comments are in order: the critical masses increase with the increase of the hyperon couplings. This increase can be as large as 0.3 - 0.4 $M_\odot$ when $x_{\sigma}$ changes from 0.6 to 0.8; the critical mass is also dependent on the particle content, namely of the strangeness content, and this explains the different relative positions for the different bag pressures of the QMC result which essentially corresponds to $x_{\sigma}=0.7$. Due to the fact that the EoS for QMC is very soft, the hyperon onset occurs at quite high densities and therefore the critical mass is always quite high for this model. The critical mass increases with the bag constant because a larger bag constant corresponds to a stiffer quark EoS and therefore the phase transition to the quark phase will occur at larger densities. When the critical mass hadronic star is converted to a black hole, this is indicated in Tables \[tabgm1\], \[tabgm3\], and \[tabqmc\] with a entry BH, in the columns for $M_{fin}$ and $E_{conv}$ (no energy will be radiated as soon as the star pass the event horizon). Notice that, in the case of the GM3 model with $x_{\sigma}=0.6$ and $B = 150$ MeV/fm$^3$ there is no entry for the critical mass value (and for $M_{cr}^b$, $M_{fin}$ and $E_{conv}$) since in this case the nucleation time of the maximum mass hadronic star ($M_{HS,max}$) is much larger than one year ([*i.e.*]{} the star is metastable with a [*life-time*]{} comparable or much higher than the age of the universe).
We observe from the results in tables \[tabgm1\], \[tabgm3\], that increasing the value of the hyperon coupling constants (for fixed $B$ and $\sigma$) reduces the central hyperon fraction ($f_{Y,cr}$) of the critical mass star, and increases the energy released during the conversion into a quark or hybrid star (for those configurations which will not form a black hole).
In Fig. \[profile\], we show the internal composition for the hadronic star with a gravitational mass $M= 2.081 M_\odot$ and radius $R = 12.6$ km, obtained using the GM1 parametrization with $x_{\sigma} = 0.8$. This star corresponds to the critical mass configuration when we consider $B = 150$ MeV/fm$^3$ and $\sigma = 30$ MeV/fm$^2$ (see table \[tabgm1\]). As we see, this star has a considerable central hyperon fraction ($f_{Y,cr} = 0.299$) and a wide hyperonic matter core which extend up to $R_Y \sim 8.7$ km. On the top of this core, one has a nuclear matter layer ($R_Y \leq r \leq R_{crust}$) with a thickness of about 3.4 km. The stellar crust extends from $R_{crust}$ up to $R$.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
![The internal composition for an hadronic star with gravitational mass $M = 2.081 M_\odot$ and radius $R = 12.6$ km, obtained with the GM1 equation of state with $x_{\sigma} = 0.8$. This star corresponds to the critical mass configuration when we consider $B = 150$ MeV/fm$^3$ and $\sigma = 30$ MeV/fm$^2$ (see table \[tabgm1\]). $R_Y$ is the radius of the hyperonic matter stellar core. The nuclear matter layer extends between $R_Y$ and $R_{crust}$. The crust extends between $R_{crust}$ and $R$. []{data-label="profile"}](fig7.eps "fig:"){height="8.5cm"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --
It has been argued by several authors [@afo86; @mad88; @ob89; @cf91] that if strange quark matter (SQM) is absolutely stable [@bod71; @witt84], then all compact stars are likely to be strange stars. The argument in favor of this thesis is the following: if the interstellar medium is sufficiently contaminated by [*quark nuggets*]{} ([*i.e.*]{} lumps of SQM), then the presence of a single quark nugget in the interior of a “normal” neutron star (hadronic star) is sufficient to trigger the conversion of the star to a strange star [@witt84; @afo86]. Likely the quark nugget contamination of the interstellar medium is the result of the merging of strange stars in binary systems [@mad88; @mad06]. Under these conditions, compact star progenitors could capture a quark nugget during their [*lives*]{} ([*i.e.*]{} during the various nuclear burning stages of the stellar evolution). Thus, according to this argument, a strange quark seed will be present in all new born compact stars, and thus the conversion to a strange star will happen immediately, without a metastable hadronic star being formed first. This is a plausible scenario, however it would be relevant only for a few of the stellar models considered in our work, [*i.e.*]{} those relative to the value $B = 75$ MeV/fm$^3$ for the bag constant (see Tables \[tabgm1\], \[tabgm3\], and \[tabqmc\]) for which SQM is absolutely stable, and will not have any effect upon the existence of metastable hadronic compact stars in all the other cases considered in the present work. The magnitude of the flux of quark nuggets in the interstellar medium (which is a crucial quantity for the validity of the scenario of ref.s [@afo86; @mad88; @ob89; @cf91]) has been estimated [@mad88] making the assumption that all pulsars exhibiting glitches must be “normal” neutron stars (hadronic stars), not strange stars. This assumption is based primarily on the nearly total lack of models for the glitch phenomenon with strange stars (see anyhow ref. [@horv04; @xu]), while such models have been quite successfully developed in the case of hadronic stars (see [*e.g.*]{} ref. [@ai75; @eb88; @leb93; @jon97]). However, recent studies have established the possibility of an inhomogeneous crystalline color superconducting phase (LOFF phase) in the interior of strange stars (see [@mrs07; @cn04] and references therein quoted), or the likely existence of a SQM crystalline crust in strange stars [@jrs06]. These late theoretical developments rise the possibility to explain pulsar glitches with strange star based models, and thus require as useful a recalculation of the astrophysical limits of the flux of quark nuggets. The scenario discussed in the present work is an alternative to the scenario [@afo86; @mad88; @ob89; @cf91] according to which all compact stars are strange stars, which requires that SQM is absolutely stable.
Conclusions {#sec:conclusions}
===========
It has been recently shown [@be03; @bo04; @drago04; @lug05; @blv07] that pure hadronic compact stars, above a threshold value of their gravitational mass, are metastable to the conversion to quark stars. In this work we have done a systematic study of the metastability of pure hadronic compact stars using different relativistic hadronic models for the equation of state of hadronic dense matter. In particular, we have used and compared the quark-meson coupling (QMC) model with those for the Glendenning–Moszkowski parametrization of the non-linear Walecka model (NLWM). In the case of the QMC model, we have obtained that the region of metastability of pure hadronic stars is very narrow. For the GM model, we have investigated the effect of the hyperon couplings on the critical mass of the hadronic star sequence and on the stellar conversion energy. We have found that increasing the value of the hyperon coupling constants shifts the bulk transition point for quark deconfinement to higher densities, increasing the value of the critical mass for the hadronic stellar sequence, and thus makes the formation of quark stars less likely. The nucleonic EoS for QMC is very soft and therefore the onset of hyperons occurs at quite high densities, which gives rise to large critical masses. The conversion to a quark star will occur only for a small value of the bag constant. Finally we point out that both QMC and GM1 with the largest values of the hyperon-meson couplings predict [*limiting masses*]{} [@bo04] which may be as high as 1.9 - 2.1 $M_\odot$.
These values would be able to describe highly-massive compact stars, such as the one associated to the millisecond pulsars PSR B1516+02B [@frei07a], and nearly the one in PSR J1748-2021B [@frei07b].
[ccccccccccccccc]{} & & & & &&\
\
& & & & & & & & & & & & & &\
\
0.6 & 75 &1.630 & 1.968 & 1.326 & 1.454 & 0.079 & 1.254 & 128.4 && 1.471& 1.630& 0.147 & 1.387 & 149.4\
& 85 &1.542 & 1.812 & 1.447 & 1.596 & 0.134 & 1.385 & 110.4 && 1.540& 1.711& 0.201 & 1.479 & 125.5\
&100&1.457 & 1.661 & 1.598 & 1.789 & 0.261 & BH & BH && 1.658& 1.865& 0.332 & BH & BH\
&150&1.447 & 1.601 & 1.770 & 2.010 & 0.527 & BH & BH & & 1.790& 2.036& 0.637 & BH & BH\
\
0.7 & 75 & 1.630 & 1.968 &1.442 & 1.595 &0.059 &1.361 & 144.8& & 1.602 &1.794 &0.119&1.507 &169.6\
& 85 & 1.542& 1.812 & 1.584 & 1.764 &0.111 & 1.509 & 134.2 & & 1.685 &1.892 &0.170 & BH& BH\
&100& 1.457& 1.661 & 1.724 & 1.950 &0.197 &BH & BH & & 1.791 & 2.036 & 0.253& BH& BH\
&150& 1.518& 1.686 & 1.905 & 2.188 & 0.370&BH & BH & &1.931 & 2.223 & 0.403 &BH & BH\
\
0.8 & 75& 1.630& 1.968 & 1.592 & 1.782 & 0.044& 1.498 & 167.9 && 1.763 & 2.000 &0.095 & BH & BH\
& 85 & 1.542 & 1.812 & 1.735 & 1.954 & 0.085& BH & BH && 1.841 & 2.091 &0.132 & BH & BH\
&100 & 1.457 & 1.661 & 1.879 & 2.152 & 0.152& BH & BH &&1.946 & 2.243 & 0.194 & BH & BH\
& 150 & 1.518& 1.686 & 2.054 & 2.391 & 0.275& BH & BH && 2.081 & 2.429 & 0.299& BH & BH\
\[tabgm1\]
[ccccccccccccccc]{} & & & & &&\
\
& & & & & & & & & & & & & &\
\
0.6 & 75 & 1.630& 1.968 & 1.237 & 1.351 & 0.092 & 1.175 & 111.6 && 1.269 & 1.389 & 0.110 & 1.204 & 115.4\
& 85 & 1.543& 1.812 & 1.350 & 1.482 & 0.165 & 1.298 & 91.3 && 1.362 & 1.497 & 0.178 & 1.310 & 92.8\
&100 & 1.465& 1.673 & 1.461 & 1.626 & 0.307 & 1.431 & 54.9 && 1.469 & 1.636 & 0.316 & 1.438 & 55.8\
&150 & 1.487& 1.658 & — & — & — & — & — && — & — & — & — & —\
\
0.7 & 75 & 1.630 & 1.968 & 1.373 & 1.510 & 0.078 & 1.297 & 136.4 && 1.402 & 1.545 & 0.091 &1.324 & 140.4\
& 85 & 1.543 & 1.812 & 1.511 & 1.680 & 0.157 & 1.447 & 113.2 && 1.541 & 1.717 & 0.182 & 1.475 & 117.6\
&100 & 1.465 & 1.673 & 1.610 & 1.806 & 0.253 & BH & BH && 1.645 & 1.851 & 0.295 & BH & BH\
&150 & 1.495 & 1.667 & 1.716 & 1.945 & 0.419 & BH & BH && 1.723 & 1.956 & 0.444 & BH & BH\
\
0.8 & 75 & 1.630 & 1.968 & 1.574 & 1.759 & 0.076 & 1.482 & 165.2 && 1.611 & 1.806 & 0.092 & 1.516 & 170.7\
& 85 & 1.543 & 1.812 & 1.694 & 1.913 & 0.143 & BH & BH && 1.744 & 1.979 & 0.181 & BH & BH\
&100 & 1.465 & 1.673 & 1.771 & 2.014 & 0.204 & BH & BH && 1.802 & 2.057 & 0.234 & BH & BH\
&150 & 1.495 & 1.668 & 1.848 & 2.119 & 0.295 & BH & BH && 1.856 & 2.131 & 0.309 & BH & BH\
\[tabgm3\]
[cccccccccccccc]{} & & & &&\
\
& & & & & & & & & & & & &\
\
75 & 1.630 & 1.968 & 1.587 & 1.768 & 0.044&1.488 &176.8 && 1.694 & 1.903 & 0.090 & 1.585 & 195.8\
85 & 1.530 & 1.793 & 1.705 & 1.917 & 0.096& BH & BH && 1.768 & 1.998 & 0.145 & BH & BH\
100 &1.454 & 1.656 & 1.790 & 2.027 & 0.168& BH & BH && 1.830 & 2.080 & 0.211 & BH & BH\
150 &1.479 & 1.638 & 1.898 & 2.171 & 0.352& BH & BH && 1.909 & 2.187 & 0.377 & BH & BH\
\[tabqmc\]
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was partially supported by FEDER/FCT (Portugal) under the projects POCI/FP/63918/2005 and PTDC/FIS/64707/2006 and by the Ministero dell’Università e della Ricerca (Italy) under the PRIN 2005 project [*Theory of Nuclear Structure and Nuclear Matter*]{}.
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J. Madsen, arXiv:astro-ph/0612740. J.E. Horvath, arXiv:astro-ph/0404324. R.X. Xu, [*Astrophys. J.*]{} [**596**]{} (2003) L59; R.X. Xu, and A.Z. Zhou, arXiv:astro-ph/0411018.
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[^1]: We have found a typo in one of the GM EoS coupling constants in the code used by the authors of ref. [@bo04]. In the present calculations, we have corrected this typo and we have increased the numerical accuracy of our code. This justifies the small differences between the present results and those reported in ref. [@bo04].
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SOPA Sponsors Received 4X in Contributions from Hollywood as from Silicon Valley - luigi
http://maplight.org/content/72896
======
MarkPNeyer
this has nothign to do with sopa, do not read it.
[https://plus.google.com/107304794162956058165/posts/bRpzedGR...](https://plus.google.com/107304794162956058165/posts/bRpzedGRihy)
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Serotonin abnormalities in Engrailed-2 knockout mice: New insight relevant for a model of Autism Spectrum Disorder.
Autism spectrum disorder (ASD) is a congenital neurodevelopmental behavioral disorder that appears in early childhood. Recent human genetic studies identified the homeobox transcription factor, Engrailed 2 (EN2), as a possible ASD susceptibility gene. En2 knockout mice (En2-/-) display subtle cerebellar neuropathological changes and reduced levels of tyrosine hydroxylase, noradrenaline and serotonin in the hippocampus and cerebral cortex similar to those ones which have been observed in the ASD brain. Furthermore other similarities link En2 knockout mice to ASD patients. Several lines of evidence suggest that serotonin may play an important role in the pathophysiology of the disease. In the present study we measured, by using an HPLC, the 5-HT levels in different brain areas and at different ages in En2-/- mice. In the frontal and occipital cortex, the content of 5HT was reduced in En2-/- 1 and 3 months old mice; in 6 month old mice, the difference was still present, but it was not statistically significant. The 5-HT content of cerebellar cortex was significantly reduced at 1 month old but significantly high when the KO mice reached 3 months of age. The increase was present even at 6 months of age. A similar trend was highlighted by SERT immunolabeling in En2-/- mice compared to control in the same areas and age analyzed. Our findings, in agreement with the current knowledge on the 5-HT system alterations in ASD, confirm the early neurotransmitter deficit with a late compensatory recovery in En2 KO-mice further suggesting that this experimental animal may be considered a good predictive model for the human disease. |
Influence of age on mianserin pharmacokinetics.
The pharmacokinetics of Mianserin (MIA) after acute administration have been studied in nine volunteers, divided into two groups according to age. The subjects were given a single oral dose of 30 mg MIA. The plasma peak time was shorter in the younger subjects than in the older. In general, the concentrations of MIA in the plasma were higher in the older subjects than in the younger, and in the former group there was no relationship between administered dose (mg/kg) and plasma levels. The area under curve, volume of distribution and clearance were significantly different in the two groups. The side effects (both incidence and type) differed in the two groups. There were no significant changes in blood pressure, either supine or standing. The sedative effect was more marked in the young than in the elderly subjects. A relationship between drowsiness and MIA plasma levels was observed only in the younger subjects. |
# 前端每周清单第 7 期：Next 2.0 发布，Safari 10.1 新增系列重要特性，Vue.js 2.2 完整 API 手册
`前端` `前端每周清单`
[前端每周清单](http://www.infoq.com/cn/FE-Weekly)专注前端领域内容，分为新闻热点、开发教程、工程实践、深度阅读、开源项目、巅峰人生等栏目。关注【前端之巅】微信公众号(ID：frontshow)，及时获取前端每周清单。
## 新闻热点
`国内国外，前端最新动态`
- [《Safari 10.1 发布，新增众多重要 Web 特性》](https://parg.co/bC2)：近日随着 iOS 10.3 与 macOS Sierra 10.12.4 的版本发布，系统内置的 Safari 10.1 增加了许多重要的 Web 特性支持与性能提升，包括 CSS Grid 布局、fetch、IndexDB 2.0、Custom Elements、Form Validation、Media Capture 等等。( https://parg.co/bC2 )
- [《Next.js 2.0 发布》](https://zeit.co/blog/next2)：近日 Next 2.0 正式发布，它为我们提供了便捷的快速开发 React 服务端渲染的工具，使得开发者能够专注于业务开发。在 2.0 版本中它提供了自定义路由、自定义服务端渲染代码、组件内 CSS、Prefetching 等等多个新特性。( https://zeit.co/blog/next2 )
- [《echarts v3.5 发布：新增日历坐标系、坐标轴指示器；同时统计扩展 v1.0 发布》](https://parg.co/bhh)：在 echarts 新发布的 3.5 版本中，新增了日历坐标系，增强了坐标轴指示器。同时，echarts 统计扩展 1.0 版本发布了。日历坐标系用于在日历中绘制图表，坐标轴指示器方便用户观察数据内容，统计扩展是一个专门用来进行数据分析的工具。( https://parg.co/bhh )
- [《Relay 1.0.0 发布》](https://twitter.com/leeb/status/847635878486745088)：近日 Relay 1.0.0-alpha 发布，提供了许多新的特性。作为 Facebook 发布的全特性 GraphQL 客户端，其能够在低配的移动设备上构建高性能、复杂可扩展的应用。( https://twitter.com/leeb/status/847635878486745088 )
## 开发教程
`步步为营，掌握基础技能`
- [《使用 Vue.js 与 Electron 构建桌面问卷应用》](https://parg.co/bQ3)：本文介绍了如何利用 Vue.js 与 Electron 来构建简单的桌面问卷应用，作者首先介绍了如何使用 vue-cli 创建简单的 Web 项目，然后讨论了如何将项目运行在 Electron 中，最后阐述了如何将应用整体打包发布。( https://parg.co/bQ3 )
- [《Progressive Web Apps：响应式 Web 设计的延伸》](https://julian.is/article/progressive-web-apps/)：本文是对于 Progressive Web Apps 概念、设计理念与简单实践的介绍，作者介绍了 PWA 应用应该具备的基本特性、性能与体验上的要求以及如何将现有站点转化为 PWA 的简单实践。( https://julian.is/article/progressive-web-apps/ )
- [《2017 简明 React 入门指南》](https://parg.co/bCx)：本文是针对那些熟悉 jQuery 与传统 JavaScript 开发的前端工程师准备的现代 React 开发入门指南，其包括了环境配置、create-react-app 使用、学习资料、应用编写与发布等等章节。( https://parg.co/bCx )
- [《8 个可能你没考虑过关于 CSS 的知识》](https://parg.co/bhl)：不同的技术学习曲线可能相差甚远，而 CSS 的初学者则会觉得相当容易入手，但是深入之后可能发现连居中都不甚容易。本文作者是个深度 CSS 热爱者，他从自己多年的经验介绍了 CSS 中的垂直居中、100% 属性、float、z-index 等等多个细节知识点。( https://parg.co/bhl )
- [《React Bits》](https://github.com/vasanthk/react-bits)：一本关于 React 设计模式、技术与技巧的书，涵盖了常见的 React 应用开发中的设计模式、需要规避的反模式、处理 UX 变种、性能调试与样式处理等等。( https://github.com/vasanthk/react-bits )
- [《实例讲解 CSS 盒模型》](https://parg.co/bhN)：有经验的前端开发者都知道 HTML 中的布局就是盒套盒，而本文则是以某个街区的例子深入浅出地讲解 CSS 中的盒模型。( https://parg.co/bhN )
## 工程实践
`立足实践，提示实际水平`
- [《12 个精妙的 JavaScript 代码片》](https://parg.co/bhH)：本文作者分享了十二个非常不错的 JavaScript 代码片，这些代码片能够帮你优化现有代码，让代码更加地赏心悦目。( https://parg.co/bhH )
- [《Node.js 应用监控实践指南》](https://parg.co/bhb)：本文介绍生产环境下 Node.js 应用监控实践指南，包括了监控的意义、监控的对象、目前开源的监控解决方案以及一些 SaaS 解决方案等。( https://parg.co/bhb )
- [《使用 Faker.js 为 Node.js 应用创建模拟数据》](https://parg.co/bhU)：在应用开发中我们往往会头疼于如何构建大量的随机数据，特别是那些符合某些固定模式的数据，我们可能会要用这些数据仿制 RESTful 接口、进行单元测试等等。而 Faker.js 则为我们提供了这样的随机数据生成器。( https://parg.co/bhU )
- [《Vue.js 2.2 完整 API 清单》](https://parg.co/bhC)：本文是 Vue.js 2.2 中完整的 API 介绍，可以作为手册随时查阅 。( https://parg.co/bhC )
- [《JavaScript 中构建响应式引擎》](https://parg.co/bhR)：本系列文章介绍了如何在 JavaScript 中构建高性能的响应式引擎，对于有兴趣了解 MobX 底层原理的同学来说也是个不错的教程，目前包含了对于可观测对象的构造解释、属性推导与依赖追踪等内容( https://parg.co/bhR )
## 深度阅读
`深度思考，升华开发智慧`
- [《深入浅出构建简单的 Chess AI》](https://parg.co/bCw)：本文作者介绍了如何基于 JavaScript 构建一个国际象棋的 AI，虽然不属于前端开发范畴，不过还是蛮有意思的一篇文章。本文主要包括移动生成、棋盘可视化、位置评估、基于 Minimax 算法的搜索树、Alpha-beta 修剪等等。( https://parg.co/bCw )
- [《构建高性能扩展与折叠动画》](https://parg.co/bCz)：本文以菜单伸缩动画为例，介绍如何构建高性能扩展与折叠动画。较简单但是性能有缺陷的方式譬如修改元素宽高或者使用 clip 变换属性；而本文主要是由 CSS3 的 scale 变换来实现菜单的扩展与折叠，其为了保证菜单按钮的视觉效果与整体的平滑缩放还使用了所谓的对冲缩放技巧。( https://parg.co/bCz )
- [《基于 ReactNaive 与 Uber 工程基础构建 UberEATS》](https://eng.uber.com/ubereats-react-native/)：本文是 UberEATS 的工程师团队介绍的他们基于 Uber 原工程架构与 ReactNative 实现应用的工程实践；包括了构建迁移路径、应用架构定义、自动更新、测试与静态类型检测等等。( https://eng.uber.com/ubereats-react-native/ )
- [《利用机器学习优化网站性能》](https://parg.co/bhQ)：本文是从浅显机器学习的角度来考虑如何优化网站性能，可能从专业服务端机器学习的角度来看并不复杂，但是从前端网站构建的角度来看也是蛮有意思的。本文包括了记录网站点击与提交信息、利用 AWS S3 进行模型训练、利用 UI 进行可视化展示等等。( https://parg.co/bhQ )
- [《Chrome 中 Preload、Prefetch 以及 优先级介绍》](https://parg.co/bhM)：本文是 Google Chrome 团队的 Addy Osmani 对 Chrome 中的 Preload、Prefetch 以及抓取优先级的介绍，并且介绍了在网站性能优化中对于 Preload 与 Prefetch 的使用技巧和 HTTP/2 带来的服务端推送如何协同使用等内容。( https://parg.co/bhM )
## 开源项目
`乐于分享，共推前端发展`
- [《jsinspect》](https://github.com/danielstjules/jsinspect)：jsinspect 提供了方便的命令行工具或者构建插件来检测代码库中的复制粘贴或者结构相似的代码片，以方便开发者在优化过程中完成代码的重构。( https://github.com/danielstjules/jsinspect )
- [《Reactide》](https://github.com/reactide/reactide)：Reactide 是首个面向 React Web 应用开发 IDE，其基于 Electron 提供了跨平台的特性。Reactide 允许我们像传统开发那样打开单个文件就可以完成预览，并且提供了便捷的组件导入、格式化等功能。该项目仍处于积极的开发中，可以拭目以待。( https://github.com/reactide/reactide )
- [《marky》](https://github.com/nolanlawson/marky)：marky 是基于 performance.mark/measure 封装的高性能 JavaScript 计时器，其相较于`console.time()`以及`console.timeEnd()`具有更好地性能表现，相较于简单的`Date.now()`具有更好地准确度。( https://github.com/nolanlawson/marky )
- [《Service Worker Mock》](https://parg.co/bCD)：在 PWA 应用的开发中对于 Service Worker 的测试一直比较麻烦，每个文件都可能通过`self.addEventListener`产生副作用，并且 Service Worker 的运行环境也迥异于正常的 Web 或者 Node 环境。而本包则是通过注入伪造的 Service Worker 环境来方便测试。( https://parg.co/bCD )
- [《Public JSON APIs》](https://github.com/toddmotto/public-apis)：本仓库列举了许多 Web 开发中用到的公开的 JSON 接口，包括信息检索、机器学习、工具使用等等多个方面。( https://github.com/toddmotto/public-apis )
- [《generator-ngx-app》](https://github.com/angular-starter-kit/generator-ngx-app)：Angular 4 商业级应用项目生成器，其包括了 angular-cli 提供的现代工具与工作流，以及来自于社区的最佳实践、可扩展的基础模板以及较好地学习曲线。( https://github.com/angular-starter-kit/generator-ngx-app )
## 巅峰人生
- [《WWW 之父 Tim Berners-Lee 获图灵奖》](https://parg.co/bhv)：美国计算机学会(ACM)宣布将 2016 年的图灵奖授予万维网(WWW)的发明者 Tim Berners-Lee。他将获得由 Google 赞助的一百万美元奖金。ACM 提到他的获奖理由是，“inventing the World Wide Web, the first web browser, and the fundamental protocols and algorithms allowing the web to scale.”。( https://parg.co/bhv)
## 前端之巅
「前端之巅」是 InfoQ 旗下关注前端技术的垂直社群，加入前端之巅学习群请关注「前端之巅」公众号后回复“加群”。投稿请发邮件到 [email protected]，注明“前端之巅投稿”。
|
Bionova Omegasure Syrup, 200ml
Vitamin E mainly acts as an antioxidant in your body, combating harmful substances called free radicals. Vitamin E’s antioxidant effects protect body’s cells from death or DNA damage, and also helps your body to utilize vitamin K and form red blood cells. Flaxseed oil serves as a source of omega-3 fatty cids, particularly alpha-linonlenic acid, or ALA. Body converts this ALA into docosahexaenoic acid, or DHA, and eicosapentaenoic acid, or EPA -- two fats that benefit cardiovascular health and play a role in healthy brain function. Omega-3 fatty acids help your body to control inflammation, along with promoting overall health. Flaxseed oil is often manufactured with vitamin E because the vitamin helps keep the oil from spoiling when exposed to heat, light and oxygen. |
The Quick Guide to Bangkok’s Chatuchak Weekend Market if You’re Short on Time…
If you want to check out Chatuchak (or Jatujak also JJ Market; Thai: จตุจักร) in Bangkok and you don’t have a lot of time, I’m going to show you where you should go to pick up unique hand made Thai souvenirs, eat delicious Thai street food and of course shop to your hearts content.
