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Q: Выделение строки в подстроке WPF Допустим есть datagrid. В нем есть некоторое ключевое поле, по которому я делаю фильтрацию, считайте поиск. Механизм не важен. В момент фильтрации. Мне необходимо подсветить каждой строке грида, подстроку по которой идет поиск.
допустим ищем по подстроке три
**три**жды
с**три**жка
с**три**ж
Взял реализацию которая описана в данной статьи : Выделение строки в подстроке
Данный метод идет в лоб. У нас грубо говоря будет 3 строки и 3 textbox. Это есть не очень хорошо. В моих условиях, в datagrid будет находиться порядка 80т-150т объектов.
Позже мной были найдены интересные вещи в xaml. В практически каждом контроле есть такая вещь как <Run/>. И сам по себе textbox можно описать через несколько Run'ов, и они будут линейно отображаться. Сама соль в том что Run'у можно повесить любой атрибут, например цвет фона. По идее можно описать конвертер, который берет строку которая находиться в textbox и подстроку поиска, задать ей фоновый цвет, и сформировать визуальное представление, данный способ мне кажется более гибким и красивым. Прошу помощи, может кто уже реализовывал похожие вещи!
A: Если кому интересно, вот реализация :
public class HighlightingTextBlock : TextBlock
{
#region DependencyProperty
public static readonly DependencyProperty SearchStringDependencyProperty =
DependencyProperty.Register("SearchString", typeof (string), typeof (HighlightingTextBlock),
new FrameworkPropertyMetadata(string.Empty)
{
DefaultUpdateSourceTrigger = UpdateSourceTrigger.PropertyChanged,
PropertyChangedCallback = PropertyChangedCallback,
});
public static readonly DependencyProperty TextDependencyProperty =
DependencyProperty.Register("Text", typeof (string), typeof (HighlightingTextBlock),
new FrameworkPropertyMetadata(string.Empty)
{
DefaultUpdateSourceTrigger = UpdateSourceTrigger.PropertyChanged,
PropertyChangedCallback = PropertyChangedCallback,
});
private static void PropertyChangedCallback(DependencyObject _dependencyObject,
DependencyPropertyChangedEventArgs _dependencyPropertyChangedEventArgs)
{
if (_dependencyObject is HighlightingTextBlock)
((HighlightingTextBlock) _dependencyObject).RedrawControl();
}
#endregion
#region Properties
public string Text
{
get { return (string) GetValue(TextDependencyProperty); }
set { SetValue(TextDependencyProperty, value); }
}
public string SearchString
{
get { return (string) GetValue(SearchStringDependencyProperty); }
set { SetValue(SearchStringDependencyProperty, value); }
}
#endregion
private void RedrawControl()
{
if (!string.IsNullOrWhiteSpace(Text) && !string.IsNullOrEmpty(SearchString))
{
if (!Text.Contains(SearchString))
{
ReturnEmpty();
return;
}
string[] splitedBaseText = Regex.Split(Text, string.Format(@"({0})", SearchString),
RegexOptions.IgnoreCase);
if (splitedBaseText.Any())
{
Inlines.Clear();
foreach (var splited in splitedBaseText)
{
if (splited == SearchString.ToLower())
{
Inlines.Add(new Run(splited) {Background = new SolidColorBrush(Colors.Aqua)});
}
else
{
Inlines.Add(new Run(splited));
}
}
}
else
{
ReturnEmpty();
}
}
else
{
ReturnEmpty();
}
}
private void ReturnEmpty()
{
if (!string.IsNullOrWhiteSpace(Text))
{
Inlines.Clear();
Inlines.Add(new Span(new Run(Text)));
}
else
{
Inlines.Clear();
Inlines.Add(new Span(new Run(string.Empty)));
}
}
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 3,023
|
Q: C# check if object is of a type known only at runtime Why won't the following work?
if(!(obj is (DataGridView1.SortedColumn.ValueType)))
I get an error that a type is expected. Doesn't ValueType return a type?
A: ValueType is not an actual class, right? It's a property that returns a Type. So to figure this out at runtime, you need to say:
if(!(DataGridView1.SortedColumn.ValueType.IsAssignableFrom(obj.GetType())))
A: Change your code to
if(obj.GetType() != DataGridView1.SortedColumn.ValueType)
Edit Updated code to fix typos
A: No, the ValueType property is an instance of the Type class. It's not a type itself.
Try this:
if (!(obj.GetType() == DataGridView1.SortedColumn.ValueType))
However, this won't account if obj is a derived type, so if you need that, you'll have to get a little fancier.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,479
|
Borrowing money is important if you need money for extra working capital. This can help your business grow and operate more efficiently. The problem is when to get one. Here are some insights on when the best time is to gain the most benefit from a commercial loan.
A commercial loan is a type of loan that is only given to businesses and not to individuals. Commercial loans are of varied kinds which can be borrowed from different kinds of lenders. You can have different loan terms and different loan amounts. You can use the money as working capital to hire employees, purchase inventory, or make investments for your business growth. You should already know your loan options before you choose one.
There are companies that simply use their savings account or their personal line of credit to fund their business. Doing this will not establish credit in your company's name. Your business credit rating can increase depending on the credit accounts established for your company. If you apply for a commercial loan, then what this means is that your business is responsible for handling this kind of credit. So make your first loan even if you don't need it as yet just to establish credit for your company because you will never know when you will need it most. Borrow, then, a modest amount and pay it promptly and you then have access to credit when you need it.
If you are to manage a thriving business, then it is important to manage your cash flow. No matter how profitable your business is, not being able to handle your bills or pay it on time means failure for your business. Access to extra cash gives you money savings in the long run. Your savings can buy new equipment, more inventory to qualify for discount and hire new employees. One aspect of managing cash flow is managing credit. Hiring more employees and buying more equipment can help you produce extra revenue. You need a commercial loan to achieve.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,793
|
Q: Applecare Validity for Macbook in Turkey I am getting conflicting answers (even from the Apple itself) about the validity of the applecare plan for Macbook Pro (purchased from US) in Turkey. I know that there are Apple stores in Istanbul and that Apple gives technical assistance there. However, some says that applecare is not valid in Turkey. They say that in order to benefit from the applecare plan I should either mail my computer to US or bring it to US or perhaps to some nearby EU country where the plan is valid. On the contrary some says that applecare is completely valid for MacBooks in Turkey (though not for Iphones, Ipads, etc.). I would be grateful if I can have the correct answer and, if the plan is valid in Turkey then how the process works. Thank you.
A: Apple products have a global warranty, so you should be covered.
You need to contact Apple directly - someone there will be able to organise something for you. You may need to provide proof of purchase, so have that ready (scan it in if you haven't got a digital copy).
For your info, below is an experience that may shed some light on this for you.
I recently helped a lady who purchased a new iPhone from a local online store. About 10 months later the iPhone developed a fault, and while it was definitely under warranty, it took a while to establish this fact because the serial number came up as having been purchased in Thailand, and because it was dated about 7 weeks earlier than when she had purchased it.
In this case, she needed to show her purchase receipt to correct the commencement date for her warranty, as it was originally showing as only 8 days remaining compared to the two months she thought she had left. It did get all sorted, but the issue had to get escalated before it was resolved.
So, if it was me, I would visit Apple's contact page for Turkey and contact them directly via phone to discuss.
A: While I understand that it has been quite a while since this question has been asked/answered, my wife and I just had this exact situation. Maybe this will help someone who comes across a search for this question.
We have a 2018 MacBook Pro that had 2 issues: 1) a recalled SSD and 2) a failed battery. We had purchased the MacBook in California last summer with AppleCare and have now moved to Istanbul.
We took the MacBook into the Apple Store in the Zorlu Center mall after making a Genius Bar appointment. They looked up the serial number and it was completely repaired in just a few days.
They replaced the entire lower 1/2 of the case and it even came back (thankfully) with a US keyboard. They even had personnel that spoke English, as I'm still struggling with Turkish.
Super service!
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,234
|
set -e
: ${RTCS_DEST_DIR:="$HOME/rt/riak_cs"}
echo "Making $(pwd) the current release:"
cwd=$(pwd)
echo -n " - Determining version: "
if [ -f $cwd/dependency_manifest.git ]; then
VERSION=`cat $cwd/dependency_manifest.git | awk '/^-/ { print $NF }'`
else
VERSION="$(git describe --tags)-$(git branch | awk '/\*/ {print $2}')"
fi
echo $VERSION
cd $RTCS_DEST_DIR
echo " - Resetting existing $RTCS_DEST_DIR"
git reset HEAD --hard > /dev/null 2>&1
git clean -fd > /dev/null 2>&1
rm -rf $RTCS_DEST_DIR/current
mkdir $RTCS_DEST_DIR/current
cd $cwd
echo " - Copying devrel to $RTCS_DEST_DIR/current"
cp -p -P -R dev $RTCS_DEST_DIR/current
echo " - Writing $RTCS_DEST_DIR/current/VERSION"
echo -n $VERSION > $RTCS_DEST_DIR/current/VERSION
cd $RTCS_DEST_DIR
echo " - Reinitializing git state"
git add --all .
git commit -a -m "riak_test init" --amend > /dev/null
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,383
|
Q: Undefined index... but it says it's defined in the debugger (EDIT: I solved my issue! Though I still don't understand the situation I see in the debugger. See my answer for more details)
(TL;DR: index is always undefined when used with a certain array. Doubt that would be enough info, but maybe for someone who's experienced this before.)
So basically, I'm using an array in javascript, and I started noticing some odd behaviour, so I went to the debugger, and I found that a defined variable representing the index was being treated as undefined. It's ONLY the case with this specific array, and it's index. I don't get errors saying that it's undefined, but when I look in the debugger, it says it's undefined when I hover over the variable in the array call (but it's defined if I hover over it anywhere before the array call), and I'm getting bugs that make it clear that the array is not being used properly. It makes absolutely no sense to me, but maybe someone's encountered a similar issue.
Take this example of code, It's drawing a tilemap layer for my MapRenderer class. The culprit here is "this.Map.layers". When I go into this function in the debugger, layerIndex is defined if I hover over the function parameter, but if I hover over it on the array call, it says it's undefined, and the whole logic breaks.
DrawLayer(ctx, camPos, layerIndex)
{
// Get the map/tile position based on the camera position, to decide which tile to start drawing.
var mapCamPos = new Point(Math.floor(camPos.x/TILESIZE),
Math.floor(camPos.y/TILESIZE));
// Get the max tile position based on camera position, to decide where to stop drawing.
var camPosLimit = new Point(Math.ceil(this.DrawSize.x/TILESIZE)+mapCamPos.x,
Math.ceil(this.DrawSize.y/TILESIZE)+mapCamPos.y);
// loop through all tiles we need to draw using rows and columns.
for(var row=mapCamPos.y;row<this.Map.layers[layerIndex].height&&row<=camPosLimit.y;row++)
{
for(var col=mapCamPos.x;col<this.Map.layers[layerIndex].width&&col<=camPosLimit.x;col++)
{
var currentTileID = this.GetTileID(layerIndex, row, col);
if (currentTileID >= 0 && !isNaN(currentTileID))
{
var drawPos = new Point(((col*TILESIZE)-camPos.x), ((row*TILESIZE)-camPos.y));
this.Spritesheet.PlayFrame(currentTileID);
this.Spritesheet.Draw(ctx, drawPos);
}
}
}
}
This is happening in many instances of my code wherever I'm using that array. I want to add how this started, because all of this logic was working previously. I had my tilemap working with multiple csv files, which I loaded as 2d arrays into an array. Today, I decided to switch it all to use one json file, as it is simply cleaner (one file rather than one csv per map layer), and I can add extra properties and stuff in the future rather than just having the tileIDs. So, in the above example, this.Map gets initialized through an ajax call(using jquery) to read the json file, before DrawLayer ever gets called. Still, I don't see why this would cause this. Doing "mapRenderer.Map.layers" in the console tells me that it's a normal array, and when I try calling it normally from the console, it works fine. I'm so confused at this issue. I had literally the same function before and it worked, just that my array has changed a bit(it used to be just "this.Layers", instead of "this.Map.layers"), but it's still a normal array... I don't see why it would behave so differently just because it was generated via json...
Any help or explanations would be greatly appreciated, thanks.
A: I still don't understand the situation I see in the debugger, maybe it's a firefox bug, or feature I don't understand. But I managed to fix my issue. It was a basic logic bug: I'm using the "Tiled" map editor, and when you export those maps to CSVs, the tile IDs are zero-based, meaning empty tiles are -1. When you export to json, they aren't zero-based, meaning empty tiles are 0, which I failed to notice, and this was the root of all my current issues. If anyone can explain why the firefox debugger might say defined variables are "undefined" when you hover over them, that would still be good to know.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 3,044
|
'Where is Australia?' China makes a bold play for the south Pacific's 'Treasure Islands'
Bougainville independence
By Ben Bohane
Bougainville: Chinese agents have offered MPs in the Solomon Islands $SD1 million ($A200,000) in bribes to switch diplomatic allegiance from Taiwan to Beijing, according to one Solomons politician.
It comes as China is proposing up to $1 billion in infrastructure and resource extraction in what it calls an "early harvest" operation to extend its influence in the Pacific.
China is also showing considerable interest in Bougainville, offering $US1 billion ($1.47 billion) worth of infrastructure and seeking Bougainville's mineral wealth in "collateral" as part of the deal.
The allegations of bribery in the Solomons come from the premier of the island of Malaita, Daniel Suidani, who says he rejected the offer to switch diplomatic recognition, but isn't sure if the nation's 50 MPs and other premiers have done the same. The offer of payment was disclosed in the Solomon Star newspaper in September, citing anonymous sources.
Under Mr Suidani's leadership, Malaita has told the central government of the Solomons that it wants to stick with Taiwan and there have even been threats of secession.
The Chinese Communist Party is aggressively seeking influence and resources wealth in the Pacific through its Belt and Road Initiative, and the latest developments come only a few years after Australia spent more than $1 billion and a decade stabilising the Solomons with the RAMSI mission, which ended that nation's civil war.
China's already active diplomacy and investment in the region could extend to assisting an independent Bougainville. Credit:Ben Bohane
Among the projects China is interested in are the Gold Ridge mine, the Tulagi port and other strategic areas on Guadalcanal, the Solomons' main island.
Also revealed is the scale of a Chinese master plan for Bougainville as it heads towards a referendum on independence from Papua New Guinea. It's expected a clear majority of Bougainvilleans will vote in the next few weeks for independence, and if PNG's Parliament ratifies the outcome, then Bougainville is on track to become the newest nation in our region since Timor Leste.
A public presentation by former Bougainville Revolutionary Army general Sam Kauona, filmed by a crew from 60 Minutes in recent weeks, shows Mr Kauona unfurling a large satellite map of Bougainville with Chinese script highlighting proposed bridges, a port at Torokina on the west coast (where US forces landed in World War II), a highway linking Buka town in the north with Buin in the south, an airport and a luxury resort, among other proposed developments. Mining is almost certainly part of the deal.
"This is the first holistic offer, which has come from China," Mr Kauona told attending ward councillors and MPs. "Where is Australia and the US and Japan? Earlier this year I met representatives from Fortescue mining but I have been waiting 10 months for them to make a commitment," Mr Kauona said.
Former Bougainville Revolutionary Army general and now a leading independence figure, Sam Kauona. Credit:Ben Bohane
"So far they are keeping me at arm's length, so we don't know if they are genuine. At the moment China's offer is plan A."
Mr Kauona, one of the leading candidates to become next president of Bougainville, says he is not necessarily "pro-China", and wants companies who can help partner in nation-building. "We are a treasure island. As we move towards independence we want partners who are interested in helping us build a nation, not just exploit our resources.
"It seems that Western banks are closed to us but eastern ones are open. China has made a proposal and we are still waiting to hear from our Australian and American friends if they have any proposals."
If PNG seriously delays or does not ratify the vote, there is the prospect of a Bougainville unilateral declaration of independence that could well be recognised by some Pacific countries – and China.
Bougainville Revolutionary Army guerillas above the Panguna copper-gold mine in 1994. Credit:Ben Bohane
As Bougainville's war for independence was beginning in 1989, Mr Kauona, then a young officer in the PNG defence forces, was being trained in explosives by the Australian Defence Force in Portsea, Victoria. He left Australia and switched sides, running the naval blockade of his island via the "back door" of the Solomons and taking command of a nascent guerilla force.
Successfully resisting the PNG defence forces takeover of Bougainville, he later brought the rank-and-file revolutionary army soldiers into the peace process when women chiefs had had enough of the war.
Yet despite bringing his fighters into a lasting peace process and working towards economic development, Australia denied him a visa for more than 20 years, only recently allowing him and his wife Josie visas. But the damage was done. In recent years he says he was forced to conduct most of his business meetings in Singapore and Hong Kong.
Although still well disposed to Australia, he is angry at his treatment and wants Australia to apologise to him and to Bougainville for its role in the war, at a time when reconciliations between Papua New Guinea Defence Force and Bougainville Revolutionary Army soldiers have already begun.
Former PNG Prime Minister Sir Rabbie Namaliu and former PNGDF Commander Brigadier-General Jerry Singirok apologised for their roles in the war last week in emotional ceremonies where former foes shook hands, chewed betelnut and declared "no more war".
Mr Kauona, speaking in the Bougainville capital Arawa, said he feels responsible for helping rebuild Bougainville and is open to offers from anywhere.
"As Commander of the BRA I was responsible for the destruction of infrastructure on my island during the war, so I have a responsibility to rebuild it now. I am very open to Australia and the US but they are not offering any integrated development plans for Bougainville."
Mr Kauona says he doesn't want to be caught up in a big power competition.
"Don't talk to me about military bases and this or that strategy. Come and talk to me about economic development for our people and how you will help us with nation-building".
Ben Bohane is a Vanuatu-based photojournalist who has covered the Pacific for 25 years. He is the recipient of this years' inaugural Walkley/Sean Dorney grant for Pacific journalism.
CCP influence
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,484
|
Given a string s and a non-empty string p, find all the start indices of p's anagrams in s.
Strings consists of lowercase English letters only and the length of both strings s and p will not be larger than 20,100.
The order of output does not matter.
## Code
```JAVA
public class Solution {
public List<Integer> findAnagrams(String s, String p) {
List<Integer> result = new ArrayList<Integer>();
if(s==null || p==null || s.length() < p.length()) return result;
int[] charTable = new int[256];
for(char c: p.toCharArray()){
charTable[c]++;
}
int left=0, right=0, count=p.length();
while(right<s.length()) {
if (charTable[s.charAt(right)] >= 1){
count--;
}
charTable[s.charAt(right)]--;
right++;
if (count==0) {
result.add(left);
}
if ( right - left == p.length() ){
if (charTable[s.charAt(left)] >= 0) {
count++;
}
charTable[s.charAt(left)]++;
left++;
}
}
return result;
}
}
```
## Thinkings
1. sub-searching problem, slide window algorithm
2. ```for(char c: p.toCharArray()){
charTable[c]++;
}``` used for construct charTable or you call it hash
3. the idea of count is like to maintain how many you still expect
4. 2 entering condition inside the while
5. charTable not only used for maintain the used info, all other not used element also have minus(-) number indicate their not in p
6. matching process is reversed, like using p to match s.
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 2,392
|
"The process of norm erosion started decades ago — long before Trump descended an escalator to announce his presidential candidacy," they write.
At the root, they say, is racism, particularly racism inspired by passage of the 1965 Civil Rights Act, which empowered black Americans to vote. Couple that with the passage the same year of landmark immigration reform, which empowered non-white Americans and created an animus among white Americans that has triggered a series of moves to counter diversity and preserve white influence.
President Richard Nixon's so-called Southern strategy aimed at exploiting the anger white Southerners felt at changes in their states. That helped turn the South from a Democratic stronghold to a place where the statewide election of a Democrat, such as Sen. Doug Jones in Alabama, becomes a national story.
At the root, they say, is racism, particularl ...[text shortened]... ection of a Democrat, such as Sen. Doug Jones in Alabama, becomes a national story.
Norms are what have sustained American democracy "in ways we have come to take for granted." They identify two in particular: "mutual toleration," or the understanding among competing parties and politicians that they are legitimate rivals rather than existential enemies; and "forbearance," or the understanding among politicians that just because they technically have the power to do something doesn't mean they ought to use it. The erosion of these two norms can lead to a partisan death spiral. The authors argue that Trump has tried to eviscerate both.
We have seen this before Trump in, for example, the Republicans' willingness to filibuster anything and everything in the Senate (thus imposing an at least extraconstitutional and probably unconstitutional 3/5 requirement in that body to pass legislation) whereas the tactic had been rarely used in the past.
I don't really think that Trump himself is much a threat to our democracy; he is too much of a clown (and largely perceived as such) to be awarded such a standing. There are numerous threats to our democracy such as voter suppression, gerrymandering, campaign contribution bribes, etc. etc. etc. but I have a fair degree of confidence that these can be overcome.
Ignoring the effects of racism in American history is beyond ignorant.
Norms are what have sustained American democracy "in ways we have come to take for granted." They identify two in particular: "mutual toleration," or the understanding among competing parties and politicians that they are legitimate rivals rather than existential enemies; and "forbearance," or t ...[text shortened]... bution bribes, etc. etc. etc. but I have a fair degree of confidence that these can be overcome.
The mutual toleration and forbearance issues are discussed in this transcript of an interview.
LEVISKY: The rules themselves, particularly in a very simple, short Constitution like that of the United States, can never get a - can never fully guide behavior. And so our behavior needs to be guided by informal rules, by norms. And we focus on two of them in particular - what we call mutual toleration, which is really, really fundamental in any democracy, which is simply that among the major parties, there's an acceptance that their rivals are legitimate, that we may disagree with the other side. We may really dislike the other side. But at the end of the day, we recognize publicly - and we tell this to our followers - that the other side is equally patriotic, and that it can govern legitimately. That's one.
The other one is what we call forbearance, which is restraint in the exercise of power. And that's a little bit counterintuitive. We don't usually think about forbearance in politics, but it's absolutely central. Think about what the president can do under the Constitution. The president can pardon anybody he wants at any time. The president can pack the Supreme Court. If the president has a majority in Congress - which many presidents do - and the president doesn't like the makeup of the Supreme Court, he could pass a law expanding the court to 11 or 13 and fill with allies - again, he needs a legislative majority - but can do it. FDR tried.
I do not see respect for these two principles as valued on this forum. Rather, those who do show respect, are attacked.
Flinging the overused, battered cliche'd race card around (ala' D64) detracts from actual cases of racism.
LEVISKY: The rules themselves, particularly in a very simple, short Constitution like that of the Unit ...[text shortened]... r these two principles as valued on this forum. Rather, those who do show respect, are attacked.
People don't fear and loathe one another over taxes or health care. As political scientists have shown, the roots of today's polarization are racial and cultural. Whereas 50 years ago both parties were overwhelmingly white and equally religious, advances in civil rights, decades of immigration and the migration of religious conservatives to the Republican Party have given rise to two fundamentally different parties: one that is ethnically diverse and increasingly secular and one that is overwhelmingly white and predominantly Christian.
These are excellent, though depressing, observations.
|
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"redpajama_set_name": "RedPajamaC4"
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| 6,489
|
Q: Store dbcc checkdb result to a file I would like to run dbcc checkdb on weekly basis and wanted to store that result in a .txt file. Please let me know how could I export the data from ssms to txt file.
Thanks in advance!!
A: You can save the script into .sql file and execute it using sqlcmd utility.
for example: exec xp_cmdshell 'SQLCMD -i "D:\ChdbQuery.sql" -o "D:\dbcheckpubs.txt"'
only need to change path, server name & authentication appropriately
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 1,180
|
{"url":"https:\/\/brilliant.org\/problems\/an-interesting-integral\/","text":"# An interesting integral\n\nCalculus Level 2\n\n$\\large \\int_0^\\infty\\frac{\\ln\\frac{1+x^{11}}{1+x^3}}{(1+x^2)\\ln x}\\, \\mathrm dx = \\, ?$\n\nThe integral above has a closed form. Evaluate this integral and give your answer to three decimal places.\n\n\u00d7","date":"2018-12-15 10:21:02","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.924691915512085, \"perplexity\": 1662.4232483060925}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-51\/segments\/1544376826842.56\/warc\/CC-MAIN-20181215083318-20181215105318-00535.warc.gz\"}"}
| null | null |
Leptotarsus constrictus är en tvåvingeart som först beskrevs av Frederick Askew Skuse 1890. Leptotarsus constrictus ingår i släktet Leptotarsus och familjen storharkrankar. Inga underarter finns listade i Catalogue of Life.
Källor
Storharkrankar
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\section{Introduction}
If an operator $P$ is conjugated to a unitary operator, then it is uniformly bounded in the sense that $\sup_{n\in \mathbf{Z}} \|P^n\|$ is finite. The classical 1947 article by B.~Sz\H{o}kefalvi-Nagy~\cite{Sz-Nagy47} establishes the converse. A remarkable feature of Sz.-Nagy's short proof is that it uses the Banach--Mazur ``generalised limits''.
\smallskip
More generally, a representation $\pi$ of a group $G$ on a Hilbert space $V$ is called \emf{unitarisable} if there is an invertible operator $T$ of $V$ such that $T\pi(g) T^{-1}$ is unitary for all $g\in G$. In that case, $\pi$ is necessarily \emf{uniformly bounded} in the sense that $\sup_{g\in G} \|\pi(g)\|$ is finit
. Both J.~Dixmier~\cite{Dixmier50} and M.~Day~\cite{Day50} noticed that the very proof of Sz.-Nagy establishes that every amenable group is \emf{unitarisable}, meaning that all its uniformly bounded representations are unitarisable. Indeed, a group is \emf{amenable} by definition if it admits an invariant mean, \emph{i.e.} a generalised Banach limit
\medskip
J.~Dixmier asked in~\cite[\S5]{Dixmier50} whether unitarisability characterises amenability; the present note contributes to this question. For more background, we refer to~\cite{PisierLNM}.
\medskip
A property very much opposed to amenability is the non-vanishing of the first \emf{$L^2$-Betti number} $\betti$ (to be briefly recalled below; for a detailed discussion, see~\cite{Eckmann00, Lueck}).
\begin{thm}\label{thm:betti}
Let $G$ be a residually finite group. If~$\betti(G) > 0$, then $G$ is not unitarisable.
\end{thm}
A similar property is that the \emf{cost} $\mathscr{C}$ studied in~\cite{Gaboriau00} be larger than one. As suggested by M.~Ab\'ert, the arguments leading to the previous result have a parallel with cost.
\begin{thm}\label{thm:cost}
Let $G$ be a finitely generated residually finite group. If~$\mathscr{C}(G) > 1$, then $G$ is not unitarisable.
\end{thm}
In fact, Dixmier first asked whether any group at all fails to be unitarisable; this was answered in 1955 when Ehrenpreis--Mautner~\cite{Ehrenpreis-Mautner} showed that the complementary series of $\mathrm{SL}_2(\mathbf{R})$ can be extended to uniformly bounded representations that are not unitarisable. A detailed treatment was given by Kunze--Stein~\cite{Kunze-Stein}.
By general properties of unitarisability, the existence of any non-unitarisable group implies that the free group $F_2$ is non-unitarisable (see~\cite{PisierLNM}). Very explicit non-unitarisable representations of $F_2$ were constructed in the eighties~\cite{Mantero-Zappa83,Pytlik-Swarc,Bozejko87}. It follows by induction of representations that any group containing $F_2$ as a subgroup is non-unitarisable.
\medskip
Until now, there was no example of non-unitarisable group not containing $F_2$. In fact, even the existence non-amenable groups without $F_2$ subgroup was a long-standing open problem in group theory, not solved until the 1980's~\cite{Olshanskii80,Adyan83}.
\bigskip
We aim to construct non-unitarisable representations under weaker assumptions than the existence of a free subgroup. A result of Gaboriau--Lyons~\cite{Gaboriau-Lyons}, notably using~\cite{Hjorth_attained} and~\cite{Pak-Smirnova-Nagnibeda}, provides an $F_2$-action on the Bernoulli percolation of any non-amenable countable group $G$ in such a way that $F_2$ can be thought of as a ``random subgroup'' of $G$, even when $G$ has no actual such subgroup. It was suggested in~\cite{MonodICM} (Problem~N) to apply an induction procedure for specific representations of random subgroups in order to answer Dixmier's question. In fact, a first use of~\cite{Gaboriau-Lyons} towards a cohomological question asked in~\cite[\S10]{Johnson} can be found in~\cite[\S5]{MonodICM} and a second use is the ergodic-theoretical result~\cite{Epstein07}.
\smallskip
We shall follow the above strategy, using the language of random forests. A \emf{forest} on a group $G$ is a subset $F\subseteq G\times G$ such that the resulting graph $(G, F)$ has no cycles. The collection $\mathscr{F}_G$ of all forests on $G$ is a closed $G$-invariant subspace of the compact $G$-space of all subsets of $G\times G$ if we consider the usual product topology (\emph{i.e.} pointwise convergence) and the left diagonal $G$-action. A \emf{random forest} is a $G$-invariant Borel probability measure on $\mathscr{F}_G$. By $G$-invariance, the expected degree of a vertex in a random forest does not depend on the vertex; we call it the \emf{expected degree $\deg(\mu)$ of the random forest $\mu$}. Similarly, we define the \emf{width} $\mathrm{width}(\mu)$ as the number of vertices that neighbour a given vertex with positive probability. We shall be interested in forests with finite width. Of course, one has $\deg(\mu)\leq \mathrm{width}(\mu)$.
\begin{thm}\label{thm:forests}
Let $G$ be a unitarisable group. Then the quantity
$$\frac{\deg(\mu)^2}{\mathrm{width}(\mu)}$$
is bounded uniformly over all random forests $\mu$ (of finite width) defined on all countable subgroups of $G$.
\end{thm}
\begin{remark}
We only made the countability assumption in order to have a metrisable space of forests on which the probability is defined. This is an inessential restriction; in any case, unitarisability is a countably determined property~\cite[0.10]{Pisier_survey}. Notice also that all trees in a forest of finite width are countable.
\end{remark}
Using known estimates on specific random forests, Theorem~\ref{thm:forests} implies the following statement, wherein the \emf{rank} $\mathrm{rk}(H)$ denotes the minimal number of generators of a group $H$.
\begin{thm}\label{thm:betti_bis}
Let $G$ be a unitarisable group. Then the quantities
$$\frac{\big(\betti(H)\big)^2}{\mathrm{rk}(H)}, \kern10mm \frac{\big(\mathscr{C}(H)\big)^2}{\mathrm{rk}(H)}$$
are bounded uniformly over all finitely generated subgroups $H$ of $G$.
\end{thm}
We asked D.~Osin whether one knows examples of groups without non-Abelian free subgroup and violating the above bound involving $\beta^1$. It turns out that D.~Osin can construct \emph{torsion} groups with this property (using among others~\cite{Peterson-Thom}); for this and more, we refer to the forthcoming~\cite{Osin09}. Thus, Osin's examples allow to deduce the following from Theorem~\ref{thm:betti_bis}.
\begin{cor}\label{cor:torsion}
There exist non-unitarisable torsion groups.
\end{cor}
We shall begin by proving Theorem~\ref{thm:forests} in Section~\ref{sec:forests}. This result makes it desirable to investigate general constructions of forests with large expected degree. Indeed, Theorems~\ref{thm:betti} and~\ref{thm:cost} will be deduced by considering specific models of random forests and using known estimates for their degrees. In Section~\ref{sec:betti}, we include an expository account of the required properties of the free uniform spanning forest and reduce Theorem~\ref{thm:betti} to Theorem~\ref{thm:forests}. The reduction of Theorem~\ref{thm:cost} to Theorem~\ref{thm:forests} in Section~\ref{sec:cost} follows similar lines but using the minimal spanning forest. Strictly speaking, one could reduce Theorem~\ref{thm:betti} to Theorem~\ref{thm:cost} except for the finite generation issue discussed in Section~\ref{sec:further}; we prefer to present a more detailed account of the relation between $L ^2$-Betti numbers and forests and be more concise in Section~\ref{sec:cost}.
Section~\ref{sec:further} discusses the context and further directions of research; we point out for instance that \emph{any} non-amenable finitely generated group admits a random forest with non-trivial (\emph{i.e.}~$>2$) expected degree.
\subsection*{Acknowledgements}
It is a pleasure to thank the following colleagues: Gilles Pisier first mentioned Dixmier's problem to us; Adrian Ioana found a mistake in an earlier draft; Wolfgang L\"uck helped out with a reference. Special thanks to Mikl\'os Ab\'ert for suggesting to use the cost and to Denis Osin for providing the groups mentioned in Corollary~\ref{cor:torsion}.
\section{Forests and Littlewood}\label{sec:forests}
\begin{flushright}
\begin{minipage}[t]{0.5\linewidth}\itshape\small
---?`Usted sin duda querr\'a ver el jard\'in? $[\ldots]$\\
---?`El jard\'in?\\
---El jard\'in de los senderos que se bifurcan.\footnotemark
\end{minipage}
\footnotetext{J.~L.~Borges, \emph{El jard\'in de senderos que se bifurcan}
(\emph{The Garden of Forking Paths}), 1941.%
}
\end{flushright}
\medskip
We follow Serre's conventions~\cite{Serre77} for graphs, which are thus pairs $(V,E)$ of vertex and edge sets with structural maps $E\to V, e\mapsto e_\pm$ and $E\to E, e\mapsto \bar e$. Recall that the underlying ``geometric'' edges consist of pairs of opposed edges $e,\bar e$. In the case of simple graphs, \emph{i.e.} without loops or multiple geometric edges (such as forests), one shall always consider $E$ as a subset of $V\times V$ invariant under the canonical involution and not meeting the diagonal. Recall also that an \emf{orientation} is a fundamental domain for the involution in $E$.
Given a group $G$, we define the space $\mathscr{G}_G$ of all (simple) graphs on $G$ as the subset $\mathscr{G}_G\subseteq 2^{G\times G}$ of all subsets $E\subseteq G\times G$ defining a simple graph $(G, E)$. The space $2^{G\times G}$ is compact for the product topology and has a natural $G$-action by left multiplication; since $\mathscr{G}_G$ is closed and invariant, it is itself a compact $G$-space. A \emf{random graphing} of $G$ is a $G$-invariant probability measure on $\mathscr{G}_G$.
We now consider the closed $G$-invariant subspace $\mathscr{F}_G\subseteq\mathscr{G}_G$ of forests and recall from the Introduction that a \emf{random forest} is a random graphing supported on $\mathscr{F}_G$. We shall not be interested in the forest of width zero. We denote by $\mathscr{F}_G^+$ the set of all orientations of all forests and view it as a closed $G$-invariant subspace of the compact $G$-space of subsets of $G\times G$. There is a canonical $G$-equivariant quotient map $\mathscr{F}_G^+\to\mathscr{F}_G$.
\begin{example}
Suppose that $S\subseteq G$ is a subset freely generating a free subgroup. Then we obtain a forest $F\in\mathscr{F}_G$ by $F=\big\{(g, g') : g^{-1} g' \in S\cup S^{-1}\}$. This forest is $G$-fixed and hence is a (deterministic) random forest.
\end{example}
\begin{example}\label{ex:UST}
Suppose that $G$ is finite and already endowed with a graph structure $(G,E)$. The uniform measure on the set of all spanning trees of $(G,E)$ is a random forest. Aside from the notion of $G$-invariance, this random forest makes sense for any finite graph $(G,E)$ and is called the \emf{uniform spanning tree}; it will be encountered again in Section~\ref{sec:betti}.
\end{example}
Given a random forest $\mu$ on a group $G$, we denote by $f_\mu(g)$ the probability that $g\in G$ is neighbouring the identity $1\in G$. In other words, $f_\mu(g) = \mu \{F\in\mathscr{F}_G : (1,g)\in F\}$. If $\mu$ has finite width, then $f_\mu$ is a finitely supported function.
We now recall the definition of the $T^1$-norm on the space $\mathbf{C}[G]$ of finitely supported functions and refer to~\cite{PisierLNM} for details and context. Given $f\in \mathbf{C}[G]$, on considers all pairs $f^\pm$ of functions $G\times G\to \mathbf{C}$ such that
$$f(g^{-1} g') = f^+(g,g') + f^-(g,g') \kern 5mm \forall\, g,g'\in G.$$
The norm $\|f\|_{T^1(G)}$ is the infimum of all such pairs $f^\pm$ of the expression
$$\sup_{g\in G}\sum_{g'\in G} |f^+ (g,g')| + \sup_{g\in G}\sum_{g'\in G} |f^- (g',g)|.$$
The completion of $\mathbf{C}[G]$ for this norm is a Banach space denoted by $T^1(G)$ that can be realised as functions on $G$. Such functions are called \emf{Littlewood functions} (see \emph{e.g.}~\cite{Varopoulos74, Bozejko87, Bozejko-Fendler91}) in reference to classical harmonic analysis~\cite{Littlewood30}.
\begin{prop}\label{prop:estimates}
Let $\mu$ be a random forest of finite width on a countable group. Then
$$\|f_\mu\|_{T^1(G)} \leq 2 \kern5mm\text{and}\kern5mm \|f_\mu\|_{\ell^2(G)}\geq \frac{\deg(\mu)}{\sqrt{\mathrm{width}(\mu)}}.$$
\end{prop}
This proposition is a concrete way to carry over to random forests the geometric aspects of a construction for free groups from~\cite{Bozejko-Fendler91}, in accordance with the ideas exposed in the Introduction.
\begin{proof}[Proof of Proposition~\ref{prop:estimates}]
The second inequality is a straightforward application of the Cauchy--Schwarz inequality: setting $S=\{g : f_\mu(g)>0\}$, we have $|S|=\mathrm{width}(\mu)$ and hence
$$\deg(\mu)\ =\ \sum_{g\in G} f_\mu(g)\ =\ \sum_{g\in G} f_\mu(g)\cdot 1_S(g)\ \leq\ \sqrt{\mathrm{width}(\mu)} \,\|f_\mu\|_{\ell^2(G)},$$
as claimed.
\smallskip
We now focus on the first inequality.
Let $\{g_n\}_{n\in \mathbf{N}}$ be an enumeration of the group $G$. We define a Borel section $\mathscr{O}:\mathscr{F}_G\to \mathscr{F}_G^+$ as follows. For a forest $F$ and $(g, g')\in F$, let $n$ be the first integer such that $g_n$ belongs to the tree containing $(g, g')$. We then declare that $(g, g')$ belongs to $\mathscr{O}(F)$ if $g'$ lies between $g$ and $g_n$ in that tree; otherwise, $(g', g)\in \mathscr{O}(F)$.
We now define two functions $f_\mu^\pm$ on $G\times G$ by
$$f_\mu^+ (g,g') = \mu \big\{ F\in \mathscr{F}_G : (g,g')\in \mathscr{O}(F) \big\}, \kern5mm f_\mu^-(g,g') = f_\mu^+(g',g).$$
The sum $f_\mu^+ (g,g') + f_\mu^-(g,g')$ is $\mu \big\{ F\in \mathscr{F}_G : (g,g')\in F \big\}$ by the definition of an orientation. Since $\mu$ is $G$-invariant, this quantity depends only on $g^{-1} g'$ and thus coincides with $f_\mu(g^{-1} g')$. Therefore, in view of the definition of $T^1(G)$, it remains to justify
$$\sup_{g\in G}\sum_{g'\in G} f_\mu^+ (g,g') \leq 1.$$
Fix thus any $g\in G$. Given a forest $F$, there is at most one $g'\in G$ such that $(g,g')\in\mathscr{O}(F)$. Indeed, the integer $n$ introduced in the definition of $\mathscr{O}$ is uniquely determined by $g$ and thus $g'$ can only be the first step towards $g_n$ from $g$, unless $g=g_n$ in which case there is no such $g'$. Therefore $\sum_{g'} f_\mu^+ (g,g')$ is a sum of measures of disjoint subsets of $\mathscr{F}_G$ and hence is bounded by $\mu(\mathscr{F}_G)=1$.
\end{proof}
The space $T^1(G)$ is directly related to uniformly bounded representations:
\begin{prop}\label{prop:inclusion}
If $G$ is unitarisable, then there is a constant $K$ such that
$$\|\cdot\|_{\ell^2(H)}\ \leq\ K\, \|\cdot\|_{T^1(H)}$$
holds for all subgroups $H<G$.
\end{prop}
Observe that the juxtaposition of Propositions~\ref{prop:inclusion} and~\ref{prop:estimates} establishes Theorem~\ref{thm:forests}.
\begin{proof}[Proof of Proposition~\ref{prop:inclusion}]
The fact that unitarisability implies $T^1(G)\subseteq \ell^2(G)$ was established in~\cite[2.3(i)]{Bozejko-Fendler91}, see also Remark~2.8 in~\cite{PisierLNM}. We sketch the main idea for convenience. First, any $T^1$-function gives rise to a uniformly bounded representation on $\ell^2(G)\oplus \ell^2(G)$ by twisting the (diagonal) regular representation with the derivation given by the commutator between the regular representation and kernel operator defined by $f^+$ (using $f^-$ yields the same derivation up to a sign since $f^+ + f^-$ is $G$-invariant). If $G$ is unitarisable, this construction implies that $T^1(G)$ is contained in the space $B(G)$ of matrix coefficients of unitary representations on $G$. Then the stronger conclusion $T^1(G)\subseteq \ell^2(G)$ is obtained by a cotype argument.
Next, we claim that this inclusion is continuous. This follows from the closed graph theorem; indeed, the diagonal in $T^1(G)\times \ell^2(G)$ is closed since it is closed for the weaker topology of pointwise convergence (the latter being Hausdorff).
To conclude the proof, it suffices to show that for all subgroups $H<G$ the canonical inclusion map $\mathbf{C}[H]\to \mathbf{C}[G]$ extends to an isometric map $T^1(H)\to T^1(G)$ since the analogous statement for $\ell^2(H)\to \ell^2(G)$ is obvious. Following~\cite[2.7(ii)]{PisierLNM}, we choose a set $R\subseteq G$ of representatives for $G/H$; we arrange that $R$ contains the identity. Given $f\in T^1(H)$, we still write $f:G\to \mathbf{C}$ for the function extended by zero outside $H$. Let $f^\pm$ be any pair of functions $H\times H\to\mathbf{C}$ as required by the definition of $T^1(H)$. We now extend the definition of $f^\pm$ to functions $G\times G\to \mathbf{C}$ by setting
$$f^\pm(g, g')\ =\
\begin{cases}
f^\pm(h, h') & \text{if $g=rh, g'=rh'$ for $r\in R$ and $h, h'\in H$,}\\
0 & \text{otherwise.}
\end{cases}$$
The definition is well-posed since $R$ maps injectively to $G/H$. This construction witnesses that $f\in T^1(G)$ with $T^1(G)$-norm bounded by $\|f\|_{T^1(H)}$; the reverse inequality is immediate.
\end{proof}
\section{First \texorpdfstring{$L^2$}{L2}-Betti number}\label{sec:betti}
\begin{flushright}
\begin{minipage}[t]{0.63\linewidth}\itshape\small
To achieve this wonder, electricity is the one and only means. Inestimable good has already been done by the use of this all powerful agent, the nature of which is still a mystery.\footnotemark
\end{minipage}
\footnotetext{N.~Tesla, \emph{The transmission of electrical energy without wires as a means for furthering Peace}, 1905.}
\end{flushright}
\medskip
In 1847, G.~Kirchhoff~\cite{Kirchhoff} proved that given a unit electric current between the endpoints of an edge~$e$ in a finite graph, the current flowing through~$e$ equals the (counting) probability that~$e$ belongs to the uniform spanning tree as introduced in Example~\ref{ex:UST}. There is a well-known connection between currents and combinatorially harmonic functions: see H.~Weyl~\cite{Weyl23} or B.~Eckmann~\cite{Eckmann45VD} and~\cite{Eckmann45} pp.~247--248. This is the starting point for the relation between random forests and the first $L^2$-Betti number that emerged from the work of R.~Pemantle~\cite{Pemantle91}, D.~Gaboriau~\cite{Gaboriau05} and R.~Lyons, exposed in~\cite{Lyons-PeresBOOK}. We shall present just what we need in our setting and refer to~\cite{Lyons-PeresBOOK} and~\cite{Benjamini-Lyons-Peres-Schramm01} for much more material.
\medskip
Let $H$ be a countable group and $S\subseteq H$ some finite subset. Consider the graph $\mathfrak{g}=(H, E)$ obtained by assigning a geometric edge (\emph{i.e.} two opposed elements of $E$) between $h, h'\in H$ whenever $h^{-1} h'$ is in $S\cup S^{-1}$. Recall that when $S$ generates $H$, the graph $\mathfrak{g}$ is called a \emf{Cayley graph} for $H$. The left $H$-action preserves the graph structure and we shall investigate random forests arising as subgraphs of $\mathfrak{g}$. Given an enumeration of $H$, let $\mathfrak{g}_n$ be the subgraph of $\mathfrak{g}$ spanned by the first~$n$ elements in $H$. R.~Pemantle~\cite{Pemantle91} showed that the uniform spanning tree measure on $\mathfrak{g}_n$ converges weakly to a measure on $\mathscr{F}_H$. Indeed, it suffices essentially to prove that the probability of the elementary event that a given edge $e$ belongs to a tree in $\mathfrak{g}_n$ (with $n$ large enough to ensure $e\in\mathfrak{g}_n$) is non-increasing in $n$. In view of Kirchhoff's result, this monotonicity follows from Rayleigh's principle stating that added edges can only reduce the current through a given edge. The resulting measure on the space of subgraphs is supported on $\mathscr{F}_G$ since the latter is closed; it is called the \emf{free uniform spanning forest}. (Notice that finite trees can and generally do get disconnected in the limit.) The monotonicity implies in particular that the limit measure does not depend on the enumeration and hence is group-invariant. Much information about this measure can be found in~\cite{Pemantle91, Benjamini-Lyons-Peres-Schramm01, Lyons-PeresBOOK}.
\smallskip
Let $\ell^2_{\mathrm{alt}}(\mathfrak{g})$ be the space of $L^2$-functions on $E$ that change sign under the involution $e\mapsto \bar e$ (\emph{i.e.} ``$1$-forms''). Define the elementary edge function $\chi_e:=\delta_e - \delta_{\bar e}$, where $\delta$ is the Dirac mass. Denote by $d:\ell^2(H)\to\ell^2_{\mathrm{alt}}(E)$ the combinatorial derivative (coboundary) defined by $df(e)=f(e_+) - f(e_-)$ and by $d^*$ its adjoint. Let $\ell^2_\bigstar(\mathfrak{g}) \subseteq \ell^2_{\mathrm{alt}}(\mathfrak{g})$ be the closure of $d\ell^2(H)$ and $\ell^2_\bigcirc(\mathfrak{g})\subseteq \ell^2_{\mathrm{alt}}(\mathfrak{g})$ the closed span of all cycles (\emph{i.e.} sums $\sum_i\chi_{e_i}$ for sequences $\{e_i\}$ forming cycles). We make the corresponding definitions for the graphs $\mathfrak{g}_n$. The latter being finite, linear algebra provides the orthogonal decomposition $\ell^2_{\mathrm{alt}}(\mathfrak{g}_n) = \ell^2_\bigstar(\mathfrak{g}_n) \oplus\ell^2_\bigcirc(\mathfrak{g}_n)$. The failure of this relation for a general infinite graph $\mathfrak{g}$ is crucial below. Equally important is the fact that whilst $\ell^2_{\mathrm{alt}}(\mathfrak{g}_n)$ and $\ell^2_\bigcirc(\mathfrak{g}_n)$ clearly densely exhaust $\ell^2_{\mathrm{alt}}(\mathfrak{g})$ and $\ell^2_\bigcirc(\mathfrak{g})$ as $n\to\infty$, the corresponding circumstance does not hold for $\ell^2_\bigstar$. (This is the key difference between the present model of \emph{free} random forests and the so-called \emph{wired} case where the finite approximations $\mathfrak{g}_n$ are defined differently.)
\smallskip
We record the following result stated (with all necessary indications for the proof) in the current version of Chapter~10 of the book in progress~\cite{Lyons-PeresBOOK}.
\begin{prop}\label{prop:FUSF}
If $S$ generates $H$, then the expected degree of the free uniform spanning forest is at least $2 \betti(H)$.
\end{prop}
In fact, the exact value $2 \betti(H)+2$ is given in~\cite{Lyons-PeresBOOK}, compare Remark~\ref{rem:ExactValue} below.
\begin{proof}[Proof Proposition~\ref{prop:FUSF}]
We can assume $S=S^{-1}$ and $1\notin S$ without affecting the statement, so that the neighbours of $1$ in $\mathfrak{g}=(H, E)$ are exactly $S$. Given an edge $e$ and $n$ large enough, denote by $i_n(e)$ the probability that $e$ (or rather the corresponding geometric edge) is in the uniform spanning tree of $\mathfrak{g}_n$. We need to prove
$$\sum_{s\in S} \lim_{n\to\infty} i_n(e_s) \ \geq\ 2 \betti(H), \kern5mm \text{wherein}\kern5mm e_s:=(s, 1).$$
By definition, the first $L^2$-Betti number $\betti(H)$ is the von Neumann dimension of the first $L^2$-cohomology of $H$. The dimension is not affected by passing to the Hausdorff quotient called the \emph{reduced} $L^2$-cohomology. The latter admits a Hodge--de~Rham decomposition which realises the first reduced $L^2$-cohomology of the finitely generated group $H$ as the space
$$\mathscr{D}_\mathfrak{g}\ :=\ \Big(\ell^2_\bigstar(\mathfrak{g}) \oplus\ell^2_\bigcirc(\mathfrak{g})\Big)^\perp\ \subseteq\ \ell^2_{\mathrm{alt}}(E)$$
of coboundaries of harmonic functions on vertices~\cite[\S1.1.4]{Lueck}. (In other words $\mathscr{D}_\mathfrak{g}$ is the space of differentials of harmonic Dirichlet functions, which is isomorphic to the quotient of harmonic Dirichlet functions by the constants.) As for the von Neumann dimension, we recall that for a closed invariant subspace $W<\ell^2(H)$ it is given explicitly by $\pi_W(\delta_1)(1)$, where $\pi_W:\ell^2(H)\to W$ is the orthogonal projection. Combining this with the canonical isometric $H$-identification $\ell^2_{\mathrm{alt}}(E) \cong \oplus_{s\in S} \ell^2(H \cdot e_s)$ determined by $\chi_{e_s}\mapsto2\delta_{e_s}$, one has
$$\betti(H)\ =\ \frac12 \sum_{s\in S} \pi_{\mathscr{D}_\mathfrak{g}}(\chi_{e_s})(e_s),$$
where now $\pi_{\mathscr{D}_\mathfrak{g}} : \ell^2(E)\to \mathscr{D}_\mathfrak{g}$. A hurried reader may as well skip the above paragraph and take this identity as \emph{ad hoc} definition of $\betti$.
In view of Kirchhoff's laws, the current on $\mathfrak{g}_n$ yielding unit flow between the endpoints of an edge $e$ is $\pi_{\ell^2_\bigstar(\mathfrak{g}_n)}(\chi_e)$ (see \emph{e.g.}~\cite{Eckmann45} p.~248). Therefore, Kirchhoff's characterisation~\cite{Kirchhoff} in terms of the uniform spanning tree shows $i_n(e) = \pi_{\ell^2_\bigstar(\mathfrak{g}_n)}(\chi_e)(e)$. Recalling that $\ell^2_\bigcirc$, but not $\ell^2_\bigstar$, is compatible with the exhaustion, we obtain
\begin{multline*}
\sum_{s\in S} \lim_{n\to\infty} i_n(e_s)\ =\ \sum_{s\in S} \pi_{\ell^2_\bigcirc(\mathfrak{g})^\perp}(\chi_{e_s})(e_s)\ =\ \sum_{s\in S} \pi_{\mathscr{D}_\mathfrak{g}\oplus\ell^2_\bigstar(\mathfrak{g})}(\chi_{e_s})(e_s)\\
=\ \sum_{s\in S} \pi_{\mathscr{D}_{\mathfrak{g}}}(\chi_{e_s})(e_s) + \sum_{s\in S} \pi_{\ell^2_\bigstar(\mathfrak{g})}(\chi_{e_s})(e_s).
\end{multline*}
We know already that the first summand equals $2 \betti(H)$. In order to conclude the proof, it remains only to justify that the function $f:=\pi_{\ell^2_\bigstar(\mathfrak{g})}(\chi_{e_s})$ is non-negative at $e_s$. This is the case since (i)~$\chi_{e_s}(e_s)=1$, (ii)~$f$ and $\chi_{e_s}$ are alternating and (iii)~orthogonality imposes $\|f-\chi_{e_s}\|\leq \|\chi_{e_s}\|=2$.
\end{proof}
\begin{remark}\label{rem:ExactValue}
The expected degree is $2 \betti(H)+2$. Indeed, the second summand in the proof above is the expected degree of the \emph{wired} uniform spanning forest on $\mathfrak{g}$ for reasons entirely similar to the above, namely because the exhaustion defining this other model is compatible with~$\ell^2_\bigstar$. On the other hand, it is shown in~\cite{Benjamini-Lyons-Peres-Schramm01} that this expected degree is two, using a different characterisation of the wired forest \emph{via} an algorithm of D.~Wilson~\cite{Wilson96}.
\end{remark}
We are now ready to complete the reduction of Theorem~\ref{thm:betti} to Theorem~\ref{thm:forests}. Recall that the \emf{rank} $\mathrm{rk}(H)$ is the minimal number of generators of a group $H$.
\begin{proof}[Proof of Theorem~\ref{thm:betti_bis}, first bound]
Let $H<G$ be a finitely generated subgroup with a generating set $S$ of size $\mathrm{rk}(H)$. The corresponding free uniform spanning forest $\mu$ on $H$ satisfies $\mathrm{width}(\mu)\leq \mathrm{rk}(H)$. Therefore, Proposition~\ref{prop:FUSF} shows that Theorem~\ref{thm:forests} yields the desired bound.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:betti}]
Let $G$ be any residually finite group with $\betti(G)>0$. Since $G$ is the union of the directed set of all its finitely generated subgroups, Theorem~7.2(3) in~\cite{Lueck} provides us with a finitely generated subgroup $G_0<G$ with $\betti(G_0)>0$. Strictly speaking, one needs to express $G$ as a directed union of \emph{infinite} subgroups in order to apply \emph{loc. cit.}; this is not a restriction since if no such family existed, then $G$ would be amenable as directed union of finite groups, contradicting $\betti(G)>0$ (Theorem~0.2 in~\cite{Cheeger-Gromov}).
The group $G_0$ is still residually finite; we shall use the weaker property that $G_0$ admits finite index subgroups of arbitrarily large index. Notice that for all finite index subgroups $H<G_0$, the quantities $\mathrm{rk}(H)$ and $\betti(H)$ are finite. Moreover, denoting by $[G_0:H]$ the index, one has
$$\betti(H) = [G_0:H]\, \betti(G_0) \kern5mm\text{and}\kern5mm \mathrm{rk}(H) \leq [G_0:H]\, \mathrm{rk}(G_0).$$
The above equality is a basic property of $L^2$-Betti numbers~\cite[1.35(9)]{Lueck} whilst the inequality is a (non-optimal) consequence of the Reidemeister--Schreier algorithm, see \emph{e.g.} Proposition~4.1 of~\cite{Lyndon-Schupp} (in fact the quantity $\mathrm{rk}-1$ is sub-multiplicative).
Since $[G_0:H]$ is unbounded, the above (in)equalities violate the first bound of Theorem~\ref{thm:betti_bis}.
\end{proof}
\section{Cost}\label{sec:cost}
This section will be more concise since we shall deduce Theorem~\ref{thm:cost} from our Theorem~\ref{thm:forests} in very much the same way as we did above for Theorem~\ref{thm:betti}.
\medskip
The \emf{cost} $\mathscr{C}(G)$ of a countable group $G$ is a numerical invariant extensively studied by D.~Gaboriau~\cite{Gaboriau00} (and suggested by G.~Levitt~\cite[p.~1174]{Levitt95}). It is defined as the infimum over all free probability-preserving $G$-actions and over all families of partial isomorphisms generating the resulting orbit equivalence relation of the sum of the measures of the domains of the partial isomorphisms.
We shall not use this definition, but rather the following alternative definition: The cost $\mathscr{C}(G)$ is the infimum of half the expected degree over all connected random graphings of $G$. The equivalence of the definitions is proved \emph{e.g.} in Proposition~29.5 of~\cite{Kechris-Miller04} (where the definition of the degree differs by a factor~$2$).
\medskip
We now proceed to recall another family of models of random forests, namely the \emf{free minimal spanning forests}; first studied on $\mathbf{Z}^d$ in~\cite{Alexander-Molchanov, Alexander95}, it received a general treatment in~\cite{Lyons-Peres-Schramm06}. Let $H$ be a group generated by a finite set $S=S^{-1}\not\ni 1$ and let $\mathfrak{g}$ be the corresponding Cayley graph $\mathfrak{g}=(H, E)$ (as in Section~\ref{sec:betti}). The free minimal spanning forest associated to this choice $\mathfrak{g}$ is the random graphing of $H$ obtained by assining weights on the (geometric) edges of $\mathfrak{g}$ independently and deleting every edge that has maximal weight in some cycle. We shall need the following fact due to R.~Lyons.
\begin{prop}\label{prop:FMSF}
Let $\mu$ be the above random forest. Then $\deg(\mu)\geq 2 \mathscr{C}(H)$.
\end{prop}
\begin{proof}
For any $0\leq p \leq 1$, let $\mu_p$ be the random graphing obtained by adding to the $\mu$-random forest each edge of $E$ with probabiliy $p$ independently (thus $\mu_p$ is the union of $\mu$ and of the Bernoulli $p$-percolation random graphing on $\mathfrak{g}$). According to Theorem~1.3 in~\cite{Lyons-Peres-Schramm06}, $\mu_p$ is almost surely connected whenever $p>0$. However, we have by construction
$$\deg(\mu_p)\ \leq\ \deg(\mu) + |S|\cdot p.$$
Letting $p$ tend to zero, the statement follows from the characterisation of $\mathscr{C}(H)$ recalled above.
\end{proof}
Now the reduction of Theorem~\ref{thm:cost} to Theorem~\ref{thm:forests} proceeds exactly along the lines of the arguments given for Theorem~\ref{thm:betti} in Section~\ref{sec:betti}: First, Proposition~\ref{prop:FMSF} applied to finitely generated subgroups $H$ of $G$ shows that Theorem~\ref{thm:forests} yields the second bound of Theorem~\ref{thm:betti_bis}. Then, one considers finite index subgroups $H<G$ of arbitrarily large index and argues as for Theorem~\ref{thm:betti}, using this time the relation
$$\mathscr{C}(H)-1\ =\ [G:H]\,(\mathscr{C}(G) -1),$$
which is Theorem~3 in~\cite{Gaboriau00}. There is no need here to choose a subgroup $G_0$ since $G$ was assumed finitely generated from the outset.
\section{Further considerations}\label{sec:further}
\subsection{}
We begin with a few remarks about the relation between Theorems~\ref{thm:betti} and~\ref{thm:cost}. For any infinite countable group $G$, one has $\mathscr{C}(G)-1\geq\betti(G)$ (this follows from Corollaire~3.23 in~\cite{GaboriauL2}). A well-known question is whether equality holds. Thus, in the special case of finitely generated groups, Theorem~\ref{thm:cost} is \emph{a priori} stronger than Theorem~\ref{thm:betti}. For general countable groups, one would need the fact that a directed union of cost one groups still has cost one; this is not in the literature (a partial result is Lemme~VI.25 in~\cite{Gaboriau00}), though D.~Gaboriau has orally communicated us a proof. Moreover, the second bound of Theorem~\ref{thm:betti_bis} implies the first.
\smallskip
As for the two types of forests used on finitely generated groups in the reduction of these two theorems to Theorem~\ref{thm:forests}, it is a general fact that on the same Cayley graph, the free \emph{minimal} spanning forest has expected degree bounded below by its \emph{uniform} analogue, see Corollary~1.4 in~\cite{Lyons-Peres-Schramm06}.
\subsection{}\label{sec:WMSF}
As mentioned in the Introduction, \itshape any non-amenable finitely generated group $G$ admits a random forest of expected degree~$>2$\upshape. Indeed, let $S=S^{-1}\not\ni 1$ a finite generating set. For an integer $k$, consider the $k$th product graph $\mathfrak{g}^{[k]}$ associated to the Cayley graph $\mathfrak{g}$, recalling that it consists of the graph on $G$ where edges correspond to $k$-paths in $\mathfrak{g}$. Strictly speaking, it is a multi-graph, but any forest on $\mathfrak{g}^{[k]}$ can be considered as a forest on the Cayley graph associated to $S^k$. Using spectral isoperimetric estimates, it is proved in~\cite{Pak-Smirnova-Nagnibeda} that the Bernoulli percolation on $\mathfrak{g}^{[k]}$ satisfies $p_c<p_u$ when $k$ is large enough, where $p_c, p_u$ are respectively the critical probability and the uniqueness probability (see~\cite{Lyons-PeresBOOK} for more background). By Proposition~1.7 in~\cite{Lyons-Peres-Schramm06}, this implies that the free minimal spanning forest differs from its \emf{wired} analogue, which implies that the former has higher expected degree by Proposition~3.5 \emph{loc.\ cit}. We recall here that the wired minimal forest is defined exactly as in Section~\ref{sec:cost}, except that one deletes an edge if it has maximal weight even in a cycle ``through infinity'', which is just a bi-infinite path (our reference is still~\cite{Lyons-Peres-Schramm06}). Summing up, it remains only to prove that the expected degree of the wired minimal spanning forest is at least~$2$. In fact, it is exactly~$2$ in the Cayley graph case at hand, see Theorem~3.12 in~\cite{Lyons-Peres-Schramm06}. The above reasoning can be extracted from the arguments of~\cite{Gaboriau-Lyons}.
\subsection{}
It would be desirable to have examples of (residually finite) groups $G$ with $\betti(G)>0$ or $\mathscr{C}(G)>1$ but not containing $F_2$. We would expect such examples to exist, be it only because the non-vanishing of $\betti$ is a measure-equivalence invariant by~\cite{GaboriauL2}, and $\mathscr{C}>1$ is so by definition; it seems unlikely that the containment of $F_2$ should be preserved. Interestingly, it is established in~\cite{Peterson-Thom} that for a torsion-free group satisfying a weaker form of the Atiyah conjecture, $\betti>0$ implies the existence of a free subgroup $F_2$. In view of the measure-equivalence invariance of $\betti>0$, one can ask if this statement should be considered as evidence against the Atiyah conjecture. On the other hand, an indication of perhaps surprisingly strong restrictions given by additional algebraic assumptions is M.~Lackenby's result~\cite{Lackenby_detecting} that implies in particular that \emph{residually-$p$-finite finitely presented} groups with $\betti>0$ contain~$F_2$.
\subsection{}
Let $G$ be a group generated by a finite set $S$ and let $\mathfrak{g}$ be the corresponding Cayley graph. Theorem~\ref{thm:forests} is an incentive to find random forests in $\mathfrak{g}$ with large expected degree (compared to the size of $S$). One immediate restriction is given by the \emf{vertex isoperimetric constant} of $\mathfrak{g}$, namely the infimum $i_V(\mathfrak{g})$ of the ratio $|\partial_V \mathfrak{h}|/|\mathfrak{h}|$, where $\mathfrak{h}$ ranges over all finite subgraphs and $\partial_V$ denotes the vertex-boundary. Indeed, one verifies that the expected degree of any random forest on $\mathfrak{g}$ is bounded by $1+i_V(\mathfrak{g})/2$. (For the edge-isoperimetric constant, this inequality occurs in~\cite{Lyons-Pichot-Vassout08}.) One can increase at will $i_V$ for any non-amenable graph by replacing $S$ with high powers of that set (as in~\cite{Pak-Smirnova-Nagnibeda}, see~\ref{sec:WMSF} above), but this procedure affects also the denominator in Theorem~\ref{thm:forests}.
\smallskip
Whilst an application of the Hall marriage lemma and of a Cantor--Bernstein argument shows that $\mathfrak{g}$ contains a forest of $n$-regular trees whenever $n\leq i_V(\mathfrak{g})$, there is no indication that there should be a $G$-invariant measure on the space of such forests.
\subsection{}
Let $G$ be a finitely generated group. Rather than residual finiteness, the proof of Theorem~\ref{thm:betti} (and thus also of Theorem~\ref{thm:cost}) actually uses the existence of infinitely many finite quotients of $G$, or equivalently of some infinite sequence $\{H_n\}$ of finite index subgroups $H_n< G$ which we may assume nested. The Reidemeister--Schreier algorithm quoted earlier shows that the limit
$$\lim_{n\to\infty} \frac{\mathrm{rk}(H_n) - 1}{[G:H_n]}$$
exists; it was introduced in~\cite{Lackenby05} as the \emf{rank gradient}. The \emf{absolute rank gradient} of~\cite{Abert-Nikolov07HEEGARD} is the infimum of the above ratio over all finite index subgroups of $G$.
\smallskip
Does the existence of an infinite sequence with positive rank gradient imply that there are random forests $\mu$ on (subgroups of) $G$ with unbounded ratio $\deg(\mu)^2/\mathrm{width}(\mu)$~? Are there such forests at least when $G$ has positive \emph{absolute} rank gradient?
\smallskip
It follows from the definitions that both $\betti(G)$ and $\mathscr{C}(G)$ are bounded by $\mathrm{rk}(G)$; therefore, the multiplicativity of $\betti$ and $\mathscr{C}$ (as recalled in earlier sections) imply that both are lower bounds for the absolute rank gradient. The results of Ab\'ert--Nikolov~\cite{Abert-Nikolov07} suggest some similarity of the rank gradient with the behaviour of these invariants. Moreover, in~\cite{Abert-Nikolov07HEEGARD}, Ab\'ert--Nikolov express the rank gradient of certain chains $\{H_n\}$ as the cost of a specific $G$-action attached to the chain. This result gives added interest to the \emph{fixed price question} which asks whether all relations produced by a given countable group have same cost~\cite{Gaboriau00}.
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Michelle decided to organize a club at her school to assist the girls she'd seen in the photos, initially by holding bake sales to help pay education costs. As the club expanded to include more than 100 members in three branches, it obtained nonprofit status and tackled a variety of ambitious projects. So far, Michelle's "Together to Empower" organization has not only raised funds to help 550 Guatemalan girls go to school, but also made it possible for several Ugandan women to learn business skills and entrepeneurship, organized a two-week computer coding camp for 25 girls, hosted a fundraising banquet to promote safe health practices for African women, and held tech workshops to teach 50 girls how to develop mobile computer applications. The group is now preparing to publish an art book depicting various artists' interpretations of what it means to be a woman.
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The real Sylvia Plath
Her newly published, unexpurgated journals reveal the poet's true demons -- and support a little-known theory about what drove her to suicide. First of two parts.