Because JJ Market is such a huge sprawling weekend market you can easily run around like a headless chicken. And who wants to get lost and miss the good parts? Plus if you’re going in on a hot day it won’t be a lot of fun exploring the shops and stalls that makes the city of Bangkok so unique.
Lots of local designers open shop in Zone #2…
The biggest weekend market in Thailand is organized and separated into zones. Below you will find one of the best maps of Chatuchak Weekend Market I found on the internet.
But Here’s a Few Tips You Want to Know Before You Go
It’s best to visit Chatuchak around 10am when most of the shops open. Plus it’ll be a bit cooler and less crowded.
Best time to get a discount is when the shop just opened. Thais are very superstitious and believe the first sale of the day is the most important. They will take the cash you give them and rub it on their products.
Hydrate, hydrate and hydrate more. If you can’t get to Chatuchak early in the day it can get very hot. Small bottled waters cost 10THB so there is no reason to not get plenty of fluids.
The Do Not Miss Areas inside JJ Market
Remember, Chatuchak is separated into zones, each specializing on the items and goods sold. If you don’t know where you’re going it’s very easy to get lost as every nook and corner begins to look same. You can get plenty of maps online or at Thai tourist info centers. That said, here are the must see zones I highly recommend if you don’t have a lot of time to explore JJ Market:
Zone #2 is the bohemian/hipster chic area of the market. Once you take a look at this section and a look at all the others you will not only see zone #2 is different but also feels different!
Zone #5 and #6 is the used clothes section but you can also find some new clothing items too. You will see brand name jeans for sale in that area though I can not confirm if the jeans are genuine or not.
The best map of Bangkok’s Chatuchak Weekend Market I ever found on the internet…
As you can see on the map, zones #2, #5 and #6 are all on the same side of the market. If you walked from zone #2 to zone #6 it would only take 30 minutes that is if you don’t stop and check out the goodies. Good luck on that.
I recommend starting off at zone #2 simply because the Kamphaeng Phet MRT subway station is practically right next to zone #2. Just be sure to use subway station Exit 2.
This is exit 2 of the Kamphaeng Phet subway station, very close to zone #2…
Get Ready to Shop and Drop in Bangkok
Zone #2 near has gone through a sort of transformation. It’s like a hipsters’ paradise filled with small little boutique shops started by enterprising Thai designers that live in Bankgok.
Now if you take a good look at the map below, you will see many other sections also selling clothes and Thai handicraft, particularly right in the center of Chatuchak.
Zone 2 has a different feel and vibe…
Zone #2 is the top spot for finding Thai handmade souvenirs in Chatuchak at reasonable prices. It’s also the best place to buy unique graphic t-shirts and even shoes, both brand new or second hand. I know a lot of Japanese heading over to JJ Market looking for vintage clothing items to resell back in their home country.
The best map of Bangkok’s Chatuchak Weekend Market I ever found on the internet…
But Be Prepared, Cause There’s Going to Be a Lot of People
Zone #2 is perhaps one of the most visited section of Chatuchak Weekend Market. Thai locals have a voracious appetite for the new. They love to seek the unique and share it on Facebook. And that’s how really zone #2 became so popular through word of mouth.
Thai hand made items can be found in zone #2…
Prices are higher in zone 2 but not by much…
Lots of graphic t-shirts can be found in Chatuchak…
Zone 2 has a lot of one of a kind boutique shops…
Many smaller shops selling stone bracelets and necklaces too…
My favorite t-shirt shops are in zone 2…
Seems like white t-shirts are popular in Bankgok….
Many boutique shops in Chatuchak follow fashion trends…
Now I’m not saying the rest of Chatuchak is not worth visiting. No way. I love the whole entire market. It’s just that if you don’t have a lot of time and you don’t want to sweat buckets when it’s hot you just want to see the best part. And the zone #2 is where you’ll want to go if you don’t have a lot of time to visit Chatuchak Market.
Lots of fusion grilled food found in zone 2
Coffee and cakes are very popular in Bangkok…
Many fruit drink stands are found as well along zone 2…
And of course one of Bangkok’s famous dish is represented, sticky rice mango..
There’s just so many unique little treasures located in Chatuchak’s Zone #2 that I don’t even know how I can ever list them all. So it’s best if you go there and check them out for yourself. You just might find something you’re going to fall in love with. And I recommend that if you see it, it’s best to just buy it. As long as it fits in your budget of course. Because the worst thing you can do is getting back on that plane and regretting you never bought it.
Finally items in zone 2 are a bit more expensive compared to the rest of JJ Market because the items are a little more unique. You are allowed to bargain but please don’t be so aggressive. And don’t try and low ball the shop keepers because they do work hard and have bills to pay.
If You Love Denim You’ll love Zone #5 and #6
In this zone you will find used clothing, particularly denim. You can also find stalls selling brand name jeans as well with labels. Now I don’t know if they are fakes or not but the prices are not so expensive. Even though Bangkok is hot and humid through the year Thai locals love denim wear. New or used it doesn’t matter. If it looks good on them and the price is right they have to have a pair of jean pant or jacket.
Prices for clothes in this zone are reasonable though there is space to get discounts as long as you buy more items.
Zone #5 and #6 is a well known location inside Chatuchak for denim wear…
You can spend hours inside looking at selections of new and used clothings
Many used shoes are repaired and sold…
Even old used sneakers are cleaned for resale…
There are a lot of military style clothing sold inside Chatuchak…
Denim is always popular for Thai people…
Military fashion is also popular with Thais…
Local Thais love shopping in zone #5 and #6…
Even used soccer jerseys are sold in zone #5…
If it’s worn and used you’ll probably find it at Chatuchak…
But you’re also going to see a lot of military styled clothing, shoes (new and used) and all sorts of other second hand clothing items. You might even find brand name clothing too. But again, if they are fakes I honestly wouldn’t know.
Getting a Quick Bite in Chatuchak Market
If you still have some time for a quick bite to eat, there’s a small shop selling to die for Chinese styled roast pork. It’s one of the best roast pork I’ve ever had. Even better than the ones I’ve had in Hong Kong! If you don’t like pork they also sell awesome fried chicken wings!
This shop is located on zone #19, Soi 7/1, about a 5 minute walk from zone #2…
These chunks of Chinese roast pork belly are melt in your mouth good!…
I can’t believe Chatuchak Market has the best Chinese roast pork…
Chatuchak is an incredible market filled with lots to see, eat and of course shop. But if you don’t have a lot of time to spend there then just checking out the zones I’ve mentioned on this blog is going to give you the full experience to remember in Bangkok.
24 Comments
Most definitely you will find all types of tableware in Chatuchak. Head to the southern tip of Chatuchak that’s where you will find all of it. I also recommend heading over past JJ Mall there is another outdoor shop, very big selling even more plates of all shapes and sizes.
Do you know what section we could look for the stainless crafted cutlery? We bought some cool sets there years ago and would like another set. Often 6 pieces in one set and you can buy 6 or how ever many settings you want
Not worth to visit at all. Price wise around the same in partunam market or can be more expensive here. Beware of the weight on the snack item, I was cheated on the weight that I bought and is more expensive!!!
Hello Chris! Although you can find antiques in Chatuchak it’s best to be sure you’re really buying an “Antique” Lots of items are purposely made to look aged. Great blog article by the way, I enjoyed reading it!
Hello Joyce, I can understand your confusion. There is JJ Mall, which is a building near Chatuchak Weekend Market which many confuse both as one in the same. So, JJ Mall is opened 7 days a week from 10am(ish) to 9pm(ish).
Then there’s Chatuchak Weekend Market which is also called “JJ” Market opens only on weekends from 10am(ish) to 9pm(ish).
There are no exact hours as each stall is independently owned. Some shops can close earlier, some shops close later.
Hello Warren,
Was wondering if you have came across any shops selling ukelele at Chatuchak Weekend Market? If so, where is the location of the shop?
I saw a photo of a shop selling ukele but had been unable to find more details of the shop.
Thank you in advance.
MY
Might be more convenient to buy it at a shopping mall. The cost might not be big. But unless you’re buying a bunch of other items then I guess it would make sense to go instead of only buying one shirt only. For clothing head to the center of Chatuchak Weekend Market, on the map it’s dark pink area.I don’t have any shop recommendations but shops often move around so you’ll have to just walk around. Have fun!
My Thai wife and I bought some handcrafted framed wall art in a stall in section 20. A sign hanging nearby said Section 20 Soi 5 & 6. The stuff was a huge hit among her Thai friends and we had offers to buy it from us at a nice profit. If I could find the vendor I would buy a bunch more and have them shipped to us. Can anybody help us locate the vendor? |
-60259.0113 rounded to zero dps?
-60259
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-3596000
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9.9
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-0.0000021
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-4940.3
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159100000
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16470000
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60.2
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-532400
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-0.001103
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3500
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-228300000
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5695900000
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-21590000
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3000
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65.19
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13.75
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30
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-64440000
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0.11
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-5.723
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-6674000
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15200000
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0
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1070000
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-107.636
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0.0000863
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-0.0007
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-42.3
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554000000
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-1.656
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664
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1000000
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0.000049
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4385200000
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20000
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0.0000326
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-10.09
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-50395
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0.00104
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-0.047
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9156
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-6700000
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15062
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-7770000
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-800000
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-14.07
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0
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-1310000
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2600000
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-7000
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2291800000
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-0.017
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0.11
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-290
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1.653
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85.773
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-3170
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-0.02898
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217.02
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-0.0805
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13198000000
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920000
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871.5
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-0.07
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0.002086
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24000000
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2000
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79500000
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-100000
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0.0033
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-8000000
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1.0325
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4700
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2463000000
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30
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0.29
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-40
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0.001
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-2547.9
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150.57
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290
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-0.0000028
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1990.4
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1716.9
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0.0004983
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489000
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-559500000
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-0.000004
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0.000007
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0.0000011
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-0.0036
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-0.000487
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-11
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-3180000
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300
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-19000000
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-0.006932
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0.0002197
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0.0885
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-0.0000017
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28.6
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0.4
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0.000536
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-0.0000025
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220
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-0.000271
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-0.02051
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90030000
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91730
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-89000000
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-56599000
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-4865
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-0.000056
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2511000
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-2194
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4.8
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-0.0015326
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-22.5
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-7652000
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13
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2500
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136000
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-6980000
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0.00024
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-76000
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0.033971
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-0.0071
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1500
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-0.006
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-259000
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-0.001866
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-28
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200
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0.00038
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257200000
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-7.48
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-971500000
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0.001354
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720
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287200000
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1390000
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-363570000
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-17100000
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-187600
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-0.0013
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3794000000
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-0.014146
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-6830000
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-0.0000435
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8000
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320400000
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-0.011
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-11000
Round -0.00105201447 |
[Advance on research of gene expression during spermiogenesis at transcription level].
After meiosis, round spermatid develops into mature sperm through metamorphosis. During this stage, most cytoplasm in the germ cell is gradually lost. The histones associated with chromatin are replaced by transition proteins and eventually transformed into protamines. Thus, the spermatid chromatin is stringently packaged and highly concentrated. It was thought that the transcription activity of spermatid is lost and RNAs are absent in spermatid. Nevertheless, many types of transcripts are detected in recent years, including the transcripts needed during chromatin repackaged and some small RNAs, etc. Because histones in the nuclear are not replaced entirely, and there are some active sites on the chromatin, we conjectured that spermatid has some transcription activity, and this activity is regulated by hormone and epigenetic modification. These RNAs may be the residues in the spermatogenesis, or timely expressed during spermiogenesis. A deep study on gene transcription in spermiogenesis will help understand the genetic characteristics and provide the theoretic basis for reproductive control using male gamete. This article reviewed recent advances in spermiogenesis at gene transcription level and proposed the future research directions. |
The Japanese women’s clothing fashion brand WEGO has announced that they will be collaborating with Persona 5 for items that will be going on sale on March 16, 2018.
Items that will be released consist of a unisex hoodie which comes in two colors, the T-shirt Futaba Sakura wears, and a Morgana plush toy backpack.
The clothing will be available for purchase on the WEGO web store, as well as physical locations in Japan.
Layered T-Shirt
Hoodie (Black / Red)
Morgana Plush Toy Backpack
Product Details
Unisex Hoodie (Black, Red / Medium, Large): 3,590 yen + tax (3,877 yen including tax)
3,590 yen + tax (3,877 yen including tax) Layered T-Shirt (Black / One size fits all): 3,590 yen + tax (3,877 yen including tax)
3,590 yen + tax (3,877 yen including tax) Morgana Plush Toy Backpack (Black): 4,990 yen + tax (5,389 yen including tax)
— WEGO |
Letters to the Editor: Tom W. Davis
Published in the Tacoma WeeklyWednesday, 12 December 2012
Dear Editor ,
I read “Council should avoid Citizens United” (staff editorial, TW 11/23). I agree on your stance about Arizona and some other state issues; however, I feel it is most important Tacoma City Council cannot have its hands tied by not being able to address issues important to the citizens of Tacoma. We must not make them myopic and unaware of trends and matters that could or do affect us now or in the future. I applaud the council for keeping aware of all issues affecting our citizens and issues that could affect us, in order that we can start thinking what is right and what is wrong, in advance of entertainment news, which is about 99 percent nonsense and non-factual. Folks must support issues and an uninformed electorate is democracy lost. I am a supporter of newsprint, as good factual news will keep us a free people.
Regarding the U.S. Supreme Court’s 2010 decision in Citizens United v. Federal Election Commission: This decision has been described by most all Republican and Democrat lawyers to be unconstitutional. No one, except big money union bosses, who get to spend more of the member dues and travel more, and of course our do nothing Legislature in Washington agree this is a monumental decision.
Our legislators have spent a year collecting money from lobbyists, traveling to announce their positions and line up their parties in their states, while not doing a darn thing to solve this nation’s problems. It is in fact a monumental blunder by the Supreme Court that should be reversed. |
The teleprompters at President Barack Obama’s inaugural address were still powering down when British literary heavyweight Jonathan Raban deemed Obama "the best writer to occupy the White House since Lincoln."
Raban was hardly alone in this enthusiasm. Later that year, Rocco Landesman, chairman of the National Endowment for the Arts, told a conference of (sigh!) grant writers, “ This is the first president that actually writes his own books since Teddy Roosevelt and arguably the first to write them really well since Lincoln.”
Never mind that Landesman, an Obama appointee, overlooked Grant, Wilson, Hoover, and Nixon among other book writers or that Lincoln never wrote a book. The point was made: Obama was Lincoln’s literary heir apparent.
In a speech to an SRO crowd of historians in late 2010, Harvard’s James Kloppenberg upped the adulatory ante. He described the president as "gifted," a "genius," a man of "exceptional intelligence," one who writes "brilliantly and poignantly.” Obama, as Kloppenberg saw it, was a “true intellectual” in a class with Adams, Jefferson, Madison, Wilson and, yes, Abraham Lincoln. According to the New York Times, the crowd greeted his extended gush “with prolonged applause.”
Abraham Lincoln knew something about madness. If he feared anything more than “a human form with reason fled,” the subject of an 1846 poem, it was “the wild and furious passions” of a mob. One could only imagine what he would have thought of a mania that not only carried a pretender like Barack Obama to the White House but that also, fantastically, lifted him to the literary heights.
Thanks to Fred Kaplan’s insightful 2008 book, Lincoln: the Biography of a Writer, the reader has a sense of how Lincoln might have responded. In Stephen Douglas, after all, Lincoln faced an opportunistic Illinois Senator whose rhetoric combined “ linguistic shiftiness, hyperbole, and disregard for the integrity of fact.” He knew the type. As Kaplan shows, only Lincoln’s relentless self-improvement as a thinker and writer enabled him, finally, to prevail.
That Lincoln could write at all was unusual for his time and place. His father, a hardscrabble farmer on civilization’s edge, could not write and saw no reason for his son to learn. Young Abe simply willed himself. True, he did not have the distractions that plagued the young Obama--say, TV’s Brady Bunch or body surfing at Waikiki—but he did have distractions of his own. Backbreaking fieldwork comes to mind. Still, Lincoln persisted.
His friends noticed. However cursory his schooling, Lincoln was “always ahead of all the classes he ever was in,” said one. He was “exceedingly studious” said another. He would write whenever he could, sometimes just on chalkboard if there was no paper. Even as a boy, he wrote about things of consequence—temperance, slavery, cruelty to animals, American history. By nineteen, said a friend, Lincoln was “the best penman [writer] in the Neighborhood.”
No one attests to Obama’s early intellect or industry. Sympathetic biographer David Remnick tells us that he was an “unspectacular” student in his two years at Columbia University and at every stop before that going back to grade school. A Northwestern University prof who wrote a letter of reference for Obama tells Remnick, “I don’t think [Obama] did too well in college.”
Despite his luxurious education—Punahou, Occidental, Columbia, Harvard Law—Obama has left a surprisingly slim paper trail, much slimmer, in fact, than that of the largely self-schooled Lincoln. Still, despite Obama’s dogged efforts to bury his academic past, a few prose artifacts have surfaced, the earliest being an 1800-word article in the Columbia University’s weekly news magazine, Sundial.
Remnick describes the March 1983 article, “Breaking The War Mentality,” as “muddled.” He is being kind. If the average citizen need not overly trouble himself with issues of syntax and grammar, a senior at an Ivy League university is expected to. Yet “Breaking” is so far below the Ivy norm that it raises serious questions about Obama’s admission to Columbia, let alone his rapid literary ascent in the years to follow.
In the article, Obama celebrates “the flowering of the nuclear freeze movement.” He then questions “whether this upsurge comes from young people's penchant for the latest 'happenings' or from growing awareness of the consequences of nuclear holocaust.”
“Upsurge,” of course, is the wrong word. “Happenings” should be singular, and even then it sounds like something Mike Brady would have said to Greg or Marcia, but, to answer Obama’s question, the “upsurge” had its origins in the devious imagination of the KGB. Its agents orchestrated the movement to discourage President Reagan from deploying Pershing Missiles in Western Germany. Despite the contributions of “useful idiots” like Obama, they failed.
To be sure, millions of well meaning progressives were likewise duped, but few among them expressed their thoughts quite so ungrammatically. In at least five sentences, for instance, Obama cannot get the subject and verb to agree. This one nicely captures both the article’s botched grammar and its baffling logic:
The belief that moribund institutions, rather than individuals are at the root of the problem, keep SAM's energies alive.
“SAM” stands for “Students Against Militarism.” In the previous paragraph, Obama warned his readers about “the relentless, often silent spread of militarism in the country.” In this paragraph, military institutions are said to be near death. What exactly energizes SAM is far from clear. Again, too, there is an agreement issue. The sentence should read, “The belief . . . keeps SAM's energies alive.”The random use of commas throws everything off. This is typical Obama circa 1983.
The young Lincoln had one advantage over Obama. Although elected president 100 years to the month before Obama was conceived, Lincoln had richer influences. Kaplan cites Gray, Addison, Cicero, Lord Mansfield, Lord Chesterton, and especially Shakespeare and the Bible.
Obama ignored the western canon and looked instead to radical anti-imperialists like Frantz Fanon and Malcolm X, communists like Langston Hughes and Richard Wright, and Stalin-loving fellow travelers like W.E.B. DuBois. In his 1995 memoir, Dreams From My Father, Obama gives no suggestion that this reading was a mere phase in his development. He moved on to no new school, embraced no new worldview.
Lincoln was a little more than a year older than Obama when he penned his first surviving prose piece. As a youthful candidate for the state legislature, he addressed the citizens of Sangamo County. The following excerpt gives a hint of his wit and humility, two qualities that seem to have escaped Obama. Observe, too, that Lincoln’s subjects and verbs all agree.
Considering the great degree of modesty which should always attend youth, it is probable I have already been more presuming than becomes me. However, upon the subjects of which I have treated, I have spoken as I thought. I may be wrong in regard to any or all of them; but holding it a sound maxim, that it is better to be only sometimes right, than at all times wrong, so soon as I discover my opinions to be erroneous, I shall be ready to renounce them.
Five years would pass before the best writer since Lincoln put anything else in print. In 1988, likely to pad his resumé, Obama wrote an essay titled “Why Organize?” for a publication called Illinois Issues. Like the Sundial article, this effort showed not a hint of style, sophistication, or promise.
The article repeats what was emerging as Obama’s signature blunder, the failure to get subjects and verbs to see eye-to-eye. Obama also was developing a distinctive way of letting phrases dangle in space, for instance:
Facing these realities, at least three major strands of earlier movements are apparent.
Nothing works here. “Facing these realities” modifies nothing. “Strands.” In any case, do not “face reality.” People do. This is normative Obama circa 1988. He is now 27 years-old.
Unlike Obama, Lincoln tirelessly practiced his craft, and it showed. As a 28 year-old, he addressed the Young Men's Lyceum of Springfield, Illinois, on the subject of “our political institutions” and here specifically on the mythic histories that once sustained the nation:
They were a forest of giant oaks; but the all-resistless hurricane has swept over them, and left only, here and there, a lonely trunk, despoiled of its verdure, shorn of its foliage; unshading and unshaded . . .
Lincoln could be funny as well as poetic. A few years later, he described a political opponent’s preening at a local dance with such wicked precision that the satire very nearly led to a duel:
. . . and there was this same fellow Shields floatin' about on the air, without heft or earthly substances, just like a lock of cat fur where cats had been fighting.
Obama has no gift for the humorous or the poetic. As in his 1990 letter to the Harvard Law Record defending affirmative action, the unaided Obama prefers to pontificate--awkwardly, illogically, and often ungrammatically.
In the letter’s very first sentence Obama leads with his signature failing: his inability to make subject and verb agree.
“Since themerits of the Law Review's selection policy has been the subject of commentary for the last three issues,” writes Obama, “I'd like to take the time to clarify exactly how our selection process works.” This is one of at least four sentences with an agreement problem.
In this letter, Obama admits to having “undoubtedly benefited” from affirmative action programs, including perhaps the Law Review’s. What he refuses to concede, even to see, is that affirmative action does not bestow talent. As Lincoln knew, only God could bestow that, and only hard work could polish it.
Just four years later, of course, Barack Obama would magically find his mojo and write Dreams from My Father, a book the estimable Joe Klein of Time Magazine would call “the best-written memoir ever produced by an American politician.” Just as incredibly, no one in the literary establishment doubts he wrote every word of it. I do, and in my book, Deconstructing Obama, I explain why. The evidence overwhelms the objective observer.
Inexplicably, for a full decade after Dreams, this young literary lion confined his outsized talents to a semi-regular offering in the neighborhood newspaper, the Hyde Park Herald. If he wrote a single inspired or imaginative sentence in any of these columns, I was unable to find it. Articles headlined “Post Office Should Deliver Answers” and “Getting the Lead Out of Our Children” do not hold much promise in any case.
As an Illinois politician on the rise, Lincoln used his gifts for slightly bigger things, like ending slavery and preserving the union. In an 1854 speech at Peoria, he summarized the cause:
Let north and south—let all Americans—let all lovers of liberty everywhere—join in the great and good work. If we do this, we shall not only have saved the Union; but we shall have so saved it, as to make, and to keep it, forever worthy of the saving.
On his own, Obama could never write a sentence this elegant or consequential. To be sure, his 2004 Democratic Convention speech wowed the crowd, but it was a crowd suffering from what George Bush might have called “the soft bigotry of low expectations.” Even here, Obama advanced no cause but his own.
The same committee that wrote the convention speech wrote Obama’s 2006 book, Audacity of Hope. In the acknowledgment section of Audacity, Obama lists an astonishing 24 people who provided “invaluable suggestions.” (Lincoln did his own research.) Remnick generously called Audacity a “shrewd candidate’s book.” Bill Ayers, the primary craftsman behind Dreams, more accurately called it a “political hack book.”
“We can be certain that he wrote every word to which his name is attached,” says Kaplan of Lincoln. Obama only pretended to the same. "I've written two books," he told a crowd of teachers in Virginia in July of 2008. "I actually wrote them myself."
He did no such thing. And now in his hour of crisis, Obama lacks the confidence, the character and the very word power that Lincoln developed through years of strenuous work and honest self-assessment.
Crises are hard on poseurs. Obama has no better angels to summon, no last full measure of devotion to evoke, and try as he might to bind up the nation's wounds, he will never do so by saying, “It was Bush’s fault.” |
Glossopharyngeal neuralgia as onset of multiple sclerosis.
Glossopharyngeal neuralgia is a painful condition, affecting the ninth cranial nerve, rarely described in the course of multiple sclerosis. Here we describe a case of multiple sclerosis presenting with glossopharyngeal neuralgia. We suggest the presence of demyelinating areas at the nerve root entry zone as principal trigger mechanism. |
Account
Npower parent calls pan-Euro media pitch
German electricity supplier RWE to launch first-ever review across the group, putting the UK incumbent, Vizeum, on alert
Npower: handed its digital media business to Maxus and its ad account to McCann Manchester
Npower’s parent company, RWE, is set to kick off its first-ever group-wide pitch as it reviews its media agencies across Europe.
The review, which is being handled by RWE without an intermediary, is in its early stages, with a tender due to be sent out in October. It is the first time that the German electricity supplier has issued a group-wide brief and comes at a time when it is looking to cut costs across the business.
Last week, RWE announced plans to cut its shareholder dividend and accelerate cost savings as it warned that the earning prospects for its conventional power-generation business had deteriorated.
In the UK, the review will include npower’s £5 million offline media business only and the incumbent, Vizeum, will be invited to repitch. An npower spokeswoman said other country reviews could include digital media and would be decided on a case-by-case basis.
Earlier this year, npower handed Maxus its digital media planning and buying account after a competitive pitch against six agencies.
The company has also appointed McCann Manchester as its lead creative agency. The shop is currently working on a marketing drive to convey its "customer-centric" approach. Npower previously worked with VCCP Blue.
The npower spokeswoman said: "We can confirm that npower, along with our parent company RWE, have put a group-wide media tender in place.
"This is the first time we have worked with our European partners on a marketing tender process and comes at a time when we are reviewing efficiencies across our business to benefit our customers, who are facing an upward pressure on bills."
RWE becomes the latest energy supplier to review its media agency requirements, with results on the EDF Energy and SSE pitches due in the coming months. |
Sky releases backup point guard Carter
Former Texas A&M guard Sydney Carter was released by the Chicago Sky. Carter, the Sky's 2012 draft pick, played in just one WNBA game for the Sky.Associated Press/file
By Daily Herald News Services
After playing in just one WNBA game, guard Sydney Carter has been released by the Chicago Sky, team officials announced.
Carter, a 5-foot 6-inch guard, was the Sky's 27th pick of the 2012 Draft from Texas A&M, and she was signed after leading scorer Epiphanny Prince suffered a foot injury. Playing nine minutes at Minnesota on June 23, Carter contributed 4 points and 1 steal.
"Sydney has continued to impress me in her short time with us," said Pokey Chatman, Sky head coach and general manager Pokey Chatman. "She will remain on my shortlist of quality players."
Chicago begins a four-game homestand at 11:30 a.m. today when it hosts the Indiana Fever. Then the team hosts the Phoenix Mercury on June 29, the Atlanta Dream on July 1 and the New York Liberty on July 6.
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If a comment violates these standards or our terms of service, click the X in the upper right corner of the comment box. To find our more, read our FAQ. |
New investor, the Accident Compensation Corporation of New Zealand (ACC), also joins the round.
Funding will support continued Electron vehicle production expansion, new launch sites and three new major research and development programs.
Huntington Beach, CA – November 15, 2018 – US orbital launch provider Rocket Lab has closed a Series E financing round of $140 million (USD). The funding round closed last month, prior to the launch of the successful mission ‘It’s Business Time,’ and was led by existing investor Future Fund, with strong participation from current investors including Greenspring Associates, Khosla Ventures, Bessemer Venture Partners, DCVC (Data Collective), Promus Ventures and K1W1. New investor ACC also contributed to the round. The Series E round close brings total Rocket Lab funding to date to more than $288 million (USD), with the company now soaring past its previous $1 billion-plus (USD) valuation.
“It has been a big year for Rocket Lab with two successful missions to orbit and another about to roll out to the pad, but it’s even more significant for the global small satellite industry that now has a fully commercial, dedicated ride to space,” said Rocket Lab CEO and founder Peter Beck. “This funding also enables the continued aggressive scale-up of Electron production to support our targeted weekly flight rate. It will also see us build additional launch pads and begin work on three major new R&D programs."
The round also follows the opening of Rocket Lab’s new mass production facility for the Electron vehicle last month, as well as the announcement confirming the location of Rocket Lab’s second orbital launch site. Construction has now begun on Rocket Lab Launch Complex 2, which is based within the Mid-Atlantic Regional Spaceport at NASA Wallops Flight Facility in Virginia, USA. Launch Complex 2 can support monthly orbital launches from US soil, and is designed specifically to serve the responsive space needs of government customers. Between the two Rocket Lab launch complexes, the company can support up to 130 orbital missions per year.
“Rocket Lab is a truly remarkable company. We were fortunate enough to follow Rocket Lab’s journey from their initial engine program to the first launch, the factory scale-up and now the start of commercial operations, all of which happened in record time,” said Sven Strohband, Partner and CTO of Khosla Ventures and Rocket Lab Board member. “We are proud to continue to support Peter and the amazing Rocket Lab team in their journey to open space for all and are excited about the next chapter.”
Matt Ocko – Managing Partner, DCVC (Data Collective)
“We were privileged to lead Rocket Lab’s prior round, and continue to be committed strongly to this superb team in this round and beyond,” said Matt Ocko, Managing Partner of DCVC (Data Collective) and Rocket Lab board member. “We saw proof of Rocket Lab’s transformative cost and frequency of launch just in the number of DCVC’s own next-gen ‘scale-out space’ companies enabled by Rocket Lab, let alone the hundreds more lined up for this disruptive orbital access capability.”
Mike Collett – Founder and Managing Partner, Promus Ventures
“There are few companies that have set out to achieve something as big as the Rocket Lab team and accomplished their goal as quickly as they have. The exciting part of the story is there is so much more left to write. Promus Ventures is privileged to be able to back Rocket Lab and its talented global team and platform.”
David Cowan – Partner, Bessemer Ventures Partners
“We see hundreds of start-ups gearing up to replace billion-dollar mainframes in Geosynchronous Orbit with constellations of cheaper, faster, better micro-sats,” said David Cowan of Bessemer Venture Partners, who serves on Rocket Lab’s board. “Rocket Lab alone now provides the predictable, frequent and affordable launch capability critical for this new ecosystem.”
Sir Stephen Tindall – Founder, K1W1
“The team at Rocket Lab has been inspirational and have now proved themselves to be the only operational private small launch provider globally. Their contribution to the NZ tech sector will continue to increase as they ramp up the launch frequency.” |
The Submachine (meaning "submerged machines") games is a series of point and click, "escape from the room" puzzle games where the player must find objects and clues scattered throughout various settings, figure out what to do with them, and apply them to certain objects and circumstances to open up more areas of each game and to hopefully eventually beat the games.
Basically all the player (who is never shown onscreen) has to do is use the mouse to search for onscreen items and clues and click on them in order to place them somewhere or store them into their inventory for later use, or to press a button or move a lever, switch or other device to perform a function that will help the player solve the mysteries of the game.
The Loop
This Submachine entry is totally different from the previous two games in the series.
It isn't made clear how the player arrives in this area as it is. There are almost no items to collect at all in this game; just one at the beginning and another one at the very end. Like the name implies, it's a giant loop: the player is in an unending loop of corridors, as they can move straight up, down, left or right, and the corridors will never end. This is also the only Submachine game where the player can die at the end if they perform the wrong action in the final area.
Supposedly in a showdown against the Submachine itself, the player must solve a series of puzzles in this maze of corridors. Some involve manipulating various machines, levers and buttons found in the corridors or on machines, and others are in regards to the map itself that is usually found not far from the player's starting point in a level. Some of the puzzles that are needed to solve to advance to the next level include moving to certain coordinates and activating a lever found there, reproducing patterns projected on a wall with devices, and turning all levers in a level, among others.
The game ends by the player either incorrectly taking the wrong action in the last level and dying, or by collecting one last object and taking it to a statue (?), which warps them to The Lab...Submachine 4.
Notes
This is the only Submachine game ever made so far where it has passwords as a save feature, as the other Submachine games either don’t have a save feature or they are automatic. |
2 So.3d 132 (2009)
STATE of Florida, Petitioner,
v.
Joshua MESHELL, Respondent.
No. SC08-903.
Supreme Court of Florida.
January 22, 2009.
*133 Bill McCollum, Attorney General, Tallahassee, FL, Kristen L. Davenport and Wesley Heidt, Assistant Attorneys General, Daytona Beach, FL, for Petitioner.
James S. Purdy, Public Defender, and Nancy Ryan, Assistant Public Defender, Seventh Judicial Circuit, Daytona Beach, FL, for Respondent.
POLSTON, J.
Petitioner State of Florida argues that the Fifth District Court of Appeal in Meshell v. State, 980 So.2d 1169 (Fla. 5th DCA 2008), erred in holding that Respondent Joshua Meshell's convictions for lewd and lascivious battery, under section 800.04(4), Florida Statutes (2006), for vaginal penetration or union (Count 1) and for oral sex (Count 3) violated double jeopardy. Because these are distinct criminal acts, we agree with the State that there is no double jeopardy violation.
Although the Fifth District reversed the trial court's judgment, holding that pursuant to its precedent the convictions for both Counts 1 and 3 violated double jeopardy, the district court noted that its ruling was inconsistent with various Florida district court of appeal rulings relating to the analogous sexual battery statute, section 794.011, Florida Statutes (2006). Meshell, 980 So.2d at 1170. The different sex acts proscribed in the sexual battery statute, ruled as distinct criminal acts for double jeopardy purposes, are the same sex acts as those proscribed in the lewd and lascivious battery statute. Accordingly, in its decision, the Fifth District certified the following question to be of great public importance:
ARE THE SEX ACTS PROSCRIBED BY SECTIONS 794.011 AND 800.04(4), FLORIDA STATUTES, PROPERLY VIEWED AS "DISTINCT CRIMINAL ACTS" FOR DOUBLE JEOPARDY PURPOSES, SO THAT A DEFENDANT CAN BE SEPARATELY CONVICTED FOR EACH DISTINCT ACT COMMITTED DURING A SINGLE CRIMINAL EPISODE?
*134 Id. at 1175.[1] Because the Fifth District only had section 800.04(4) at issue before it, and ruled only on that statute, we limit our review to the certified question as it pertains to section 800.04(4),[2] and answer it affirmatively.
I. BACKGROUND
Over the weekend of December 19-21, 2006, Joshua Meshell, age twenty-three, engaged in various sexual acts with a thirteen-year-old female. The State charged Meshell with five counts of lewd and lascivious battery in violation of section 800.04(4). Of these, the first three occurred at approximately the same time on December 19:(1) Meshell "did with his penis penetrate or have union with the vagina of [the victim];" (2) Meshell "did with his mouth have union with the vagina of [the victim];" and (3) Meshell "did with his penis have union with the mouth of [the victim]." After the jury returned a guilty verdict for all counts but Count 2, the trial judge sentenced Meshell to ten years in prison.
On appeal, Meshell challenged the constitutionality of his convictions for Counts 1 and 3. Meshell, 980 So.2d at 1171. Specifically, Meshell argued that double jeopardy prohibited his conviction and sentences for these two acts because the record did not reflect a "temporal break" sufficient for him to form a new criminal intent. Id. The Fifth District agreed, holding that its prior opinion in Capron v. State, 948 So.2d 954 (Fla. 5th DCA 2007), along with this Court's decision in State v. Paul, 934 So.2d 1167 (Fla.2006), requires a "temporal break." Id. at 1171, 1174. Therefore, the Fifth District reversed Meshell's conviction as to Count 3. Id.
However, the Fifth District noted that its ruling is inconsistent with well-settled precedent holding that sexual acts prohibited in the sexual battery statute, section 794.011, are distinct criminal acts so that separate convictions for each of the various acts do not violate double jeopardy. Id. at 1172. Distinct acts of sexual battery do not require a "temporal break" between them to constitute separate crimes. Id.
II. NO DOUBLE JEOPARDY VIOLATION BECAUSE DISTINCT ACTS
As the Fifth District noted, in cases of sexual battery, Florida courts have focused on whether the acts forming the basis of the charges are "distinct." For example, in Duke v. State, 444 So.2d 492, 493 (Fla. 2d DCA), approved, 456 So.2d 893 (Fla. 1984), the Second District Court of Appeal reviewed two convictions for attempted sexual battery: one attempted anal penetration and one attempted vaginal penetration. The two attempts occurred within seconds of each other. Id. at 494. The defendant argued that both acts collectively constituted one violation of the statute and that, as a result, double jeopardy barred his two convictions. The Second District, however, disagreed. Id. Upon inspecting the definition of sexual battery in section 794.011, which defines anal and vaginal penetration separately, the Second District found:
As the statute indicates, each act is a sexual battery of a separate character and type which logically requires different elements of proof. Clearly, penetration of the vagina and penetration of the anus are distinct acts necessary to *135 complete each sexual battery. Therefore, notwithstanding the short interval of time involved here, we believe each act is a separate criminal offense.
Id. (emphasis provided). Because the acts were distinct criminal acts, double jeopardy did not bar two convictions.
Similarly, in Begley v. State, 483 So.2d 70, 74 (Fla. 4th DCA 1986), the Fourth District Court of Appeal addressed Begley's claims that his separate sentences for attempted sexual battery for intercourse, attempted sexual battery for cunnilingus, and sexual battery for fellatio were invalid because the State failed to prove that three sexual acts were separate transactions. The Fourth District ruled that they were separate because each required different elements of proof, quoting section 775.021(4), Florida Statutes (1983), which provides that separate criminal offenses in the course of one criminal transaction or episode are separate criminal offenses. Id.
In Saavedra v. State, 576 So.2d 953, 954 (Fla. 1st DCA 1991), approved, 622 So.2d 952 (Fla.1993), the defendant was convicted of, among other things, three counts of sexual battery. Saavedra argued that double jeopardy precluded separate convictions and sentences because the underlying acts were of the same type and committed against the same victim. Id. at 956. While the First District Court of Appeal ultimately found that sufficient time existed between the acts for Saavedra to form a new criminal intent, the First District also acknowledged the significance of other critical factors:
The sexual battery statute may be violated in multiple, alternative ways, i.e., "oral, anal, or vaginal penetration by, or union with, the sexual organ of another or the anal or vaginal penetration of another by any other object." § 794.011(1)(g) Fla. Stat. (1987). Sexual battery of a separate character and type requiring different elements of proof warrant multiple punishments. See Duke v. State, 444 So.2d 492 (Fla. 2nd DCA) (vaginal penetration followed a moment later by anal penetration), aff'd, 456 So.2d 893 (Fla.1984); Grunzel v. State, 484 So.2d 97 (Fla. 1st DCA 1986) (cunnilingus followed a few seconds later by vaginal intercourse); Begley v. State, 483 So.2d 70 (Fla. 4th DCA 1986) (attempted vaginal intercourse, attempted cunnilingus, fellatio, committed over two week period); Bass v. State, 380 So.2d 1181 (Fla. 5th DCA 1980) (oral sex followed by rape). However, the fact that the same victim is sexually battered in the same manner more than once in a criminal episode by the same defendant does not conclusively prohibit multiple punishments. Spatial and temporal aspects are equally as important as distinctions in character and type in determining whether multiple punishments are appropriate.
Id. at 956-57 (emphasis provided) (footnote omitted); see also Gill v. Sec'y, Dep't of Corrections, No. 8:04-cv-140-T-23MAP, 2008 WL 906647, *27 (M.D.Fla. Mar.31, 2008) (ruling that Florida law did not support petitioner's argument that double jeopardy was violated because "the alleged placing of Gill's penis in or in union with [M.H.]'s vagina and anus during the same time or criminal episode constitutes only an alternative means of committing a single or the same crime" (citing Saavedra, 576 So.2d at 956-57, and Schwenn v. State, 898 So.2d 1130, 1132 (Fla. 4th DCA 2005))).
We agree that sexual acts of a separate character and type requiring different elements of proof, such as those proscribed in the sexual battery statute, are distinct criminal acts that the Florida Legislature has decided warrant multiple punishments. *136 See § 775.021(4)(a), Fla. Stat. (2006) ("Whoever, in the course of one criminal transaction or episode, commits an act or acts which constitute one or more separate criminal offenses, upon conviction and adjudication of guilt, shall be sentenced separately for each criminal offense; and the sentencing judge may order the sentences to be served concurrently or consecutively. For the purposes of this subsection, offenses are separate if each offense requires proof of an element that the other does not, without regard to the accusatory pleading or the proof adduced at trial.") (codification of the test in Blockburger v. United States, 284 U.S. 299, 52 S.Ct. 180, 76 L.Ed. 306 (1932)); see also Paul, 934 So.2d at 1171-72 ("The prevailing standard for determining the constitutionality of multiple convictions for offenses arising from the same criminal transaction is whether the Legislature `intended to authorize separate punishments for the two crimes.'") (quoting M.P. v. State, 682 So.2d 79, 81 (Fla.1996)).
Significantly, the same sexual acts proscribed in the sexual battery statute are also proscribed in the lewd and lascivious battery statute, under which Meshell was charged. Lewd and lascivious battery is defined as, among other things, "sexual activity with a person 12 years of age or older but less than 16 years of age." § 800.04(4)(a), Fla. Stat. (2006). "Sexual activity," in turn, means "oral, anal, or vaginal penetration by, or union with, the sexual organ of another or the anal or vaginal penetration of another by any other object." § 800.04(1)(a), Fla. Stat. (2006). Likewise, "sexual battery" is defined as "oral, anal, or vaginal penetration by, or union with, the sexual organ of another or the anal or vaginal penetration of another by any other object." § 794.011(1)(h), Fla. Stat. (2006); see also Williams v. State, 957 So.2d 595, 599 (Fla.2007) ("The definitions of `sexual battery' in chapter 794 and `sexual activity' in chapter 800 are identical, both described in pertinent part as `oral, anal, or vaginal penetration by, or union with, the sexual organ of another or the anal or vaginal penetration of another by any other object.'"). Because the definitions of the proscribed sexual acts are identical, the same double jeopardy analysis for the sexual battery also applies to the lewd and lascivious battery statute.
III. CONCLUSION
We hold that the sex acts proscribed in section 800.04(4) (oral, anal, or vaginal penetration) are of a separate character and type requiring different elements of proof and are, therefore, distinct criminal acts. Thus, punishments for these distinct criminal acts do not violate double jeopardy. Paul, 934 So.2d at 1172 n. 3 ("Of course, if two convictions occurred based on two distinct criminal acts, double jeopardy is not a concern." (citing Hayes v. State, 803 So.2d 695, 700 (Fla. 2001))).
Because the oral sex described in Count 3 is a criminal act distinctively different from the vaginal penetration or union in Count 1, there is not a double jeopardy violation. Therefore, we quash the decision of the Fifth District in Meshell and remand with directions to reinstate the convictions and sentences as originally imposed by the trial court.
It is so ordered.
QUINCE, C.J., WELLS, PARIENTE, and LEWIS, JJ., and ANSTEAD, Senior Justice, concur.
CANADY, J., concurs in result only with an opinion.
CANADY, J., concurring in result only.
I concur that the Fifth District Court of Appeal's decision should be quashed and *137 the case remanded with directions to reinstate the convictions and sentences imposed by the trial court. In my view, a variation in the character and type of the proscribed sex acts committed in a single episode is not necessary for the imposition of more than one punishment for lewd and lascivious battery. The instant case, of course, does not address the circumstance where in the course of a single episode, the defendant has committed more than one criminal sex act of the same type against the same victim. Our decision should not be read as denying that separate instances of the same type of criminal sex act in a single episode may be punishable as separate offenses.
NOTES
[1] We have jurisdiction. See art. V, § 3(b)(4), Fla. Const.
[2] See McEnderfer v. Keefe, 921 So.2d 597, 597 n. 1 (Fla.2006) (declining to address issues not directly addressed by the district court).
|
Endovascular repair of abdominal aortic aneurysm: current status.