By Kate Moses
Published May 30, 2000 7:30PM (EDT)
It's the tally of "my lusts and my little ideas," wrote 17-year-old Sylvia Plath of the journals in which she confessed her judgments, her "test tube infatuations," her story notes, her cake baking, her dreams and her fears from the age of 12 until days before her death by her own hand at the age of 30. Plath's characterization of her journal stands in stunning contrast to the monumentally revealing document she created: more than a thousand pages scattered through various handwritten notebooks, diaries, fragments and typed sheets, the sum of it an extraordinary record of what she called the "forging of a soul," the creation of a writer and a woman whose many veils and guises have succeeded in forestalling anyone from knowing who she really was, despite her lifelong quest to discover the answer for herself.
"You walked in, laughing, tears welling confused, mingling in your throat. How can you be so many women to so many people, oh you strange girl?" Plath asked herself in the summer of 1952 when she was about to enter her junior year at Smith College in Northampton, Mass. Now, with the English publication of Plath's unabridged journals this spring, we are closer than ever to knowing the real identity of this disappointed wife and bereaved daughter, this suicidal mother of two, this poet of electrically charged perceptions and amplified imagination, this woman "enigmatical/shifting my clarities," this Lady Lazarus who evolved out of her own inner torment, the record of which now opens fully, or almost, before us.
The publication of these journals is a watershed event. They allow us, for the first time, to see this dazzlingly, maddeningly fragmented woman as an integrated being. The Plath that emerges here is paradoxically at once saner -- less a creature of willful mental excess -- and more buffeted by forces beyond her control. Those forces, it seems tragically clear, were not just familial, but chemical. Almost from the day she died, readers and scholars, faced with the huge, faceless enigma of her suicide, have been perplexed and thwarted by Plath's mental condition. The unabridged journals and other new information, some of it reported here for the first time, lend credence to a little-noticed theory that Sylvia Plath suffered not just from some form of mental illness (probably manic depression) but also from severe PMS.
The idea that Plath's demons had a biological basis, far from being reductive, only increases her stature as a poet and a human being. She wrested her art from great darkness.
In the fall of 1962, during the final flood of creativity that preceded her death by a few months, Sylvia Plath alluded to her first suicide attempt in "Daddy," now her most widely recognized poem. "At twenty I tried to die," she wrote, "...But they pulled me out of the sack,/ And they stuck me together with glue." Four decades since Plath killed herself on the morning of February 11, 1963, it seems more accurate to say that she's been stuck back together with paper. Tons and tons of paper: her own posthumously published poetry collection, the fierce and mythic "Ariel," an encoded autobiography which indeed, as she predicted, made her name; the softened "corrective" of the dutiful, chirpy "Letters Home" edited by her mother, Aurelia Plath; her Pulitzer-prize winning "Collected Poems," which builds inexorably from polite surface poise to crackling, incinerating force; a smattering of fairly neutral stories and telling journal fragments in "Johnny Panic and the Bible of Dreams"; and her journals, published in heavily edited form in 1982 that, depending on whose side you were on, made Plath appear either mad or victimized.
All of Plath's work, including her three additional poetry collections, remains in print. But even more voluminous is the critical response her writings have generated -- about a dozen biographies and "recollections" and hundreds of articles, critical studies and cultural commentaries.
What's most noticeable about the veritable industry of books and articles about Plath is that none of them succeed in creating an integrated portrait of their subject. She is variously portrayed as a fragile, brilliant immigrant's daughter scarred by overarching ambition and her father's early death; a righteous proto-feminist shrugging off husband, children and the crippling reins of culturally prescribed domesticity; an unreasonable perfectionist whose outrageous demands alienated everyone who crossed her path; a devoted wife and mother shattered by her idolized husband's betrayal; and an unbalanced artist who would use and sacrifice everything, including her own life, to serve her art.
By her own admission Plath was a woman of many masks, someone who felt it necessary to reveal only facets of herself in any given situation, social or professional. Her husband, the late British Poet Laureate Ted Hughes, wrote in the introduction to her 1982 journals, "I never saw her show her real self to anybody -- except, perhaps, in the last three months of her life."
Hughes, of course, has been the central figure and object of suspicion, even persecution, in the vitriolic 40-year-old controversy regarding the "real" Sylvia Plath. In the summer of 1962, the Hughes' marriage broke down when Plath discovered that Hughes was having an affair. According to Hughes' infrequent comments regarding his relationship with Plath, theirs had been a mutually creative, valuable symbiosis from the very start: "Our minds soon became two parts of one operation," he told the Paris Review in 1995.
But things went very wrong, as his 1998 poetry collection addressed to Plath, the international bestseller "Birthday Letters," attests. When they separated traumatically in September 1962 after six years of marriage, the couple were parents of a 2-year-old daughter, Frieda, and an 8-month-old baby son, Nicholas; Hughes moved to London, while Plath remained with the children at their house in the English countryside. With only sporadic childcare and often ill with fevers, flu and infections, Plath wrote the bulk of the "Ariel" poems in a seven-week rush during the pre-dawn hours before her children awoke. When Plath died, she was still legally married to Hughes, and the responsibility of conducting her literary estate fell to him. In 1969, Hughes' lover, Assia Wevill, mimicked Plath's suicide by gassing herself as well as the young daughter, Shura, whom she shared with Hughes. Hughes wrote to Plath biographer Anne Stevenson in 1989, "... I saw quite clearly from the first day that I am the only person in this business who cannot be believed by all who need to find me guilty."
He was right. As Hughes slowly released her posthumously published works -- which succeeded in winning for her an enormous readership as well as entry into the canon of American 20th century poetry, status she had decidedly not held during her lifetime -- he was viciously attacked by scholars and critics, feminists in particular, who read the blistering "Ariel" poems and later the judiciously pruned 1982 journals as an indictment against him. He was controlling, egotistical, faithless and selfish; he had tried to shame Plath, a poetic genius, into sewing on his buttons.
Hughes has since been consistently criticized for his "censoring" and "stifling" of Plath through his editorial decisions, which notably included trimming and reordering the "Ariel" manuscript, thereby changing its tone and theme from one of transformative rebirth to one of inevitable self-destruction, and his most condemned deed of all, destroying Plath's final journal from the last three months of her life. "I did not want her children to have to read it," Hughes wrote in his introduction to the journals in 1982. Another journal, covering late 1959 through the fall of 1962, or the pivotal "Ariel" period, was said by Hughes to have "disappeared," though it "may ... still turn up."
Hughes' actions -- destroying or losing Plath's final journals and rearranging "Ariel" -- represent a crux of moral ambiguity that readers and scholars have battled over for decades. Did his actions simply reflect, as he consistently maintained, his obligations toward his children? Or were they motivated by self-interest -- an emotion which under the circumstances could be considered reasonable?
It is hard not to feel sympathy for a man who famously wrote of the lost journals, "In those days I regarded forgetfulness as an essential part of survival." Yet it is undeniable that by destroying them Hughes forever silenced the record of the process he considered so essential to Plath's poetic achievement, and to Plath herself, of whom he wrote in 1971, "I feel a first and last obligation to her."
Since the late 1970s, Hughes had maintained that all of Plath's writings, no matter how private, were vital insofar as they shed light on the "true" Sylvia Plath. Plath's central project and problem, Hughes believed, was the creation of herself. He likened Plath's creative process to an alchemical one in which her immature writings, her highly mannered early poetry and the stiff stories into which she desperately tried to breathe life were "like impurities thrown off from the various stages of the inner transformation, by-products of the internal work." "Ariel" and the related final poems, by dramatic contrast, were the voice of her true self, "the proof," he wrote in the 1982 journal's foreword, "that it arrived. All her other writings, except these journals, are the waste products of its gestation." According to Hughes, the journals were Plath's private record of her many camouflages, the stylistic personalities she tried on, the identities and defenses she assumed. The journals reveal "the day to day struggle with her warring selves."
By 1998, Hughes had come to defer to the judgment of his children, who no longer needed his protection, about publishing the journals. "This was really Frieda's and Nicholas' decision in conjunction with their father," said Karen Kukil, editor of the unabridged journals, in a recent interview with Salon. Frieda Hughes called Kukil, curator of Smith College's 4,000-page Plath collection since 1990, in the spring of 1998 to ask Kukil to edit a complete, unexpurgated volume of all of her mother's journals in the Smith library.
When news broke earlier this year that the British publisher Faber & Faber intended to release those unabridged journals, the announcement engendered a flurry of speculation about what other Plath bombshells might be in the offing. Perhaps the disappeared journal would emerge, or more likely, all of the imagined juicy details of insufferable husbandly domination and adulterous calumny that Hughes had witheld from the journals in 1982 to save his own reputation. Hughes' admission that he'd destroyed the journal had predictably nurtured the assumption among his critics that the editing of the journals had been for his own benefit, rather than to eliminate what Frances McCullough, editor of the 1982 journals, characterized as the less relevant material as well as "the nasty bits" that would have caused unnecessary pain or embarrassment to Plath's surviving relatives, friends and colleagues.
Earlier this month, Faber & Faber released those journals in Britain (the American edition will appear this fall from Anchor Books). Unlike the 1982 journals, which were shaved down to about a third of their actual volume, Faber's "unabridged" edition brings together every extant journal from 1950 onward. (The famously missing journal from 1959-1962 isn't included.) The Faber edition is a meticulous preservation of Plath's misspellings, grammar, spot illustrations, capitalization and punctuation, and an absolutely faithful rendering of her words -- pure, unadulterated Sylvia Plath for the first time.
The unabridged journals include material that vindicates both the anti- and pro-Hughes camps. More importantly, they give Plath's readers their first-ever opportunity to experience the uncensored breadth of Plath's imagination in its richest medium, the private testing ground of her relentlessly self-reflective artistry. As the anti-Hughes camp had always protested, they contain material with scholarly rather than merely prurient value. But it is also obvious that much of the deleted material was justifiably censored to spare the feelings of Plath's friends and family.
The volume includes in their entirety Plath's two consecutive journals from 1957 to 1959, when Plath returned with Hughes from England to teach miserably for a year at Smith followed by a year spent living in Boston, where she resumed psychoanalysis with Ruth Barnhouse Beuscher, who had treated Plath during her recovery from the 1953 suicide attempt. It was a time of revisiting old ghosts and old haunts. Plath uncovered first her scornful disdain for her Smith friends and colleagues ("Botany professors forking raw tongue with dowdy seat-spread wives" is one of her milder observations), and second her deep hatred and resentment of her "vampire" mother, whose death in 1994 presumably made publication of this vitally illuminating portion of the journals palatable to the Plath estate.
The unabridged journals confirm the anti-Hughes camp's assumption that Hughes censored details about himself, but his elisions appear to be dictated by a concern for basic privacy rather than the need to conceal damning information. Nothing about Hughes that is new to the unabridged journals reveals him as any worse than he already had allowed himself to be seen in earlier books. It's easy, though, to imagine why anyone, especially England's future poet laureate, might have wanted to censor his wife's nattering on about his "delicious skin smells," infrequent hair washing and "hairy belly."
To be sure, all of the major themes of the journals were present in the 1982 journals -- among them, Plath's precocious and unwavering ambition as a writer, which drove her mercilessly toward artistic growth and publication; her boy-crazy social whirl in college and her attendant preoccupation with the limitations of marriage and gender roles in the cramped cultural mind of the '50s; the familial demons of her childhood -- her father's death from a complication of diabetes when she was 8, and her conflicted relationship with her widowed mother; the emotional, psychological and artistic enormity of her relationship with Hughes; and most compelling, her indefatigible struggle to wrestle control over her chaotic emotional life, what Hughes 20 years ago called "her will to face what was wrong in herself, and to drag it out into examination, and to remake it."
And yet the 1982 journals didn't feel whole. Despite Hughes' stated intentions, Plath still seemed vague and fragmented, her poems only dimly illuminated. The 1982 journals felt figuratively as well as literally elliptical, and into those ellipses could be injected all sorts of strange and dark and terrible fantasies, possibly stranger and darker than the truth. "More terrible," the Plath of "Stings" might say, "than she ever was."
It's not the "true self" of Sylvia Plath that comes rushing at you with vivid immediacy -- at least not the true self as Hughes defined it, a Plath distilled into pure, ferocious, luminous essence. Nor is it the vague, half-glimpsed Sylvia Plath of the earlier journals, whose longings and crises and furies didn't quite add up. Instead, it is the IMAX version of Sylvia Plath who appears from the very first pages of the journals -- the exaggerated, high-voltage, bigger-than-life personality and imagination that no one, not a single one of her detractors or friends, has denied was consistently evident (if frequently hard to take) in the flesh.
This feverish Sylvia Plath floods the reader's senses as her own were flooded throughout her life: on wave after wave of ecstatic or crashing experience, on sparkling details she seems helpless, at every moment, to ignore. "Eyes pulled up like roots" is how the poet Anne Carson characterized Plath, and the image carries its shock of authenticity. "I've talked to alumni who knew Plath," says Kukil, "and they say that everything she did was at the same intense level. Everything she did, she experienced to the hilt." "It's getting so I live every moment with terrible intensity," she wrote to pen pal Ed Cohn in 1950.
Twenty years ago, it may have seemed to Hughes and McCullough that preserving Plath's rush of quotidian detail -- the icebox cheesecakes she immortalized, the epiphany over a story in Cosmopolitan magazine that gave her the idea to write "The Bell Jar" ("I must write one about a college girl suicide ... There is an increasing market for mental-health stuff."), her obsessive bemusement about dog shit, the noting of the cold water and salt in which were soaked the sheets bloodied by her newborn son's afterbirth, the 54 descriptions of what the moon looked like that minute -- would diminish the impact of her unique genius in the journals rather than enhance it.
The opposite is true: It is the most ordinary details of Plath's daily life that now give her such astonishing depth and balance and make her seem, within the thrum of her intensity, refreshingly sane and vibrant. Teeming as they are with prescient observations and, as Plath puts it, "foolishness," the unabridged journals are no less her artistic "Sargasso" for the jumble of her "gabbling" -- they are, in fact, more so. Plath's is a personality integrated by cumulative effect. The details pull forward not just toward the poems, but toward a fuller and more distinct picture of the woman who wrote them: They add immeasurably to Plath's artistic and psychological stature.
Even so, there are many passages whose previous excisions are understandable, lines and whole entries redolent with the whiff of taboo of one kind or another. Hilarious as it is to envision now, no doubt Hughes didn't relish the idea of letting it be known that Plath had in 1958 -- after he'd won the attention of W.H. Auden, Stephen Spender and Marianne Moore with his first book -- entered their poems in jingles contests run by food companies: "the dole pineapple & heinz ketchup contests close this week, but the French's mustard, fruit-blended oatmeal & slenderella & Libby-tomato juice contests don't close till the end of May. We stand to win five cars, two weeks in Paris, a year's free food, and innumerable iceboxes & refrigerators and all our debts paid. Glory glory." Some of the 1982 cuts were simply Plath's caustic sniping and thinly disguised jealousies -- there is a wonderfully sulky account of a lunch with fellow poets, drooling unattractive babies, and spilled tea that ends "Too much salt in a fruit salad. We ate, grumpily, and left."
Much has been made of the journal episode of May 19 to 22, 1958, in which Plath records her shock and disgust at her discovery of Hughes' feet of clay. On that day, her last day of teaching at Smith, Plath and Hughes had made plans to meet after her last class. When Hughes didn't show, Plath had "an intuitive vision" that she would see him walking with a college girl on the campus; not only was she right, but the girl literally ran away and Hughes made no attempt to introduce her. Because the 1982 version of the journals left quite enough material to make Hughes look like a cad if not a downright adulterer and further piqued suspicions by inserting numerous [OMISSION] flags that glowed malignantly within the passage, many readers and critics have understandably assumed that the elisions would point directly to Hughes' infidelity.
Instead, the reinstated omissions make clear that what really upset Plath was Hughes' open display of vanity -- that on her special day, he put his own ego (only figuratively stroked by the fleeing, thick-legged co-ed in Bermuda shorts) ahead of hers. Hughes, "whose vanity is not dead, but thrives," "a liar and a vain smiler," definitely comes out looking all too human, but the edited version had made him seem truly sinister. It's ironic that in this memorable instance Hughes cut references to his vanity (and his saggy pants and greasy hair and the universally condemnable smarminess of his "heavy ham act ... 'Let's make up'") presumably in order to assuage his self-regard, and yet by doing so he planted in the minds of Plath's readership the seeds of his early-and-often abuse of Plath's faith in him.
The journals were Plath's magic cauldron, the receptacle where she stewed the observations that would help her give shape to her life in its myriad desired guises. It can be seen burbling away in her eavesdropping on an adult cocktail party at the summer home of the Mayos, a family for whom she worked as a mother's helper during the summer of 1951: "What were they talking about? What was the subtle line that marked you from entering a group such as this? ... I can hear the voices coming up to me, laughter, raveled words. Up here, on the second floor porch, the air blurs the syllables and continuity of conversation like sky-writing ..."
Other previously omitted passages illuminate Plath's apprenticeship in her life as well as her art to the degree that their previous removal now seems peculiarly shortsighted. Among the themes fleshed out by the unabridged journals are Plath's ongoing struggles with the concept of marriage, which she both feared as stultifying to her creativity and desired for its sexual and emotional intimacy.
Related to that is her "hatred" of men, oft-cited by critics. That hatred now appears more accurately as an envy borne of the frustratingly confining '50s-era sexual mores that made it impossible for Plath to seek the experiences she wanted, to be as sexually free in her thought and actions as men could be. Plath also easily articulates the polarity between her desire to mother versus her protectiveness of her professional ambition -- belying the theory circulated in some circles that Plath's ambivalence toward motherhood was not quite normal.
The unexpurgated 1957 to '59 entries reveal the depth of Plath's awkwardness with people, as opposed to the outward "golden girl" gaiety typically ascribed to her. While teaching at Smith, Plath instituted a program to compel herself to interact. "People: eyes & ears not shut, as they are now," she coached herself, "I apart, aware of apartness & a strange oddity that makes my coffee-shop talk laughable -- we are inviting people to dinner: four a week, 16 a month: I shall not go sick or nervous or over-effusive ..."
Throughout the early years of the journals, Plath's lack of experience is sometimes cringingly obvious, her early attempts at hammering the episodes of her life into fictional or poetic shape hilariously sophomoric. During her college years, Plath often recorded her life in scenes addressing herself as "you" or in a frequently self-congratulatory third person: "Outwardly, all one could see on passing by is a tan, long-legged girl in a white lawn chair, drying her light brown hair ... Tonight she will dress in the lovely white sharkskin hand-me-down dress of last summer's employer and gaze winningly at her entranced Princeton escort ..." On the occasion of the end of a brief infatuation, Plath threw herself with full intensity into a melodramatic chunk of doggerel:
The slime of all my yesterdays
Rots in the hollow of my skull:
And if my stomach would contract
Because of some explicable phenomenon
Such as pregnancy or constipation
I would not remember you.
She was not unaware of her early failures. In fact, wherever the craft of writing was at issue Plath was notoriously hard on herself. But what the young Plath lacked in experience she made up for in imagination and most decidedly in will. At 18, she scolded herself: "I am a victim of introspection. If I have not the power to put myself in the place of other people, but must be continually burrowing inward, I shall never be the magnanimous creative person I wish to be. Yet I am hypnotized by the workings of the individual, alone, and am continually using myself as a specimen." Her journals are rife with her exhortations to get over herself and get on with the work beyond. "God, to lift up the lid of heads," she bemoans in 1958.
And yet despite her constant efforts to "flay" herself into the writer she knew she could be, the most fluid writings in Plath's journals are those in which she is unself-consciously subjective, getting straight to the business of telegraphing her thoughts and feelings without sculpting them into something suitable for the Saturday Evening Post, the Christian Science Monitor, or -- the twin heights of her literary Olympus -- the New Yorker and Ladies' Home Journal.
During a grim winter afternoon at Smith during her teaching year, Plath has coffee alone in the coffee shop of her youth and notices "music souping from jukebox, melancholy, embracing." On a trip to Paris in 1956, Plath writes of walking along the Seine's right bank when a masher in a "lowslung" black car "oozed alongside while he begged me to come for a ride." And three months later, on her honeymoon in Spain, every detail of her notes shimmers with sensory vividness. This makes a perplexing contrast to the handful of short stories she fretted over from that time.
A particularly terrible story idea is the one for "The Day of Twenty-four Cakes," the plot of which emerged during the weeks prior to the dread Smith teaching year, a time when Plath sensed the creative silence her return home was going to impose on her. In the breathless paragraph that outlines the story (Plath characterizes the potential audience as "Either Kafka lit-mag serious or SATEVEPOST aim high"), Plath's heroine sounds like nothing less than a naked reflection of her own desperation: "Wavering between running away or committing suicide: stayed by need to create an order: slowly, methodically begins to bake cakes, one each hour, calls store for eggs, etc. from midnight to midnight. Husband comes home: new understanding."
Plath's stilted admonishments to herself to lift up the world in tweezers and examine it from every angle, to make it "gem-like", "jewel-like", "diamond-edged," "diamond faceted," "jewelled," "gem-bright", "glittering" could not bully her work into taking on those qualities. And yet those qualities, so evident in her later poetry, were quite obviously within her grasp. Her innate gifts, ultimately imposed successfully on her poetry, do indeed exist like gems buried in their crudest form in the journals. In the unintentionally funny 1952 passage "... night thickening, congealing around her in her loneliness and longing like an imprisoning envelope of gelatin ..." one can hear the echo of 1962's "A Birthday Present," in which she repurposed the word "congeal" to much better effect:
... It breathes from my sheets, the cold dead center
Where spilt lives congeal and stiffen to history.
Perhaps the most exciting aspect of a close reading of Plath's journals is the thrill of watching the laboratory of her mind at work, watching her coax her raw materials toward their concentrated final form. And knowing that once she got her "self" going -- her electrified intellect, that piercing imagination -- that she would unleash the unstoppable poetic force of a runaway train. Yet until the point when her true self took flight in "Ariel," Plath was plagued by the "fatal" feeling that "I write as if an eye were upon me." That eye may now be ours, the audience she literally dreamed of, but while Plath was alive, the unabridged journals make agonizingly clear, the eye was her mother's.
Plath's real feelings about her mother are no longer cushioned by careful edits that subvert her sharp opinions. It is no longer a matter of Dr. Beuscher giving Plath "permission to hate your mother" or Plath admitting hatred "for ... all mother figures." Plath unhesitatingly states that she hates -- as well as pities and desires the approval of -- her mother, and in turn feels her mother's envy and lack of unconditional love. "What to do with her, with the hostility, undying, which I feel for her? I want, as ever, to grab my life from out under her hot itchy hands. My life, my writing, my husband, my unconceived baby."
Aurelia Plath had no self; she lived for and through her children. From Sylvia Plath's infancy, her primary parent's selflessness gave Plath no model for a self that could maintain its autonomy or exist beyond meeting other people's needs. What Plath had instead was one big boundariless, free-floating ego, a self utterly dependent on the inflation by the selfless parent, and all psychic roads, ultimately, led right back to Sylvia. Plath spent her entire adult life trying to trace the ego boundaries for herself that her mother neglected to impose. "She is, in many ways, like an empty vessel," Perloff said of Plath in an interview with Salon. "It's really no wonder that she erupted with all these strong feelings and reactions, the guilt and the rage and the incredible hatred that comes out, first, in 'The Bell Jar.'"
Plath understood that her mother lived vicariously through her daughter and her daughter's achievements, and that Plath's own 1953 breakdown and suicide attempt was in large part a reaction to her unhealthy "union" with her mother: "I lay in bed when I thought my mind was going blank forever and thought what a luxury it would be to kill her, to strangle her skinny veined throat which could never be big enough to protect me from the world. But I was too nice for murder. I tried to murder myself: to keep from being an embarrassment to the ones I loved and from living myself in a mindless hell ... I'd kill her, so I killed myself."
Not that critics and readers hadn't already suspected as much. In 1979 the literary critic Marjorie Perloff, author of some of the most influential articles on Plath, made the point that the shallow perfection of Plath's early work and her later metamorphosis into the writer of the inimitable "Ariel" poems was traceable to Plath's struggle to shrug off the burden of pleasing her mother, who had forfeited her own life for her two children, Sylvia and Warren. The deal, as Sylvia came to understand it, was that in return for their mother's uncomplaining slave labor -- their mother's life -- the children would feed back accomplishments. Plath became an achievement junkie, living for two and never sure of her mother's love.
Given Plath's awareness of her uncomfortable "osmosis" with her mother, it must have been horrifying for her, as Perloff points out, to realize that during the summer of 1962 "she had become ... a 'widowed' young mother with very slender financial means -- in short, she had become her mother. Even the sex of her two children -- first a girl, then a boy -- repeated the Sylvia-Warren pattern. Only now, one gathers, did Sylvia fully grasp the futility of her former goals. And so she had to destroy the 'Aurelia' in herself ... In the demonic Ariel poems, she could finally vent her anger, her hatred of men, her disappointment in life. 'Dearest Mother' now becomes the dreaded Medusa."
Her poetry leaves no doubt that Plath was indeed also obsessed with her father, but the trail of crumbs left in the journals leads elsewhere: Plath, who never failed to pointedly examine her own motivations, appears markedly resigned to her longing for her father. "My obsession with my father," she says; "it hurts, father, it hurts, oh father I have never known." You might say she "gets" her longing for her father, as she "gets" her fury at her mother.
What seems the most logical explanation for Plath's enigmatic relationship with her parents is not that one or the other was her demon, but that due to circumstance she remained psychologically dependent on and victimized by both of them. Her father's death left her not only with a hoard of unresolved grief, but it also left her defenseless against her mother's unintended vampirish harm. She had only her mother to rely on until she began a second symbiotic relationship with Hughes. Plath's depressions and rages, her restlessness and feeling of entrapment seem appropriate reactions, at least to a degree, to her family situation.
What is still hard for many of her readers to believe is that such an intuitive, perceptive and nuanced person as Sylvia Plath, who had at her disposal so many interior tools to understand her own traumas, would ultimately self-destruct. Yet the journals show, now more than ever, the extent to which she grappled helplessly with her high-strung emotional life, how tortured she was by her own intensity despite her desire to cultivate her "weirdness" and transform it into art. What is most constant about her inconstant emotions is her attempts to wrestle them down, to find a plane on which she could exist in relative psychic comfort.
There is a palpable urgency, even a poignant heroism, to Plath's mission to understand -- and to control by sheer self-discipline -- her uncontrollable moods. The 1982 journals were not lax in highlighting this theme; "God, is this all it is," Plath wrote in 1950, "the ricocheting down the corridor of laughter and tears? Of self-worship and self-loathing? Of glory and disgust?" And in 1951: "I have the choice of being constantly active and happy or introspectively passive and sad." And in 1958: "I have been, and am, battling depression. It is as if my life were magically run by two electric currents: joyous positive and despairing negative -- which ever is running at the moment dominates my life, floods it."
Numerous times after her marriage Plath warned herself to learn to manage her own emotions, to keep her problems to herself, to "not tell Ted" despite her all-consuming neediness and her sense of his soothing effect on her nerves; in the unabridged journals, ironically just a month before the disillusioning May 1958 co-ed incident, Plath wrote of Hughes, "He is ... my pole-star centering me steady & right."
Despite Plath's brittle hope that determination alone could steer her ungovernable emotions, the real key to her lifelong struggle with her mind may lie in a little-noticed medical theory -- one that does not just shed light on her poetic obsessions, but that allows us to see something few have observed in the life of this scrutinized, tortured, impossible, frighteningly brilliant writer: courage.
Part 2 of "The real Sylvia Plath": Did PMS kill Plath?
Hear Sylvia Plath read "November Graveyard" and other poems.
Hear actress Frances McDormand read from Sylvia Plath's "The Bell Jar."
Kate Moses
Kate Moses is the author of "Wintering: A Novel of Sylvia Plath" (St. Martin's.) She was the co-founder, with Camille Peri, of Salon's "Mothers Who Think" site, and she and Peri also co-edited the award-winning book "Mothers Who Think: Tales of Real-Life Parenting." She lives in San Francisco.
MORE FROM Kate Moses
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"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,003
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Q: Java scheduled job stop for sometime then fired all at once I have two classes:
*
*Main class to run a scheduled job:
public class TimerTest {
public static void main(String[] args){
try {
File logFile = new File("timer.log");
TestTimerJob job = new TestTimerJob(logFile);
Timer timer = new Timer();
timer.scheduleAtFixedRate(job, 0, Integer.parseInt(args[0]) * 60 * 1000);
} catch (Exception e) {
System.out.println(e.getMessage());
}
}
}
*Job class simply write current datetime to a file (timer.log)
public class TestTimerJob extends TimerTask {
private File logFile;
public TestTimerJob(File logFile) {
this.logFile = logFile;
}
@Override
public void run() {
try {
DateFormat df = new SimpleDateFormat("yyyy/MM/dd HH:mm:ss");
Calendar cal = Calendar.getInstance();
BufferedWriter writer = new BufferedWriter(new FileWriter(this.logFile, true));
writer.write(df.format(cal.getTime()));
writer.newLine();
writer.close();
} catch (Exception e) {
System.out.println(e.getMessage());
}
}
}
Then i run the program with command:
java -cp Test.jar test.TimerTest 1
which means schedule above job to run every 1 minute.
I let it run for about 12 hours straight and when look at timer.log i found this:
2015/04/02 07:32:17
2015/04/02 07:33:17
2015/04/02 07:34:17
2015/04/02 07:35:17
2015/04/02 07:36:17
2015/04/02 07:37:17
2015/04/02 07:38:17
2015/04/02 07:39:17
2015/04/02 07:40:17
2015/04/02 07:41:17
2015/04/02 07:42:17
2015/04/02 08:02:16
2015/04/02 08:02:16
2015/04/02 08:02:16
2015/04/02 08:02:16
2015/04/02 08:02:16
2015/04/02 08:02:16
2015/04/02 08:02:16
2015/04/02 08:02:16
2015/04/02 08:02:16
2015/04/02 08:02:16
2015/04/02 08:02:16
2015/04/02 08:02:16
2015/04/02 08:02:16
2015/04/02 08:02:16
2015/04/02 08:02:16
2015/04/02 08:02:16
2015/04/02 08:02:16
2015/04/02 08:02:16
2015/04/02 08:02:16
2015/04/02 08:02:17
2015/04/02 08:03:17
2015/04/02 08:04:17
2015/04/02 08:05:17
2015/04/02 08:06:17
2015/04/02 08:07:17
2015/04/02 08:08:17
Somehow it paused at 2015/04/02 07:42:17 then resumed at 2015/04/02 08:02:16 and run all the missfired schedules at once.
Can you explain what is this problem and how to fix it? I'm using jdk 1.6.0_34. Sorry for my bad english.
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"redpajama_set_name": "RedPajamaStackExchange"
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Ceylonhängpapegoja (Loriculus beryllinus) är en fågel i familjen östpapegojor inom ordningen papegojfåglar.
Utseende
Ceylonhängpapegoja är en liten (14 cm), grön papegojfågel. Hanen har röd näbb, röd övergump och rödfärgad hjässa. Vidare syns en orangefärgad anstrykning på manteln. Liknande blåstrupig hängpapegoja har just en blå strupe, men saknar den röda hjässan och är grön på manteln.
Utbredning och systematik
Fågeln är endemisk för Sri Lanka. Den behandlas som monotypisk, det vill säga att den inte delas in i några underarter.
Status och hot
Arten har ett rätt litet utbredningsområde, men det finns inga tecken på vare sig några substantiella hot eller att populationen minskar. Utifrån dessa kriterier kategoriserar IUCN arten som livskraftig (LC).
Noter
Externa länkar
Bilder och filmer på Internet Bird Collection
Läten på xeno-canto.org
Östpapegojor
Fåglar i orientaliska regionen
Endemiska fågelarter på Sri Lanka
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,092
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Dogs really are man's best friend and following the Kennel Club's nationwide search to find hero dogs that have had a life-changing impact on people's lives, the five finalists will be at Crufts this year. Ann Evans tells us more.
Amongst the many displays and competitions that will take place at this year's Crufts, will be the Kennel Club's Friends for Life competition which celebrates dogs that have had a major impact on people's lives.
In January the Kennel Club launched their nationwide search to find these hero dogs. The public were asked to nominate a dog that had changed the life of its owner, or another person and deserved recognition for it.
More than 300 owners entered their pets into the 2018 Crufts Hero Dog Awards telling wonderful stories of a dog's bond with its owner and its unfailing help, companionship and support to a person or people who need their help. Each of the finalists were chosen from five distinct categories and entered into a public vote to name the top hero dog.
The winner will receive £5,000 from The Kennel Club Charitable Trust for the dog charity of their choice, and other finalists will also receive £1,000 towards their own canine charities. Voting remains open until midday this Sunday, 11th March 2018.
Sir Jack Spratticus the Border Terrier and Vanessa Holbrow from Burnham on Sea, Somerset.
Waffle and Sarah Mohammadi from Hayes, West London.
Taz the Collie and Gayle Wilde from Lanarkshire.
Clare Syvertsen and Griffin from Northolt.
Buttons the Shih Tzu and Hannah Gates from High Wycombe.
Crufts takes place at the Birmingham NEC from Thursday 8th March to Sunday 11th March.
Photos are courtesy of The Kennel Club PA.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 7,976
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\section{Introduction}\label{sec_IntMR}
The Hill-Schr\"{o}dinger operators
\begin{equation*}
S(q)u:=-u''+q(x)u,\quad u\in \mathrm{Dom}\left(S(q)\right)
\end{equation*}
with real-valued 1-periodic distributional potentials $q(x)\in H_{1\mbox{-}per}^{-1}(\mathbb{R})$ are well defined on the
Hilbert space $L^{2}(\mathbb{R})$ in the following \textit{equivalent} basic ways \cite{MiMl6}:
\begin{itemize}
\item as form-sum operators;
\item as quasi-differential operators;
\item as limits of operators with smooth 1-periodic potentials in the norm resolvent sense.
\end{itemize}
The operators $S(q)$ are lower semibounded and self-adjoint on the Hilbert space $L^{2}(\mathbb{R})$. Their spectra are absolutely continuous and have a band and gap structure as in the classical case of $L_{1\mbox{-}per}^{2}(\mathbb{R})$-potentials \cite{HrMk, Krt, DjMt3, MiMl6}.
The object of our investigation is the behaviour of the lengths of spectral gaps. Under the assumption
\begin{equation}\label{eq_12_IntMR}
q(x)=\sum_{k\in \mathbb{Z}}\widehat{q}(k)e^{i k 2\pi x}\in
H_{1\mbox{-}per}^{-1+}(\mathbb{R},\mathbb{R}),
\end{equation}
that is
\begin{equation*}
\sum_{k\in \mathbb{Z}}(1+2|k|)^{2s}|\widehat{q}(k)|^{2}<\infty\quad \forall s>-1,
\quad\text{and}\quad \mathrm{Im}\,q(x)=0,
\end{equation*}
we will prove many terms asymptotic estimates for the lengths
$\{\gamma_{n}(q)\}_{n=1}^{\infty}$ and midpoints
$\{\tau_{n}(q)\}_{n=1}^{\infty}$ of spectral gaps of the
Hill-Schr\"{o}dinger operators $S(q)$ (Theorem \ref{th_12_IntMR}). These
estimates enable us to establish relationship between the rate of
\textit{decreasing}/\textit{increasing} of the lengths of spectral gaps
and the \textit{regularity} of the singular potentials (Theorem
\ref{th_18_IntMR} and Theorem~\ref{th_19_IntMR}).
It is well known that if the potentials
\begin{equation*}\label{eq_14_IntMR}
q(x)=\sum_{k\in \mathbb{Z}}\widehat{q}(k)e^{i k 2\pi x}\in L_{1\mbox{-}per}^{2}(\mathbb{R},\mathbb{R}),
\end{equation*}
i.e., when
\begin{equation*}
\sum_{k\in \mathbb{Z}}|\widehat{q}(k)|^{2}<\infty,\quad\text{and}\quad \mathrm{Im}\,q(x)=0,
\end{equation*}
then the Hill-Schr\"{o}dinger operators $S(q)$ are lower semibounded and self-adjoint operators on the Hilbert space
$L^{2}(\mathbb{R})$ with absolutely continuous spectra which have a zone structure \cite{DnSch2, ReSi4}.
Spectra $\mathrm{spec}\,(S(q))$ are defined by the location of the endpoints
$\{\lambda_{0}(q),\lambda_{n}^{\pm}(q)\}_{n=1}^{\infty}$ of spectral gaps which satisfy the following
inequalities:
\begin{equation*}
-\infty<\lambda_{0}(q)<\lambda_{1}^{-}(q)\leq\lambda_{1}^{+}(q)<
\lambda_{2}^{-}(q)\leq\lambda_{2}^{+}(q)<\cdots\,.
\end{equation*}
Moreover, for even/odd numbers $n\in \mathbb{Z}_{+}$ the endpoints of spectral gaps are eigenvalues of the
periodic/semiperiodic problems on the interval $[0,1]$:
\begin{align*}
S_{\pm}(q)u & :=-u''+q(x)u=\lambda u, \\
\mathrm{Dom}(S_{\pm}(q)) & :=\left\{u\in H^{2}[0,1]\left|\, u^{(j)}(0)=\pm\, u^{(j)}(1),\, j=0,1\right.\right\}\equiv H_{\pm}^{2}[0,1].
\end{align*}
Spectral bands (stability or tied zones),
\begin{equation*}
\mathcal{B}_{0}(q):=[\lambda_{0}(q),\lambda_{1}^{-}(q)],\qquad
\mathcal{B}_{n}(q):=[\lambda_{n}^{+}(q),\lambda_{n+1}^{-}(q)],\quad n\in
\mathbb{N},
\end{equation*}
are characterized as a locus of those real $\lambda\in \mathbb{R}$ for which all solutions of the equation $S(q)
u=\lambda u$ are bounded. On the other hand, spectral gaps (instability or forbidden zones),
\begin{equation*}
\mathcal{G}_{0}(q):=(-\infty,\lambda_{0}(q)),\qquad
\mathcal{G}_{n}(q):=(\lambda_{n}^{-}(q),\lambda_{n}^{+}(q)),\quad n\in
\mathbb{N},
\end{equation*}
are a locus of those real $\lambda\in \mathbb{R}$ for which any nontrivial solution of the equation $S(q) u=\lambda u$
is unbounded.
Due to Marchenko and Ostrovskii \cite{MrOs} the endpoints of spectral gaps of the Hill-Schr\"{o}dinger operators $S(q)$
satisfy the asymptotic estimates
\begin{equation}\label{eq_18_IntMR}
\lambda_{n}^{\pm}(q)=n^{2}\pi^{2}+\widehat{q}(0)\pm \left|\widehat{q}(n)\right|+h^{1}(n),\quad n\rightarrow\infty.
\end{equation}
As a consequence, for the lengths of spectral gaps,
\begin{equation*}
\gamma_{n}(q):=\lambda_{n}^{+}-\lambda_{n}^{-},\quad n\in \mathbb{N},
\end{equation*}
the following asymptotic formulae are fulfilled,
\begin{equation}\label{eq_20_IntMR}
\gamma_{n}(q)=2\left|\widehat{q}(n)\right|+h^{1}(n),\quad n\rightarrow\infty.
\end{equation}
Hochstadt \cite{Hchs} ($\Rightarrow$) and Marchenko, Ostrovskii \cite{MrOs}, McKean, Trubowitz \cite{McKTr} ($\Leftarrow$) proved that the potential $q(x)$ is an infinitely differentiable function if and only if the lengths of spectral gaps $\{\gamma_{n}(q)\}_{n=1}^{\infty}$ decrease faster than arbitrary power of $1/n$:
\begin{equation*}\label{eq_21.3_IntMR}
q(x)\in C_{1\mbox{-}per}^{\infty}(\mathbb{R},\mathbb{R})\Leftrightarrow
\gamma_{n}(q)=O(n^{-k}),\; n\rightarrow\infty\quad \forall k\in \mathbb{Z}_{+}.
\end{equation*}
Marchenko and Ostrovskii \cite{MrOs} discovered that
\begin{equation}\label{eq_22_IntMR}
q(x)\in H_{1\mbox{-}per}^{k}(\mathbb{R},\mathbb{R})\Leftrightarrow
\{\gamma_{n}(q)\}_{n=1}^{\infty}\in h^{k},\quad k\in \mathbb{Z}_{+}.
\end{equation}
The relationship \eqref{eq_22_IntMR} was extended by Kappeler, Mityagin \cite{KpMt} ($\Rightarrow$) and Djakov, Mityagin
\cite{DjMt1} ($\Leftarrow$) (see also the survey \cite{DjMt2} and the references therein) on the case of symmetric,
monotonic, submultiplicative and strictly subexponential weights $\Omega=\left\{\Omega(n)\right\}_{n\in \mathbb{Z}}$:
\begin{equation*}
q(x)\in H_{1\mbox{-}per}^{\Omega}(\mathbb{R},\mathbb{R})\Leftrightarrow \{\gamma_{n}(q)\}_{n=1}^{\infty}\in h^{\Omega}.
\end{equation*}
P\"{o}schel \cite{Psch} proved the latter statement in a quite different way.
Here and throughout the remainder of the paper we use the complex Hilbert spaces $H_{1\mbox{-}per}^{w}(\mathbb{R})$ (as well as $H_{\pm}^{w}[0,1]$) of 1-periodic functions and distributions defined by means of their Fourier coefficients:
\begin{align*}
f(x) & =\sum_{k\in \mathbb{Z}}\widehat{f}(k)e^{i k 2\pi x}\in H_{1\mbox{-}per}^{w}(\mathbb{R}) \Leftrightarrow \left\{\widehat{f}(k)\right\}_{k\in \mathbb{Z}}\in h^{w}, \\
h^{w} & =\left\{a=\left\{a(k)\right\}_{k\in \mathbb{Z}}\,\left|\,\|a\|_{h^{w}}=\left(\sum_{k\in \mathbb{Z}}w^{2}(k)|a(k)|^{2}\right)^{1/2}<\infty\right.\right\}.
\end{align*}
Basically we use the power weights
\begin{equation*}
w_{s}=\left\{w_{s}(k)\right\}_{k\in \mathbb{Z}}:\qquad w_{s}(k):=(1+2|k|)^{s},\quad s\in \mathbb{R}.
\end{equation*}
The corresponding spaces we denote as
\begin{equation*}
H_{1\mbox{-}per}^{w_{s}}(\mathbb{R})\equiv H_{1\mbox{-}per}^{s}(\mathbb{R}),\;
H_{\pm}^{w_{s}}[0,1]\equiv H_{\pm}^{s}[0,1],\quad\text{and}\quad h^{w_{s}}\equiv h^{s},\; s\in \mathbb{R}.
\end{equation*}
For more details, see Appendix.
\section{Main results}\label{ssec_DsPt_IntMR}
As we already remarked, under the assumption \eqref{eq_12_IntMR} the Hill-Schr\"{o}dinger operators $S(q)$ are lower semibounded and
self-adjoint on the Hilbert space $L^{2}(\mathbb{R})$. Their spectra are absolutely continuous and have a classical zone structure \cite{HrMk, Krt, DjMt3, MiMl6, MiSb}.
Using the results of the papers \cite{KpMh, Mhr}, the Isospectral Theorem \ref{th_30_Prf} and \cite[Theorem C]{MiMl6} we
prove uniform many terms asymptotic estimates for the lengths of spectral gaps $\{\gamma_{n}(q)\}_{n=1}^{\infty}$ and their
midpoints $\{\tau_{n}(q)\}_{n=1}^{\infty}$,
\begin{equation*}
\tau_{n}(q):=\frac{\lambda_{n}^{+}(q)+\lambda_{n}^{-}(q)}{2},\quad n\in \mathbb{N}.
\end{equation*}
\begin{theorem}[\cite{MiMl3, Mlb2}]\label{th_12_IntMR}
Let $q(x)\in H_{1\mbox{-}per}^{-\alpha}(\mathbb{R},\mathbb{R})$, $\alpha\in [0,1)$. Then for any $\varepsilon>0$ uniformly on the bounded sets of distributions $q(x)$ in the corresponding Sobolev spaces $H_{1\mbox{-}per}^{-\alpha}(\mathbb{R})$ the lengths
$\{\gamma_{n}(q)\}_{n=1}^{\infty}$ and midpoints $\{\tau_{n}(q)\}_{n=1}^{\infty}$ of spectral gaps of the Hill-Schr\"{o}dinger
operators $S(q)$ for $n\geq n_{0}\left(\|q\|_{H_{1\mbox{-}per}^{-\alpha}(\mathbb{R})}\right)$ satisfy the following asymptotic
formulae:
\begin{align}
& \gamma_{n}(q)=2\left|\widehat{q}(n)\right|+h^{1-2\alpha-\varepsilon}(n), \label{eq_24_IntMR} \\
& \tau_{n}(q)=n^{2}\pi^{2}+\widehat{q}(0)+h^{1-2\alpha-\varepsilon}(n). \label{eq_26_IntMR}
\end{align}
\end{theorem}
\begin{corollary*}[\cite{MiMl3, Mlb2}]\label{cr_16_IntMR}
Let $q(x)\in H_{1\mbox{-}per}^{-\alpha}(\mathbb{R},\mathbb{R})$ with $\alpha\in [0,1)$. Then for any $\varepsilon>0$ uniformly by $q(x)$ for the endpoints of spectral gaps of the Hill-Schr\"{o}dinger operators $S(q)$ the following asymptotic estimates are
fulfilled:
\begin{equation*}\label{eq_28_IntMR}
\lambda_{n}^{\pm}(q)=n^{2}\pi^{2}+\widehat{q}(0)\pm\left|\widehat{q}(n)\right|+h^{1-2\alpha-\varepsilon}(n).
\end{equation*}
\end{corollary*}
Now, we can describe a bilateral relationship between the rate of
decreasing/increasing of the lengths of spectral gaps
$\{\gamma_{n}(q)\}_{n=1}^{\infty}$ and the regularity of the
potentials $q(x)$ in the refined scale.
Let
\begin{equation*}
w_{s,\varphi}=\left\{w_{s,\varphi}(k)\right\}_{k\in \mathbb{Z}}:\qquad w_{s,\varphi}(k):=(1+2|k|)^{s}\,
\varphi(|k|),\quad s\in \mathbb{R},\; \varphi\in \mathrm{SV},
\end{equation*}
where $\varphi$ is a slowly varying on $+\infty$ in a sense of Karamata function \cite{Snt}. It means that it is a
positive, measurable on $[A,\infty)$, $A>0$ function obeying the condition
\begin{equation*}
\lim_{t\rightarrow +\infty}\frac{\varphi(\lambda t)}{\varphi(t)}=1,\quad \lambda>0.
\end{equation*}
For example,
\begin{equation*}
\varphi(t)=(\log t)^{r_{1}}(\log\log t)^{r_{2}}\ldots (\log\ldots\log t)^{r_{k}}\in \mathrm{SV},\quad
\{r_{1},\ldots,r_{k}\}\subset \mathbb{R},\; k\in \mathbb{N}.
\end{equation*}
The H\"{o}rmander spaces
\begin{equation*}
H_{1\mbox{-}per}^{w_{s,\varphi}}(\mathbb{R})\equiv H_{1\mbox{-}per}^{s,\varphi}(\mathbb{R})\simeq H^{s,\varphi}(\mathbb{S}),\quad \mathbb{S}:=\mathbb{R}/2\pi\mathbb{Z},
\end{equation*}
and the weighted sequence spaces
\begin{equation*}
h^{w_{s,\varphi}}\equiv h^{s,\varphi}
\end{equation*}
form the refined scales:
\begin{align}
H_{1\mbox{-}per}^{s+\varepsilon}(\mathbb{R})&\hookrightarrow H_{1\mbox{-}per}^{s,\varphi}(\mathbb{R})\hookrightarrow H_{1\mbox{-}per}^{s-\varepsilon}(\mathbb{R}), \label{eq_27.1_IntMR} \\
h^{s+\varepsilon}&\hookrightarrow h^{s,\varphi}\hookrightarrow h^{s-\varepsilon}, \hspace{90pt} s\in \mathbb{R},\,
\varepsilon>0,\,\varphi\in \mathrm{SV},
\label{eq_27.2_IntMR}
\end{align}
which in a general situation were studied by Mikhailets and Murach \cite{MiMr}.
The following statements show that the sequence $\{\gamma_{n}(q)\}_{n=1}^{\infty}$ has the same behaviour as the Fourier
coefficients $\{\widehat{q}(n)\}_{n=-\infty}^{\infty}$ with respect to the refined scale $\{h^{s,\varphi}\}_{s\in
\mathbb{R},\varphi\in \mathrm{SV}}$.
\begin{theorem}\label{th_18_IntMR}
Let $q(x)\in H_{1\mbox{-}per}^{-1+}(\mathbb{R},\mathbb{R})$. Then
\begin{equation*}\label{eq_30_IntMR}
q(x)\in H_{1\mbox{-}per}^{s,\varphi}\left(\mathbb{R},\mathbb{R}\right)\Leftrightarrow
\{\gamma_{n}(q)\}_{n=1}^{\infty}\in h^{s,\varphi},\quad s\in (-1,0], \varphi\in \mathrm{SV}.
\end{equation*}
\end{theorem}
Note that the H\"{o}rmander spaces $H_{1\mbox{-}per}^{s,\varphi}(\mathbb{R})$ with $\varphi\equiv 1$ coincide with the Sobolev spaces,
\begin{equation*}
H_{1\mbox{-}per}^{s,1}(\mathbb{R})\equiv H_{1\mbox{-}per}^{s}(\mathbb{R}),\quad\mbox{and}\quad h^{s,1}\equiv h^{s},\quad s\in \mathbb{R}.
\end{equation*}
\begin{corollary*}[\cite{MiMl3, Mlb2}]
Let $q(x)\in H_{1\mbox{-}per}^{-1+}(\mathbb{R},\mathbb{R})$, then
\begin{equation}\label{eq_31_IntMR}
q(x)\in H_{1\mbox{-}per}^{s}\left(\mathbb{R},\mathbb{R}\right)\Leftrightarrow
\{\gamma_{n}(q)\}_{n=1}^{\infty}\in h^{s},\quad s\in (-1,0].
\end{equation}
\end{corollary*}
Theorem \ref{th_18_IntMR} together with \cite[Theorem 1.2]{KpMt}, and the properties \eqref{eq_27.1_IntMR} and \eqref{eq_27.2_IntMR},
involve the following extension of the Marchenko-Ostrovskii Theorem \eqref{eq_22_IntMR}.
\begin{theorem}\label{th_19_IntMR}
Let $q(x)\in H_{1\mbox{-}per}^{-1+}(\mathbb{R},\mathbb{R})$. Then
\begin{equation*}
q(x)\in H_{1\mbox{-}per}^{s,\varphi}\left(\mathbb{R},\mathbb{R}\right)\Leftrightarrow
\{\gamma_{n}(q)\}_{n=1}^{\infty}\in h^{s,\varphi},\quad s\in (-1,\infty), \varphi\in \mathrm{SV}.
\end{equation*}
In particular,
\begin{equation*}
q(x)\in H_{1\mbox{-}per}^{s}\left(\mathbb{R},\mathbb{R}\right)\Leftrightarrow
\{\gamma_{n}(q)\}_{n=1}^{\infty}\in h^{s},\quad s\in (-1,\infty).
\end{equation*}
\end{theorem}
\begin{remark*}\label{rm_20_IntMR}
In the preprint \cite{DjMt3} the authors announced without a proof the more general statement:
\begin{equation*}
q(x)\in H_{1\mbox{-}per}^{\widehat{\Omega}}(\mathbb{R},\mathbb{R})\Leftrightarrow
\{\gamma_{n}(q)\}_{n=1}^{\infty}\in h^{\widehat{\Omega}},\quad
\widehat{\Omega}=\left\{\frac{\Omega(n)}{1+2|n|}\right\}_{n\in \mathbb{Z}},
\end{equation*}
where the weights $\Omega=\{\Omega(n)\}_{n\in \mathbb{Z}}$ are supposed to be symmetric, monotonic, submultiplicative and
strictly subexponential ones. This result contains the limiting case
\begin{equation*}
q(x)\in H_{1\mbox{-}per}^{-1}\left(\mathbb{R},\mathbb{R}\right)\setminus H_{1\mbox{-}per}^{-1+}\left(\mathbb{R},\mathbb{R}\right).
\end{equation*}
An implication
\begin{equation*}
q(x)\in H_{1\mbox{-}per}^{-1}\left(\mathbb{R},\mathbb{R}\right)\Rightarrow
\{\gamma_{n}(q)\}_{n=1}^{\infty}\in h^{-1}
\end{equation*}
was proved in the paper \cite{Krt}.
\end{remark*}
\section{Proofs}\label{sec_Prf}
Spectra of the Hill-Schr\"{o}dinger operators $S(q)$, $q(x)\in
H_{1\mbox{-}per}^{-1}\left(\mathbb{R},\mathbb{R}\right)$ are defined by the endpoints
$\{\lambda_{0}(q),\lambda_{n}^{\pm}(q)\}_{n=1}^{\infty}$ of spectral gaps. The endpoints as in the case of
$L_{1\mbox{-}per}^{2}(\mathbb{R})$-potentials satisfy the inequalities:
\begin{equation*}
-\infty<\lambda_{0}(q)<\lambda_{1}^{-}(q)\leq\lambda_{1}^{+}(q)<
\lambda_{2}^{-}(q)\leq\lambda_{2}^{+}(q)<\cdots\,.
\end{equation*}
For even/odd numbers $n\in \mathbb{Z}_{+}$ they are eigenvalues of the periodic/semiperiodic problems on the interval
$[0,1]$ \cite[Theorem C]{MiMl6},
\begin{equation*}
S_{\pm}(q)u=\lambda u.
\end{equation*}
The operators
\begin{align*}
& \hspace{15pt} S_{\pm}u\equiv S_{\pm}(q)u:=D_{\pm}^{2}u+q(x)u, \hspace{125pt}\mbox{} \\
& \bullet\; D_{\pm}^{2}:=-d^{2}/dx^{2},\; \mathrm{Dom}\,(D_{\pm}^{2})=H_{\pm}^{2}[0,1]; \\
& \bullet\; q(x)=\sum_{k\in \mathbb{Z}}\widehat{q}(k)\,e^{i\,k 2\pi x}\in H^{-1}_{+}\left([0,1],\mathbb{R}\right); \\
& \bullet\; \mathrm{Dom}\left(S_{\pm}(q)\right)=\left\{u\in H_{\pm}^{1}[0,1]\,\left|\, D_{\pm}^{2}u+q(x)u
\in L^{2}(0,1)\right.\right\},
\end{align*}
are well defined on the Hilbert space $L^{2}(0,1)$ as lower semibounded, self-adjoint form-sum operators, and they
have the pure discrete spectra
\begin{equation*}
\mathrm{spec}\left(S_{\pm}(q)\right)=\left\{\lambda_{0}[S_{+}(q)],\; \lambda_{2n-1}^{\pm}[S_{-}(q)],\;
\lambda_{2n}^{\pm}[S_{+}(q)]\right\}_{n=1}^{\infty}.
\end{equation*}
In the papers \cite{MiMl3, Mlb2, MiMl4, MiMl5} the authors meticulously investigated the more general
periodic/semiperiodic form-sum operators
\begin{equation*}
S_{m,\pm}(V):=D_{\pm}^{2m}\dotplus V(x),\quad V(x)\in H_{+}^{-m}[0,1],\; m\in \mathbb{N},
\end{equation*}
on the Hilbert space $L^{2}(0,1)$.
So, we need to find precise asymptotic estimates for the operators $S_{\pm}(q)$ eigenvalues. It is quite difficult problem
as the form-sum operators $S_{\pm}(q)$ are not convenient for investigation. We also cannot apply
approach developed by Savchuk and Shkalikov (see the survey \cite{SvSh} and the references therein) considering the
operators $S_{\pm}(q)$ as quasi-differential ones as the periodic/semiperiodic boundary conditions are not strongly
regular by Birkhoff. Therefore we propose an alternative approach which is based on isospectral transformation of the problem.
Kappeler and M\"{o}hr \cite{KpMh, Mhr} investigated the second order differential operators $L_{\pm}(q)$, $q(x)\in
H_{+}^{-1}\left([0,1],\mathbb{R}\right)$ (in general, with complex-valued potentials) defined on the \textit{negative}
Sobolev spaces $H_{\pm}^{-1}[0,1]$,
\begin{equation*}
L_{\pm}\equiv L_{\pm}(q):=D_{\pm}^{2}+q(x),\quad \mathrm{Dom}\,(L_{\pm}(q))=H_{\pm}^{1}[0,1].
\end{equation*}
They established that the operators $L_{\pm}(q)$ with $q(x)\in H_{+}^{-\alpha}\left([0,1],\mathbb{R}\right)$,
$\alpha\in [0,1)$ have the real-valued discrete spectra
\begin{equation*}
\mathrm{spec}\left(L_{\pm}(q)\right)=\left\{\lambda_{0}[L_{+}(q)],\; \lambda_{2n-1}^{\pm}[L_{-}(q)],\;
\lambda_{2n}^{\pm}[L_{+}(q)]\right\}_{n=1}^{\infty}
\end{equation*}
such that
\begin{equation*}
\left|\lambda_{n}^{\pm}[L_{\pm}(q)]-n^{2}\pi^{2}-\widehat{q}(0)\right|\leq C n^{\alpha},
\quad n\geq n_{0}\left(\|q\|_{H_{+}^{-\alpha}[0,1]}\right).
\end{equation*}
More precisely, for the values
\begin{align*}
\gamma_{n}[L_{\pm}(q)] & :=\lambda_{n}^{+}[L_{\pm}(q)]-\lambda_{n}^{-}[L_{\pm}(q)],\quad n\in \mathbb{N}, \\
\tau_{n}[L_{\pm}(q)] & :=\frac{\lambda_{n}^{+}[L_{\pm}(q)]+\lambda_{n}^{-}[L_{\pm}(q)]}{2},\quad n\in \mathbb{N}
\end{align*}
they proved the following result.
\begin{proposition}[Kappeler, M\"{o}hr \cite{KpMh, Mhr}]\label{pr_26_Prf}
Let $q(x)\in H_{+}^{-\alpha}\left([0,1],\mathbb{R}\right)$, and $\alpha\in [0,1)$. Then for any $\varepsilon>0$ uniformly on the bounded sets
of distributions $q(x)$ in the Sobolev spaces $H_{+}^{-\alpha}[0,1]$ for the operators $L_{\pm}(q)$ values
$\left\{\gamma_{n}[L_{\pm}(q)]\right\}_{n=1}^{\infty}$ and $\left\{\tau_{n}[L_{\pm}(q)]\right\}_{n=1}^{\infty}$ for
$n\geq n_{0}\left(\|q\|_{H_{+}^{-\alpha}[0,1]}\right)$ the following asymptotic estimates are fulfilled:
\begin{align*}
i) \hspace{15pt} & \left\{\min_{\pm}\left|\gamma_{n}[L_{\pm}(q)]\pm 2\sqrt{\left(\widehat{q}+
\omega\left)(-n)\right(\widehat{q}+\omega\right)(n)}\right|\right\}_{n\in \mathbb{N}}
\in h^{1-2\alpha-\varepsilon}, \hspace{70pt}\mbox{} \\
ii) \hspace{15pt} & \tau_{n}[L_{\pm}(q)]=n^{2}\pi^{2}+\widehat{q}(0)+h^{1-2\alpha-\varepsilon}(n),
\end{align*}
where the convolution
\begin{equation*}
\left\{\omega(n)\right\}_{n\in \mathbb{Z}}\equiv \left\{\frac{1}{\pi^{2}}\sum_{k\in \mathbb{Z}\setminus\{\pm n\}}
\frac{\widehat{q}\,(n-k)\widehat{q}(n+k)}{n^{2}-k^{2}}\right\}_{n\in \mathbb{Z}}\in
\begin{cases}
h^{1-\alpha}, & \alpha\in [0,1/2), \\
h^{3/2-2\alpha-\delta}, & \alpha\in [1/2,1)
\end{cases}
\end{equation*}
with any $\delta>0$ (see the Convolution Lemma \cite{KpMh, Mhr}).
\end{proposition}
\begin{remark*}\label{rm_28_Prf}
In the papers \cite{Mlb1, MiMl1, MiMl2, Mlb2} more general operators
\begin{equation*}
L_{m,\pm}(V):=D_{\pm}^{2m}+ V(x),\quad V(x)\in H_{+}^{-m}[0,1],\; m\in \mathbb{N}
\end{equation*}
on the spaces $H_{\pm}^{-m}[0,1]$ were studied. In particular, the analogue of Proposition \ref{pr_26_Prf} was proved.
\end{remark*}
The following statement is an essential point of our approach.
\begin{theorem}[Isospectral Theorem \cite{MiMl3, Mlb2}]\label{th_30_Prf}
The operators $S_{\pm}(q)$ and $L_{\pm}(q)$ are isospectral ones:
\begin{equation*}
\mathrm{spec}\left(S_{\pm}(q)\right)=\mathrm{spec}\left(L_{\pm}(q)\right).
\end{equation*}
\end{theorem}
\begin{proof}
The injections
\begin{equation*}
\mathrm{spec}\left(S_{\pm}(q)\right)\subset \mathrm{spec}\left(L_{\pm}(q)\right)
\end{equation*}
are obvious since
\begin{equation*}
S_{\pm}(q)\subset L_{\pm}(q).
\end{equation*}
Let prove the inverse injections
\begin{equation*}
\mathrm{spec}\left(L_{\pm}(q)\right)\subset
\mathrm{spec}\left(S_{\pm}(q)\right).
\end{equation*}
Let $\lambda\in \mathrm{spec}\left(L_{\pm}(q)\right)$, and $f$ be a correspondent eigenvector or rootvector. Therefore
\begin{equation*}
\left(L_{\pm}(q)-\lambda Id\right)f=g,\quad f,g\in
\mathrm{Dom}\left(L_{\pm}(q)\right)=H_{\pm}^{1}[0,1],
\end{equation*}
where $f$ is an eigenfunction if $g=0$, and a rootvector if $g\neq 0$.
So, we have got
\begin{equation*}
L_{\pm}(q)f=\lambda Id f+g\in H_{\pm}^{1}[0,1],
\end{equation*}
i.e.,
\begin{equation*}
L_{\pm}(q)f=D_{\pm}^{2}f+q(x)f\in L^{2}(0,1).
\end{equation*}
Thus we have proved that $f\in \mathrm{Dom}\left(S_{\pm}(q)\right)$. In the case when $f$ is a rootvector ($g\neq
0$) in a similar fashion we show that $g\in \mathrm{Dom}\left(S_{\pm}(q)\right)$ also. Continuing this process until necessary (draw attention that it is finite as the eigenvalue $\lambda$ has a finite algebraic multiplicity) we
obtain that all correspondent to $\lambda$ eigenvectors and rootvectors belong to the operators $S_{\pm}(q)$ domains
$\mathrm{Dom}\left(S_{\pm}(q)\right)$. Consequently we can conclude that
\begin{equation*}
\lambda\in \mathrm{spec}\left(S_{\pm}(q)\right),
\end{equation*}
and the required injections
\begin{equation*}
\mathrm{spec}\left(L_{\pm}(q)\right)\subset \mathrm{spec}\left(S_{\pm}(q)\right)
\end{equation*}
have been proved.
The proof is complete.
\end{proof}
Now, Theorem \ref{th_12_IntMR} follows from Proposition \ref{pr_26_Prf}, the Isospectral Theorem \ref{th_30_Prf} and
\cite[Theorem~C]{MiMl6}, since
\begin{align*}
\widehat{q}(n) & =\overline{\widehat{q}(-n)},\quad n\in \mathbb{Z}, \\
\omega(n) & =\overline{\omega(-n)},\quad n\in \mathbb{Z},
\end{align*}
and as a consequence
\begin{equation*}
\min_{\pm}\left|\gamma_{n}(q)\pm 2\sqrt{\left(\widehat{q}+\omega\left)(-n)\right(\widehat{q}+
\omega\right)(n)}\right|=\left|\gamma_{n}(q)-2\left|\left(\widehat{q}+\omega\right)(n)\right|\right|.
\end{equation*}
The proof of Theorem \ref{th_12_IntMR} is complete.
To prove Theorem \ref{th_18_IntMR} we firstly prove its Corollary. The formula \eqref{eq_31_IntMR} follows from \cite[Corollary 0.2 (2.6)]{KpMh}, the Isospectral Theorem \ref{th_30_Prf} and \cite[Theorem C]{MiMl6}. Also it can be proved directly as well as \cite[Corollary~0.2~(2.6)]{KpMh} using the estimates \eqref{eq_24_IntMR}.
Further, to prove Theorem \ref{th_18_IntMR} it is sufficient to apply the asymptotic estimates \eqref{eq_24_IntMR}, the
properties \eqref{eq_27.1_IntMR} and \eqref{eq_27.2_IntMR} of the refined scales, and the formula \eqref{eq_31_IntMR}:
\begin{align*}
&\mbox{\textbullet}\; & q\in H_{1\mbox{-}per}^{s,\varphi}\left(\mathbb{R},\mathbb{R}\right)
&\overset{\eqref{eq_27.1_IntMR}}{\Longrightarrow} q\in
H_{1\mbox{-}per}^{s-\delta}\left(\mathbb{R},\mathbb{R}\right),\;\delta>0\overset{\eqref{eq_24_IntMR}}{\Longrightarrow}
\gamma_{n}=2\left|\widehat{q}(n)\right|+h^{1+2(s-\delta)-\varepsilon}(n) \\
&&& \overset{\eqref{eq_27.2_IntMR}}{\Longrightarrow}\gamma_{n}=2\left|\widehat{q}(n)\right|+h^{s,\varphi}(n)\Longrightarrow
\{\gamma_{n}(q)\}_{n=1}^{\infty}\in h^{s,\varphi}; \\
&\mbox{\textbullet}\; & \{\gamma_{n}(q)\}_{n=1}^{\infty}\in h^{s,\varphi}&\overset{\eqref{eq_27.2_IntMR}}{\Longrightarrow}
\{\gamma_{n}\}_{n=1}^{\infty}\in h^{s-\delta},\;\delta>0\overset{\eqref{eq_31_IntMR}}{\Longrightarrow}
q\in H_{1\mbox{-}per}^{s-\delta}\left(\mathbb{R},\mathbb{R}\right) \\
&&& \overset{\eqref{eq_24_IntMR}}{\Longrightarrow} \gamma_{n}=2\left|\widehat{q}(n)\right|+h^{1+2(s-\delta)-\varepsilon}(n)
\overset{\eqref{eq_27.2_IntMR}}{\Longrightarrow}\gamma_{n}=2\left|\widehat{q}(n)\right|+h^{s,\varphi}(n) \\
&&& \Longrightarrow \{\widehat{q}(n)\}_{n\in \mathbb{Z}}\in h^{s,\varphi}(n).
\end{align*}
Remark that due to arbitrary choice of $\delta>0$ and $\varepsilon>0$ we may choose them such that
\begin{equation*}
1+s-2\delta-\varepsilon>0.
\end{equation*}
The proof of Theorem \ref{th_18_IntMR} is complete.
Now, we are ready to prove Theorem \ref{th_19_IntMR}.