Endovascular aneurysm surgery (EVAR) was introduced a decade ago. Early results are promising, however, there remain concerns regarding the longer-term durability of this technique. Consequently, the national multi-centre EVAR trial has been commenced to define the role of endovascular surgery in the management of abdominal aortic aneurysm. Successful EVAR requires accurate pre-operative assessment of aneurysm morphology. Current stent-grafts allow 60% of all infra-renal AAA to be treated. Reduced physiological stress and low peri-operative morbidity and mortality rates have been demonstrated with this technique when compared to open repair. Endoleak is an Achilles heel of EVAR, although in itself does not accurately predict outcome. First and second generation devices are estimated to have a 1% per year risk of rupture. Increased understanding of the issues surrounding aneurysm morphology and successful stent-grafting have allowed a major reduction of early type I endoleak. Late endoleak and graft migration remain problematic. Type I and III endoleaks are risk factors for subsequent rupture although the significance of type II endoleak remains uncertain. More robust indicators of outcome success/failure are required so that follow-up may be rationalised. |
require 'one_gadget/gadget'
# https://gitlab.com/david942j/libcdb/blob/master/libc/glibc-2.20-8.fc21.i686/lib/i686/nosegneg/libc-2.20.so
#
# Intel 80386
#
# GNU C Library (GNU libc) stable release version 2.20, by Roland McGrath et al.
# Copyright (C) 2014 Free Software Foundation, Inc.
# This is free software; see the source for copying conditions.
# There is NO warranty; not even for MERCHANTABILITY or FITNESS FOR A
# PARTICULAR PURPOSE.
# Compiled by GNU CC version 4.9.2 20150212 (Red Hat 4.9.2-6).
# Available extensions:
# The C stubs add-on version 2.1.2.
# crypt add-on version 2.1 by Michael Glad and others
# GNU Libidn by Simon Josefsson
# Native POSIX Threads Library by Ulrich Drepper et al
# BIND-8.2.3-T5B
# RT using linux kernel aio
# libc ABIs: UNIQUE IFUNC
# For bug reporting instructions, please see:
# <http://www.gnu.org/software/libc/bugs.html>.
build_id = File.basename(__FILE__, '.rb').split('-').last
OneGadget::Gadget.add(build_id, 248462,
constraints: ["ebx is the GOT address of libc", "[esp+0x28] == NULL"],
effect: "execve(\"/bin/sh\", esp+0x28, environ)")
OneGadget::Gadget.add(build_id, 248464,
constraints: ["ebx is the GOT address of libc", "[esp+0x2c] == NULL"],
effect: "execve(\"/bin/sh\", esp+0x2c, environ)")
OneGadget::Gadget.add(build_id, 248468,
constraints: ["ebx is the GOT address of libc", "[esp+0x30] == NULL"],
effect: "execve(\"/bin/sh\", esp+0x30, environ)")
OneGadget::Gadget.add(build_id, 248475,
constraints: ["ebx is the GOT address of libc", "[esp+0x34] == NULL"],
effect: "execve(\"/bin/sh\", esp+0x34, environ)")
OneGadget::Gadget.add(build_id, 248510,
constraints: ["ebx is the GOT address of libc", "[eax] == NULL || eax == NULL", "[[esp]] == NULL || [esp] == NULL"],
effect: "execve(\"/bin/sh\", eax, [esp])")
OneGadget::Gadget.add(build_id, 248511,
constraints: ["ebx is the GOT address of libc", "[[esp]] == NULL || [esp] == NULL", "[[esp+0x4]] == NULL || [esp+0x4] == NULL"],
effect: "execve(\"/bin/sh\", [esp], [esp+0x4])")
OneGadget::Gadget.add(build_id, 416308,
constraints: ["ebx is the GOT address of libc", "eax == NULL"],
effect: "execl(\"/bin/sh\", eax)")
OneGadget::Gadget.add(build_id, 416309,
constraints: ["ebx is the GOT address of libc", "[esp] == NULL"],
effect: "execl(\"/bin/sh\", [esp])")
|
Q:
interestOps throws IllegalArgumentException
I wand send a message to all User in map.
for (User u : _userMap.values()) {
u.getMessages().add(data);
u.getKey().interestOps(SelectionKey.OP_WRITE);
}
but whene I run this function I see
Exception in thread "main" java.lang.IllegalArgumentException
this line make error
u.getKey().interestOps(SelectionKey.OP_WRITE);
getKey() returns SelectionKey, getMessages returns ArrayList, data is a byte[] array with message I read using channel.read(buffer);
MORE INFO:
In a constructor I make Selector
_selector = Selector.open();
I run server
public void startServer() throws IOException {
while (true) {
_selector.select();
Iterator<SelectionKey> keys = _selector.selectedKeys().iterator();
while (keys.hasNext()) {
SelectionKey key = keys.next();
keys.remove();
if (!key.isValid())
continue;
if (key.isAcceptable())
accept(key);
else if (key.isReadable())
read(key);
else if (key.isWritable())
write(key);
}
}
}
I accept connection
private void accept(SelectionKey key) throws IOException {
ServerSocketChannel serverChannel = (ServerSocketChannel) key.channel();
SocketChannel channel = serverChannel.accept();
channel.configureBlocking(false);
User u = new User(key);
_userMap.put(channel, u);
channel.register(_selector, SelectionKey.OP_READ);
}
In read function whene I read message I have this for each loop. But whene is one user and I move line with interestOps just behind loop it works.
//u.getKey().interestOps(SelectionKey.OP_WRITE);
}
key.interestOps(SelectionKey.OP_WRITE);
Full read and write function:
private void read(SelectionKey key) throws IOException {
SocketChannel channel = (SocketChannel) key.channel();
ByteBuffer buffer = ByteBuffer.allocate(2048);
int read = -1;
try {
read = channel.read(buffer);
} catch (Exception e) {
e.printStackTrace();
}
if (read == -1) {
_userMap.remove(channel);
channel.close();
key.cancel();
return;
}
byte[] data = new byte[read];
System.arraycopy(buffer.array(), 0, data, 0, read);
/// WYSyŁA DO WSZYSTKICH. usunąć
for (User u : _userMap.values()) {
u.getMessages().add(data);
u.getKey().interestOps(SelectionKey.OP_WRITE);
}
//key.interestOps(SelectionKey.OP_WRITE);
///////
}
private void write(SelectionKey key) throws IOException {
SocketChannel channel = (SocketChannel) key.channel();
ArrayList<byte[]> msg = _userMap.get(channel).getMessages();
Iterator<byte[]> i = msg.iterator();
while (i.hasNext()) {
byte[] item = i.next();
i.remove();
channel.write(ByteBuffer.wrap(item));
}
key.interestOps(SelectionKey.OP_READ);
}
SOLUTION:
I can't answer my own question now, so put it here:
SelectionKey in accept method is a little handicapped. I tried to replace it with new key in read method and it works. So in User class I don't keep SelectionKey var any more, now I keep SocketChannel. SocketChannel have keyFor method, so whene I have selector I can get key
u.getChannel().keyFor(_selector).interestOps(SelectionKey.OP_WRITE);
A:
Sounds like it's probably exactly as documented:
Throws
IllegalArgumentException - If a bit in the set does not correspond to an operation that is supported by this key's channel, that is, if set & ~(channel().validOps()) != 0
It's hard to know why that's the case without knowing more about the channel in question though...
|
Introduction {#s1}
============
A growing interest in animal welfare, both in the scientific community and by the public, has recently drawn attention to animal needs. For the sake of the animals and to improve the quality and reproducibility of results, laboratory animals' well-being (psychological and physical health) needs to be particularly cared for in both husbandry and experimental setting (i.e. breeding, housing, manipulation and experimental procedures) \[[@r31]\]. Although somewhat subjective and not standardized, evaluation of animal welfare is usually based on several parameters including: growth and reproductive parameters (e.g. body weight, food intake, fertility rate), physiological measures (e.g. hormones), behavioural profile (e.g. day-light activity cycles, emotionality, cognition, stereotypical behaviours) and experimenter's subjective evaluation based on his/her experience and knowledge of the biology of the species. Reports in the literature on the effects of novel techniques to improve welfare and assessment of good practices are limited to few journals and poorly acknowledged in the methods section \[[@r33]\].
It is now recognized that animal welfare depends on the possibility to express species-specific behaviours \[[@r1]\] and can be strongly compromised in socially and environmentally deprived conditions. To provide proper conditions, personnel training and husbandry facilities for laboratory animals have been improved focusing on sanitation, nutritional needs and enrichment items. All international and national regulations for the accommodation and care of animals used for experimental and other scientific purposes are along this line. The European Directive (2010/63/EU) on the protection of animals used for scientific purposes states that "all animals shall be provided with space of sufficient complexity to allow expression of a wide range of normal behaviour" (article 33, annex III, 3.3. Housing and enrichment) and, in particular, "bedding materials or sleeping structures adapted to the species shall always be provided"(article 33, annex III, 3.6. Resting and sleeping areas). Differently shaped objects can be introduced into the animals' cages to provide secure locations (i.e. refuges). A protected area can help the individual to temporarily conceal from stress-full stimulations (e.g. aggressive cage-mates, experimenter handling, sudden external noises).
The introduction of objects in the animals' home cage has already been realized for experimental purposes. Differently shaped objects, together with larger cages and social groups, represent a well known experimental condition firstly recognized by Hebb \[[@r9]\] and then developed by Rosenzweig \[[@r20]\] in laboratory rodents and named "environmental enrichment". Environmental enrichment (EE) was in fact classically defined \[[@r9], [@r20]\] as a complex sensory-motor stimulation that provides the animal with an increased opportunity of physical exercise, learning experience and social interactions. Till now many studies have demonstrated that EE induces neurobehavioral changes through changes in gene expression \[[@r17], [@r18], [@r21], [@r22], [@r26]\]. This experimental condition, highly variable across studies, once applied to animals previously reared in standard husbandry, is able to revert and/or prevent pathological conditions resulting from genetic, environment and pharmacological insults \[[@r13]\]. The results of these studies emphasized thus the importance and the potential therapeutic effects of complex environments and, at the same time, the under-stimulating/deprived conditions that have been applied till now to laboratory animals. The opposite experimental condition, being reared from birth in EE and moved to standard (deprived) housing thereafter, resulted in behavioural abnormalities \[[@r12]\].
The relative roles of the increased physical and/or social complexity have been scarcely investigated and it is not possible to extrapolate from studies in the literature the importance of refuges only, to EE induced neurobiological changes. The great variability and complexity that characterize EE studies in terms of additional space, objects and partners provided, are hardly replicable and difficult to be reproduced in laboratories' and breeders' animal facilities \[[@r5], [@r28], [@r32]\]. In order to harmonize animal housing conditions, several mouse/rat houses, serving as possible nests/refuges, have been produced by different suppliers to easily improve animal welfare and several studies have been conducted to evaluate the animals' preference \[[@r14], [@r23], [@r29]\]. The red transparent mouse house was not always the preferred item by all the strains tested but it is one of the most used in large animal facilities.
The aim of this study was to evaluate whether the presence of the mouse house in the home cage, induced modifications on basic growth and behavioural parameters from adolescence to adulthood. We decided to conduct this study on young animals to give them time to recover from transfer and habituate to the new housing environment, before starting experiments. The mouse houses we used were autoclavable and re-usable, had more than one entry/exit, were large enough to allow more animals to fit inside. These red houses provided the animal a place to hide, as the mice perceive them as being dark and give them a better control of the physical and social environment. Moreover the red transparent material allows the experimenter continuous monitoring of the animals even inside the house. Specifically, two main questions were addressed: 1) Were these objects used by the animals as refuges? 2) Did the introduction of secure refuges modify the behavioural and physiological profiles of the animals? This study was carried out on young males and females of two inbred mouse strains, the BALB and the B6, never exposed to such houses before in their lives. BALB and B6 mice were selected because they are commonly used as reference strains in biomedical research. We found that the use of the houses differed between the two strains, was sex dependent and no major effects were observed on growth and emotionality.
Materials and Methods {#s2}
=====================
Subjects and housing
--------------------
C57BL/6JOlaHsd and BALB/cOlaHsd male and female mice (Harlan, San Pietro al Natisone, UD, Italy) were used in this study. After their arrival, at five weeks of age, animals were weighed, individually ear tagged and housed in groups of four same-sex mice in 1264C Eurostandard Type II cages (26.7 × 20.7 × 14.0 cm - floor area 370 cm², Tecniplast, Buguggiate, VA, Italy) with bedding (Scobis 2, Mucedola, Italy), food (Diet 2018, Harlan, Italy) and water available *ad libitum*. A total of 96 subjects to form 6 cages × strain × sex were used. Cages were cleaned once a week. According to the animal breeder, animals had never been exposed to a mouse house. The animal room had a controlled 12-h light cycle (lights on at 0700 h), lux level (on average 100 lux), temperature (21 ± 1°C) and relative humidity (50 ± 5%). Animals were subjected to experimental protocols approved by the Veterinary Department of the Italian Ministry of Health, and experiments were conducted according to the ethical and safety rules and guidelines for the use of animals in biomedical research provided by the relevant Italian law and European Union Directive (Italian Legislative Decree 26/2014 and 2010/63/EU) and the International Guiding Principles for Biomedical Research involving animals (Council for the International Organizations of Medical Sciences, Geneva, CH). All adequate measures were taken to minimize animal pain or discomfort.
Mouse house
-----------
A red transparent plastic triangular-shaped Mouse House™ (150 × 110 × 77 h mm, Tecniplast, Buguggiate, VA, Italy, [Fig 1C](#fig_001){ref-type="fig"}Fig. 1.Experimental procedure. (A) Time line of experimental procedure. Salient events along over a month of study: body weight and food intake measures, behavioural observation periods, cage changes and videorecording times at different time points. Age (weeks) of the animals is also indicated. (B) Depiction of different time points/conditions at cage changing times when mice were observed and behaviours scored for 10 min. (C) Mouse house.) was introduced in half of the home cages (always in the same position in one of the corners) after a period of 10 days of acclimatization in our animal house facility. The mouse house has two entrances, one on a side with a small tunnel (50 × 25 × 25 h mm) and one on the flat top (60 × 45 × 45 mm). No additional nest material was provided in the cages.
Experimental procedure
----------------------
From their arrival the animals were monitored for food intake and body weight. After ten days from arrival, animals in the different experimental conditions, with and without mouse house, were observed on different occasions during the study period: 1) at specific time points during cage cleaning/mouse house cleaning; 2) continuously for eight days (4 + 4 days) during the first 2 weeks starting the day after the mouse house introduction; 3) in a 5 min open field test to evaluate emotionality after habituation to the mouse house. See [Fig. 1A](#fig_001){ref-type="fig"} for time line of experimental procedure.
Body weight and food intake
---------------------------
Once a week, at the time of cage cleaning, each animal was weighed. The total amount of food, pellet consumed before mouse house presentation (week -1) and at the end of behavioural observations (week 3), were measured for each cage.
Behavioural observations at cage or mouse house cleaning
--------------------------------------------------------
Each cage was video-recorded (DCR-TRV320E PAL; SONY) for 10 min right after cage or mouse house cleaning procedures ([Fig. 1](#fig_001){ref-type="fig"}). Four conditions/time points were considered: (a) first presentation of a clean mouse house in the home cage (Day 1), (b) right after cage cleaning (Day 12), the mouse house is not changed, (c) presentation of a clean mouse house in the dirty home cage (Day 18), (d) presentation of a clean mouse house in a clean cage (Day 32). Due to a technical problem the video recording of a cage on day 12 was missing. Video files were analysed by an expert observer. The number of mice (max 4) displaying specific behaviours directed either towards or outside the mouse house were simultaneously scored for each cage/condition by observational sampling every 15 seconds. Behaviours directed to the house: approach, contact, inside (in), through top or side entrances (doors), on top of the mouse house (on), digging toward the mouse house (dig), i.e. "burying behaviour". Behaviours not directed to the mouse house (outside): general activity outside the mouse house such as grooming, climbing to the roof of the cage, fighting, feeding or drinking. No fighting events however occurred during the observation sessions.
Behavioural observations during habituation to the mouse house
--------------------------------------------------------------
Behavioural observations were also conducted on four continuous days/week during the first two weeks of habituation to the mouse house (Days 2−5 and 8−11, [Fig. 1A](#fig_001){ref-type="fig"}). Two daily 30-min sessions of observations were conducted in the animal facility room (from 0830 h to 0900 h and from 1230 h to 1300 h) without moving the cages from their racks standing at a distance of 60−80 cm from the cages. The experimenters started collecting data 10 min after entering the room to habituate the animals to the their presence. Behavioural sampling consisted of recording the behaviour of the four animals of each cage simultaneously once every 5 min for a total of six observations × session, 12 observations a day, for a total of 12 × 8 observations per cage. The following behaviours for each subject were recorded on check sheets as resting (IN or OUT) or active in contact with the house. No fighting events occurred during the observation sessions.
Emotionality
------------
Emotionality was measured in an open field test (5 min) between days 19 and 26 after mouse house introduction ([Fig. 1A](#fig_001){ref-type="fig"}). Each mouse was introduced in a clean squared arena (60 × 60 ×25 h cm) and left free to explore for a 5 min period. Each session was video recorded and the percentage of time and number of entries in the centre of the arena were used as dependent variables using an automated video-tracking system (Smart, PANLAB, Cornella, Spain).
Statistical analysis
--------------------
The sum of frequencies of single behaviours per mouse was calculated for each cage across four days/conditions ([Fig. 1B](#fig_001){ref-type="fig"}) and during the 8 days of observation pooling AM and PM data (Days 2−5 and 8−11, [Fig. 1A](#fig_001){ref-type="fig"}). Cluster analysis of variables (oblique principal components) \[[@r7]\], principal component analysis (PCA) \[[@r19]\], and inferential analyses (Chi-square and ANOVA) were performed using the SAS System (Cary, NC). Statistical significance was set at *P*\<0.05.
Multidimensional statistics (variable clustering, PCA) had as variables the cages and as statistical units the behavioural measures, allowing for a well-conditioned data structure (statistical units outnumbering variables) and, analogously to the approach used in gene expression microarray \[[@r19]\] pattern recognition, instead of separately considering the different behavioural aspects had as main focus the discrimination between an invariant behavioural pattern (or size component) and specific between profiles differences (shape components), classically registered by minor components \[[@r10]\].
The cages are considered as specific profiles in terms of behavioural variables and their mutual correlations gave rise to both cage clustering and PCA structure. The interpretation of the components was based upon the measures with the highest scores on the different components \[[@r19]\] while the inferential statistics were based upon chi-square (relation between cage cluster and strain and time respectively) and ANOVA having as source of variation strain and time and as dependent variable component loadings. Repeated measures (RM) ANOVA, performed on data from the eight observations (this in the usual mode having cages as statistical units and behavioural descriptors as variables) had time (RM) and strain, sex and house presence as sources of variation.
Results {#s3}
=======
Body weight and food intake
---------------------------
The presence of the mouse house did not affect body weight and food intake in either males or females of both strains ([Fig. 2](#fig_002){ref-type="fig"}Fig. 2.Body weight and food intake. (A-B) Body weight and (C-D) food intake, in respectively male and female mice of BALB and B6 strains with and without a mouse house measured before and after mouse house introduction. Data are expressed as mean ± SEM.). No effect of strain, mouse house presence or their interaction was detected in male's body weight increase. A significant time effect is evident during development (*F*~5.222~=872.64, *P\<*0.001) with a significant strain × time effect (*F*~5.220~=18.67, *P\<*0.01) with B6 males showing, in comparison with BALB males, lower body weight upon arrival and similar weight at the end of the experiment. No other significant effects emerged.
Females' body weight increase is shown in [Fig. 2B](#fig_002){ref-type="fig"}. The three-way ANOVA indicates significant effect of strain (*F*~1.220~=55.00, *P*\<0.001), time (*F*~5.220~=444.72, *P\<*0.001) and time × strain (*F*~5.220~=4.10, *P*\<0.01). The presence of the mouse house did not significantly affect female body weight gain, neither as main nor in interaction with other independent variables.
Food intake measured per cage during the week before mouse house introduction (week -1) and on week 3 ([Figs. 2C and 2D](#fig_002){ref-type="fig"}) did not seem to be affected by the mouse house.
Behavioural observations at cage or mouse house cleaning
--------------------------------------------------------
Mouse behaviour during the routine cage cleaning was observed to evaluate if the animals habituate to the mouse house. Salient behaviours directed outside or towards the mouse house were scored ([Fig. 3](#fig_003){ref-type="fig"}Fig. 3.Behaviours observed right after cage changing. Percentage of behaviours performed outside or directed to the house in respectively (A-C) male and (B-D) female BALB and B6 mice observed at cage or mouse house changing times. (A-B) Day 1 and Day 18 clean mouse house in a dirty home cage. Day 12, dirty (C) or Day 32, clean (D) mouse house in a clean cage.). In general no differences in behavioural profile was observed between males and females regardless of strain in the four experimental conditions/time points ([Figs. 4](#fig_004){ref-type="fig"}Fig. 4.Behavioural profiles on day 1 vs day 18: clean house in a dirty cage. Percentage of behaviours scored in respectively (A) male and (B) female BALB and B6 mice observed upon first exposure to a clean mouse house (Day1) and to same cage/house condition on Day 18 of exposure. The percentage of behaviours performed outside the house (grooming, climbing to the roof of the cage, fighting, feeding or drinking) and the percentage of behaviours directed to the house (approach, contact, digging, on top, inside, through the doors) are shown. and [5](#fig_005){ref-type="fig"}Fig. 5.Behavioural profiles on day 12 vs. day 32: dirty vs clean house in a clean cage. Percentage of behaviours scored in respectively (A) male and (B) female BALB and B6 mice observed upon exposure in a clean cage, to a dirty mouse house (Day12) and to a clean house on Day 32 of experiment. The percentage of behaviours performed outside the house (grooming, climbing to the roof of the cage, fighting, feeding or drinking) and the percentage of behaviours directed to the house (approach, contact, digging, on top, inside, through the doors) are shown.). However, different behavioural profiles depending on the strain or the experimental condition, emerged from the variable cluster analysis (Oblique Principal Components). The clustering is driven by the construction of cluster made of variables (behavioural profiles relative to different cages) maximally correlated among them and independent from the cages pertaining to other classes. Cages clustered in four groups ([Supplementary Table 1](#pdf_001){ref-type="supplementary-material"}) with a 94.8% of total variation explained. The obtained clustering was demonstrated to have a statistically significant relation with both strain and day/condition ([Supplementary Table 2](#pdf_001){ref-type="supplementary-material"}; Strain: χ^2^=8.716, *P*\<0.05; Day: χ^2^=55.38, *P*\<0.0001).
Having demonstrated the general relation between 'behavioural profile' and both strain and time in an unsupervised way (we only based on general resemblance between profiles as emerging from cage clustering) and the inferential analysis was applied *a posteriori* on the obtained 'natural' clusters, we go more in depth into the interpretation of the nature of the differences.
We compared mouse behaviour on DAY 1 (first exposure) with DAY 18, when clean mouse houses were introduced in dirty home cages ([Figs. 3A--3B](#fig_003){ref-type="fig"}, males and females respectively). All mice directed between 40 and 60% of their behaviours to the mouse house on DAY 1 with BALB mice spending slightly more time with the house. A detailed behavioural profile ([Fig. 4](#fig_004){ref-type="fig"}) showed that B6 mice approached the house from a distance whereas BALB mice touched, climbed on top or even entered the house within the first 10 min of exposure. On DAY 18 ([Figs. 3A and 3B](#fig_003){ref-type="fig"}), contrary to B6 mice, BALB males and females increased the frequency of interaction with the mouse house (about 80%) making contact, climbing, entering, or staying inside the mouse house, therefore reducing the approach behaviour ([Fig. 4](#fig_004){ref-type="fig"}).
When a mouse house was introduced in a clean cage ([Figs. 3C and 3D](#fig_003){ref-type="fig"}), the behaviour of both mouse strains was very similar when the house was dirty (DAY 12). However, on DAY 32, when both cages and houses were clean, the two strains seemed to behave differently towards the house with BALB showing about 60% of behaviours directed to the house, compared to about 40% in B6 mice. The presence of the clean mouse house in a clean cage (DAY 32) did not affect BALB mice behaviour compared to DAY 12, whereas B6 mice spent more time outside. However in B6, after 32 days of exposure to the house, direct physical contact with the clean house was observed with almost no approach from a distance ([Fig. 5](#fig_005){ref-type="fig"}).
In conclusion, B6 mice appeared to be more 'cautious' than BALB every time they were exposed to a clean mouse house, even after 32 days of familiarization with this object.
In order to have a more comprehensive view of the above description, a PCA was computed over the data set, again having cages as variables and behavioural measures as statistical units. This data organization will lead the loadings to play the role of dependent variables for Analysis of Variance and component scores as the descriptors of the relative weight, of each specific behaviour, in the component meaning \[[@r4], [@r19]\].
PCA gave rise to a three-component solution accounting for 97.9% of the total variance ([Supplementary Table 3](#pdf_001){ref-type="supplementary-material"}). Factor 1 accounted for 84.07% of the variance which is clearly a size component \[[@r10]\] having all positive (and near to unity) loadings and thus representing the commonality among all the cages. The subsequent analysis of the scores ([Supplementary Table 4](#pdf_001){ref-type="supplementary-material"}) showed for Factor 1, variables related to "general activity" both towards and outside the house. It is worth noting that variables like "out" and "house" pointing to an opposite direction of motion have the same score sign as for Factor 1 giving a proof of concept of the interpretation of Factor 1 as "general activity". This interpretation is reinforced by a relatively high score of the "active" variable. The presence of a common "size component" \[[@r4], [@r10]\] accounting for more than 80% total information is the image in light of the presence of a "strong invariant core" of behavioural profile across different cages and thus giving a proof-of-concept of the reliability of the proposed strategy.
Factor 2 (9.45% of variance) is a shape component (both positive and negative loadings with cages), this implying we are no more measuring the "commonality" (components are each other independent by construction) but the profile differences between cages. Factor 2 showed a positive and high score mainly for the behaviours directed toward the house while outside had a high but negative score with Factor 2 ([Supplementary Table 4](#pdf_001){ref-type="supplementary-material"}). This allows us to interpret this Factor has a preference toward "near house" activity with respect to outside and could be interpreted as an "interest for the house" factor ([Fig. 6A](#fig_006){ref-type="fig"}Fig. 6.Principal Components Analysis: Factor 2 and Factor 3 loadings. (A) Loadings for Factor 2 (9.45% of variance) that was interpreted as an "interest in the house" factor. \#\#\# *P*\<0.0005 BALB vs B6, ANOVA. Data, from [Supplementary Table 3](#pdf_001){ref-type="supplementary-material"}, are expressed as mean ± SEM (n=3 cages per sex/strain). (B) Loadings for Factor 3 (4.38% of variance) that was interpreted as 'habituation/approach to the house' and varied over time, \*\*\**P*\<0.0001, ANOVA. Each point represents one cage (data from [Supplementary Table 3](#pdf_001){ref-type="supplementary-material"}).).
Factor 3 (4.38% of variance) is again a shape component (both positive and negative loadings), and showed higher scores for "approach" behaviours and high but negative score for behaviours "in" the house ([Supplementary Table 4](#pdf_001){ref-type="supplementary-material"}). We interpreted Factor 3 as a sort of 'habituation/approach to the house' that varied over time ([Fig. 6B](#fig_006){ref-type="fig"}).
An ANOVA performed on the Factors showed a significant difference between strains for Factor 2 ("interest to the house", *F*~1.43~=16.18, *P*\<0.0005; [Fig. 6A](#fig_006){ref-type="fig"}) and among days for Factor 3 ("habituation/approach to the house", *F*~1.46~=41.15, *P*\<0.0001, [Fig. 6B](#fig_006){ref-type="fig"}). This suggests that BALB and B6 have a different "interest" in the house and that in general the 'approach' or interaction with the house changes over time. Interestingly, a statistically significant difference due to sex was never observed.
Behavioural observations during habituation to the mouse house
--------------------------------------------------------------
In general, it was observed that the presence of the house did not affect the total amount of home cage resting behaviour ([Figs. 7A and 7B](#fig_007){ref-type="fig"}Fig. 7.Resting and contact with house behaviours. Frequency of resting behaviours (A-B) and contact with the mouse house (C) observed in male and female mice of BALB and B6 strains during the two weeks of habituation to the mouse house. Resting behaviour frequency was scored when occurring either in or out of the house (A-B). Data are expressed as mean (± SEM) frequency of contacts per week (C).). However, strains and sex showed different preference in the use of the house for resting. In particular, BALB male mice, but not females, preferred to sleep inside the house (ca 70% vs. 0%). On the other hand, B6 male mice rested inside the house about 25% of the total frequency of resting behaviours compared to about 50% in females. The RM ANOVA showed that BALB rested more than B6 mice (*F*~1.18~=6.78, *P*\<0.05). It is of note that resting behaviour showed a very high frequency in general since animals were observed during the light phase of the light/dark cycle. As for the contact with the house ([Figs. 7C and 7D](#fig_007){ref-type="fig"}), the RM ANOVA showed a significant interaction (sex × strain × week: *F*~1.8~=6.86, *P*\<0.05). Only two groups of mice (BALB males and B6 females) increased their contact with the house between the two weeks of habituation. On the other hand, BALB female mice inside or in contact with the house showed no change in frequency between the two weeks of habituation and as for B6 males, their contact with the house was even reduced from the first to the second week.
Emotionality
------------
Exposure to the mouse house did not affect emotionality as measured by number of entries and percentage of time spent in the centre of the open field ([Fig. 8](#fig_008){ref-type="fig"}Fig. 8.Emotionality (open field test). Percentage of time spent (A) or number of entries (B) in the centre of the open field in male and female mice of BALB and B6 strains exposed (House) or not exposed (No House) to a mouse house for about three weeks. Data are expressed as mean ± SEM.). A 3-way ANOVA indicated a statistically significant effect for the factor strain only for the number of entries in the centre of the arena (*F*~1.59~=35.67, *P*\<0.0001). Neither sex, nor house factors, significantly affected mouse behaviour in the open field and no interaction reached statistical significance.
Discussion {#s4}
==========
This study demonstrates that the presence of a mouse house in the home cage did not interfere with main developmental and behavioural parameters of BALB and B6 mice. Additionally, both strains habituated to the house in about a week. Even if a general invariance of the behavioural profiles was observed pointing to a high level of commonality (accounting for around 80% of total variance) across cages, we were able to single out subtle but statistically significant differences between strains as for the use of house.
The presence of the mouse house did not affect body weight gain or food intake, neither in males nor in females of both mouse strains ([Fig. 2](#fig_002){ref-type="fig"}). Some differences observed between strains or sexes in food intake were within the expected range at that age \[[@r15]\]. This result is particularly relevant considering that animals in this study were observed during development, from young age (five weeks) to adulthood (10 weeks), and moving developing animals from the supplier breeding colony to the new facility, may have a strong impact on animal physiology and behaviour.
Behavioural observations conducted during cage/house changing, indicated that BALB mice made contact with the house much more than B6, especially from the very first day of exposure ([Fig. 4](#fig_004){ref-type="fig"}). Contrary to what is expected on the basis of literature, that considers B6 less anxious than BALB mice in several behavioural tests \[[@r2], [@r6], [@r11]\], the B6 seemed more "cautious" toward the house from the beginning (high approach behaviour on Day 1) and almost 'indifferent' at the end of the experiment. These mice showed no approach and a low percentage of behaviours directed to the house in comparison with the BALB mice ([Figs. 3](#fig_003){ref-type="fig"}, [4](#fig_004){ref-type="fig"}, [5](#fig_005){ref-type="fig"}). Other strain differences were also observed in the use of the house ([Fig. 7](#fig_007){ref-type="fig"}): BALB male mice preferred to sleep inside the house more than B6 males, suggesting that they consider it as a protected area. We had confirmation from the supplier that animals of this study had never been exposed to the house before. Therefore we can state that they habituated to it in about one week and from the second week onward they showed a stable "interaction" with the mouse house.
Sex differences were observed in the use of the house during habituation. In particular, BALB males' contact with the house increased overtime while BALB females did not use it, even for sleeping, possibly reflecting lower anxiety levels compared to males. High variability observed in B6 behaviour did not allow a clear interpretation, but our data suggest a time × sex opposite trend in this strain ([Fig. 7](#fig_007){ref-type="fig"}). Interestingly, not so many sex differences in behaviours directed to the house were observed during cage/house changing ([Figs. 4](#fig_004){ref-type="fig"}, [5](#fig_005){ref-type="fig"}, [6](#fig_006){ref-type="fig"}). It should be noted however that active behaviours (contact with the house) occurred in a very low frequency compared to resting behaviours ([Fig. 7](#fig_007){ref-type="fig"}), due to the fact that behavioural observations were conducted during the light phase of the light/dark cycle, as already mentioned.
Mice emotionality, after three weeks from mouse house supplementation was also evaluated using the open field test ([Fig. 8](#fig_008){ref-type="fig"}). A lower number of entries in the central part of the apparatus was considered as an index of higher emotionality during exploration of a new environment \[[@r2]\]. Our findings confirmed that BALB mice entered less frequently in the centre of the arena as reported extensively in the literature to support strain differences in emotionality \[[@r8], [@r27]\]. Similar results were reported elsewhere \[[@r24]\] indicating that BALB mice housed with a shelter did not change their strain-dependent behavioural phenotype.
Data discussed above seem to be discordant with the idea of "anxious" BALB as reported here (open field test) and in the literature. However, this finding stresses the importance of considering the context while measuring emotionality. In effect, BALB mice general activity was reported higher than the B6 ones, when measured in the home cage \[[@r3], [@r25]\], but lower in an unfamiliar environment \[[@r6]\]. Limits and inconsistence of association between standard anxiety tests (open-field, elevated plus-maze and emergence tests) in mice has been discussed \[[@r11]\] and an attempt to correlate behaviour with physiological parameters (heart rate, body temperature) failed to reach successful results in BALB mice \[[@r6]\]. B6 seemed more "cautious" and dug more on day 1. Digging in this case could be considered as an avoidance behaviour \[[@r16]\] or alternatively as a strain specific behaviour by which B6 have been considered "burrowers" as opposed to BALB considered instead as "surface nestlers" \[[@r30]\].
According to the EU Directive, cage enrichment should be adopted by all animal facilities and thus represents the new standard for all laboratory animals. It is thus important that breeders and laboratory animal facilities agree to use similar cage enrichment, such as a refuge/nest, to reduce stress and time of habituation when animals are transferred among facilities. Transfer of animals between laboratories for scientific purposes is a very common procedure due to the greater interdisciplinary approach of current research that needs multiple competence, not easily available in a single institution. The introduction of a mouse house, such as the one described here could represent an easy and relatively inexpensive way to standardize cage environments among animal facilities. Therefore, the mouse house inserted in the home cage from the arrival of the animals, is part of the physical environment after a few days and does not elicit additional investigation/exploration, also when removed and substituted with a clean one. This would reassure scientists that this implementation does not modify the animal behavioural profile and body weight gain, in comparison with animals tested in the absence of the mouse house.
New regulations on the use of animals for scientific purposes recommend and promote the use of alternative models, without imposing an *a priori* definition of standardized environment, and it stresses the importance of providing adequate housing conditions to improve animal welfare. However, some concern was raised especially within the behavioural neuroscience community, about the effect of arbitrary cage enrichment on variability among laboratories, and reproducibility of experimental findings \[[@r5], [@r28], [@r32]\].
Based on our findings, these are a few suggestions for scientists facing the adoption of housing supplementation in their animal facilities to improve animal welfare. Firstly, be informed about the housing conditions animals had before arriving at the facility, i.e. whether any kind of environmental enrichment/supplementation was used. As far as we know, suppliers do not use any type of mouse house, and if they use environmental enrichment in breeding colonies, most often they limit it to nest paper. It would be a good practice for animal suppliers to communicate if and which type of housing supplementation is used in their facilities. Secondly, according to our results, one week was sufficient in five week-old mice, to habituate to a mouse house. Thirdly, the presence of the mouse house did not modify the home cage resting and feeding profile of our experimental groups. Fourthly, even if a difference in mouse house use is reported in this study between strains and sexes, no objection for the use of the house can be raised, as it did not alter typical behavioural profiles.
Supplementary Material
======================
###### Supplement table
The authors would like to thank M. Di Virgilio for technical assistance, G. Di Fonzo for help in reviewing the English and S. Macri' for critical reading of the manuscript.
|
Known coffee machines generally comprise a cap intended to receive a capsule containing a dose of the drink to be infused. The cap is taken, manually or automatically, to a percolating head comprising a bored needle, which allows the injection of water and/or of steam into the capsule.
One of the disadvantages of the existing machines is the fact that it is not possible to adapt the concentration of the drink prepared as the capsules all contain the same dose of the product to be infused. The only parameter which the user can vary is the quantity of water injected into the capsule. Thus, if the user desires a concentrated drink in a cup of large capacity, he will have to use two capsules of product in succession in order to obtain the desired concentration. |
Activation of two different kinds of neurons is necessary for appetitive and aversive memory recall in crickets, say researchers BMC Biology who blocked octopaminergic (OA-ergic) and dopaminergic (DA-ergic) transmission and found that this resulted in the inability to recall pleasant and unpleasant memories, respectively.
Makoto Mizunami (now at Hokkaido University, Japan) led a team of researchers from Tohoku University, Japan, who carried out the tests. He said, "This is the first study to suggest that classical conditioning in insects involves neural mediation between an originally neutral stimulus and a pleasant or unpleasant stimulus and the activation of these neural responses for memory recall. Such neural responses are often called cognitive processes in classical conditioning in higher vertebrates".
Mizunami and his colleagues previously reported that, in crickets, OA-ergic neurons and DA-ergic neurons convey signals about reward and risk, respectively. In this report, they found that blockers of synaptic transmission from OA-ergic and DA-ergic neurons prevented the insects from recalling which stimuli were related to the reward, and, therefore, could be approached, and which stimuli were related to the risk, so should be avoided. According to Mizunami, "These findings are not consistent with conventional neural models of classical conditioning in insects. Instead, we suggest that the cognitive account of classical conditioning proposed for higher vertebrates is applicable to insects".
Source: BioMed Central |
---
category: packages
---
## ui-menu
[![npm][npm]][npm-url]
[![build-status][build-status]][build-status-url]
[![MIT License][license-badge]][LICENSE]
[![Code of Conduct][coc-badge]][coc]
A dropdown menu component.
### Components
The `ui-menu` package contains the following:
- [Menu](#Menu)
### Installation
```sh
yarn add @instructure/ui-menu
```
### Usage
```js
import React from 'react'
import { Menu } from '@instructure/ui-menu'
const MyMenu = () => {
return (
<Menu>
<Menu.Item value="foo">Foo</Menu.Item>
<Menu.Item value="bar">Bar</Menu.Item>
</Menu>
)
}
```
[npm]: https://img.shields.io/npm/v/@instructure/ui-menu.svg
[npm-url]: https://npmjs.com/package/@instructure/ui-menu
[build-status]: https://travis-ci.org/instructure/instructure-ui.svg?branch=master
[build-status-url]: https://travis-ci.org/instructure/instructure-ui "Travis CI"
[license-badge]: https://img.shields.io/npm/l/instructure-ui.svg?style=flat-square
[license]: https://github.com/instructure/instructure-ui/blob/master/LICENSE
[coc-badge]: https://img.shields.io/badge/code%20of-conduct-ff69b4.svg?style=flat-square
[coc]: https://github.com/instructure/instructure-ui/blob/master/CODE_OF_CONDUCT.md
|
eplacement from kggggkgggkgggkkgkg.
1/51
Four letters picked without replacement from {l: 5, y: 2, d: 2, a: 9}. Give prob of sequence yyld.
1/3672
Four letters picked without replacement from {f: 5, n: 3, c: 1}. Give prob of sequence cnnf.
5/504
Four letters picked without replacement from {h: 2, b: 4, s: 2}. Give prob of sequence sbbh.
1/35
Calculate prob of sequence nkkl when four letters picked without replacement from kylyyykkknym.
1/990
Two letters picked without replacement from olzkklh. What is prob of sequence zh?
1/42
What is prob of sequence xm when two letters picked without replacement from {f: 1, x: 3, m: 10, y: 2}?
1/8
Three letters picked without replacement from eneennneeneenennneee. What is prob of sequence nnn?
7/95
Four letters picked without replacement from {h: 1, g: 1, x: 3, r: 1, y: 4, f: 2}. Give prob of sequence fhgx.
1/1980
Two letters picked without replacement from hhahhhhahhhrhr. What is prob of sequence ar?
2/91
Three letters picked without replacement from {g: 4, e: 11, x: 4}. What is prob of sequence eee?
55/323
Three letters picked without replacement from {b: 4, l: 2, a: 2, v: 8}. Give prob of sequence blb.
1/140
Three letters picked without replacement from hxxhxwxdhxhhhhdu. What is prob of sequence xxx?
1/56
Four letters picked without replacement from {u: 2, z: 3, r: 1, c: 4, f: 2, j: 2}. Give prob of sequence uurc.
1/3003
What is prob of sequence fc when two letters picked without replacement from dddzdddddgfczddf?
1/120
Four letters picked without replacement from {f: 12, x: 7}. What is prob of sequence xxff?
77/1292
Four letters picked without replacement from lleallllmanlllaaa. What is prob of sequence nlaa?
3/952
What is prob of sequence qeoo when four letters picked without replacement from jocqeeejeoqa?
2/1485
Three letters picked without replacement from {t: 10, z: 9}. Give prob of sequence zzz.
28/323
Calculate prob of sequence cd when two letters picked without replacement from ddcdccoc.
3/14
Three letters picked without replacement from svvvsvv. What is prob of sequence svs?
1/21
What is prob of sequence fghj when four letters picked without replacement from hfghfhhffhhjkhhhhh?
11/18360
Three letters picked without replacement from {d: 9, r: 2}. What is prob of sequence ddd?
28/55
Calculate prob of sequence lw when two letters picked without replacement from lwwllwlwlll.
14/55
Calculate prob of sequence uuy when three letters picked without replacement from {y: 1, h: 2, u: 2, m: 1}.
1/60
Two letters picked without replacement from jgqjgqqjgggbgqqg. Give prob of sequence bq.
1/48
Three letters picked without replacement from ksdpedr. Give prob of sequence kre.
1/210
Two letters picked without replacement from avgggagvvaagaggva. Give prob of sequence ag.
21/136
Two letters picked without replacement from {l: 4, f: 2, r: 2, g: 2, m: 1, y: 1}. What is prob of sequence gg?
1/66
Calculate prob of sequence ndd when three letters picked without replacement from nmmndnwwnnndwwwdn.
7/680
What is prob of sequence jr when two letters picked without replacement from rrrrjjrrjrrr?
9/44
Three letters picked without replacement from oootm. Give prob of sequence mot.
1/20
What is prob of sequence kku when three letters picked without replacement from ssskksskksskssuss?
1/204
Three letters picked without replacement from {i: 12, h: 7}. Give prob of sequence hhi.
28/323
Three letters picked without replacement from iilsiliiliisfili. What is prob of sequence fsi?
3/560
Calculate prob of sequence qi when two letters picked without replacement from {i: 4, q: 3}.
2/7
Calculate prob of sequence vttv when four letters picked without replacement from {r: 4, v: 4, t: 6}.
15/1001
Two letters picked without replacement from ttrttttttttttttttrtt. Give prob of sequence tt.
153/190
Calculate prob of sequence lpun when four letters picked without replacement from puunlnpx.
1/210
Two letters picked without replacement from {n: 7, m: 1, q: 6, w: 4}. Give prob of sequence nw.
14/153
Calculate prob of sequence ffaa when four letters picked without replacement from {z: 4, f: 5, m: 2, a: 2, c: 1}.
5/3003
Two letters picked without replacement from yjyyjjuuy. Give prob of sequence uy.
1/9
Three letters picked without replacement from dddssddsdssssddd. Give prob of sequence dds.
3/20
What is prob of sequence qq when two letters picked without replacement from qytqqqyy?
3/14
What is prob of sequence jkj when three letters picked without replacement from jjvvvjjvjkj?
1/33
Two letters picked without replacement from {p: 6, e: 7}. Give prob of sequence pp.
5/26
Three letters picked without replacement from ezecccccece. What is prob of sequence eee?
4/165
What is prob of sequence bj when two letters picked without replacement from {j: 1, b: 4, v: 2, d: 8}?
2/105
Calculate prob of sequence bq when two letters picked without replacement from fbdaafddqff.
1/110
What is prob of sequence ff when two letters picked without replacement from {f: 4}?
1
What is prob of sequence nm when two letters picked without replacement from {n: 1, m: 1, k: 1}?
1/6
Four letters picked without replacement from bwyzwwzrsyrb. Give prob of sequence swzb.
1/990
Calculate prob of sequence oa when two letters picked without replacement from {a: 5, o: 5, w: 8}.
25/306
Two letters picked without replacement from {q: 2, c: 4, r: 1, k: 7, a: 1}. Give prob of sequence rk.
1/30
Calculate prob of sequence adj when three letters picked without replacement from iajiaiididid.
1/220
Calculate prob of sequence abf when three letters picked without replacement from {i: 1, a: 8, f: 1, z: 4, b: 1}.
4/1365
Two letters picked without replacement from lbboobllooblloo. Give prob of sequence bo.
4/35
What is prob of sequence ccz when three letters picked without replacement from xccxxkkcnzckkckz?
1/84
Two letters picked without replacement from pcccppcpcqv. Give prob of sequence cq.
1/22
Four letters picked without replacement from ijjijqa. What is prob of sequence jijq?
1/70
Two letters picked without replacement from izzlilzllszalflzllll. Give prob of sequence il.
1/19
Calculate prob of sequence iw when two letters picked without replacement from {j: 1, o: 3, w: 1, i: 2}.
1/21
Two letters picked without replacement from {y: 11, h: 3}. Give prob of sequence hy.
33/182
Two letters picked without replacement from {i: 1, o: 7}. Give prob of sequence io.
1/8
Four letters picked without replacement from bbbb. What is prob of sequence bbbb?
1
Four letters picked without replacement from ccjcjwwjcjj. Give prob of sequence wcwc.
1/330
Calculate prob of sequence iiq when three letters picked without replacement from {i: 2, q: 1, b: 2, j: 1}.
1/60
Four letters picked without replacement from {z: 2, o: 7, k: 4}. What is prob of sequence ozoz?
7/1430
What is prob of sequence lou when three letters picked without replacement from uloll?
1/20
Calculate prob of sequence cooo when four letters picked without replacement from {i: 9, c: 5, o: 2}.
0
What is prob of sequence ulu when three letters picked without replacement from {l: 3, e: 7, u: 4}?
3/182
What is prob of sequence lx when two letters picked without replacement from {x: 2, l: 5}?
5/21
Calculate prob of sequence ro when two letters picked without replacement from {q: 2, r: 6, o: 9}.
27/136
Calculate prob of sequence xm when two letters picked without replacement from {l: 11, x: 2, u: 1, m: 2}.
1/60
Four letters picked without replacement from iimmmmmmim. What is prob of sequence immm?
1/8
Four letters picked without replacement from poopepyyepoqkepepoy. Give prob of sequence yqkp.
1/5168
Calculate prob of sequence ttgt when four letters picked without replacement from ttttttttttttgtttgtt.
5/57
What is prob of sequence ttll when four letters picked without replacement from vvltltlvl?
1/126
Calculate prob of sequence llbr when four letters picked without replacement from {y: 4, r: 4, b: 4, l: 3}.
4/1365
Three letters picked without replacement from kyyttitkpycytc. Give prob of sequence ykk.
1/273
Calculate prob of sequence dwbj when four letters picked without replacement from {w: 1, j: 1, b: 4, d: 1, h: 2}.
1/756
Four letters picked without replacement from {f: 10, u: 4}. What is prob of sequence ufuu?
10/1001
Two letters picked without repla |
Q:
How to draw 3D curves by SciPy?
I draw a 2D curve with the code
c = 11
x = np.arange(0, 5, 0.1)
y = np.exp(c)/x
plt.plot(x,y)
How can I draw a series of the x,y curves while the z axis is c? The first line will be changed to
c = np.arange(1, 70, 1)
How can I draw the 70 x,y curves along the z axis?
A:
You could use matplotlibs Axes3D, a tutorial can be found here:
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
c = np.arange(1, 10, 1) # made this 10 so that the graph is more readable
x = np.arange(0, 5, 0.1)
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
for i in c:
y = np.exp(i) / x
ax.plot(x, y, i)
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.set_zlabel("z")
plt.show()
Which gives the figure:
|
Q:
Redirect non-www to www with aws elastic beanstalk
I'm using Elastic Beanstalk and I've followed the instructions to deploy my app using the express web server as follow:
http://docs.aws.amazon.com/elasticbeanstalk/latest/dg/create_deploy_nodejs_express.html
This setup uses nginx and route 53.