At first, note that from \cite[Theorem 1.2]{KpMt} we get the following asymptotic formulae for the lengths of spectral gaps:
\begin{equation}\label{eq_40_Prf}
\gamma_{n}(q)=2\left|\widehat{q}(n)\right|+h^{1+s}(n)\quad\text{as}\quad q(x)\in
H_{1\mbox{-}per}^{s}\left(\mathbb{R},\mathbb{R}\right),\;s\in [0,\infty),
\end{equation}
which for the integer numbers $s\in \mathbb{Z}_{+}$ were proved by Marchenko and Ostrovskii \cite{MrOs}.
Using \eqref{eq_31_IntMR}, \eqref{eq_40_Prf} and \eqref{eq_22_IntMR} it is easy to prove the relationship
\begin{equation}\label{eq_42_Prf}
q(x)\in H_{1\mbox{-}per}^{s}\left(\mathbb{R},\mathbb{R}\right)\Leftrightarrow
\{\gamma_{n}(q)\}_{n=1}^{\infty}\in h^{s},\quad s\in (-1,\infty).
\end{equation}
\textit{Sufficiency} of Theorem \ref{th_19_IntMR}. Let $q(x)\in
H_{1\mbox{-}per}^{s,\varphi}\left(\mathbb{R},\mathbb{R}\right)$. If $s\in (-1,0]$ then due to Theorem \ref{th_18_IntMR} we
obtain that $ \{\gamma_{n}(q)\}_{n=1}^{\infty}\in h^{s,\varphi}$. If $s>0$ then
\begin{align*}
q(x)\in H_{1\mbox{-}per}^{s,\varphi}\left(\mathbb{R},\mathbb{R}\right)&\overset{\eqref{eq_27.1_IntMR}}{\hookrightarrow}
H_{1\mbox{-}per}^{s-\delta}\left(\mathbb{R},\mathbb{R}\right),\;\delta>0\overset{\eqref{eq_40_Prf}}{\Longrightarrow}
\gamma_{n}(q)=2\left|\widehat{q}(n)\right|+h^{1+s-\delta}(n) \\
& \overset{\eqref{eq_27.2_IntMR}}{\Longrightarrow}\gamma_{n}(q)=2\left|\widehat{q}(n)\right|+h^{s,\varphi}(n)
\Longrightarrow \{\gamma_{n}(q)\}_{n=1}^{\infty}\in h^{s,\varphi}.\hspace{50pt}\mbox{}
\end{align*}
Sufficiency is proved.
\textit{Necessity} of Theorem \ref{th_19_IntMR}. Let suppose that $\{\gamma_{n}(q)\}_{n=1}^{\infty}\in h^{s,\varphi}$. If
$s\in (-1,0]$ then from Theorem \ref{th_18_IntMR} we have that $q(x)\in
H_{1\mbox{-}per}^{s,\varphi}\left(\mathbb{R},\mathbb{R}\right)$. If $s>0$ then
\begin{align*}
\{\gamma_{n}(q)\}_{n=1}^{\infty}\in h^{s,\varphi}&\overset{\eqref{eq_27.2_IntMR}}{\hookrightarrow}h^{s-\delta},\;\delta>0
\overset{\eqref{eq_42_Prf}}{\Longrightarrow} q(x)\in H_{1\mbox{-}per}^{s-\delta}\left(\mathbb{R},\mathbb{R}\right)
\overset{\eqref{eq_40_Prf}}{\Longrightarrow}\gamma_{n}(q)=2\left|\widehat{q}(n)\right|+h^{1+s-\delta}(n) \\
& \overset{\eqref{eq_27.2_IntMR}}{\Longrightarrow}\gamma_{n}(q)=2\left|\widehat{q}(n)\right|+h^{s,\varphi}(n)
\Longrightarrow q(x)\in H_{1\mbox{-}per}^{s,\varphi}\left(\mathbb{R},\mathbb{R}\right).
\end{align*}
Necessity is proved.
The proof of Theorem \ref{th_19_IntMR} is complete.
\section{Concluding remarks}
In fact we can prove the following result: if $q(x)\in H_{1\mbox{-}per}^{-1+}\left(\mathbb{R},\mathbb{R}\right)$
and
\begin{align*}
(1+2|k|)^{s}& \ll w(k)\ll (1+2|k|)^{1+2s},\quad s\in (-1,0], \\
(1+2|k|)^{s}& \ll w(k)\ll (1+2|k|)^{1+s},\quad s\in [0,\infty),
\end{align*}
then
\begin{equation*}
q(x)\in H_{1\mbox{-}per}^{w}\left(\mathbb{R},\mathbb{R}\right)\Leftrightarrow
\{\gamma_{n}(q)\}_{n=1}^{\infty}\in h^{w}.
\end{equation*}
This result is not covered by the theorems of the preprint
\cite{DjMt3}, because it does not require the weight function to
be monotonic and submultiplicative.
\section*{Appendix}
The complex Sobolev spaces $H_{1\mbox{-}per}^{s}(\mathbb{R})$, $s\in \mathbb{R}$ of 1-periodic func\-tions and dis\-tri\-butions
over the real axis $\mathbb{R}$ are defined by means of their Fourier coefficients,
\begin{align*}
H_{1\mbox{-}per}^{s}(\mathbb{R}) & :=\left\{f=\sum_{k\in \mathbb{Z}}\widehat{f}(k)e^{i k2\pi
x}\left|\;\parallel f\parallel_{H_{1\mbox{-}per}^{s}(\mathbb{R})}<\infty\right.\right\}, \\
\parallel f\parallel_{H_{1\mbox{-}per}^{s}(\mathbb{R})} & :=\left(\sum_{k\in \mathbb{Z}}
\langle 2k\rangle^{2s}|\widehat{f}(k)|^{2}\right)^{1/2},\quad \langle k\rangle:=1+|k|, \\
\widehat{f}(k) & :=\langle f,e^{i k2\pi x}\rangle_{L_{1\mbox{-}per}^{2}(\mathbb{R})},\quad k\in \mathbb{Z}.
\end{align*}
By $\langle \cdot,\cdot\rangle_{L_{1\mbox{-}per}^{2}(\mathbb{R})}$
we denote a sesqui-linear form pairing the dual spaces
$H_{1\mbox{-}per}^{s}(\mathbb{R})$ and
$H_{1\mbox{-}per}^{-s}(\mathbb{R})$ with respect to $L_{1\mbox{-}per}^{2}(\mathbb{R})$, which (the sequi-linear form
$\langle \cdot,\cdot\rangle_{L_{1\mbox{-}per}^{2}(\mathbb{R})}$) is
an extension by continuity of the
$L_{1\mbox{-}per}^{2}(\mathbb{R})$-inner product \cite{Brz, GrGr}:
\begin{equation*}
\langle f,g\rangle_{L_{1\mbox{-}per}^{2}(\mathbb{R})}:=\int_{0}^{1}f(x)\overline{g(x)}\,dx
=\sum_{k\in \mathbb{Z}}\widehat{f}(k)\overline{\widehat{g}(k)}\quad
\forall f,g\in L_{1\mbox{-}per}^{2}(\mathbb{R}).
\end{equation*}
It is useful to notice that
\begin{equation*}
H_{1\mbox{-}per}^{0}(\mathbb{R})\equiv L_{1\mbox{-}per}^{2}(\mathbb{R}).
\end{equation*}
By $H_{1\mbox{-}per}^{s+}(\mathbb{R})$, $s\in \mathbb{R}$ we denote an inductive limit of the Sobolev spaces
$H_{1\mbox{-}per}^{t}(\mathbb{R})$ with $t>s$,
\begin{equation*}
H_{1\mbox{-}per}^{s+}\left(\mathbb{R}\right):=\bigcup_{\varepsilon>0}H_{1\mbox{-}per}^{s+\varepsilon}\left(\mathbb{R}\right).
\end{equation*}
It is a topological space with an inductive topology.
In a similar fashion the Sobolev spaces $H_{\pm}^{s}[0,1]$, $s\in \mathbb{R}$ of
1-periodic/1-semiperiodic functions and distributions over the interval $[0,1]$ are defined:
\begin{align*}
H_{\pm}^{s}[0,1] & :=\left\{f=\sum_{k\in \Gamma_{\pm}}\widehat{f}\left(\frac{k}{2}\right)e^{i k\pi
x}\left|\;\parallel f\parallel_{H_{\pm}^{s}[0,1]}<\infty\right.\right\}, \\
\parallel f\parallel_{H_{\pm}^{s}[0,1]}&:=\left(\sum_{k\in \Gamma_{\pm}}
\langle k\rangle^{2s}\left|\widehat{f}\left(\frac{k}{2}\right)\right|^{2}\right)^{1/2},\quad \langle k\rangle=1+|k|, \\
\widehat{f}\left(\frac{k}{2}\right) &:=\langle f(x),e^{i k\pi x}\rangle_{\pm}, \quad k\in \Gamma_{\pm}.
\end{align*}
Here
\begin{align*}
\Gamma_{+} \equiv 2\mathbb{Z} & :=\left\{ k\in\mathbb{Z}\; \left|\;k\equiv 0\; (\mathrm{mod}\,2)\right.\right\}, \\
\Gamma_{-} \equiv 2\mathbb{Z}+1 & :=\left\{ k\in\mathbb{Z}\; \left|\;k\equiv 1\; (\mathrm{mod}\,2)\right.\right\},
\end{align*}
and $\langle\cdot,\cdot\rangle_{\pm}$ are sesqui-linear forms pairing the dual spaces
$H_{\pm}^{s}[0,1]$ and $H_{\pm}^{-s}[0,1]$ with respect to $L^{2}(0,1)$, which (the sesqui-linear forms $\langle\cdot,\cdot\rangle_{\pm}$) are
extensions by continuity of the $L^{2}(0,1)$-inner product \cite{Brz, GrGr}:
\begin{equation*}
\langle f,g\rangle_{\pm}:=\int_{0}^{1}f(x)\overline{g(x)}\,dx =\sum_{k\in
\Gamma_{\pm}}\widehat{f}\left(\frac{k}{2}\right)\overline{\widehat{g}\left(\frac{k}{2}\right)}\quad \forall f,g\in L^{2}(0,1).
\end{equation*}
It is obvious that
\begin{equation*}
H_{+}^{0}[0,1]\equiv H_{-}^{0}[0,1]\equiv L^{2}(0,1).
\end{equation*}
We say that 1-periodic function or distribution $f(x)$ is \textit{real-valued} if $\mathrm{Im}\,f(x)=0$. Let us remind
that
\begin{equation*}
\mathrm{Re}\,f(x):=\frac{1}{2}(f(x)+\overline{f(x)}),\quad
\mathrm{Im}\,f(x):=\frac{1}{2i}(f(x)-\overline{f(x)}),
\end{equation*}
(see, for an example, \cite{Vld}). In terms of the Fourier coefficients we have
\begin{equation*}
\mathrm{Im}\,f(x)=0\Leftrightarrow\widehat{f}(k)=\overline{\widehat{f}(-k)},\quad k\in \mathbb{Z}.
\end{equation*}
Set
\begin{align*}
H_{1\mbox{-}per}^{s}(\mathbb{R},\mathbb{R}) & :=\left\{f(x)\in H_{1\mbox{-}per}^{s}(\mathbb{R}) \left|\,\mathrm{Im}\,f(x)=0
\right.\right\}, \\
H_{1\mbox{-}per}^{s+}(\mathbb{R},\mathbb{R}) & :=\left\{f(x)\in H_{1\mbox{-}per}^{s+}(\mathbb{R}) \left|\,\mathrm{Im}\,f(x)=0
\right.\right\}, \\
H_{\pm}^{s}\left([0,1],\mathbb{R}\right) & :=\left\{f(x)\in H_{\pm}^{s}[0,1] \left|\,\mathrm{Im}\,f(x)=0
\right.\right\}.
\end{align*}
Also we will need the Hilbert sequence spaces
\begin{equation*}
h^{s}\equiv h^{s}\left(\mathbb{Z},\mathbb{C}\right),\quad s\in \mathbb{R}
\end{equation*}
of (two-sided) weighted sequences,
\begin{align*}
h^{s} & :=\left\{a=\left\{a(k)\right\}_{k\in \mathbb{Z}}\,\left|\,\|a\|_{h^{s}}:=
\left(\sum_{k\in \mathbb{Z}}\langle k\rangle^{2s}|a(k)|^{2}\right)^{1/2}<\infty\right.\right\},\quad \langle
k\rangle=1+|k|.
\end{align*}
Note that
\begin{equation*}
h^{0}\equiv l^{2}\left(\mathbb{Z},\mathbb{C}\right),
\end{equation*}
and
\begin{equation*}
a=\left\{a(k)\right\}_{k\in \mathbb{Z}}\in h^{s},\;s\in \mathbb{R}\qquad\Rightarrow\qquad a(k)=o(|k|^{-s}),
\quad k\rightarrow \pm\infty.
\end{equation*}
|
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}
| 9,097
|
New York State Thruway (I-90)
To Interstate-81 South
To Interstate-88 East
Exit 15 off of I-88
Left onto Foster/Lettis Highway. It will cross Main Street at its first major intersection, becoming Maple Street.
At the first light on Maple Street, make a left onto Center Street. At the end of Center Street, make a right onto West Street.
Turn right at the SUNY Oneonta sign onto Ravine Parkway and into the campus and look for the sign for the Welcome Center.
FROM THE CAPITAL DISTRICT AND NORTH
New York State Thruway (I-90) West to Exit 25A.
Take I-88 West to Exit 15. Make a right at the end of ramp onto Foster/Lettis Highway.
As the Foster/Lettis Highway crosses Main Street at its first major intersection, it becomes Maple Street in residential Oneonta.
At the first light on Maple Street, make a left onto Center Street.
At the end of Center Street, make a right onto West Street.
Allow about 90 minutes from Albany
Take Route 23 West to Oneonta.
At the first light past the Southside Mall, make a right onto Foster/Lettis Highway.
FROM THE SOUTH AND POINTS EAST
New York State Thruway - 87 (toll) or the Palisades Parkway and New York Routes 6 and 17.
We assume you will know what routes to take to get to either of these two points. Either route will bring you to the Harriman exit where the choice is yours:
New York State Thruway route
Continue north to the Catskill exit. Look for signs for New York Route 23 west and stay on that until you reach Oneonta (about 90 minutes). At the first light past the Southside Mall, make a right onto Foster/Lettis Highway.
Alternate route: Route 17 to Roscoe and 50 miles to Oneonta
This is a tricky route to Oneonta but probably the most popular with our students.
Take Route 17 West to Roscoe (Exit 94) and make a left at the end of the exit ramp.
You will be on NY Route 206 all the way but may notice signs indicating New York Route 30 and County Routes 91 (Sullivan) and 7 (Delaware). Don't be alarmed; this is O.K.
Continue on Route 206 to Walton. Turn right at the first caution light and continue to next light. Turn left onto Route 206 (now also NY Route 10) and within 40 YARDS be ready to make a quick right at the next light onto Townsend St. At this point you have left Route 206. Continue to the end of the road and make a left on your way to Franklin.
You will be on County Route 21 (note: county signs are much smaller in size and not easily seen.) Approximately 1 mile outside of Walton, the road divides...stay right.
Entering Franklin, you will make a right turn off of County 21 onto NY Route 357 and on into the village of Franklin.
Route 357 brings you to New York Route 28. Turn left and you're almost with us. Follow Route 28 to Oneonta and follow the signs to campus.
|
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| 2,592
|
{"url":"https:\/\/mathoverflow.net\/questions\/243232\/stabilizers-of-pairs-of-ternary-quadratic-forms","text":"# Stabilizers of pairs of ternary quadratic forms\n\nLet $A,B$ be two ternary quadratic forms with real coefficients, given by symmetric matrices\n\n$$\\displaystyle 2A = \\begin{pmatrix} 2a_{11} & a_{12} & a_{13} \\\\ a_{12} & 2a_{22} & a_{23} \\\\ a_{13} & a_{23} & 2a_{33} \\end{pmatrix}, 2B = \\begin{pmatrix} 2b_{11} & b_{12} & b_{13} \\\\ b_{12} & 2b_{22} & b_{23} \\\\ b_{13} & b_{23} & 2b_{33} \\end{pmatrix}.$$\n\nLet $V_\\mathbb{R}$ denote the 12 dimensional real vector space of $(A,B)$ over $\\mathbb{R}$, and let $G(\\mathbb{R}) = \\operatorname{GL}_2(\\mathbb{R}) \\times \\operatorname{SL}_3(\\mathbb{R})$. Let $(r,g)$ be an element of $G(\\mathbb{R})$, where\n\n$$\\displaystyle r = \\begin{pmatrix} r_1 & r_2 \\\\ r_3 & r_4 \\end{pmatrix}.$$\n\nThen $(r,g)$ acts on $(A,B) \\in V_\\mathbb{R}$ by sending $(A,B)$ to $$(r_1 (gAg^T) + r_2 (gBg^T), r_3 (gAg^T) + r_4 (gBg^T)).$$\n\nBhargava, in his paper The density of discriminants of quartic rings and fields, stated that the stabilizer in $G(\\mathbb{R})$ of an element $(A,B) \\in V_\\mathbb{R}$ has order 24 if $A,B$ have four common real zeroes over $\\mathbb{P}^2$, order 8 if they have exactly one pair of real zeroes over $\\mathbb{P}^2$, and order 4 if they have no common real zeroes. He simply said that \"one easily checks\" that this is the case. Can anyone give an explanation as to why this should be easy to see, and a proof of why it's true?\n\nThe symmetry groups are taken modulo $\\{ \\pm 1 \\}$, so we work projectively in ${\\rm PGL}_2({\\bf R}) \\times {\\rm SL}_3({\\bf R})$. We must assume that we're in the generic case that $A,B$ span a two-dimensional space of conics that vanish on four distinct points of ${\\bf P}^2$ in general linear position (i.e., no three on a line). Then over ${\\bf C}$ the stabilizer is always $S_4$, because it is known (and easy) that ${\\bf PGL}_3$ acts simply-transitively on ordered four-tuples in general linear position. But over ${\\bf R}$ we must use permutations that commute with complex conjugation. This conjugation acts on the four points as the identity, a simple transposition, or a double transposition according as four, two, or none of them are real. The commutators have orders $24$, $4$, and $8$ respectively, as claimed.","date":"2022-07-05 03:55:20","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9049572944641113, \"perplexity\": 130.3073204936543}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656104512702.80\/warc\/CC-MAIN-20220705022909-20220705052909-00120.warc.gz\"}"}
| null | null |
Carmakers Refrain From Price Hike Despite Surge In Key Input Costs
May 15,2018 Read News PRINT
May 15,2018 Read News
Car buyers have a reason to cheer — at least temporarily. In a departure from the previous trend, majority of passenger vehicle makers have refrained from going for an immediate price hike even as prices of key raw materials including steel, aluminium, copper, have risen sharply in the recent past. This, despite most companies seeing a good volume run, which in turn give them enough power to hike prices.
Passenger vehicles sales in India rose 7.8 per cent to 3.2 million in 2018-19, according to Society of Indian Automobile Manufacturer (Siam). But most including volume carmakers Maruti Suzuki India and Hyundai Motor India have chosen to absorb input cost increase for the time being.
Price changes in the car market are governed by the price leader. Maruti, the maker of Baleno and Brezza models, hasn't increased car prices yet, setting precedence. January was the last month when Maruti and others increased prices as they entered the new model year.
Prices of hot-rolled (HR) steel has gone up 7.2 per cent from the March quarter till date, aluminium and nickel have gone up 19.4 per cent and 5.2 per cent, respectively, in the same period, according to Bloomberg. For the full year ended March, prices of HR steel and aluminium have gone up 22 per cent and 2.6 per cent, respectively, over a year ago-period.
In an earnings call earlier this month, Ajay Seth, chief financial officer at Maruti said even as increasing commodity prices have been bothering for some time, the company would go for a price increase only when it's a must. So far, cost control measures coupled with a better product mix - higher contribution of pricier models in its overall sales mix - has helped the company keep prices in check. "We do not want to exploit this opportunity just because customers like our vehicles and there is a waiting time," said Seth, adding, "We go for a price increase only if it becomes extremely necessary."
Hyundai, the second-largest carmaker in India, and Honda Cars India also have not passed on the effects of commodity price increase. Rakesh Srivastava, director sales and marketing at Hyundai India, declined to comment. A spokesperson at Honda also declined to comment.
The only companies in the mass segment that have passed on the cost increase are Tata Motors, which took a price hike of up to Rs 60,00 across its product line up last month, and Mahindra & Mahindra that has announced hiking prices by 1.43 percentage points across the range.
"It's a fine balance between demand and profitability. While everyone has their own reason to hike prices or not, most are refraining because there is momentum on the ground, and companies are looking at profitability through volume rather than just pricing," said Veejay Nakra, senior vice-president, sales and marketing, automotive sector, Mahindra & Mahindra. He said M&M tried striking a balance between the two by taking a small hike of 1.5 per cent across its passenger vehicle models.
Toyota Kirloskar Motor, too, has increased prices but for a different reason. This is to offset the increase in Customs duty on completely-knocked-down kits. N Raja, deputy managing director, Toyota Kirloskar, said, "With monsoon setting in, the June quarter is typically slow. We can't take frequent price increases and disturb the apple cart that is why most of the manufacturers are holding back price hikes."
Mahantesh Sabarad, head retail research at SBICAP Securities, said price change is also a function of the level of inventory at a company. The best time to increase prices is not when there is high demand because you end up disturbing volumes; you do it when there is low demand. "The principle of economics doesn't always work in real market condition," he said, adding that a price hike in a slow market induces demand as people tend to advance purchase to avail discounts on the billed amount.
Both Maruti and Hyundai are operating at optimal capacity utilisation and low inventory levels, this has helped these companies control and absorb costs said Hitesh Goel, analyst at Kotak Institutional Equities.
car price hike
toyota kirloskar
Society of Indian Automobile Manufacturer
hyundai motors
|
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| 8,147
|
Love Is Stronger Than Fear with Amy Julia Becker
S3 E17 | How the Church Can Support School Reform with Nicole Baker Fulgham
October 27, 2020 Nicole Baker Fulgham Season 3 Episode 17
Nicole Baker Fulgham
Do we, as a society, truly think that every child can succeed in school? Nicole Baker Fulgham, president and founder of The Expectations Project, talks with Amy Julia about the societal expectations for children in schools, the inequity within public education, and how to mobilize the church to work towards education reform.
Nicole Baker Fulgham (PhD, UCLA) is the president and founder of The Expectations Project and author of Educating All God's Children and Schools in Crisis.
Follow Nicole:
Twitter: @nicolebfulgham
Connect with The Expectations Project:
Website: expectations.org
Resources for advocacy and service
Facebook: @TheExpectationsProject
Instagram: @hopeforstudents
Twitter: @expectproject
"My faith pushes me on the issues of justice and equity and serving those who have the least."
"I believe as a Christian that God doesn't differentiate academic potential between black and brown and white and Asian and rich and poor kids...Being in the classroom didn't take away my firm belief in that. I just had to figure out: how do we collectively get there?"
"How do we fix the system so that we can support teachers differently, support schools and families differently, so that we can unleash that God-given potential in every kid?"
"Do we, as a society, truly think that every child can achieve? And when I say every I mean every. The child whose parents are incarcerated. The child who is homeless. The child who just immigrated here and is six years old and has to learn English first. The child whose family is struggling economically…I'm not sure that we deep down believe that about everybody….If we did, we would be investing in kids differently in our country."
"We always want to support individuals, but we also have to look at not just the educational system that's impacting them—institutional racism, all those things—but also those other systems outside of school."
On the Podcast:
The Expectations Project
Penny's diagnosis of Down syndrome
Thank you to Breaking Ground, the co-host for this podcast.
White Picket Fences, Season 3 of Love is Stronger Than Fear, is based on my book White Picket Fences, and today we are talking about chapter 11. Check out free RESOURCES—action guide, discussion guides—that are designed to help you respond. Learn more about my writing and speaking at amyjuliabecker.com.
Note: This transcript is generated using speech recognition software and does contain errors. Please check the corresponding audio before quoting in print.
0 (3s):
Hi friends. I'm Amy Julia Becker and this is love is stronger than Fear a podcast about pursuing hope and healing in the midst of social division in the season were talking about my book White Picket Fences and today's episode takes a look at the themes of chapter 11, which has called possibilities. My guest today is Nicole Fulgham Nicole has been a champion of education Reform for many years now. And she approaches the problems of inequity in education here in America, as a black Christian woman with personal experiences and, and understanding of policy and those aspects of who she is all combined offer so much insight into what is needed in our schools today.
0 (47s):
I love the way Nicole underlines the importance of Expectations when it comes to our kids and our schools and the importance of community engagement and the importance of believing in our children in the importance of policy changes. And it's this insistence on all of these things in the ways that we can be involved in making our schools better for all kids, they make me really excited for you to meet her. Today all right. Well, my guest today is Nicole Fulgham founder and president of The Expectations Project, which I'm going to have her tell you all about, but first I just want to say, Nicole welcome.
1 (1m 28s):
Thank you so much. I appreciate you inviting me. I look forward to the conversation
For me to, I have followed your work for a long time, many, many years, but I'm sure that there are listeners who don't know who you are, what you do and who will be interested, especially this moment. I think having some information as well as guidance and how to think about and participate in education and public education in the United States, we'll be really helpful. So I would love to just start by asking you to tell us a little bit about yourself and how you came to found The Expectations Project in what it is.
1 (2m 4s):
Sure, sure. Happy to. So I grew up in Detroit, Michigan in, in the city in what I would probably call a working class neighborhoods. So we had a few lower middle-class families working class for indefinitely, a fair amount of the people who lived in poverty. And so I saw education in inequality. You kind of outside of my front door every day, growing up my parents or a working class family, but were able to really work hard and sacrifice a lot of things to send my brother and I to a school is outside of our neighborhood. So starting from preschool, my mother majored in early childhood education in college. So she definitely knew what she was looking for for us, the patient and wrestled with whether or not to send us to our neighborhood public schools and ultimately decided to send us to an urban, the Lutheran school were schools that you, as the neighborhoods like Detroit became increasingly African-American a lot of private schools fled in the schools stayed.
And so they were like a kind of the next step up from the neighborhood public schools in terms of honestly quality. So there are good, but they weren't, you know, sort of this like super like Tony, you know, and to have private suburban school, but it was arguably safer. Umm, and there were something a bit more than we got, which was a huge sacrifice for my parents to do that. And I am forever grateful but sad that they had to make that choice. Right. Anyway. So at the time that I got to high school, I went to, I got accepted into an exam public high school in Detroit called a Renaissance high school. And I saw the difference between what I was getting from my friends on my block in K through eight. But I definitely saw the difference in high school.
I had been prepared both by my School, but also my parents to be able to get into this very competitive public high school in Detroit. And from day one, they were talking to us about, you know, sat prep and you know, taking AP classes. They assume we were all going to college because we were quote best of the best of my kids in Detroit and all 99% of us went on to college and you know, were all still friends on Facebook because I'm old Today and people are, you know, attorneys and physicians and you know, running engineering firms and sort of this amazing success of a mostly black kids from Detroit, all of them definitely not wealthy.
But the difference is all of my friends who didn't get into that high school, which was most of the kids are on my block. I went to the neighborhood school, which I had no AP classes at that time. No one to talk to my friends a lot about college, the entire time that we were in high school. And so about half of them dropped out my brother and I are two of three kids in our neighborhood that we know of that went on to a four-year college. And the crazy thing is that they were just as smart as I was. I am convinced smart and curious has unlimited potential, but we are going to two very different school systems within the Detroit public schools. So that really changed because of the way I looked at the world early on the other piece of information.
And this is a lot of background about how I got here, but I think it's good to hear people's stories. And so indulge me if you will. Absolutely. That's what I want it on it. Yeah. I mean, at the same time I was attending a church, a church with my, my family as a, as an AME church, African Methodist Episcopal church, which has a rich tradition in activism and sort of engaging in civil rights issues, you know, since its founding splitting off from the Methodist church that wouldn't allow, we were, you know, right after slavery, wouldn't allow black people to sit in at the same views as the other who are white. And so the church began. And so growing up, I saw this very large influential church repeatedly hold elected officials accountable to the things they say they're going to do when they kind of came through the congregation to shake hands and you know, wink, wink, we'd like your vote kind of thing there, pastor.
It was like, absolutely. Yeah, come on through. And they would have them in the next six months in front of the entire congregation asking them. And so you said you were going to do these three things. You haven't done them yet. When are you going to do them? And they like consistently called out the leadership in this city, black, white, verbal, it didn't matter. They were like you said, you are going to do it and he didn't do it. So we are going in and I'll start to protest or we are going to call your office. We are going to bring you back in front of this church that you promised these things too. And so for me, I saw that as an extension of my faith. Like my bait pushes me because it was always on the issues of like justice and equity and serving those who have the least. So those two experiences, the School inequality in growing up in a church that really put that in practice really led me to the path that I'm on today after, you know, being a public school teacher for a few years, getting my PhD in education policy because I was trying to figure out how do you change this system to make them have kids and Andy over the years is I just wanted to have to really marry these two ideas of what would it look like to have people of faith incredibly focused from there, faith lens.
And I am a Christian from a biblical lens focused on, of course we have a system that is right now, educating different groups, have kids based on race and class to different outcomes. And as people of faith that shouldn't be because we assume all kids have potential. What would it look like to mobilize that group of faith motivated folks and to have them push their left, that officials kind of on mass, right? To change the policies, the system and the practices, and to demand that they fix this system year after year after year more equally reflects the potential of all of that. That's what I do with The Expectations Project.
Thank you so much for giving us that story. I want to add one more piece to it or ask you to speak to one more piece of it, which is your own experiences teaching in Compton, I believe right out of college. Is that right with Teach for me?
Yeah, I did. Yes. I had joined teach for America M right out of college and definitely learned a lot. I think my students, I'm not sure who taught a few more in a way with a first year teachers. And I taught for a few years and taught fifth grade. And you know, that was definitely one of those life changing moments because it made me really put into practice, all the things I said, I believe, right. I said, well, you know, growing up in Detroit, I know my friends or just as smart as I am, like, why aren't we pushing them? And the same way I'm getting pushed, right? Like I believe in like, God, doesn't differentiate academic potential between black and Brown or white and Asian are rich and poor kids.
Like I'm not going to day right. First off. But then when you're on this side of the classroom to have to actually prove that to be the case with it changes your perspective. It doesn't take away my firm belief in that I just had to figure out how do we collectively get there? And, you know, it was a really hard, I was very fortunate to be placed that at a school that has some veteran teachers who we're, you know, at that time near the age of my parents who are quite frankly, and my grandparents took me under their wing and they were willing to let me kind of like shadowed them in the classroom for them pick their brain CRI with them. A lot of them are Christians. They prayed with me right outside of the classroom in a legal way, but this amazing community that they taught me how to engage and get to know my students, families.
And so my family's became partners and their kids' education. And I just was kind of naive enough to think like, actually we should be able to do this. And so we were able to be really honest with kids and families about where they were academically. And then in fact, like two thirds of my kids were reading at, you know, probably a second grade level or below in fifth grade, some work at functionally illiterate and the same with math really below grade level. And so I was honest with them, with their families, not in a blaming way, but just we've got work to do. And so people got on board and I got lots of volunteers from my church at the time community members to help us do extra tutoring because my kids needed more time to catch up.
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But it really was this amazing thing. It was, I mean, I'm glossing over it, you know, a lot's of like tears prayer's, you know, mistakes failure's. But at the end of the year, my fifth graders on a standardized test, it had the most academic growth of any fifth grade class in Compton for a brand new teacher is sort of crazy. And I don't say that to say like, Oh my gosh, I was like an amazing teacher because I probably wasn't. Right. I was just sort of like naive enough to like try a bunch of things. And I had the energy of, you know, I don't know a journal, right? So it was 80 hours a week. Things that in reality are just not sustainable to be a teacher that way for 30 years. It's not.
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So I don't judge other teachers who aren't doing those things, but it was this beautiful thing to see. So many of my students in their families see, Oh my gosh, I've had students say, I didn't think I was smart until this year. And that is like what our role is supposed to be. And so for me, it's like, well then how do we, as part of our life as a classroom, was it that's one classroom? How do we fix the system? Right. That it can allow more of that to happen, to support teachers differently, to deport support schools and families differently. So we can unleash that God given potential in every kid.
Yeah. I So resinate with that. One of the reasons I wanted to have you on the show is because I've been walking through during this podcast season, my book White Picket Fences and the chapter that we're kind of on is called possibilities. And it's a chapter that I wrote about our daughter Penney and the possibility's that opened up for her as a child with a disability within the school system. And I've just been so struck as I've watched her earlier this week, we had a PPT meeting for her, and then you got this team of people and I'm just so aware of how much they believe in her. They believe in her abilities and they certainly are going to make accommodations based on her needs.
And they're going to give her the support that she needs, but they really do believe in her. And in watching that and in watching the difference, honestly, she's had teachers before were in a dance class, for example, she's gone had the same class, but different teachers and one teacher will write me and say, penny got tired after 20 minutes and she couldn't continue the class. And then I'll talk to the other teacher. Who's teaching the same lesson on another day. And, and she'll say no to a penny has never gotten tired and it really came down to you believe she can do it. So I just saw it in, I re-read your book in preparing for this interview today, which I'll just mention here for people who are interested called Educating all God's children.
And I was struck by that same thing of just like what you just said about the student who was like, I never believed I was smart. Like I never knew that. And you believed that, right. You actually believed what you had said, theoretically must be true about your neighborhood. It's like, okay, I'm going to actually live that out. And so I wanted to ask you to, just to talk a little bit as I think about not just the, you know, kids with disabilities, but other kids who are on the margins for other reasons, and who are vulnerable, especially when it comes to education for other reasons, what do you think are like societal expectations are for those kids and how do we begin to change the expectations? Right. I'm thinking of The Expectations Project and why you even called it.
That definitely that, and that's exactly the reason, because for me, it was such a differentiator at that school high school that I attended, you know, honestly, like I think we, if we dug deep down in sort of honestly, and like a, the depth of our soul, I say we, as a society, we want to ask ourselves, do we truly think that every child can achieve, right? And then when I say every I mean every like The child, whose parents are incarcerated The child who is homeless The child who just immigrated here and is six years old and has to learn English first, right. The child whose family is struggling economically, who doesn't have, you know, the clean clothes that we, you know, sort of a think that I should have whatever the thing is.
I don't think we do. And, and it's hard to acknowledge that as a country, because at this wonderful likes sort of a bumper sticker slogan write, like all kids can achieve, like everyone has potential and is great, but like, we need to do the deep soul searching and say like, it's okay to kind of acknowledge, like, I'm not sure that I believe that. Right. And it is that we don't 'cause we see so many messages reinforcing that, right? Like we don't see messages reinforcing these, you know, superstar stories have kids with every like seeming, you know, every possible disadvantage is succeeding.
They will seem like almost one in a million. Right. And, and for good reason, like it's harder. Right? Like, I don't think, I mean, I'm very clear that, you know, the child that I'm raising, you know, is going to have different advantages even than I did write it as a, as a youngster in, so like I'm IX, there's a, there's an expectation people probably have of her, you know, race aside she's African-American as a My, but there, there are some assumptions, right. That she'd be able to achieve, but I'm just not sure that we deep down believed that about everybody. And the reason I'd say that is because if we did, we would be investing in kids differently in our country. Like we just like we are.
And I totally agree. I I'm thinking of, I remember it was when penny was really little that I read the line, the soft bigotry of low expectations.
And in fact it was like a punch in the gut. Right. Like it resonates. And that's the thing, like, I wrestled with that as a teacher and I came in with absolutely. But when you get in the thick of it, right. And you're like, Oh my gosh, like I know that. And I've had students who had some, you know, they experienced trauma. Some of my students, you know, during the school year with family things that happened. And you're like, wow, okay. So that's just happened. And I'm asking you to focus on fractions on Monday morning. And I know that it happened in your house over the weekend. Right. It's not that I, I still had to help them focus on fractions. Right. Because that was the, long-term what they needed. But I also, can you do that and make sure that the family has the social services or counseling support they need through the school or through the County.
Right. Like both things can be true. But what we often do is in not just because we are biased or like it's our own bigotry or prejudices, it's this, it's out of like sympathy that we can sort of fall into this. Like, Oh my gosh. But I couldn't expect them to do that because there's so much, and it's like, get them the help they may need. But also like, they still need to learn fractions because like that matters too. And like, who are we to say that they don't get the right to learn it, but, but again, like at the end of the day, if we truly believe that every kid would succeed, you wouldn't see, you go into schools now, you know, suburban and urban schools and just the facilities, everything is so dramatically different.
You wouldn't see that type of inequality. If we truly believe that every kid can achieve it, you just wouldn't. Yeah.
And again, my like little, you know, a Petri dish has been my own household as far as watching the same, be true when it comes to our daughter, because there again can be low expectations and high expectations. And she's got similar to what you described in terms of your kids, like a ton of social support from her family and community that plenty of kids don't have, but there is still that sense of if our attitude towards her is that we believe in her certainly in terms of her potential to learn, but also her potential to contribute. Because I think that's part of it too, is like, what if we assume that she has someone who is going to give to our society and we don't know exactly how, but what if we assume that about every kid, all the kids that you described as well, who I think we often assume are going to just take from our society, which is absolutely not true and does not honor the image of God in them whatsoever.
And I wanted to ask you about, you have written about a history of how Christians have been involved in educational initiatives in Reform in the past. So I'd love for you, if you can't just do a sketch it a little bit of what the history is, but then also talk about the contemporary moment, our churches, our Christians involved in public education reform and why or why not so past and present. Could you just speak to that a bit? Sure.
Yeah. I mean, it's so, so many of the, the public schools are at the beginning of schools in America were largely Protestants present in communities that were founding them. There was, there was a, a, a deep belief that, you know, Educating young people, particularly a lot of these Protestant communities reached out to immigrants, right? That, that was an important part of their calling. Now, granted, they were doing it because they wanted to make sure that people knew Jesus, right. That was the main, the rest of it. Like oftentimes the educational texts where, you know, religious documents on. So we can, we can discuss that at some other point, but there was also a belief that people needed to be literate and contributing members of society, which that part, I definitely, you know, adhere to that over time, that lessened right as schools became like truly public.
And then there was the massive debate over a prayer in schools, et cetera, the place where I would say like the conservative Christian White conservative Christians really fell down on this promise was when schools were being desegregated in the sixties. And that is this really, really disturbing moment for a lot of, of, of more conservative white denominations that started all of these quote unquote Christian Academy's. And I say quote-unquote because like the root was pretty wicked. Y many of them were started. I'm sure they'll talk to you this, but the route was, it was, it was pretty foul starting. These are all white Christian academies to ensure that there are white children, didn't have to go to school with black and Brown kids.
And so that is, you know, one of those parts of our history that we don't like to talk about it as much, but it's part of our story. It also leads to why we've had, you know, in some communities resistance when churches now write, want to get involved in this is changing a bit, I would say, but there are communities that remember that history. And so I'm more skeptical right. Of, of Christians wanting to get involved. I also think, as I say, in the, in the book there in some communities, Christians are known more for getting involved in public schools to push sort of religious issues and to be sort of counter cultural, whether it's, you know, a protest and removal of prayer and Schools or questions about, you know, curriculum dealing with sexuality issues or what not.
And, you know, my take is if that's your, because by all means like a champion your costs. I think the sad part is that we are often known more for that in connection to the schools, then Being partner that wants to come in and say, Oh my gosh, kids are struggling with the families are struggling. How do we support this school? I would say the good news. It was like that tide has turned, I would say in a lot of communities in the last, you know, 10 or 15 years. So we do see so many more congregations that are taking on trying to develop really honest symbiotic partnerships with public schools and just coming in and saying, Hey, do you want to be helpful? What do you need? Which has always the best way to be a partner in any context from my experience, which has been exciting.
So I think where were we are still growing? Is that last step of how do we sort of over time a limit as that? I like to say it eliminate the ne the need for every Church to need to tutor our kids, to get them ready for kindergarten, because there's no pre-school in our community that is affordable for families to saying like, actually, if we would organize advocate and speak truth to power to get the affordable quality pre-school for everybody, like long-term, that's going to impact thousands of kids instead of the 20 kids where tutoring, right. Thousands of kids will then have what they need to be ready for kindergarten. We can maybe still tutor to, to like, make it even better, but it's the systemic change.
The system, the policy's the practice is the funding mechanisms. That's a place where I think we're on the cutting edge. Umm, and can step more into that space. And the last thing I'll say is when Christians on math and I say this the broad strokes, let me caveat caveat. There are plenty of Christian denominations that have been doing this work for years. They typically are denomination's of color, particularly black communities engaged in education policy, you know, from the civil rights movement. On so I want to make sure I acknowledge that history, but from a broad strokes contemporary space, I think there's a lot of us who have been involved in policies around vouchers for a private schools for obvious reason, right? Lots of us to send our kids to a private or Christian schools or have them in our church's.
And so that's a self-interest we have, and also, you know, potentially a belief that parents should be able to be free to choose with with government tax dollars, where are they send their children So but lots of, lots of opportunity for us to, to engage in the, the policy reform. And I think we shouldn't, we shouldn't let those opportunities go by when this system is so broken and so not reflective of the kingdom of God for, for all of his kids. Yeah. So I had a of questions based
On what you just said. And one is just that I feel like, especially in Christian circles, there's a tension that comes up between this idea of like individual responsibility and structural concerns. So our schools are failing and our kids falling behind because of systemic racism and structural inequality or are they failing? And school is failing kids falling behind because individuals aren't doing what they should be doing. And I hear this debate come up in a different language and in different ways, but, and I tend to think it's a false dichotomy, but I'd still love to hear just what you have to say to Christians about the relationship between individual responsibility.
And this is more a structural or a systemic problem that we see in various ways, but especially in schools.
Oh, that's a great question. It, it is a constant ongoing discussion in part because there are, there are truths on both sides. Right. And, and I liked in this, I'm sure it was for some people who, when they talk to me about these issues, I tend to see the, the gray in between usually because things are not as stark as we want to kind of put people in the bucket, you believe this or that. And it's like, well, if only we were that simple. All right. Yeah. So, so on this one, I think there is, there is space and movement needed on both sides. I think when I look at what is going to move the needle more for a larger number of people, I look at it from a systems perspective, right?
Again, that analogy, I get a whiff, we can tutor kids' for the next one, a hundred years and tutor 20 kids to get them ready for kindergarten are going to get quality pre-school for an entire state that didn't have it. They had no funding for poor families previously. And then it's like, Oh, a a hundred thousand kids now. And it does it make the policy perfect. Write all sorts of implementation challenges. We all know that, right. But it's, it's a, a root it's getting at the root issue is where do I like to come at things from personally? So are there places where individual families may need more support? Of course, like, you know, we are both parents. We know that there is always, it feels like more we can do to support our own in this work.
But I also recognize the putting all of that responsibility on an individual families in it's, usually parents, they get blamed. I hear this a lot. What if the parents cared more about education for the kids cared more of X, Y, and Z happening. And its like, well, so that's try to put ourselves in the position of a family who is struggling to make ends meet whether it's a single, a two parent family, grandma, auntie whomever, if they are working, you know, two or three jobs, right. Don't have paid time off to go up to school. So we can meet with the teacher at two o'clock in the afternoon, which is a privilege that a lot of families don't have, like let's start linking all of these systems together outside of school are impacting an individual families, options and choices.
Right. And how is that impacting? Not how much they care, but what they're able to do and put into practice that fully reflects how much they actually do care because I have never met up here and he's like, I could care less if my kid graduates from high school. Like, I mean, I haven't met that person. So that's why for me like, yes, we always want to support individuals, but we also have to look at not just the educational system that's impacting them and the institutional racism, all of those things, but also the other systems outside of school that are impacting their ability to kind of execute on all of the beliefs that they have about why education matters so much. I mean, I think we see that now in a time of COVID like loud and clear, right?
The fact that we all of a sudden had to start doing virtual learning and there were parents who did not have a high speed internet and didn't have a laptop for their kids. Like millions of the parents don't have to have that. Right. And I think they're going back to this a little, if you can, parents who have two, you know, whose kids, schools aren't opening because of, you know, spikes and COVID and they have to stay home and do learning if the child has seven in is on zoom calls. But my mom and dad drive as a bus, a city bus, and they have to go out to drive that bus, to get the paycheck, to pay the rent, like tell me how that's going to work. Right. So it's like that. So those are the types of that, that I hope people can continue to connect and our churches to realize like, Oh, there's, there is the system thing that actually matters to you.
Yeah. I've actually one of the things that has gotten me particularly engaged in education these days has been recognizing last March that my kids were able to go immediately on to zoom for a couple of different reasons. One, they had been issued Chromebooks by their Schools. So there were already familiar as we were there, teachers too, we had multiple rooms in our home where we could spread out and Each be on different devices and it wasn't perfect. It's like, Oh, we have an old laptop and sometimes zoom crashes or somebody on a phone, but nevertheless, they were able to do it. And I was so aware that tens of thousands of kids just in my home state of Connecticut were not able to keep learning.
And it was like, well, whose fault is that? And certainly not the children's fault. Yeah. So I feel like this moment has just really like been like a magnifying glass are a spotlight on the inequities that we're already present, but it has certainly drawn my attention even more to it that said one thing I wanted to ask you about. So I think the were talking before this episode will actually air and it's the day before the final presidential debate. So I've watched the first debate. Yeah, can't wait. And the VP debate. But what I noticed in the two debates that have already been happened and the one that's happening tomorrow, I looked up the topics like education is not mention, I think Kamala Harris like made a reference to the fact that there were kids who are not learning, but it was like a passing comment.
And again, I looked at the list of topics for tomorrow is debate and education is not on the docket. So especially in this moment, I'm just curious, do you have an opinion on like, why are our national leaders are not talking about education and reforming education and like actually helping to change this as well?
Yeah. You know, it is, it is tragic, right. Because I feel like we went through this moment, right. The four schools were in this school year was as opposed to begin when life that was like, you know, the topic does you're right. Everyone from, you know, the president of the United States on down, you know, had opinions on, you know, what school should be doing in the open up or not. And so we had this like little window where I was like, Oh my gosh, it's actually coming front and center. So it was really only like a political sort of discussion. Right. And kind of a political point scoring for it for a national dialog. So I think we're back to this place now where people are just kind of hunkering down and you know, it's just incredibly complicated.
And my theory is that no one actually knows what to really do about it. Right. So that dilemma I gave, you know, a few moments about the little child who has to go to school virtually who's parents have to leave the house to work. Like I have asked that question in so many spaces have like policymakers think tanks and like, so it was anyone figuring out where that kid's gonna, what that is going to do all day. But I wouldn't want to my seven year old, a year old at home zooming by herself. Right. Like that's not a day like a depressing, a whole host of things. Right. And it's one of those things where like, no one's sort of knows. And I almost feel like right now everyone recognizes that is such a, S spiraling, downward, like a challenge that has multiple layers that I almost feel like people have kind of given up on the public sphere or even talking about it because I feel like we're almost resound.
We are almost given up on this idea that we can get it right this year. And it's a moment that we know very clear that certain groups of kids are going to come out fine. You know, parents who have the ability to, you know, pay for private tutors are a pandemic pot or whatever. Right. And, and no judgement on that, right. People are going to do what they're going to do for their kids. It's just that we do, we know that the gap's are going to be wider. And I, I just have a sneaking suspicion that no one wants to talk about it for that reason. And Oh, by the way, it's always like the left behind topic that no one wants to talk about it in any presidential election here. And sadly like, this is, this is another one. And it's, you know, were just shortsighted honestly, as a nation.
Like we want to talk about the topic and not realize like, if we don't fix this particular problem, like we're losing the generations of kids in our country, like stature and ability to be affective as a nation. It, you know, literally like, you know, as one of my friends says, it's like a national security issue. Right. If we don't think about it globally, like, and we don't fix this and have a completely like massive, you know, uneducated population of, of young people are in this country, like we are going to have major problems, but again, we'd have to have the longitudinal view to really take that on. And we just historically haven't as a country.
Yeah. I think that's all true where it's always been left behind it. And then we've got the current moment where it's like, I really don't want to talk about that because its just a hard and a bad situation right now. I want it back up a little bit and ask you to, just to describe what The Expectations Project actually does. I realized, I didn't ask you to elaborate on how your organization functions and what you're doing to really be a game changing force in this whole equation.
Yeah. I'm So we are the nation's largest network of faith motivated education equity advocates, which is fun to be able to say after laboring over this 11 organizations since 2012. Yeah. That's amazing. And it's really exciting. So our work is to educate, equip, and then mobilize faith constituent's so people of faith around the country to take tangible actions, to hold elected officials accountable, to push them for policy change on the issues that matter most for, for equity for all kids, it sounds like a mouthful, but it really, when you boil it down, it is right now we have a network of about 50,000 individual people of faith who are doing all sorts of things that are advocacy action.
So e-mailing members of Congress are the state house at our school board on something. They want to see a change making phone calls, letter writing campaigns, petition signing, doing public comment, you know, used to be in person at school board meetings now is virtually on pieces of legislation in issues. And so that's sort of the action piece, right? And so those things might seem miniscule, but they actually add up into real change, like any elected official. If they get, especially at the local level, a a a hundred people calling them about an issue, there are like, Oh shoot, I guess I should pay attention to that. And so we identify places where they're are relevant issues or policies coming up that have the potential to be game changer.
So we did a whole campaign at Indiana a few years ago on getting state funding for low income families to have preschool in this. You had mentioned a couple of times already and there wasn't any. And so we believe that in other organizations did as well that if we can rally people to keep speaking about this, obviously our folks who are talking about it from there, their faith and values perspective as well. And if we get the attention of the people that have to make the decisions and you know, for lack of a better word, pester them enough and really help them see our perspective that laws and practices will change. And that did in Indiana, they passed the first ever a piece of state legislation to allow low-income families to access funding for preschool.
And so it changed the game for tens of thousands of families, right? We were doing that on school discipline issues. When we see black kids are four times as likely to get suspended from schools nationwide. And so we're digging into why is that the case and our, how do we flip that to have more restorative justice practices, which aligned very much for our beliefs as Christian's. And so we are, you know, having our advocates, our are taking action, you know, phone calls, emails, petitions, et cetera, to really shift the conversation around that. So when policies come up for a vote, whether it's, you know, eliminating the practice of suspending kindergarteners and preschoolers, which is amazing to me that that actually happens, but I'm and putting in place funding and teacher training for restorative justice, which is helping kids like have still have consequences, but in a loving, like let's restore the relationship's and get to the root of why the child is misbehaving.
That's a very different than simply suspending them. And so that's the work that we do. It involves a lot of educating people on issues. So we do lots of trainings and now they're all, you know, webinars, of course we create a small group Bible studies on various topics, prayer guides for congregations to, to use a sermon talking points for pastors, right. To get the education out there. And then we have these tangible ways that our constituents can get involved to really change what's happening in their local community. And you know, it just this past summer, I guess the last like four or five months when COVID, we shifted a lot of our work to that, that piece, we've had our constituents have taken over 64,000 individual actions in the last few months on a variety of education issues.
And they've seen, you know, that they've gotten the attention of their elected officials and just the way that we'd hope they would. It's a very exciting, we have lots of power and voice lets use it.
And so if someone wants to get involved, would that be as an individual, as a church community? Like what, what would that look like if someone's listening to this and wants to know either broadly, like I just want to get involved in changing education in America. What can I do or specifically I want to get involved in The Expectations Project what would that look like?
Absolutely. So I'm, if you want it to get involved with the Expectations project, go to our website expectations.org and sign up for our email list. We don't spam you. I promise we're a small organization. You're not going to get 5,000 emails a week from us, but we will then get you connected to ways that you can take action in your local community. It's we make it as easy for you as possible because we know everyone's busy, but you can find a way to, you know, attend a webinar or if you want to or read more about a policy and then take some very specific, fairly easy to do actions locally. So we encourage that as one step. We also have resources for congregations that we want to learn more. I think I referenced, you know, small group Bible studies, which will be a virtual now I'm sure for many congregations weighs for congregation's to kind of self-assess how equity has happening in their local, a local communities.
And then come to a conclusion on, you know, one to three issues, which one do they want to do more to educate their own congregation about. We're also happy to be a resource to talk that through. We do lots of, you know, a quick phone calls in consultancies with congregations who we're like, we want to do this. We're not sure what to do next. So that's part of what we do as well.
Yeah, that's awesome. Well, I will make sure that just that information gets into the show notes so that people can have follow up because I have certainly found as I have become more engaged even just in the past six months in issues around educational justice here in Connecticut, that really, my knowledge was much broader than it was deep. So I knew a lot about kind of education and America and it's like, well, what about education in your town in New York County? And you know, the local places where you actually could make a difference and certainly your congregation can make a difference because as much as I bemoan, like I really wished that our national leaders were talking about and making reforms when it comes to education the same time it's on the local level in terms of town by town and state, by state where a lot of that change and transformation really can happen.
So thank you for being a leader in that.
No, that's exactly right. And I know the national, you know, politics, you know, that takes up a lot of oxygen. You do it, it has for the last several years, but you're exactly right. You know, so even if things are happening at the national level, you know, that our constituents are, you know, sometimes frustrated about, you know, depending on the issue we always had to pivot pull back. So that actually doesn't preclude us from looking at what's happening in our state, our city, our district, because it's honestly easier to move the needle because you're like a bigger fish in a smaller pond, right? Like a a hundred people are calling their state representative on an issue actually will get their attention because they don't often get that type of, of attention if a, a a hundred people to try to email the, you know, the secretary of education, or there are a Senator in Congress, a lot harder to get their attention.
You are going to need like a thousand people, right. Or 10,000. Right. And so like local politics matter and or it should say it not even in politics, local policies and decision-makers matter.
All right. Well, thank you so much for the work that you are doing. It's really exciting to hear about how that has grown and obviously it to even get a sense of the thousands and tens of thousands of people who really do care about these issues. And I hope that just having had this conversation, they'll be a few more who get on board. Yeah. I'm really grateful for all the work you do and for your time today.
Thank you. It was my pleasure. Thanks for M having this conversation and for, to, to keep us in the conversation, go in and anyone who has, you know, questions or wants to connect to us, please feel free to do so. We definitely want to be a resource for all who were interested. Partly you tell us your website one more time. Yes. It's a, Expectations with an s.org.
Awesome. All right. Thanks. Nicole and I look forward to talking to you again sometime. All right. Take care. Thanks so much for listening to love is stronger than fear. I do want to let you know about what's coming. We have a couple of really fantastic upcoming episodes, and I'm sure you don't want to miss them. I certainly don't. I'm going to be talking with Andy crouch next week about love and power and vulnerability. It's going to come out. It's going to be released on election day, even though we will record that conversation beforehand. And I think it won't be a helpful guide to what promises to be an intense day for our nation. So please join me for that. And then the following week, I'll be talking with David Bailey.
Some of you may remember him from the first episode of this season, and we will be talking about what faithful love justice and reconciliation work looks like. And by then will know who will know something about this election. And we will be talking about what love and justice and reconciliation work looks like no matter what is happening in the political sphere. I'll tell you this for two reasons to give you a sense of where we're headed, but also as a prompt, if you have not subscribed to this podcast, please do please share it with friends. Please rate it, review it. All of that will help more people become aware of what's going on here, because these are conversations for people who care about nuanced arguments, who care about compassion, who care about working together across dividing lines.
And if you are a listener already, and then I suspect you now, more people who care about these things and who would benefit from these conversations as I certainly do a week after week. Thanks so much again for listening and for being here. And I do, as you go into your day today, I hope and pray that you will carry with you. The peace that comes from believing
That love is stronger than fear.
All content © 2022 Love Is Stronger Than Fear with Amy Julia Becker.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,689
|
Q: Difference between a simplicial complex and a graph My mathematical sophistication is pretty low. Looking for an intuitive but accurate explanation of the different between a simplicial complex and a graph (a set of vertices and set of edges).
A: A graph (unoriented) is a one-dimensional simplicial complex. More-dimensional simplicial complexes are sometimes referred to as "hypergraphs".
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,864
|
package xp;
import java.util.InputMismatchException;
import java.util.Scanner;
public class Q6 {
public static void main(String[] args) {
int num;
Scanner keyboard = new Scanner(System.in);
boolean invalid = true;
System.out.println("Question 6: Sum");
do {
try {
System.out.print("Enter an integer: ");
num = keyboard.nextInt();
invalid = false;
int temp, sum = 0;
while (num != 0) {
temp = num % 10;
sum += temp;
num /= 10;
}
System.out.println("Sum of digits: " + sum);
} catch (InputMismatchException e) {
System.out.println("Invalid! Input isn't an integer!");
keyboard.nextLine();
}
} while (invalid);
keyboard.close();
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,472
|
"The Dramatis Personae of Our Lives": On "A Bountiful Harvest: The Correspondence of Anthony Hecht and William L. MacDonald"
By Patrick Kurp
FRIENDSHIPS ARE AUTONOMOUS REGIONS with their own languages, folkways, and treacheries. Old friends are telepathic and get each other's jokes. What they don't say is heavy with significance. It's not always a case of "opposites attract," nor are friends necessarily "two peas in a pod." Rather, long-lived friendships, more than some marriages, embody the nation's motto: e pluribus unum.
When Anthony Hecht and William MacDonald first met in Rome in 1954, both were young, unknown, and unproven. Hecht, 31, had just published his first poetry collection, A Summoning of Stones, and was on a Guggenheim Fellowship; MacDonald, 33, was on a Rome Prize Fellowship at the American Academy. Hecht went on to become one of the leading postwar American poets, and MacDonald would occupy a comparable place in the field of architectural history. A Bountiful Harvest, edited by the British publisher Philip Hoy, documents their 36-year correspondence. That may sound dry and academic, especially as both men were, in fact, academics. But it isn't. Seldom has a collection of letters read so consistently laugh-out-loud funny, before turning unexpectedly sad. Hecht and MacDonald were men with well-exercised comic senses, not afraid to be ridiculous, whimsical, scatological, or scathingly critical of acquaintances and public figures. Their letters are filled with puns, put-ons, mock pedantry, and even a protracted exchange of Polish jokes. Both possessed a gift for inspired Monty Python–esque silliness.
Hecht and MacDonald were also strongly competitive in an adolescent sort of way. Each tried to outdo the other in the outlandishness of the letterheads on their stationary, even enlisting friends to scout for exotically unlikely sources. MacDonald types on paper from the Byzantine Institute, Twentieth Century Fox Film Corporation, and the Tom Sawyer Motor Inn in Albany, New York. Hecht replies with letterheads from Dubai InterContinental Hotel, Fudan University in Shanghai, and Croisière de Musique à bord Mermoz. They address and sign their letters and postcards with goofy pseudonyms: Admiral Dewey, Milton of Saudi Arabia, Walter Ego, Irving of Arimathaea, Comrade General Ivan Ivanovich, Ethelred the Moderately Well-Prepared, Timon of Brooklyn, and, with a nod to W. C. Fields, A. Pismo Clam. Hecht and MacDonald, serious and scholarly men, never lost their giddy sophomoric streak.
Hoy's editing is heroic. He reproduces 403 of the 440 known surviving letters and postcards, and annotates them sparingly, enough to orient readers who are likelier to know more about Hecht than MacDonald. He divides the collection into three chronologically arranged sections and provides useful biographical introductions to each, as well as a chronology and glossary of names. We follow the men through marriages, divorces, births of children, job changes, disputes with editors and department chairs, honors, and illnesses. In 1981, MacDonald is diagnosed with myasthenia gravis, a chronic autoimmune, neuromuscular disease without a cure. Typically, he jokes about his illness: "Serious. Ho-hum." Hecht had never heard of the disorder and thinks MacDonald is kidding. A telephone conversation follows, and Hecht apologizes. With touching thoroughness, he studies his friend's disease, speaks with doctors, and, characteristically, looks up myasthenia gravis in the latest supplement of the Oxford English Dictionary. In a letter from February 1982, Hecht, at his wife's suggestion, invites MacDonald (who teaches at Smith College and lives in Northampton, Massachusetts) to move into their house in Rochester, New York: "You could, therefore, have a suite of small rooms of your own, including, of course, a private bath. And there would always be someone around to help or drive or attend in whatever way." MacDonald declines, but writes, "Friends, old friends, good friends, are the most precious things in life. They really are the only people who can help. The children want to, often, but through no fault of their own they can't in any serious way."
A Bountiful Harvest reads like an epistolary novel, complete with drama and comedy, subplots and ancillary characters. But even before I started reading, I noticed the correspondence had ended abruptly in 1990. I knew Hecht lived until 2004, and the dust jacket informs us that MacDonald died in 2010. What happened to end their friendship? Diplomatically, without taking sides, Hoy explains in his afterword: "The answer is that the friends had a serious falling out." Not surprisingly for two deeply learned men, it was triggered by a difference of opinion over a bit of Roman historical/architectural arcana.
In September 1990, Hecht and his wife hosted a dinner party for MacDonald and a friend of their son. Conversation centered on Trajan's Column in Rome, completed around AD 113. The column is the central image in Hecht's 1971 poem "The Cost," later placed as the first poem in Millions of Strange Shadows (1977). A young Italian couple circles the monument on their motor scooter. The column's 600-foot spiraling frieze recounts Trajan's bloody campaigns against the tribes of Dacia. The lovers are oblivious to this ancient history: "All of that youth and purpose is, of course, / No more than so much dust." But the speaker forgives the young couple their heedlessness in the final stanza:
And why should they take thought
Of all that ancient pain,
The Danube winters, the nameless young who fought,
The blood's uncertain lease?
Or remember that that fifteen-year campaign
Won seven years of peace?
During the dinner, MacDonald informed Hecht that the Dacian Wars had lasted two or three years, at most, and not 15. "If he was shocked by what MacDonald told him," Hoy writes, "he was also at a loss to understand why his friend hadn't said anything about the matter before." When asked, MacDonald said he had remained quiet because he didn't think the error compromised the intent of the poem. Hecht was outraged and dismayed. Hoy explains, "For [Hecht], a poem that is trying get at the truth about something, and whose argument depends upon statements advanced as historically accurate, cannot survive unscathed the discovery that those statements are false. It belittled the importance of poetry to suppose otherwise."
Hecht's wife told Hoy he stayed up all night after the dinner, searching his library for the source of information about the "fifteen-year campaign." Though partially vindicated when he located it, Hoy writes, "[H]e also realized that the source was an unreliable one, and that he'd been unwise to rely on it." The following day, Hecht telephoned MacDonald and, Hoy tells us, "[T]he conversation did not go well. Indeed, it seems to have gone quite badly, and when the call came to an end, it did so with both men feeling upset and aggrieved." Hecht's final letters to MacDonald are missing. A postcard from MacDonald dated November 13, 1990, contains a single sentence: "I suggest a long moratorium, after which we might talk if both of us wish to." At this point, the sense that we are reading a story, and that a denouement may be at hand, grows stronger. On February 1, 1992, MacDonald writes that he is about to undergo triple bypass surgery. "Your letter has been put aside, unopened," he says, "with other things to be attended to later on." The following April, MacDonald writes, "Thanks for your note of long ago. I am almost fully reanimated, as the medics say, and am back at work at last. It was in some ways a horrible experience, brightened by the care of the hospital nurses and staff." No more letters or cards follow. That's the end.
By the time readers learn of the permanent rupture in the friendship, they may recall that Hecht, in the preceding 500 pages, has never written in detail to MacDonald about writing poetry. Hecht was in his poetic prime, composing the work collected in Millions of Strange Shadows and The Venetian Vespers (1979), his most accomplished and lasting work. For Hecht, poetry seems to have occupied a private, heavily guarded realm — hardly an aberration for a poet. By sharing "The Cost" with MacDonald, Hecht had allowed his friend into that near-sacred realm; by not telling Hecht about the error when he had the opportunity, MacDonald had profaned it, or so Hecht chose to think. We'll never know if Hecht ever swallowed his pride and forgave MacDonald, or if MacDonald could have accepted such forgiveness. But one may hope so. In "Sarabande on Attaining the Age of Seventy-Seven," published in 2000, Hecht characteristically refused to exempt himself from moral strictures:
The dramatis personae of our lives
Dwindle and wizen; familiar boyhood shames,
The tribulations one somehow survives,
Rise smokily from propitiatory flames
Of our forgetfulness until we find
It becomes strangely easy to forgive
Even ourselves with this clouding of the mind,
This cinerous blur and smudge in which we live.
Patrick Kurp is a writer living in Houston, and the author of the literary blog Anecdotal Evidence.
A Bountiful Harvest
The Correspondence of Anthony Hecht and William L. MacDonald
By Philip Hoy
The Waywiser Press
"The Exceptional Man": Rereading Richard Wilbur
A Negative Freedom: Thirteen Poets on Formal Verse
A Lifetime in Poetry: Marvin Bell on Iowa and the "Dead Man" Poems
By Loren Glass
"Literature with a Capital L": On Arthur Krystal's "This Thing We Call Literature"
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,708
|
package org.neo4j.driver.testutil;
import java.util.Collections;
import java.util.HashSet;
import java.util.Map;
import java.util.Set;
import java.util.stream.Collectors;
import org.junit.jupiter.api.extension.AfterEachCallback;
import org.junit.jupiter.api.extension.BeforeEachCallback;
import org.junit.jupiter.api.extension.ExtensionContext;
import org.neo4j.driver.Bookmark;
import org.neo4j.driver.Query;
import org.neo4j.driver.Record;
import org.neo4j.driver.Result;
import org.neo4j.driver.Session;
import org.neo4j.driver.Transaction;
import org.neo4j.driver.TransactionCallback;
import org.neo4j.driver.TransactionConfig;
import org.neo4j.driver.TransactionWork;
import org.neo4j.driver.Value;
/**
* A little utility for integration testing, this provides tests with a session they can work with.
* If you want more direct control, have a look at {@link DatabaseExtension} instead.
*/
public class SessionExtension extends DatabaseExtension implements Session, BeforeEachCallback, AfterEachCallback {
private Session realSession;
@Override
public void beforeEach(ExtensionContext context) throws Exception {
super.beforeEach(context);
realSession = driver().session();
}
@Override
public void afterEach(ExtensionContext context) {
if (realSession != null) {
realSession.close();
}
}
@Override
public boolean isOpen() {
return realSession.isOpen();
}
@Override
public void close() {
throw new UnsupportedOperationException("Disallowed on this test session");
}
@Override
public Transaction beginTransaction() {
return realSession.beginTransaction();
}
@Override
public Transaction beginTransaction(TransactionConfig config) {
return realSession.beginTransaction(config);
}
@Override
@SuppressWarnings("deprecation")
public <T> T readTransaction(TransactionWork<T> work) {
return realSession.readTransaction(work);
}
@Override
@SuppressWarnings("deprecation")
public <T> T readTransaction(TransactionWork<T> work, TransactionConfig config) {
return realSession.readTransaction(work, config);
}
@Override
public <T> T executeRead(TransactionCallback<T> callback, TransactionConfig config) {
return realSession.executeRead(callback, config);
}
@Override
@SuppressWarnings("deprecation")
public <T> T writeTransaction(TransactionWork<T> work) {
return realSession.writeTransaction(work);
}
@Override
@SuppressWarnings("deprecation")
public <T> T writeTransaction(TransactionWork<T> work, TransactionConfig config) {
return realSession.writeTransaction(work, config);
}
@Override
public <T> T executeWrite(TransactionCallback<T> callback, TransactionConfig config) {
return realSession.executeWrite(callback, config);
}
@Override
@SuppressWarnings("deprecation")
public Bookmark lastBookmark() {
return realSession.lastBookmark();
}
@Override
@SuppressWarnings("deprecation")
public Set<Bookmark> lastBookmarks() {
Bookmark bookmark = lastBookmark();
if (bookmark == null || bookmark.isEmpty()) {
return Collections.emptySet();
} else if (bookmark.values().size() == 1) {
return Collections.singleton(bookmark);
} else {
return bookmark.values().stream().map(Bookmark::from).collect(Collectors.toCollection(HashSet::new));
}
}
@Override
public Result run(String query, Map<String, Object> parameters) {
return realSession.run(query, parameters);
}
@Override
public Result run(String query, Value parameters) {
return realSession.run(query, parameters);
}
@Override
public Result run(String query, Record parameters) {
return realSession.run(query, parameters);
}
@Override
public Result run(String query) {
return realSession.run(query);
}
@Override
public Result run(Query query) {
return realSession.run(query.text(), query.parameters());
}
@Override
public Result run(String query, TransactionConfig config) {
return realSession.run(query, config);
}
@Override
public Result run(String query, Map<String, Object> parameters, TransactionConfig config) {
return realSession.run(query, parameters, config);
}
@Override
public Result run(Query query, TransactionConfig config) {
return realSession.run(query, config);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,264
|
\section{Introduction}
The existing classification scheme of supernovae \citep[SNe,][]{filippenko_1997, Gal-Yam2017} has been successful in sorting out the majority of objects discovered, in a way that facilitates their study or practical use, e.g. as standard candles \citep{1998AJ....116.1009R, Perlmutter_1999}. It does that by relying mainly on their spectra.
The current classification scheme divides SNe to Type I and Type II, comprised of SNe lacking H-lines and containing them, respectively. The reason for this is mostly historical \citep{minkowski1941}, and modern understanding distinguishes between two other categories based on the progenitor star and the underlying physical explosion process. The first class is thermonuclear SNe, including mostly members of the spectroscopic Type Ia group, that were found \citep{2011Natur.480..344N} to be a result of a white dwarf (WDs) progenitor, exploding in a manner still actively researched \citep{Ia_research1,Ia_research2}. Spectroscopically, normal Type Ia SNe are identifiable by the strong {\ion{Si}{2}~6355\AA} line seen in peak spectra; though this is less clear in some sub-types. The rarer group of Ca-rich Type I SNe has also been associated with long-lived, WD progenitors \citep{Perets2010,De2020}. The second category is core-collapse SNe, originating from massive stars whose self-gravity ceases to be supported by nuclear fusion in their core. These are further divided into the H-rich Types II and IIn, the latter distinguished by their narrow Balmer emission lines ascribed to circumstellar material (CSM), and the stripped-envelope SN Types, which are the explosive deaths of massive stars which have already lost their outer hydrogen shell prior to explosion. These include Types Ib and Ic, the latter differing by the lack of He in peak spectra (in particular, the {5876\AA}, {6678\AA}, and {7065\AA} lines). Type IIb SNe exhibit hydrogen lines only in early spectra, which later disappear, indicating a much thinner layer of H in their progenitor star than those of normal Type II SNe. He-poor stripped-envelope SNe with broad spectral lines belong to Type Ic-BL, corresponding to explosions with larger kinetic energy per unit mass, and hence higher ejecta velocity -- up to about threefold compared to the $\sim 10^4 \,\mathrm{km/s}$ of normal Type Ic SNe. While these are the main, most frequently observed Types, others exist, including stripped-envelop SNe with CSM interaction of Types Ibn \citep[e.g.,][]{Pastorello2008} and Icn \citep{Gal-Yam2021}, superluminous SNe \citep[e.g.,][]{Gal-Yam2019}, as well as numerous subdivisions of the Types mentioned here. A recent review about the current classification scheme of SNe could be found in \citet{Gal-Yam2017}.
In practice, an experienced individual could examine a spectrum of some SN, mainly by looking at emission or absorption lines, determine its type and enter it to some database. The problem of subjectivity that could possibly emerge in this human classifier scenario has been treated by developing tools such as \texttt{Superfit} \citep{superfit} or \texttt{SNID} \citep{snid}, both relying on a template bank of SN spectra to which a query spectrum is compared. Considerations such as redshift, reddening through Milky Way or host dust, photometric calibration of the spectrum and contamination by host lines are present in these methods as fit parameters or have their effects diminished by smoothening or continuum estimation and subtraction. These methods, as opposed to human classification, standardize the treatment of different SNe such that different reddening or contamination conditions have less impact on the classification, and they are also more objective, being independent of the user, his attention to details or other forms of human error. The use of such automatic tools also streamlines the task of classification of large amounts of data. Another quantitative approach to classification, of Type I SNe, also relying on peak spectra, was presented by \citet{sun2017}. This classification uses the estimation of the depths of a line around {6150\AA} attributable to \ion{Si}{2} or H, depending on the SN, and of {\ion{O}{1}~7774\AA} and achieves significant separation between the main Type I SN subtypes (Ia, Ib and Ic), as well as between different Ia subtypes.
However, one drawback in the presently available schemes and methods which still awaits remedy is that they do not rely on the entirety of information available for a SN, but on a single spectrum of it, i.e. from a single epoch. This could also lead to ambiguity, this time not a subjective, user-dependent one, but the mere fact that the current scheme could assign different types to the same object when faced with different spectra of it. An example is Type IIb, which includes core-collapse SNe exhibiting H in early spectra but not in later ones. \citet{phys_mot} have approached the study of this type-continuum by distinguishing between 4 sub-types ranging from Ib to IIb based on the $\mathrm{H}\alpha$ emission and absorption lines. They also treated the continuum between the He-poor stripped core-collapse SNe Types Ic and Ic-BL, the latter being defined by significantly higher explosion velocities than the former, evident in the breadth of spectral lines. This difference in velocities is, just as the Ib--IIb case, not only a continuous parameter discretized into two groups, but also a time-dependent property, as they show that it changes throughout the lifetime of the transient. They do this by examining in practice the related (though only weakly correlated) property of feature count, and divide the continuum into 5 subtypes depending on the average of the feature count over a certain time interval. Examples for He-poor stripped core-collapse events of uncertain classification are SN2004aw \citep{04aw} and SN2016coi \citep{2018MNRAS.473.3776K}. Additionally, a certain confusion in the classification of stripped core-collapse SNe as Ib or Ic exists. One reason for it is the presence of He, observed in some events initially classified as Type Ic (e.g., SN2007gr, SN2009jf). This has been perceived as a flaw in the current classification scheme, rather than an incomplete physical understanding of the objects \citep{on_Ibc,09jf_07gr,snid}.
There exists then a certain need for a classification scheme both quantitative and taking into account the entire \emph{spectral-temporal} energy distribution of a SN (i.e., the spectral flux $f(t,\lambda)$ along both wavelength and time) in order to provide a more definite answer about the degree of similarity between, and thus the proper taxonomic sorting of SNe. Such a method could perhaps further unveil some structure in the space of SNe, possibly in the form of a few hidden parameters controlling their properties. Those might, ideally, be later ascribed to physical properties through correlation analysis.
\subsection{Machine Learning Methods in Astronomy}
Various methods of data analysis have been used in astronomy, the emphasis being put in recent years on machine learning methods. A few examples for uses in the context of SN classification are \citet{sasdelli+2014_Ia_metric,sasdelli+2016_Ia_autoencoder,Williamson_2019}. \citet{fremling2021sniascore} presents a binary deep-classifier named \texttt{SNIascore}, which succeeds in classifying SNe as Type Ia with very low error rates. See also \citet{baron2019machine} for a useful review on machine learning methods in astronomy, including those used in this work. The question of quantitative SN classification could be approached by means of some machine learning algorithm which would be trained on feature vectors describing a set of SNe. This training set should then ideally consist of well-sampled SNe, whose classification and comparison to other objects is well-understood. After training, the algorithm should be able to provide new insight about this taxonomy of and similarity between objects when input SNe from or outside of the training set. The method we use in this work is an unsupervised Random Forest (RF) serving as a metric \citep{shi_horvath_2006}, which has been found viable in defining a distance between spectra \citep{baron_rfalg,reis2018a}. Section~\ref{subsec:rf} describes the algorithm and its application.
\section{Objective}
In this work, we present a method to characterize, or classify, SNe based on their spectral-temporal energy distribution by means of a data-driven embedding in an abstract metric space. We compare the spectroscopic types in the current scheme -- as determined by a human expert after examining the spectral sequence available for the SN in question -- with the resulting embedding.
Other than this comparison, we examine possible Type-continua or new subtypes which may arise from the embedding.
\section{Method}
\subsection{Estimating the Spectral-Temporal Energy Distribution of a SN}\label{subsec:pycoco}
\begin{figure*}
\myplot{1}{07af_PyCoCo_complete.jpg}
\caption{An example of the output (left) and input (right) of \texttt{PyCoCo} for SN2007af. The interpolated, calibrated and dereddened rest-frame flux function $f(t, \lambda)$ can be seen on the left, and the input spectral and photometric data is on the right. Note that the colors on the right panel differ from those on the left as the input spectra and photometry are not yet calibrated or dereddened.}
\label{fig:pycoco_example}
\end{figure*}
The raw data of each SN, i.e. its spectra and multi-band photometry, are processed with \texttt{PyCoCo}, a program for template creation of spectral energy distribution of SNe developed by \citet[][hereafter V19]{Vincenzi_2019}. It uses Gaussian Process Regression to interpolate the flux in the intervals between the input spectra, with the photometry data used both for flux calibration of the spectra and as data points. \texttt{PyCoCo} also corrects the flux for Milky Way extinction and host extinction\footnote{Host uncorrected output is also produced, though in this work we only use the host corrected versions.} values the user inputs, and removes tellurics and host lines, as well as transforming the data into the rest frame of the SN. The final output from \texttt{PyCoCo} is the (normalized) rest-frame spectral energy distribution $f(t, \lambda)$ of the input SN on some grid of time and wavelength.
We keep the power $n$ of the light-curve rising fit in \texttt{PyCoCo}, $f \propto (t-t_0)^n$, (V19, Eq.~2) as a free parameter for all the SNe we apply it on, and we also choose to not extend and oversample the light-curves at late times, as our analysis concentrates on relatively early times (see next section). As the majority of \texttt{PyCoCo} outputs used in this work were generated by V19 (see Section~\ref{sec:data}), it should be mentioned that they only left $n$ as a free parameter in cases where the photometric data shortly after explosion was good enough to allow a reasonable fit. When not, $n$ was fixed to a value, which depends on whether the SN is H-rich (Types II and IIn) or not (the rest of the types), as indicated by the user. This is the only place where input regarding classification is used in \texttt{PyCoCo}. It should also be noted that a different light curve rise model was used for the data sets of Type IIb SN1993J, SN2011dh, SN2011fu, and SN2013df generated by V19, in order to take into account their double early peaks. This is also true for SN2006aj, but in this case its double peak is likely a result of its unusually early detection, rather then an actual peculiarity \citep{06aj}, so we omit the first peak data for this object. Readers are referred to Section~2.1 in V19 for details about the light-curve fits in \texttt{PyCoCo}.
The code used in this section is available on GitHub\footnote{\url{https://github.com/ofek-b/PyCoCo_templates}}. See Figure~\ref{fig:pycoco_example} for an example of the process.
\subsection{Preparation of Feature Vectors}\label{subsec:prep}
We use an algorithm from \citet{spectres}\footnote{\url{https://github.com/ACCarnall/spectres}} to rebin in wavelength and perform linear interpolation in time to obtain the fluxes on a uniform grid $T \times \Lambda$ over all SNe, where
\begin{center}
$T = $ 0 d to 50 d at 1 d intervals\\
$\Lambda = $ 4000 {\AA} to 8000 {\AA} at 40 {\AA} intervals.
\end{center}
Times are counted since the estimated explosion of each template. We choose this wavelength interval $\Lambda$ as it is covered available photometric data for almost all SNe in our data set, making the output of \texttt{PyCoCo} more credible. The time interval $T$ is chosen so it includes times where most SNe are not too faint.
As others have done (e.g., \citealt{sasdelli+2014_Ia_metric}, or less directly in \citealt{snid}), we use the wavelength derivative of the logarithm of the flux of every SN,
$$
\tilde{f}(t,\lambda) \equiv \frac{\mathrm{d}}{\mathrm{d}\lambda}\mathrm{log}f(t,\lambda),
$$
instead of the flux itself, in order to emphasize spectral lines and diminish the effect of the unaccounted for reddening. The data is then sorted in an $N$-row matrix $X$, where each row $X_i$ corresponds to the $i$-th SN in our data set:
$$
X_i = (\tilde{f}(t_j, \lambda_j))_{(t_j, \lambda_j) \in T \times \Lambda}.
$$
In order to account for some missing data points we then use the Expectation Maximization Principle Component Analysis (EMPCA) algorithm presented in \citet{empca}\footnote{\url{https://github.com/sbailey/empca}} on our feature matrix $X$ to produce a principal components matrix, $P_C$, which is then immediately transformed back to the feature space to produce $X_C$, now without missing data. The number of principal components we choose is $C = 50$, which preserves 93\% of the variance.
\subsection{Unsupervised Random Forest as a Metric}\label{subsec:rf}
\begin{figure*}
\myplot{0.7}{example_bintree.pdf}
\caption{An illustration of a trained binary Decision Tree, multiple different instances of which comprise an RF. The input object propagates according to its features \textsf{f}$_i$.}
\label{fig:tree}
\end{figure*}
We now train an unsupervised RF on the data in $X_C$ and use it as a metric on the set of SNe as described by the same matrix $X_C$. This section describes this process.
RF is originally a supervised machine learning classifier \citep{ho95}. A brief description of it follows, regarding only binary classification as relevant for this work. First, we describe binary Decision Trees. A Decision Tree is a supervised classifier by itself, structured as a tree graph -- a directed graph with a single start node, several terminal nodes and several "generations" of intermediate nodes in between, where every intermediate node has exactly one arrow pointing to it from its parent and exactly two arrows which it points to its children. When an object is queried into the Decision Tree, each non-terminal node $k$ hands it over to one of its two child nodes, chosen according to the truth value of a condition of the form $f_{i_k} < c_k$, where $f_{i_k}$ is the value of the $i_k$-th feature of the Tree input, and $c_k$ is a real number. The training process consists of the simultaneous input of all objects in the training set. Before propagating a set of objects which has arrived to any node $k$, the training algorithm chooses the parameters $i_k$ and $c_k$ providing the best separation between the known labels for the objects. The definition of goodness of separation is a matter of choice (a so-called hyperparameter). When all of the objects which arrived in a node are of the same label, this node is set as a terminal node and is assigned the label. Training ceases when all objects arrived at terminal nodes. Classifying a queried object occurs by propagating it through the trained Decision Tree and returning the label corresponding to the terminal node it reaches. Note that Decision Trees and their training process are deterministic. See Figure~\ref{fig:tree} for an illustration and the classification scheme for Type~I~SNe presented in \citet[][Section~4.2]{sun2017} for a working example.
An RF consists of multiple Decision Trees. Every Tree is (i) only trained on a random subset of the training set, and (ii) only allowed to use a random subset of the features. The class assigned to a queried object is determined by a majority vote of all the Decision Trees in the Forest. An RF is thus an ensemble learning method, aggregating the results of the Trees to achieve better accuracy and robustness. In order to use an RF as a metric, one performs the following \emph{unsupervised} training process. The training data set in question is used to create a synthetic data set of the same size. The samples in the synthetic data set have the same number of features as in the original data set and the same marginal distributions for each feature, but zero correlations between features. Supervised training follows, as described above, using the real and synthetic sets, where the real set is assigned the label \texttt{real} and the synthetic set is assigned the label \texttt{synth}. Note that these labels have nothing to do with SN types or with the data itself in any way, and they merely differentiate between the real, correlation exhibiting data, and the synthetic, uncorrelated data. After training, the similarity returned for two queried objects is the fraction of Trees in the Forest for which the two objects end up in the same terminal node, out of only the Trees which assign label \texttt{real} for both objects (i.e., the correct label, as one tries to find the similarity measure between two real objects). The metric, distance or dissimilarity (used interchangeably in this work) is obtained by subtracting the similarity from unity.
This similarity measure thus learns the most prominent correlations in the training data, and measures to which extent do two given objects share such correlations. One such correlation could be, for instance, the presence of all three strongest He lines in a spectrum, as opposed to only some of them.
The algorithm we use was written by \citet{baron_rfalg}\footnote{\url{https://github.com/dalya/WeirdestGalaxies}}. The number of Trees used is 2000, and the separability criterion is the Gini Impurity. The output of our analysis is the distance matrix for our SN data set, with entries between 0 and 1.
The code used in Sections~\ref{subsec:prep} and \ref{subsec:rf}, as well as in the following visualization of the results is available on GitHub\footnote{\url{https://github.com/ofek-b/spectra_in_time}}.
\section{Data}\label{sec:data}
Alongside the description of the \texttt{PyCoCo} code, V19 also includes the code output for 67 core-collapse SNe, listed in Table~2 of their publication. Our data set is built upon those SNe, with some exclusions and type changes of SNe listed in Table~\ref{tab:vincmod}. We extend this data set by adding the SNe listed in Table~\ref{tab:mydata}. The final list of 82 SNe used in this work is given in Table~\ref{tab:alldata}, along with the Data Quality Index (DQI) described next. This data set will be made available upon request.
Two SN Types missing from the V19 set, which we also did not add, are Ca-rich SNe and H-poor superluminous SNe (SLSN-I). In the case of Ca-rich SNe, the reason is the poor data sets available for those objects. In the case of SLSN-I, peak light often occurs late, and outside our temporal window, such that our selected time grid would miss important information about those objects, the same holds also for SN1987A-like Type II SNe. One could envision including these groups once more data are available, and including a temporal renormalization for SLSNe-I and 87A-like events.
\begin{table*}
\centering
\caption{The SNe from V19 which were excluded from the training set in this work or whose assigned spectroscopic type was changed after examining their spectral series.\label{tab:vincmod}}
\begin{tabular}{lll}
\tableline
Name & Type in V19 & Comments\\
\tableline
\textit{Excluded:}\\
SN1987A & 87A-like & long rise time with peak outside our temporal range\\
SN2005bf & Ib & very anomalous, double-peaked light curve \citep{05bf}\\
SN2008D & Ib & highly extinguished \citep{rabinak_waxman}; unique early time data\\
SN2011bm & Ic & very wide light curve \citep{11bm}\\
SN2016bkv & II & faint, unusually long light curve, weaker lines \citep{16bkv}\\
SN2008in & II & ${\rm DQI} < 0.8$ \\
SN2009dd & II & ${\rm DQI} < 0.8$ \\
SN2013fs & II & added again manually for the analysis in Section~\ref{sec:displ}. appears on Table~\ref{tab:mydata}. \\
SN2008aq & IIb & ${\rm DQI} < 0.8$ \\
SN2009ip & IIn & peculiar object, may not be a SN \citep{09ip}\\
SN2011ht & IIn & ${\rm DQI} < 0.8$ \\
\textit{Type changed:}\\
SN2009jf & Ib & changed to Ic, see \citet{Gal-Yam2017}\\
SN2010al & IIn & changed to Ibn \citep{10al}\\
SN2008fq & IIn & changed to II\\
SN2007pk & IIn & changed to II\\
\tableline
\end{tabular}
\end{table*}
\begin{table}
\centering
\caption{The SNe used for training in this work in addition to those from V19.\label{tab:mydata}}
\begin{tabular}{lllllll}
\tableline
Name & z & Type & Photometry & \# Spec. & $E(B-V)_\mathrm{host}$ & Ref. \\
\tableline
SN2013fs & 0.011855 & II & $UVW2,UVM2,UVW1,U,B,V,R,I$ & 23 & 0.015 & (1-3) \\
SN2017gpn & 0.007388 & IIb & $B,V,g,r,i$ & 8 & 0.000 & (4) \\
SN2006el & 0.017000 & IIb & $B,V,R,r,i$ & 6 & 0.081 & (5-9) \\
SN1996cb & 0.002372 & IIb & $B,V,R$ & 8 & 0.120 & (5, 7, 10-11) \\
SN2015ap & 0.011400 & IIb & $B,V,u,g,r,i$ & 17 & 0.000 & (4-5) \\
SN2010jl & 0.010700 & IIn & $UVW2,UVM2,UVW1,U,B,V,u,r,i$ & 3 & 0.022 & (12-16) \\
SN2011fe & 0.000804 & Ia & $UVW2,UVM2,UVW1,U,B,V,R,I,r$ & 27 & 0.014 & (13, 17-28) \\
SN2012fr & 0.004000 & Ia & $UVW2,UVM2,UVW1,U,B,V,R,I,g,r,i$ & 22 & 0.000 & (13, 22, 24, 29-31) \\
SN2017erp & 0.006174 & Ia & $UVW2,UVM2,UVW1,U,B,V,R,I$ & 11 & 0.097 & (13, 20, 22, 26, 32) \\
SN2005cf & 0.006461 & Ia & $UVW2,UVM2,UVW1,U,B,V,R,I$ & 23 & 0.100 & (13, 26, 33-36) \\
SN2012ht & 0.004000 & Ia & $UVW2,UVM2,UVW1,U,B,V,R,I,g,r,i$ & 14 & 0.000 & (13, 22, 37-39) \\
SN2011by & 0.002843 & Ia & $UVW2,UVM2,UVW1,U,B,V,R,I$ & 11 & 0.039 & (13, 17, 20, 26, 40-41) \\
SN2009ig & 0.008770 & Ia & $UVW2,UVM2,UVW1,U,B,V,R,I$ & 12 & 0.000 & (13, 17, 20, 22, 26, 42-43) \\
SN2007af & 0.005464 & Ia & $UVW2,UVM2,UVW1,U,B,V,R,I$ & 14 & 0.130 & (13, 26, 34, 44-47) \\
SN2004dk & 0.005200 & Ib & $B,V,R,I$ & 8 & 0.180 & (7-8, 48-50) \\
SN2015ah & 0.016000 & Ib & $B,V,u,g,r,i,z$ & 9 & 0.020 & (4-5) \\
SN2016jdw & 0.018900 & Ib & $u,g,r,i,z,c,o$ & 5 & 0.000 & (4) \\
SN2017bgu & 0.008000 & Ib & $u,g,r,i,z,c,o$ & 7 & 0.020 & (4) \\
SN1999ex & 0.011414 & Ibc & $U,B,V,R,I$ & 5 & 0.028 & (51-52) \\
SN2006lc & 0.016200 & Ibc & $B,V,g,r,i,z$ & 6 & 0.298 & (5-7, 13, 53-56) \\
SN2019deh & 0.054690 & Ibn & $g,r,i,z$ & 9 & 0.224 & (9, 57) \\
SN2019aajs & 0.035800 & Ibn & $g,r,i,z$ & 4 & 0.224 & (9, 57) \\
SN2019iep & 0.057000 & Ibn & $g,r$ & 7 & 0.224 & (9, 57) \\
SN2014L & 0.008029 & Ic & $U,B,V,R,I$ & 13 & 0.630 & (5, 58) \\
SN2016coi & 0.003600 & Ic-BL & $UVW2,UVM2,UVW1,U,B,V,R,I,g,r,i$ & 48 & 0.000 & (13, 59-61) \\
SN2003jd & 0.018860 & Ic-BL & $B,V,R,I$ & 10 & 0.100 & (5-7, 62) \\
\tableline
\end{tabular}
\tablerefs{(1) \cite{2016MNRAS.459.3939V}; (2) \cite{2017NatPh..13..510Y}; (3) \cite{2016PASA...33...55C}; (4) \cite{2019MNRAS.485.1559P}; (5) \cite{2019MNRAS.482.1545S}; (6) \cite{2014ApJS..213...19B}; (7) \cite{2014AJ....147...99M}; (8) \cite{2011ApJ...741...97D}; (9) median $E(B-V)_\mathrm{host}$ value, \cite{2016MNRAS.458.2973P}; (10) \cite{1999AJ....117..736Q}; (11) \cite{2001AJ....121.1648M}; (12) \cite{2012AJ....144..131Z}; (13) \cite{2014Ap&SS.354...89B}; (14) \cite{2016MNRAS.456.2622J}; (15) \cite{2014ApJ...797..118F}; (16) \cite{2012AJ....143...17S}; (17) \cite{2016PASP...128...961}; (18) \cite{2013A&A...554A..27P}; (19) \cite{2016ApJ...820...67Z}; (20) \cite{2019MNRAS.490.3882S}; (21) \cite{2013NewA...20...30M}; (22) \cite{2020MNRAS.492.4325S}; (23) \cite{2012ApJ...752L..26P}; (24) \cite{2017MNRAS.472.3437G}; (25) \cite{2014MNRAS.439.1959M}; (26) \cite{2012MNRAS.425.1789S}; (27) \cite{2011Natur.480..344N}; (28) \cite{2015MNRAS.446.3895F}; (29) \cite{2013ApJ...770...29C}; (30) \cite{2018arXiv180310095C}; (31) \cite{2014AJ....148....1Z}; (32) \cite{2019ApJ...877..152B}; (33) \cite{2009ApJ...697..380W}; (34) \cite{2010ApJS..190..418G}; (35) \cite{2007A&A...471..527G}; (36) \cite{2007MNRAS.376.1301P}; (37) \cite{2018PASP..130f4101V}; (38) \cite{2014ApJ...782L..35Y}; (39) \cite{2018arXiv180906381B}; (40) \cite{2013MNRAS.430.1030S}; (41) \cite{2020MNRAS.491.5991F}; (42) \cite{2012ApJ...744...38F}; (43) \cite{2012ApJ...749...18B}; (44) \cite{2007ApJ...671L..25S}; (45) \cite{2011Sci...333..856S}; (46) \cite{2010ApJ...721.1608B}; (47) \cite{2012AJ....143..126B}; (48) \cite{2017PASP..129e4201S}; (49) \cite{2008ApJ...687L...9M}; (50) \cite{2008A&A...488..383H}; (51) \cite{2002AJ....124..417H}; (52) \cite{2002AJ....124.2100S}; (53) \cite{2017arXiv170707616S}; (54) \cite{2011A&A...526A..28O}; (55) \cite{2014arXiv1401.3317S}; (56) \cite{2018A&A...609A.135S}; (57) Kool et al. in preparation; (58) \cite{2018ApJ...863..109Z}; (59) \cite{2019ApJ...883..147T}; (60) \cite{2018MNRAS.478.4162P}; (61) \cite{2018MNRAS.473.3776K}; (62) \cite{2008MNRAS.383.1485V}}
\end{table}
\begin{table}
\centering
\caption{All SNe used for training in this work and the DQI calculated for them.\label{tab:alldata}}
\begin{tabular}{lp{15cm}}
\tableline
Type & Names (DQI) \\
\tableline
II & ASASSN14jb (0.91), ASASSN15oz (0.89), SN1999em (0.98), SN2004et (0.83), SN2005cs (0.96), SN2007od (0.91), SN2007pk (0.92), SN2008bj (0.83), SN2008fq (0.95), SN2009N (0.84), SN2009bw (0.83), SN2009ib (0.87), SN2009kr (0.90), SN2012A (0.97), SN2012aw (1.00), SN2013ab (0.99), SN2013am (0.97), SN2013by (0.95), SN2013ej (1.00), SN2013fs (1.00), SN2014G (1.00), SN2016X (1.00) \\
IIb & SN1993J (1.00), SN1996cb (0.87), SN2006T (0.95), SN2006el (0.84), SN2008ax (1.00), SN2008bo (0.91), SN2011dh (1.00), SN2011ei (0.95), SN2011fu (1.00), SN2011hs (0.96), SN2013df (0.85), SN2015ap (0.99), SN2016gkg (0.95), SN2017gpn (0.97) \\
IIn & SN2006aa (0.92), SN2010jl (0.90) \\
Ia & SN2005cf (1.00), SN2007af (0.93), SN2009ig (0.91), SN2011by (0.99), SN2011fe (0.99), SN2012fr (1.00), SN2012ht (0.99), SN2017erp (0.99) \\
Ib & SN1999dn (0.90), SN2004dk (0.87), SN2004gq (0.92), SN2004gv (0.90), SN2005hg (0.95), SN2006ep (0.86), SN2007Y (0.96), SN2007uy (0.83), SN2009iz (0.91), SN2015ah (0.92), SN2016jdw (0.90), SN2017bgu (0.86), iPTF13bvn (1.00) \\
Ibc & SN1999ex (0.91), SN2004gt (0.80), SN2006lc (0.87), SN2012au (0.81) \\
Ibn & SN2010al (0.94), SN2019aajs (0.88), SN2019deh (0.98), SN2019iep (0.97) \\
Ic & SN1994I (0.96), SN2004aw (0.85), SN2004fe (0.96), SN2007gr (0.96), SN2009jf (0.94), SN2013ge (0.99), SN2014L (0.94) \\
Ic-BL & SN1998bw (0.89), SN2002ap (0.99), SN2003jd (0.89), SN2006aj (0.93), SN2007ru (0.88), SN2009bb (0.92), SN2012ap (0.91), SN2016coi (0.99) \\
\tableline
\end{tabular}
\end{table}
For the SNe we added, spectra were taken from WISeREP\footnote{\url{https://wiserep.weizmann.ac.il}} \citep{wiserep} and photometry from the Open Supernova Catalog\footnote{\url{https://sne.space}} \citep{osc}, where available. The values for Milky Way reddening $E(B-V)_\mathrm{MW}$ were taken from \citet{ebvmw}. The host reddening estimations $E(B-V)_\mathrm{host}$ were taken from the provided references. When no estimation could be found, the median value for the SN type was taken.
\subsection{Data Quality Index}
The quality of the output of \texttt{PyCoCo} depends on the coverage of the spectral-temporal region $T\times\Lambda$ by the spectroscopic and photometric data of each object. A Data Quality Index (DQI) between 0 and 1 is thus calculated for each SN in the following manner. For each point on the $T\times\Lambda$ grid, the distance to the nearest input data point (spectroscopic or photometric) is calculated as the usual Euclidean distance, where the time and wavelength axes are normalized by their total length. The fraction of grid points which are closer than 0.1 to a data point is defined as the DQI. We only included SNe with ${\rm DQI} \geq 0.8$ in our analysis. The impact of this choice is minimal (the removal of 4 objects; Table~\ref{tab:vincmod}).
\section{Results}
After applying the EMPCA algorithm, the fraction of data variance explained is $93\%$. Figure~\ref{fig:dissim} shows the distance matrix $D$ resulting from an unsupervised RF as described. It is not completely symmetrical due to computation errors -- the median absolute symmetric difference of off-diagonal terms is $0.005$ -- so we enforce symmetry by using $\frac{1}{2}(D+D^\mathrm{T})$ as the dissimilarity matrix. A certain degree of block-diagonality is noticeable, expressing agreement with the current classification scheme for these well-documented SNe.
\begin{figure*}
\myplot{1.1}{distmatrix.png}
\caption{The resulting dissimilarity matrix after applying the RF dissimilarity measure on the data set. Colors represent dissimilarity, are in linear scale and only vary between the 1st and 50th percentile.}
\label{fig:dissim}
\end{figure*}
\subsection{Visualization of The Metric Space}\label{sec:vis}
For visualisation of this dissimilarity matrix inside a low-dimensional Euclidean space while trying to preserve structure, we use t-Distributed Stochastic Neighbor Embedding (tSNE), developed by \citet{tsne}. The visualization in 3d space is shown in Figure~\ref{fig:tsne_3d}, with colors representing the spectroscopic type assigned by a human expert after manually inspecting the evolution of spectra available for the SN. Examining the visualization one can notice the aforementioned block-diagonality in the form of well-separated clusters which are relatively homogeneous in type. We remind the reader that the spectroscopic types represented by the colors were not used in the analysis which led to the dissimilarity matrix or the visualization (the only exception is mentioned in Section~\ref{subsec:pycoco}).
Another way to visualize the results is to look at the Minimum Spanning Tree (MST) of the fully-connected graph defined by the distance matrix. The MST is a fully-connected subgraph minimal in the sum of edge weights\footnote{The MST is generally not unique, though it is unique in this case.} (here -- distances) and is shown in Figure~\ref{fig:mst}. The algorithm by \citet{yamada}\footnote{\url{https://github.com/dakota-hawkins/yamada}} was used. The concept of MST in the context of an abstract dissimilarity space could be employed, as was done by \citet{sequencer}, to extract sequences from the data, which could later be compared with physical parameters.
\begin{figure*}
\makebox[\textwidth][c]{\includegraphics[trim=30 20 0 40, clip]{3d.pdf}}
\myplot{0.8}{corner.pdf}
\caption{Embedding of the dissimilarity matrix in Figure~\ref{fig:dissim} by tSNE with parameters \texttt{perplexity = 10, learning\_rate = 10}, shown in 3D (top) and in a corner plot (bottom). The KL divergence with respect to the dissimilarity matrix is $0.20$. The colors represent the types as assigned for the spectral series by a human expert. The axes represent the coordinates of the embedded Euclidean space. A rotatable version of the top panel is available in the online version and under \url{https://github.com/ofek-b/spectra_in_time/blob/rf_dlog_no_pca/3d_paper.html}. The rotation is about the dim 3 axis. The speed and direction of the rotation can be controlled by the buttons or by dragging the slider.}
\label{fig:tsne_3d}
\end{figure*}
\begin{figure*}
\myplot{1}{mst.pdf}
\caption{The MST of the graph of SNe defined by the dissimilarity matrix. Note that edge lengths are arbitrary.}
\label{fig:mst}
\end{figure*}
Looking at Figure~\ref{fig:tsne_3d}, we see several remarkable features. Objects of Type Ibn are close to those of Type IIn probably due to sharing narrow features. In contrast, the proximity of Type Ic-BL objects to the Type II cluster is unexpected. A possible explanation could be that the features those two groups share with one another are the dominance of the continuum.
One notices the very close pair of objects SN2015ap (IIb, purple) and SN2004gq (Ib, blue). A possible explanation is that SN2015ap, although exhibiting H-absorption initially, loses them at a relatively early stage, $\sim 10$ days after explosion, leaving no H evident in its spectrum. Another object to be noticed is SN2004gt (Ibc, cyan) in between the main Ib and Ic clusters, which fits previous classification \citep{04gt_ibc}. As for SN2007Y (main Ib cluster, blue), its initial classification was Ib despite H$\alpha$ absorption lines observed at early times and it has been claimed that IIb would be a better classification \citep{late_IIb,07Y,IIb_07Y}. The embedding suggests that it is part of the main Ib cluster, though it is on its edge.
As could also be observed in the tSNE visualization, the Type~Ic SN2009jf and SN2007gr (two dark green points in the upper main Ic cluster) are very close by, which fits the fact that they are almost identical spectroscopically in the optical and IR wavelengths throughout their evolution \citep{09jf_07gr}. They are also close to the Ib cluster, with the He-poor SN2007gr slightly differing from SN2009jf which shows stronger He lines, leading \citet{09jf_07gr} to classify as Ib, though they acknowledge the possibility that a Ic could be a better choice. Likewise, \citet{on_Ibc} have found that the data of SN2007gr, classified there as Type Ic, may be well explained by a He-rich model, but with weak lines. These authors conclude from their analysis of Type Ib and Ic SNe that the property usually assumed to define Type Ic event, the lack of He, is justified for very few, if any, objects. Indeed, a continuum such as the one visualized in Figure~\ref{fig:tsne_3d}, may be a more suitable way to understand those objects.
\subsection{The Split of Type~Ib SNe}
It can be noticed in the visualizations that some Type~Ib SNe are set apart from the main cluster formed by this type. The SNe in the main cluster are SN2004gv, SN2007Y, SN2006ep, SN2015ah, SN1999dn, iPTF13bvn, SN2005hg and SN2009iz. The offset Type~Ib SNe are SN2004gq, SN2007uy, SN2016jdw, SN2017bgu, SN2004dk and SN2015ap, which is of Type~IIb but still included in this list (see Section~\ref{sec:vis}). Figure~\ref{fig:Ib_split} shows a comparison between the mean spectra of these two groups.
The figure suggests that this split could be correlated with the SN expansion velocity, as the features in the mean spectrum of the offset cluster are wider and bluer, especially before- and at peak. To validate this effect, we examine the raw spectra of each event near peak (i.e., prior to any processing of the data). We find that the average expansion velocities derived from the location of the absorption minimum of the He I lines at $\lambda\lambda5876,6678,7065$\AA\ are $9700\pm1000$ and $12300\pm1200$\,km\,s$^{-1}$ for the main and offset clusters respectively. The velocities were measured using the dedicated tool on WISeREP. This supports the results of our analysis, and suggests that there may be a split between populations of low-velocity and high-velocity SNe Ib. This of course brings to mind the well-known separation between spectroscopically normal SNe Ic and broad-line, high-velocity SNe Ic-BL (e.g., \citealt{Modjaz2016}, \citealt{Gal-Yam2017}, \citealt{phys_mot}), though for SNe Ib the velocity spread between normal and high-velocity events may be less extreme, making this division less obvious.
\begin{figure*}
\myplot{1}{main_ib_vs_offset.png}
\caption{Comparison between the mean spectra as a function of time from explosion of Type~Ib SNe from the main cluster and of the rest of the events (see text) in the offset sub-cluster. The flux $f$ (right panel) and its derivative $\tilde{f}$ (left panel) are used for the comparison. One can notice (more clearly on the left hand side) that the spectral features are narrower, more prominent and have lower expansion velocity, especially before and at peak, for SNe that belong to the main cluster than for those from the offset one.}
\label{fig:Ib_split}
\end{figure*}
\subsection{An Application to SNe With Less Data} \label{sec:displ}
While the SNe used for training are some of the best observed objects of their types, we wish to test to which extent is the embedding of a new SN with lower amount of data useful. Figure~\ref{fig:degraded} shows the effect of only including two spectra in the input to \texttt{PyCoCo} on the displacement of the embedding relative to the "true" one produced by using all of the data for the SN. We test this on the Type~Ia SN2005cf and the Type~II SN2013fs. We do not degrade the photometric data in this test. It can be seen that at least for those SNe, using only two spectra does not affect much the quality of embedding. This is highly relevant since future data sets from ongoing and future surveys such as the Zwicky Transient Facility (ZTF; \citealt{Bellm2019, Graham2019}) and in the future the Rubin Observatory include high-quality multicolor light curves. Spectroscopy is often obtained as soon as possible after discovery (e.g., \citealt{Gal-Yam2011}) and again around peak, so a typical future data set may likely include this combination of two spectra and good light curves.
\begin{figure*}
\plottwo{05cf_degraded.pdf}{13fs_degraded.pdf}
\caption{Embedding SN2005cf (Type Ia) and SN2013fs (Type II) with degraded data in a metric space trained on a set not containing these SNe. After training, each SN was embedded in the space using only a subset of the data. The degraded subsets are the \texttt{PyCoCo} output for the entire photometric data, but only two spectra: one at the B-band maximum and the other at the time given on the x-axis. The y-axis is the distance of the embedded degraded form from the respective cluster (Type~Ia or Type~II SNe), in units of the specific cluster diameter. One can see that the degraded data sets remain in the close vicinity of the mother cluster.}
\label{fig:degraded}
\end{figure*}
\section{Conclusion}
We present a method for quantitative comparison of SNe by means of an unsupervised embedding in a metric space. This data-driven method identifies the important correlations/spectral features evident in the spectral-temporal energy distributions it is trained on. Other methods which have been used for quantitative classification of SNe \citep[e.g.,][]{sun2017,phys_mot} can be understood as supervised decision trees constructed by experienced astrophysicists rather than by an automated training process. Our approach is in principal similar, but chooses an unsupervised method in order to detect the most important correlations -- deemed as such by an objective measure -- in a given training set. The method succeeds in defining some clusters, which mostly agree with the known spectroscopic Types, and in visualizing the IIb-Ib-Ic-Ic-BL continuum. There, objects like SN2007Y, SN2009jf and SN2015ap could get a quantitative answer regarding their Type by this method, while the classical element-based classification is not as conclusive.
One result our method produces is the possible split of Type Ib SNe into two subgroups, of normal and higher-velocity events, perhaps mimicking the more prominent division between spectroscopically normal SNe Ic and SNe Ic-BL. As discussed earlier, this separation could be of a discrete nature and should be investigated further (preferably with a larger sample) to further validate or dismiss its existence and test the underlying differences in the physics of the explosions.
Preliminary tests show that even a small number of spectra, when combined with multi-band photometric measurements, suffice for obtaining a relatively accurate embedding. This could make the method useful in the context of surveys such as those to be conducted by the Rubin Observatory.
The application as a classifier for less typical SNe should be approached with caution, as the training set needs to contain a wide enough selection of SNe (which in turn need to have enough data) in order to be able to produce a meaningful result.
\begin{acknowledgments}
\section*{Acknowledgments}
We thank Dalya Baron, Ido Irani, Barak Zackay and the ZTF collaboration for useful advice, and an anonymous referee for a helpful review.
\software{This work made use of the Python packages \texttt{NumPy} \citep{numpy}, \texttt{Matplotlib} \citep{matplotlib}, \texttt{scikit-learn} \citep{scikit-learn}, \texttt{pandas} \citep{pandas} and \texttt{NetworkX} \citep{networkx}.}
\end{acknowledgments}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,584
|
{"url":"http:\/\/tex.stackexchange.com\/questions\/3413\/does-anything-like-a-pdfxelatex-exist","text":"# Does anything like a \u201cpdfXeLaTeX\u201d exist?\n\nI have generally been using `pdfLaTeX` to typeset my documents. I recently heard about `XeTeX` which supposedly is the same thing, except has better support for things like unicode and fonts.\n\nIs there a tool which operates as XeTeX does but which allows a direct PDF translation, rather than going through intermediate stages?\n\n-\n\n-\nCan you tell Lualatex that a document uses Unicode as its character encoding, without using inputenc? \u2013\u00a0 Charles Stewart Sep 24 '10 at 7:08\nYes, you should not use inputenc (or anything else) if you're using UTF8 encoded input. For fontencoding, use EU2 fontenc or the fontspec package. \u2013\u00a0 topskip Sep 24 '10 at 7:18\nThat was really not subtle, Patrick ;-) \u2013\u00a0 Arthur Reutenauer Sep 24 '10 at 16:58\n@Charles: LuaTeX expects UTF-8 input, full stop. Well, that's not the whole story, as Patrick says, but that's how LuaTeX is supposed to ve used. \u2013\u00a0 Arthur Reutenauer Sep 24 '10 at 17:00\n\nXeLaTeX outputs a PDF by default. Yes, it does use `xdvipdfmx` along the way, but why should that bother you? No DVI file is left behind.\n\n-\nOne problem is that routing the typesetting through `xdvipdfmx` precludes the use of pdfTeX-like enhancements such as those provided by the `microtype` package. `microtype` is currently adding support for LuaTeX. \u2013\u00a0 Sharpie Sep 23 '10 at 18:49\nIt's unclear to me that the intermediate level is the problem. Anyway, `microtype` is also adding (partial) support for XeTeX... See my question here. tex.stackexchange.com\/questions\/2986\/\u2026 \u2013\u00a0 frabjous Sep 23 '10 at 19:45\n`microtype` has added LuaTeX support long ago, and a preliminary version that works with XeTeX is available from xetex.tk \u2013\u00a0 Philipp Sep 23 '10 at 19:54\nI stand corrected. Sorry for the noise. \u2013\u00a0 Sharpie Sep 23 '10 at 20:51\nWhat you say is true for the TeX engine, but XeTeX uses an extended DVI format (en.wikipedia.org\/wiki\/XeTeX), so I expect them to have added PNG support to that format. \u2013\u00a0 Blaisorblade Sep 30 '10 at 16:36","date":"2014-12-18 22:28:38","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9056206345558167, \"perplexity\": 4164.9406193000395}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-52\/segments\/1418802768034.59\/warc\/CC-MAIN-20141217075248-00055-ip-10-231-17-201.ec2.internal.warc.gz\"}"}
| null | null |
{"url":"https:\/\/gamedev.stackexchange.com\/questions\/90776\/libgdx-how-do-i-get-the-translation-and-rotation-of-a-node-from-a-model-instanc","text":"# Libgdx, how do i get the translation and rotation of a node from a model instance?\n\nI have a model instance of a character, and a model instance of a sword weapon. THe character has a node on his right hand so he can \"carry\" the sword. Right now I use this code so the sword's transform follows that of the node.\n\nweaponModelInstance.transform.set(modelInstance.transform).mul(weaponAttachmentNode.globalTransform);\nweaponModelInstance.transform.rotate(Vector3.Z, -90); \/\/ adjustment for blender coordinates\nworld.modelBatch.render(weaponModelInstance, world.environment);\n\n\nthis works, except that the model instance gets scaled to the scale of the node, which i do not want, i want the weapon to keep its scale. I've tried doing the following code to fix this but I end up getting weird (and wrong) rotations for the sword\n\nweaponModelInstance.transform.set(modelInstance.transform).mul(weaponAttachmentNode.globalTransform);\nweaponModelInstance.transform.set(\nweaponModelInstance.transform.getTranslation(new Vector3()),\nweaponModelInstance.transform.getRotation(new Quaternion()),\nnew Vector3(1, 1, 1)\n);\nweaponModelInstance.transform.rotate(Vector3.Z, -90);\nworld.modelBatch.render(weaponModelInstance, world.environment);\n\n\ndo anyone know how to properly copy the world translation and rotation from the node, but not its scale?\n\nYou will need to normalize the rotation matrix to remove its scale.\n\nThis can be done by normalizing all 3 rows of the transform basis matrix.\n\nnew btMatrix3x3(\ntransform.getBasis().getRow(0).normalize()\n, transform.getBasis().getRow(1).normalize()\n, transform.getBasis().getRow(2).normalize()\n);\n\n\nor\n\nnew btMatrix3x3(\ntransform.getBasis()[0].normalize()\n, transform.getBasis()[1].normalize()\n, transform.getBasis()[2].normalize()\n);\n\n\nThis new matrix contains only the rotation without scaling.\n\nIf you want to extract the scaling instead:\n\nnew btVector3(\ntransform.getBasis()[0].length()\n, transform.getBasis()[1].length()\n, transform.getBasis()[2].length()\n);\n\n\u2022 Alternatively, you could use libgdx's Matrix4#getRotation(Quaternion, boolean), the boolean argument being a \"normalizeRotation\" flag that, when set to true, tells libgdx to perform this operation for you. \u2013\u00a0Chris Bode May 2 '15 at 21:11","date":"2020-09-30 20:23:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4748522639274597, \"perplexity\": 1921.57809947245}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-40\/segments\/1600402127397.84\/warc\/CC-MAIN-20200930172714-20200930202714-00544.warc.gz\"}"}
| null | null |
Q: Could not import utils ModuleNotFoundError:No module named 'utils.general' I am trying to run my code and i facing a strange error when i tried to import utils , i have already appended directory using sys.path.insert but still my program is throwing errors
import numpy as np
import matplotlib.image as mpimg
import matplotlib.pyplot as plt
import cv2
import argparse
from glob import glob
import os
import sys
sys.path
sys.path.insert(0, '/home/Documents/idr/code')
#import utils.utils as utils
import utils.general as utils
def get_Ps(cameras,number_of_cameras):
Ps = []
for i in range(0, number_of_cameras):
P = cameras['world_mat_%d' % i][:3, :].astype(np.float64)
Ps.append(P)
return np.array(Ps)
and the error that i am incurring is
import utils.general as utils
ModuleNotFoundError: No module named 'utils.general'
in my utils directory the i have a file with general.py but still when i try to import , i am getting an error. please suggest me where i was going wrong.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,482
|
<!DOCTYPE html>
<html>
<head>
<script src="/resources/testharness.js"></script>
<script src="/resources/testharnessreport.js"></script>
</head>
<body>
<script type="module">
import {Result} from '/gen/media/midi/midi_service.mojom.m.js';
import {MockMIDIService} from './resources/mock-midiservice.js';
import {setMidiPermission} from './resources/permissions-helper.js';
const mock = new MockMIDIService();
promise_test(async t => {
await setMidiPermission({}, 'granted');
mock.setStartSessionResult(Result.INITIALIZATION_ERROR);
return promise_rejects_dom(t, 'InvalidStateError',
navigator.requestMIDIAccess());
}, 'initialization failure causes requestMIDIAccess to fail');
</script>
</body>
</html>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,500
|
Minor league baseball returned to Reno this year in the form of the Reno Aces of the Pacific Coast League. Reno built a brand-new ballpark for the Aces. Barbara and I love ballparks, minor league baseball, and road trips, so we drove to Reno last Saturday to watch the Las Vegas 51s play the Aces. The ballpark is gorgeous, and the Aces are very popular: an hour before game time, when the ballpark gates opened, thousands of people were already lined up to get in on a blustery night. The ballpark is downtown, near Harrah's, wedged nicely between the train tracks and the Truckee River. There is no parking at the park: fans park at downtown casinos and office buildings, then walk over. There's a big retail development under construction next to the ballpark, and a riverwalk is also being built. It's a great scene and I hope it will be even nicer in a year or two when the surrounding area is built up.
The most unique and bizarre ballpark feature shows up during the 7th inning stretch. A giant inflated talking baseball rises up over the hitter's eye in center field, holding on with giant inflated hands, and proceeds to talk to the crowd -- as his mouth moves -- and lead the fans in the traditional singing of Take Me Out to the Ball Game.
This is just as bizarre and wonderful as it looks.
The huge outfield led to numerous fly ball doubles.
Outfield berm seating. Lights are drag race theme, with flags of other Pacific Coast League teams. When the Aces score, the lights flash and strobe.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,610
|
A couple of weeks back, whilst working on some building some internal management apps, I finally got around to implementing the Devise Google Authenticator gem into a rails app outside of its own testing app. During this process I realised that I hadn't correctly updated some of the extension's code to properly work with the Devise 2.0 release, in particular the changes to the migration schema. A few amendments, a push or two and version 0.3.3 was now available.
So far though, we've only had a few queries come in. But, to try and capture them in a more appropriate place I've started a Google Groups which, if you wish, you can sign up to and post queries. Or, if it's easier, just hit us up on twitter: @xntrik or @asteriskinfosec.
Asterisk is happy to announce the release of their first (beta) Ruby Gem. The "devise_google_authenticator" gem is a Devise Extension that integrates Google's 2nd Factor Authenticator into Devise's authentication scheme. It's not designed to replace the existing password scheme (database_authenticatable), but it's ideal to provide a second factor authentication mechanism from your smart phone (Android, Blackberry, iOS).
Is based on a modularity concept: use just what you really need.
After you register your user (after clicking Sign Up), you should be displayed with a QR Code. Simply add this to your Google Authenticator app on your phone, enable the authenticator, close down your browser (to clear your session), revisit the website and after you sign in, you'll be prompted for your one time password.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,891
|
import os
import ctypes
from ctypes import *
from ctypes.util import find_library
# Types
CTMfloat = c_float
CTMint = c_int32
CTMuint = c_uint32
CTMcontext = c_void_p
CTMenum = c_uint32
# Constants
CTM_API_VERSION = 0x00000100
CTM_TRUE = 1
CTM_FALSE = 0
# CTMenum
CTM_NONE = 0x0000
CTM_INVALID_CONTEXT = 0x0001
CTM_INVALID_ARGUMENT = 0x0002
CTM_INVALID_OPERATION = 0x0003
CTM_INVALID_MESH = 0x0004
CTM_OUT_OF_MEMORY = 0x0005
CTM_FILE_ERROR = 0x0006
CTM_BAD_FORMAT = 0x0007
CTM_LZMA_ERROR = 0x0008
CTM_INTERNAL_ERROR = 0x0009
CTM_UNSUPPORTED_FORMAT_VERSION = 0x000A
CTM_IMPORT = 0x0101
CTM_EXPORT = 0x0102
CTM_METHOD_RAW = 0x0201
CTM_METHOD_MG1 = 0x0202
CTM_METHOD_MG2 = 0x0203
CTM_VERTEX_COUNT = 0x0301
CTM_TRIANGLE_COUNT = 0x0302
CTM_HAS_NORMALS = 0x0303
CTM_UV_MAP_COUNT = 0x0304
CTM_ATTRIB_MAP_COUNT = 0x0305
CTM_VERTEX_PRECISION = 0x0306
CTM_NORMAL_PRECISION = 0x0307
CTM_COMPRESSION_METHOD = 0x0308
CTM_FILE_COMMENT = 0x0309
CTM_NAME = 0x0501
CTM_FILE_NAME = 0x0502
CTM_PRECISION = 0x0503
CTM_INDICES = 0x0601
CTM_VERTICES = 0x0602
CTM_NORMALS = 0x0603
CTM_UV_MAP_1 = 0x0700
CTM_UV_MAP_2 = 0x0701
CTM_UV_MAP_3 = 0x0702
CTM_UV_MAP_4 = 0x0703
CTM_UV_MAP_5 = 0x0704
CTM_UV_MAP_6 = 0x0705
CTM_UV_MAP_7 = 0x0706
CTM_UV_MAP_8 = 0x0707
CTM_ATTRIB_MAP_1 = 0x0800
CTM_ATTRIB_MAP_2 = 0x0801
CTM_ATTRIB_MAP_3 = 0x0802
CTM_ATTRIB_MAP_4 = 0x0803
CTM_ATTRIB_MAP_5 = 0x0804
CTM_ATTRIB_MAP_6 = 0x0805
CTM_ATTRIB_MAP_7 = 0x0806
CTM_ATTRIB_MAP_8 = 0x0807
# Load the OpenCTM shared library
if os.name == 'nt':
_lib = WinDLL('openctm.dll')
else:
_libName = find_library('openctm')
if not _libName:
raise Exception('Could not find the OpenCTM shared library.')
_lib = CDLL(_libName)
if not _lib:
raise Exception('Could not open the OpenCTM shared library.')
# Functions
ctmNewContext = _lib.ctmNewContext
ctmNewContext.argtypes = [CTMenum]
ctmNewContext.restype = CTMcontext
ctmFreeContext = _lib.ctmFreeContext
ctmFreeContext.argtypes = [CTMcontext]
ctmGetError = _lib.ctmGetError
ctmGetError.argtypes = [CTMcontext]
ctmGetError.restype = CTMenum
ctmErrorString = _lib.ctmErrorString
ctmErrorString.argtypes = [CTMenum]
ctmErrorString.restype = c_char_p
ctmGetInteger = _lib.ctmGetInteger
ctmGetInteger.argtypes = [CTMcontext, CTMenum]
ctmGetInteger.restype = CTMint
ctmGetFloat = _lib.ctmGetFloat
ctmGetFloat.argtypes = [CTMcontext, CTMenum]
ctmGetFloat.restype = CTMfloat
ctmGetIntegerArray = _lib.ctmGetIntegerArray
ctmGetIntegerArray.argtypes = [CTMcontext, CTMenum]
ctmGetIntegerArray.restype = POINTER(CTMuint)
ctmGetFloatArray = _lib.ctmGetFloatArray
ctmGetFloatArray.argtypes = [CTMcontext, CTMenum]
ctmGetFloatArray.restype = POINTER(CTMfloat)
ctmGetNamedUVMap = _lib.ctmGetNamedUVMap
ctmGetNamedUVMap.argtypes = [CTMcontext, c_char_p]
ctmGetNamedUVMap.restype = CTMenum
ctmGetUVMapString = _lib.ctmGetUVMapString
ctmGetUVMapString.argtypes = [CTMcontext, CTMenum, CTMenum]
ctmGetUVMapString.restype = c_char_p
ctmGetUVMapFloat = _lib.ctmGetUVMapFloat
ctmGetUVMapFloat.argtypes = [CTMcontext, CTMenum, CTMenum]
ctmGetUVMapFloat.restype = CTMfloat
ctmGetNamedAttribMap = _lib.ctmGetNamedAttribMap
ctmGetNamedAttribMap.argtypes = [CTMcontext, c_char_p]
ctmGetNamedAttribMap.restype = CTMenum
ctmGetAttribMapString = _lib.ctmGetAttribMapString
ctmGetAttribMapString.argtypes = [CTMcontext, CTMenum, CTMenum]
ctmGetAttribMapString.restype = c_char_p
ctmGetAttribMapFloat = _lib.ctmGetAttribMapFloat
ctmGetAttribMapFloat.argtypes = [CTMcontext, CTMenum, CTMenum]
ctmGetAttribMapFloat.restype = CTMfloat
ctmGetString = _lib.ctmGetString
ctmGetString.argtypes = [CTMcontext, CTMenum]
ctmGetString.restype = c_char_p
ctmCompressionMethod = _lib.ctmCompressionMethod
ctmCompressionMethod.argtypes = [CTMcontext, CTMenum]
ctmCompressionLevel = _lib.ctmCompressionLevel
ctmCompressionLevel.argtypes = [CTMcontext, CTMuint]
ctmVertexPrecision = _lib.ctmVertexPrecision
ctmVertexPrecision.argtypes = [CTMcontext, CTMfloat]
ctmVertexPrecisionRel = _lib.ctmVertexPrecisionRel
ctmVertexPrecisionRel.argtypes = [CTMcontext, CTMfloat]
ctmNormalPrecision = _lib.ctmNormalPrecision
ctmNormalPrecision.argtypes = [CTMcontext, CTMfloat]
ctmUVCoordPrecision = _lib.ctmUVCoordPrecision
ctmUVCoordPrecision.argtypes = [CTMcontext, CTMenum, CTMfloat]
ctmAttribPrecision = _lib.ctmAttribPrecision
ctmAttribPrecision.argtypes = [CTMcontext, CTMenum, CTMfloat]
ctmFileComment = _lib.ctmFileComment
ctmFileComment.argtypes = [CTMcontext, c_char_p]
ctmDefineMesh = _lib.ctmDefineMesh
ctmDefineMesh.argtypes = [CTMcontext, POINTER(CTMfloat), CTMuint, POINTER(CTMuint), CTMuint, POINTER(CTMfloat)]
ctmAddUVMap = _lib.ctmAddUVMap
ctmAddUVMap.argtypes = [CTMcontext, POINTER(CTMfloat), c_char_p, c_char_p]
ctmAddUVMap.restype = CTMenum
ctmAddAttribMap = _lib.ctmAddAttribMap
ctmAddAttribMap.argtypes = [CTMcontext, POINTER(CTMfloat), c_char_p]
ctmAddAttribMap.restype = CTMenum
ctmLoad = _lib.ctmLoad
ctmLoad.argtypes = [CTMcontext, c_char_p]
ctmSave = _lib.ctmSave
ctmSave.argtypes = [CTMcontext, c_char_p]
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,337
|
Q: Prevent user to stop a macro (Excel VBA) I would like to prevent a user from stopping a macro during saving.
I have found the following code :
Application.EnableCancelKey = xlDisabled
It works if you hit the ESC key once, but if you hold the key a long time you can stop the macro.
I tried Application.EnableCancelKey = xlErrorHandler as follow :
Application.EnableCancelKey = xlErrorHandler
On Error GoTo errHandler:
ActiveWorkbook.Save
exitHere:
Exit Sub
errHandler:
msgbox"something"
Resume exitHere
Sometime it works perfectly, but sometimes if I hit the ESC key at the perfect moment and for a few seconds, I am able to stop the macro.
Do you know if there is a way to inactive ESC key for real?
A: I've never done this, but this is how I would start from your code:
On Error GoTo errHandler:
Application.EnableCancelKey = xlErrorHandler
ActiveWorkbook.Save
Application.EnableCancelKey = xlInterrupt
exitHere:
Exit Sub
errHandler:
If MsgBox("something", vbOKCancel) = vbCancel Then
Application.EnableCancelKey = xlInterrupt
Exit Sub
End If
Resume
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,712
|
typedef struct stringpool {
uint32_t size, next;
char pool[];
} stringpool;
// API methods
// public: create a stringpool
void stringpool_init(stringpool* p);
// public: add a string, returning an int
uint32_t stringpool_add(stringpool* p, const char* s);
// public: does this stringpool need to be increased?
int stringpool_needs_bump(stringpool* p);
// public: increase the size of the stringpool
void stringpool_bump_size(stringpool* p);
// public: given an id, return the string
char* stringpool_lookup(stringpool* p, uint32_t id);
// public: returns the byte size of the pool
uint32_t stringpool_size(stringpool* p);
// public: returns the initial byte size for an empty pool
uint32_t stringpool_initial_size();
// public: returns the byte size for the next larger version of a pool
uint32_t stringpool_next_size(stringpool* p);
#endif
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,423
|
package org.semagrow.querylog.api;
/**
* Created by angel on 10/20/14.
*/
public class QueryLogException extends Exception {
public QueryLogException(Exception e) {
super(e);
}
public QueryLogException(String msg) { super(msg); }
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,840
|
{"url":"http:\/\/www.physicsforums.com\/showpost.php?p=94265&postcount=6","text":"View Single Post\n P: 280 Gravity was mentioned. I can help out with that. Gravity is one of the four forces, the other three are the strong nuclear force, the weak nuclear force and the electrostatic force. Gravity is the least well understood of these forces, it is a force that mysteriously arises as a result of a body having mass, although mass is pretty mysterious and there are a few theories floating around about that. Anyway, you probably don't want to know all this so lets jump into the physics of it. Gravity is always an attractive force (although I've just created a thread in this very forum where it suggests to me that it isn't). To fit in with electrostatics we say that an attractive force is always negative. The reason for this is because protons and electrons attract, if you multiply the charges +1 * -1 you get -1, therefore the attractive force is negative. If you have a 1000Kg car, the Earth pulls down on it with a force of 1000g N where g is the graviational field strength, this is just what the acceleration due to gravity would be if the ground suddenly weren't there. g can also be thought of as the force in Newtons that acts vertically downwards on a body of 1Kg. There are some more complicated equations to deal with the forces of attraction between two masses for a certain separation of their centres and the change in gravitational potential when moving from one point in a gravitational filed to another. The equations for the two I've just mentioned are: F = - G*m1*m2 -------- r2 V = - G*m1 ------ r Where G is the gravitational constant 6.67*10-11 If you are dealing with stuff near the Earth's surface then not much of this will have helped yet, I'm very sorry but there isn't really much to do about gravity near the Earth's surface other than constant acceleration and potential energy changes. Since you did mention energy, I thought that I might show the equation for potential energy changes near the Earth: E = mg[del]h It's considerably simpler that the other formulae and the mass of the Earth doesn't need to be included in the formula as it is assumed that g will stay constant. My description of gravity and the laws thereof has probably been a little bit patchy because I don't have anything to bo specific about, so like chroot said, if you want better explanations then you're gonna need to post an example of the kind of question that you're having problems with.","date":"2014-03-08 19:44:13","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7020878195762634, \"perplexity\": 227.62493122589876}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-10\/segments\/1393999661726\/warc\/CC-MAIN-20140305060741-00074-ip-10-183-142-35.ec2.internal.warc.gz\"}"}
| null | null |
Getting Ready to buy Easter Candy? Here's Where to get Sponge Candy!
Community Outraged & Wanting to help – After Special Bike was STOLEN!
We started Totally Buffalo two years ago and what a ride it has been!
We started the site/blog to highlight the positive people, places and things in Buffalo and Western New York. We've published 1,254 stories over the past 24 months and we've loved every minute of it!
We feel so blessed to live in such an amazing community of people who truly care about others. Do we have problems? Of course we do. Do we have a lot of work to do? Oh yes we do. But, we are moving in the right direction and that is a start!
We've been happy to share stories that bring attention to folks and organizations who need help. We've been happy to be the middleman – bringing the generous Western New Yorkers a way to help. It's been amazing!
But, we wanted to do more.
We receive many emails from people, families, charities that need help. We bring their need to our readers and it's usually a perfect match that gets the job done.
We knew we wanted to start a charity at some point – but my gosh, there are so many out there. We didn't want to step on any toes. We have very close relationships with many local charities and help them whenever and however we can.
My family – all of us – wanted to do something in memory of our dad – the most generous man I've ever known. So that is what we are doing.
One thing we've learned over time is how difficult the holidays can be for folks. Most of us enjoy the family time, the spirituality, the food, the drinks, the gifts – the love. But, for many – holidays can be the most difficult of times. Holidays can be lonely. Holidays can be scary and frustrating and sad. So, that is where our thinking went.
We decided to start a charity to help others around the holidays. All of the holidays.
We're calling it Totally Buffalo's Hope for the Holidays. We are in the process of forming a 501c3 – we hoped to be all set by now – but the Government shutdown stalled us a bit. We are almost there.
We are extremely excited about this project that will bring hope for the holidays to many!
Sure, the obvious ones are Thanksgiving and Christmas where we can help bring food to tables and gifts to homes. We can help those who are lonely by showing them we care. We have big plans. But, it's more than that. We will help all year round! We'll bring Valentine's to sick children, Easter baskets to those who may not get them, we'll help veterans and their families on Memorial Day, Veterans Day and the 4th of July. We'll help get kids ready for back to school with supplies they need. We've got plans! We'll also shower moms on Mother's Day and dads on Father's Day. We will deliver gifts to babies born at our local hospitals. We will help bring happiness and joy to sick children and the elderly on their birthdays. We'll cover each and every holiday!
We have an amazing board in place to help find and fill the need in our community!
Obviously, we can't do it alone! We need help from you. We need to know who needs our help. If you know of a person or family going through a tough time, please send us the information.
We also need to raise money! We will be announcing a few fundraisers very soon – some super fun, super cool ways to make a difference!!
You can also make a donation to our paypal account HERE.
We will have more ways to donate once our website is complete!!!
Also – follow us on Facebook and share our posts and our page! If you'd like to help with fundraisers, let us know! We want this to be a community effort and together, we will bring hope to many.
Following my heart with my husband and four daughters. An Emmy Award winning journalist lucky enough to work in television & radio for 20 years -seeing wonderful places, meeting great people and telling their stories.
Following my heart with my husband and four daughters. An Emmy Award winning journalist lucky enough to work in television & radio for 20 years - seeing wonderful places, meeting great people and telling their stories.
Welcome to Totally Buffalo! My name is Mary Friona-Celani and I am the creator of this site! After spending 20 years in the Buffalo media, I moved on to focus on my beautiful family. Now, the time has come for a new project. I was ready for something new. Something mine.
I am born and raised in WNY and my pride runs deep. My husband, Scott and I are very happy to be raising our four daughters here.
Copyright © 2016 Totally Buffalo. All rights reserved.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,693
|
The life and music of Neil Diamond will be celebrated in a production coming to the Embassy Theatre, in Skegness.
A Beautiful Noise will be at the Grand Parade venue this Friday, November 23, at 7.30pm.
Although Neil Diamond's touring days are over, A Beautiful Noise will take audiences on a journey through the life and music of Neil Diamond.
As a lifelong fan of Neil Diamond, Fisher Stevens is delighted to relive Neil's music.
Accompanied by a live band including brass, string and backing singers, the show will include hits such as Sweet Caroline, Song Sung Blue, Cracklin' Rosie, and Forever In Blue Jeans.
Fisher Stevens said: "It is my privilege to celebrate that talent and take the audience on a journey".
"A Beautiful Noise is an exuberant celebration of a legendary talent and is packed with all the classic hits!
"I am really excited to be bringing our brand new version of the A Beautiful Noise to Skegness.