Everything works well, but now I'm trying to redirect from non-www/non-https URLs to "https://www.domain.com" (always https with www).
I've seen different solutions out there that either aren't working or seem hacky. What's the proper way to do this from the aws console?
Thanks a lot!
A:
You can setup a S3 bucket that redirects naked domain to www. It is explained here.
http://docs.aws.amazon.com/AmazonS3/latest/dev/website-hosting-custom-domain-walkthrough.html
You can redirect http to https by using Cloudfront. You can read more information here.
http://docs.aws.amazon.com/AmazonCloudFront/latest/DeveloperGuide/using-https.html
You can setup the webserver on your EC2 instances to redirect as well, but that requires that you setup up your SSL certificate as well. It is easier to let AWS handle that with Cloudfront.
You are probably using Apache so it would be something like this.
NameVirtualHost *:80
<VirtualHost *:80>
ServerName mysite.example.com
DocumentRoot /usr/local/apache2/htdocs
Redirect permanent / https://mysite.example.com/
</VirtualHost>
<VirtualHost _default_:443>
ServerName mysite.example.com
DocumentRoot /app/directory/
SSLEngine On
# etc...
</VirtualHost>
Then setup your SSL certificate with LetsEncrypt in your deploy script.
|
Field
One or more example embodiments of inventive concepts relate to memory controllers and methods of controlling memories, and more particularly, to memory controllers and flash memory reading methods that use various decoding methods according to threshold voltage distributions of flash memory cells.
Description of Conventional Art
A conventional semiconductor memory device detects and corrects data errors using encoding and decoding technology based on error correction codes. One example of an error correction code is a low density parity check (LDPC) code, which is an error correction code that uses a probability-based repeated calculation. However, decoding time in a reading process of a flash memory using such an LDPC code may be relatively lengthy. |
(ShardManager).GetShardsForSentinels
Description
*** This method is specific to Dropbox zookeeper-managed memcache ***
This returns a (shard id -> (connection, list of items)) mapping for
the requested sentinel items. Sential items that do not belong to any
shard are mapped to shard id -1. |
Seven people have been arrested in connection with the death of a Delhi Police head constable during clashes over the new citizenship law in northeast district last month, police said on Thursday.
Head constable Ratan Lal died of bullet injuries in the clashes in Gokalpuri on February 24.
"Seven people have been arrested in Ratan Lal's case. It emerged during investigation that the spot where the incident took place was the venue for an anti-CAA protest. In the fateful day, a conspiracy was hatched to attack police," a senior Delhi Police official said. |
Elizabeth Taylor arrives for a play in Los Angeles in this December 1, 2007 file photo REUTERS/Mario Anzuoni/Files
Friends and family of Elizabeth Taylor may get unexpected visitors at her funeral — picketers from the Westboro Baptist Church.
Margie Phelps, daughter of the hate group’s leader Fred Phelps, tweeted Wednesday that they would picket Taylor’s funeral, most likely because of Taylor’s gay friends and humanitarian work against AIDS. Phelps also tweeted several attacks on Taylor, including “RIP Elizabeth Taylor is in hell as sure as you’re reading this & getting mad as a wet hen. She should’ve obeyed God. Too late!”
Taylor was a prominent AIDS activist, starting the American Foundation for AIDS Research after the 1985 death of Rock Hudson. In 25 years, the foundation raised over $100 million. (via TMZ)
(More on TIME.com: See Elizabeth Taylor’s life in pictures)
(More on TIME.com: See the top 10 gratuitously provocative acts) |
n the Lady Shark’s first game of the morning they faced the Busan International Foreign School (BIFS) Bears. The Bears started the game with a full court press and really pressured our girls, making it very difficult to advance the ball up the court. BFS struggled to break the pressure and were dominated by BIFS, losing by a score of 6-25! BFS had 32 turnovers in the game and with that many turnovers it is easy to see how BIFS pulled away to win by 19 points. For the game, our BFS Lady Sharks made just 3 shots out of 12 attempts for a total of 6 points, while the Bears made 12 shots out of 21 shooting attempts. Although we lost this game we were confident that we could earn a rematch against BIFS in the Final if we could win our next two games.
In the second game of the day the Lady Sharks faced Handong International School (HIS). Our girls played much better and outscored HIS 29-8 for the win. The Sharks made 13 field goals and 1 three point shot out of a total of 44 attempts (29 points), while holding HIS to just 3 for 27 shooting the basketball and two free throws to round out their scoring at 8 points.
In their third game of the day our Lady Sharks faced the Daegu International School (DIS) Jets. In this game we continued our hot shooting by making 9 out of 18 shots in the first half, including 5 three pointers by Daniella; propelling the Sharks out in front at halftime by a score of 23-5. DIS made just 1 shot out of 5 shooting attempts to go along with 3 free throws prior to halftime. In the second half the Jets made 3 baskets and a free throw to round out their scoring at 12 points, while BFS scored 2 baskets in the second half to finish out their scoring for a final result of BFS 27 – DIS 12.
Having won the previous two games our girls had now earned their rematch with BIFS in the SKAC HS Girls Basketball Championship Game. Knowing we would face tremendous pressure, we knew ball control and ball security was going to be the difference in the game. The game started similar to the first round game whereby BIFS pressured and scored early, hitting the first two baskets of the game before the Sharks broke the ice and hit their first field goal to make the score 2-4 about midway through the first half. After more tough defense on both sides Daniella hit a 3 point shot to give the Sharks their first lead of the game at 5-4. BFS picked up another basket and a free throw, and BIFS also scored a basket to close out the halftime score at 8 to 6 in favor of the BFS Lady Sharks.
In the second half Jihong scored an early bucket to push the lead in favor of BFS to 10-6. BIFS hit a free throw, then another basket to close the gap to 10-9 with 11 minutes remaining in the championship game. It was at this point in the game that our Lady Sharks showed their true grit and determination, with three successive baskets to push the Shark’s ahead to 16-9. BIFS answered with a bucket to cut the lead to 16-11, then Daniella lifted a high-soft-floater from around 22′ that hit nothing but the bottom of the net “SWISH”, another 3 point basket that pushed BFS out in front for good at 19-11 with roughly 5 minutes to play. In the end the BFS Lady Sharks would simply not be denied the championship trophy; closing out the game to win by a final score of 23 – 16!!
But… what about the boys?
The boys basketball team journeyed to the BIFS Bear Cave for the final competition of the SKAC season. Fate would have it that the BFS Sharks’ first game was against the Jets of Daegu. The Sharks’ defense was ferocious, holding DIS to only 15 points! The offense was slow to get the engine running though, and the final score was 8 to 15 in favor of the Jets. In their second game of the day, the Sharks were face to face with HIS, who would go on to win the championship. The BFS Sharks proved in the first half that they can be a force to be reckoned with! Going into halftime, BFS was only 6 points behind the future champs. Maintaining a physical and tenacious defense proved to be too much for BFS in the end, and the final score was 13 to 38. In their final game of the season, the Sharks were physically superior to their opponent, KFS. They left it all on the court, racking up seven charges taken on defense! KFS made some big three point baskets in the closing minutes and ended up winning 25 to 40.
The BFS boys team has so much potential, and they have their eyes set on the SKAC championship. The team will continue to practice and work hard to improve in preparation for the KISAC tournament in Jeju. |
from django.db import models
from pydis_site.apps.api.models.mixins import ModelReprMixin
class DocumentationLink(ModelReprMixin, models.Model):
"""A documentation link used by the `!docs` command of the bot."""
package = models.CharField(
primary_key=True,
max_length=50,
help_text="The Python package name that this documentation link belongs to."
)
base_url = models.URLField(
help_text=(
"The base URL from which documentation will be available for this project. "
"Used to generate links to various symbols within this package."
)
)
inventory_url = models.URLField(
help_text="The URL at which the Sphinx inventory is available for this package."
)
def __str__(self):
"""Returns the package and URL for the current documentation link, for display purposes."""
return f"{self.package} - {self.base_url}"
class Meta:
"""Defines the meta options for the documentation link model."""
ordering = ['package']
|
993 S.W.2d 171 (1999)
Jeffrey EDWARDS, Appellant,
v.
The STATE of Texas, Appellee.
No. 08-97-00518-CR.
Court of Appeals of Texas, El Paso.
March 18, 1999.
Rehearing Overruled June 23, 1999.
*174 Charles Louis Roberts, El Paso, for Appellant.
Jaime E. Esparza, District Attorney, Tom A. Darnold, Assistant District Attorney, El Paso, for the State.
Before Panel No. 1 LARSEN, McCLURE, and CHEW, JJ.
OPINION
SUSAN LARSEN, Justice.
This is an appeal from a conviction for four counts of indecency with a child. The appellant, Jeffrey Edwards, entered an open guilty plea to the trial court, and the court sentenced him to five years confinement in the Texas Department of Criminal JusticeInstitutional Division. We affirm the trial court's judgment.
FACTS
On September 10, 1997, Edwards pleaded guilty to four counts of indecency with a child. The trial court admonished Edwards regarding the range of punishment and Edwards' right to a jury trial. In response to the court's questioning, Edwards stated that he understood these issues. Edwards further responded that he understood there was no punishment recommendation from the State, and that the trial court could sentence him to any punishment within the punishment range. Edwards affirmed that no one had forced him into entering his plea, nor had anyone promised him anything to cause him to enter the plea. He professed that he was satisfied with his counsel's representation and that he felt his counsel had adequately investigated and prepared his case. Moreover, Edwards confirmed that he had read and signed the plea papers in the case which, among other things, stated that there was no punishment recommendation and that no punishment recommendation was binding upon the court. Edwards also signed written admonishments which confirmed that he understood there was no plea bargain recommendation, that he had not been threatened or promised anything to get him to make a guilty plea, and that he was mentally competent and knew what he was doing by entering a plea.
During the State's punishment presentation, Paul Strelzin, principal of the high school the victim attended and where Edwards worked as a trainer, testified that Edwards had voluntarily given him a handwritten statement detailing one of the offenses. Strelzin related that Edwards was crying and "extremely emotional" after giving him the statement. Strelzin was "incredulous" over the incident because he had never had a problem with Edwards before. According to Strelzin, Edwards had been "an outstanding trainer ... *175 [w]ell qualified, capable of handling his job."
Edwards also addressed the court during punishment. He apologized to the court, the victim, and the victim's family. Stating, "I don't know why I did that. I will never do it again," Edwards asked for the forgiveness of both the court and the victim's family.
After receiving a five year sentence, however, Edwards filed a motion for new trial contending that he was not competent at the time of his plea, that he pleaded guilty only because of inaccurate advice and improper pressure from his trial counsel, and that his trial counsel had otherwise rendered ineffective assistance. At the hearing on Edwards' motion, psychologist Dr. Karen Gold testified about her post-guilty plea evaluation of Edwards and her conclusions regarding Edwards' mental competence. Dr. Gold concluded that Edwards suffered from a variety of psychological disorders, most importantly a "Cluster C personality disorder." According to Dr. Gold, Edwards' personality disorder caused him to take responsibility for things that have gone wrong when confronted, even if he had nothing to do with the problem. Moreover, Dr. Gold opined that Edwards is incapable of thinking independently, has difficulty handling himself with authority figures, and will avoid any kind of hostility. Edwards' disorder was so severe, according to Dr. Gold's evaluation, that Edwards would even jump off an overpass on to Interstate 10 "if you applied enough pressure and insisted that he do it...." In Dr. Gold's estimation, Edwards would "instantly capitulate" to an attorney's advice to plead guilty if the attorney pressured him. Edwards' father and girlfriend also testified at the hearing and both agreed that Edwards had difficulty standing up for himself and would often acquiesce to avoid confrontation. Edwards' father further testified that when he visited with Edwards approximately a week before Edwards' guilty plea, Edwards was "strung out" and so nervous and scared that he was unable to hold a conversation.
Edwards testified that despite his repeated protestations of innocence, his trial counsel had insisted that he plead guilty. Edwards' trial counsel allegedly assured him that the judge and the district attorney had agreed to set his punishment at five years deferred adjudication probation in return for his guilty plea. Additionally, Edwards and his girlfriend testified that trial counsel refused to take Edwards' calls and even threatened Edwards with jail time if Edwards persisted in calling. Finally, Edwards claimed that he made the written statement to Strelzin under duress from an assistant principal who told him exactly what to write.
The trial court denied Edwards' request for a new trial and Edwards appeals from that decision and from his original guilty plea with ten issues.
INVOLUNTARY PLEA
In his first six issues, Edwards contends that his guilty plea was involuntary, and that the trial court erred in failing to grant him a new trial based on the involuntariness of his plea. He makes three arguments in support of these contentions: (1) he was incompetent at the time of his plea; (2) he was threatened by his trial counsel to plead guilty; and (3) he was induced to plead guilty by improper promises from his trial counsel. We will analyze Edwards' first six issues in terms of the three arguments he makes in support of them.
1. Competence
A defendant may challenge his competency to stand trial in a motion for new trial.[1] When raising the competency issue in this manner, a defendant may present evidence regarding his competency developed after conviction as Edwards *176 has done in this case.[2] Because the motion for new trial hearing occurs after sentencing and not "during trial,"[3] the trial court applies the traditional standard used to determine whether to grant a motion for new trial. In other words, the trial court considers all the evidence presented, judges the credibility of the witnesses, and resolves conflicts in the evidence.[4]
In evaluating Edwards' claim that the trial court erroneously denied his motion for new trial, we consider all of the competency evidence presented at the motion for new trial hearing and reverse the trial court only if it abused its discretion.[5] We apply this standard because, at this stage of the proceeding, the trial court was determining whether Edwards' incompetency impugned the integrity of its judgment during trial.[6] The trial court, having observed Edwards both at trial and at the motion for new trial hearing, is in the best position to make this determination.[7]
Under the Texas Code of Criminal Procedure, a defendant is incompetent to stand trial if he does not have either: (1) sufficient present ability to consult with his lawyer with a reasonable degree of rational understanding; or (2) a rational as well as factual understanding of the proceedings against him.[8] Edwards' evidence in this case did not establish either of these factors. Rather, it established that Edwards would not get into a confrontation and would back down under pressure. Specifically, Edwards' claim is that given his particular personality disorder, he was unable to overcome the pressure brought to bear by his attorney's insistence that he plead guilty. Thus, the evidence does not indicate that Edwards' disorder rendered him unable to consult with an attorney with a reasonable degree of understanding nor does it establish that Edwards did not have the ability to understand the proceedings against him.
Even if Edwards' evidence did tend to establish legal incompetence, the trial court was entitled to balance that evidence against the record from Edwards' guilty plea, including the trial court's own observations of Edwards at the time of the plea.[9] Evidence tending to negate Edwards' theory was presented at the punishment stage. The trial court had before it Strelzin's testimony that Edwards had admitted the offense at a time prior to Edwards' contact with the allegedly overbearing attorney. Additionally, the record from the guilty plea and the punishment hearing in this case shows that Edwards coherently answered questions presented by the trial court. He gave a lucid apology to the court and to the victim's family. This evidence tends to establish that Edwards understood the proceedings and the nature of the allegations against him. Both the trial court and Edwards' attorney specifically stated that they found Edwards to be competent. Given the evidence available from both the motion for new trial hearing and the original guilty plea, we find that the trial court was within its discretion to deny Edwards a new trial based upon incompetence.
2. Threats and Promises
Edwards' next two arguments center on alleged threats and promises made by his trial counsel to induce his *177 guilty plea. Specifically, Edwards contends that his trial counsel assured him that the judge and the district attorney had agreed to allow him to serve a deferred adjudication probation in return for his guilty plea. Moreover, Edwards claims that his trial counsel told him to plead guilty "or else." The law regarding guilty pleas is firmly established. A guilty plea must be freely, voluntarily, and knowingly made on the part of the defendant.[10] This requirement assures that each defendant who pleads guilty to a criminal offense does so with a full understanding of the charges and the consequences of his plea.[11]
When the record shows that the trial court properly admonished the defendant, as it does here, however, it presents a prima facie showing that the guilty plea was knowing and voluntary, and the burden then shifts to the defendant to establish that he or she did not understand the consequences of the plea.[12] Although Edwards testified that he pleaded guilty because his counsel threatened him and promised him deferred adjudication probation, the record reflects that the trial court admonished Edwards that there was no punishment recommendation from the State. The trial court cautioned Edwards that, without any recommendation, Edwards' punishment could be anywhere within the range of punishment "all the way up to 20 years confinement in the State Penitentiary." Moreover, the trial court reminded Edwards of a plea offer Edwards had rejected and specifically told Edwards that his sentence could be harsher than the rejected plea offer. During his plea, Edwards acknowledged that no one had promised him anything to cause him to enter his plea, nor had anyone forced him to plead guilty. Finally, Edwards affirmatively stated that he was satisfied with his counsel's representation.
This court was presented with a similar situation in White v. State.[13] The appellant in White, like Edwards, testified at his plea hearing that he had not been promised anything in return for his plea. Also like Edwards, the appellant in White later testified that he was promised something in exchange for his guilty plea. We found, however, that the appellant in White failed to prove his counsel's erroneous promises prejudiced his defense for three reasons. First, the appellant stated at the time of his plea that he was not entering his plea on the basis of a promise. Second, the appellant was properly admonished by the trial court. Third, the appellant signed a statement at the time of his plea indicating he was satisfied with his defense counsel's representation.[14] In this case, Edwards affirmed at the time of his plea that he was not entering his plea based on any promises or threats. Similarly to the appellant in White, Edwards was properly admonished in this case, and he stated to the trial court that he was satisfied with his counsel's representation. We therefore find the circumstances in this case almost identical to the circumstances in White, and we find that Edwards has similarly failed to demonstrate involuntariness in his plea.
3. Conclusion: Involuntariness
Having found that the trial court did not err in denying Edwards a new trial based on incompetence or based on involuntariness of his plea, we overrule Edwards' first six issues for review.
INEFFECTIVE ASSISTANCE OF COUNSEL
In his seventh through tenth issues, Edwards contends that he did not receive *178 effective assistance of counsel, and that the trial court erred in failing to grant him a new trial based on ineffective assistance. In issues seven and eight, Edwards challenges his trial counsel's performance prior to and during his guilty plea. In issues nine and ten, Edwards challenges his trial counsel's performance at punishment stage.
1. Counsel's Performance: Guilty Plea
The United States Supreme Court established a two-prong test for analyzing an ineffective assistance of counsel claim in Strickland v. Washington.[15] The Strickland test also provides the standard of review when there is a challenge to a guilty plea based on ineffective assistance of counsel.[16] Under the first prong, appellant must demonstrate that his counsel's performance fell below an objective standard of reasonableness.[17] Once appellant meets this burden, he must next show that there is a reasonable probability that, but for counsel's errors, he would not have entered his pleas and would have insisted on going to trial.[18] In this case, Edwards relies on his counsel's failure to properly advise him regarding his guilty plea and his counsel's failure to discover and raise the competence issue to establish that counsel's performance fell below an objective standard of reasonableness.
We do not find that the record necessarily establishes trial counsel's failure to properly advise Edwards regarding his plea. Although Edwards testified at the hearing on his motion for new trial that his trial counsel incorrectly promised him deferred adjudication probation and improperly threatened him with jail time unless he pleaded guilty, these allegations are contradicted by Edwards' own responses to the trial court's questioning at the time of his guilty plea. Edwards confirmed his understanding that the trial court could sentence him to any punishment within the punishment range. Moreover, Edwards acknowledged that his guilty plea was not the result of any promise or threat. Accordingly, the record is conflicting at best. Allegations of ineffective assistance will be sustained only if they are firmly founded in the record.[19] We are hard pressed to find Edwards' allegations "firmly founded" in this record. We therefore overrule those portions of Edwards' seventh and eighth issues asserting ineffective assistance for failure to properly advise Edwards on his guilty plea and complaining of the trial court's failure to order a new trial based on those assertions.
Edwards also charges that his trial counsel was ineffective for failure to raise Edwards' incompetence. He maintains that had counsel only spent more time with him and conducted a more thorough investigation into his background, his personality disorder would have become apparent. Edwards contends that trial counsel could have raised Edwards' incompetence as a defense if counsel had discovered it. As we have already determined, however, the personality disorder Edwards claims to have, while unfortunate, does not render him legally incompetent to understand the proceedings or to consult with his attorney with a reasonable degree of rational understanding.
*179 In any event, the record does not firmly establish that trial counsel performed deficiently by failing to discover Edwards' disorder. The record reveals that trial counsel had contacted at least thirty of Edwards' friends, colleagues, and former co-workers prior to the guilty plea. In addition, trial counsel represented that he talked to Edwards' father for four hours.[20] Edwards did not present any evidence that any of these people either informed trial counsel of Edwards' personality disorder, or could have so informed counsel if questioned sufficiently. Moreover, Edwards did not present any evidence of any person trial counsel might have contacted, but failed to contact, who could have shed light on Edwards' disorder. There is also no evidence that Edwards himself ever attempted to apprise his trial counsel of his disorder.
We note also that the trial court had an opportunity to observe Edwards during the guilty plea, punishment evidence, and during Edwards' articulate statement during punishment in which he apologized to the court and to the victim's family. The trial court was able to observe Edwards further during his testimony at the new trial hearing. At this hearing, the trial court could have observed Edwards when he stood up for himself by disagreeing with the prosecutor when she cross-examined him. These in-court observations of Edwards under pressure gave the trial court an opportunity to gauge the degree to which Edwards' personality disorder manifested itself and the ease (or difficulty) with which trial counsel may have discovered it.
Accordingly, we find that the record as a whole supports a conclusion that trial counsel conducted a reasonable investigation into Edwards' background. It further supports the conclusion that counsel's failure to discover a specific personality disorder despite an otherwise thorough investigation did not, in itself, render the investigation unreasonable. We therefore find that the trial court did not abuse its discretion in refusing Edwards a new trial based on ineffective assistance during the guilty plea and we overrule the remainder of Edwards' seventh and eighth issues for review.
2. Counsel's Performance: Punishment
In his ninth and tenth issues, Edwards contends that trial counsel was ineffective at the punishment stage for failure to present live witnesses and again for failure to present evidence of his personality disorder. The standard of review applied to determine if an accused was denied effective assistance of counsel in the punishment phase of a non-capital trial is whether the accused received reasonably effective assistance based upon the totality of the representation.[21] In applying the standard, we are required to review the full scope of counsel's assistanceincluding representation, performance, and deliveryto determine the quality of the assistance actually rendered.[22] Appellant's right to reasonably effective counsel does not entitle him to errorless counsel whose effectiveness is judged through hindsight.[23] Furthermore, reasonably effective assistance can include strategies fostered by trial counsel which, without the distorting effects of hindsight, might not have been advanced by appellate counsel.[24] Even *180 though the appellant need not meet the more stringent two-pronged Strickland standard applied in other situations, the appellant does shoulder the burden of establishing that the totality of the assistance actually rendered was not reasonably effective.
Edwards first asserts that trial counsel failed to render reasonably effective assistance because he failed to call one of the approximately thirty persons he contacted during his investigation and instead attempted to introduce letters from each of them. The State objected to the letters as hearsay and the trial court excluded them from evidence. Edwards maintains that this incident constitutes ineffective assistance for two reasons. First, trial counsel failed to call known defense witnesses to testify. Second, trial counsel demonstrated a lack of basic knowledge of the law by attempting to introduce hearsay evidence. Viewing the full scope of counsel's assistance, however, we do not find that these alleged deficiencies brought counsel's performance below the level of reasonable effectiveness.