"We really had a fantastic time performing our previous production of the Neil Diamond Story at the Embassy Theatre last year.
Tickets, priced at £24.50 or £26.50, from www.embassytheatre.co.uk or the box office on 01507 613100.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,184
|
Located in Central District, this hotel is within a 5-minute walk of Johor Bahru City Square and Komtar JBCC. KSL City Mall is 2.8 mi (4.6 km) away. JB Sentral is 16 minutes by foot.
All 104 rooms offer complimentary wireless Internet access, LCD TVs with cable channels, and ceiling fans. Other amenities available to guests include showers, desks and free toiletries.
T-Hotel Johor Bahru offers 104 forms of air-conditioned accommodation with free toiletries and a ceiling fan. 24-inch LCD televisions come with premium cable channels. Bathrooms include a shower. Guests can surf the web using complimentary wireless Internet access. An iron/ironing board and hairdryers can be requested. Housekeeping is provided on a daily basis.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,840
|
\section{Introduction}
Bangla is the mother language of
Bangladesh and the 7th most widely spoken language in the world. There are more than 200 million native Bangla speakers. It is the official language of Bangladesh and several Indian states including West Bengal, Tripura, Assam and Jharkhand. Bangla is also the official language of Sierra Leone a West African country. With the rapid adoption of technology in different sectors in these regions, recognizing handwritten Bangla characters is an important challenge to overcome. While there has been great successes in the application of machine learning tools for the English language, the same level of effectiveness is not observed for Bangla. One of the many reasons for this is the lack of a single comprehensive dataset which covers the frequently used Bangla characters. There are existing data sets which cover either just the Bangla numerals, or just the Bangla characters, or just the Bangla compound characters. While it is possible to combine them to form a unified data set, the incovenience faced by the researchers stem from the lack of consistency in the data presentation of the different data sets.
BanglaLekha-Isolated is the first of a chain of datasets being released which aims to foster Bangla handwriting related research by:
\begin{itemize}
\item Providing a large dataset suitable for machine learning applications which include the most frequently used Bangla characters covering Bangla numerals, basic characters and compound characters.
\item Provide a suitably pre-processed version of the dataset to reduce the time between data set acquisition and reporting results.
\item Provide multiple labels per character/character group to facilitate research in:
\begin{itemize}
\item Automatic recognition certain characteristics of the writer (Age, gender, location etc)
\item Automatic assessment of handwriting quality and methods of giving useful feedback.
\end{itemize}
\end{itemize}
The BanglaLekha-Isolated dataset contains smaples of 84 different Bangla handwritten numerals, basic characters and compound characters. A comparison with the two other popular sources of Bangla handwriting related datasets (CMATERDB\cite{CMATERDB} and the ISI Handwriting datasets\cite{ISI}) are given in Table \ref{TABLE}.
The BanglaLekha-Isolated dataset consists total of 166,105 square images (while preserving the aspect ratio of the characters), each containing a sample of one of 84 different Bangla characters. The number of samples in each class are almost equal, which is not the case in some of the other datasets (e.g. CMATERDB Compound character set). The 84 characters classes contain 10 numerals, 50 basic characters and 24 frequently used compound characters. Some samples images of the dataset are shown in Figure \ref{fig:test}.
\begin{figure}
\centering
\begin{subfigure}{.2\textwidth}
\centering
\includegraphics[width=.9\linewidth]{IMG_1.png}
\caption{Basic}
\label{fig:sub1}
\end{subfigure}%
\begin{subfigure}{.2\textwidth}
\centering
\includegraphics[width=.9\linewidth]{IMG_2.png}
\caption{Numeral}
\label{fig:sub2}
\end{subfigure}
\begin{subfigure}{.2\textwidth}
\centering
\includegraphics[width=.9\linewidth]{IMG_3.png}
\caption{Compound}
\label{compound}
\end{subfigure}
\caption{Sample Images of BanglaLekha-Isolated}
\label{fig:test}
\end{figure}
\begin{table}
\caption{Number of images in different datasets}
\label{TABLE}
\begin{center}
\begin{tabular}{ |p{3cm}||p{3cm}|p{3cm}|p{3cm}| }
\hline
Type & CMATERDB\textsuperscript{1}\cite{CMATERDB} &ISI Dataset\textsuperscript{2} \cite{ISI}&BanglaLekha-Isolated\\
\hline
Basic Character & 15,103 &30,966& 98,950\\
\hline
Numerals & 6,000 & 23,299 &19,748\\
\hline
Compound Characters & 42,248 & None & 47,407\\
\hline
\end{tabular}
\end{center}
\begin{tablenotes}
\item[a] \textsuperscript{1} CMATERDB dataset has 3 different datasets for basic characters,numerals and compound characters.
\item[b] \textsuperscript{2} ISI dataset has two different dataset for basic characters and numerals.
\end{tablenotes}
\end{table}
\section{Data Collection and Pre-processing}
This dataset was collected from literate native Bangla speakers of different regions and with age range between 4 to 27. A small fraction of the samples were collected from individuals with physical disabilities. Each individual was supplied with a form similar to the one shown in Figure \ref{p_map}. For a wider distribution of handwriting quality, samples were collected specific time constraints (10 Minutes, 5 Minutes, 2 Minutes). Each subject also gave information about her/his age, gender, and district he lives in.
\begin{figure}
\centering
\includegraphics{Forms.jpg}
\caption{Form that was used for collecting dataset. \label{p_map}}
\end{figure}
The images that are present in the dataset were pre-processed in the following ways:
\begin{itemize}
\item Foreground and background were inverted so that images have a black background with the letter drawn in white.
\item Noise removal was attempted by using the median filter.
\item An edge thickening filter was applied.
\item Images were resized to be square in shape with appropriate padding applied to preserve the aspect ratio of the drawn character.
\end{itemize}
\section{Possible Uses of BanglaLekha-Isolated Dataset}
Our dataset can be used for handwritten character recognition, which is obvious, but there are some more features that can be used for research purpose using our dataset. As it is already mentioned in Section 1 that it is possible to work on automatic recognition on certain characteristics of the writer such as age,gender, location etc. These informations can be used even for forensic purposes.
\section{Naming Convention}
Each and every sample of BangLekha-Isolated dataset has an unique form ID by which the age, gender, district, and Institute of the participants can be identified. So, a 22 characters long form Id was proposed, where first 2 digit is for the district, then the next four digits is for Institution, the next one digit is for gender and the next two is for age, again the next four is for date and the last four is used for form serial number and every information (digit part) is separated by an underscore. For example- $$ \textbf{01\_0001\_0\_09\_1016\_0001}$$ is a unique form id and here 01 means the participant is form Comilla, 0001 means participant is from Ispahani School and College, then 0 means the participant is a male and 1016 means the participant filled up the form in October of 2016 and 0001 is the form serial number. So whenever one used any character form this dataset (around 1,68,000 data), he/she can get the information (age, gender, district, etc.) of the participants.
\section{Marking}
All the 2000 forms that were collected were marked by 3 native Bangla speakers using the criteria set by a handwriting specialist. The judgment on the mark is based on:
\begin{itemize}
\item Shape of the characters
\item Clarity of the image
\item Appropriate use of matra (A horizontal straight line put over the consonants and some vowels of the Bengali alphabet)
\item Subjective evaluation based on beauty of letters
\end{itemize}
The marks are also provided with the dataset in a separate spreadsheet.
\section{Conclusion}
BanglaLekha-Isolated dataset aims for creating new scopes for researchers who are interested in working on Bangla handwritten characters. The dataset is available in \cite{BANGLALEKHA} \href{http://www.banglalekha.org}. This report documents the initial release of the data set. As more refinements are done and/or new data sets are collected, this report will be updated as appropriate.
\addtolength{\textheight}{-2.725cm}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,724
|
\section{Introduction}
\IEEEPARstart{T}{echnology-scaling} trend in recent years has led to emerging challenges for the traditional SRAM-based
large $Last$-$Level~Caches$ (LLCs)
including unreliability, leakage power, and low density
in multi- and many-core processors
~\cite
itrs,ogden2017impact, farbeh2016floating, ghaemi2019sleepy}.
To tackle these challenges, extensive ongoing industrial and academic research efforts have focused on replacing SRAMs with emerging \textit{$Non$}-$Volatile$~$Memories$ (NVMs)~\cite{wu2016temperature,Chen2016,ghaemi2019sleepy,vatajelu2017challenges, salkhordeh2016operating}.
According to the recent industrial reports, \textit{$Spin$}-\textit{$Transfer~Torque$} \textit{$Magnetic~Random$} \textit{$Access~Memories$} (STT-MRAMs) are the most promising technologies to substitute SRAMs in
LLCs
~\cite{wu2016temperature, wang2016memres,Chintaluri2015, imani2016approximate, kim2016exploration}.
STT-MRAM caches benefit from non-volatility, higher-density, near-zero leakage power, and immunity to radiation-induced particle strikes~\cite{zuo2018write,salkhordeh2019analytical,mittal2015survey, farbeh2018cache}.
However, STT-MRAM reliability is a major challenge for its applicability in
LLCs.
\textit{$Retention~failure$} (i.e., the content of a cell flips during its idle time)~\cite{Chen2016, kim2016exploration, Chintaluri2015}, \textit{$read~disturbance$} (i.e., unintentional flip of a cell due to applying read current during a read access)~\cite{fong2014failure,aliagha2019react,kang2015yield, wang2015selective}, and \textit{$write~failure$} (i.e., unsuccessful write operation due to inability of a cell to switch) ~\cite{farbeh2016floating,eli-aspdac,choi2017nvm} are the main sources of errors in STT-MRAM
LLCs.
Stochastic switching behavior of magnetic field direction in STT-MRAM cells is the source of the mentioned error types~\cite{zhao2012spin, farbeh2016floating, lakys2012self}.
Besides the device-level parameters and characteristics of STT-MRAM cells, the rates of these errors are affected by two major sources: a) system-level parameters, e.g., workloads, which affect the memory content and access patterns \cite{salkhordeh2018reca,bor2017tdsc,eli-aspdac, kishani2018modeling} and
b) physical parameters, which are $Process~Variations$ (PVs) that differently (sometimes oppositely) affect these error rates~\cite{imani2016approximate, kim2016exploration, kang2015reconfigurable, 12-EDCC-sun2012process}.
From the workload perspective, the diversity in cache access patterns and the content of data blocks in different workloads play an important role in the rate of the retention failure, read disturbance, and write failure.
Retention failure probability in cache blocks with larger gap between consecutive accesses is higher, whereas more frequently-accessed blocks experience lower failure probability.
However, the occurrence probability of read disturbance and write failure per unit of time is higher for more frequently-accessed blocks.
On the other hand, the rate of read disturbance and write failure directly depends on the content of data blocks.
Read disturbance error is unidirectional and only $unintentional$ ${1\rightarrow 0}$ transitions are probable.
Write failure, on the other hand, is only probable when the content of the original and updated values are different in the cache.
Therefore, read disturbance and write failure rates are affected by the number of `1's in a block and the hamming distance between the content of a cache block and the updated data block, respectively.
In addition to the system workloads, PVs can significantly change the error rates by deviating the physical parameters of a cell from their nominal values.
A deviation changes the rate of the mentioned error types differently and even in opposite directions~\cite{12-EDCC-sun2012process,kang2015reconfigurable, zhao2012spin}.
Considering an STT-MRAM cache as an array of STT-MRAM cells, the overall retention failure, read disturbance, and write failure rates depend on the interaction of process variations effects among the STT-MRAM cells.
Although the error rate of a single STT-MRAM cell can be estimated using its physical parameters,
the vulnerability estimation of the entire STT-MRAM cache requires system-level analysis considering the workloads behaviors as well as PV effects.
A recent study has reported only the rate of write failure and read disturbance in STT-MRAM cache and shown that these rates are strongly workload-dependent~\cite{eli-aspdac, EDCC}.
To the best of our knowledge, none of the previous studies neither have addressed the workloads- and PVs-dependent vulnerability of STT-MRAM caches considering all types of errors, nor reported the cache reliability considering the conflicting sources of errors.
Therefore, to achieve an acceptable level of reliability in STT-MRAM cache with affordable costs, it is necessary to \textbf{a)} estimate the vulnerability of the cache to each error type as well as the total cache vulnerability considering the conflicting dependencies of error types to different factors and \textbf{b)} utilize effective mitigation techniques for each error type according to its contribution in the total cache vulnerability.
In this paper, we propose a system-level framework for early exploration of STT-MRAM cache reliability, which considers \textbf{a)} workload-dependent cache access patterns and data content, \textbf{b)} PV affected STT-MRAM cell vulnerability to errors, and \textbf{c)} inter-correlation between the rate of three-mentioned errors, i.e., retention failure, read disturbance, and write failure, for reliability exploration.
To this end, we first formulate the vulnerability of STT-MRAM caches to each error type by extrapolating the error probability of PV affected cache blocks from nominal error probability of a single STT-MRAM cell.
Then, we deduce the error probability of a PV affected STT-MRAM cache from the blocks error probability.
Our formulation is then extended to integrate probabilities of the three differently-originated and exclusively-occurred error types into a single cache vulnerability equation.
The inputs of our architecture-level formulas are twofold: 1) device-level parameters of single STT-MRAM cell, e.g., thermal stability factor, critical switching current, tunneling spin polarization, and write/read current, which are derived from several industrial and technical reports, 2) system-level parameters derived from workload behavior, e.g., frequency and sequence of read and write operations, data content, and cache block idle intervals.
We present a framework integrated with gem5 full-system simulator~\cite{gem5} to extract and analyze the reliability characteristics of STT-MRAM LLCs based on our formulations.
This framework provides a high flexibility in terms of STT-MRAM cells physical and circuit-level parameters as well as cache configurations.
For a careful evaluation of the STT-MRAM cache vulnerability, the rate calculation for each error type is implemented in the gem5 based on a) the number of accesses, b) idle times, c) data content, and d) cell transitions.
The evaluations have been conducted for an STT-MRAM
LLC
shared across cores of a quad-core processor running a set of multi-programmed workloads from SPEC CPU2006 benchmark suite~\cite{spec2006}.
The results show that the rate of each error type as well as the total error rate significantly varies for different workloads.
The workload dependency of the errors resulted in a minimum of 6.8x variation in the rate of three types of errors.
This value is 32.0x for total cache failure probability, on average.
Our observations also reveal that PVs increase the occurrence probability of all error types by more than 8x for all workloads.
The \textbf{main}~\textbf{contributions} of this paper are as follows:
\begin{enumerate}
\item This is the first study that formulates the vulnerability of STT-MRAM caches to retention failure, read disturbance, and write failure errors based on the workload-dependent cache behavior and by extrapolating the nominal error rate of a single STT-MRAM cell.
\item We formulate the total reliability of STT-MRAM caches by proposing an approach to integrate the cache unreliability per unit of time (for retention failure), per read access (for read disturbance), and per write access (for write failure) into a unified cache failure probability.
\item We investigate the effects of PVs on the rate of all three error types as well as the total error rate in STT-MRAM cache. To the best of our knowledge, this is the first study that investigates the system-level reliability impacts of STT-MRAM physical parameters affected by PVs.
\item We present a framework integrated with gem5 full-system simulator~\cite{gem5} to extract and analyze the reliability characteristics of STT-MRAM
LLCs
based on our formulations. This framework provides a high flexibility in terms of STT-MRAM cells physical and circuit-level parameters as well as cache configurations.
The proposed framework supports both perpendicular and in-plane STT-MRAM technologies.
\item We investigate the dependency between workloads behavior and the rate of errors in a STT-MRAM
LLC
shared across cores of a quad-core processor.
Our study reveals that retention failure, read disturbance, and write failure vary by 25.4x, 6.8x, and 15.1x in different workloads, respectively.
A 32.0x variation is also observed for the total error rate including all types of errors.
These observations indicate that the cache access pattern and data content not only extremely differentiate the rate of errors, but also their effect on the rate of three error types is significantly different.
\item We investigate the effects of PVs on the
LLC
error rates and show that PVs increase the rate of retention failure, read disturbance, and write failure by 32.5x, 9.0x, and 16.7x, respectively.
In addition, the total error rate is increased by 6.5x.
These observations reveal that the susceptibility of the error types to PVs is highly different and the effect of PVs on the cache reliability is mostly determined by an error type with the highest contribution in the total error rate.
\end{enumerate}
The rest of this paper is organized as follows. Section II describes the basics of STT-MRAM technology and its reliability challenges. In Section III, the observations and motivations for this work are discussed. The details of the proposed formulations and framework are presented in Section IV. Section V gives the simulation setup and evaluation results. A discussion on the existing reliability improvement techniques and suggested guidelines based on our study is presented in Section VI. Finally, we conclude the paper in Section VII.
\section{Preliminaries}
\label{sec:PRELIMINARIES}
In recent years, beside STT$-$MRAM,~\textit{Spin}-\textit{Orbit}~\textit{Torque}~\textit{Magnetic}~\textit{RAM} (SOT$-$MRAM) is introduced as an alternate generation of MRAM technology.
SOT$-$MRAM is based on the three-terminal MTJ and uses \textit{Spin} \textit{Hall} \textit{Effect} to switch.
It tries to overcome high write latency and energy of STT$-$MRAM by separating the read path from the write path.
This separation also avoids read disturb in the cell. Despite of these advantages, unlike STT$-$MRAM that requires only one access transistor for both read and write, SOT$-$MRAM needs two transistors for read and write operations.
This makes the STT$-$MRAM denser than SOT$-$MRAM.
On the other hand, STT$-$MRAM memory is more mature and is designed and manufactured by big companies such as Toshiba~\cite{Toshiba}, TSMC~\cite{tsmc}, and Samsung~\cite{samsung} as a prototype and is also used in recent commercial products. Although SOT$-$MRAM has some advantages over STT$-$MRAM and investigating its reliability characteristics can be an interesting research track, it takes a long way for SOT$-$MRAM to be commercialized and this study focuses only on STT$-$MRAM technology. In this section, we explain the structure of STT-MRAM cells and the mechanisms of read and write operations in STT-MRAM cells. Then, we focus on the sources of errors in STT-MRAM cells and discuss the dependency of the rate of these errors to STT-MRAM cells configuration.
\subsection{STT-MRAM Cell Basics}
An STT-MRAM cell comprises a storage component, named \textit{$Magnetic~Tunnel~Junction$} (MTJ), to store data values, and an NMOS access transistor.
The structure of this cell, known as 1T1J STT-MRAM, is shown in Fig.~\ref{fig:1}(a).
An MTJ consists of two ferromagnetic layers separated by a thin oxide barrier layer~\cite{wu2016temperature, farbeh2018cache, Eli-TC}.
These layers as depicted in Fig.~\ref{fig:1}(b) are \textit{$reference$}, \textit{$free$}, and $tunnel~barrier$.
The barrier layer made of crystallized $Magnesium~oxide$ (MgO) insulates the electrons movement from the reference layer to the free layer and vice versa.
The reference layer has a fixed magnetic field direction, while the magnetic field direction of the free layer can be changed by applying a magnetic force \cite{khvalkovskiy2013basic,14-EDCC-apalkov2013spin, kang2015yield}.
\begin{figure}[t]
\centering
\subfloat[]{\includegraphics[width=0.35\linewidth]{Fig1--a-new1}
\hspace{5pt}
\subfloat[]{\includegraphics[width=0.59\linewidth]{0,1-value-in-STT-inplane}
\caption{STT-MRAM cell schematic: (a) 1T1J STT-MRAM cell structure and (b) MTJ low and high resistance states.}\vspace{-10pt}
\label{fig:1}
\end{figure}
As the STT-MRAM technology is based on magnetic charge instead of electrical charge, the magnetic field direction of the MTJ ferromagnetic layers determines the cell content. The relative direction of magnetic field in the free and reference layers (anti-parallel or parallel directions) causes two different resistances, i.e., $R_{High}$ and $R_{Low}$ ($R_{H}$ and $R_{L}$), in MTJ as shown in Fig.~\ref{fig:1}(b).
$R_{H}$ and $R_{L}$ represent high and low MTJ resistances, respectively, and are distinguished based on the voltage value between the $bit~line$ (BL) and $source~line$ (SL) terminals on a read operation. $Tunneling~Magneto~Resistance$ (TMR) ratio parameter, which is defined as $TMR$ = ($R_{H}$ - $R_{L}$)/$R_{L}$, shows the ratio between these resistances and indicates the cell state~\cite{15-EDCC-zhao2011design, zhao2012spi
}.
When the MTJ is on parallel (or anti-parallel) state, the MTJ resistance is low (or high), which represents that `0' (or `1') logic is stored in the cell. The MTJ states and its logic values are depicted in Fig.~\ref{fig:1}(b).
\subsection{Read and Write Operations}
As mentioned, the resistance of MTJ ($R_H$ or $R_L$) shows the value in an STT-MRAM cell. To read the stored value in a cell, first the $word~line$ (WL) is set to turn on the access transistor. Then, a small read current ($I_{read}$) should be applied to the STT-MRAM cell~\cite{kim2016exploration, itrs}.
In (\ref{eq:1}) and (\ref{eq:2}), the {$bit~line~voltage$} ($V_{BL}$) in a 1T1J STT-MRAM cell is shown.\vspace{-15pt}
\begin{flalign}
\label{eq:1}
\resizebox{.55\linewidth}{!}{$ V_{BL-Low} = {I_{read}\times(R_L+R_{NMOS})}$} \phantom{\hspace{1.2cm}}
\end{flalign}
\begin{flalign}
\label{eq:2}
\resizebox{.55\linewidth}{!}{$ V_{BL-High} = {I_{read}\times(R_H+R_{NMOS})}$} \phantom{\hspace{1.2cm}}
\end{flalign}
Where, $V_{BL-Low}$ and $V_{BL-High}$ are the bit line voltages when the MTJ is at low and high resistance states, respectively.
In these equations, $R_L$ and $R_H$ are the low and high MTJ resistance, respectively, $R_{NMOS}$ is the resistance of NMOS access transistor, and $I_{read}$ is read current~\cite{14-zazad-eken2014novel, eli-date}.
By applying $I_{read}$ to an STT-MRAM cell, a voltage is generated between the bit line and source line.
By comparing the $V_{BL}$ with a {$reference~voltage$} ($V_{REF}$), the MTJ resistance state can be read out~\cite{ mittal2017survey, 15-EDCC-zhao2011design}.
As shown in Fig. \ref{fig:basics}(a) (Fig. \ref{fig:basics}(b)), if the sensed value is higher (lower) than the reference voltage, it means that the resistance of MTJ is low (high) and the cell contains `0' (`1') value.
Write operation is more complicated than read operation due to requiring to change the MTJ resistance.
MTJ resistance changes if the magnetic field direction of the free layer flips.
To this end, a write current is applied to the source line or bit line to write `0' or `1', as shown in Fig. \ref{fig:basics}(c) and Fig. \ref{fig:basics}(d), respectively.
Consequently, electrons spin in the free layer orient in the same direction or opposite direction of the reference layer magnetic field, based on the direction of the applied current.
This phenomenon causes a spin-polarized current.
When the amount of spin-polarized current exceeds a threshold value, the magnetic field direction of the free layer switches.
This is the time when the MTJ content flips and a value is written into the cell.
Switching the magnetic field direction from anti-parallel to parallel (or from parallel to anti-parallel) leads electrons to flow from the reference layer to the free layer (or vice versa)~\cite
mittal2017survey, 15-EDCC-zhao2011design,kang2014variation, farbeh2018cache}.
\begin{figure}[t
\centering
\subfloat[]{\includegraphics[width=0.47\linewidth]{Fig3--11a}
\vspace{10pt}
\hspace{10pt}
\subfloat[]{\includegraphics[width=0.45\linewidth]{Fig3--11b}}\vspace{-3pt
\hspace{10pt}
\subfloat[]{\includegraphics[width=0.45\linewidth]{Fig3--11c}
\hspace{10pt}
\subfloat[]{\includegraphics[width=0.45\linewidth]{Fig3--11d}}\vspace{-3pt
\caption{STT-MRAM read and write operations: (a) reading `0', (b) reading `1', (c) writing `0', and (d) writing `1'.}\vspace{-10pt}
\label{fig:basics}
\end{figure}
\subsection{STT-MRAM Reliability}
Reliability of STT-MRAM cells is threatened by retention failure, read disturbance, and write failure. $Retention~failure$ occurs when a cell is idle (a cell that is not being read nor written) and its content flips stochastically. $Read~disturbance$ occurs during a read operation when the content of a cell changes unintentionally. A $write~failure$ occurs when the content of a cell is not switched by the current applied during the write operation.
The origin of all these errors is the stochastic switching behavior of STT-MRAM cells~\cite{Pajouhi2016JETC,Na-TCAS-II-16,Ran2016JSA, eli-date}.
Read disturbance and write failure are the main reliability threats in 22nm technology node~\cite{naeimi2013intel}.
While the process technology scales down, it is predicted that the retention failure probability increases exponentially, read disturbance probability increases in lower rate than retention failure probability, whereas the rate of write failure decreases.
Therefore, the total error rate considering all three sources of errors is significantly increased by technology downscaling, in which the retention failure is the main contributor.
Because of a retention failure, the content of an idle STT-MRAM cell flips accidentally without passing any current through it.
The retention failure probability in time interval \textit{t} is calculated according to (\ref{eq:3})~\cite{15-EDCC-zhao2011design, naeimi2013intel}.
\begin{flalign}
\label{eq:3}
\resizebox{.8\linewidth}{!}{$ P_{Retention-Failure} = 1- exp({-t }\times {exp(-\Delta)})$
\end{flalign}
Where, $t$ is the cell idle time and $\Delta$ is $thermal~stability~factor$ of a STT-MRAM cell.
The retention failure probability increases in larger values of $t$ and/or smaller values of $\Delta$.
The thermal stability factor of a cell is calculated according to (\ref{eq:4})~\cite{naeimi2013intel}.\vspace{-10pt}
\begin{flalign}
\label{eq:4}
\resizebox{.18\linewidth}{!}{$\Delta = \frac{E_{b}}{ K T}$}
\end{flalign}
Where, \textit{E$_b$} is barrier energy, \textit{K} is Boltzmann constant, and \textit{T} is temperature in Kelvin.
During a read operation in a STT-MRAM cell, content of the cell can change unintentionally due to the read current passing through the cell, causing a read disturbance error. The direction of the read current, which flows through the MTJ layers in a read operation, is the same as the direction of the current for writing either `1' or `0' in a write operation.
The current in read operation is much lower than the current in the write operation~\cite{farbeh2016floating}.
However, this low current can cause an unintentional flip in the cell during a read operation and erroneously change its content.
The switching probability of an STT-MRAM cell during a read operation is according to (\ref{eq:5})~\cite{lakys2012self, eli-date, naeimi2013intel}
\begin{equation}
\begin{multlined}
\label{eq:5
\shoveright[2cm]{P_{Read-Disturbance}=1- exp(\frac{-t_{read}}{\tau}\times}\\
\shoveleft[2.9cm]{exp(\frac{\Delta(I_{read}-I_{C_0})}{I_{C_0}}))}
\end{multlined}
\end{equation}
Where, \textit{$\tau$} is attempt period equal to 1ns, \textit{I$_{read}$} is read current, \textit{$I_{C_{0}}$} is critical switching current, \textit{t$_{read}$} is read pulse duration, and $\Delta$ is thermal stability factor.
Write failure as another reliability challenge occurs when the magnetic field direction of the free layer in MTJ cannot be switched during the interval where write pulse is applied \cite{Pajouhi2016JETC, eli-date}.
The MTJ switching time depends on various parameters, e.g., MTJ switching current, process variations, thermal fluctuations, and switching pulse width. The occurrence probability of a write failure for a STT-MRAM cell is according to (\ref{eq:6})~\cite{wu2016temperature, 15-EDCC-zhao2011design, 14-zazad-eken2014novel}.
\begin{equation}
\begin{multlined}
\label{eq:6
P_{Write-Failure} = exp( -t_{write}\times \\
\shoveright[1cm]{\frac{2 \times \mu_{\beta}\times p\times(I_{write}-I_{C_0})}{c+\log_{e}(\pi^2\times\Delta/4)\times (e\times m\times (1+p^2))})}
\end{multlined}
\end{equation}
Where, \textit{I$_{write}$} is write current, \textit{c} is Euler constant, \textit{e} is electron charge, \textit{m} is magnetic momentum of the free layer, \textit{p} is tunneling spin polarization, \textit{$\mu$$_{\beta}$} is Bohr magneton, \textit{t$_{write}$} is write pulse duration, and $\Delta$ is thermal stability factor.
Beside write failure, retention failure, and read disturbance errors, there are three other factors that may affect the STT-MRAM error rate as follows: \textbf{a)}~\textbf{false}~\textbf{read} in which the sense amplifier circuit is unable to correctly determine the content of STT-MRAM cell. This error is because of peripheral CMOS circuitry and it is not specific to STT-MRAM technology; \textbf{b)}~\textbf{limited} \textbf{endurance}, which is due to oxide barrier breakdown of STT-MRAM cells. Although some studies addressed this challenge, several recent industrial reports have illustrated an unlimited endurance for STT-MRAM cells compared to the ten-year lifetime of digital system; and \textbf{c)}~\textbf{radiation-induced}~\textbf{transient}~\textbf{errors} in peripheral CMOS circuitry, which is not related to STT-MRAM technology. While retention failure, read disturbance, and write failure are inter-correlated in STT-MRAM cell, limited endurance is no longer a reliability concern and false read and transient error have never been known as a source of STT-MRAM vulnerability.
\section{Motivation}
The reliability challenge of STT-MRAM cells due to retention failure, read disturbance, and write failure becomes more threatening when considering process variations.
On the other hand, workloads and data patterns affect the rates of these errors.
In the following, we first discuss the major reliability challenges by exploring the workloads effects on the error rates. Then, we explore the effects of process variations on these workload-dependent errors.
\subsection{Effects of Workloads and Data Patterns}
Retention failure rate depends on the data access patterns of the workload in a system.
For example, retention failure probability for a cache block not accessed for a long time is much higher than a cache block frequently accessed in a short interval.
On the other hand, a retention failure in a cache block is masked if the block is overwritten before it is read.
Hence, the errors occurred in intervals between a read/write access and its subsequent write access has no effect in data integrity and these intervals are beyond of vulnerable times.
As a result, the retention failure rate varies for different workloads, different phases of a single workload, and even different cache blocks in a given time slot.
As shown in Fig. \ref{fig:basics}(a), the read current applied to STT-MRAM cells for read operation is always in a predetermined direction (from SL to BL or vice versa). In other words, read current is predeterminedly in the direction of writing either `0' or `1'.
Therefore, only STT-MRAM cells containing `1' (or `0') are vulnerable to read disturbance and the other cells containing `0' (or `1') are not affected.
This $unidirectional$ read current makes the occurrence probability of read disturbance in a cache block to be content-dependent.
On the other hand, the total read disturbance rate in a cache during a workload execution depends on the number of read accesses to the cache.
Obviously, both the content of cache blocks (the number of `0's and `1's) and the number of read accesses can have a large variation for different workloads.
Therefore, besides its dependency to physical characteristics of STT-MRAM cells and the circuit-level parameters of the cache, the read disturbance rate is highly workload-dependent.
Same as the rate of retention failure and read disturbance, write failure rate is highly affected by workloads behavior.
On a write operation in a cache block, a subset of STT-MRAM cells that their content need to switch are vulnerable to write failure, since it is probable that these cells may not switch during applying the write current.
No write failure is probable in other cells, since their content is the same as the updated value.
In this regard, the write failure rate depends on not only the number of write requests, but also the similarity between the content of cache blocks and the value of updated data.
In addition, for cells that need to switch on a write operation, the probability of unsuccessful switching in ${0\rightarrow 1}$ transitions is by more than two orders of magnitude higher than in ${1\rightarrow 0}$ transitions\cite{ZAZADTPDS, Na-TCAS-II-16}.
This $asymmetric$ write failure rate for the directions of cell switching makes the unreliability of STT-MRAM caches even more content-dependent.
Based on the above discussions, the total error rate of STT-MRAM cache, which consists of retention failure, read disturbance, and write failure strongly depends on \textbf{a)} data patterns, \textbf{b)} cell contents, and \textbf{c)} data access times.
Fig. \ref{fig:motive} shows variations in an L2-cache access patterns for a set of workloads from SPEC CPU2006 benchmark suite~\cite{spec2006}.
Number of `1's read from block during the workload execution affect the read disturbance probability, number of ${0\rightarrow 1}$ and ${1\rightarrow 0}$ transitions affects the write failure probability, and the idle intervals excluding the intervals before a write operation affect the retention failure probability.
The results illustrate significant variations for all of these parameters in different workloads.
This observation indicates a strong workload-dependency of not only the rate of all error types, but also the contribution of each type of error in total error rate.
\begin{figure}[t
\centering
\includegraphics[width=1\linewidth]{Fig--motive1}\vspace{-10pt}
\caption{Workload dependency of parameters affecting retention failure, read disturbance, and write failure rate.}\vspace{-10pt}
\label{fig:motive}
\end{figure}
\subsection{Effects of Process Variations}
Similar to other memory technologies, STT-MRAM technology is affected by process variations. These variations are exacerbated by downscaling the technology node~\cite{imani2016approximate,kim2016exploration, kang2015reconfigurable, 12-EDCC-sun2012process}.
PVs in the MTJ ferromagnetic layers and oxide barrier layer as well as in NMOS access transistor deviate the STT-MRAM parameters from their nominal values.
Thermal stability factor ($\Delta$), read and write currents ($I_{read}$ and $I_{write}$), critical switching current ($I_{C_{0}}$), magnetic moment of MTJ free layer ($m$), and tunneling spin polarization ($p$) are among the major parameters affected by PVs.
In the following, we explain how PVs affect the STT-MRAM cache reliability.
As observed in (\ref{eq:4}), the occurrence probability of retention failure depends on a cell idle time and its thermal stability factor.
Although the former is determined by only the workload behavior and architectural properties of the cache, the latter is highly under PVs.
$\Delta$ of a cell is deviated from its nominal value by altering the cell barrier energy ($E_b$) affected by PVs.
According to (\ref{eq:4}), the retention failure probability exponentially depends on the exponential of $\Delta$.
Therefore, even a minor deviation in $\Delta$ of a cell excessively changes the retention failure probability.
A recent study shows that the retention time of STT-MRAM cells in a memory cell can vary from a few minutes to several years\cite{nair2017vaet}.
Similar to the retention failure probability, $\Delta$ exponentially affects an exponential function that generates the read disturbance probability, based on (\ref{eq:6}).
Hence, a small decrease in $\Delta$ due to PVs extremely increases the read disturbance probability.
Deviated from its nominal value, a small decrease in critical switching current ($I_{C_{0}}$) has the same effect as $\Delta$ deviation on the read disturbance probability.
The effect of increase in read current ($I_{read}$) on increase in read disturbance probability is the same as the effect of decease in $\Delta$ and $I_{C_{0}}$.
Therefore, all parameters in (\ref{eq:6}) that are under PVs, i.e., $\Delta$, $I_{C_{0}}$, and $I_{read}$, affect the read disturbance probability through the exponent of an exponential function.
Write current ($I_{write}$), critical switching current ($I_{C_{0}}$), thermal stability factor ($\Delta$), magnetize moment of MTJ free layer ($m$), and tunneling spin polarization ($p$) are the parameters under PVs that affect write failure rate.
According to (\ref{eq:6}), a rise in $I_{C_{0}}$, $\Delta$, $m$, and $p$ values due to PVs increase the write failure probability exponentially.
On the other hand, this probability increases by decreasing $I_{write}$.
In summary, as the cell contents, number of accesses, time interval between accesses, and PVs determine the cache error rate, we need to investigate the reliability of STT-MRAM caches in the presence of PVs and different workloads.
$\Delta$, $I_{C_{0}}$, $I_{read}$, $I_{write}$, $m$, and $p$ are the main parameters in STT-MRAM cells affected by PVs and can significantly change the cache reliability in the following aspects:
\begin{itemize}
\item An increase (decrease) in $\Delta$ increases (reduces) the write failure rate in one hand and reduces (increases) the retention failure and read disturbance rates on the other hand.
\item An increase (decrease) in $I_{C_{0}}$ increases (decreases) the write failure rate and decreases (increases) the read disturbance rate. The value of $I_{C_{0}}$ has no effect on retention failure rate.
\item An increase (decrease) in $I_{read}$ increases (decreases) the read disturbance rate without affecting the other two error types.
\item An increase (decrease) in $I_{write}$ decreases (increases) the write failure rate and has no effect on the other two error types.
\item An increase (decrease) in $m$ and $p$ increases (decreases) the write failure rate and has no effect on the others
\end{itemize}
\section{Proposed Formulations and Framework}
Retention failure, read disturbance, and write failure rates, which are affected by workloads and process variations cannot be measured through the cells physical characteristics.
In other words, the reliability of a STT-MRAM cache needs system-level investigations as well as circuit-level information.
The vulnerability of STT-MRAM cache to different error types requires a system-level exploration through a joint consideration of device-, circuit-, architecture-, and application-level effects.
Moreover, the reliability estimation of STT-MRAM cache requires the integration of the cache vulnerability to all error types, which their rates are differently and even oppositely affected from device- to application-level parameters.
This work presents a comprehensive reliability formulation of STT-MRAM cache by jointly considering all three error sources, i.e., write failure, read disturbance, and retention failure. In other words, our formulations are an integration of write failure, read disturbance, and retention failure rates. The total cache error rate under each of these error sources is derived from error probability of all cache blocks. Error probability of each cache block is the summation of all STT-MRAM cells of the block. The input parameters of all the proposed reliability formulas are the error rate of a single STT-MRAM cell. These parameters are derived practically from real experiments in several studies and the formulations are adjusted to match the experimental results.
Early reliability exploration of STT-MRAM caches at design time is a necessity for not only confirming the applicability of STT-MRAM technology as a reliable alternative to SRAM technology, but also to design cost-efficient error-tolerant caches.
In this section, we first present our formulations for the vulnerability of STT-MRAM caches to each error type.
Then, the formulations for total cache vulnerability, which is the integration of cache vulnerability to all error types is presented.
Finally, our framework, as an extension to gem5 full-system simulator~\cite{gem5}, for reliability exploration is detailed.
\setcounter{secnumdepth}{4}
\subsection{Cache Vulnerability Formulations}
The vulnerability of STT-MRAM caches to error sources depends on \textbf{a)} the number and sequence of read and write accesses, \textbf{b)} the interval between accesses, and \textbf{c)} the content of stored and updated data value.
On the other hand, PVs affect the cache vulnerability by changing the STT-MRAM physical parameters and complicate the vulnerability formulations.
For the sake of simplicity and generality, we first formulate the cache vulnerability without considering PVs and then extend our formulations by including the PV effects, in the following subsections.
\vspace{5pt}
\subsubsection{\textbf{Cache Vulnerability Formulation without PVs}}
The probability of retention failure, read disturbance, and write failure for a single STT-MRAM cell was given in (\ref{eq:3}), (\ref{eq:5}), and (\ref{eq:6}), respectively, for a single interval/access.
We extrapolate these equations to calculate error probabilities for an $N$-$bit$ cache block in a single interval/access and extend it for error probability of a block during workloads execution.
Finally, the vulnerability of the cache is formulated using the error probability of all blocks.
\vspace{5pt}
\paragraph{\textbf{Retention Failure Probability}}
The retention failure probability of an $N$-$bit$ cache block in an idle interval $t$ is according to (\ref{eq:7}).\vspace{-3pt}
\begin{equation}
\begin{multlined}
\label{eq:7}\hspace{-1cm}
P_{RF,blk,t}=1- $$\prod_{i=0}^{N-1} (1- P_{RF, bit_{i},t})$$\\
\shoveleft[0.5cm]{=1- $$\prod_{i=0}^{N-1} (exp({-t }\times {exp(-\Delta)})$$}\\
\shoveleft[0.5cm]{=1- exp({-t }\times {exp(-\Delta)})^ {N}}\\
\shoveleft[0.09cm]{=1- exp({-N\times t }\times {exp(-\Delta)})}
\end{multlined}
\end{equation}
Where, $P_{RF,blk,t}$ is probability of retention failure (RF) for the block and $P_{RF,bit_i,t}$ is the retention failure probability for a single cell given in (\ref{eq:3}).
A block is erroneous if a retention failure occurs in at least one out of its $N$ cells.
In other words, a block is error-free if all of its cells are error-free.
Since the cache blocks are assumed to be accessed in the block granularity, the idle intervals are equal for all cells and the retention failure probability of all cells are the same.
On the other hand, the probability of no retention failure occurrence in $N$ bits for interval $t$ is equivalent to that probability in a single bit for interval $N\times t$.
Therefore, the retention failure probability is simplified according to (\ref{eq:7}).
A cache block experiences multiple idle intervals during a workload execution.
The retention failure probability for a block in total execution time is according to (\ref{eq:8}). \vspace{-3pt}
\begin{equation}
\begin{multlined}
\label{eq:8
P_{RF,blk,total}=1-$$\prod_{j=1}^{T} (1- P_{RF, blk,t_{j}})$$\\
\shoveleft[2cm]{=1-$$\prod_{j=1}^{T} (exp({-N\times t_{j} }\times {exp(-\Delta)})$$}\\
\shoveleft[2cm]{=1-exp({-N}\times $$\sum_{j=1}^{T} t_{j}\times {exp(-\Delta)})}
\end{multlined}
\end{equation}
Where, $T$ is the total number of vulnerable idle intervals for the block and $P_{RF,blk,t_j}$ is the retention failure probability given in (\ref{eq:7}).
Vulnerable idle intervals of a block are idle intervals ended by a read access.
Retention failures occurred in idle intervals ended by a write access are overwritten.
Fig. \ref{fig:4} depicts vulnerable intervals of a block for a sample sequence of write/read accesses.
The retention failure rate of a block in total intervals is equivalent to that probability for a single interval that its duration is the summation of all intervals.
Using (\ref{eq:8}), the retention failure probability of the cache in total execution time is calculated according to (\ref{eq:9}).
\begin{equation}
\begin{multlined}
\label{eq:9
P_{RF,cache}=1- $$\prod_{b=0}^{B-1} (1- P_{RF, blk_{b},total})$$\\
\shoveleft[0.1cm]{=1- $$\prod_{b=0}^{B-1} (exp({-N}\times $$\sum_{j=1}^{T_{b}} t_{b,j}\times {exp(-\Delta)}))$$}\\
\shoveleft[0.1cm]{=1- exp({-N}\times $$\sum_{b=0}^{B-1} $$\sum_{j=1}^{T_{b}} t_{b,j}\times {exp(-\Delta)})}
\end{multlined}
\end{equation}
Where, $B$ is the total number of blocks in the cache and $T_b$ is total intervals of block $b$.
The retention failure of the cache is equivalent to the retention failure of a single block that its interval is the summation of all intervals of all blocks.
\vspace{5pt}
\paragraph{\textbf{Read Disturbance Probability}}
Based on the design time configurations, read disturbance error is probable in memory cells containing either logic value `1' or `0'.
In this study, we assume that the direction of read current is the same as the current for writing `0', and only STT-MRAM cells containing `1' are vulnerable to read disturbance.
It is noteworthy that the direction of read disturbance has no effect on our formulations.
Using the read disturbance probability for a single cell in (\ref{eq:5}), we derive the read disturbance probability for an $N$-$bit$ cache block in a single read access, according to (\ref{eq:10})
\begin{equation}
\begin{multlined}
\label{eq:10
P_{RD,blk,RA}=1- $$\prod_{i=0}^{N-1}(1- P_{RD,bit_{i},RA})\\
\shoveleft[0.3cm]{=1- $$\prod_{i=0}^{N-1}(1- bit_{i}\times P_{RD,cell,RA})}\\
\shoveleft[0.3cm]{=1-{(1- P_{RD,cell,RA}})^{n}} \\
\shoveleft[0.005cm]{=1- (exp(\frac{-t_{read}}{\tau}\times exp(\frac{\Delta(I_{read}-I_{C_0})}{I_{C_0}})))^n}\vspace{2pt}
\end{multlined}
\end{equation}
Where, $RD$ and $RA$ stands for Read Disturbance and Read Access, respectively.
$P_{RD,bit_{i},RA}$ is the read disturbance probability of $bit_i$ in the block, which is equal to zero if $bit_i$= `0' and is equal to $P_{RD,cell,RA}$ given in (\ref{eq:5}), otherwise.
Assuming $n$ bits out of the N-bit block contains `1', the read disturbance probability of a block in a single read access is equivalent to read disturbance probability in a cell containing `1' for $n$ read accesses.
It is noteworthy that a block is erroneous if read disturbance occurs in at least one of its $N$ bits.
\begin{figure}[t
\centering
\includegraphics[width=1\linewidth]{interval}\vspace{-8pt
\caption{Vulnerable intervals ($t_{vul.}$) of cache blocks to retention failure occurrence.}\vspace{-5pt}
\label{fig:4}
\end{figure}
A cache block is read for several times with different contents during a workload execution.
The read disturbance probability for a block in total execution time is according to (\ref{eq:11}).
\begin{equation}
\begin{multlined}
\label{eq:11
P_{RD,blk, total}=1- $$\prod_{i=1}^{R}(1- P_{RD,blk,RA_{i}})\\
\shoveleft[0.001cm]{=1- $$\prod_{i=1}^{R}(1- P_{RD,cell,RA})^{n_i}} \\
\shoveright[2cm]{=1- (exp(\frac{-t_{read}}{\tau}\times exp(\frac{\Delta(I_{read}-I_{C_0})}{I_{C_0}})))^{(\sum_{i=1}^{R} n_{i})}}
\end{multlined}
\end{equation}
Where, $R$ is the total number of read accesses to the block, $P_{RD,blk,RA_{i}}$ is the read disturbance probability in a single read access $RA_{i}$, and $n_i$ is the number of `1's in the block for access $RA_{i}$.
Read disturbance probability for a block in $R$ accesses, in which $n_i$ bits contain `1' for ${i}\textsuperscript{th}$ access is equivalent to read disturbance probability in a cell containing `1' and is read for $\sum_{i=1}^{R} n_{i}$ times.
Using the read disturbance probability of a cache block, the read disturbance probability of the whole cache in total execution time is calculated according to (\ref{eq:12}). \vspace{-5pt}
\begin{equation}
\begin{multlined}
\label{eq:12
P_{RD,cache}=1- $$\prod_{b=0}^{B-1}(1- P_{RD,blk_{b},total})\\
\shoveleft[1.6cm]{=1- $$\prod_{b=0}^{B-1}((1- P_{RD,cell,RA})^{n_i})^{(\sum_{i=1}^{R_{b}} n_{b,i})}} \\
\shoveleft[1.6cm]{=1- (exp(\frac{-t_{read}}{\tau}\times}\\
\shoveleft[1.7cm]{exp(\frac{\Delta(I_{read}-I_{C_0})}{I_{C_0}})))^{(\sum_{b=0}^{B-1}\sum_{i=1}^{R_{b}} n_{b,i})}}
\end{multlined}
\end{equation}
Where, $P_{RD,cache}$ is equivalent to read disturbance probability of a cell containing `1' after reading it for the total number of `1's during execution time.
\vspace{5pt}
\paragraph{\textbf{Write Failure Probability}}
In a write operation, write failure is probable in STT-MRAM cells that their content requires flipping.
On the other hand, the write failure probability for ${0\rightarrow1}$ transitions is different from that in ${1\rightarrow 0}$ transitions.
Therefore, the write failure probability for a cache block in a single write operation depends on not only the number of switching required, but also the number of required switching in each direction.
Having the write failure probability for a single cell in single write access, the write failure probability for a cache block in a write access is according to (\ref{eq:13}).\vspace{-5pt}
\begin{equation}
\begin{multlined}
\label{eq:13
P_{WF,blk,WA}=1- $$\prod_{i=0}^{N-1}(1- P_{WF,bit_{i},WA}) \\
\shoveleft[2cm]{=1- (1- P_{WF,cell_{0 \rightarrow1}})^{n_{0\rightarrow1}}\times}\\
\shoveleft[2cm]{(1- P_{WF,cell_{1\rightarrow 0}})^{n_{1\rightarrow 0}}}
\end{multlined}
\end{equation}
Where, $WF$ and $WA$ stand for Write Failure and Write Access, respectively, $P_{WF,bit_{i},WA}$ is the write failure probability in $bit_i$ of the block, and
$P_{WF,cell_{0\rightarrow1}}$ and $P_{WF,cell_{1\rightarrow0}}$ are the write failure probability for ${0\rightarrow1}$ and ${1\rightarrow0}$ transitions, respectively.
$n_{0\rightarrow 1}$ and $n_{1\rightarrow 0}$ are the number of ${0\rightarrow 1}$ and ${1\rightarrow 0}$ transitions, respectively.
The write failure probability in a block is equivalent to this probability when trying to flip a cell containing `1' for $n_{1\rightarrow 0}$ times and a cell containing `0' for $n_{0\rightarrow 1}$ times.
For a cache block written for several times during a workload execution, the total write failure probability can be derived from (\ref{eq:13}), as shown in (\ref{eq:14}).\vspace{-4pt}
\begin{equation}
\begin{multlined}
\label{eq:14
P_{WF,blk,total}=1- $$\prod_{i=1}^{W}(1- P_{WF,blk,WA_{i}})\\
\shoveleft[1cm]{=1- $$\prod_{i=1}^{W}((1- P_{WF,cell_{0\rightarrow1}})^{{(n_{0\rightarrow1})}_i}}\\
\shoveleft[1.3cm]{\times(1- P_{WF,cell_{1\rightarrow0}})^{{(n_{1\rightarrow0})}_i})}\\
\shoveleft[1cm]{=1-(1- P_{WF,cell_{0\rightarrow1}})^{(\sum_{i=1}^{W} {(n_{0\rightarrow1})}_i)}}\\
\shoveleft[0.8cm]{\times (1- P_{WF,cell_{1\rightarrow0}})^{(\sum_{i=1}^{W} {(n_{1\rightarrow0})}_i)}}\vspace{-3pt}
\end{multlined}
\end{equation}
Where, $W$ is the total number of write accesses to the block, $P_{WF,blk,WA_{i}}$ is the write failure probability in write access $i$, in which the number of required $0\rightarrow 1$ and $1\rightarrow 0$ transitions are equal to ${(n_{0\rightarrow 1})}_i$ and ${(n_{1\rightarrow 0})}_i$, respectively.
Write failure probability for a block during the total execution time is equivalent to this probability when trying to flip a cell containing `0' for ${\sum_{i=1}^{W} {(n_{0\rightarrow1})}_i}$ times and a cell containing `1' for ${\sum_{i=1}^{W} {(n_{1\rightarrow0})}_i}$ times.
The write failure probability of the whole cache in total execution time is derived from this probability for the blocks, according to (\ref{eq:15}).\vspace{-10pt}
\begin{equation}
\begin{multlined}
\label{eq:15}\vspace{6pt
P_{WF, cache}=1- $$\prod_{b=0}^{B-1}(1- P_{WF,blk_b,total})\\
\shoveleft[1.5cm]{=1-$$\prod_{b=0}^{B-1}((1- P_{WF,cell_{0\rightarrow1}})^{(\sum_{i=1}^{W_b} {(n_{0\rightarrow1})}_{b,i})}}\vspace{6pt}\\
\shoveleft[1.8cm]{\times (1- P_{WF,cell_{1 \rightarrow0}})^{(\sum_{i=1}^{W_b} {(n_{1\rightarrow0})}_{b,i})}}\vspace{6pt}\\
\shoveleft[1.5cm]{=1-(1- P_{WF,cell_{0\rightarrow1}})^{(\sum_{b=0}^{B-1} \sum_{i=1}^{W_b} {(n_{0\rightarrow1})}_{b,i})}\vspace{6pt}\\
\shoveleft[1.5cm]{\times (1- P_{WF,cell_{1\rightarrow0}})^{(\sum_{b=0}^{B-1} \sum_{i=1}^{W_b} {(n_{1\rightarrow0})}_{b,i})}}}
\end{multlined}
\end{equation}
Where, $B$ is the number of blocks in the cache, $P_{WF,blk_b,total}$ is the write failure probability for block $b$ in total execution time, $W_b$ is the number of write accesses to block $b$, and ${(n_{0\rightarrow 1})}_{b,i}$ and ${(n_{1\rightarrow 0})}_{b,i}$ are the number of $0\rightarrow 1$ and $1\rightarrow 0$ transitions in write access $i$ to block $b$, respectively.
Write failure probability in the whole cache is equivalent to this probability in a single cell when trying to flip its content from `1' to `0' and from `0' to `1' for $T_1$ and $T_2$ times, respectively, in which $T_1$ and $T_2$ are the total number of $0\rightarrow 1$ and $1\rightarrow 0$ transitions in the cell during workloads execution, respectively.
\vspace{5pt}
\subsubsection{Cache Vulnerability Formulation with PV Effects}
As observed in above formulations, the probability of errors in STT-MRAM caches depends on both workloads behavior and the physical characteristics of STT-MRAM cells.
By considering PVs in our formulations, the error probability of different cache blocks and even that of different cells in a block differs for the same access pattern and cells content.
In the following, we formulate the errors probability by including PV effects.
\vspace{5pt}
\paragraph{\textbf{PV Affected Retention Failure Probability}}
As observed in (\ref{eq:3}), the only parameter affected by PVs is the thermal stability factor ($\Delta$) in STT-MRAM cells.
The retention failure probability of a cache block for a single idle interval $t$ is according to (\ref{eq:16}).\vspace{-1pt}
\begin{equation}
\begin{multlined}
\label{eq:16} \vspace{8pt}
P_{RF,blk,t,PV}=1-$$\prod_{i=0}^{N-1}(1-P_{RF,bit_i,t,PV})\\
\shoveleft[1.5cm]{=1-$$\prod_{i=0}^{N-1} (exp({-t\times exp(-{\Delta}_i)}))$$}
\end{multlined}
\end{equation}
Where, $P_{RF,bit_i,t,PV}$ is the retention failure probability of bit $i$ of the block in interval $t$ affected by PVs and $\Delta_i$ is the thermal stability factor of that bit, which can be different from that of other bits.
The retention failure probability for a block in total execution time is according to (\ref{eq:17}).\vspace{-1pt}
\begin{equation}
\begin{multlined}
\label{eq:17}\vspace{8pt
P_{RF,blk,total, PV}=1- $$\prod_{j=1}^{T} (1- P_{RF, blk,t_{j}, PV})$$\\
\shoveleft[1cm]{=1- $$\prod_{j=1}^{T} ($$\prod_{i=1}^{N-1}(exp(-t_{j} \times exp(-{\Delta}_i)))$$)$$}\vspace{8pt}\\
\shoveleft[0.7cm]{=1- $$\prod_{i=0}^{N-1}($$\sum_{j=1}^{T} t_{j}\times exp(exp(-{\Delta}_i)))}
\end{multlined}
\end{equation}
Where, $T$ is the total number of idle intervals during the workload execution, $t_j$ is the duration of ${j}\textsuperscript{th}$ interval, and $P_{RF,blk,t_{j},PV}$ is the retention failure probability of the block in interval $t_j$.
The retention failure probability of a cell in the block for $T$ intervals is equivalent to this probability for the cell in a single interval that its duration is the summation of all intervals.
The retention failure probability for the total cache is given in (\ref{eq:18}).\vspace{-1pt}
\begin{equation}
\begin{multlined}
\label{eq:18}\vspace{8pt
P_{RF,cache,PV}= 1- $$\prod_{b=0}^{B-1} (1- P_{RF, blk_{b},total,PV})$$\\
\shoveright[5cm]{=1- $$\prod_{b=0}^{B-1}($$\prod_{i=0}^{N-1}($$\sum_{j=1}^{T_{b}} t_{b,j}\times exp(exp(-{\Delta}_{b,i})))$$)$$}
\end{multlined}
\end{equation}
Where, $P_{RF,blk_{b},total,PV}$ is the retention failure probability of block $b$ in total execution time, ${\Delta}_{b,i}$ is thermal stability factor for cell $i$ in block $b$, and $t_{b,i}$ is the interval $i$ of block $b$
\vspace{5pt}
\paragraph{\textbf{PV affected Read Disturbance Probability}}
As observed in (\ref{eq:5}), PVs affect thermal stability factor ($\Delta$), read current ($I_{read}$), and critical switching current ($I_{C_{0}}$).
The read disturbance probability of a cache block for a single read access is according to (\ref{eq:19}). \vspace{-1pt}
\begin{equation}
\begin{multlined}
\label{eq:19}\vspace{8pt
P_{RD,blk,RA,PV}= 1- $$\prod_{i=0}^{N-1}(1- P_{RD,bit_{i},RA,PV})\vspace{8pt}\\
\shoveleft[0.06cm]{=1- $$\prod_{i=0}^{N-1}(exp(\frac{-t_{read}}{\tau}\times exp(\frac{{\Delta}_i ({(I_{read})}_{i}-{(I_{C_{0}})}_{i})}{{(I_{C_{0}})}_{i}})))^{bit_{i}}}\vspace{8pt}\\
\shoveright[3cm]{=1- $$\prod_{i=0}^{N-1}(exp(\frac{-bit_{i}\times t_{read}}{\tau}\times exp(\frac{{\Delta}_i ({(I_{read})}_{i}-{(I_{C_{0}})}_{i})}{{(I_{C_{0}})}_{i}})))}
\end{multlined}
\end{equation}
Where, $P_{RD,~bit_{i},RA,PV}$ is the read disturbance probability of bit $i$ in the block, $bit_i$ is value of that bit, and $\Delta_i$, (${I_{read}})_{i}$, and ${(I_{C_{0}})}_{i}$ are thermal stability factor, read current, and critical switching current for $bit_i$ for a block, respectively.
Since the error is only probable in the bits containing `1', we put the value of bit $i$ ($bit_i$) in the exponent of read disturbance probability of that bit to eliminate the contribution of bits containing `0' in the read disturbance probability.
The read disturbance probability of a cache block for total read accesses is given in (\ref{eq:20}).\vspace{-1pt}
\begin{equation}
\begin{multlined}
\label{eq:20}\vspace{6pt
P_{RD,blk,total,PV}= 1- $$\prod_{j=0}^{R}(1- P_{RD,blk,RA_{j},PV})\vspace{6pt}\\
\shoveleft[0.1cm]{=1- $$\prod_{j=1}^{R}($$\prod_{i=0}^{N-1}(exp(\frac{-bit_{j,i}\times t_{read}}{\tau}\times }\vspace{6pt}\\
\shoveleft[0.5cm]{exp(\frac{{\Delta}_i ({(I_{read})}_{i}-{(I_{C_{0}})}_{i})}{{(I_{C_{0}})}_{i}}))))}\vspace{6pt}\\
\shoveleft[0.1cm]{=1-$$\prod_{i=0}^{N-1}(exp(\frac{-\sum_{j=1}^{R} bit_{i,j}\times t_{read}}{\tau}}\vspace{6pt}\\
\shoveright[2cm]{\times exp(\frac{{\Delta}_i ({(I_{read})}_{i}-{(I_{C_{0}})}_{i})}{{(I_{C_{0}})}_{i}})))}
\end{multlined}
\end{equation}
Where, $R$ is the total number of read accesses, $P_{RD,blk,RA_{j},PV}$ is the read disturbance probability in read access $j$, and $bit_{j,i}$ is the value of bit $i$ of the block in read access $j$.
Finally, we calculate the read disturbance probability for the whole cache using (\ref{eq:21}). \vspace{-1pt}
\begin{equation}
\begin{multlined}
\label{eq:21}\vspace{6pt
P_{RD,cache,PV}=1-$$\prod_{b=0}^{B-1}(1- P_{RD,blk_{b},total,PV})\vspace{6pt}\\
\shoveleft[0.7cm]{=1- $$\prod_{b=0}^{B-1}($$\prod_{i=0}^{N-1}(exp(\frac{-\sum_{j=1}^{R} bit_{b,i,j}\times t_{read}}{\tau}\times}\vspace{6pt}\\
\shoveleft[1cm]{exp(\frac{{\Delta}_{b,i} ({(I_{read})}_{b,i}-{(I_{C_{0}})}_{b,i})}{{(I_{C_{0}})}_{b,i}}))))}
\end{multlined}
\end{equation}
Where, $P_{RD,blk_{b},total,PV}$ is the read disturbance probability of block $b$ in total execution time.
\vspace{5pt}
\paragraph{\textbf{PV Affected Write Failure Probability}}
As observed in (\ref{eq:6}), PVs affect the write failure probability by changing several parameters to STT-MRAM cells.
The write failure probability in a cache block for single write access is according to (\ref{eq:22}). \vspace{-10pt}
\begin{equation}
\begin{multlined}
\label{eq:22}\vspace{6pt
P_{WF,blk,WA,PV}=1- $$\prod_{i=0}^{N-1}(1- P_{WF,bit_{i},WA,PV})\vspace{6pt} \\
\shoveleft[0.1cm]{=1- $$\prod_{i=1}^{N-1}((1- P_{WF,{(cell_i)}\textsubscript{$0\rightarrow1$}})^{{(bit_{i})}\textsubscript{$0\rightarrow1$}}}\vspace{6pt}\\
\shoveright[1.1cm]{\times(1- P_{WF,{(cell_i)}\textsubscript{$1\rightarrow0$}})^{{(bit_{i})}\textsubscript{$1\rightarrow0$}})}
\end{multlined}
\end{equation}
Where, $P_{WF,bit_{i},WA,PV}$ is the write failure probability for bit $i$ in the block, which its value depends on the direction of transition required for that bit in this write operation.
$P_{WF,{(cell_{i})}\textsubscript{$0\rightarrow 1$}}$ and $P_{WF,{(cell_{i})}\textsubscript{$1\rightarrow0$}}$ are the write failure probability in $cell_i$ for ${0\rightarrow1}$ and ${1\rightarrow0}$ transitions, respectively.
The value of ${(bit_i)}\textsubscript{$0\rightarrow1$}$ or ${(bit_i)}\textsubscript{$1\rightarrow0$}$ is one if this bit needs a ${0\rightarrow1}$ or ${1\rightarrow0}$ transition, respectively
The write failure probability of a cache block for all write accesses in total workload execution is according to (\ref{eq:23}). \vspace{-1pt}
\begin{equation}
\begin{multlined}
\label{eq:23}\vspace{6pt
P_{WF,blk,total,PV}=1- $$\prod_{j=1}^{W}(1- P_{WF,blk,WA_{j},PV})\vspace{6pt}\\
\shoveleft[0.1cm]{=1- $$\prod_{j=1}^{W}($$\prod_{i=0}^{N-1}((1- P_{WF,(cell_{i})\textsubscript{$0\rightarrow1$}}))^{{(bit_{j,i})}\textsubscript{$0\rightarrow1$}}}\vspace{6pt}\\
\shoveleft[0.5cm]{\times(1- P_{WF,(cell_{i})\textsubscript{$1\rightarrow0$}}))^{{(bit_{j,i})}\textsubscript{$1\rightarrow0$}})}\\
\shoveleft[0.1cm]{=1- $$\prod_{i=0}^{N-1}((1- P_{WF,{(cell_{i})}\textsubscript{$0\rightarrow1$}})^{(\sum_{j=1}^{W} {(bit_{i,j})}\textsubscript{$0\rightarrow1$})}}\vspace{6pt}\\
\shoveleft[0.1cm]{\times (1- P_{WF,{(cell_{i})}\textsubscript{$1\rightarrow0$}})^{(\sum_{j=1}^{W} {(bit_{i,j})}\textsubscript{$1\rightarrow0$})})}
\end{multlined}
\end{equation}
Where, $W$ is the total number of write accesses to the block, $P_{WF,blk,WA_i,PV}$ is the write failure probability in write access $i$ given in (\ref{eq:22}), and ${(bit_{j,i})}\textsubscript{$0\rightarrow1$}$ and ${(bit_{j,i})}\textsubscript{$1\rightarrow0$}$ are one if bit $i$ requires a ${0\rightarrow1}$ or ${1\rightarrow0}$ transition in write access $j$, respectively.
Finally, we formulate write failure probability in the whole cache in total execution time according to (\ref{eq:24}). \vspace{-1pt}
\begin{equation}
\begin{multlined}
\label{eq:24}\vspace{6pt
P_{WF,cache,PV}= 1- $$\prod_{b=0}^{B-1}(1- P_{WF,blk_{b},total, PV})\vspace{6pt}\\
\shoveleft[0.001cm]{=1- $$\prod_{b=0}^{B-1}$$\prod_{i=0}^{N-1}((1- P_{WF,{(cell_{b,i})}\textsubscript{$0\rightarrow1$}})^{(\sum_{j=1}^{W_{b}} {{(bit_{b,i,j})}\textsubscript{$0\rightarrow1$}})}}\vspace{6pt}\\
\shoveright[1.2cm]{\times (1- P_{WF,{(cell_{b,i})}\textsubscript{$1\rightarrow0$}})^{(\sum_{j=1}^{W_{b}} {{(bit_{b,i,j})}\textsubscript{$1\rightarrow0$}})})}
\end{multlined}
\end{equation}
Where, $P_{WF,blk_{b},total,PV}$ is the write failure probability of block $b$, $P_{WF,{(cell_{b,i})}\textsubscript{$0\rightarrow1$}}$ and $P_{WF,{(cell_{b,i})}\textsubscript{$1\rightarrow0$}}$ are the write failure probabilities in cell $i$ of block $b$ in ${0\rightarrow1}$ and ${1\rightarrow0}$ transitions, respectively, and the value of ${(bit_{b,i,j})}\textsubscript{$0\rightarrow1$}$ (${(bit_{b,i,j})}\textsubscript{$1\rightarrow0$}$) is one if bit $i$ of block $b$ needs ${0\rightarrow1}$ (${1\rightarrow0}$) transition in write access $j$.
\subsection{Total Cache Failure Probability
Earlier, we formulated the occurrence probability of each error type in an STT-MRAM cache during workload execution time.
To measure the cache reliability, we need to calculate the total cache failure probability considering all sources of errors.
A STT-MRAM cache block is always vulnerable to \textit{only} one source of errors at any instant of time during the block lifetime.
A block either is being read and vulnerable to read disturbance, is being written and vulnerable to write failure, or is idle and vulnerable to retention failure.
On the other hand, for a read/write access to a block, only the accessed block is vulnerable to read disturbance/write failure, while all other blocks are vulnerable to retention failure, simultaneously.
Therefore, for a STT-MRAM cache with $B$ blocks, in any instant of time, either all blocks are only vulnerable to retention failure or one block is vulnerable to read disturbance/write failure while the remaining $B-1$ blocks are vulnerable to retention failure.
Calculating the total cache failure probability requires integrating all probable errors occurred in the cache originated from three exclusive sources with overlapping occurrence time.
Retention failure is calculated in unit of time, write failure is formulated based on the number of write accesses, and read disturbance is a function of the number of read accesses.
Integration of all error types requires transforming their probability into a same unit of measurement.
To this aim, we reformulate the probability of retention failure, read disturbance, and write failure in unit of time.
Using the total retention failure, read disturbance, and write failure probabilities of the cache given in (\ref{eq:9}), (\ref{eq:12}), and (\ref{eq:15}), respectively, the reliability in accordance with retention failure, read disturbance, and write failure are derived according to (\ref{eq:25}), (\ref{eq:26}), and (\ref{eq:27}), respectively.\vspace{-1pt}
\begin{equation}
\label{eq:25
{R_{RF}(t)=exp(-N \times \frac{\sum_{b=0}^{B-1} \sum_{i=1}^{T_b} t_{b,i}}{t_{exe}}\times exp(-\Delta)) }
\end{equation}
\begin{equation
\begin{multlined}
\label{eq:26
{R_{RD}(t)=exp(\frac{-t_{read}}{\tau} \times} {exp(\frac{\Delta (I_{read}-{I_{C_0}})}{I_{C_0}}))^{\frac {(\sum_{b=0}^{B-1} \sum_{i=1}^{R_b}n_{b,i})}{t_{exe}}}}
\end{multlined}
\end{equation}
\begin{equation}
\begin{multlined}
\label{eq:27
{R_{WF}(t)=(1-P_{WF, cell_{1\rightarrow0}})^{\frac{(\sum_{b=0}^{B-1} \sum_{i=1}^{W_b} {(n_{1\rightarrow0})\textsubscript {b,i}})}{t_{exe}}}\times} \\
\shoveleft[0.85cm]{(1-P_{WF, cell_{0\rightarrow1}})^{\frac{(\sum_{b=0}^{B-1} \sum_{i=1}^{W_b} {(n_{0\rightarrow1})\textsubscript {b,i}})}{t_{exe}}}}
\end{multlined}
\end{equation}
Where, $R_{RF}(t)$, $R_{RD}(t)$, and $R_{WF}(t)$ are the occurrence probability of retention failure, read disturbance, and write failure per unit of time, respectively, and $t_{exe}$ is the total execution time for which the vulnerable interval $t_{b,i}$, total number of reads for each block ($R_b$), number of `1's in a cache read access ($n_{b,i}$), total number of writes for a cache block ($W_b$), and total number of ${0\rightarrow1}$ and ${1\rightarrow0}$ transitions for each write (${(n_{0\rightarrow1})}\textsubscript{$b,i$}$ and ${(n_{1\rightarrow0})}\textsubscript{$b,i$}$) are calculated.
The final step is to calculate the total cache failure probability per unit of time by integrating the above three reliability equations, according to (\ref{eq:28}). \vspace{-10pt}
\begin{equation}
\label{eq:28
P_{TF/t}=1- (R_{RF}(t)\times R_{RD}(t)\times R_{WF}(t))
\end{equation}
The effects of PVs is not included in the above formulations.
To calculate the cache reliability in terms of retention, read disturbance, and write failure considering PVs, we simply use PV affected probability given in (\ref{eq:18}), (\ref{eq:21}), and (\ref{eq:24}), instead of probabilities in (\ref{eq:9}), (\ref{eq:12}), and (\ref{eq:15}), respectively, for calculating the total cache failure probability.
The parameter $P_{TF}/t$ contains all factors that affect the reliability of an STT-MRAM cache.
It integrates all conflicting physical parameters as well as cache access patterns and data contents.
$P_{TF}/t$ is a key parameter for early reliability exploration of STT-MRAM caches in various scenarios of different device-level to system-level configurations.
An error in a cache block, regardless of its source (retention failure, write failure, and read disturbance) is manifested through a read operation. Therefore, all three sources of error contribute in the probability of reading an erroneous cache block. A block is error-free on a read operation if \textit{none} of the three types of errors occurs. In our evaluations, the occurrence probability of error in a block is calculated on each read operation on that block by integrating the occurrence probability of retention failure during its latest idle interval, write failure in the most recent write operation, and read disturbance in the current read operation. Therefore, while both retention and write failures are coupled with a read disturbance in their subsequent read operation, each of these three error sources has its own contribution in total error probability of a block. In this regards, we \textit{not}~\textit{only} report the total error rate of cache blocks, but also the \textit{breakdown} of the three error sources.
\subsection{Architectural Aspects of Proposed Framework
An early reliability exploration considering device- and circuit-level parameters affected by PVs as well as architecture- and system-level configurations and behaviors requires two main steps.
First, modeling an STT-MRAM cache considering STT-MRAM cells physical characteristics and circuit-level cache configurations.
Second, calculating the reliability metrics, i.e., $R_{RF}(t)$, $R_{RD}(t)$, $R_{WF}(t)$, and $P_{TF/t}$, using the proposed formulations for the target application.
To this aim, we develop a framework that includes all necessary components for reliability exploration and integrate it to gem5 full-system simulator \cite{gem5}.
This framework keeps track of runtime cache accesses to extract the required information for cache block, including content of each block cell at read and write accesses, content of updated data at write accesses, and the intervals between accesses.
The framework imports the physical- and circuit-level parameters as well as different PVs models for STT-MRAM cells.
Using the mentioned inputs, the framework calculates the rate of each error per cache block and the whole cache as well as the total error rate per block and the whole cache.
This framework is also configurable to report the instantaneous error rates and the rates for any time frame.
Fig. \ref{fig:6} shows an abstract view of gem5 simulator and the components of the reliability exploration framework appended to gem5.
The boxes in gray are our framework components.
The physical characteristics of STT-MRAM cells including both nominal and PV affected values as well as the circuit-level configurations of the cache for read/write accesses are included to gem5 at configuration time and the simulator is rebuilt for the simulation.
At runtime, the cache blocks and access requests are monitored and the error probabilities and rates are calculated based on their corresponding formulas.
By the end of simulation, the reliability statistics are reported besides the performance statistics.
\begin{figure}[t
\centering
\includegraphics[width=1\linewidth]{flowchart-amir1}\vspace{-4pt
\caption{Flow of integrating the proposed framework with gem5 simulator.}\vspace{-15pt}
\label{fig:6}
\end{figure}
\section{Evaluation Methodology}
In this section, we experimentally illustrate the dependency of STT-MRAM cache reliability to the workloads behaviors and quantitatively show the extreme error rate variation across different workloads.
In addition, the effects of PVs on the cache reliability are investigated and the observations are discussed.
We integrate our proposed framework with gem5 full-system simulator~\cite{gem5}.
A quad-core processor is modeled, which consists of dedicated instruction- and data- cache per core and a shared L2-cache.
The L2-cache blocks are made of STT-MRAM cells that their detailed parameters are given in Table I and we conduct our reliability evaluations based on this cache configuration.
These parameters are reported in 32nm technology node. To take the PV effects into account, we consider the Gaussian distribution for each PV affected parameter that its mean value is according to Table \ref{table:0}.
The standard deviation for all PV affected parameters is 0.05 of their nominal value, according to \cite{Eken-DAC-2016}.
The system configuration details are given in Table \ref{table:1}.
Workloads are 18 multi-programs generated by combining the workloads in SPEC CPU2006 benchmark suite~\cite{spec2006}.
The details of these combinations are shown in Table \ref{table:2}.
The first one billion instructions in each workload execution are skipped for warm-up phase and the results are reported for the next four billion instructions.
In the following, we first evaluate the effect of workloads on each error probability, separately, with no PV consideration.
Then, the effects of PVs on the probability of each error type are investigated.
In the last subsection, we present the total cache failure probability for all workloads \textit{without} and \textit{with} PVs consideration as well as the contribution of each error type in the total failure probability.
The failure probabilities are reported in microsecond as a unit of time.
\begin{table}[t
\centering\vspace{-1pt}
\caption{STT-MRAM cell parameters and their values}\vspace{-10pt}
\includegraphics[width=1\linewidth]{parameter4}
\label{table:0}\vspace*{-15pt}
\end{table}
\begin{table}[t
\centering\vspace{-1pt}
\caption{Details of processor and caches configuration}\vspace{-10pt}
\includegraphics[width=0.9\linewidth]{TABLE1----new
\label{table:1}\vspace*{-10pt}
\end{table}
\begin{table}[t
\centerin
\caption{Combinations of multi-programmed workloads for quad-core processor}\vspace{-10pt}
\includegraphics[width=0.85\linewidth]{TABLE2--}
\label{table:2}\vspace*{-15pt}
\end{table}
\subsection{Workload Effects on Error Probabilities}
The effects of workloads on error rates are twofold: \textbf{a)} the pattern of cache requests, and \textbf{b)} the content of stored and updated data blocks.
The rate of read disturbance depends on both the number of read accesses and the content of the read blocks; the rate of write failure depends on both the number of write accesses and the previous and current content of written blocks; and finally, the rate of retention failure depends on the number of read/write accesses, the intervals between accesses to blocks, and the combination of read and write accesses to a block.
For a deep analysis and a more detailed investigation, we present both the error probabilities considering all effects of workloads (access sequences and blocks contents) and the worst case error probability based on the cache access patterns.
In the worst case error probability for read disturbance and write failure, it is assumed that all block cells are vulnerable regardless of their content.
Therefore, the effect of data patterns is excluded in this case.
In the worst case error probability for retention failure, all idle intervals are included, whereas the intervals between a read and its subsequent write access are excluded when considering only vulnerable intervals.
Reporting error probabilities in both mentioned schemes unfolds the breakdown of workloads effects on cache reliability.
\begin{figure}[t
\centering
\includegraphics[width=1\linewidth]{WC-retention}\vspace{-8pt
\caption{Effects of workloads behavior on retention failure probability with/without excluding the intervals that errors are masked.}\vspace{-10pt}
\label{fig:7}
\end{figure}
Fig. \ref{fig:7} shows the retention failure probability for all workloads.
The average error probability varies from 7.1$\times$$10^{\text{-8}}$ in $Mix1$ workload to 1.8$\times$$10^{\text{-6}}$ in $Mix5$ workload, indicating 25.4x variation between the error probabilities of different workloads.
The average probability of retention failure is 2.6$\times$$10^{\text{-7}}$.
By including the intervals between the read and its subsequent write access, the error probability is increased by more than 3 orders of magnitude, on average.
The variation between error probabilities of different workloads in the worst case is due to various execution times.
This variation is much lower than the time considering only vulnerable intervals.
\begin{figure}[t]\vspace{-13pt}
\centering
\includegraphics[width=1\linewidth]{WC-read}\vspace{-8pt}
\caption{Effects of workloads behavior on read disturbance probability with/without considering the cache blocks content (content-dependent vs. worst case error rate).}\vspace{-10pt}
\label{fig:9-rate}
\end{figure}
\begin{figure}[t
\centering
\includegraphics[width=1\linewidth]{WC-write-HF}\vspace{-8pt}%
\caption{Effects of workloads behavior on write failure probability with/without considering the required transitions on a write access (content-dependent vs. worst case error rate).}\vspace{-12pt}
\label{fig:7write}
\end{figure}
The read disturbance probability for all workloads is shown in Fig. \ref{fig:9-rate}.
A large variation is observed in both content-dependent and worst case error probabilities.
The content-dependent error probability varies from 4.7$\times$$10^{\text{-5}}$ in $Mix15$ workload to 3.2$\times$$10^{\text{-4}}$ in $Mix14$ workload, indicating 6.8x variations, while the worst case error probability varies from 2.7$\times$$10^{\text{-4}}$ in $Mix6$ workload to 1.3$\times$$10^{\text{-3}}$ in $Mix1$ workload, indicating 4.8x variation.
From another perspective, the content of the data not only causes a large variation in error probabilities, as observed in content-dependent error probabilities, but also significantly reduces the error probability, as the large gap between content-dependent and worst case probabilities confirms this fact.
On average, the content-dependent error probability is by 4.2x lower than the worst case error probability, which is due to large number of `0's in cache blocks.
However, in some workload, e.g., $Mix12$, $Mix13$, and $Mix14$, the content-dependent error probability is close to that in the worst case.
\begin{figure}[t]\vspace{-10pt}
\centering
\includegraphics[width=1\linewidth]{PV-retention11}\vspace{-8pt}%
\caption{Retention failure probability without/with considering PV effects.}\vspace{-5pt}
\label{fig:9}
\end{figure}
Fig. \ref{fig:7write} depicts the write failure probabilities for the workloads.
Assuming all block cells require a ${0\rightarrow1}$ transition, the worst case error probability is 2.7$\times$$10^{\text{-3}}$, on average.
This value varies from 7.7$\times$$10^{\text{-4}}$ in $Mix6$ workload to 6.7$\times$$10^{\text{-3}}$ in $Mix1$ workload.
Including the data content, the average error probability varies from 3.9$\times$$10^{\text{-5}}$ in $Mix6$ workload to 5.9$\times$$10^{\text{-4}}$ in $Mix1$ workload, which indicates a 15.1x variation.
\begin{figure}[t
\centering
\includegraphics[width=1\linewidth]{PV-read11}\vspace{-10pt}%
\caption{Read disturbance probability without/with considering PV effects.}\vspace{-13pt}
\label{fig:10}
\end{figure}
\subsection{PV Effects on Error Probabilities}
\begin{figure}[b]\vspace{-10pt}
\centering
\includegraphics[width=1\linewidth]{PV-write11}\vspace{-8pt}%
\caption{Write failure probability without/with considering PV effects.}\vspace{-7pt}
\label{fig:11}
\end{figure}
The effect of PVs on the vulnerability of a single cell can easily be analyzed based on the deviations in the physical cell parameters.
However, due to dependency of error probabilities to the workloads behavior, the effect of PVs on the cache error probabilities needs system-level exploration.
In the following, we report the probability of cache error type in the presence of PVs.
Fig. \ref{fig:9} shows the retention failure probability for all workloads.
On average, the error probability is increased by 32.5x.
Regarding retention failure probability, thermal stability factor ($\Delta$) is the only parameter affected by PVs.
Although PVs reduce the retention failure probability for some STT-MRAM cells and increase it for other cells, the total effect of PVs on retention failure probability of the cache result in a significant increase in its probability for all workloads.
The effects of PVs on read disturbance probability is shown in Fig. \ref{fig:10}.
On average, the error probability is increased by 9.0x when considering the PV effects.
The results show that the effects of PVs on error probability are almost similar for all workloads.
Read current ($I_{read}$), critical switching current ($I_{C_{0}}$), and thermal stability factor ($\Delta$) are under PVs and can oppositely affect the vulnerability of an STT-MRAM cell to read disturbance.
Although the PVs can increase the error probability in some cells and decrease in other cells, the overall effect of PVs on STT-MRAM cache is to significantly increase the read disturbance error probability.
\begin{figure*}[t]\vspace{-10pt}
\centering
\includegraphics[width=1\linewidth]{Total-PV-new11}\vspace{-8pt}%
\caption{Effect of workloads on total failure probability without/with considering PV effects.}\vspace{-10pt}
\label{fig:12}
\end{figure*}
\begin{figure*}[b]\vspace{-10pt}
\centering
\subfloat[]{\includegraphics[width=1\linewidth]{EDCC-extention2}}\vspace{-8pt
\hfill
\subfloat[]{\includegraphics[width=1\linewidth]{EDCC-extention1}}\vspace{-3pt
\caption{Breakdown of total error rate to retention failure, read disturbance, and write failure (a) retention failure probability in only vulnerable intervals, (b) retention failure probability in all intervals.
\label{fig:13}
\end{figure*}
The probability of write failure affected by PVs in conjunction to that without PVs is depicted in Fig. \ref{fig:11}.
The results show an average of 16.7x increase in write failure probability when considering the PV effects.
The effect of PVs on a single STT-MRAM cell is either to increase or decrease the write failure probability depending on the deviations of thermal stability factor ($\Delta$), write current ($I_{write}$), and critical switching current ($I_{C_{0}}$) from their nominal values.
However, the overall consequence of PVs on the write failure of a STT-MRAM cache is to significantly increase the probability of this error.
\vspace{-8pt}
\subsection{Total Error Rate}
Fig. \ref{fig:12} depicts the total cache failure probability including retention failure, read disturbance, and write failure rates, without and with PVs consideration.
On average, the total failure probability is 3.1$\times$$10^{\text{-4}}$ and is increased to 2.0$\times$$10^{\text{-3}}$ when considering PV effects.
This observation implies more than 6.5x increase in total failure probability by the PVs.
Without considering PVs, the minimum and maximum failure probabilities are observed in $Mix10$ and $Mix12$, respectively, which indicates a large gap between the corner cases.
The maximum failure probability observed in $Mix12$ workload is by 32.0x higher than the minimum failure probability observed in $Mix10$ workload.
The variation between the failure probabilities of workloads when considering PV effects is much lower than that of not including PVs.
The maximum failure probability, which is observed in $Mix14$ workload (4.5$\times$$10^{\text{-3}}$) is by 6.8x higher than the minimum failure probability, which is observed in $Mix15$ workload (6.6$\times$$10^{\text{-4}}$).
The source of large values in total cache failure probability is because of high write failure and read disturbance probabilities, as observed in Fig. \ref{fig:10} and Fig. \ref{fig:11}.
The read disturbance and write failure probabilities of the cache are affected by the circuit- and physical-level parameters that determine the error rate of a single STT-MRAM cell.
Error rate is exponentially affected by changing these parameters, e.g., write current and pulse width, read current and pulse width, and $\Delta$, as previously observed in (\ref{eq:5}) and (\ref{eq:6}).
The goal of this study is \textit{not} to focus on the absolute error rate of STT-MRAM caches, which is strongly technology-dependent and is largely affected by design- and manufacturing-time parameters adjustment.
Our goal, regardless of the per-cell error rate of the STT-MRAM cache, which is fixed and measurable through manufacturing process, is to demonstrate that: \textbf{a)} total cache error rate is largely affected by workloads behavior and a significant variation exists in different workloads; \textbf{b)}~\textit{none} of the three sources of errors can be ignored, since they have considerable contribution in total cache error rate; \textbf{c)} the workload behavior causes a large variation in the contribution of three sources of errors in total error rate.
Fig. \ref{fig:13} shows the breakdown of total failure probability for all workloads in two scenarios for retention failure.
As mentioned, retention failure occurred in an interval between a read/write access followed by a write access is overwritten by an updated data and can be ignored.
Fig. \ref{fig:13}(a) and Fig. \ref{fig:13}(b) depict the breakdown of total failure probability when excluding the mentioned intervals in retention failure and when including them, respectively.
According to Fig. \ref{fig:13}(a), read disturbance and write failure contribute in the majority of total failure probability, whereas the average contribution of retention failure is only 1.2\%.
Due to negligible contribution of retention failure in this scenario, we depict the results in both linear and logarithmic scales.
On average, 54.6\% of total failure probability is due to read disturbance while write failure causes 44.2\% of errors.
A maximum of 71.6\% contribution for write failure is observed in $Mix1$ workload, while the minimum value is less than 0.05\% in $Mix10$ workload.
The contribution of read disturbance varies from 28.4\% in $Mix1$ workload to 98.2\% in $Mix10$ workload.
A significantly larger variation is observed in the contribution of retention failure, in which the minimum and maximum values are 0.003\% and 15.7\% in $Mix12$ and $Mix15$ workloads, respectively.
It is noteworthy that the results are for an L2-cache and the contribution of retention failure will significantly increase in lower-level caches, e.g., L3-caches, due to lower frequency of read/write accesses and larger cache size.
Fig. \ref{fig:13}(b) depicts the breakdown of total failure probability considering all idle intervals of cache blocks in retention failure probability calculation.
The contribution of retention failure is as high as 73.1\%, on average, while read disturbance and write failure contribute for 12.2\% and 14.7\% of total errors, respectively.
The significant increase in the contribution of retention failure compared with that in Fig. \ref{fig:13}(a) implies that the cache blocks spend a large fraction of their idle time in intervals that are waiting to be written.
These intervals are the times between a read followed by a writeback access or eviction of the blocks.
Read disturbance contributes from 3.7\% in $Mix5$ workload to as high as 59.0\% in $Mix12$ workload.
The minimum and maximum value for write failure contribution are 0.002\% and 25.6\% in $Mix10$ and $Mix1$ workloads, respectively.
\section{Discussion and Guidelines}
Several techniques have been presented in the literature to overcome each source of errors in STT-MRAM memories.
Increasing write current and write pulse width, decreasing the thermal stability factor, reading and verifying data after each write (write-read-verify), and employing ${Error}$-${Correcting~Codes}$ (ECCs) are the technique to overcome the write failures~\cite{nowak2016dependence,sun-TMAG-12,sun2011design,lakys2012self,Pajouhi2016JETC,naeimi2013intel,ZAZADTE,ZAZADTPDS,ahn2013selectively}.
Reducing read current and read pulse width, increasing thermal stability factor, overwriting after each read, and employing ECCs are among the techniques that tackle read disturbance~\cite{wang2015selective,Ran2016JSA,Na-TCAS-II-16, maddah2015cafo,14-zazad-eken2014novel}.
Vulnerability to retention failure is reduced by increasing thermal stability factor, employing ECCs, and memory scrubbing\cite{naeimi2013intel,smullen2011relaxing,mittal2017survey}.
There are three challenges in designing a reliable STT-MRAM cache based on the existing techniques.
First, these techniques overcome the errors independently and three error mitigation techniques need to be employed for three sources of errors.
Without including the workload-dependency of each error type and its contribution in total error rate, each technique should be configured for the worst case error rate.
This pessimistic assumption leads to an overdesigned cache with significant protection overheads.
For example, several studies presented ECCs for each error type and to overcome all three sources of errors, three kinds of coding need to be employed for each cache block.
Second, techniques for protecting against one error may increase the rate of other error type(s).
Reduction in $\Delta$ for decreasing write failure increases the rate of read disturbance and retention failure.
Increasing $I_{write}$ and $t_{write}$ worsens the probability of MTJ barrier breakdown in addition to imposing significant energy consumption and delay overheads.
Reducing $I_{read}$ and $t_{read}$ increases the sensing error probability.
In addition, memory scrubbing increases the rate of read disturbance.
Third, some of these techniques are in contradiction to each other and cannot be employed jointly.
Increasing $\Delta$ of STT-MRAM cells for reducing read disturbance and retention failure rate is in contrast with the technique of decreasing $\Delta$ for reducing write failure rate.
Overwriting a data block after each read for correcting read disturbance errors cannot be integrated with write-read-verify technique.
Our formulations and framework help to estimate the cache reliability before and after employing error mitigation techniques and to determine the level of protection required for each error type based on its effects on the total error rate.
Based on our observations, there is a large difference between the contribution of error types in total cache error rate and even in the rate of each error in various scenarios.
These observations provide a roadmap to optimize the given budget of the cache protection for maximizing the reliability or to achieve the required level of reliability with the minimum cost.
They can also help to design cache configurations that adaptively utilize error mitigation techniques at runtime based on the current behavior of different sources of errors.
\section{Conclusions}
Unreliability of emerging STT-MRAM caches due to retention failure, read disturbance, and write failure is a major design concern.
Vulnerability of cache blocks to these sources of errors strongly depends on not only process variations, but also the cache access patterns and data content.
On the other hand, the rate of different errors is differently and even oppositely affected by a workload behavior and process variations.
This paper formulated the vulnerability of STT-MRAM caches to all error types considering the conflicting effects of both PVs and workloads behavior on the rate of different errors.
These formulations provide an analytical reliability exploration environment for STT-MRAM caches.
In addition, we developed a system-level framework to empirically investigate the reliability of STT-MRAM caches for different device- to system-level configurations and various PVs and workload considerations.
We integrated our framework with gem5 full-system simulator and investigated the effects of PVs and workloads on the cache reliability.
The results for a shared L2-cache in a quad-core processor showed that PVs increase the cache error rate by 6.5x, on average, while this rate varies for an average of 32.0x in different workloads.
In addition, a large diversity is observed in the contribution of each error type in total vulnerability of STT-MRAM cache.
Our formulations and framework help system architects to optimize the error mitigation techniques based on each error behavior and to design cost-efficient highly reliable STT-MRAM caches.
\bibliographystyle{IEEEtran}
|
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| 1,578
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Il Museo dell'Opera Pia Purgatorio ad Arco è un museo di Napoli situato all'interno della chiesa di Santa Maria delle Anime del Purgatorio ad Arco.
Il museo espone per lo più oggetti liturgici, databili dal XVI al XIX secolo. Inoltre sono visibili anche paramenti, calici, libri, suppellettili delle celebrazioni ed alcune tele del Seicento napoletano. Fanno parte del museo dell'Opera anche gli spazi sottostanti la chiesa
Descrizione
Il museo è allestito negli ambienti interni della chiesa, più precisamente nella sacrestia e nell'oratorio dell'Immacolata. Il piccolo museo fu realizzato sotto la volontà del professor Nicola Spinosa, sopraintendente per il polo museale di Napoli, con l'intento di dare una migliore collocazione agli oggetti esposti, fino a quel tempo conservati all'interno degli armadi della sacrestia.
Chiesa
Nella sacrestia presenti arredi mobiliari della prima metà dell'Ottocento, eseguiti da Michele Guggenberger su disegno di Michelangelo del Gaiso. Gli argenti ivi esposti sono invece databili dal Seicento all'Ottocento e sono per lo più calici, una croce d'altare, una pisside, turiboli, lampade ed ostensori.
Sia nella sacrestia che nell'oratorio sono altresì esposti alcuni dipinti del Seicento e Settecento, come una copia del dipinto di Luis de Morales raffigurante la Madonna della Purità, oggi conservato presso gli ambienti interni della basilica di San Paolo Maggiore di Napoli. Altri dipinti presenti sono un Sant'Aniello che scaccia i Saraceni da Napoli di Fabrizio Santafede, un San Sebastiano curato da un angelo attribuito a Giacomo Farelli, unAssunta di Paolo De Matteis, un San Gennaro in gloria attribuito ad Alessio D'Elia ed unImmacolata autografa e datata 1748 di Michelangelo Buonocore (allievo del sopracitato De Matteis).
Ipogeo
Sulla sinistra del portale della chiesa, da una botola si accede ad un ambiente sotterraneo delle stesse dimensioni della chiesa costituente un luogo di culto anteriore alla chiesa delle Anime del Purgatorio. Il pavimento è stato realizzato dai fratelli Giuseppe e Donato Massa mentre l'altare in piperno risale al XVIII secolo. A sinistra vi è un corridoio decorato con teschi che dà accesso alla tomba di Giulio Mastrilli. Questo secondo ambiente sotterraneo fungeva da ipogeo, con teschi e spazi destinati alla sepoltura di corpi umani. Uno di questi teschi richiama il ricordo dell'anima di Lucia, morta in un naufragio insieme al suo sposo: a questa figura viene richiesta grazie e intercessioni e vengono offerti fiori e foto dei familiari come ex voto.
Bibliografia
A.V., Il Purgatorio ad Arco - La chiesa l'ipogeo il museo, Altrestampa
Voci correlate
Chiesa di Santa Maria delle Anime del Purgatorio ad Arco
Collegamenti esterni
Purgatorio ad Arco
Purgatorio ad Arco
Chiesa di Santa Maria delle Anime del Purgatorio ad Arco
Opera Pia
|
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{"url":"https:\/\/www.hackmath.net\/en\/math-problem\/689?tag_id=7","text":"# Root\n\nThe root of the equation\n$(x-10)2 +10 = x2 +61x$\nis:\n\n\u00ab Correct result\n\n#### Solution:\n\n$(x-10)^2 +10 = x^2 +61x \\ \\\\ x = \\dfrac{ 10^2 +10 } { 2\\cdot 10 +61 } \\ \\\\ x = 1.36 \\ \\\\ x > 0$\n\nCheckout calculation with our calculator of quadratic equations.\n\nOur examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!\n\nLeave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):\n\nBe the first to comment!\n\nTips to related online calculators\nLooking for help with calculating roots of a quadratic equation?\nDo you have a linear equation or system of equations and looking for its solution? Or do you have quadratic equation?\n\n## Next similar math problems:\n\n1. A rectangle 2\nA rectangle has a diagonal length of 74cm. Its side lengths are in ratio 5:3. Find its side lengths.\nFind the radius of the circle with area S = 200 cm\u00b2.\n3. Secret treasure\nScouts have a tent in the shape of a regular quadrilateral pyramid with a side of the base 4 m and a height of 3 m. Determine the radius r (and height h) of the container so that they can hide the largest possible treasure.\n4. Before yesterday\nHe merchant adds a sale sign in his shop window to the showed pair of shoes in the morning: \"Today by p% cheaper than yesterday. \" After a while, however, he decided that the sign saying: \"Today 62.5% cheaper than the day before yesterday\". Determine the\n5. Geometric progressiob\nIf the sum of four consective terms of geometric progression is 80 and arithmetic mean of second and fourth term is 30 then find terms?\n6. Square side\nIf we enlarge the square side a = 5m, its area will increase by 10,25%. How many percent will the side of the square increase? How many percent will it increase the circumference of the square?\n7. Conical bottle\nWhen a conical bottle rests on its flat base, the water in the bottle is 8 cm from it vertex. When the same conical bottle is turned upside down, the water level is 2 cm from its base. What is the height of the bottle?\n8. The product\nThe product of a number plus that number and its inverse is two and one-half. What is the inverse of this number\n9. Rectangle field\nThe field has a shape of a rectangle having a length of 119 m and a width of 19 m. , How many meters have to shorten its length and increase its width to maintain its area and circumference increased by 24 m?\n10. Three parallels\nThe vertices of an equilateral triangle lie on 3 different parallel lines. The middle line is 5 m and 3 m distant from the end lines. Calculate the height of this triangle.\n11. Flowerbed\nWe enlarge the circular flower bed, so its radius increased by 3 m. The substrate consumption per enlarged flower bed was (at the same layer height as before magnification) nine times greater than before. Determine the original flowerbed radius.\n12. Sides of right angled triangle\nOne leg is 1 m shorter than the hypotenuse, and the second leg is 2 m shorter than the hypotenuse. Find the lengths of all sides of the right-angled triangle.\n13. Equation 23\nFind value of unknown x in equation: x+3\/x+1=5 (problem finding x)\n14. Medians in right triangle\nIt is given a right triangle, angle C is 90 degrees. I know it medians t1 = 8 cm and median t2 = 12 cm. .. How to calculate the length of the sides?\n15. Faces diagonals\nIf the diagonals of a cuboid are x, y, and z (wall diagonals or three faces) respectively than find the volume of a cuboid. Solve for x=1.2, y=1.7, z=1.45\n16. Two chords\nCalculate the length of chord AB and perpendicular chord BC to circle if AB is 4 cm from the center of the circle and BC 8 cm from the center of the circle.\n17. Reciprocal value\nHow do I calculate a number x that is 9 greater than its reciprocal (1\/x)?","date":"2020-02-19 22:56:13","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 2, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5159900784492493, \"perplexity\": 1027.9850791989506}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875144429.5\/warc\/CC-MAIN-20200219214816-20200220004816-00033.warc.gz\"}"}
| null | null |
package org.optaplanner.examples.examination.domain;
import com.thoughtworks.xstream.annotations.XStreamAlias;
import org.optaplanner.examples.common.domain.AbstractPersistable;
@XStreamAlias("Room")
public class Room extends AbstractPersistable {
private int capacity;
private int penalty;
public int getCapacity() {
return capacity;
}
public void setCapacity(int capacity) {
this.capacity = capacity;
}
public int getPenalty() {
return penalty;
}
public void setPenalty(int penalty) {
this.penalty = penalty;
}
public String getLabel() {
return Long.toString(id);
}
@Override
public String toString() {
return Long.toString(id);
}
}
|
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"redpajama_set_name": "RedPajamaGithub"
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class ScopedLeakSanitizerDisabler {
public:
ScopedLeakSanitizerDisabler() { __lsan_disable(); }
~ScopedLeakSanitizerDisabler() { __lsan_enable(); }
private:
DISALLOW_COPY_AND_ASSIGN(ScopedLeakSanitizerDisabler);
};
#define ANNOTATE_SCOPED_MEMORY_LEAK \
ScopedLeakSanitizerDisabler leak_sanitizer_disabler; static_cast<void>(0)
#define ANNOTATE_LEAKING_OBJECT_PTR(X) __lsan_ignore_object(X);
#else
#define ANNOTATE_SCOPED_MEMORY_LEAK ((void)0)
#define ANNOTATE_LEAKING_OBJECT_PTR(X) ((void)0)
#endif
#endif // BASE_DEBUG_LEAK_ANNOTATIONS_H_
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 3,577
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{"url":"https:\/\/math.libretexts.org\/Courses\/Monroe_Community_College\/MTH_104_Intermediate_Algebra\/7%3A_Rational_Expressions_and_Functions\/7.2%3A_Multiply_and_Divide_Rational_Expressions","text":"\n# 7.2: Multiply and Divide Rational Expressions\n\n\nSummary\n\nBy the end of this section, you will be able to:\n\n\u2022 Determine the values for which a rational expression is undefined\n\u2022 Simplify rational expressions\n\u2022 Multiply rational expressions\n\u2022 Divide rational expressions\n\u2022 Multiply and divide rational functions\n\nBefore you get started, take this readiness quiz.\n\n1. Simplify: $$\\frac{90y}{15y^2}$$.\nIf you missed this problem, review [link].\n2. Multiply: $$\\frac{14}{15}\u00b7\\frac{6}{35}$$.\nIf you missed this problem, review [link].\n3. Divide: $$\\frac{12}{10}\u00f7\\frac{8}{25}$$.\nIf you missed this problem, review [link].\n\nWe previously reviewed the properties of fractions and their operations. We introduced rational numbers, which are just fractions where the numerators and denominators are integers. In this chapter, we will work with fractions whose numerators and denominators are polynomials. We call this kind of expression a rational expression.\n\nRATIONAL EXPRESSION\n\nA rational expression is an expression of the form $$\\frac{p}{q}$$, where p and q are polynomials and $$q\\neq 0$$.\n\nHere are some examples of rational expressions:\n\n$\u2212\\frac{24}{56} \\qquad \\frac{5x}{12y} \\qquad \\frac{4x+1}{x^2\u22129} \\qquad \\frac{4x^2+3x\u22121}{2x\u22128}\\nonumber$\n\nNotice that the first rational expression listed above, $$\u2212\\frac{24}{56}$$, is just a fraction. Since a constant is a polynomial with degree zero, the ratio of two constants is a rational expression, provided the denominator is not zero.\n\nWe will do the same operations with rational expressions that we did with fractions. We will simplify, add, subtract, multiply, divide and use them in applications.\n\n# Determine the Values for Which a Rational Expression is Undefined\n\nIf the denominator is zero, the rational expression is undefined. The numerator of a rational expression may be 0\u2014but not the denominator.\n\nWhen we work with a numerical fraction, it is easy to avoid dividing by zero because we can see the number in the denominator. In order to avoid dividing by zero in a rational expression, we must not allow values of the variable that will make the denominator be zero.\n\nSo before we begin any operation with a rational expression, we examine it first to find the values that would make the denominator zero. That way, when we solve a rational equation for example, we will know whether the algebraic solutions we find are allowed or not.\n\nDETERMINE THE VALUES FOR WHICH A RATIONAL EXPRESSION IS UNDEFINED.\n\n1. Set the denominator equal to zero.\n2. Solve the equation.\n\nEXAMPLE $$\\PageIndex{1}$$\n\nDetermine the value for which each rational expression is undefined:\n\n\u24d0 $$\\frac{8a^2b}{3c}$$ \u24d1 $$\\frac{4b\u22123}{2b+5}$$ \u24d2 $$\\frac{x+4}{x^2+5x+6}$$.\n\nThe expression will be undefined when the denominator is zero.\n\n$$\\begin{array} {ll} &\\frac{8a^2b}{3c} \\\\ \\begin{array} {l} \\text{Set the denominator equal to zero and solve} \\\\ \\text{for the variable.} \\end{array} &3c=0 \\\\ &c=0 \\\\ &\\frac{8a^2b}{3c}\\text{ is undefined for }c=0 \\end{array}$$\n\n$$\\begin{array} {ll} &\\frac{4b-3}{2b+5} \\\\ \\begin{array} {l} \\text{Set the denominator equal to zero and solve} \\\\ \\text{for the variable.} \\end{array} &2b+5=0 \\\\ &2b=-5 \\\\ &b=-\\frac{5}{2} \\\\ & \\\\ &\\frac{4b-3}{2b+5} \\text{ is undefined for }b=-\\frac{5}{2} \\end{array}$$\n\n$$\\begin{array} {ll} &\\frac{x+4}{x^2 + 5x + 6} \\\\ \\begin{array} {l} \\text{Set the denominator equal to zero and solve } \\\\ \\text{for the variable.} \\end{array} &x^2+5x+6=0 \\\\ &(x+2)(x+3)=0 \\\\ &x+2=0\\text{ or }x+3=0 \\\\ &x=-2\\text{ or }x=-3 \\\\ & \\\\ &\\frac{x+4}{x^2+5x+6}\\text{ is undefined for }x=-2\\text{ or }x=-3 \\end{array}$$\n\nEXAMPLE $$\\PageIndex{2}$$\n\nDetermine the value for which each rational expression is undefined.\n\n\u24d0 $$\\frac{3y^2}{8x}$$ \u24d1 $$\\frac{8n\u22125}{3n+1}$$ \u24d2 $$\\frac{a+10}{a^2+4a+3}$$\n\n\u24d0 $$x=0$$ \u24d1 $$n=\u2212\\frac{1}{3}$$\n\u24d2 $$a=\u22121,a=\u22123$$\n\nEXAMPLE $$\\PageIndex{3}$$\n\nDetermine the value for which each rational expression is undefined.\n\n\u24d0$$\\frac{4p}{5q}$$ \u24d1 $$\\frac{y\u22121}{3y+2}$$ \u24d2 $$\\frac{m\u22125}{m^2+m\u22126}$$\n\n\u24d0 $$q=0$$ \u24d1 $$y=\u2212\\frac{2}{3}$$\n\u24d2 $$m=2,m=\u22123$$\n\n# Simplify Rational Expressions\n\nA fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator. Similarly, a simplified rational expression has no common factors, other than 1, in its numerator and denominator.\n\nSIMPLIFIED RATIONAL EXPRESSION\n\nA rational expression is considered simplified if there are no common factors in its numerator and denominator.\n\nFor example,\n\n$\\begin{array} {l} \\frac{x+2}{x+3} \\text{ is simplified because there are no common factors of } x+2 \\text{ and }x+3. \\\\ \\frac{2x}{3x} \\text{ is not simplified because x is a common factor of }2x\\text{ and }3x. \\\\ \\end{array} \\nonumber$\n\nWe use the Equivalent Fractions Property to simplify numerical fractions. We restate it here as we will also use it to simplify rational expressions.\n\nEQUIVALENT FRACTIONS PROPERTY\n\nIf a, b, and c are numbers where $$b\\neq 0,c\\neq 0,$$\n\n$\\text {then } \\frac{a}{b}=\\frac{a\u00b7c}{b\u00b7c} \\text{ and } \\frac{a\u00b7c}{b\u00b7c}=\\frac{a}{b}\\nonumber$\n\nNotice that in the Equivalent Fractions Property, the values that would make the denominators zero are specifically disallowed. We see $$b\\neq 0,c\\neq 0$$ clearly stated.\n\nTo simplify rational expressions, we first write the numerator and denominator in factored form. Then we remove the common factors using the Equivalent Fractions Property.\n\nBe very careful as you remove common factors. Factors are multiplied to make a product. You can remove a factor from a product. You cannot remove a term from a sum.\n\nRemoving the x\u2019s from $$\\frac{x+5}{x}$$ would be like cancelling the 2\u2019s in the fraction $$\\frac{2+5}{2}!$$\n\nHow to Simplify a Rational Expression\n\nEXAMPLE $$\\PageIndex{4}$$\n\nSimplify: $$\\frac{x^2+5x+6}{x^2+8x+12}$$\n\nEXAMPLE $$\\PageIndex{5}$$\n\nSimplify: $$\\frac{x^2\u2212x\u22122}{x^2\u22123x+2}$$.\n\n$$\\frac{x+1}{x\u22121},x\\neq 2,x\\neq 1$$\n\nEXAMPLE $$\\PageIndex{6}$$\n\nSimplify: $$\\frac{x^2\u22123x\u221210}{x^2+x\u22122}$$.\n\n$$\\frac{x\u22125}{x\u22121},x\\neq \u22122,x\\neq 1$$\n\nWe now summarize the steps you should follow to simplify rational expressions.\n\nSIMPLIFY A RATIONAL EXPRESSION.\n\n1. Factor the numerator and denominator completely.\n2. Simplify by dividing out common factors.\n\nUsually, we leave the simplified rational expression in factored form. This way, it is easy to check that we have removed all the common factors.\n\nWe\u2019ll use the methods we have learned to factor the polynomials in the numerators and denominators in the following examples.\n\nEvery time we write a rational expression, we should make a statement disallowing values that would make a denominator zero. However, to let us focus on the work at hand, we will omit writing it in the examples.\n\nEXAMPLE $$\\PageIndex{7}$$\n\nSimplify: $$\\frac{3a^2\u221212ab+12b^2}{6a^2\u221224b^2}$$.\n\n$$\\begin{array} {ll} &\\frac{3a^2\u221212ab+12b^2}{6a^2\u221224b^2} \\\\ & \\\\ & \\\\ \\begin{array} {l} \\text{Factor the numerator and denominator,} \\\\ \\text{first factoring out the GCF.} \\end{array} &\\frac{3(a^2\u22124ab+4b^2)}{6(a^2\u22124b^2)} \\\\ & \\\\ &\\frac{3(a\u22122b)(a\u22122b)}{6(a+2b)(a\u22122b)} \\\\ & \\\\ \\text{Remove the common factors of }a\u22122b\\text{ and }3. &\\frac{\\cancel{3}(a\u22122b)\\cancel{(a\u22122b)}}{\\cancel{3}\u00b72(a+2b)\\cancel{(a\u22122b)}} \\\\ &\\frac{a\u22122b}{2(a+2b)} \\end{array}$$\n\nEXAMPLE $$\\PageIndex{8}$$\n\nSimplify: $$\\frac{2x^2\u221212xy+18y^2}{3x^2\u221227y^2}$$.\n\n$$\\frac{2(x\u22123y)}{3(x+3y)}$$\n\nEXAMPLE $$\\PageIndex{9}$$\n\nSimplify: $$\\frac{5x^2\u221230xy+25y^2}{2x^2\u221250y^2}$$.\n\n$$\\frac{5(x\u2212y)}{2(x+5y)}$$\n\nNow we will see how to simplify a rational expression whose numerator and denominator have opposite factors. We previously introduced opposite notation: the opposite of a is $$\u2212a$$ and $$\u2212a=\u22121\u00b7a$$.\n\nThe numerical fraction, say $$\\frac{7}{\u22127}$$ simplifies to $$\u22121$$. We also recognize that the numerator and denominator are opposites.\n\nThe fraction $$\\frac{a}{\u2212a}$$, whose numerator and denominator are opposites also simplifies to $$\u22121$$.\n\n$\\begin{array} {ll} \\text{Let\u2019s look at the expression }b\u2212a. &b\u2212a \\\\ \\text{Rewrite.} &\u2212a+b \\\\ \\text{Factor out }\u20131. &\u22121(a\u2212b) \\nonumber\\end{array}$\n\nThis tells us that $$b\u2212a$$ is the opposite of $$a\u2212b$$.\n\nIn general, we could write the opposite of $$a\u2212b$$ as $$b\u2212a$$. So the rational expression $$\\frac{a\u2212b}{b\u2212a}$$ simplifies to $$\u22121$$.\n\nOPPOSITES IN A RATIONAL EXPRESSION\n\nThe opposite of $$a\u2212b$$ is $$b\u2212a$$.\n\n$\\frac{a\u2212b}{b\u2212a}=\u22121 \\quad a\\neq b\\nonumber$\n\nAn expression and its opposite divide to $$\u22121$$.\n\nWe will use this property to simplify rational expressions that contain opposites in their numerators and denominators. Be careful not to treat $$a+b$$ and $$b+a$$ as opposites. Recall that in addition, order doesn\u2019t matter so $$a+b=b+a$$. So if $$a\\neq \u2212b$$, then $$\\frac{a+b}{b+a}=1$$.\n\nEXAMPLE $$\\PageIndex{10}$$\n\nSimplify: $$\\frac{x^2\u22124x\u221232}{64\u2212x^2}$$\n\n Factor the numerator and the denominator. Recognize the factors that are opposites. Simplify.\n\nEXAMPLE $$\\PageIndex{11}$$\n\nSimplify: $$\\frac{x^2\u22124x\u221252}{5\u2212x^2}$$\n\n$$\u2212\\frac{x+1}{x+5}$$\n\nEXAMPLE $$\\PageIndex{12}$$\n\nSimplify: $$\\frac{x^2+x\u22122}{1\u2212x^2}$$.\n\n$$\u2212\\frac{x+2}{x+1}$$\n\n# Multiply Rational Expressions\n\nTo multiply rational expressions, we do just what we did with numerical fractions. We multiply the numerators and multiply the denominators. Then, if there are any common factors, we remove them to simplify the result.\n\nMULTIPLICATION OF RATIONAL EXPRESSIONS\n\nIf p, q, r, and s are polynomials where $$q\\neq 0$$, $$s\\neq 0$$, then\n\n$\\frac{p}{q}\u00b7\\frac{r}{s}=\\frac{pr}{qs}\\nonumber$\n\nTo multiply rational expressions, multiply the numerators and multiply the denominators.\n\nRemember, throughout this chapter, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example, x\\neq 0,x\\neq 0,x\\neq 3,x\\neq 3, and x\\neq 4.x\\neq 4.\n\nEXAMPLE $$\\PageIndex{13}$$: How to Multiply Rational Expressions\n\nSimplify: $$\\frac{2x}{x^2\u22127x+12}\u00b7\\frac{x^2\u22129}{6x^2}$$.\n\nEXAMPLE $$\\PageIndex{14}$$\n\nSimplify: $$\\frac{5x}{x^2+5x+6}\u00b7\\frac{x^2\u22124}{10x}$$.\n\n$$\\frac{x\u22122}{2(x+3)}$$\n\nEXAMPLE $$\\PageIndex{15}$$\n\nSimplify: $$\\frac{9x^2}{x^2+11x+30}\u00b7\\frac{x^2\u221236}{3x^2}$$.\n\n$$\\frac{3(x\u22126)}{x+5}$$\n\nMULTIPLY RATIONAL EXPRESSIONS.\n\n1. Factor each numerator and denominator completely.\n2. Multiply the numerators and denominators.\n3. Simplify by dividing out common factors.\n\nEXAMPLE $$\\PageIndex{16}$$\n\nMultiply: $$\\frac{3a^2\u22128a\u22123}{a^2\u221225}\u00b7\\frac{a^2+10a+25}{3a^2\u221214a\u22125}$$.\n\n$$\\begin{array} {ll} &\\frac{3a^2\u22128a\u22123}{a^2\u221225}\u00b7\\frac{a^2+10a+25}{3a^2\u221214a\u22125} \\\\ & \\\\ \\begin{array} {ll} \\text{Factor the numerators and denominators} \\\\ \\text{and then multiply.} \\end{array} &\\frac{(3a+1)(a\u22123)(a+5)(a+5)}{(a\u22125)(a+5)(3a+1)(a\u22125)} \\\\ & \\\\ \\begin{array} {l} \\text{Simplify by dividing out} \\\\ \\text{common factors.} \\end{array} &\\frac{\\cancel{(3a+1)}(a\u22123)\\cancel{(a+5)}(a+5)}{(a\u22125)\\cancel{(a+5)}\\cancel{(3a+1)}(a\u22125)} \\\\ & \\\\ \\text{Simplify.} &\\frac{(a\u22123)(a+5)}{(a\u22125)(a\u22125)} \\\\ & \\\\ \\text{Rewrite }(a\u22125)(a\u22125)\\text{ using an exponent.} &\\frac{(a\u22123)(a+5)}{(a\u22125)^2} \\end{array}$$\n\nEXAMPLE $$\\PageIndex{17}$$\n\nSimplify: $$\\frac{2x^2+5x\u221212}{x^2\u221216}\u00b7\\frac{x^2\u22128x+16}{2x^2\u221213x+15}$$.\n\n$$\\frac{x\u22124}{x\u22125}$$\n\nEXAMPLE $$\\PageIndex{18}$$\n\nSimplify: $$\\frac{4b^2+7b\u22122}{1\u2212b^2}\u00b7\\frac{b^2\u22122b+1}{4b^2+15b\u22124}$$.\n\n$$\u2212\\frac{(b+2)(b\u22121)}{(1+b)(b+4)}$$\n\n# Divide Rational Expressions\n\nJust like we did for numerical fractions, to divide rational expressions, we multiply the first fraction by the reciprocal of the second.\n\nDIVISION OF RATIONAL EXPRESSIONS\n\nIf p, q, r, and s are polynomials where $$q\\neq 0$$, $$r\\neq 0$$, $$s\\neq 0$$, then\n\n$\\frac{p}{q}\u00f7\\frac{r}{s}=\\frac{p}{q}\u00b7\\frac{s}{r}\\nonumber$\n\nTo divide rational expressions, multiply the first fraction by the reciprocal of the second.\n\nOnce we rewrite the division as multiplication of the first expression by the reciprocal of the second, we then factor everything and look for common factors.\n\nEXAMPLE $$\\PageIndex{19}$$: How to Divide Rational Expressions\n\nDivide: $$\\frac{p^3+q^3}{2p^2+2pq+2q^2}\u00f7\\frac{p^2\u2212q^2}{6}$$.\n\nEXAMPLE $$\\PageIndex{20}$$\n\nSimplify: $$\\frac{x^3\u22128}{3x^2\u22126x+12}\u00f7\\frac{x^2-4}{6}$$.\n\n$$\\frac{2(x^2+2x+4)}{(x+2)(x^2\u22122x+4)}$$\n\nEXAMPLE $$\\PageIndex{21}$$\n\nSimplify: $$\\frac{2z^2}{z^2\u22121}\u00f7\\frac{z^3\u2212z^2+z}{z^3+1}$$.\n\n$$\\frac{2z}{z\u22121}$$\n\nDIVISION OF RATIONAL EXPRESSIONS\n\n1. Rewrite the division as the product of the first rational expression and the reciprocal of the second.\n2. Factor the numerators and denominators completely.\n3. Multiply the numerators and denominators together.\n4. Simplify by dividing out common factors.\n\nRecall from Use the Language of Algebra that a complex fraction is a fraction that contains a fraction in the numerator, the denominator or both. Also, remember a fraction bar means division. A complex fraction is another way of writing division of two fractions.\n\nEXAMPLE $$\\PageIndex{22}$$\n\nDivide: $$\\frac{\\frac{6x^2\u22127x+2}{4x\u22128}}{\\frac{2x^2\u22127x+3}{x^2\u22125x+6}}$$.\n\n$$\\begin{array} {ll} &\\frac{\\frac{6x^2\u22127x+2}{4x\u22128}}{\\frac{2x^2\u22127x+3}{x^2\u22125x+6}} \\\\ & \\\\ \\text{Rewrite with a division sign.} &\\frac{6x^2\u22127x+2}{4x\u22128}\u00f7\\frac{2x^2\u22127x+3}{x^2\u22125x+6} \\\\ & \\\\ \\begin{array} {l} \\text{Rewrite as product of first times reciprocal} \\\\ \\text{of second.} \\end{array} &\\frac{6x^2\u22127x+2}{4x\u22128}\u00b7\\frac{x^2\u22125x+6}{2x^2\u22127x+3} \\\\ & \\\\ \\begin{array} {l} \\text{Factor the numerators and the} \\\\ \\text{denominators, and then multiply.} \\end{array} &\\frac{(2x\u22121)(3x\u22122)(x\u22122)(x\u22123)}{4(x\u22122)(2x\u22121)(x\u22123)} \\\\ & \\\\ \\text{Simplify by dividing out common factors.} &\\frac{\\cancel{(2x\u22121)}(3x\u22122)\\cancel{(x\u22122)}\\cancel{(x\u22123)}}{4\\cancel{(x\u22122)}\\cancel{(2x\u22121)}\\cancel{(x\u22123)}} \\\\ \\text{Simplify.} &\\frac{3x\u22122}{4} \\end{array}$$\n\nEXAMPLE $$\\PageIndex{23}$$\n\nSimplify: $$\\frac{\\frac{3x^2+7x+2}{4x+24}}{\\frac{3x^2\u221214x\u22125}{x^2+x\u221230}}$$.\n\n$$\\frac{x+2}{4}$$\n\nEXAMPLE $$\\PageIndex{24}$$\n\nSimplify: $$\\frac{\\frac{y^2\u221236}{2y^2+11y\u22126}}{\\frac{2y^2\u22122y\u221260}{8y\u22124}}$$.\n\n$$\\frac{2}{y+5}$$\n\nIf we have more than two rational expressions to work with, we still follow the same procedure. The first step will be to rewrite any division as multiplication by the reciprocal. Then, we factor and multiply.\n\nEXAMPLE $$\\PageIndex{25}$$\n\nPerform the indicated operations: $$\\frac{3x\u22126}{4x\u22124}\u00b7\\frac{x^2+2x\u22123}{x^2\u22123x\u221210}\u00f7\\frac{2x+12}{8x+16}$$.\n\n Rewrite the division as multiplication by the reciprocal. Factor the numerators and the denominators. Multiply the fractions. Bringing the constants to the front will help when removing common factors. Simplify by dividing out common factors. Simplify.\n\nEXAMPLE $$\\PageIndex{26}$$\n\nPerform the indicated operations: $$\\frac{4m+4}{3m\u221215}\u00b7\\frac{m^2\u22123m\u221210}{m^2\u22124m\u221232}\u00f7\\frac{12m\u221236}{6m\u221248}$$.\n\n$$\\frac{2(m+1)(m+2)}{3(m+4)(m\u22123)}$$\n\nEXAMPLE $$\\PageIndex{27}$$\n\nPerform the indicated operations: $$\\frac{2n^2+10n}{n\u22121}\u00f7\\frac{n^2+10n+24}{n^2+8n\u22129}\u00b7\\frac{n+4}{8n^2+12n}$$.\n\n$$\\frac{(n+5)(n+9)}{2(n+6)(2n+3)}$$\n\n# Multiply and Divide Rational Functions\n\nWe started this section stating that a rational expression is an expression of the form $$\\frac{p}{q}$$, where p and q are polynomials and $$q\\neq 0$$. Similarly, we define a rational function as a function of the form $$R(x)=\\frac{p(x)}{q(x)}$$ where $$p(x)$$ and $$q(x)$$ are polynomial functions and $$q(x)$$ is not zero.\n\nRATIONAL FUNCTION\n\nA rational function is a function of the form\n\n$R(x)=\\frac{p(x)}{q(x)}\\nonumber$\n\nwhere $$p(x)$$ and $$q(x)$$ are polynomial functions and $$q(x)$$ is not zero.\n\nThe domain of a rational function is all real numbers except for those values that would cause division by zero. We must eliminate any values that make q(x)=0.q(x)=0.\n\nDETERMINE THE DOMAIN OF A RATIONAL FUNCTION.\n\n1. Set the denominator equal to zero.\n2. Solve the equation.\n3. The domain is all real numbers excluding the values found in Step 2.\n\nEXAMPLE $$\\PageIndex{28}$$\n\nFind the domain of $$R(x)=\\frac{2x^2\u221214x}{4x^2\u221216x\u221248}$$.\n\nThe domain will be all real numbers except those values that make the denominator zero. We will set the denominator equal to zero , solve that equation, and then exclude those values from the domain.\n\n$$\\begin{array} {ll} \\text{Set the denominator to zero.} &4x^2\u221216x\u221248=0 \\\\ \\text{Factor, first factor out the GCF.} &4(x^2\u22124x\u221212)=0 \\\\ &4(x\u22126)(x+2)=0 \\\\ \\text{Use the Zero Product Property.} &4\\neq 0\\quad x\u22126=0\\quad x+2=0 \\\\ \\text{Solve.} &\\hspace{24mm}x=6\\qquad x=\u22122 \\\\ &\\text{The domain of }R(x)\\text{ is all real numbers} \\\\ &\\text{where }x\\neq 6\\text{ and }x\\neq \u22122 \\end{array}$$.\n\nEXAMPLE $$\\PageIndex{29}$$\n\nFind the domain of $$R(x)=\\frac{2x^2\u221210x}{4x^2\u221216x\u221220}$$.\n\nThe domain of $$R(x)$$ is all real numbers where $$x\\neq 5$$ and $$x\\neq \u22121$$.\n\nEXAMPLE $$\\PageIndex{30}$$\n\nFind the domain of $$R(x)=\\frac{4x^2\u221216x}{8x^2\u221216x\u221264}$$.\n\nThe domain of $$R(x)$$ is all real numbers where $$x\\neq 4$$ and $$x\\neq \u22122$$.\n\nTo multiply rational functions, we multiply the resulting rational expressions on the right side of the equation using the same techniques we used to multiply rational expressions.\n\nEXAMPLE $$\\PageIndex{31}$$\n\nFind $$R(x)=f(x)\u00b7g(x)$$ where $$f(x)=\\frac{2x\u22126}{x^2\u22128x+15}$$ and $$g(x)=\\frac{x^2\u221225}{2x+10}$$.\n\n$$\\begin{array} {ll} &R(x)=f(x)\u00b7g(x) \\\\ & \\\\ &R(x)=\\frac{2x\u22126}{x^2\u22128x+15}\u00b7\\frac{x^2\u221225}{2x+10} \\\\ & \\\\ \\text{Factor each numerator and denominator.} &R(x)=\\frac{2(x\u22123)}{(x\u22123)(x\u22125)}\u00b7\\frac{(x\u22125)(x+5)}{2(x+5)} \\\\ & \\\\ \\text{Multiply the numerators and denominators.} &R(x)=\\frac{2(x\u22123)(x\u22125)(x+5)}{2(x\u22123)(x\u22125)(x+5)} \\\\ & \\\\ \\text{Remove common factors.} &R(x)=\\frac{\\cancel{2}\\cancel{(x\u22123)}\\cancel{(x\u22125)}\\cancel{(x+5)}}{\\cancel{2}\\cancel{(x\u22123)}\\cancel{(x\u22125)}\\cancel{(x+5)}} \\\\ & \\\\ \\text{Simplify.} &R(x)=1 \\end{array}$$\n\nEXAMPLE $$\\PageIndex{32}$$\n\nFind $$R(x)=f(x)\u00b7g(x)$$ where $$f(x)=\\frac{3x\u221221}{x^2\u22129x+14}$$ and $$g(x)=\\frac{2x^2\u22128}{3x+6}$$.\n\n$$R(x)=2$$\n\nEXAMPLE $$\\PageIndex{33}$$\n\nFind $$R(x)=f(x)\u00b7g(x)$$ where $$f(x)=\\frac{x^2\u2212x}{3x^2+27x\u221230}$$ and $$g(x)=\\frac{x^2\u2212100}{x^2\u221210x}$$.\n\n$$R(x)=\\frac{1}{3}$$\n\nTo divide rational functions, we divide the resulting rational expressions on the right side of the equation using the same techniques we used to divide rational expressions.\n\nEXAMPLE $$\\PageIndex{34}$$\n\nFind $$R(x)=\\frac{f(x)}{g(x)}$$ where $$f(x)=\\frac{3x^2}{x^2\u22124x}$$ and $$g(x)=\\frac{9x^2\u221245x}{x^2\u22127x+10}$$.\n\n$$\\begin{array} {ll} &R(x)=\\frac{f(x)}{g(x)} \\\\ \\text{Substitute in the functions }f(x),\\space g(x). &R(x)=\\frac{\\frac{3x^2}{x^2\u22124x}}{\\frac{9x^2\u221245x}{x^2\u22127x+10}} \\\\ & \\\\ \\begin{array} {l} \\text{Rewrite the division as the product of} \\\\ f(x)\\text{ and the reciprocal of }g(x). \\end{array} &R(x)=\\frac{3x^2}{x^2\u22124x}\u00b7\\frac{x^2\u22127x+10}{9x^2\u221245x} \\\\ & \\\\ \\begin{array} {l} \\text{Factor the numerators and denominators} \\\\ \\text{and then multiply.} \\end{array} &R(x)=\\frac{3\u00b7x\u00b7x\u00b7(x\u22125)(x\u22122)}{x(x\u22124)\u00b73\u00b73\u00b7x\u00b7(x\u22125)} \\\\ & \\\\ \\text{Simplify by dividing out common factors.} &R(x)=\\frac{\\cancel{3}\u00b7\\cancel{x}\u00b7\\cancel{x}\\cancel{(x\u22125)}(x\u22122)}{\\cancel{x}(x\u22124)\u00b7\\cancel{3}\u00b73\u00b7\\cancel{x}\\cancel{(x\u22125)}} \\\\ & \\\\ &R(x)=\\frac{x\u22122}{3(x\u22124)} \\end{array}$$\n\nEXAMPLE $$\\PageIndex{35}$$\n\nFind $$R(x)=\\frac{f(x)}{g(x)}$$ where $$f(x)=\\frac{2x^2}{x^2\u22128x}$$ and $$g(x)=\\frac{8x^2+24x}{x^2+x\u22126}$$.\n\n$$R(x)=\\frac{x\u22122}{4(x\u22128)}$$\n\nEXAMPLE $$\\PageIndex{36}$$\n\nFind $$R(x)=\\frac{f(x)}{g(x)}$$ where $$f(x)=\\frac{15x^2}{3x^2+33x}$$ and $$g(x)=\\frac{5x\u22125}{x^2+9x\u221222}$$.\n\n$$R(x)=\\frac{x(x\u22122)}{x\u22121}$$\n\n# Key Concepts\n\n\u2022 Determine the values for which a rational expression is undefined.\n1. Set the denominator equal to zero.\n2. Solve the equation.\n\u2022 Equivalent Fractions Property\nIf a, b, and c are numbers where $$b\\neq 0$$, $$c\\neq 0$$, then $$\\frac{a}{b}=\\frac{a\u00b7c}{b\u00b7c}$$ and $$\\frac{a\u00b7c}{b\u00b7c}=\\frac{a}{b}$$.\n\u2022 How to simplify a rational expression.\n1. Factor the numerator and denominator completely.\n2. Simplify by dividing out common factors.\n\u2022 Opposites in a Rational Expression\nThe opposite of $$a\u2212b$$ is $$b\u2212a$$.\n$$\\frac{a\u2212b}{b\u2212a}=\u22121 \\qquad \\qquad a\\neq b$$\nAn expression and its opposite divide to $$\u22121$$.\n\u2022 Multiplication of Rational Expressions\nIf p, q, r, and s are polynomials where $$q\\neq 0$$, $$s\\neq 0$$, then\n$$\\hspace{40mm} \\frac{p}{q}\u00b7\\frac{r}{s}=\\frac{pr}{qs}$$\n\u2022 How to multiply rational expressions.\n1. Factor each numerator and denominator completely.\n2. Multiply the numerators and denominators.\n3. Simplify by dividing out common factors.\n\u2022 Division of Rational Expressions\nIf p, q, r, and s are polynomials where $$q\\neq 0$$, $$r\\neq 0$$, $$s\\neq 0$$, then\n$$\\hspace{40mm} \\frac{p}{q}\u00f7\\frac{r}{s}=\\frac{p}{q}\u00b7\\frac{s}{r}$$\n\u2022 How to divide rational expressions.\n1. Rewrite the division as the product of the first rational expression and the reciprocal of the second.\n2. Factor the numerators and denominators completely.\n3. Multiply the numerators and denominators together.\n4. Simplify by dividing out common factors.\n\u2022 How to determine the domain of a rational function.\n1. Set the denominator equal to zero.\n2. Solve the equation.\n3. The domain is all real numbers excluding the values found in Step 2.\n\n## Practice Makes Perfect\n\nDetermine the Values for Which a Rational Expression is Undefined\n\nIn the following exercises, determine the values for which the rational expression is undefined.\n\n\u24d0 $$\\frac{2x^2}{z}$$\n\u24d1 $$\\frac{4p\u22121}{6p\u22125}$$\n\u24d2 $$\\frac{n\u22123}{n^2+2n\u22128}$$\n\n\u24d0 $$z=0$$ \u24d1 $$p=\\frac{5}{6}$$\n\u24d2 $$n=\u22124,n=2$$\n\n\u24d0 $$\\frac{10m}{11n}$$\n\u24d1 $$\\frac{6y+13}{4y\u22129}$$\n\u24d2 $$\\frac{b\u22128}{b^2\u221236}$$\n\n\u24d0 $$\\frac{4x^2y}{3y}$$\n\n\u24d1 $$\\frac{3x\u22122}{2x+1}$$\n\u24d2 $$\\frac{u\u22121}{u^2\u22123u\u221228}$$\n\n\u24d0 $$y=0$$ \u24d1 $$x=\u2212\\frac{1}{2}$$\n\u24d2 $$u=\u22124,\\frac u=7$$\n\n\u24d0 $$\\frac{5pq^2}{9q}$$\n\u24d1 $$\\frac{7a\u22124}{3a+5}$$\n\u24d2 $$\\frac{1}{x^2\u22124}$$\n\nSimplify Rational Expressions\n\nIn the following exercises, simplify each rational expression.\n\n$$\u2212\\frac{44}{55}$$\n\n$$\u2212\\frac{4}{5}$$\n\n$$\\frac{56}{63}$$\n\n$$\\frac{8m^3n}{12mn^2}$$\n\n$$\\frac{2m^2}{3n}$$\n\n$$\\frac{36v^3w^2}{27vw^3}$$\n\n$$\\frac{8n\u221296}{3n\u221236}$$\n\n$$\\frac{8}{3}$$\n\n$$\\frac{12p\u2212240}{5p\u2212100}$$\n\n$$\\frac{x^2+4x\u22125}{x^2\u22122x+1}$$\n\n$$\\frac{x+5}{x\u22121}$$\n\n$$\\frac{y^2+3y\u22124}{y^2\u22126y+5}$$\n\n$$\\frac{a^2\u22124}{a^2+6a\u221216}$$\n\n$$\\frac{a+2}{a+8}$$\n\n$$\\frac{y^2\u22122y\u22123}{y^2\u22129}$$\n\n$$\\frac{p^3+3p^2+4p+12}{p^2+p\u22126}$$\n\n$$\\frac{p^2+4}{p\u22122}$$\n\n$$\\frac{x^3\u22122x^2\u221225x+50}{x^2\u221225}$$\n\n$$\\frac{8b^2\u221232b}{2b^2\u22126b\u221280}$$\n\n$$\\frac{4b(b\u22124)}{(b+5)(b\u22128)}$$\n\n$$\\frac{\u22125c^2\u221210c}{\u221210c^2+30c+100}$$\n\n$$\\frac{3m^2+30mn+75n^2}{4m^2\u2212100n^2}$$\n\n$$\\frac{3(m+5n)}{4(m\u22125n)}$$\n\n$$\\frac{5r^2+30rs\u221235s^2}{r^2\u221249s^2}$$\n\n$$\\frac{a\u22125}{5\u2212a}$$\n\n$$\u22121$$\n\n$$\\frac{5\u2212d}{d\u22125}$$\n\n$$\\frac{20\u22125y}{y^2\u221216}$$\n\n$$\\frac{\u22125}{y+4}$$\n\n$$\\frac{4v\u221232}{64\u2212v^2}$$\n\n$$\\frac{w^3+21}{6w^2\u221236}$$\n\n$$\\frac{w^2\u22126w+3}{6w\u22126}$$\n\n$$\\frac{v^3+125}{v^2\u221225}$$\n\n$$\\frac{z^2\u22129z+20}{16\u2212z^2}$$\n\n$$\\frac{\u2212z\u22125}{4+z}$$\n\n$$\\frac{a^2\u22125z\u221236}{81\u2212a^2}$$\n\nMultiply Rational Expressions\n\nIn the following exercises, multiply the rational expressions.\n\n$$\\frac{12}{16}\u00b7\\frac{4}{10}$$\n\n$$\\frac{3}{10}$$\n\n$$\\frac{3}{25}\u00b7\\frac{16}{24}$$\n\n$$\\frac{5x^2y^4}{12xy^3}\u00b7\\frac{6x^2}{20y^2}$$\n\n$$\\frac{x^3}{8y]$$\n\n$$\\frac{12a^3b}{b^2}\u00b7\\frac{2ab^2}{9b^3}$$\n\n$$\\frac{5p^2}{p^2\u22125p\u221236}\u00b7\\frac{p^2\u221216}{10p}$$\n\n$$\\frac{p(p\u22124)}{2(p\u22129)}$$\n\n$$\\frac{3q^2}{q^2+q\u22126}\u00b7\\frac{q^2\u22129}{9q}$$\n\n$$\\frac{2y^2\u221210y}{y^2}+\\frac{10y+2}{5}\u00b7\\frac{y+5}{6y}$$\n\n$$\\frac{y\u22125}{3(y+5)}$$\n\n$$\\frac{z^2+3z}{z^2\u22123z\u22124}\u00b7\\frac{z\u22124}{z^2}$$\n\n$$\\frac{28\u22124b}{3b\u22123}\u00b7\\frac{b^2+8b\u22129}{b^2\u221249}$$\n\n$$\\frac{\u22124(b+9)}{3(b+7)}$$\n\n$$\\frac{72m\u221212m^2}{8m+32}\u00b7\\frac{m^2+10m+2}{4m^2\u221236}$$\n\n$$\\frac{5c^2+9c+2}{c^2\u221225}\u00b7\\frac{c^2+10c+25}{3c^2\u221214c\u22125}$$\n\n$$\\frac{(c+2)(c+2)}{(c\u22122)(c\u22123)}$$\n\n$$\\frac{2d^2+d\u22123}{d^2\u221216}\u00b7\\frac[d^2\u22128d+16}{2d^2\u22129d\u221218}$$\n\n$$\\frac{2m^2\u22123m\u22122}{2m2+7m+3}\u00b7\\frac{3m^2\u221214m+15}{3m^2+17m\u221220}$$\n\n$$\\frac{(m\u22123)(m\u22122)}{(m+4)(m+3)}$$\n\n$$\\frac{2n^2\u22123n\u221214}{25\u2212n^2}\u00b7\\frac{n^2\u221210n+25}{2n^2\u221213n+21}$$\n\nDivide Rational Expressions\n\nIn the following exercises, divide the rational expressions.\n\n$$\\frac{v\u22125}{11\u2212v}\u00f7\\frac{v^2\u221225}{v\u221211}$$\n\n$$\u2212\\frac{1}{v+5}$$\n\n$$\\frac{10+w}{w\u22128}\u00f7\\frac{100\u2212w^2}{8\u2212w}$$\n\n$$\\frac{3s^2}{s^2\u221216}\u00f7\\frac{s^3\u22124s^2+16s}{s^3\u221264}$$\n\n$$\\frac{3s}{s+4}$$\n\n$$\\frac{r^2\u22129}{15}\u00f7\\frac{r^3\u221227}{5r^2+15r+45}$$\n\n$$\\frac{p^3+q^3}{3p^2+3pq+3q^2}\u00f7\\frac{p^2\u2212q^2}{12}$$\n\n$$\\frac{4(p^2\u2212pq+q^2)}{(p\u2212q)(p^2+pq+q^2)}$$\n\n$$\\frac{v^3\u22128w^3}{2v^2+4vw+8w^2}\u00f7\\frac{v^2\u22124w^2}{4}$$\n\n$$\\frac{x^2+3x\u221210}{4x}\u00f7(2x^2+20x+50)}$$\n\n$$\\frac{x\u22122}{8x}$$\n\n$$\\frac{2y^2\u221210yz\u221248z^2}{2y\u22121}\u00f7(4y^2\u221232yz)$$\n\n$$\\frac{\\frac{2a^2\u2212a\u221221}{5a+20}}{\\frac{a^2+7a+12}{a^2+8a+16}}$$\n\n$$\\frac{2a\u22127}{5}$$\n\n$$\\frac{\\frac{3b^2+2b\u22128}{12b+18}}{\\frac{3b^2+2b\u22128}{2b^2\u22127b\u221215}}$$\n\n$$\\frac{\\frac{12c^2\u221212}{2c^2\u22123c+1}}{\\frac{4c+4}{6c^2\u221213c+5}}$$\n\n$$3(3c\u22125)$$\n\n$$\\frac{\\frac{4d^2+7d\u22122}{35d+10}}{\\frac{d^2\u22124}{7d^2\u221212d\u22124}}$$\n\nFor the following exercises, perform the indicated operations.\n\n$$\\frac{10m^2+80m}{3m\u22129}\u00b7\\frac{m^2+4m\u221221}{m^2\u22129m+20}\u00f7\\frac{5m^2+10m}{2m\u221210}$$\n\n$$\\frac{4(m+8)(m+7)}{3(m\u22124)(m+2)}$$\n\n$$\\frac{4n^2+32n}{3n+2}\u00b7\\frac{3n^2\u2212n\u22122}{n^2+n\u221230}\u00f7\\frac{108n^2\u221224n}{n+6}$$\n\n$$\\frac{12p^2+3p}{p+3}\u00f7\\frac{p^2+2p\u221263}{p^2\u2212p\u221212}\u00b7\\frac{p\u22127}{9p^3\u22129p^2}$$\n\n$$\\frac{(4p+1)(p\u22124)}{3p(p+9)(p\u22121)}$$\n\n$$\\frac{6q+3}{9q^2\u22129q}\u00f7\\frac{q^2+14q+33}{q^2+4q\u22125}\u00b7\\frac{4q^2+12q}{12q+6}$$\n\nMultiply and Divide Rational Functions\n\nIn the following exercises, find the domain of each function.\n\n$$R(x)=\\frac{x^3\u22122x^2\u221225x+50}{x^2\u221225$$\n\n$$x\\neq 5$$ and $$x\\neq \u22125$$\n\n$$R(x)=\\frac{x^3+3x^2\u22124x\u221212}{x^2\u22124}$$\n\n$$R(x)=\\frac{3x^2+15x}{6x^2+6x\u221236}$$\n\n$$x\\neq 2$$ and $$x\\neq \u22123$$\n\n$$R(x)=\\frac{8x^2\u221232x}{2x^2\u22126x\u221280}$$\n\nFor the following exercises, find $$R(x)=f(x)\u00b7g(x)$$ where $$f(x)$$ and $$g(x)$$ are given.\n\n$$f(x)=\\frac{6x^2\u221212x}{x^2+7x\u221218}$$\n$$\\quad g(x)=\\frac{x^2\u221281}{3x^2\u221227x}$$\n\n$$R(x)=2$$\n\n$$f(x)=\\frac{x^2\u22122x}{x^2+6x\u221216}$$\n$$\\quad g(x)=\\frac{x^2\u221264}{x^2\u22128x}$$\n\n$$f(x)=\\frac{4x}{x^2\u22123x\u221210}$$\n$$\\quad g(x)=\\frac{x^2\u221225}{8x^2}$$\n\n$$R(x)=\\frac{x+5}{2x(x+2)}$$\n\n$$f(x)=\\frac{2x^2+8x}{x^2\u22129x+20}$$\n$$\\quad g(x)=\\frac{x\u22125}{x^2}$$\n\nFor the following exercises, find $$R(x)=f(x)g(x)$$ where $$f(x)$$ and $$g(x)$$ are given.\n\n$$f(x)=\\frac{27x^2}{3x\u221221}$$\n$$\\quad g(x)=\\frac{3x^2+18x}{x^2+13x+42$$\n\n$$R(x)=\\frac{3x(x+7)}{x\u22127}$$\n\n$$f(x)=\\frac{24x^2}{2x\u22128}$$\n$$\\quad g(x)=\\frac{4x^3+28x^2}{x^2+11x+28}$$\n\n$$f(x)=\\frac{16x^2}{4x+36}$$\n$$\\quad g(x)=\\frac{4x^2\u221224x}{x^2+4x\u221245}$$\n\n$$R(x)=\\frac{x(x\u22125)}{x\u22126}$$\n\n$$f(x)=\\frac{24x^2}{2x\u22124}$$\n$$\\quad g(x)=\\frac{12x^2+36x}{x^2\u221211x+18}$$\n\n## Writing Exercises\n\nExplain how you find the values of x for which the rational expression $$\\frac{x^2\u2212x\u221220}{x^2\u22124}$$ is undefined.\n\nExplain all the steps you take to simplify the rational expression $$\\frac{p^2+4p\u221221}{9\u2212p^2}$$.\n\n\u24d0 Multiply $$\\frac{7}{4}\u00b7\\frac{9}{10}$$ and explain all your steps.\n\u24d1 Multiply $$\\frac{n}{n\u22123}\u00b7\\frac{9}{n+3}$$ and explain all your steps.\n\u24d2 Evaluate your answer to part \u24d1 when $$n=7$$. Did you get the same answer you got in part \u24d0? Why or why not?\n\n\u24d0 Divide $$\\frac{24}{5}\u00f76$$ and explain all your steps.\n\u24d1 Divide $$\\frac{x^2\u22121}{x}\u00f7(x+1)$$ and explain all your steps.\n\u24d2 Evaluate your answer to part \u24d1 when $$x=5$$. Did you get the same answer you got in part \u24d0? Why or why not?\n\n## Self Check\n\n\u24d0 After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.\n\n\u24d1 If most of your checks were:\n\n\u2026confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!\n\n\u2026with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential - every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?\n\n\u2026no - I don\u2019t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.\n\n# Glossary\n\nrational expression\nA rational expression is an expression of the form $$\\frac{p}{q}$$, where p and q are polynomials and $$q\\neq 0$$.\nsimplified rational expression\nA simplified rational expression has no common factors, other than 1, in its numerator and denominator.\nrational function\nA rational function is a function of the form $$R(x)=\\frac{p(x)}{q(x)}$$ where $$p(x)$$ and $$q(x)$$ are polynomial functions and $$q(x)$$ is not zero.","date":"2021-10-27 18:36:42","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9363029599189758, \"perplexity\": 619.161424019289}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323588242.22\/warc\/CC-MAIN-20211027181907-20211027211907-00278.warc.gz\"}"}
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package org.springframework.boot.autoconfigure.web.servlet;
import java.time.Duration;
import java.util.LinkedHashMap;
import java.util.Locale;
import java.util.Map;
import org.springframework.boot.context.properties.ConfigurationProperties;
import org.springframework.boot.context.properties.DeprecatedConfigurationProperty;
import org.springframework.boot.context.properties.IncompatibleConfigurationException;
import org.springframework.http.MediaType;
import org.springframework.util.Assert;
import org.springframework.validation.DefaultMessageCodesResolver;
/**
* {@link ConfigurationProperties properties} for Spring MVC.