Edwards contends that the defense witnesses counsel failed to call would have testified to his good character in mitigation of punishment. It is undisputed that the witnesses were not called, but the record reflects that counsel deduced evidence of Edwards' good character from a State witness. Paul Strelzin, who was called by the State, testified in response to trial counsel's questioning that he had "tremendous admiration for [Edwards]," and that Edwards "had the concern of the welfare of the students, of the school, of the staff, of the community" in mind as he went about his training duties. Trial counsel elicited from Strelzin that Strelzin had been "totally in shock" over the allegations against Edwards because Edwards had been a "man that I [Strelzin] rely upon." Finally, trial counsel extracted from Strelzin that he "had a great deal of admiration for him [Edwards] then, still do." Thus the record reflects that trial counsel effectively presented testimony of Edwards' good reputation and character in mitigation of punishment. Furthermore, there is no indication that any of the thirty witnesses Edwards claims should have been called on his behalf would have been any more helpful to mitigation of Edwards' punishment than Strelzin. Accordingly, we find that Edwards has failed to demonstrate that the quality of the totality of the assistance actually rendered to him at punishment fell below the standard of reasonable effectiveness. We therefore overrule those parts of Edwards' ninth and tenth issues complaining of ineffective assistance for failure to present additional character witnesses, and of the trial court's failure to order a new trial on that ground.
Finally, Edwards contends that his trial counsel was ineffective because he failed to discover and present evidence of Edwards' personality disorder in mitigation of punishment. For the same reasons we found trial counsel reasonably effective despite his failure to discover and present evidence of incompetence at the guilty plea stage, we also find counsel rendered reasonably effective assistance despite the failure to discover and present evidence of personality disorders in mitigation of punishment. Accordingly, we overrule the remainder of Edwards' ninth and tenth issues.
CONCLUSION
Having considered and overruled each of Edwards' ten issues on appeal, we affirm the judgment of the trial court.
NOTES
[1] See Brown v. State, 960 S.W.2d 772, 778 (Tex.App.Dallas 1997, pet. ref'd).
[2] Id.
[3] TEX.CODE CRIM. PROC. ANN. art. 46.02, § 2(b) (Vernon 1979).
[4] Brown, 960 S.W.2d at 778; Hafford v. State, 864 S.W.2d 216, 217 (Tex.App.Beaumont 1993, no pet.).
[5] Lewis v. State, 911 S.W.2d 1, 7 (Tex.Crim. App.1995).
[6] Brown, 960 S.W.2d at 778.
[7] Id.
[8] TEX.CODE CRIM. PROC. ANN. art. 46.02, § 1(a) (Vernon 1979).
[9] See Brown, 960 S.W.2d at 778 (trial court, having observed defendant both at trial and at motion for new trial hearing, is in best position to determine competence).
[10] Ex parte Evans, 690 S.W.2d 274, 276 (Tex. Crim.App.1985); White v. State, 892 S.W.2d 223, 226 (Tex.App.El Paso 1995, no pet.).
[11] Basham v. State, 608 S.W.2d 677, 678 (Tex.Crim.App.1980); White, 892 S.W.2d at 226.
[12] Hernandez v. State, 885 S.W.2d 597, 601 (Tex.App.El Paso 1994, no pet.); Smith v. State, 857 S.W.2d 71, 73 (Tex.App.Dallas 1993, pet. ref'd).
[13] 892 S.W.2d 223 (Tex.App.El Paso 1995, no pet.).
[14] Id. at 227.
[15] 466 U.S. 668, 104 S.Ct. 2052, 80 L.Ed.2d 674 (1984).
[16] Hill v. Lockhart, 474 U.S. 52, 58, 106 S.Ct. 366, 370, 88 L.Ed.2d 203, 210 (1985); Ex parte Adams, 707 S.W.2d 646, 649 (Tex.Crim. App.1986); Larson v. State, 759 S.W.2d 457, 460 (Tex.App.Houston [14th Dist.] 1988, pet. ref'd), cert. denied, 490 U.S. 1008, 109 S.Ct. 1646, 104 L.Ed.2d 161, pet. reh'g denied, 490 U.S. 1085, 109 S.Ct. 2112, 104 L.Ed.2d 672 (1989).
[17] Strickland, 466 U.S. at 687-88, 104 S.Ct. at 2065, 80 L.Ed.2d at 693.
[18] Id. at 694, 104 S.Ct. at 2068, 80 L.Ed.2d at 697; White, 892 S.W.2d at 228.
[19] Welch v. State, 908 S.W.2d 258, 261 (Tex. App.El Paso 1995, no pet.).
[20] This representation was disputed at the motion for new trial hearing. Trial counsel's contact with thirty of Edwards' acquaintances, however, was undisputed.
[21] Ex parte Walker, 794 S.W.2d 36, 37 (Tex. Crim.App.1990); Ex parte Duffy, 607 S.W.2d 507, 516 (Tex.Crim.App.1980); Jaile v. State, 836 S.W.2d 680, 683, 686-87 (Tex.App.El Paso 1992, no pet.).
[22] Walker, 794 S.W.2d at 37.
[23] See Stafford v. State, 813 S.W.2d 503, 506 (Tex.Crim.App.1991); Ex parte Cruz, 739 S.W.2d 53, 58 (Tex.Crim.App.1987); Jaile, 836 S.W.2d at 687.
[24] See Stafford, 813 S.W.2d at 506.
|
- 15*w + 21*w - 1. Is r(9) a multiple of 28?
False
Let f(q) = -q**2 - 9*q - 9. Suppose -7 = 2*j - j. Let y be f(j). Suppose -2*b + p = -4*b + 64, b - 10 = y*p. Does 19 divide b?
False
Let w(k) = k**3 - 3*k**2 - 2*k - 4. Let z be w(4). Is 15 a factor of (z/(-6))/((-18)/2403)?
False
Let u be -3*1*16/12. Let w be (-55)/(-33) - u/(-6). Does 6 divide 130/(-5)*w/(-2)?
False
Let b = 10836 - 6452. Is b a multiple of 32?
True
Let a = 997 - 529. Is 12 a factor of a?
True
Suppose 5*j - 6 = 6*j. Does 24 divide (9/j)/3*-33*6?
False
Does 5 divide 1737/5 - ((-24)/(-60) + -1)?
False
Let b(n) = -n**2 + 7*n - 5. Let h be b(5). Suppose -4*v - h*j = 84 - 282, -3*j = 4*v - 202. Is v a multiple of 17?
False
Let s(j) = j**3 - 7*j**2 - 7*j - 8. Let t be s(8). Let y be (2 + t)/((-8)/(-364)). Let z = y + -35. Is 28 a factor of z?
True
Suppose 18*m = -d + 23*m + 183, 2*m = -4. Is 10 a factor of d?
False
Suppose -4*m - 52 = -256. Let g = -10 + m. Is 11 a factor of g?
False
Does 50 divide (2962/(-10))/(((-117)/45)/13)?
False
Let d(p) = p + 71. Let z be d(-7). Let s = z + -40. Is s a multiple of 6?
True
Suppose -3*n + 0*k - 63 = 3*k, -22 = 2*n - 2*k. Let r = n + 44. Is r a multiple of 6?
False
Let w = 7 - 7. Suppose 0*a - a + 2 = w. Suppose 2*n + a*n + 3*h = 233, h + 253 = 4*n. Is n a multiple of 27?
False
Let t = 26 - 54. Does 9 divide ((-10)/3)/(t/378)?
True
Suppose -343 = -5*o - c, 5*c - 190 - 29 = -3*o. Is 17 a factor of o?
True
Let b(v) = -v**3 + 17*v**2 - 4*v - 3. Is 16 a factor of b(16)?
False
Suppose 3*f + l - 23 = 0, -3*f - l = 3*l - 38. Let q(j) = -10*j - 8. Let x be q(f). Let y = x + 108. Is 20 a factor of y?
True
Let q(o) = -11*o - 1 + 13 + o + o**2. Let m be q(10). Is 11 a factor of m/24 - (-174)/4?
True
Suppose 4*g - 399 = -h, 0*g + 495 = 5*g + 2*h. Does 3 divide g?
False
Let d(j) = -706*j + 42. Is d(-1) a multiple of 43?
False
Let h(x) = 22*x + 0*x**3 + 14 - x**3 - 6*x**2 + 4*x**2 + 17*x**2. Is h(16) a multiple of 22?
True
Does 73 divide 0 + 212 - (-140)/20?
True
Suppose -2186 = -13*f - 314. Is 9 a factor of f?
True
Suppose 0 = -3*h + 5*x + 5, -5*h + 0*x + 4*x = -4. Let f = -3 - -4. Does 14 divide f/(1/28) - h?
True
Let p(k) = 8*k**2 + 9*k - 119. Is p(9) a multiple of 47?
False
Let c = 571 + -402. Does 36 divide c?
False
Let w(d) = -2*d**2 - 9*d + 7. Let q be w(6). Let o = -257 - q. Let z = 258 + o. Is 32 a factor of z?
False
Suppose -5*n + 4*h + 57 + 40 = 0, -h = 5*n - 82. Suppose 15*i + 80 = n*i. Is i a multiple of 8?
True
Let h be (-4)/(-3)*(-6)/(-4). Suppose -5 - 31 = -h*y. Is y a multiple of 10?
False
Suppose -3*m - 4*m - 196 = 0. Let q = m - -13. Let r = 58 + q. Does 17 divide r?
False
Let c(o) = -99*o - 269. Does 12 divide c(-11)?
False
Let s = 11 - 11. Suppose -4*y = -0*y + 4*n - 688, s = -y - 4*n + 178. Suppose -3*g + 1 = -y. Is 19 a factor of g?
True
Let k = 4530 + -2363. Does 71 divide k?
False
Let v(o) = 14*o - 63. Does 74 divide v(26)?
False
Let r(o) = -o - 1. Let y(x) = 8*x - 3. Let v(u) = -2*r(u) + y(u). Does 4 divide v(2)?
False
Let j(l) = l**3 + 2*l**2 - 1. Let v be j(-1). Let w be (2 + -6 + v)*-1. Suppose w*y = -x + 11, 7*y = 3*y - 4*x - 4. Is y even?
True
Is 25 a factor of (-3)/(((-6)/400)/((-2)/(-8)))?
True
Let q(w) be the first derivative of -10*w + 8 - 9/2*w**2. Does 15 divide q(-6)?
False
Suppose -12 = 3*o - 90. Let s = o + -24. Is s even?
True
Let n(h) = 10*h + 28. Is 13 a factor of n(4)?
False
Let o(v) = v**2 + 18*v - 4 - 1 - 26*v + 0*v**2. Is o(-5) a multiple of 15?
True
Let s be 2/((-92)/(-100) - 1). Let p(h) = -61*h**3 - 2*h**2 + 1. Let l be p(-1). Let j = s + l. Is j a multiple of 17?
False
Suppose 1355 = 35*n - 30*n. Is n a multiple of 8?
False
Let w(y) = 2*y**2 - 21*y + 105. Is 19 a factor of w(30)?
False
Suppose 270 = 4*f - 458. Is f a multiple of 14?
True
Let s = 8 - -2. Suppose p - 5*t = s - 42, -t + 136 = -5*p. Is 3 a factor of 1/(-3) - 333/p?
True
Suppose -52*y = -56*y + 36. Let s(c) = 2*c + 12. Does 30 divide s(y)?
True
Suppose -3*y = 2*y - 15. Let i(o) = y*o**3 + 3*o**3 + o**3 - o**3. Does 2 divide i(1)?
True
Suppose 3*x = x + 8. Suppose -4*v - 92 = -k, -8 = 3*v + x. Suppose 5*u = u + k. Does 8 divide u?
False
Let u be (6/(-4))/(3/(-4)). Suppose 5*s = -5*t + 15, 3*t + 6 = 4*t + u*s. Suppose 3*w + 2*d - 143 = t, 3*d - 303 = -4*w - 113. Is 11 a factor of w?
False
Let v = 34 - 62. Let b = v - -53. Suppose -b = -c + 35. Does 15 divide c?
True
Let r = 401 - -812. Is r a multiple of 41?
False
Suppose -2*a - a - 22 = 2*w, 4*w = -2*a - 44. Is 6 a factor of w/2*(4 - 10)?
False
Let w = -642 + 1453. Is 37 a factor of w?
False
Let c be (8/(-16))/(1/(-48)). Let q(b) = -b**3 + 8*b**2. Let g be q(8). Suppose g = -z - z + c. Is z a multiple of 10?
False
Suppose 0*y = 5*y - 10. Let t be 1/1*10/2. Suppose y*w - 119 = 3*o - 35, t*w - 210 = -4*o. Does 26 divide w?
False
Suppose 3*v - 483 = -3*x, 4*x = 4*v + 3*x - 619. Does 7 divide v?
False
Let f(a) = -4*a + 4. Let z(g) = g - 1. Let q(m) = 2*f(m) + 9*z(m). Let i(j) = -9*j - 7. Let p(y) = i(y) + 6*q(y). Is 4 a factor of p(-6)?
False
Suppose 2*f - 23 = -3*v, 2*f + 4 - 12 = 0. Suppose 3*g - 77 = -v*a, 5 = -3*a - 1. Suppose g = 4*d - 3*d. Does 11 divide d?
False
Let z(m) = -m**3 + 20*m**2 + 8*m + 27. Is z(20) a multiple of 31?
False
Suppose -919 = 18*l - 9559. Is l a multiple of 4?
True
Suppose 0 = -5*t + 3*x + 2*x + 355, 3*x - 59 = -t. Suppose -4*j = -t - 248. Is j*(-2 + -1 - -4) a multiple of 20?
False
Let o(m) = -5*m**3 - 3*m**2 - m - 1. Let l be o(-2). Let a = l + -29. Suppose -i = -a*i - 86. Does 21 divide i?
False
Let c(p) = 17*p**2 + 5*p + 4. Let a(h) = h**2 - 1. Let i(b) = 2*a(b) + c(b). Let f be i(-2). Let l = f + 1. Does 17 divide l?
False
Suppose -48*o = -60*o + 8136. Is o a multiple of 53?
False
Let v = -1104 + 2554. Does 46 divide v?
False
Let q be 0/3*-1 - -3. Suppose -25 = 5*g, -4*r - g - q*g = -152. Suppose 0 = -2*u - 3*j + r - 0, -54 = -3*u - j. Is u a multiple of 5?
False
Let a(u) = 2*u - 3. Let k(o) = -10*o + 14. Let i(x) = -14*a(x) - 3*k(x). Let p be i(1). Suppose -p*r + 3*r - 63 = 0. Is 22 a factor of r?
False
Let u be (-6)/(-9)*12/(-8). Let c be 2 + (-4 - 1/u). Let f(b) = -50*b**3 - b**2 - 2*b - 1. Is f(c) a multiple of 20?
False
Is 25 a factor of 6 - (3 + (-201 - -4))?
True
Let r(o) be the second derivative of o**4/12 + 7*o**2 - 24*o. Is 10 a factor of r(6)?
True
Let t(q) = -q**3 - 9*q**2 - 15*q - 36. Is t(-12) a multiple of 24?
True
Let g(t) = t**3 + 13*t**2 + 11*t - 10. Let x = -13 + 3. Does 45 divide g(x)?
True
Suppose 5*r - 8 = 7*r. Is ((-34)/8 - r/16) + 151 a multiple of 21?
True
Is 16 a factor of 10660/18 - 3/27*2?
True
Let m(z) = -z. Let n(f) = 2*f**2 - f - 5. Let a(g) = 3*m(g) + n(g). Is 24 a factor of a(10)?
False
Let b = -23 + 12. Let y = b + 6. Is 16 a factor of (-109)/y - (-4)/20?
False
Let y be ((-20)/6)/(2/(-3)). Let m(f) = 2*f + 42. Let d be m(-19). Suppose -3*r - y*j = -29, -d*j + 28 = 4*r - r. Does 3 divide r?
False
Let q = 431 + 94. Suppose -11*j = -6*j - q. Does 21 divide j?
True
Does 19 divide 133/(-6*4/(-24))?
True
Let k be 18/3 - (-2)/(-2). Let u(t) = 4*t**3 - 2*t**2 + 2*t - 8. Let l be u(3). Suppose 0 = k*o - l - 17. Does 6 divide o?
False
Let s(p) = 2*p - 24. Let c be s(12). Does 43 divide (-43)/(c - (-4 - -5))?
True
Is 2 a factor of 44/3*(-33)/(-11)?
True
Is 4/(-12) - (-1 + (-916)/12) a multiple of 8?
False
Suppose -2*r = -8*a + 7*a + 66, 2*a + r = 127. Does 8 divide a?
True
Suppose 38*l = -5901 + 32311. Is 6 a factor of l?
False
Let r = 667 + -303. Is 43 a factor of r?
False
Let a be 1 - 6*(-3)/(-6). Let c be 462/8 - (-3)/(-36)*-3. Let j = c - a. Is j a multiple of 15?
True
Let d be 389*(6 + -1 - 4). Let c = d - 277. Does 8 divide c?
True
Suppose -m + 36 = 1. Is 2 + m/(4 + -3) a multiple of 37?
True
Let q = -91 - -88. Is -33*((-6)/(-3) + q) a multiple of 3?
True
Let v(p) = -3*p**3 + 6 - 8*p**2 - 5*p**2 + 2*p**3 + 24*p**2. Let j be v(11). Is 3 - (-57 + j/2) a multiple of 15?
False
Suppose -3*v + 10 = 2*v. Let l be (-30)/(-20) - (-3)/v. Suppose p = -l, -3*b + 3*p = -p - 225. Is b a multiple of 17?
False
Let t be 2/1 + 0/(-13). Suppose -29 = -t*z - 5. Let n(q) = q**2 - 12*q + 14. Is 7 a factor of n(z)?
True
Let l be ((-172)/16)/((-3)/(-12)). Let p = -23 - l. Does 16 divide p?
False
Suppose -28 = 2*x - j, -2*x - 3*j = -4*x - 32. Does 21 divide - |
Yesterday a freight forwarding partner, Flexport, reported that the U.S. started hiking tariffs on Chinese goods from 10% to 25% without any formal notice from the U.S. Trade Representative. Customs Brokers prepared Harmonized Tariff Schedule (HTS) entries on Chinese goods through Customs and Border Patrol (CBP) and noticed a 25% tariff was included on goods arriving May 10th or later.
Large and small businesses could be impacted greatly by the preemptive hike in tariffs depending on the HTS codes used to identify their products. Without an official announcement many companies may not be able to make adjustments in time to protect their margins.
On May 6th, the U.S. Trade Representative, announced that plans were being made to increase tariffs on goods subject to Tranche 3 of the China 301 Tariffs from 10% to 25% on May 10th and later. However, due to a lack of an official notice posted in the Federal Register, many companies will undoubtedly be caught by surprise.
The Tweet Heard Around the World
On May 5th, President Trump posted a tweet threatening that tariffs on Chinese goods were to increase from 10% to 25% on May 10th. However, there is an official process that typically must be conducted before changes can be made to the HTS. It appears that this process has been bypassed. |
1. Introduction {#sec1-materials-11-00019}
===============
One focus of telemedicine is to provide health care in out-of-hospital settings. To that end, long-term remote patient monitoring, i.e., recording of vital signs like respiratory rate, electrocardiogram (ECG) or electrodermal activity (EDA), and interactive services for patient care have become available in recent years. Such developments can improve the quality of a patient's treatment by providing immediate access to healthcare. In particular, areas with high patient-to-doctor distance can benefit from such concepts \[[@B1-materials-11-00019]\]. However, even medically well-supplied areas can benefit, because long-term monitoring adds valuable information, whe compared to intermittent physical examinations by medical doctors. Besides, health care expenses can be reduced by gaining vital parameters remotely, without consulting a doctor \[[@B2-materials-11-00019]\].
The recording of biopotentials, i.e., electrical signals, is one essential aspect of remote monitoring systems \[[@B3-materials-11-00019],[@B4-materials-11-00019]\]. To monitor biopotentials in telemedical settings, skin electrodes are used. Typically, skin electrodes can be roughly classified into wet and dry electrodes \[[@B5-materials-11-00019]\]. The conventional Ag/AgCl wet electrodes are widely used in the clinical environment for measuring biopotentials, and possess the benefits of simplicity, low cost, and reliability. However, they have some limitations. The conductive gel dries over time, and is vulnerable to perspiration \[[@B6-materials-11-00019]\]. This hampers their application in long-term recordings because of their decreasing signal quality over time. Moreover, cleaning and skin preparation, which are necessary in order to yield the best possible conduction, can be time-consuming, particularly when using multiple electrodes. Finally, allergic dermatitis has been reported when using gel electrodes \[[@B7-materials-11-00019]\].
Dry electrodes have been developed to derive biopotentials without any need for skin preparation and conductive gel by using a benign metal. Although some patients developed a metal allergy \[[@B8-materials-11-00019],[@B9-materials-11-00019]\], dry electrodes can generally be assumed to improve patient comfort and ensure long-term applicability. However, such electrodes have the disadvantages of higher electrode-to-skin impedance, and are sensitive to movement artifacts \[[@B4-materials-11-00019],[@B6-materials-11-00019]\]. Furthermore, metal electrodes in combination with perspiration may degrade or lead to incompatibilities due to galvanic processes that generate unexpected ions \[[@B8-materials-11-00019]\]. Various designs have been developed to yield similar (electrical) properties to wet electrodes. Novel designs for measuring ECG or electroencephalogram (EEG) signals include needle electrodes \[[@B10-materials-11-00019]\], dry polymer electrodes \[[@B11-materials-11-00019],[@B12-materials-11-00019]\], so-called "epidermal electrodes" \[[@B13-materials-11-00019]\], and even textile-based dry electrodes \[[@B14-materials-11-00019],[@B15-materials-11-00019]\]. The latter are an essential component in the emerging field of adaptable, skin-mounted and wearable electronics \[[@B3-materials-11-00019],[@B16-materials-11-00019]\]. The challenge in the development of wearable skin electrodes is to fabricate lightweight, non-interfering, easy-to-handle, biocompatible, flexible electrodes that are capable of delivering good signal quality \[[@B5-materials-11-00019],[@B17-materials-11-00019]\].
This paper pursues an alternative approach for future healthcare applications: an on-demand electrode-skin contact module. As shown in [Figure 1](#materials-11-00019-f001){ref-type="fig"}, this comprises a polymeric substrate, electrodes, and an insulation layer, which typically separates the electrodes from skin contact. In case of measurement, the expandable pressure chamber establishes electrode-skin contact by inflating the chambers. After measurement or between measurement intervals, the pressure can be released or the chamber can be evacuated, retracting the electrode from the skin. This approach possesses three advantages: Firstly, the electrode being actively pressed onto the skin ensures good electrical contact. Secondly, with regard to long-term applications, the electrode does not have permanent skin contact, as there are reported issues of rashes during continuous skin contact. The effective electrode-skin contact time is reduced to the time needed for the measurement (seconds), and the electrode is subsequently removed from the skin. If another measurement is needed, contact can be established again. Finally, the module is fully printed, which carries with it all of the advantages of additive manufacturing, including low cost, quickly up scalable and individual fabrication. For instance the shape and number of electrodes is quickly adapted to new requirements.
This paper demonstrates a first set-up of a pneumatic driven electrode module together with a comparison of screen and inkjet printing on 3D printed substrates to fully additively manufacture electro-pneumatic functions, which can be further applied for designing e.g., microfluidics components such as valves or pumps.
2. Additive System Integration {#sec2-materials-11-00019}
==============================
The expendable electrode module was fully additively manufactured using fused deposition modeling (FDM) 3D printing, screen printing, and inkjet printing.
The manufacturing process of the module is shown in [Figure 2](#materials-11-00019-f002){ref-type="fig"}. In the first step, (a), the basic module, including pressure chamber and flexible membrane, was printed. The printer Mankati Fullscale XT Plus (Shanghai, China) works with a filament of 2.85 mm in diameter. The flexible filament used was PolyFlex™ from Polymaker LLC (Shanghai, China), and is available in black and white color. The material is thermoplastic polyurethane (TPU) with following properties \[[@B18-materials-11-00019]\]:Shore hardness 95ATensile strength (29 ± 2.8) MPaElongation at break (330.1 ± 14.9) %.
The filament was printed at 230 °C, at a speed of 45 mm/s and a set layer height of 100 µm.
The module is 4 mm high, 40 mm long and 22 mm wide. The setting of the distance between the nozzle and the build plate is the main determinant of the membrane thickness, and was set manually before each print by visual inspection when printing the test structures surrounding the actual model. An arithmetic average surface roughness of about R~a~ = 1.5 µm was achieved by printing on an adhesive foil. The conical shape of the pressure chamber is essential for successful 3D printing without supporting structures. The printer closes the chamber layer by layer, just a little at a time, until it fully closes on top. In the next step, (b), the printing of the electrodes was accomplished by using one of two technologies: screen printing and inkjet printing. For screen printing ([Figure 2](#materials-11-00019-f002){ref-type="fig"}b.1), polymeric silver paste C5029 from DuPont de Nemours GmbH (Hamm, Germany) and an MPM SPT printer (ITW EAE, Glenview, IL, USA ) was used. The paste was cured for 60 min at 60 °C in a hot air oven according the manufacturer's process description. The layer height was, on average, 23.1 µm, and the average surface roughness was R~a~ = 4.6 µm. For inkjet printing ([Figure 2](#materials-11-00019-f002){ref-type="fig"}b.2), NPS-JL silver ink from Harima Chemicals (Tokyo, Kantō, Japan) and a Pixdro LP50 printer (Eindhoven, The Netherlands) with a Dimatix Cartridge system were used. The paste was cured for 60 min at 120 °C in a hot air oven, and has been proven suitable for flexible application in the literature \[[@B19-materials-11-00019]\]. Two layers were printed, with a resulting average layer height of 1.9 µm and R~a~ = 0.7 µm. The area of the electrodes was 12.6 mm^2^, and two pads were connected to produce a redundant contact. The last step ([Figure 2](#materials-11-00019-f002){ref-type="fig"}c) was again FDM printing of the insulation layer with a thickness of 0.7 µm using PolyFlex™ on top of the electrodes. Here, the nozzle distance was set not to damage the previously printed electrodes. The connection pads were left uncovered by this layer in order for the electrodes to be able to contact the copper wires, using conductive adhesive H20E from Epoxy Technology Inc. (Billerica, MA, USA). For this first state, the connections were isolated using polymeric tape.
3. Methods of Characterization {#sec3-materials-11-00019}
==============================
The electrode module was characterized regarding its suitability for potential long-term skin contact applications. In doing so, both inkjet and screen printing will be considered as possible methods for electrode printing. For practical reasons, and for better illustration, the module with screen-printed electrodes was printed with black filament ([Figure 3](#materials-11-00019-f003){ref-type="fig"}a), and the inkjet-printed module with white filament ([Figure 3](#materials-11-00019-f003){ref-type="fig"}b).
3.1. Characterization of Dilatation {#sec3dot1-materials-11-00019}
-----------------------------------
The dilatation of the membrane depends on the pressure applied to the chamber, the thickness of the membrane, and its material properties, as well as the influence of the conductive layer on it. The latter has not been taken into account in this first feasibility study. Due to the manufacturing process, the membrane thickness is subject to variation. To access the pressure chambers of the module from the outside, syringe needles have been used. The connection seals are airtight, and pressure can be applied through syringes of 50 mL volume. The leakage was determined to be 2.4% over a period of time of 60 s at a pressure of 75 kPa. In [Figure 4](#materials-11-00019-f004){ref-type="fig"}a, the schematic cross-section shows a dilated membrane if the left chamber is pressured. The dilatation was measured using a laser profilometer µscan from NanoFocus AG (Oberhausen, Germany) with the chromatic sensor CLA. The pressure was measured with an analog manometer from Festo AG (Esslingen am Neckar, Germany) with graduation of 100 mbar and a digital DP200 from Mecotec (Gembloux, Belgium). After the experiment, the membrane thicknesses were determined by cutting them out and measuring with µscan. Furthermore, concerning the first reliability forecast, 5 modules with (319 ± 30) µm membrane thickness were tested with 300 pressure and release cycles with a maximum pressure of 100 kPa. At the beginning, and after every 100 cycles, the dilatation was measured at peak pressure using the laser profilometer µscan. The pressure higher than the intended operation pressure was chosen for further skin contact measurements.
3.2. Electrodermal Activity Measurement {#sec3dot2-materials-11-00019}
---------------------------------------
For electrical characterization of the electrode module, we exemplarily performed measurements of the electrodermal activity (EDA). EDA is a widely used index for sympathetic nerve activity, which reflects its actions on sweat glands. The more active the nervous system, the higher the skin conductivity, due to its having a higher sweat secretion \[[@B20-materials-11-00019]\].
For this first feasibility study of an alternative approach for an electrode module, we measured 5 healthy subjects. Therefore, both electrode modules were tested at the forearm by pressing them on using approximately 1 N force. The modules were connected to a GSR Amp (ADInstruments, Sydney, Australia), which uses low, constant-voltage AC excitation (22 mV rms @ 75 Hz), thus reducing the electrode polarization artifacts found in DC systems. For reference, the standard MLT116F finger electrodes (stainless steel, electrode area 480 mm^2^, ADInstruments, Sydney, Australia) were attached to the middle and ring finger. For further reference, stainless steel electrodes with an area of 31 mm^2^ were used in the same order at the same measuring spot on the forearm. When measuring, pressure was applied to the pressure chambers of the electrode modules, alternating between 75 kPa (skin contact realized---conduction phase) and −10 kPa (no skin contact---non conduction phase). Modules with membrane thicknesses of (319 ± 30) µm were used, which ensured a proper contact at 75 kPa, while at the same time being far from plastic deformation stress. To assess the reproducibility and stability of the contact, we realized 4 conduction phases (3 times for 10 s and a final phase of 40 s) separated by non-conduction phases of 10 s each. The time was measured using a stopwatch. Steel and finger electrodes were removed manually from the skin with a similar frequency to that at which pressure was released from the screen-printed and inkjet-printed electrodes, in order to evaluate the possible variations in signal course.
4. Results {#sec4-materials-11-00019}
==========
4.1. Dilatation {#sec4dot1-materials-11-00019}
---------------
The 3D-printed electrode module was characterized with regard to its suitability for skin electrodes with on-demand contact. [Figure 5](#materials-11-00019-f005){ref-type="fig"}a shows the screen-printed (left) and the inkjet-printed (right) electrode modules with an applied underpressure of −10 kPa, which leads to indrawn electrodes (-z-direction) and hence the maximum distance between skin and electrodes. In [Figure 5](#materials-11-00019-f005){ref-type="fig"}b, the modules are depicted with an overpressure of 75 kPa in the pressure chambers. The dilated membrane lifts the electrodes beyond the isolation layer in the z-direction, and is able to establish a contact. The gap shown between substrate and isolation layer indicates alignment issues in the FDM printer.
[Figure 6](#materials-11-00019-f006){ref-type="fig"} shows the dilatation of different membrane thicknesses when the pressure is first increased from 0 kPa to 150 kPa, and afterwards decreased to −10 kPa. During one cycle, only minor viscoelastic behavior of the TPU occurred. This can be noticed in the hysteresis of the dilatation curve, indicated by black arrows, which shows pressure and release direction. The dilatation when increasing pressure (lower curve), compared to the decreasing pressure process (upper curve), is less. The membranes with lower thickness show irreversible deformation at a pressure of above 100 kPa, so the maximum utilized pressure in the measurement of those membranes was lower. Furthermore, a close relation between membrane thickness and dilatation could be observed (the thinner the membrane, the more dilatation at lower pressure). For instance, a 154 µm membrane showed a dilatation of 1.62 mm at 100 kPa. A correlation between membrane thickness and hysteresis was not found after one cycle. As expected, the thickest membrane (406 µm) showed the lowest maximum dilatation. To estimate possible security issues for the electrode module resulting from, e.g., the bursting of the membranes, a pressure of up to 240 kPa was applied. Membranes with 332 µm thickness dilated 1.85 mm at this pressure, but showed major viscoelastic properties when decreasing. Membranes with lower thicknesses were irreversibly plastically deformed, but did not burst.
In [Figure 7](#materials-11-00019-f007){ref-type="fig"}, the dilatation in the z-direction at a pressure of 100 kPa is shown as a function of pressure and release cycles. With increasing cycle numbers, the dilatation also rose. Over 300 cycles, the dilatation rose approximately 5.9%.
4.2. Electrodermal Activity Measurement {#sec4dot2-materials-11-00019}
---------------------------------------
The electrical performance of the electrode modules was characterized using EDA measurement (ADInstruments, Sydney, Australia). In [Figure 8](#materials-11-00019-f008){ref-type="fig"}, the pressure-dependent conductance during on-skin measurements is shown. At a pressure of −10 kPa, no conductance was measured, due to indrawn electrodes. At 12 kPa of pressure, the electrode-skin contact was established, and 75 kPa was the maximum achievable pressure with this set up. Increasing the pressure within the module increases the electrode area, which is pressed onto the skin. In contrast to the soft skin, when pressing the electrodes on rigid surfaces, the pressure dependency was not noticeable. Because of the pressure, the electrode area doesn't cling to the rigid surface, and hence the contact area doesn't rise.
[Figure 9](#materials-11-00019-f009){ref-type="fig"} shows the EDA measurements for each type of electrode, averaged over all 5 subjects. The finger electrode showed highest conductivity, because fingers and palms contain a higher density of sweat glands compared to the forearm, and the electrode area was the largest. Taking into account that the stainless steel electrode, with the second-largest electrode area, showed the second-highest conductivity, the achieved results are reasonable. The electrode modules both showed lower conductivity in comparison, with the screen-printed electrode being the lowest. [Table 1](#materials-11-00019-t001){ref-type="table"} shows the mean value, standard deviation and variation coefficient for each type of electrode over all four conduction phases. While the absolute amplitude of the finger electrodes was about 10 times higher than the other electrodes, the signal course was qualitatively similar in all 4 types of electrodes. Variation coefficients showed higher variations in the conduction phase for screen-printed, inkjet-printed, and stainless steel electrodes compared to finger electrodes. The longer the conduction, the more the conductance for screen-printed, inkjet-printed and stainless steel electrodes rises (conductance in phase 4 is 11%--23% higher than in phase 1--3), whereas this phenomenon is not observable for the finger electrodes (3%).
5. Discussion {#sec5-materials-11-00019}
=============
This paper presents an alternative approach to yielding electrode-skin contact for future healthcare applications. The presented electrode module was fully printed: substrate, electrodes, and isolation layer. Two different technologies for electrode fabrication, screen- and inkjet-printing, were considered. The module was mechanically and electrically characterized, and showed good performance. The repeatability and reliability, especially of the viscoelastic behavior of the TPU after more than 300 cycles, has to be further evaluated. For this feasibility study, the pressure and release cycles indicated a moderate growing dilatation of 5.9%, which is not expected to have a major impact on possible future applications. This would ensure skin contact. Furthermore, both of the electrode modules, the screen-printed and inkjet-printed, were tested for electrodermal activity measurement and compared to standard electrodes. The difference can be found mainly in the amplitude and the rising peak value during the conductance phase. The reason for the low conductivity can be found in the rising resistance in the conductive tracks, when dilating the membrane. Screen-printed electrodes consist of metal particles in a polymeric matrix that move apart during bending and stretching, leading to increased resistance. Sintered silver ink is likely to produce micro cracks when stretching, but usually shows good performance when bending \[[@B21-materials-11-00019]\], due to low layer heights, and hence less internal stress. The surface roughness, which increases the effective electrode area, was higher for screen-printed electrodes, and was therefore unlikely to contribute a dominant effect to the EDA results obtained. The characterization of resistance behavior when inflating the module will be the content of future works.
Rising peak values in all electrodes apart from the finger electrodes might be attributed to the softer tissue at the forearm. Repeated application of pressure and the electrode's contact might lead to some tissue adaptation to the electrode's shape. Due to the greater size and different tissue properties, this effect is likely to vanish at the finger. However, our physiological measurements had a limited extent and were intended to prove feasibility only. Future works will include more elaborate testing, including physiological stimuli, in order to characterize the sensitivity of our electrode modules. Further, future work will characterize the reproducibility of the manufacturing process in detail. All in all, the on-demand electrode module represents a low-cost design, with all the advantages of 3D printing, and therefore might be suitable for consideration for disposable electrodes or consumer electronics.
This work was supported by the European Social Fund (ESF) and the Free State of Saxony.
Martin Schubert conceived, designed, and performed experiments and the article about the electrode module. Paul Wolter performed the manufacturing, system integration and measurements. Karlheinz Bock contributed to the conceptual design of experiments, the article and the discussion on findings. Martin Schmidt accomplished the physiological measurements using different electrodes, performed signal analyses and contributed to the submitted manuscript. Hagen Malberg contributed to the conceptual design of the physiological measurements and the discussion on the findings. Sebastian Zaunseder developed the conceptual design for the physiological measurements, accompanied them and the signal analyses and contributed to the submitted manuscript.
The authors declare no conflict of interest.
{#materials-11-00019-f001}
{#materials-11-00019-f002}
{#materials-11-00019-f003}
{#materials-11-00019-f004}
######
(**a**) View of the screen-printed (**left**) and inkjet-printed (**right**) electrodes with an applied pressure of (**a**) −10 kPa and indrawn membrane (-z-direction); and (**b**) 75 kPa and dilated membrane.
![Membrane dilatation of various membrane thicknesses at different pressures (Data from \[[@B21-materials-11-00019]\]).](materials-11-00019-g006){#materials-11-00019-f006}
{#materials-11-00019-f007}
![Pressure-dependent conductance on skin of the screen-printed electrode module on one subject (Data from \[[@B21-materials-11-00019]\]).](materials-11-00019-g008){#materials-11-00019-f008}
{#materials-11-00019-f009}
materials-11-00019-t001_Table 1
######
Mean value, standard deviation and variation coefficient of the EDA measurement.
Electrode Conductance Average in µS Standard Deviation in µS Variation Coefficient
------------------ --------------------------- -------------------------- -----------------------
Screen printed 0.205 0.030 0.146
Inkjet printed 0.552 0.064 0.116
Stainless steel 0.754 0.097 0.128
Finger electrode 15.6 1.27 0.082
|
Rape and kidnapping charges have been dropped against two former New York City police officers who admitted to having sex with a teenager while she says she was handcuffed in the back of their police van.
Former NYPD narcotics detectives Eddie Martins and Richard Hall quit their jobs two months after the incident in Brooklyn on September 15, 2017 when the officers arrested Anna Chambers, then 18, for marijuana charges.
Authorities said the pair told the teenager they would let her go if she had sex with them.
The two men later confessed to having intercourse with Chambers while on duty, but have argued the sex was consensual, according to investigators.
Prosecutors in Brooklyn, New York officially dropped rape and kidnapping charges against NYPD narcotics detectives Richard Hall (left) and Eddie Martins (right) on Wednesday
Chambers has said she was forcibly raped while in handcuffs, but declined to testify against her alleged attackers during trial, according to the New York Post.
Prosecutors originally charged both officers with felony rape and kidnapping, which carries a maximum prison sentence of 25 years.
On Wednesday, however, only felony bribery and official misconduct charges were presented in Brooklyn Supreme Court. The officers now face just two-and-a-half to seven years behind bars if convicted.
Anna Chambers (pictured), now 20, did not attend her alleged attackers hearing in Brooklyn Supreme Court on Wednesday
Chambers was 18 when she said she was raped while handcuffed in the back of a police van
The district attorney's office argued that rape would not have been an appropriate charge and that securing a conviction would have been too challenging.
'We believe – as a newly-created statute recognizes – that any sexual contact between police officers and a person in their custody should constitute a crime,' a spokesperson from the Brooklyn DA’s office said Wednesday.
'However that was not the law at the time of the incident. Because of this and because of unforeseen and serious credibility issues that arose over the past year and our ethical obligations under the rules of professional conduct, we are precluded from proceeding with the rape charges.'
The 'credibility issues' prosecutors referenced were raised by Martins and Hall's defense attorneys who previously requested Chambers be charged with perjury for allegedly lying under oath about the details of her encounter.
Hall (pictured) and Martins were charged with felony bribery and official misconduct
Martins (right) and Hall now face two-and-a-half to seven years behind bars if convicted
Chambers claimed the attack took place during a long drive from where she was arrested to her home, but GPS data collected from the officers' cell phone didn't support her version of events, the New York Times reported.
During her previous testimony before a grand jury, Chambers also said one of the officers commanded her to expose her breast.
That claim, defense attorneys argued, didn't match the story Chambers told during a deposition hearing when she mentioned she had hidden cocaine in her bra.
Martin’s lawyer, Mark Bederow, was pleased with the DA's decision.
'It’s a good day for us,' he said. 'They dismissed the indictment because they don’t believe her. It’s not personal. Cases have to be brought with credible evidence. This wasn’t. The DA agreed.'
Chambers and her attorney Michael David were disappointed.
'She's fed up,' David said of Chambers who did not attend the hearing Wednesday.
Hall (left) and Martins (right) were seen leaving Brooklyn Supreme Court after their hearing on Wedneday
'It’s just outrageous. It was a clear-cut case. She was kidnapped. There was DNA evidence... You can’t have consent, when you have two cops on duty. These are two cops over six feet, over two hundred pounds. She is 5-foot-2, 90 pounds. They have guns, they have handcuffs. You can’t have consent under those circumstances.'
Despite the reduced charges, Hall’s attorney Peter Guadagnino said the two ex-cops aren't happy with the way their criminal cases have progressed.
'They were dedicated police officers,' Guadagnino said. 'They still feel hurt by them having to leave their position now a year and a half later.'
Chambers and her attorney Michael David, pictured in September 2018, were disappointed
Martins and Hall resigned from their positions with the NYPD three days before they were scheduled to face a disciplinary trial in November of 2017.
NYPD Commissioner James P. O'Neil said: 'When a member of the N.Y.P.D. is indicted on serious charges like these, it tarnishes all of the admirable things accomplished by other, good officers every day in neighborhoods across New York City,' later adding that had the charges against the officers been affirmed at the internal trial, he 'would have fired them immediately.' |
› Money Boxes Excellent Way to Instill the Value of Managing and Saving Money
Money Boxes Excellent Way to Instill the Value of Managing and Saving Money
Thu, 08 Jan 2015
Teaching children and teenagers how to spend and save their money is crucial to their future. Knowing how to manage whatever kind of money they receive will help prepare them to be wise with their finances as they grow older and more independent. If you want to instill these kinds of principles in the minds and hearts of your children, introduce them to money boxes and piggy banks at an early age. When saving is taught and handled properly, it can truly impact the way your child manages their finances as they grow older.
As soon as your child can count, teach them about the importance of managing their money because if you don't TV and the internet will do it for you either directly or indirectly. Taking an active role in teaching them about the value and importance of budgeting and saving will go a long way as they develop into responsible adults. Communicating these principles to your children is not hard if you know how. Some of the most important lessons that keeping money boxes and piggy banks will teach your children include the value of money, spending it wisely, saving, it, and making it grow.
Giving them a money box as a birthday or Christmas present is a good way to introduce budgeting and money management to them. Letting your children manage their own allowance, for instance, is a good way to start them off with saving and money management. Making them earn their allowance is also a good way to teach them about the value of money by making them help with chores. Teaching them how to spend their money wisely and set aside whatever amount they can to save are also very important principles to impart in their young age. |
Tuesday, 31 January 2012
Handmade buttons ... ...
Buttons are great, and you'd be surprised how easy it is to make your
own, I'm not talking about the plastic ones, I'm talking chunky wood
buttons, great as a feature on a knitted hat or some other item.
It's easy to do, you need a saw (a hack saw is easy to use)
Bit battered, but works okay
You'll also need a drill of some kind and a drill bit (I normally use a 3mm bit)
My post drill.
I use a post drill for making the holes in my buttons, but if you don't
have one a cordless drill will work just as well, but be careful when
drilling, and if you don't have an electric drill then a hand drill is
just as good, but you will need a vice or some way to hold the button
still whilst drilling.
Hand drill -
You can buy these quite cheaply.
And obviously you need some kind of wood, sticks are easiest to work
with, but if they are freshly cut it's best to leave them in the house
some where to dry out a bit first.
Then all you need to do is slice your stick up into sections as thick or
thin as you want, although the thinner the sections the harder it is to
grip them for sanding.
Ready for slicing -
Gripped tightly.
Now if you don't have a vice you could use a strong clamp to hold the
wood, but I'd recommend a vice of some kind, it doesn't have to be
expensive, I have a hobby vice I use in my work room at the back of the
house, it was less than £10 and is great for small jobs like this.
Once you have some slices of wood it's on to the next step, and this is
where things get personal, it's up to you how many holes you drill,
where they are and such like, I tend to drill 2 holes in each button.
Ready for drilling -
They look a little rough, but they will turn out nice.
With holes drilled -
4 or 2 ? it's up to you.
You don't have to drill the holes first, I do things this way because if
the holes are a little rough they'll get sorted out when I finish the
buttons, saves time and extra sanding.
The next bit depends on what sort of equipment you have, if like me you
have a variety of power tools then you may well have a belt sander or
orbital sander, these will do for finishing the buttons as long as you
use a fine grit (like 180) sanding sheet/belt if not then you can use a
sheet of sand paper (again use a fine grit) and a wooden block, this
will take longer, and probably make your arms ache a little, I use my
lathe sander to finish the buttons, takes about 2 minutes (give or take)
to finish a button.
I built a sanding wheel for my lathe, I made a short video of me using
it, please remember to watch your fingers if your using a power sander
of some kind, gloves might be an idea, and a mask as things will get
dusty.
Sanding video -
Mind your fingers !
As you can see it doesn't take long, and apologies for the video
quality, it wasn't as easy as I thought sanding and filming at the same
time.
So once all the sanding is done you should have some nice smooth wooden
buttons, they might be different sizes and thicknesses, but that just
makes them more unique.
I tend to either give my buttons a coat of beeswax to protect them
without affecting the colour, or I use linseed oil, which brings out the
grain quite well, but you could wax them,varnish them, or paint them.
Finished buttons -
All done, 5 cypress & 2 pear wood.
Here's a picture of a cypress button and a pear wood button coated with linseed oil -
The pear wood is the darker one.
You can see from the picture above that you can make the buttons as
round as you like, or leave them a little less finished for a slightly
more rustic look, and if this all seems a little like hard work then I
know a chap who sells them in his shop ;-) which you can find here
(opens in new window) I'm also working on a lathe jig that will allow
me to make sets of buttons the same size and decorate the buttons with
simple designs.
Thanks, I do take a great deal of care when I'm using my tools, in 20 odd years I've only had 1 serious accident, I chopped of about 5mm of one of my thumb, just damaged the skin, so nothing lasting :-) needless to say I've not done it since. |
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<plist version="1.0">
<dict>
<key>LprojCompatibleVersion</key>
<string>106.3</string>
<key>LprojLocale</key>
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|
using System.Collections.Generic;
using System.Text.Json.Serialization;
namespace Essensoft.AspNetCore.Payment.Alipay.Domain
{
/// <summary>
/// SpiDetectionDetail Data Structure.
/// </summary>
public class SpiDetectionDetail : AlipayObject
{
/// <summary>
/// 检测结果码
/// </summary>
[JsonPropertyName("code")]
public string Code { get; set; }
/// <summary>
/// 检测内容
/// </summary>
[JsonPropertyName("content")]
public string Content { get; set; }
/// <summary>
/// 检测外部任务编号
/// </summary>
[JsonPropertyName("data_id")]
public string DataId { get; set; }
/// <summary>
/// 检测细节，检测内容可能涉及多个场景
/// </summary>
[JsonPropertyName("details")]
public List<string> Details { get; set; }
/// <summary>
/// 检测结果分类
/// </summary>
[JsonPropertyName("label")]
public string Label { get; set; }
/// <summary>
/// 检测结果信息
/// </summary>
[JsonPropertyName("msg")]
public string Msg { get; set; }
/// <summary>
/// 检测准确率
/// </summary>
[JsonPropertyName("rate")]
public string Rate { get; set; }
/// <summary>
/// 检测场景
/// </summary>
[JsonPropertyName("scene")]
public string Scene { get; set; }
/// <summary>
/// 检测建议 pass-文本正常 review-需要人工审核 block-文本违规，可以直接删除或者做限制处理
/// </summary>
[JsonPropertyName("suggestion")]
public string Suggestion { get; set; }
/// <summary>
/// 检测内部任务编号
/// </summary>
[JsonPropertyName("task_id")]
public string TaskId { get; set; }
}
}
|
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