*
* @author Phillip Webb
* @author Sébastien Deleuze
* @author Stephane Nicoll
* @author Eddú Meléndez
* @author Brian Clozel
* @since 2.0.0
*/
@ConfigurationProperties(prefix = "spring.mvc")
public class WebMvcProperties {
/**
* Formatting strategy for message codes. For instance, `PREFIX_ERROR_CODE`.
*/
private DefaultMessageCodesResolver.Format messageCodesResolverFormat;
/**
* Locale to use. By default, this locale is overridden by the "Accept-Language"
* header.
*/
private Locale locale;
/**
* Define how the locale should be resolved.
*/
private LocaleResolver localeResolver = LocaleResolver.ACCEPT_HEADER;
private final Format format = new Format();
/**
* Whether to dispatch TRACE requests to the FrameworkServlet doService method.
*/
private boolean dispatchTraceRequest = false;
/**
* Whether to dispatch OPTIONS requests to the FrameworkServlet doService method.
*/
private boolean dispatchOptionsRequest = true;
/**
* Whether the content of the "default" model should be ignored during redirect
* scenarios.
*/
private boolean ignoreDefaultModelOnRedirect = true;
/**
* Whether to publish a ServletRequestHandledEvent at the end of each request.
*/
private boolean publishRequestHandledEvents = true;
/**
* Whether a "NoHandlerFoundException" should be thrown if no Handler was found to
* process a request.
*/
private boolean throwExceptionIfNoHandlerFound = false;
/**
* Whether logging of (potentially sensitive) request details at DEBUG and TRACE level
* is allowed.
*/
private boolean logRequestDetails;
/**
* Whether to enable warn logging of exceptions resolved by a
* "HandlerExceptionResolver", except for "DefaultHandlerExceptionResolver".
*/
private boolean logResolvedException = false;
/**
* Path pattern used for static resources.
*/
private String staticPathPattern = "/**";
private final Async async = new Async();
private final Servlet servlet = new Servlet();
private final View view = new View();
private final Contentnegotiation contentnegotiation = new Contentnegotiation();
private final Pathmatch pathmatch = new Pathmatch();
public DefaultMessageCodesResolver.Format getMessageCodesResolverFormat() {
return this.messageCodesResolverFormat;
}
public void setMessageCodesResolverFormat(DefaultMessageCodesResolver.Format messageCodesResolverFormat) {
this.messageCodesResolverFormat = messageCodesResolverFormat;
}
@Deprecated
@DeprecatedConfigurationProperty(replacement = "spring.web.locale")
public Locale getLocale() {
return this.locale;
}
public void setLocale(Locale locale) {
this.locale = locale;
}
@Deprecated
@DeprecatedConfigurationProperty(replacement = "spring.web.locale-resolver")
public LocaleResolver getLocaleResolver() {
return this.localeResolver;
}
public void setLocaleResolver(LocaleResolver localeResolver) {
this.localeResolver = localeResolver;
}
@Deprecated
@DeprecatedConfigurationProperty(replacement = "spring.mvc.format.date")
public String getDateFormat() {
return this.format.getDate();
}
@Deprecated
public void setDateFormat(String dateFormat) {
this.format.setDate(dateFormat);
}
public Format getFormat() {
return this.format;
}
public boolean isIgnoreDefaultModelOnRedirect() {
return this.ignoreDefaultModelOnRedirect;
}
public void setIgnoreDefaultModelOnRedirect(boolean ignoreDefaultModelOnRedirect) {
this.ignoreDefaultModelOnRedirect = ignoreDefaultModelOnRedirect;
}
public boolean isPublishRequestHandledEvents() {
return this.publishRequestHandledEvents;
}
public void setPublishRequestHandledEvents(boolean publishRequestHandledEvents) {
this.publishRequestHandledEvents = publishRequestHandledEvents;
}
public boolean isThrowExceptionIfNoHandlerFound() {
return this.throwExceptionIfNoHandlerFound;
}
public void setThrowExceptionIfNoHandlerFound(boolean throwExceptionIfNoHandlerFound) {
this.throwExceptionIfNoHandlerFound = throwExceptionIfNoHandlerFound;
}
public boolean isLogRequestDetails() {
return this.logRequestDetails;
}
public void setLogRequestDetails(boolean logRequestDetails) {
this.logRequestDetails = logRequestDetails;
}
public boolean isLogResolvedException() {
return this.logResolvedException;
}
public void setLogResolvedException(boolean logResolvedException) {
this.logResolvedException = logResolvedException;
}
public boolean isDispatchOptionsRequest() {
return this.dispatchOptionsRequest;
}
public void setDispatchOptionsRequest(boolean dispatchOptionsRequest) {
this.dispatchOptionsRequest = dispatchOptionsRequest;
}
public boolean isDispatchTraceRequest() {
return this.dispatchTraceRequest;
}
public void setDispatchTraceRequest(boolean dispatchTraceRequest) {
this.dispatchTraceRequest = dispatchTraceRequest;
}
public String getStaticPathPattern() {
return this.staticPathPattern;
}
public void setStaticPathPattern(String staticPathPattern) {
this.staticPathPattern = staticPathPattern;
}
public Async getAsync() {
return this.async;
}
public Servlet getServlet() {
return this.servlet;
}
public View getView() {
return this.view;
}
public Contentnegotiation getContentnegotiation() {
return this.contentnegotiation;
}
public Pathmatch getPathmatch() {
return this.pathmatch;
}
@SuppressWarnings("deprecation")
public void checkConfiguration() {
if (this.getPathmatch().getMatchingStrategy() == MatchingStrategy.PATH_PATTERN_PARSER) {
if (this.getPathmatch().isUseSuffixPattern()) {
throw new IncompatibleConfigurationException("spring.mvc.pathmatch.matching-strategy",
"spring.mvc.pathmatch.use-suffix-pattern");
}
if (this.getPathmatch().isUseRegisteredSuffixPattern()) {
throw new IncompatibleConfigurationException("spring.mvc.pathmatch.matching-strategy",
"spring.mvc.pathmatch.use-registered-suffix-pattern");
}
}
}
public static class Async {
/**
* Amount of time before asynchronous request handling times out. If this value is
* not set, the default timeout of the underlying implementation is used.
*/
private Duration requestTimeout;
public Duration getRequestTimeout() {
return this.requestTimeout;
}
public void setRequestTimeout(Duration requestTimeout) {
this.requestTimeout = requestTimeout;
}
}
public static class Servlet {
/**
* Path of the dispatcher servlet. Setting a custom value for this property is not
* compatible with the PathPatternParser matching strategy.
*/
private String path = "/";
/**
* Load on startup priority of the dispatcher servlet.
*/
private int loadOnStartup = -1;
public String getPath() {
return this.path;
}
public void setPath(String path) {
Assert.notNull(path, "Path must not be null");
Assert.isTrue(!path.contains("*"), "Path must not contain wildcards");
this.path = path;
}
public int getLoadOnStartup() {
return this.loadOnStartup;
}
public void setLoadOnStartup(int loadOnStartup) {
this.loadOnStartup = loadOnStartup;
}
public String getServletMapping() {
if (this.path.equals("") || this.path.equals("/")) {
return "/";
}
if (this.path.endsWith("/")) {
return this.path + "*";
}
return this.path + "/*";
}
public String getPath(String path) {
String prefix = getServletPrefix();
if (!path.startsWith("/")) {
path = "/" + path;
}
return prefix + path;
}
public String getServletPrefix() {
String result = this.path;
int index = result.indexOf('*');
if (index != -1) {
result = result.substring(0, index);
}
if (result.endsWith("/")) {
result = result.substring(0, result.length() - 1);
}
return result;
}
}
public static class View {
/**
* Spring MVC view prefix.
*/
private String prefix;
/**
* Spring MVC view suffix.
*/
private String suffix;
public String getPrefix() {
return this.prefix;
}
public void setPrefix(String prefix) {
this.prefix = prefix;
}
public String getSuffix() {
return this.suffix;
}
public void setSuffix(String suffix) {
this.suffix = suffix;
}
}
public static class Contentnegotiation {
/**
* Whether the path extension in the URL path should be used to determine the
* requested media type. If enabled a request "/users.pdf" will be interpreted as
* a request for "application/pdf" regardless of the 'Accept' header.
*/
private boolean favorPathExtension = false;
/**
* Whether a request parameter ("format" by default) should be used to determine
* the requested media type.
*/
private boolean favorParameter = false;
/**
* Map file extensions to media types for content negotiation. For instance, yml
* to text/yaml.
*/
private Map<String, MediaType> mediaTypes = new LinkedHashMap<>();
/**
* Query parameter name to use when "favor-parameter" is enabled.
*/
private String parameterName;
@DeprecatedConfigurationProperty(
reason = "Use of path extensions for request mapping and for content negotiation is discouraged.")
@Deprecated
public boolean isFavorPathExtension() {
return this.favorPathExtension;
}
@Deprecated
public void setFavorPathExtension(boolean favorPathExtension) {
this.favorPathExtension = favorPathExtension;
}
public boolean isFavorParameter() {
return this.favorParameter;
}
public void setFavorParameter(boolean favorParameter) {
this.favorParameter = favorParameter;
}
public Map<String, MediaType> getMediaTypes() {
return this.mediaTypes;
}
public void setMediaTypes(Map<String, MediaType> mediaTypes) {
this.mediaTypes = mediaTypes;
}
public String getParameterName() {
return this.parameterName;
}
public void setParameterName(String parameterName) {
this.parameterName = parameterName;
}
}
public static class Pathmatch {
/**
* Choice of strategy for matching request paths against registered mappings.
*/
private MatchingStrategy matchingStrategy = MatchingStrategy.ANT_PATH_MATCHER;
/**
* Whether to use suffix pattern match (".*") when matching patterns to requests.
* If enabled a method mapped to "/users" also matches to "/users.*". Enabling
* this option is not compatible with the PathPatternParser matching strategy.
*/
private boolean useSuffixPattern = false;
/**
* Whether suffix pattern matching should work only against extensions registered
* with "spring.mvc.contentnegotiation.media-types.*". This is generally
* recommended to reduce ambiguity and to avoid issues such as when a "." appears
* in the path for other reasons. Enabling this option is not compatible with the
* PathPatternParser matching strategy.
*/
private boolean useRegisteredSuffixPattern = false;
public MatchingStrategy getMatchingStrategy() {
return this.matchingStrategy;
}
public void setMatchingStrategy(MatchingStrategy matchingStrategy) {
this.matchingStrategy = matchingStrategy;
}
@DeprecatedConfigurationProperty(
reason = "Use of path extensions for request mapping and for content negotiation is discouraged.")
@Deprecated
public boolean isUseSuffixPattern() {
return this.useSuffixPattern;
}
@Deprecated
public void setUseSuffixPattern(boolean useSuffixPattern) {
this.useSuffixPattern = useSuffixPattern;
}
@DeprecatedConfigurationProperty(
reason = "Use of path extensions for request mapping and for content negotiation is discouraged.")
@Deprecated
public boolean isUseRegisteredSuffixPattern() {
return this.useRegisteredSuffixPattern;
}
@Deprecated
public void setUseRegisteredSuffixPattern(boolean useRegisteredSuffixPattern) {
this.useRegisteredSuffixPattern = useRegisteredSuffixPattern;
}
}
public static class Format {
/**
* Date format to use, for example `dd/MM/yyyy`.
*/
private String date;
/**
* Time format to use, for example `HH:mm:ss`.
*/
private String time;
/**
* Date-time format to use, for example `yyyy-MM-dd HH:mm:ss`.
*/
private String dateTime;
public String getDate() {
return this.date;
}
public void setDate(String date) {
this.date = date;
}
public String getTime() {
return this.time;
}
public void setTime(String time) {
this.time = time;
}
public String getDateTime() {
return this.dateTime;
}
public void setDateTime(String dateTime) {
this.dateTime = dateTime;
}
}
/**
* Matching strategy options.
* @since 2.4.0
*/
public enum MatchingStrategy {
/**
* Use the {@code AntPathMatcher} implementation.
*/
ANT_PATH_MATCHER,
/**
* Use the {@code PathPatternParser} implementation.
*/
PATH_PATTERN_PARSER
}
/**
* Locale resolution options.
* @deprecated since 2.4.0 for removal in 2.6.0 in favor of
* {@link org.springframework.boot.autoconfigure.web.WebProperties.LocaleResolver}
*/
@Deprecated
public enum LocaleResolver {
/**
* Always use the configured locale.
*/
FIXED,
/**
* Use the "Accept-Language" header or the configured locale if the header is not
* set.
*/
ACCEPT_HEADER
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 577
|
<?php
require_once 'config.php';
require_once 'defaults.php';
$token = $_GET['id'] ? $_GET['id'] : $_POST['id'];
$renderer = $_GET['renderer'] ? $_GET['renderer'] : $_POST['renderer'];
$query = sprintf("SELECT value
FROM config
WHERE token='%s'
ORDER BY id DESC
LIMIT 1",
$token);
$retval = $dbh->query($query)->fetchColumn(0);
$retval = trim($retval) ? $retval : $default[$token];
$retval = trim($retval) ? $retval : 'Edit me!';
if ('textile' == $renderer) {
require_once './Textile.php';
$t = new Textile();
$retval = $t->TextileThis($retval);
}
print $retval;
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,500
|
announcements artists
The Booth Brothers Once Again Team With The Harper Agency
Staff April 28, 2020
The Booth Brothers Once Again Team With The Harper Agency: pictured (l-r) Michael Booth, Ed Harper and Ronnie Booth
Goodlettesville, Tennessee – One of Christian music's leading booking agencies, The Harper Agency, has announced the re-signing of popular, multi-award winning vocal group, The Booth Brothers to the company's talented artist roster. The talented trio has enjoyed an exciting career in the Gospel music genre, and Ed Harper, President of The Harper Agency, says he is extremely happy to have The Booth Brothers back under the umbrella of talented artists represented by the agency.
"We first began representing The Booth Brothers back in the fall of 2003. They have always been a vital part of our agency's family of Christian artists. Our agency has been honored to assist in taking their music to all corners of the US and other countries. It's been a great journey for many years to be a part of the growth of their music ministry."
Since bursting onto the Gospel music scene nearly three decades ago, The Booth Brothers have enjoyed tremendous success. Awards and hit songs have flowed their way in bunches, including honors for Song of the Year, Album of the Year, Trio of the Year and Male Group of the Year, just to name a few. Brothers Michael and Ronnie, with talented vocalist Paul Lancaster, delight audiences night after night with their musical excellence, refreshing humor and inspiring message.
"The Booth Brothers are an inspiration to so many," Harper shares. "Michael and Ronnie have always been encouraging to young artists, and they are willing to share their thoughts and convey wisdom in how the developing artists can grow in the marketplace in a professional manner. They have always been great cheerleaders for their peers in the industry. It's always a joy to be around those types of artists who are willing to support others. The Harper Agency is very blessed to be representing them again and play a role in the future of their touring."
Likewise, The Booth Brothers are excited to once again be represented by Ed Harper and the team at the Harper Agency.
"Over the years," Michael Booth states, "we have enjoyed a wonderful relationship with Ed, Jeff and the entire team at the Harper Agency. They have supported The Booth Brothers for a very long time, and we are thrilled to once again be represented by them."
The Harper Agency may be reached by calling 615-851-4500, via email at info@harperagency.com or visiting www.harperagency.com. Additional information regarding The Booth Brothers is available online HERE.
Did you know that you can receive the printed, full-color version of SGNScoops Magazine by subscription? Find out more HERE.
Find SGNScoops Magazine On Facebook HERE.
You can download the latest edition of SGNScoops Magazine HERE.
Read the latest edition of SGNScoops Magazine online HERE.
Listen To Today's Gospel Music HERE
Christian Musicgospel musicSouthern Gospel Musicthe booth brothersThe Harper Agency
Donnie Rabon Encourages Listeners with New Radio Single Release "God's Up To Something"
Rob Patz: God is in Control
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,326
|
var registerSystem = require('../core/system').registerSystem;
var utils = require('../utils');
/**
* Tracked controls system.
* Maintain list with available tracked controllers.
*/
module.exports.System = registerSystem('tracked-controls-webvr', {
init: function () {
var self = this;
this.controllers = [];
this.updateControllerList();
this.throttledUpdateControllerList = utils.throttle(this.updateControllerList, 500, this);
if (!navigator.getVRDisplays) { return; }
this.sceneEl.addEventListener('enter-vr', function () {
navigator.getVRDisplays().then(function (displays) {
if (displays.length) { self.vrDisplay = displays[0]; }
});
});
},
tick: function () {
if (navigator.userAgent.indexOf('Chrome') !== -1) {
// Call getGamepads for Chrome for it to update. Not sure if needed in future.
navigator.getGamepads && navigator.getGamepads();
}
this.throttledUpdateControllerList();
},
/**
* Update controller list.
*/
updateControllerList: function () {
var controllers = this.controllers;
var gamepad;
var gamepads;
var i;
var prevCount;
gamepads = navigator.getGamepads && navigator.getGamepads();
if (!gamepads) { return; }
prevCount = controllers.length;
controllers.length = 0;
for (i = 0; i < gamepads.length; ++i) {
gamepad = gamepads[i];
if (gamepad && gamepad.pose) {
controllers.push(gamepad);
}
}
if (controllers.length !== prevCount) {
this.el.emit('controllersupdated', undefined, false);
}
}
});
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,371
|
DeRozan, Spurs escape Warriors in overtime, 117-113
AP Dec 31, 2019 at 11:16p ET
SAN ANTONIO (AP) — The San Antonio Spurs and Golden State Warriors are experiencing a level of mediocrity unseen in the tenures of their veteran coaches. Gregg Popovich and Steve Kerr aren't letting their teams throw in the towel.
That was abundantly clear as they battled into overtime.
DeMar DeRozan had 24 points and the San Antonio Spurs escaped with a 117-113 overtime victory over the short-handed Golden State Warriors on Tuesday night.
San Antonio improved to 4-1 in overtime this season. Four of those games have come at home.
"It's more so familiar territory now," DeRozan said. "So, we've got out there and grind it out."
Dejounte Murray scored seven of his 15 points in overtime, as San Antonio (14-18) won for the eighth time in 13 games to finish the year below .500 for the first time since 1996.
"We're just trying to stay in the moment," Spurs forward LaMarcus Aldridge said. "Watch film and see the things that we need to keep doing better and focus on them and just build from there."
Golden State lost its second straight after a season-high four-game winning streak.
The Warriors — already without stars Stephen Curry and Klay Thompson long-term — were without starters D'Angelo Russell and Willie Cauley-Stein.
At 9-26, Golden State has already lost more games than in any of its previous five seasons under Kerr. The Warriors won 436 games and three championships in Kerr's first five seasons.
"The first five years were exceptional, they weren't normal," Kerr said. "So, I try to look at things from a realistic perspective. What's happening this year is more in line with what generally happens to NBA teams."
Still, Kerr has found a bright spot in helping resurrect the careers of Glenn Robinson III, Alec Burks and Marquis Criss.
Burks had 28 points and Robinson added 25, including 15 in the first quarter.
Draymond Green had 10 points, 10 rebounds and nine assists in 36 minutes.
"We're having a productive season; we're just not having a winning season," Kerr said. "I've enjoyed coaching this team, I just told them that. I enjoy coming in every day and working with these guys because they play hard, they play for each other and they care about that game. So, I'm not too concerned about the record."
The teams traded the lead four times and were tied twice in the final 3 minutes of regulation.
DeRozan's 17-footer with 17.1 seconds left gave San Antonio a 100-98 lead, but Robinson tied it eight seconds later with a 21-footer set up by Green's shovel pass. Aldridge missed a 15-footer at the close of regulation.
Aldridge added 17 points and 12 rebounds, and Patty Mills had 18 points.
Murray was 3 for 3 from the field in overtime, including a 3-pointer, and had a steal and an assist.
"He came through big, but I always expect Dejounte to come up big for us," Spurs guard Bryn Forbes said.
Golden State grabbed a 51-42 lead in the second quarter as San Antonio went scoreless for three minutes. Aldridge ended the drought with an off-balance, fadeaway 3-pointer as the shot clock expired.
The Warriors were 12 for 27 on 3-pointers, including consecutive 3s to open the second half and stretch their advantage to 61-53 in the third quarter.
Warriors: Kerr said Russell, who injured his right shoulder in a collision with Dallas' Luca Doncic on Saturday, will be evaluated Wednesday. … Golden State started its 16th different lineup in 35 games this season. … The Warriors have had 99 players appear in at last one game since Jan. 1, 2010. … Kerr clarified the status of Cauley-Stein, who was listed as "doubtful" due to an undisclosed illness. "Willie is not in San Antonio, so he's beyond doubtful," Kerr joked.
Spurs: Popovich and the Spurs closed the decade with 554 victories, the most by any coach and franchise in the league. Oklahoma City was second with 514 wins and Golden State third with 505. Clippers coach Doc Rivers, who previously coached in Boston, is second to Popovich with 493 victories.
Warriors: At Minnesota on Thursday night.
Spurs: Host Oklahoma City on Thursday night.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,617
|
Toddler falls out of moving car on DVP
By Sahar FatimaToronto Star
Tues., July 16, 2013
A toddler miraculously suffered only minor injuries after falling out of a moving vehicle on the Don Valley Parkway Tuesday evening.
The car was on the ramp to the southbound DVP from Lawrence Ave. E. at about 6:30 p.m. when the rear door opened and the 2-year-old girl fell out.
She was still in her car seat, leading police to believe the car seat was not attached to the car properly.
The child was taken to hospital and has since been released.
Police said it wasn't clear whether a passenger had opened the car door or if it was a mechanical issue.
The driver of the vehicle was charged.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 510
|
layout: post
title: "Bug Reopening on Firefox for Android (Before and After the New Process) [DRAFT]"
date: 2013-09-12 16:16
comments: true
categories: phd
---
In early 2011, the Mozilla Foundation [adopted a faster release process](http://arstechnica.com/information-technology/2011/03/mozilla-outlines-16-week-firefox-development-cycle/), dubbed *train model*, with new releases every 6 weeks. It wasn't until the following year, though, that Firefox for Android jumped on the bandwagon. Due to the need to rebuild the user interface, they decided to hold the switch to the train model until June, 2012. Before that, they only shipped a new version when it met quality and performance criteria -- even if this meant missing the next train/release date.
How the process change affected Firefox for Android's quality? I've computed some quality metrics (specifically metrics related to bug reopening) on its bug reports before and after the switch to the train model to try to answer this question.
We estimated that the process change occured near June 26th, 2012, when [Firefox for Android 14 was released](https://wiki.mozilla.org/Releases/Old/2012). We classified bug reports based on whether they were created before or after the date.
<!-- TODO: recompute metrics -->
Here's the computed metrics:
<!--
**Metric** | **Before** | **After**
| |
Number of issues | 2449 | 7192
Days | 1257 | 468
% verifications by test team | 79% | 55%
% of bugs reopened after FIXED | 11% | 5%
% of bugs reopened after VERIFIED | 4% | 1%
Days btw FIXED and REOPENED (avg) | 14 | 11
Days btw VERIFIED and REOPENED (avg) | 28 | 18
% of time fixing bugs | 92% | 94%
% of time bugs are latent | 4.5% | 2.2%
-->
**Metric** | **Before** | **After**
--------------------------------------|------------|----------
Number of issues | 5005 | 4636
% verifications by test team | 56% | 38%
% of bugs reopened after FIXED | 9% | 4%
% of bugs reopened after VERIFIED | 3.5% | 1.5%
Days btw FIXED and REOPENED (avg) | 12 | 14
Days btw VERIFIED and REOPENED (avg) | 24 | 18
% of time fixing bugs | 92% | 94%
% of time bugs are latent | 4.5% | 2.2%
To interpret the results, it is useful to know a little bit about the [Mozilla release process](http://mozilla.github.io/process-releases/) and [how it is reflected in bug reports](/blog/2013/09/08/mozilla-process/). For instance, keep in mind that when a bug is marked `FIXED`, its patch has undergone peer review and automated testing; when it is marked `VERIFIED`, it has additionally undergone manual integration testing.
From the table, we can see that the number of bugs reports that are reopened after being fixed dropped by more than half in the rapid release process (**TODO: why?**). The same can be said about the proportion of bugs that are reopened after being manually tested. <!-- A more dramatic change can be observed within bugs that were manually tested: now they are reopened 4x less often. -->
After a bug is considered fixed, the time it takes for people to notice that it should be reopened is called latent time. With the new process, the average latent time has increased a little, from 12 to 14 days. If we consider the lifetime of all bug reports over the two periods analyzed, 3.5% of the time used to be spent thinking that a bug was fixed when it was actually latent; now, this time was reduced to 2.5%.
A curious fact from the table regards the test team, automatically identified as the group of people whose work is specialized in manual testing (which appears in bug reports as the `VERIFIED` status). The test team used to be responsible for about 56% of the verifications; now the verification effort seems more evenly distributed, as the test team only performs 38% of the verifications. (**TODO: why?**)
## See also
* [Bringing Android Native Firefox to Beta](https://blog.mozilla.org/futurereleases/2012/01/25/bringing-android-native-firefox-to-beta/)
* [Firefox for mobile – tracking progress](http://irinasandu.com/2012/03/12/firefox-for-mobile-tracking-progress/)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,713
|
package org.apache.jackrabbit.oak.plugins.segment;
import static com.google.common.collect.Maps.newHashMap;
import static junit.framework.Assert.assertTrue;
import static org.apache.jackrabbit.oak.plugins.segment.Segment.RECORD_ALIGN_BITS;
import static org.apache.jackrabbit.oak.plugins.segment.Segment.MAX_SEGMENT_SIZE;
import static org.junit.Assert.assertFalse;
import java.util.Map;
import java.util.Map.Entry;
import java.util.Random;
import org.apache.jackrabbit.oak.plugins.segment.memory.MemoryStore;
import org.junit.Test;
public class CompactionMapTest {
public static void main(String[] args) {
// check the memory use of really large mappings, 1M compacted
// segments with 10 records each.
Runtime runtime = Runtime.getRuntime();
System.gc();
System.out.println((runtime.totalMemory() - runtime.freeMemory()) / (1024 * 1024));
CompactionMap map = new CompactionMap(100000);
SegmentTracker factory = new MemoryStore().getTracker();
for (int i = 0; i < 1000000; i++) {
if (i % 1000 == 0) {
System.gc();
System.out.println(i + ": " + (runtime.totalMemory() - runtime.freeMemory()) / (1024 * 1024) + "MB");
}
SegmentId sid = factory.newDataSegmentId();
for (int j = 0; j < 10; j++) {
RecordId rid = new RecordId(sid, j << RECORD_ALIGN_BITS);
map.put(rid, rid);
}
}
System.gc();
System.out.println("final: " + (runtime.totalMemory() - runtime.freeMemory()) / (1024 * 1024) + "MB");
}
@Test
public void testCompactionMap() {
int maxSegments = 1000;
int maxEntriesPerSegment = 10;
int seed = new Random().nextInt();
Random r = new Random(seed);
SegmentTracker factory = new MemoryStore().getTracker();
CompactionMap map = new CompactionMap(r.nextInt(maxSegments / 2));
Map<RecordId, RecordId> entries = newHashMap();
int segments = r.nextInt(maxSegments);
for (int i = 0; i < segments; i++) {
SegmentId id = factory.newDataSegmentId();
int n = r.nextInt(maxEntriesPerSegment);
int offset = MAX_SEGMENT_SIZE;
for (int j = 0; j < n; j++) {
offset = newValidOffset(r, (n - j) << RECORD_ALIGN_BITS, offset);
RecordId before = new RecordId(id, offset);
RecordId after = new RecordId(
factory.newDataSegmentId(),
newValidOffset(r, 0, MAX_SEGMENT_SIZE));
entries.put(before, after);
map.put(before, after);
assertTrue("Failed with seed " + seed,
map.wasCompactedTo(before, after));
assertFalse("Failed with seed " + seed,
map.wasCompactedTo(after, before));
}
}
map.compress();
for (Entry<RecordId, RecordId> entry : entries.entrySet()) {
assertTrue("Failed with seed " + seed,
map.wasCompactedTo(entry.getKey(), entry.getValue()));
assertFalse("Failed with seed " + seed,
map.wasCompactedTo(entry.getValue(), entry.getKey()));
}
}
/**
* Returns a new valid record offset, between {@code a} and {@code b},
* exclusive.
*/
private int newValidOffset(Random random, int a, int b) {
int p = (a >> RECORD_ALIGN_BITS) + 1;
int q = (b >> RECORD_ALIGN_BITS);
return (p + random.nextInt(q - p)) << RECORD_ALIGN_BITS;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,381
|
News Archive May 2013
Skype is being used to spread malware
Latest Email-Based Malware Infections
Facebook Accounts Hijacked
Zero-Day IE8 Exploit
Facebook App spoofing still unpatched
DDOS Services for hire
Newegg spam campaign
Skype…not so private anymore?
Chinese Hackers Compromise U.S. Weapons Designs
Skype is a proprietary voice-over-Internet Protocol service, it's a very popular online chat service which has nearly 700 million users all over the world. Recently Skype has been targeted by Cyber criminals who had plans to use this service as their malware distribution channel. Earlier this week (on October the 8th) Skype users have been warned about a new worm which spreads spam messages (for example: lol is this your new profile pic? etc.) and in such way tricks unsuspecting computer users into clicking links which lead to malicious websites distributing Trojans, ransomware infections and viruses. Most commonly links in such messages lead to Trojans which are used to steal computer user's passwords. The worst thing talking about this Skype malware is that Skype users are not aware that links posted by their contacts could lead to malicious websites or infected executable.
While social networks users have already seen such scam strategies being used in the past Skype users tend to trust links that they are given by their contacts and click on them. Such user behavior makes Skype a very wanted target for Cyber criminals to distribute their malware or ransomware infections.
To protect your PC from such security infections you should use cautious when you receive a link from your contact in Skype which looks unexpected or strange.
Furthermore you should always keep your Skype software up-to-date, and don't forget to use a legitimate antivirus and antispyware programs. While the distribution of such spam messages seems to be stopped, Cyber criminals recently started another malware distribution campaign related to Skype. This time Cyber criminals doesn't use Skype to post their deceptive messages which leads to Trojans, now they changed their tactics and they are sending email messages which states that your Skype password has been recently changed. Example of such deceptive email message:
Text in this message:
Password successfully changed
Your new Skype password has been set.
You can now view your attached call history and instructions how to change your account settings. If the changes described above are accurate,
no further action is needed. If anything doesn't look right, follow the link below to make changes: Restore password
The people at Skype
Another spam message that is being distributed by Cyber criminals looks like the one in the screenshot below.
Note that links in this particular message leads to Blackhole exploit kit.
If a computer users would click on any of the links in this spam email message their computers would get infected with Trojans. Be sure to delete such message if you have received one.
You have a new voicemail
Sign in to Skype to listen to the message.
If you no longer want to receive email alerts about new voicemails, unsubscribe now.
If you have received such email message in your inbox you shouldn't trust it. This is a deceptive message which actually were send by Cyber criminals who are hoping that you will fall for this trickery and you will open the attachment that comes with this email. If you would extract the attached zip file and double click the file that was in the archive you computer would be infected with a Trojan infection which enables Cyber criminals to remotely access your PC.
Always use cautious when opening attachments that comes with email messages.
If you have opened the attachment from this deceptive spam email you should scan your computer with legitimate antivirus and antispyware programs.
Email messages have always been a popular source of malware infections. Cyber criminals who are responsible for creating and distributing malware using infected email attachments or malicious links are striving to create a sense of urgency and make their fake email messages appear legitimate. Fake Email messages are used to trick unsuspecting PC users into clicking on the malicious links or to download infected Email attachments. Most commonly such rogue email messages exploits the names of various legitimate companies and organizations, moreover Cyber criminals are stealing the graphics of such well known brand names and embeds them in their malicious Email messages.
Recent research shows that most commonly Cyber criminals are exploiting the names of FedEx, Ups, DHL, Bank of America and Hewlett-Packard in their Email based malware attacks.
The main purpose of such rogue Email messages is to infiltrate user's computer with Trojans capable of stealing personal information, ransomware infections or fake antivirus programs.
In most cases infected Email attachments comes as .zip files, however .exe and .pdf infected files are also very common.
To create a sense of urgency Cyber criminals are using such words as "alert", "notification", "needs urgent attention", "transaction is completed" etc.
Some of the most commonly used security infections in email messages:
Blackhole Exploit Kit - Security infection which takes advantage of unpatched exploits and injects user's computer with banking Trojans, key-loggers etc.
Cridex Trojan - Security threat capable of capturing online banking credentials entered via Internet browsers, can download and execute files etc.
Dapato Trojan - Installs malicious software on the infected operating system. Commonly installs a backdoor which allows remote access to infected systems.
Zeus Trojan - Attempts to steal confidential information, online credentials and banking details
Some of the most widely spread malicious email messages from the last couple of weeks:
Fake DHL Notification:
Deceptive message which is used to trick computer users into opening an infected attachment:
DHL notification
Our companies cowrie couldn't make the delivery of parcel.
REASON: Postal code contains as error.
LOCATION OF YOUR PARCEL: New York
DELIVERY STATUS: sort order
SERVICE: One-day Shipping
NUMBER OF YOUR PARCELS: ETBAKPRSU3
FEATURES: No
Label is enclosed to the letter.
Print a label and show it at your Post office.
An additional information:
If the parches isn't received within 15 working days our company will have the right to claim compensation from you for it's keeping in the amount of $8.26 for each day of keeping of it.
You can find the information about the procedure and the conditions of parcels keeping in the nearest office.
Thank you for using our services.
DHL Global
Fake UPS "We were not able to delivery the postal package" notification:
Deceptive message which is used to trick computer users into clicking on malicious links:
UPS - Your UPS Customer Services
Good Evening, ___
Dear Customer, We were not able to delivery the postal package
Track your Shipment now!
Best wishes, Your UPS.com Customer Services.
Bank of America fake root certificate:
Fake message which is used to trick Internet users into opening an infected attachment:
Please open the attached file to install the root certificate.
Melany Montoya Forseca
TRES SVCS SPECIALIST SVC CTR
GWBO Technical Helpdesk
Bank of America N.A
Phn: 1-(888) 589-3473
melany.montoya@bankofamerica.com
This message, and any attachments, is for the intended recipient(s) only, may contain information that is privileged, confidential and/or proprietary and subject to important terms and conditions available at www.bankofamerica.com/emaildisclamer If you are not the intended recipient, please delete this message.
Bank of America fake ACH transaction notice:
Deceptive message used to trick computer users into opening an infected attachment:
ACH transaction is completed. $4439 has been successfully transferred.
If the transaction was made by mistake please contact our customer service.
Payment receipt is attached.
***This is an automatically generated email, please do not reply***
HP scanner document scam:
Fake message used to trick computer users into opening an infected attachment:
Scan from a Hewlett-Packard ScanJet #3918872
Attached document was scanned and sent to you using a Hewlett-Packard HP Officejet 85684P.
Sent by: TAMRA
Attachment Type: HTM [Internet Explorer]
Hewlett-Packard Officejet Location: machine location not set
Infected email message "explosions at Boston marathon" - spreads Trojan to gain remote access to infected computers:
As you can see from the following examples Cyber criminals are exploiting the names of well known companies and organizations as well as taking advantage of braking news.
They create their deceptive email messages to make them appear as they are legitimate and needs urgent attention.
How one can avoid email-based malware?
1. Don't open email attachment if it came from an unknown sender. The best decision would be to delete such email message.
2. If the email message was send by a trusted and known source but you didn't expect it - contact the sender to confirm that the attachment is legitimate (don't reply to the email message - use the phone or their support email).
3. Avoid opening attachments with doc, .xls, .exe .htm extensions - such files may have executable code and could potentially infect your computer with malware (prior to opening such files contact the sender and ask if the attachment is legitimate).
4. Use legitimate antivirus programs.
With over one billion registered users, Facebook has become a prime target for hackers. Microsoft has issued a warning about a new threat to the social media giant capable of hijacking individual Facebook accounts. The malware, known as Trojan:JS/Febipos.A, was first detected in Brazil and reported on Friday in a Microsoft security bulletin. The Trojan works by disguising itself as a legitimate browser extension for both Google Chrome and Mozilla Firefox. The malware even tries to update itself like a benign extension would. The program is typically installed by TrojanDropper:Win32/Febipos.A.
Once installed, the program will download an updated version of itself onto the affected machine. The infected machine will not show any obvious signs of malware. Instead, the program checks if the computer is currently logged into a Facebook account. If detection is successful, Trojan:JS/Febipos.A will attempt to download a configuration file that includes a list of commands for the fake browser plugin. The program is capable of liking pages, sharing, chatting with the account holder's friends, joining groups, and posting status updates.
The initial infection posted messages stating "15 year-old victim of bullying commits suicide after showing her breasts on Facebook." Automated comments range from (in Portuguese – English translation shown) "Sorry guys, but this is ridiculous!!!" to "I don't have a new car, I don't have spare cash, but I get really close…" according to the official Microsoft security bulletin. Variants of the original malware have already been reported. One such program includes commands to post explicit messages in Portuguese. These posts also include links to other Facebook pages.
Microsoft reported that even while analyzing the malware, a large number of likes and shares continued to be reported meaning that the program is still infecting new machines.
Specifically, in just a few hours over 400 new page "likes" were recorded. Increases in the number of shared links and comments were also noted during this time. Many of the URLs contained within the initial config file have already been blocked by Facebook and the suspect pages removed. However; those behind this malware can change the behavior of the Trojan at any time. The commands can be changed an unlimited number of times until the infection has been isolated and the source shut down. The messages, URLs, and general behavior of the program depend on the config file which is not downloaded until the malware has already embedded itself on a machine. At this time, it is not clear how the infection is spreading or the exact number of machines that have fallen victim to the Trojan.
At this time, the infection seems to be centralized in Brazil; however, Microsoft stated that the code can easily be modified to target other geographical areas as well.
Microsoft claims that the latest version of Security Essentials detects this malware. Presumably, other antivirus software definitions do as well. In any event, the best method of protection from this threat is to only download browser extensions from trusted sources such as the Chrome Web Store or Add-ons for Firefox until the source of this threat has been determined. Make sure your antivirus protection is updated regularly and look for any suspicious activity on Facebook if infection has already occurred.
Last week, Microsoft confirmed reports of a vulnerability in Internet Explorer 8 allowing remote access to the operating system in Windows XP and above. This also includes all current versions of Windows Server up to 2008 R2. The exploit relies on a previously unknown hole in Internet Explorer that allows for remote code execution (as confirmed by Microsoft). The exploit has already been used successfully against the U.S. Department of Labor website. A script on several DOL web pages redirected users to a site serving a remote access Trojan to visitors. Specifically, the attack targeted Department of Energy workers that were attempting to access the Site Exposure Matrices. These pages communicate information about toxic substances encounter at nuclear sites as well as possible side effects.
These infected machines could be located within nuclear facilities and allow the attacker the same access as the current user. The site has been cleaned up but remains offline at this time while the DOL completes its investigation into the attack.
An official statement released has stated that no loss or compromise of information is evidenced at this time.
As many as nine other sites were also targeted with most of the focus being on energy related organizations. One such company is a large European company specializing in the aerospace, defense, and security industries. Non-profit groups and institutions have also been targeted. Although no group has taken official responsibilities for the exploit, many experts believe the Chinese hacking group known as DeepPanda is at fault. If this is the work of DeepPanda, many believe that this is only a small portion of an advanced exploit kit still under development.
The exploit has already been made available for the Metasploit Framework (a penetration testing tool) and could soon make its way into the mainstream cybercrime market.
Analysis of the initial log data suggests that visitors from 37 different countries browsed compromised websites before detection.
Many (as much as71%) of these visitors were from the United States. AlienVault reported that the compromised web page used advanced reconnaissance techniques to collect information about the machines visiting the site. This information included what antivirus software was installed, browser plug-in data, the version of Adobe Flash currently installed, and the current Java version running. The code even attempts to disable certain antivirus products. The payload was also found to be encrypted using base64 in an attempt to bypass existing security measures.
Experts recommend using a different browser until this vulnerability has been patched by Microsoft.
If operating on XP, the only option is to roll back to IE7. Windows Vista and 7 users can upgrade the browser to IE9 to avoid falling victim to this exploit. Windows Server users operate in a restricted mode by default known as Enhanced Security Configuration. Restricted Zone disables scripts and ActiveX controls and will mitigate the risk to these systems. Fortunately, Microsoft's next security patch is reported to address this vulnerability. Even if the patch does not reach Tuesday's deadline, there is already a Fix it tool offered by Microsoft to temporarily fix the issue until an official security patch can be released.
Although there are many unpatched flaws in Facebook security, an especially dangerous one that remains vulnerable allows hackers to spoof the content of any Facebook app with ease. Using this exploit, hackers are able to create spoofed wall posts from any trusted Facebook app imaginable. To the user, the content looks completely legitimate but the contents of the spoofed page can be anything but harmless. The vulnerability is capable of spoofing any app but some of the common ones that have already been spoofed successfully include Saavn, Candy Crush, Spotify, and Pinterest to name a few. No matter what app is actually spoofed, the content appears to be from the specified app and typically does not raise any alarms for the user as a result.
This vulnerability relies on the way in which Facebook allows apps to automatically post content on behalf of users.
Facebook uses a publishing method known as stream.publish. This dialog contains parameters for app_id and attachment (swfsr, imgsrc, href) that can be used by hackers to inject malicious content while appearing to be from a legitimate app. Simply using the app_id value of a known Facebook application and providing attachment parameters is all that is required to perform this hack.
Each Facebook app is assigned a unique app_id parameter which can easily be injected into a malicious wall post.
Facebook does have some degree of protection in place for these types of attacks. However, Break Security has demonstrated that by simply removing the href parameter from the stream.publish dialog a hacker can bypass the security measures in place and successfully spoof the app.
Earlier this year, Facebook did update the stream.publish dialog and introduced some new parameters using Feed Dialog to publish content. The addition of these four new parameters has made the vulnerability arguably more dangerous because of the increased options available to hackers when modifying them.
The link parameter allows hackers to inject a malicious external link.
This could redirect users to a phishing site, executable file, or Zero day exploit. The picture parameter allows for spoofing content with an image. This technique will only work for spoofing wall posts as the image will not display correctly on a user's newsfeed. The caption parameter allows attackers to choose "where" the content came from even though the content could be located anywhere. For example, the post could be made to appear as though it came from Facebook.com, Zynga.com, or Ownerappdomain.com. The name parameter is responsible for the title displayed in the spoofed post.
When a Facebook user clicks on the title, they are unknowingly redirected to a malicious website automatically.
Located within the Facebook app development options is the "Stream post URL security" option. If left unchecked by a developer, the hacker can use any remotely uploaded swf file as the attachment parameter. The danger is that when clicking a link on the spoofed page, a user may inadvertently execute the swf file from an external website on his or her machine. The implications of this unpatched vulnerability include remote malware installation and social engineering of Facebook users by phishing for passwords to Facebook as well as the apps these users believe they are interacting with.
Distributed Denial of Service (DDOS) attacks are nothing new. In the past few years, there has been an increase in these still-effective attacks. Services allowing customers to rent "bot nets" for stress testing used to be found exclusively in the hacking underground. These services have recently shifted and started openly advertising across the Internet. These services, known as "booter" or "stressor" products, launch DDOS attacks against networks to supposedly test the security measures in place. These means that anyone can pay a nominal fee to rent large amounts of bandwidth and channel them against practically any server they choose. Many of these sites share the same or similar source code and although they are not capable of taking down large, well-managed networks, small to medium web hosts should take note.
It is also worth noting that although these booter services often promise customers a variety of advanced attacks including Layer 7 attacks, SYN floods, and Apache memory exhaustion, a glitch in the code often means that reflected DNS and UDP flood attacks are the only methods used against a target. Typically known for a tough stance against web-based crime, PayPal seems to be the preferred payment method for most of these services (many of which are surprisingly U.S. based).
Although PayPal denies allowing these activities to take place under its watch, recent reports suggest otherwise.
The customer data of one popular booter service, asylumstresser.com, was recently leaked and the data has been evaluated by security experts. This service was responsible for as many as 10,000 online attacks within just a few days in March. The owner of the service, reported to be a 17 year old from the United States, as recently received over $35,000 in PayPal payments as a result of his service.
Unfortunately, asylumstresser is the rule rather the exception. Using cookie cutter source code, it is relatively easy for someone to setup a similar service and hide behind disclaimers that protect the owner from litigation.
The same old "I am not responsible for what illegal actions people may use my service for" argument seems to keep these products running and the perpetrators safe.
Even though the service offers a Skype resolver service specifically designed to locate the IP address of anyone using Skype, the service purports to be only designed for stress testing systems owned or operated by the customer. Asylumstresser also has YouTube ads that talk about the service's ability to "take down" competitors servers. Like many other facets of technology, there is no clear law dictating what liability the owner of one of these sites may carry.
When recently interviewed, the administrator of this service blames poor network management on the existence of his service.
Network security experts disagree and are pushing to make these people criminally liable although to date nothing has been done to thwart these disruptive attacks from occurring. Although it is true that basic network security knowledge and proper administration can stop most of these attacks with relative ease, it does not detract from the legal implications that should be imposed upon those running the services. It has become too easy for someone with little to no programming knowledge to launch attacks against the competition.
Newegg is a popular online retailer specializing in computer parts and equipment predominantly aimed at the DIY market. Recently, a very convincing spam email campaign has been discovered that has the potential of tricking even savvy computer users. The email starts with an attention grabbing subject. "Payment charged" will usually compel a user to click on the email. Once opened, the scam does an effective job of convincing recipients that the message is authentic.
The email looks exactly like something that would be sent by Newegg and it even comes complete with unique account numbers and sales orders numbers that appear to be automatically generated for each email sent out. The spam message usually includes a dialog similar to the following:
Thank you for shopping at Newegg.com.
We are happy to inform you that your order (Sales Order Number: XXXXX) has been successfully charged to your MasterCard and order verification is now complete.
If you have any questions, please use our LiveChat function or visit or Contact Us page.
Once you know, you Newegg.
Your Newegg.com Customer Service Team
Both of the links present in the email message actually redirect users to a variety of malicious URLS. Some of the URLs that have been embedded so far include:
productsbylifestyle(dot)com/news/developing-dollar-chance(dot)php
iechomobiliy(dot)com/news/developing-dollar-chance(dot)php
rs5636000(dot)de/mechanizes/index(dot)html
Once a victim has been successfully redirected to a compromised URL, the website will attempt to gain access and drop a payload of some sort into the machine.
It appears that this email scam is relying on the Blackhole exploit kit to infect machines.
First released on an underground Russian hacking forum, the Blackhole exploit kit is typically licensed to a user with specific customizations already in place. The Blackhole exploit uses obfuscated JavaScript to determine what vulnerabilities the machine has and then uses Pony, a dropper, to fetch the required binaries for the specific exploit attempt and the payload being deployed. In this case, the payload can be anything once an exploit has been used successfully.
Trojans are a common payload as are many other forms of malware.
Many of the exploits attempted by the Blackhole kit are older browser or Java vulnerabilities that are easily blocked when the latest updates are installed on the machine. However, the exploit kit can be modified easily by the developers and may include some Zero Day exploits as well. Typically, these added features depend on the amount the customer pays for the licensing agreement.
The domains and IPs associated with this latest spam run indicate that it is part of the "Amerika" spam ring.
This scam has recently created spoofed emails appearing to originate from the Walmart.com website and PayPal as well. Other recently reported assaults have spoofed emails from the Better Business Bureau, the United States Postal Service, and the payroll company ADP. Amerika appears to be using US-based servers at first glance but recent reports indicate that it may very well be a Russian based attack.
Skype has had the VOIP market in its pocket for a long time. With over 50 million people using Skype across a variety of mediums on a daily basis, Skype's success is undisputed. Offering applications for Windows, Mac, iOS, and Android, consumers can stay connected via VOIP calls or video chat from just about anywhere in the world. The service uses a complex audio/video management system that has held a reputation for years as being very secure. According to many experts, that all changed on a fateful day in May of 2011 when Skype was acquired by Microsoft for $8.5 billion. Microsoft has made many improvements to the Skype platform since acquiring it almost two years ago. Recently, Skype for Linux was released giving Skype true cross-platform bragging rights.
Multiple updates have also been pushed to both the Windows and OSX platforms which included integration of Facebook calling.
Although perhaps not appreciated by many users, Microsoft has improved the in-app advertising campaign of the program for the free version of the program.
Consumers that have a paid subscription are not currently subjected to these advertisements but Microsoft has hinted at changing that in the near future. Another major change that Microsoft has made involves the Skype infrastructure. Originally created as a peer to peer service that was distributed amongst users, Microsoft has centralized much of the service into dedicated data centers around the world. Although this infrastructure change reduces bandwidth strain and improves stability, it is this centralization that has security experts worried about user privacy.
The fear is that Microsoft has funneled all Skype traffic through its own data centers in an attempt to intercept every conversation. The US government, operating under the protection of the Patriot Act, has been known to request access to this type of information from all major communication service providers in recent years.
An independent German security firm has confirmed that every HTTPS URL sent over Skype is actively being checked by an IP address that is registered to the US Microsoft headquarters.
Microsoft's response to this accusation is that scanning is performed to remove spam and phishing links from Skype messages using its Smart Screen algorithms. This response is not satisfactory to most considering that the Skype program actively removes these links on its own through the client interface. Even if the claims from Microsoft are true, the reality is that in order to properly sniff Skype requests, the entire message needs to be decrypted first.
Many users rely on Skype because of its "end to end" encryption methods.
Skype did offer end to end encryption when it still relied on a semi-distributed, peer to peer architecture. Unfortunately, Microsoft has taken this security measure away from users – even if they have the best intentions. It may be difficult to prove that Microsoft is working with the US government by providing message contents, however, the fact remains that the security of the service has been compromised by centralizing data transmissions through Microsoft owned data centers and the company's admission to using Smart Screen technology to sniff for malicious URLs.
Chinese cybercriminals have made the news yet again. This time it appears that they have gained access to the Department of Defense systems that govern some of the U.S. Military's weapons secrets. The Defense Science Board, a division of the Pentagon, released a confidential report that disclosed the leak and listed over two dozen weapons systems that were compromised during this attack. Even the public version of the report referred to the threat as serious and went so far as to compare the situation to the "nuclear threat of the Cold War." Although the report did not reveal all weapons systems that were compromised, an advanced version of the Patriot missile system, the F/A-18 Hornet fighter jet, and the Black Hawk helicopter were all mentioned.
Missile defense systems currently deployed by the U.S. in Asia were also cited as being compromised in the attack. Although many of these systems are well-known throughout the world already, the inner workings of the designs, many of which are still classified, are now available to the Chinese for whatever purposes they desire. Just possessing this information will save the Chinese government decades of research and billions of dollars in development costs.
Earlier this year, President Obama demanded that the Chinese stop its cyber-espionage campaign.
Apparently, the demand went unnoticed as the attacks seem to be more frequent and more severe recently. The Pentagon pointed the finger at China last month regarding a string of intrusions that occurred in 2012 aimed at uncovering the U.S. Governments foreign policies and military plans.
The exact group of people responsible for these attacks is still unknown. However, a large percentage of the recent attacks have been reportedly originating from an office building in Shanghai that is known to have ties to the People's Liberation Army.
Even scarier is the fact that the government offices affected by these attacks are often not aware that an attack has even occurred until the FBI notifies them that there was a breach.
This is a testament to the advanced techniques being used by the Chinese to gather intelligence about the U.S Government, its policies, and now even its weapons. A Chinese Defence Ministry spokesperson has claimed that the Chinese government has no involvement with this latest series of attacks; however, this is often the stance taken by the Chinese in similar matters. Despite multiple warnings in recent years, the Chinese seem determined to reduce the U.S. military advantage through these cyber-espionage tactics.
A recent National Intelligence Estimate on these activities concluded that China is actively stealing intellectual property from U.S. companies and the military.
Even the recent Zero-Day IE8 watering hole attack on the Department of Energy has been traced back to suspected Chinese hackers working under orders of the government. Although these crimes form a long pattern that has gone on for years, this attack raises the bar in the shear depth of the breach and the classified information that has been compromised as a result. Without immediate action, the U.S. may have to consider a complete overall of the security measures in place that safeguard the very military secrets that make this nation secure.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,873
|
rm thumbnails/all.gif
echo "window.thumbnails = [ " > thumbnails.js
ls -1 thumbnails | awk '{printf("%s%s%s,\n","\"",$0,"\"")}' >> thumbnails.js
echo "];" >> thumbnails.js
# create a list of images
echo "window.images = [ " > images.js
ls -1 images | awk '{printf("%s%s%s,\n","\"",$0,"\"")}' >> images.js
echo "];" >> images.js
echo "Creating gif from all the images"
cd thumbnails
convert -background black -gravity center -extent 500x400 -delay 20 *.JPG -loop 0 all.gif
cd ..
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,279
|
{"url":"https:\/\/www.jobilize.com\/online\/course\/objectives-exponents-roots-and-factorization-of-whole-by-openstax?qcr=www.quizover.com","text":"# Objectives\n\n Page 1 \/ 1\nThis module contains the learning objectives for the chapter \"Exponents, Roots, and Factorizations of Whole Numbers\" from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, jr.\n\nAfter completing this chapter, you should\n\n## Exponents and roots ( [link] )\n\n\u2022 understand and be able to read exponential notation\n\u2022 understand the concept of root and be able to read root notation\n\u2022 be able to use a calculator having the ${y}^{x}$ key to determine a root\n\n## Grouping symbols and the order of operations ( [link] )\n\n\u2022 understand the use of grouping symbols\n\u2022 understand and be able to use the order of operations\n\u2022 use the calculator to determine the value of a numerical expression\n\n## Prime factorization of natural numbers ( [link] )\n\n\u2022 be able to determine the factors of a whole number\n\u2022 be able to distinguish between prime and composite numbers\n\u2022 be familiar with the fundamental principle of arithmetic\n\u2022 be able to find the prime factorization of a whole number\n\n## The greatest common factor ( [link] )\n\n\u2022 be able to find the greatest common factor of two or more whole numbers\n\n## The least common multiple ( [link] )\n\n\u2022 be able to find the least common multiple of two or more whole numbers\n\n#### Questions & Answers\n\nwhat is variations in raman spectra for nanomaterials\nI only see partial conversation and what's the question here!\nwhat about nanotechnology for water purification\nplease someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.\nDamian\nyes that's correct\nProfessor\nI think\nProfessor\nwhat is the stm\nis there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?\nRafiq\nindustrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong\nDamian\nHow we are making nano material?\nwhat is a peer\nWhat is meant by 'nano scale'?\nWhat is STMs full form?\nLITNING\nscanning tunneling microscope\nSahil\nhow nano science is used for hydrophobicity\nSantosh\nDo u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq\nRafiq\nwhat is differents between GO and RGO?\nMahi\nwhat is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq\nRafiq\nwhat is Nano technology ?\nwrite examples of Nano molecule?\nBob\nThe nanotechnology is as new science, to scale nanometric\nbrayan\nnanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale\nDamian\nIs there any normative that regulates the use of silver nanoparticles?\nwhat king of growth are you checking .?\nRenato\nWhat fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?\nwhy we need to study biomolecules, molecular biology in nanotechnology?\n?\nKyle\nyes I'm doing my masters in nanotechnology, we are being studying all these domains as well..\nwhy?\nwhat school?\nKyle\nbiomolecules are e building blocks of every organics and inorganic materials.\nJoe\nanyone know any internet site where one can find nanotechnology papers?\nresearch.net\nkanaga\nsciencedirect big data base\nErnesto\nIntroduction about quantum dots in nanotechnology\nwhat does nano mean?\nnano basically means 10^(-9). nanometer is a unit to measure length.\nBharti\ndo you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?\nabsolutely yes\nDaniel\nhow did you get the value of 2000N.What calculations are needed to arrive at it\nPrivacy Information Security Software Version 1.1a\nGood\n7hours 36 min - 4hours 50 min","date":"2020-04-06 03:11:02","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 1, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.44580623507499695, \"perplexity\": 3166.3624539166112}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-16\/segments\/1585371612531.68\/warc\/CC-MAIN-20200406004220-20200406034720-00214.warc.gz\"}"}
| null | null |
Q: Passing extra commands to a docker image from bitbucket-piplines.yml Trying to set the default charset and collation of a mysql:5.7 docker image using Bitbucket Pipelines, the documentation is a little vague mentioning:
If you need to configure the underlying database engine further, refer to the official Docker Hub image for details.
This page that the bitbucket documentation sends you to suggests that it is possible... at least via docker:
docker run --name some-mysql -e MYSQL_ROOT_PASSWORD=my-secret-pw -d mysql:tag --character-set-server=utf8mb4 --collation-server=utf8mb4_unicode_ci
So my question is how do I pass these parameters in: --character-set-server=utf8mb4 --collation-server=utf8mb4_unicode_ci
I have seen people use command: parameter in the YML for bitbucket-pipelines however the pipeline config editor on bitbucket says it's not valid there:
definitions:
services:
mysql:
image: mysql:5.7
command: ['--character-set-server=utf8mb4', '--collation-server=utf8mb4_unicode_ci']
ports:
- "3306:3306"
variables:
MYSQL_DATABASE: $MY_DATABASE
MYSQL_ROOT_PASSWORD: $MY_PW
A: It seems that it is not possible to pass commands to containers that run as services at this point. I was able to find the schema of the YAML file that defines the pipelines (check line 365). Not only you can't set the command, but you also can't set the ports. Fortunately, 3306 is the default one.
As as workaround I'd suggest you build your own Docker image, based on the mysql:5.7 and change the CMD statement to mysqld --character-set-server=utf8mb4 --collation-server=utf8mb4_unicode_ci (you can see how the mysql image's CMD look's like from here). After that, you have to push the image to a registry to which your Bitbucket runner has access to and use this image for your pipeline.
The following Dockerfile might do the job for you:
FROM mysql:5.7
CMD ["mysqld", "--character-set-server=utf8mb4", "--collation-server=utf8mb4_unicode_ci"]
At the end, your definition will look like this:
definitions:
services:
mysql:
image: your-custom-mysql-image:5.7
variables:
MYSQL_DATABASE: $MY_DATABASE
MYSQL_ROOT_PASSWORD: $MY_PW
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 8,365
|
John Come in and take a seat, Mary. This won't take a minute.
John Well I'm sorry Mary, but I've come to the conclusion that your work's just not up to scratch. I need to see a big improvement if you want to stay here.
Mary I see. I admit I have found these first few weeks a hard, but I need time to settle in and there's a lot to learn.
Jane Oh, sorry John, I'll come back later, I didn't realize you were talking.
John That's all right, come in. I was just telling Mary that I don't think she's really up to the job here.
Jane No, I don't want to intrude on a private conversation; I'll come back later.
John No; but I have to say that if this little talk doesn't make you realize where you're going wrong, it may come to that.
Poor Mary! John has just attacked the standard of her without giving her any idea of where she is going wrong; on top of that he's repeated the criticism in front of another member of staff.
Ask her if there is anything you can do to help her improve in these areas.
Set a date for another talk in a few weeks to review her progress.
Will feel that she is getting support in her efforts to improve.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,090
|
Колонизацията на Венера представлява част от научнофантастичен сюжет, при който хора могат да бъдат заселени на планетата и да създадат постоянни или временни жилищни обекти. Мнозина считат, че колонизацията на Венера представлява неизбежна стъпка в бъдещето, когато ресурсите на Земята ще бъдат силно ограничени. Все пак заради откритието за негостоприемната повърхност на Венера, сега очакванията са насочени по-скоро към Луната и Марс.
Прилики със Земята
Венера е много близка по размер до Земята. Размерът и е около 85% от земния.
Венера е най-близката до Земята планета.
Разлики със Земята
Венера има много разлики със Земята, които ще затруднят нейната колонизация.
Атмосферното налягане на Венера е 93 пъти по-голямо от земното.
Атмосферата на Венера е изградена от въглероден диоксид и азот.
Средната температура на Венера е 463 градуса по Целзий.
Наклонът на оста на Венера е 3 градуса и заради това там няма сезони.
Един ден на Венера е равен на 121 земни дни заради бавното и въртене.
На Венера има много гъст слой облаци, които почти спират слънчевата светлина.
На Венера вали киселинен дъжд.
Венера няма магнитно поле и не спира слънчевият вятър както прави Земята.
Колонизация на космоса
Венера
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,794
|
This project will provide an empirically-supported learning progression for a key scientific practice, scientific modeling. The specific instructional materials created as part of the project can serve as a model other developers can use to design materials supporting scientific modeling and other practices. The model for educative curriculum materials as a form of teacher support can be adapted to support teacher learning about modeling or other scientific practices in other curriculum materials.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 80
|
Jane und der Drache (Originaltitel: Jane and the Dragon) ist eine kanadisch-neuseeländische 3D-Animationsserie. Die Handlung basiert auf den gleichnamigen Büchern von Martin Baynton.
Handlung
Die junge Jane wächst auf Kippernia Castle auf, wo sie als Hofdame eine strenge Ausbildung zur Ritterin durchläuft. Ihr bester Freund ist ein großer und grüner Drache, welcher trotz seiner 300 Jahre ein noch sehr junger und verspielter Zeitgenosse ist.
Produktion und Ausstrahlung
Die Serie wurde ursprünglich auf YTV in Kanada und ABC in Australien ausgestrahlt. In Deutschland wurde die Serie erstmals 2009 auf KIKA ausgestrahlt. Produziert wurde zwischen 2005 und 2006 und es sind 26 Folgen entstanden. Regie führte Mike Fallows und Motion-Capture-Regisseur war Peter Salamon.
Episodenliste
Weblinks
offizielle Webseite (archiviert)
Animationsserie
Fernsehserie (Kanada)
Fernsehserie (Neuseeland)
Fernsehserie der 2000er Jahre
Drache im Film
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,706
|
{"url":"https:\/\/www.tutorialexample.com\/php-equal-operator-vs-identical-operator-a-difference-introduction-php-tutorial\/","text":"# PHP == (Equal Operator) Vs === (Identical Operator): A Difference Introduction \u2013 PHP Tutorial\n\nBy | April 14, 2020\n\nPHP equal operator (==) and identical operator (===) are widely used in if statement, what is the difference between them. In this tutorial, we will discuss this topic for php beginners.\n\n## Equal Operator (==)\n\nYou can only compare the value of two variables.\n\nFor example:\n\n<?php\n$x = '100';$y = 100;\nif($x ==$y){\necho \"they are the same\";\n} else {\necho \"they are not the same\";\n}\n?>\n\nRun this php code, you will find $x and$y are the same. Because php will convert the type of $x and$y automatically, which means $x+$y will be 200.\n\n## Identical Operator (==)\n\nIdentical Operator not only compares the value of two variables, but also will compare the type of php variable.\n\nAs to code:\n\n<?php\n$x = '100';$y = 100;\nif($x ===$y){\necho \"they are the same\";\n} else {\necho \"they are not the same\";\n}\n?>\n\n$x and$y is not the same, because $x is string while$y is integer.\n\nCategory: PHP","date":"2021-12-07 02:07:12","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2939039468765259, \"perplexity\": 5940.740556829099}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964363332.1\/warc\/CC-MAIN-20211207014802-20211207044802-00296.warc.gz\"}"}
| null | null |
\section{Introduction}
The study of spin ladder systems \cite{rdr} has made rapid progress
since their synthesis \cite{rhatb} and identification. \cite{rrgs} Among the
problems investigated both theoretically and experimentally are
the nature of the ground state, which is dramatically different for ladders
composed of odd and even numbers of chains, the behavior of holes in ladders,
including their superconductivity, \cite{runatmk} and the properties
introduced by doping of spinless impurities. Further novel physical
phenomena are found in coupled spin ladders, where the magnetic
structure of even-chain ladders with unfrustrated coupling, is of particular
interest because the ground state is expected to be a spin liquid at
sufficiently weak coupling, but to order magnetically when the interladder
interactions are strong. This system realizes the
conditions required to investigate the QCP,
introduced by Chakravarty {\it et al.} \cite{rchn} in the context of a
nonlinear $\sigma$-model description of the two-dimensional Heisenberg
AF, and discussed extensively and more generally in Refs.
\onlinecite{rcsy} and \onlinecite{rss}. In the language of these analyses,
the interladder coupling is a parameter which tunes the system from
the renormalized classical regime of long-ranged magnetic order, through the
QCP to the one-dimensional limit where the spins are
disordered by quantum fluctuations (quantum disordered).
Such a structure is represented in three dimensions by the two-chain
ladder compound LaCuO$_{2.5}$. \cite{rht} Initial experimental studies gave
contradictory results on the nature of the ground state, as static
susceptibility measurements \cite{rht} suggested a spin liquid state with a
spin gap, while nuclear magnetic resonance (NMR) \cite{rmkiahkt} and muon spin
resonance ($\mu$SR) \cite{rkoyyakhtn} measurements indicated a transition
to a magnetically ordered phase. In a brief study of
both electronic and magnetic properties of the material, it was proposed
\cite{rnr} that the conflicting observations could be reconciled if the system
was located near the QCP of the transition between the two regimes, and
this scenario was supported by detailed numerical simulations performed by
Troyer {\it et al.} \cite{rtzu}
In this work we will analyze the properties of the system on
both sides of the critical point, employing a mean-field approximation
to the bond-operator technique which is generalized to the magnetically
ordered regime. From the evolution of the mode structure through
the transition, we may compute both the thermodynamics and the dynamical
magnetic properties of the system. In the disordered phase, the excitations
are triply degenerate magnons with a spin gap which vanishes on approach
to the QCP. In the AF ordered phase, we find that the modes evolve into
two spin-wave excitations representing rotations of the ordered moment,
accompanied by an amplitude mode corresponding to fluctuations in its
magnitude. This latter mode has a gap which grows continuously with the
moment, so it will be low-lying only near the QCP, and will develop far
from the transition into a high-lying excitation which is ignored in the
dynamics of a conventional AF.
To characterize the ordered system close to the QCP we focus on the
static susceptibility $\chi(T)$, which vanishes at $T = 0$ on approach to the
transition. The results for $\chi(T)$ in the mean-field theory agree well
both qualitatively and quantitatively by highly accurate Quantum Monte Carlo
studies carried out by Troyer {\it et al.} \cite{rtzu} Both sets of
calculations support the proposal that LaCuO$_{2.5}$ is indeed an AF close to
the QCP.
We examine the dynamic structure factor to find features associated
with the presence of an amplitude mode. This mode couples to neutrons in the
same way as the transverse excitations, and we show that it should be
observable in inelastic neutron scattering measurements on single crystals.
Raman light scattering by the magnetic excitations is also discussed, but in
this case it is difficult to find signals which may be ascribed unambiguously
to this mode.
The outline of this paper is as follows. In section II we develop the
bond-operator formalism for application in the ordered magnetic regime, to
describe the mode structure of the magnetic excitations and the
zero-temperature phase transition. In section III we discuss briefly the
statistics of the magnon excitations, in order to solve the mean-field
equations at finite temperatures and to deduce the static susceptibility on
both sides of the transition. We consider in section IV the dynamical magnetic
properties of the system in the vicinity of the QCP, and
calculate the dynamic structure factor for comparison with proposed
experiments. Section V contains our conclusions and a brief discussion.
\section{Ordered Magnetic Phase}
We begin by applying the bond-operator technique \cite{rsb,rgrs} in
the regime where the interladder superexchange coupling is sufficiently strong
to stabilize a magnetically ordered state. We will restrict ourselves to the
case of simple AF order both along and perpendicular to the ladders. However,
we shall return in section IV to a recent alternative proposition, and show
that this gives rise only to notational differences.
Following Ref. \onlinecite{rsb}, the spin degrees of freedom
on each ladder rung are represented by the bond operators
\begin{eqnarray}
| s \rangle & = & s^{\dag} | 0 \rangle = \frac{1}{\sqrt{2}} \left( | \uparrow
\downarrow \rangle - | \downarrow \uparrow \rangle \right), \label{ebor}
\nonumber \\ | t_x \rangle & = & t_{x}^{\dag} | 0 \rangle = -
\frac{1}{\sqrt{2}} \left( | \uparrow \uparrow \rangle - | \downarrow
\downarrow \rangle \right), \nonumber \\ | t_y \rangle & = & t_{y}^{\dag}
| 0 \rangle = \frac{i}{\sqrt{2}} \left( | \uparrow \uparrow \rangle +
| \downarrow \downarrow \rangle \right), \\ | t_z \rangle & = & t_{z}^{\dag}
| 0 \rangle = \frac{1}{\sqrt{2}} \left( | \uparrow \downarrow \rangle +
| \downarrow \uparrow \rangle \right), \nonumber
\end{eqnarray}
where the arrows denote the direction of the left and right spins. The
presence of a quadratic term with negative coefficient in the
transformed Heisenberg Hamiltonian ensures that the singlet operators
condense, so that one may take $\langle s_i \rangle = \overline{s}$ on each
rung. AF ordering along the ladders occurs when, in addition, one of the
triplet operators has a non-zero expectation value with alternating sign.
This we take to be the component $t_z$, which may be written as
\begin{equation}
t_{iz} = (-1)^{i_z} \overline{t} + \hat{t}_{iz} ,
\label{etc}
\end{equation}
where $\hat{t}_z$ contains fluctuations about the average value ${\overline
t}$. The dynamics of these fluctuations may not be neglected, as in the
treatment of the $s$ degree of freedom, because their dispersion is strong.
With this choice of basis states, the coexistence of a finite $\overline{t}$
with the condensed singlet $\overline{s}$ on a rung corresponds to an AF
alternation along the ladder of states $ |\uparrow \downarrow \rangle$ and
$| \downarrow \uparrow \rangle $ of fixed, staggered moment $\overline{t}$,
admixed with a singlet component of weight $\overline{s} - \overline{t}$.
In the disordered or spin-liquid regime the three triplet modes
remain degenerate and massive, a situation represented schematically in
Fig. 1(a). The spin gap vanishes as the coupling approaches the QCP, where
the three modes of the lower branch are spin-wave-like, with a linear
dispersion about the wave vector of the magnetic ordering (Fig. 1(b)). In
AF the ordered phase there is a spontaneous breaking of rotational symmetry,
which by Goldstone's theorem is accompanied by massless excitations. These
will be two spin waves, which represent rotations, or transverse oscillations,
of the finite staggered moment. The third mode corresponds to fluctuations in
the amplitude of the moment (longitudinal) and will acquire a finite gap, as
shown in Fig. 1(c). In a conventional AF system one observes only the two spin
waves, but close to the QCP the amplitude mode will also be of low energy at
the zone center.
The real-space axes, not to be confused with spin-space labels
introduced above, are chosen with ${\hat {\bf z}}$ along the ladder direction
and ${\hat {\bf x}}$ and ${\hat {\bf y}}$ the directions of
interladder coupling. Using the reduced unit cell \cite{rtzu} in the $x$-
and $y$-dimensions, but with doubling of the real-space structure in the
$z$-direction to accommodate AF ordering along the ladder, the Hamiltonian of
the three-dimensionally coupled system,
\begin{eqnarray}
H & = & J \sum_{j} {\bf S}_{l,j} {\bf .S}_{r,j} + \lambda J \sum_{j, m = l,r}
{\bf S}_{m,j} {\bf .S}_{m,j+{\hat z}} \label{essh} \nonumber \\
& & + \lambda^{\prime} J \sum_{j} \left( {\bf S}_{r,j} {\bf .S}_{l,j +
{\hat x}} + {\bf S}_{r,j} {\bf .S}_{l,j + {\hat y}} \right) ,
\end{eqnarray}
after transformation to bond operators is written as
\begin{equation}
H = H_0 + H_1 + H_2 + H_3 + H_4 ,
\label{ebosh}
\end{equation}
where the terms have the following origins.
\begin{eqnarray}
H_0 & = & J \sum_{j,\alpha} \sum_{m = 1,2} \left( - {\textstyle \frac{3}{4}}
s_{j}^{m \dag} s_{j}^{m} + {\textstyle \frac{1}{4}} t_{j,\alpha}^{m \dag}
t_{j,\alpha}^{m} \right) \label{ebosh0}
\nonumber \\ & & - \sum_{j,\alpha} \sum_{m = 1,2} \mu_{j,m} \left(
s_{j}^{m \dag} s_{j}^{m} + t_{j,\alpha}^{m \dag} t_{j,\alpha}^{m} - 1 \right)
\end{eqnarray}
contains the contribution from the ladder rung interactions, and also the
constraint which restricts the spin states on each rung to a singlet or
one of three triplets. Here $j$ is a unit cell index and $m$ an index for the
two types of rung in each cell.
\begin{equation}
H_1 = {\textstyle \frac{1}{2}} \lambda J \sum_{j,\alpha} \sum_{m = 1,2}
\left( t_{j,\alpha}^{m \dag} t_{j+{\hat z},\alpha}^{m+1} s_{j+{\hat z}}^{m+1
\dag} s_{j}^{m} + t_{j,\alpha}^{m \dag} t_{j+{\hat z},\alpha}^{m+1 \dag}
s_{j}^{m} s_{j+{\hat z}}^{m+1}
+ H.c. \right)
\label{ebosh1}
\end{equation}
and
\begin{equation}
H_2 = {\textstyle \frac{1}{4}} \lambda J \sum_{j,\alpha \ne \beta} \sum_{m =
1,2} \left( t_{j,\alpha}^{m \dag} t_{j+{\hat z},\beta}^{m+1 \dag} t_{j+{\hat
z},\alpha}^{m+1} t_{j,\beta}^{m} - t_{j,\alpha}^{m \dag} t_{j+{\hat z},
\alpha}^{m+1 \dag} t_{j+{\hat z},\beta}^{m+1} t_{j,\beta}^{m} + H.c. \right)
\label{ebosh2}
\end{equation}
are the terms quadratic and quartic in $t$ operators corresponding to ladder
leg interactions; $H_2$ may not be discarded in the ordered system because
it will contribute terms of order $\overline{t}^2$ to the dynamics of the
modes $\alpha = x,y$.
\begin{eqnarray}
H_3 & = & - {\textstyle \frac{1}{4}} \lambda^{\prime} J \sum_{j,\alpha}
\sum_{\eta = \pm {\hat x}, {\hat y}} \sum_{m = 1,2} \left( t_{j,\alpha}^{m
\dag} t_{j + \eta,\alpha}^{m} s_{j + \eta}^{m \dag} s_{j}^{m} \right.
\label{ebosh3}
\nonumber \\ & & \qquad \qquad \qquad + \left. t_{j,\alpha}^{m \dag} t_{j +
\eta, \alpha}^{m \dag} s_{j + \eta}^{m} s_{j}^{m} + H.c. \right)
\end{eqnarray}
and
\begin{eqnarray}
H_4 & = & {\textstyle \frac{1}{8}} \lambda^{\prime} J \sum_{j,\alpha}
\sum_{\eta = \pm {\hat x}, {\hat y}} \sum_{m = 1,2} \left( t_{j,\alpha}^{m
\dag} t_{j + \eta, \beta}^{m \dag} t_{j + \eta ,\alpha}^{m} t_{j,\beta}^{m}
\right. \label{ebosh4} \nonumber \\ & & \qquad \qquad \qquad - \left.
t_{j,\alpha}^{m \dag} t_{j + \eta, \alpha}^{m \dag} t_{j + \eta, \beta}^{m}
t_{j,\beta}^{m} + H.c. \right)
\end{eqnarray}
are the quadratic and quartic contributions from the interladder coupling
terms, which link rungs of the same type $m$.
In the mean-field approximation the singlet expectation value $\langle
s_{i}^m \rangle = {\overline s}$ and Lagrange multiplier $\mu_{i}^m = \mu$ are
taken to be the same on all rungs $(i,m)$, while the triplet expectation value
$\langle t_{i,z}^m \rangle = - (-1)^m {\overline t}$ alternates. The
constraint term incorporates a reduction of ${\overline s}$ due both to
the presence of the ${\overline t}$ term and to quadratic fluctuations of all
three magnon modes. A direct coupling of the longitudinal ($t_z$) fluctuations
to the magnitude of ${\overline s}$ is also relevant, and appears in the
$t_{z}^{\dag} t_z s^{\dag} s$ terms as
\begin{eqnarray}
t_{j,z}^{\dag} t_{j+{\hat z},z} s_{j}^{\dag} s_j & = & t_{j,z}^{\dag}
t_{j+{\hat z},z} \left( 1 - t_{j,z}^{\dag} t_{j,z} \right)^{1/2} \left( 1 -
t_{j+{\hat z},z}^{\dag} t_{j+{\hat z},z} \right)^{1/2} \label{emt} \nonumber
\\ & \simeq & ({\overline t} + {\hat t}_{j,z}^{\dag}) ( - {\overline t} +
{\hat t}_{j+{\hat z},z}) \left( {\overline s} - \frac{1}{2 {\overline s}}
{\hat t}_{j,z}^{\dag} {\hat t}_{j,z} \right) \left( {\overline s} -
\frac{1}{2 {\overline s}} {\hat t}_{j+{\hat z},z}^{\dag} {\hat t}_{j+{\hat
z},z} \right) \\ & = & - {\overline s}^2 {\overline t}^2 + {\overline s}^2
{\hat t}_{j,z}^{\dag} {\hat t}_{j+{\hat z},z} + {\textstyle \frac{1}{2}}
{\overline t}^2 \left( {\hat t}_{j,z}^{\dag} {\hat t}_{j,z} + {\hat
t}_{j+{\hat z},z}^{\dag} {\hat t}_{j+{\hat z},z} \right) + O({\hat t}^4) .
\nonumber
\end{eqnarray}
The first line follows from substitution of the number operators into the
constraint and the second from Eq. (\ref{etc}). In the last line, the first
term is the classical part, the second is a fluctuation term which will appear
in off-diagonal components of the Hamiltonian matrix for all leg and interrung
coupling combinations in $H$ and for all three polarizations ${\hat
t}_{\alpha}$, and the third is diagonal in the magnon operators so appears as
a mass term unique to the $t_z$ modes. Note that the mass grows with the
ordered moment ${\overline t}$, as would be expected.
The mean-field Hamiltonian in reciprocal space may be cast in the form
\begin{eqnarray}
H_{\rm m} (\mu, {\overline s}, {\overline t}) & = & N \left( - {\textstyle
\frac{3}{4}} J {\overline s}^2 + {\textstyle \frac{1}{4}} J {\overline t}^2
- \mu {\overline s}^2 - \mu {\overline t}^2 + \mu - 2 J {\overline s}^2
{\overline t}^2 (\lambda + \lambda^{\prime}) \right) \label{emfsh}
\nonumber \\ & & + \sum_{{\bf k} \alpha} \left\{ \sum_{m = 1,2} \left[
\Lambda_{\bf k}^{\alpha} t_{{\bf k} \alpha}^{m \dag} t_{{\bf k} \alpha}^{m}
+ \Delta_{\bf k}^{\alpha} \left( t_{{\bf k} \alpha}^{m \dag} t_{-{\bf k}
\alpha}^{m \dag} + t_{{\bf k} \alpha}^{m} t_{-{\bf k} \alpha}^{m} \right)
\right] \right. \\ & & + \left. \left[ \Lambda_{\bf k}^{\prime \alpha}
t_{{\bf k} \alpha}^{1 \dag} t_{{\bf k} \alpha}^{2} + \Delta_{\bf k}^{\prime
\alpha} \left( t_{{\bf k} \alpha}^{1 \dag} t_{-{\bf k} \alpha}^{2 \dag} +
t_{{\bf k} \alpha}^{1} t_{-{\bf k} \alpha}^{2} \right) \right] + [ 1
\leftrightarrow 2] \right\} , \nonumber
\end{eqnarray}
in which $N$ denotes the total number of ladder rungs, the coefficients for
the transverse modes $\alpha = \sigma \equiv (x,y)$ are
\begin{equation}
\Lambda_{\bf k}^{\sigma} = {\textstyle \frac{1}{4}} J - \mu - {\textstyle
\frac{1}{2}} \lambda^{\prime} J ({\overline s}^2 - {\overline t}^2)
(\cos k_x + \cos k_y) ,
\label{elks}
\end{equation}
\begin{equation}
\Delta_{\bf k}^{\sigma} = {\textstyle \frac{1}{4}} \lambda^{\prime} J
({\overline s}^2 + {\overline t}^2) (\cos k_x + \cos k_y) ,
\label{edks}
\end{equation}
\begin{equation}
\Lambda_{\bf k}^{\prime \sigma} = J ({\overline s}^2 - {\overline t}^2)
\cos {\textstyle \frac{1}{2}} k_z
\label{eldks}
\end{equation}
and
\begin{equation}
\Delta_{\bf k}^{\prime \sigma} = {\textstyle \frac{1}{2}} J ({\overline s}^2
+ {\overline t}^2) \cos {\textstyle \frac{1}{2}} k_z ,
\label{eddks}
\end{equation}
and for the amplitude modes $\alpha = z$,
\begin{equation}
\Lambda_{\bf k}^{z} = {\textstyle \frac{1}{4}} J - \mu + 2 J {\overline t}^2
(\lambda + \lambda^{\prime}) - {\textstyle \frac{1}{2}}
\lambda^{\prime} J {\overline s}^2 (\cos k_x + \cos k_y) ,
\label{elkz}
\end{equation}
\begin{equation}
\Delta_{\bf k}^{z} = {\textstyle \frac{1}{4}} \lambda^{\prime} J
{\overline s}^2 (\cos k_x + \cos k_y) ,
\label{edkz}
\end{equation}
and
\begin{equation}
\Lambda_{\bf k}^{\prime z} = 2 \Delta_{\bf k}^{\prime z} = {\textstyle
\frac{1}{2}} J {\overline s}^2 \cos {\textstyle \frac{1}{2}} k_z .
\label{elddkz}
\end{equation}
The part of $H_{\rm m}$ (\ref{emfsh}) quadratic in the triplet operators
is diagonalized by Bogoliubov transformation, and can be reexpressed in terms
of the appropriate quasiparticle operators $\gamma_{k \alpha}^{\pm}$ as
\begin{eqnarray}
H_{\rm m} (\mu, {\overline s}, {\overline t}) & = & N \left( - {\textstyle
\frac{3}{4}} J {\overline s}^2 + {\textstyle \frac{1}{4}} J {\overline t}^2
- \mu {\overline s}^2 - \mu {\overline t}^2 + \mu - 2 J {\overline s}^2
{\overline t}^2 (\lambda + \lambda^{\prime}) \right) \label{edmfsh} \nonumber
\\ & & - N \left({\textstyle \frac{1}{4}} J - \mu \right) - {\textstyle
\frac{1}{2}} N \left({\textstyle \frac{1}{4}} J - \mu + 2 J {\overline t}^2
(\lambda + \lambda^{\prime} \right) \\ & &
+ \sum_{{\bf k} \alpha} \sum_{\nu = \pm} \omega_{{\bf k} \alpha}^{\nu}
\left( \gamma_{{\bf k} \alpha}^{\nu \dag} \gamma_{{\bf k} \alpha}^{\nu} +
{\textstyle \frac{1}{2}} \right) .
\end{eqnarray}
In calculating the mode frequencies, all terms fourth order in ${\overline s}$
and ${\overline t}$ are found to cancel, giving the easily factorized results
\begin{equation}
\omega_{{\bf k} \sigma}^{\nu} = \left[ \left({\textstyle \frac{1}{4}} J -
\mu - 2 \nu J {\overline s}^2 a_{\bf k}^{\nu} \right) \left( {\textstyle
\frac{1}{4}} J - \mu + 2 \nu J {\overline t}^2 a_{\bf k}^{\nu} \right)
\right]^{1/2}
\label{eqsd}
\end{equation}
and
\begin{equation}
\omega_{{\bf k} z}^{\nu} = \left[ \left({\textstyle \frac{1}{4}} J -
\mu + 2 J {\overline t}^2 (\lambda + \lambda^{\prime}) \right) \left(
{\textstyle \frac{1}{4}} J - \mu + 2 J {\overline t}^2 (\lambda +
\lambda^{\prime}) - 2 \nu J {\overline s}^2 a_{\bf k}^{\nu} \right)
\right]^{1/2} ,
\label{eqzd}
\end{equation}
where
\begin{equation}
a_{\bf k}^{\pm} = \lambda \cos {\textstyle \frac{1}{2}} k_z \pm {\textstyle
\frac{1}{2}} \lambda^{\prime} ( \cos k_x + \cos k_y ) .
\label{eak}
\end{equation}
The equation for $\omega_{{\bf k} z}^{\pm}$ represents two branches for the
amplitude mode in the doubled Brillouin zone; folding back of the zone in the
$k_x$ and $k_y$ dimensions returns the four branches in the Brillouin zone of
the real material. The equation for $\omega_{{\bf k} \sigma}^{\pm}$ contains
four doubly-degenerate branches, of which the most interesting is the
lowest-lying, $\omega_{{\bf k} \sigma}^{+}$ with $k_x$ and $k_y$ in the
reduced Brillouin zone. This yields the two rotation modes, and from the
factorized form (\ref{eqsd}) it is clear that the spin-wave condition is the
same as the vanishing of the spin gap which gave the critical coupling for the
transition from the disordered side, ${\textstyle \frac{1}{4}} J - \mu = 2 J
{\overline s}^2 (\lambda + \lambda^{\prime})$.
That the massless modes described by the bond operators are indeed
true spin waves can be shown in two ways. First, considering the limit of fully
developed magnetic order, ${\overline s} = {\overline t} = {\textstyle
\frac{1}{\sqrt{2}}}$ and
\begin{eqnarray}
\omega_{{\bf k} \sigma}^{+} & = & \left({\textstyle \frac{1}{4}} J - \mu
\right) \left[ 1 - \left( \frac{a_{\bf k}^{+ 2}}{ \lambda + \lambda^{\prime}}
\right)^2 \right]^{1/2} \label{eswl} \nonumber \\ & \simeq & 2 J {\overline
s}^2 \sqrt{ (\lambda + \lambda^{\prime})} \left[ {\textstyle \frac{1}{4}}
\lambda k_{z}^2 + {\textstyle \frac{1}{2}} (k_{x}^2 + k_{y}^2) \right]^{1/2}
\end{eqnarray}
in the limit of small $k$. A textbook derivation \cite{rkqts} of the mode
spectrum for the Hamiltonian in Eq. (\ref{essh}), for a spins of arbitrary
magnitude $S$ and with four spins per unit cell, returns exactly the contents
of both lines in Eq. (\ref{eswl}), with the condition $S = {\overline s}^2 =
{\textstyle \frac{1}{2}}$, as required. More generally, in the spin basis of
Eq. (\ref{ebor}) it is straightforward to show that a spin wave, represented
by a staggered, transverse component of $S_x$ on each rung, may be represented
by the bond-operator states $- | t_x \rangle - | s \rangle$, and similarly for
$S_y$, showing its equivalence to a $t_x$ ($t_y$) bond-operator mode in the
presence of the singlet.
The average singlet and triplet occupations may be represented
by the reduced variables
\begin{equation}
d_s = \frac{2 J {\overline s}^2}{ {\textstyle \frac{1}{4}} J - \mu}
\;\;\;\;\;\;\;\; d_t \;\; = \;\; \frac{2 J {\overline t}^2}{
{\textstyle \frac{1}{4}} J - \mu} ,
\label{edst}
\end{equation}
in terms of which the mode frequencies (\ref{eqsd}) and (\ref{eqzd}) are given
by
\begin{equation}
\omega_{{\bf k} \sigma}^{\nu} = \left({\textstyle \frac{1}{4}} J -
\mu \right) \sqrt{ ( 1 - \nu d_s a_{k}^{\nu}) ( 1 + \nu d_t a_{k}^{\nu} ) }
\label{eqsdr}
\end{equation}
and
\begin{equation}
\omega_{{\bf k} z}^{\nu} = \left({\textstyle \frac{1}{4}} J -
\mu \right) \sqrt{ \left( 1 + d_t ( \lambda + \lambda^{\prime} ) \right)
\left( 1 + d_t ( \lambda + \lambda^{\prime} ) - \nu d_s a_{k}^{\nu} \right) },
\label{eqzdr}
\end{equation}
while the spin-wave condition, which is maintained everywhere in the ordered
regime, becomes
\begin{equation}
d_s = (\lambda + \lambda^{\prime})^{-1}.
\label{ecvds}
\end{equation}
By substitution into the self-consistency equation for the disordered
solution, \cite{rnr} the critical coupling $\lambda_{c}^{\prime}$ is given
implicitly by
\begin{equation}
\frac{1}{\lambda + \lambda_{c}^{\prime}} = 5 - 3 \sum_{k}^{\prime}
\left( 1 + \frac{a_k}{\lambda + \lambda_{c}^{\prime}} \right)^{-1/2} n_{\rm m}
(\omega_{\bf k}),
\label{eztcp}
\end{equation}
where in the reduced unit cell there is only one degenerate magnon branch. In
the remainder of this section we will consider the system at zero temperature,
so that the magnon thermal occupation factors $n_m (\omega_{\bf k}^{\nu})$
are unity. The full mean-field equations are
\begin{eqnarray}
\langle \frac{\partial H_{\rm m}}{\partial \mu} \rangle & = & 0 = -
{\overline s}^2 - {\overline t}^2 + 1 \label{emfem} \nonumber \\ & & \;\;
+ {\textstyle \frac{3}{2}} -
{\textstyle \frac{1}{2}} \sum_{{\bf k} \, \nu} ^{\prime} \frac{ 1 + d_t (
\lambda + \lambda^{\prime} ) - {\textstyle \frac{1}{2}} \nu d_s a_{\bf
k}^{\nu} }{ 2 \sqrt{ \left( 1 + d_t ( \lambda + \lambda^{\prime} ) \right)
\left( 1 + d_t ( \lambda + \lambda^{\prime} ) - \nu d_s a_{k}^{\nu} \right)}}
\\ & & \;\;
- \sum_{{\bf k} \, \nu} ^{\prime} \frac{ 1 + {\textstyle \frac{1}{2}} \nu d_t
a_{\bf k}^{\nu} - {\textstyle \frac{1}{2}} \nu d_s a_{\bf k}^{\nu} }{ 2 \sqrt{
( 1 - \nu d_s a_{k}^{\nu}) ( 1 + \nu d_t a_{k}^{\nu})}} \nonumber
\end{eqnarray}
\begin{eqnarray}
\langle \frac{\partial H_{\rm m}}{\partial {\overline s}} \rangle & = & 0 =
- {\textstyle \frac{3}{4}} J - \mu - 2 J {\overline t}^2 (\lambda +
\lambda^{\prime} ) \label{emfes} \nonumber \\ & & \;\;
- {\textstyle \frac{1}{2}} \sum_{{\bf k} \, \nu} ^{\prime}
\frac{ \nu J a_{\bf k}^{\nu} \left( 1 + d_t ( \lambda + \lambda^{\prime} )
\right) }{ 2 \sqrt{ \left( 1 + d_t ( \lambda + \lambda^{\prime} ) \right)
\left( 1 + d_t ( \lambda + \lambda^{\prime} ) - \nu d_s a_{k}^{\nu} \right)}}
\\ & & \;\; - \sum_{{\bf k} \, \nu} ^{\prime} \frac{ \nu J a_{\bf k}^{\nu} ( 1
+ \nu d_t a_{\bf k}^{\nu} ) }{ 2 \sqrt{ ( 1 - \nu d_s a_{k}^{\nu}) ( 1 + \nu
d_t a_{k}^{\nu})}} , \nonumber
\end{eqnarray}
and
\begin{eqnarray}
\langle \frac{\partial H_{\rm m}}{\partial {\overline t}} \rangle & = & 0 =
{\textstyle \frac{1}{4}} J - \mu - 2 J {\overline s}^2 (\lambda +
\lambda^{\prime} ) - J(\lambda + \lambda^{\prime} ) \label{emfet} \nonumber
\\ & & \;\; + {\textstyle \frac{1}{2}} \sum_{{\bf k} \, \nu} ^{\prime}
\frac{ 2 J (\lambda + \lambda^{\prime}) \left( 1 + d_t ( \lambda +
\lambda^{\prime} ) - \nu d_s a_{k}^{\nu} \right) }{ 2 \sqrt{ \left( 1
+ d_t ( \lambda + \lambda^{\prime} ) \right) \left( 1 + d_t ( \lambda +
\lambda^{\prime} ) - \nu d_s a_{k}^{\nu} \right)}} \\ & & \;\; + \sum_{{\bf k}
\, \nu} ^{\prime} \frac{ \nu J a_{\bf k}^{\nu} ( 1 - \nu d_s
a_{\bf k}^{\nu} ) }{ 2 \sqrt{ ( 1 - \nu d_s a_{k}^{\nu}) ( 1 + \nu d_t
a_{k}^{\nu})}} \nonumber ,
\end{eqnarray}
in which the first three terms on the right hand side of each expression
constitute the classical equations, and the remaining terms the quantum
corrections. The notation $\sum ^{\prime}$ denotes ${\textstyle \frac{1}{N}}
\sum$.
Unconstrained solution of these equations for $\mu$, ${\overline s}$,
and ${\overline t}$ at fixed $\lambda$ and $\lambda^{\prime} >
\lambda_{c}^{\prime}$ gives a set of parameter values which are completely
unrelated to those on the disordered side of the QCP,
meaning a first-order transition. However, such parameters correspond to
modes which are in general all massive, as the spin-wave condition is not
obeyed, and have no physical meaning in the present problem of an ordered
magnet. This discontinuity between ordered and disordered regimes in an
approach combining classical mean-field terms with quantum corrections has
been encountered previously in similar circumstances. \cite{rss,rs,rtknpc}
By the Goldstone Theorem, any breaking of a continuous symmetry must be
accompanied by the presence of zero-energy excitations, in this case
spin waves which are massless in the long-wavelength limit.
We proceed by enforcing the third mean-field equation (\ref{emfet})
classically: this is precisely the spin-wave condition Eq. (\ref{ecvds}), and
serves both to ensure that the rotation modes are massless at the zone center
and to fix the value of one of the three original variables.
The remaining two mean-field equations (\ref{emfem},\ref{emfes})
correspond to the pair solved in the disordered phase, now generalized to
include finite ${\overline t}$. These may be combined to yield a single
equation for $d_t$ as a function of $\delta \lambda^{\prime} =
\lambda^{\prime} - \lambda_{c}^{\prime}$, which as $d_t \rightarrow 0$ returns
the critical point equation (\ref{eztcp}) for $\lambda_{c}^{\prime}$.
Solution of this equation gives a weak first-order transition which may be
traced in a linear expansion about $\lambda_{c}^{\prime}$ to the fact that the
small $d_t$ term acts as a cutoff in the ${\bf k}$ summation which provides a
logarithmic contribution in three dimensions. The linearized equation has the
form
\begin{equation}
d_t ( A + B \ln d_t ) = C \delta \lambda^{\prime} ,
\label{esdtvlp}
\end{equation}
which as shown in Fig. 2, describes reentrant behavior of the onset of
magnetic order. Numerically, the value ${\overline t}_0$ (Fig. 2) where the
solution at fixed $\lambda^{\prime}$ becomes double-valued is 0.048. The
$\lambda^{\prime}$ axis is however grossly expanded, and the reentrance occurs
within the range of values of interladder coupling $ 0.118 < \lambda^{\prime}
< 0.121$. Because the first-order nature of the transition is extremely weak,
we may use the formalism developed in this section to describe the features
of the ordered magnetic system both qualitatively an on a quantitative basis.
Illustration of the zero-temperature solution for the
variation of the ordered moment with the interladder coupling is deferred to
the following section, where it is presented together with the solutions at
finite temperatures.
\section{Finite-Temperature Solution and Static Susceptibility}
\subsection*{Magnon Statistics}
To compute the properties of the coupled ladder system at all
temperatures, it is necessary first to discuss the statistics of the magnon
excitations, which are contained in the thermal occupation function $n_m
(\omega_{\bf k}^{\pm})$. \cite{rnr} The essential feature is that despite the
bosonic commutation relations \cite{rsb} of the single magnon operators,
these do not have conventional bosonic statistics because of the constraint
(\ref{ebosh0}) on their number, and this has a profound effect on the
thermodynamics of the system at intermediate and high temperatures. An
approximate approach which gives a good account of the effects of the
constrained magnon ocupation in the case of the isolated ladder (disordered
system) was introduced by Tsunetsugu and coworkers in Ref. \onlinecite{rttw},
and the reader is referred to this work for a detailed discussion. Here we
summarize the results of applying this treatment in the ordered magnetic
regime.
The following discussion is simplified by the implicit assumption of
the presence of an anisotropy field stabilizing one particular spatial
direction for the staggered moment, a point to which we shall return in more
detail below when considering the susceptibility. This results in a
small gap in the spin-wave spectrum, preventing the easy axis from reorienting
in an applied field and allowing for a reduction of the energy of one of the
modes. In an $N$-rung system (${\textstyle \frac{1}{4}} N$ unit cells) with
both rotation ($\alpha = \sigma \equiv S = \pm 1$) and amplitude ($\alpha = z
\equiv S = 0$) modes possible at the same wave vector ${\bf k}$, and in the
presence of a magnetic field $h$, the free energy per site is given by
\begin{equation}
\tilde{f} = - \frac{1}{2 \beta} \ln \left\{ 1 + z^z (\beta) + 2 \cosh(\beta h)
z^{\sigma} (\beta) \right\} ,
\label{esfe}
\end{equation}
where $z^{\alpha} (\beta) = \sum_{\bf k} ^{\prime} {\rm exp} (-\beta
\omega_{{\bf k}_j \alpha})$, is the partition function for each mode type.
The magnon thermal occupation functions per mode in zero field are
\begin{equation}
\tilde{n}_m (\omega_{{\bf k} \alpha}) = \frac{1}{{\rm e}^{-\beta \omega_{{\bf
k} \alpha}} + 3}
\label{emtofd}
\end{equation}
in the disordered phase where $\omega_{{\bf k} \alpha} = \omega_{{\bf k} z} =
\omega_{{\bf k} \sigma}$, and in the ordered phase
\begin{equation}
\tilde{n}_m (\omega_{{\bf k} \alpha}) = \frac{{\rm e}^{-\beta \omega_{{\bf k}
\alpha} }}{1 + {\rm e}^{-\beta \omega_{{\bf k} z}} + 2 {\rm e}^{-\beta
\omega_{{\bf k} \sigma} }} .
\label{emtofoa}
\end{equation}
That in the latter two equations the occupation function for a mode $z$
($\sigma$) contains a denominator dependent on the energy of a mode $\sigma$
($z)$ encodes the effects of the constraint. At high temperatures, each of
these functions approaches the limiting value of $ {\textstyle \frac{1}{4}}$
per mode. The form of the occupation function required in the mean-field
equations and denoted in Ref. \onlinecite{rnr} as $n_m (\omega_{{\bf k}
\alpha})$ is that including the zero-point term, {\it i.e.} the analog of the
bosonic ${\textstyle \frac{1}{2}} \coth \left( {\textstyle \frac{1}{2}} \beta
\omega_{{\bf k} \alpha} \right)$, and is given by $n_m (\omega_{{\bf k}
\alpha}) = {\textstyle \frac{1}{2}} + \tilde{n}_m (\omega_{{\bf k} \alpha})$.
\subsection*{Mean-Field Solutions}
Considering first the approach to the transition from the disordered
side, in Fig. 3 is shown the evolution of the spin gap $\Delta$, the minimum
of the triply degenerate massive magnon dispersion, with interladder coupling
at temperatures between 0 and $0.5J$. The zero-temperature graph is that shown
already in Ref. \onlinecite{rnr}. Solution of the mean-field equations at
finite temperatures is marginally more complicated because the initial system
of two variables may not be reduced to a single one in the same manner. As the
temperature is increased, the disordered phase becomes more robust as might be
expected, and a larger interladder coupling is required to stabilize magnetic
order. The spin gap in the isolated ladder is found also to increase with
temperature, although this rise is somewhat more linear above $T = 0.1J$ than
would be given by the simplest possible formulation, $\Delta =
\sqrt{\Delta_{0}^2 + T^2}$.
In the ordered phase, the characteristic parameter varying with the
coupling constant is ${\overline t}$, which gives the extent of staggered
moment formation. This is shown in Fig. 4 for the same temperatures. The
ordered moment rises abruptly at the transition, with no obvious indication
at any temperature of the reentrant behavior discussed in the previous
section, and then less steeply thereafter. However, the logarithmic evolution
of the ordered moment given in Eq. (\ref{esdtvlp}) is manifest in a
$\ln$-$\ln$ plot, and precludes the extraction of a power-law mean-field
exponent from data within a range $\Delta \lambda^{\prime} \simeq 0.15$ of the
transition. At high values of the interladder coupling, the ordered moment
falls short of the limiting value ${\overline t} = {\textstyle
\frac{1}{\sqrt{2}}}$ as $\lambda^{\prime} \rightarrow 1$, showing that in the
bond-operator description of the isotropic limit there remains some admixture
of rung singlets with the ordered spins, a consequence of quantum
fluctuations.
Turning now to the physical situation of a fixed interladder coupling
constant at variable temperature, it is clear from Fig. 3 that any system
will be disordered at high temperature. By solution of a pair of equations
analogous to the zero-temperature critical-point equation (\ref{eztcp}), this
phase boundary is found in the plane of $T$ and $\lambda^{\prime}$, giving the
N\'eel temperature shown in Fig. 5. Because $T_N$ in this figure is
measured in units of $J \sim 1400K$, it is evident that the real material
LaCuO$_{2.5}$, with a N\'eel temperature \cite{rmkiahkt} of 117K, is located
extremely close to the QCP (in fact at $\lambda^{\prime} = 0.127$). We will
use for illustration the interladder coupling value $\lambda^{\prime} = 0.13$,
for which the dispersion relations have already been sketched in Fig. 1(c).
The dispersion curves in the physical Brillouin zone are now shown
quantitatively in Fig. 6 for this parameter choice, at which the ordered
moment has the value ${\overline t} = 0.14$ and $T_N = 0.105J$.
\subsection*{Static Susceptibility}
The physical quantity most readily measurable which gives direct
information about the magnetic state of a spin system is the static
susceptibility $\chi(T)$, and for this reason it has been the subject of
extensive discussions \cite{rdr,rttw} and measurements in a variety of ladder
systems. \cite{rht,rahtik} The susceptibility per site is
\begin{equation}
\chi(T) = - \left. \frac{\partial^2 \tilde{f}}{\partial h^2} \right|_{h=0} ,
\label{essg}
\end{equation}
which from Eq. (\ref{esfe}) yields in the disordered phase \cite{rttw}
\begin{equation}
\chi(T) = \beta \frac{z^{\alpha} (\beta)}{1 + 3 z^{\alpha} (\beta) } .
\label{essd}
\end{equation}
This is illustrated over a wide temperature range in Fig. 7(a), where it is
clear that the interladder coupling has an effect only at low $T$, while the
remainder of the graph is well described by the numerical results for the
isolated ladder. \cite{rttw} The situation at low temperatures is displayed
in the inset, which shows that as the critical coupling is approached
$\chi(T)$ rises more rapidly with temperature. The nature of
this increase may be studied by examining the function
\begin{equation}
g(T) = T^2 \frac{\chi^{\prime} (T)}{\chi(T)},
\label{essdf}
\end{equation}
in which $\chi^{\prime}$ denotes the temperature derivative. For the
a system with spin gap \cite{rttw} $\Delta$, $\chi(T) =
T^{-\alpha} {\rm e}^{- \beta \Delta}$ and $g = \Delta - \alpha T$, while for a
power-law dependence $\chi(T) = a T^{\alpha}$, as might be expected in a
critical regime, $g = \alpha T$. Fig. 7(b) shows this function at low-$T$
for several values of the coupling constant $\lambda^{\prime}$, demonstrating
clearly the evolution of the system with interladder coupling from a spin
liquid with the spin gap $\Delta_0 \simeq 0.5J$ and power $\alpha = {\textstyle
\frac{1}{2}}$ prefactor of the isolated ladder, to a quantum critical phase
in three dimensions, where $\alpha = 2$. These results have a straightforward
interpretation from the thermal excitations of a dilute gas of (noninteracting)
triplet magnons whose dispersion is quadratic about a spin gap $\Delta$, and
becomes linear as $\Delta \rightarrow 0$. At the QCP the susceptibility is due
to excitations of spin waves, with temperature dependence determined only
by system dimensionality.
The calculation of the susceptibility in the ordered phase may proceed
in one of two ways: for an ideal system with no interaction between the spins
and the real-space or lattice coordinates, the ordered moment simply reorients
in an applied field to be perpendicular, {\it i.e.} a spin-flop transition.
For a system with an easy-axis anisotropy, the spin waves acquire a mass
proportional to the effective anisotropy field $h_a$, and this allows the
moment direction to remain stable under application of a parallel field. For
a review of both situations see Ref. \onlinecite{rkhpf}. The susceptibility
is computed in the zero-field limit, which may always be taken as an external
field smaller than the intrinsic anisotropy energy, so we restrict our
attention to the simplest case of the susceptibility contribution from normal
mode excitations for the case of pinned moment direction in a parallel field.
In addition to the part due to excitation of normal modes, the
susceptibility in the ordered phase contains a constant part due to the
presence of the finite moment, and so may be represented as
\begin{equation}
\chi(T) = \chi_0 (T) + \chi_{\rm exc} (T) .
\label{essod}
\end{equation}
$\chi_{\rm exc} (T)$ is computed by analogy with the disordered phase from the
free energy of the modes which may be excited, and is given by
\begin{equation}
\chi_{\rm exc}^{\parallel} (T) = \beta \frac{z^{\sigma} (\beta)}{1 + z^{z}
(\beta) + 2 z^{\sigma} (\beta) } .
\label{esso}
\end{equation}
Because this part has its leading contributions from the spin-wave modes with
dispersion $\omega_{{\bf k} \sigma}$, it can be expected always to have a
quadratic $T$ dependence at low temperatures. The component due to application
of a perpendicular field is similar, but slightly more involved because the
evolution of the mode $\omega_{{\bf k} z}$ with field is not linear for a
finite gap, a point which will be illustrated below. The numerator contains
also a term in $z^z (\beta)$, which is much smaller than the spin-wave
contribution when the mass of the $z$ mode is finite, and higher-order terms
which cancel at the transition. The dominant feature of the temperature
dependence remains the $T^2$ part due to the spin-wave contribution, with a
smaller prefactor. On averaging over the crystallite directions in a
polycrystalline sample, $\chi_{\rm exc} (T) = {\textstyle \frac{1}{3}}
\chi_{\rm exc}^{\parallel} (T) + {\textstyle \frac{2}{3}} \chi_{\rm
exc}^{\perp} (T)$.
Similarly, the static part in the anisotropy-pinned case may be
written as $\chi_0 (T) = {\textstyle \frac{2}{3}} \chi_{0}^{\perp}$ for a
polycrystalline sample, as only the transverse part has a finite value and
this will be observed as an average over all crystallite orientations.
$\chi_{0 \, \perp}$ may be computed in the bond-operator formalism by
considering the additional term in the mean-field Hamiltonian in the presence
of a finite magnetic field ${\bf h}$,
\begin{equation}
\sum_i {\bf h.} \left( {\bf S}_{l_i}^1 + {\bf S}_{r_i}^1 + {\bf S}_{l_i}^2 +
{\bf S}_{r_i}^2 \right) = -i \epsilon_{\alpha \beta \gamma} \sum_i \left(
t_{i \beta}^{1 \, \dag} t_{i \gamma}^1 + t_{i \beta}^{2 \, \dag} t_{i
\gamma}^2 \right) .
\label{ehmft}
\end{equation}
The evolution of the bond-operator modes of a ladder system with application
of a magnetic field will be presented elsewhere. Here
we state the result that in general a field component $h_{\alpha}$ acts to
create an ordered moment $t_m$ in the operators $t_{\beta}$ and $t_{\gamma}$,
where $\alpha \ne \beta \ne \gamma$. In the presence of a finite, staggered
$\langle t_z \rangle = {\overline t}$, application of a small field $h_x$
stabilizes a finite, staggered ordered moment $\langle - i t_y \rangle \equiv
{\overline t}_m$. One of the spin-wave modes is unaffected by the field and
the induced moment and one increases linearly with $h$, while the amplitude
mode energy rises only quadratically with $h$. We find that ${\overline t}_m
\propto {\overline t} h$, and that the magnetization per site stabilized by
a small field $h_x$ is ${\overline m}_x = - {\textstyle \frac{1}{2}}
{\overline t}_m {\overline t}$, giving for the transverse part
$\chi_{0}^{\perp} = \frac{\partial {\overline m}_x}{\partial h_x} \propto
{\overline t}^2$. As a function of interladder coupling, this contribution
appears very similar to the the results in Fig. 5 for ${\overline t}$.
$\chi_{0}^{\perp}$ vanishes approximately in the manner of a second-order
transition as the temperature is increased towards $T_N$. When the same
formalism is applied to the case of a longitudinal field, the ordered moment
is $O \left( {\overline t}_{m}^2 \right)$ and it is clear that
$\chi_{0}^{\parallel}$ is vanishing, so that the total $\chi^{\parallel}$
has $T^2$ contributions only.
In Fig. 8 is shown the full static susceptibility $\chi(T)$
(\ref{essod}) for a variety of values of
interladder coupling in the ordered phase, $\lambda^{\prime} >
\lambda_{c}^{\prime}$. The curves begin at the finite value of $\chi_0 (T=0)$
calculated in the preceding paragraphs, and their variation with temperature
is approximately quadratic because $\chi_0 (T)$ is nearly constant at low T,
particularly for values of the interladder coupling not in close proximity to
the critical point. Because the scale of $\chi_0$ is significantly smaller
than that of $\chi_{\rm exc}$, the second-order transition in $\chi_0 (T)$
at $T_N$ appears as a minor downward cusp. The results compare very well with
highly accurate Quantum Monte Carlo studies on very large systems by Troyer
{\it et al.} \cite{rtzu}, which show the transition from spin liquid to ordered
magnet occurring very close to $\lambda^{\prime} = 0.12$, and continuous
evolution of the susceptibility from an exponential form in the disordered
phase to a quadratic variation at the critical point with an additive constant
part which grows continuously on moving to the ordered side. Quantitatively,
the magnitude of the peak susceptibility is some 15\% greater in the
mean-field theory, and that of the constant parts approximately 20\%
smaller at common values of the coupling in the ordered phase. One may
fit the measured static susceptibility \cite{rht} to the form $\chi (T) = a
+ b T^2$ at low temperatures, and quadratic temperature dependence is found
\cite{rtzu} to give a good account of the data. The intrinsic susceptibility
$\chi_0$ of the coupled ladder system is very difficult to extract from the
constant $a$, as this requires detailed knowledge of core atomic
susceptibility terms, and we note only the qualitative result that all terms
involved are very small, a further indication of the proximity of the system
to the QCP. That the mean-field theory is in
such generally good agreement with the numerical results, which for an
unfrustrated spin system and with demonstrably negligible finite-size
corrections can be taken to be essentially exact, is presumed to be primarily
a consequence of the three-dimensionality of the ordered magnetic system.
\section{Dynamic Magnetic Properties}
In the preceding section we have deduced that the LaCuO$_{2.5}$
system is magnetically ordered but located close to the QCP, as a
result of which the transition temperature and ordered moment are small.
One consequence is that the magnetic modes corresponding to fluctations
in the amplitude of the ordered moment, which in a conventional magnet are
high-lying and play no role in determining the low-energy properties of the
system, should have only a small mass here. In this section
we study the dynamic magnetic properties of such a system to isolate those
features due to the presence of the amplitude mode and predict the ways in
which these may be identified definitively by experiment.
We focus directly on the dynamic structure factor measured by
neutron scattering, which may be written following Ref. \onlinecite{rstns}
in the form
\begin{eqnarray}
S^{R} ({\bf q},\omega) & = & \frac{1}{2 \pi} \sum_{\alpha \beta} \left(
\delta_{\alpha \beta} - {\hat q}_{\alpha} {\hat q}_{\beta} \right)
\sum_{i, \langle ij \rangle, \tau, \tau^{\prime}} {\textstyle \frac{1}{4}}
g_{\tau} g_{\tau^{\prime}} F_{\tau}^{*} ({\bf q}) F_{\tau^{\prime}} ({\bf q})
\label{esfm} \nonumber \\ & & \times
\int_{-\infty}^{\infty} {\rm d} t {\rm e}^{- i \omega t} \langle \exp \left(-
i {\bf q.r}_{i, \tau} (0) \right) \exp \left( i {\bf q.r}_{i + {\bf r}_{ij},
\tau^{\prime}} (t) \right) \rangle \langle S_{i + \tau}^{\alpha} (0) S_{i +
{\bf r}_{ij} + \tau^{\prime}}^{\beta} (t) \rangle ,
\end{eqnarray}
Here $i$ and $j$ denote unit cells, and ${\bf \tau}$ and ${\bf \tau^{\prime}}$
are vectors specifying the locations of the atoms in each cell.
$F_{\tau} ({\bf q})$ is the magnetic form factor which describes the spatial
extent of the spin density around the site ${\tau}$, and $g_{\tau}$ is the
Land\'e factor. In this expression the space and spin variables are partially
factorized, and each expectation value can be separated into a constant part,
corresponding to elastic scattering, and a time-dependent part responsible
for inelastic processes.
We consider first the elastic magnetic component, in order to isolate
those Bragg peaks with the highest intensity about which one may measure the
dynamical structure factor most readily, and also to discuss the nature of the
magnetically ordered state. In Ref. \onlinecite{rnr} it was assumed that the
interladder coupling in LaCuO$_{2.5}$ was AF. This
assumption was made on the basis that ferromagnetic coupling, which relies on
Hund's Rule coupling between orbitals on an atom in the superexchange path, is
generally weaker than AF coupling energy scales, and that
because the system is one of low symmetry is was likely that the latter would
have appreciable components on certain paths. In fact this assumption was not
tested in detail, and the tight-binding fit to the Local Density Approximation
(LDA) bandstructure was made using a single Wannier orbital on each Cu
site, which was taken to be a mixture (due to the low symmetry) of Cu
3$d_{x^2 - y^2}$- and 3$d_{3z^2 - r^2}$-centred orbitals in the pyramidal
CuO$_5$ system. The good
agreement obtained with this procedure appeared to justify the assumptions
made, but did not rule out alternative parameter combinations. The results of
the preceding section of this work provide a more reliable means of
estimating the magnitude of the interladder coupling. Recently a quantitative
effort has been made by Mizokawa {\it et al.} \cite{rmpc} to understand the
superexchange parameters in LaCuO$_{2.5}$. These authors deduce some of the
parameters of the electronic structure from measurements made by Cu $2p$
core-level spectroscopy, and use these in Hartree-Fock calculations.
They conclude that the magnetically ordered state of lowest
energy is that where the AF coupled spins in each ladder are ferromagnetically
coupled between the ladders, and that the magnitude of the interladder
interaction is less than $0.1J$. Superexchange estimates involving competing
exchange paths are difficult, and can involve significant errors
due to computing small differences between large numbers. Nevertheless
this is the most systematic study performed on the system to date. We note
that the results of the preceding sections are essentially independent of
the sign of the coupling ratio $\lambda^{\prime}$, which acts only to exchange
the branch indices $\nu = \pm$.
The elastic magnetic scattering is described by the static structure
factor, the time-independent part of Eq. (\ref{esfm}), and has finite
components
\begin{equation}
S_{\rm s}^{R} ({\bf G}) = N \langle S^z \rangle^2 \left( 1 - ({\bf G.{\hat
z}})_{av}^2 \right) \left[ {\textstyle \frac{1}{2}} g F ({\bf q}) \right]^2
{\cal F} ({\bf G})
\label{essff}
\end{equation}
only at the reciprocal lattice vectors ${\bf G}$, which are the magnetic
Bragg peaks.
Here ${\hat z}$ is the direction of the ordered spin moment in real space,
and is not yet known. In cuprates, the coupling of the spin system to the
lattice which determines this direction is generally very weak, although
by comparison with the parent phases of tetragonal high-temperature
superconducting materials the spins may be expected to align along either the
rungs or chains of the ladders. $\langle S^z \rangle$ is the magnitude
of the ordered moment, and for the coupling value $\lambda^{\prime} = 0.13$
is of magnitude $\sqrt{2} {\overline t} \simeq 0.2$. Thus the square of this
quantity yields only a 4\% effect, rendering the elastic scattering rather
weak close to the QCP.
\begin{equation}
{\cal F} ({\bf G}) = \sum_{{\bf \tau}, {\bf \tau^{\prime}}} (-1)^{(\tau +
\tau^{\prime})} {\rm e}^{i {\bf G.} ( {\bf \tau} - {\bf \tau^{\prime}} ) }
\label{esffg}
\end{equation}
is the structure function, obtained by summing over the sites in the unit
cell, in which the convention used with the site labels ${\tau}$ is such as to
ensure that each $\uparrow$ and $\downarrow$ spins introduce factors of the
opposite sign, corresponding to the spin density distribution around each
site. The appropriate site labeling for one plane of Cu atoms is shown in
Fig. 9(a) for the simple AF structure function, which we denote as Type I AF,
and in Fig. 9(b) for the case with ferromagnetic coupling between the AF
ladders, which we denote as Type II AF.
It is straightforward to calculate ${\cal F} ({\bf
G})$, and the results are expressed by quoting the reciprocal lattice vector
${\bf G}$ of the magnetic Brillouin zone as $(h, k, l)$, with $o$ and $e$
denoting odd and even integers respectively. For the Type I configuration
\begin{equation}
{\cal F} ({\bf G}) = \left\{ \begin{array}{cc} 64 \cos^2 G_x a_x \sin^2 G_y
a_y & (e,e,o), (o,o,o) \\ 64 \sin^2 G_x a_x \cos^2 G_y a_y & (e,o,o), (o,e,o)
\\ 0 & l \,\, {\rm even} \end{array} \right. ,
\label{esfafc}
\end{equation}
while for Type II the results are identical with [$x \leftrightarrow y$];
the displacement vector components $a_x$ and $a_y$ are shown in Fig. 9.
These quantities, normalized to unity, are shown in Fig. 10 for the two
configurations in any reciprocal-space plane of odd $l$ and for values of $h$
and $k$ between 0 and 10. It is evident that there are Bragg peaks of nearly
maximal amplitude, and so the best points in the reciprocal zone around which
to investigate the dispersion of the dynamical modes would be for example (2,
0, $o$) and (0, 1, $o$) in Type I, or (3, 0, $o$), (0, 4, $o$)
and (5, 1, $o$) in Type II. Because the two configurations differ
significantly in the locations of strong Bragg peaks, it should be possible
by diffractometry to determine the sign of the interladder superexchange.
Turning to the dynamical structure factor as it might be observed (in
a single-crystal sample) around one Bragg peak, the factor of interest is the
spin part $\int {\rm d} t {\rm e}^{i \omega t} \langle S_{\bf q}^{\alpha}
(0) S_{-{\bf q}}^{\beta} (t) \rangle$, from which the spatial dependence on
the crystal structure has been removed. In order to compute this quantity
in terms of the bond-operator eigenmodes, we take first the rung
combinations \cite{rgrs}
\begin{equation}
S_{{\bf q} \alpha}^{\pm} (t) = \sum_i {\rm e}^{i {\bf q.r}_i} \left[ S_{l_i \,
\alpha} (t) \pm S_{r_i \, \alpha} (t) \right] ,
\label{eborc}
\end{equation}
and then combine the variables on the two rungs in the reduced,
AF ordered unit cell to give
\begin{eqnarray}
S_{{\bf q} \alpha}^{+ \pm} & = & - i \epsilon_{\alpha \beta \gamma} \sum_{\bf
k} \left( t_{{\bf k} + {\bf q} \, \beta}^{1 \dag} t_{{\bf k} \, \gamma}^{1}
\mp t_{{\bf k} + {\bf q} \, \beta}^{2 \dag} t_{{\bf k} \, \gamma}^{2} \right)
\label{eboucc} \nonumber \\ S_{{\bf q} \alpha}^{- \pm} & = & {\overline s}
\left( t_{{\bf q} \, \alpha}^{1} + t_{- {\bf q} \, \alpha}^{1 \dag} \right)
\pm {\overline s} \left( t_{{\bf q} \, \alpha}^{2} + t_{- {\bf q} \,
\alpha}^{2 \dag} \right) .
\end{eqnarray}
The final expression for the structure factor
\begin{equation}
S^R ({\bf q}, \omega) = \sum_{\alpha} \sum_{\mu \nu \eta \rho = \pm}
\int_{-\infty}^{\infty} {\rm d} t {\rm e}^{i \omega t} \langle S_{{\bf q}
\alpha}^{\mu \nu} (0) S_{- {\bf q} \alpha}^{\eta \rho} (t) \rangle
\label{edsff}
\end{equation}
is a sum over the three spin indices (for unpolarized neutron scattering) and
all sign combinations of $S^{\pm \pm}$, and its evaluation is aided by
extensive cancellation of terms. For scattering studies at low temperatures,
where very few excited states are occupied and these only singly, we may
from their bosonic commutation relations \cite{rsb} compute the thermal
expectation value by summation over bosonic Matsubara frequencies. The two
types of component which emerge from Eq. (\ref{edsff}) are
\begin{eqnarray}
S^{R-} ({\bf q}, \omega) & = & \sum_{\alpha} \pi \left( {\overline s}^2 +
{\overline t}^2 \right) \left( \cosh 2 \theta_{\bf q}^{\alpha} - \sinh 2
\theta_{\bf q}^{\alpha} \right) \left[ n_{\bf q}^{\alpha} + \Theta (\omega)
\right] \delta \left( \omega_{\bf q}^{\alpha} - |\omega| \right)
\label{edsfnmc} \\ S^{R+} ({\bf q}, \omega) & = & \sum_{{\bf k} \, \alpha
\ne \beta} \left\{ \left( \cosh 2 (\theta_{{\bf k} + {\bf q}}^{\alpha}
- \theta_{\bf k}^{\beta}) + 1 \right) n_{\bf k}^{\beta} (n_{\bf k}^{\alpha} +
1) \delta \left( \omega_{{\bf k} + {\bf q}}^{\alpha} - \omega_{\bf k}^{\beta}
- \omega \right) \right. \label{edsfnms} \nonumber \\
& & + \left. {\textstyle \frac{1}{2}} \left( \cosh 2 (\theta_{{\bf k} + {\bf
q}}^{\alpha} - \theta_{\bf k}^{\beta}) - 1 \right) \left[ n_{\bf k}^{\alpha}
+ \Theta (\omega) \right] \left[ n_{\bf k}^{\beta} + \Theta (\omega) \right]
\delta \left( \omega_{{\bf k} + {\bf q}}^{\alpha} + \omega_{\bf k}^{\beta} -
|\omega| \right) \right\} ,
\end{eqnarray}
in which the hyperbolic trigonometric functions are the coefficients of the
Bogoliubov transformation, \cite{rnr} and the $n_{\bf k}$ are thermal
occupation functions. Errors due to the constraint on magnon occupations are
neglible at low temperatures, and may in fact be taken into account fully by
a more general formulation of the scattering expression, which gives different
thermal occupation functions $n_{\bf k}$.
$S^{R-} ({\bf q}, \omega)$ (\ref{edsfnmc})
appears with coefficient $\left( {\overline s}^2 + {\overline t}^2 \right)$
for contributions from the $\sigma$ components of the spins, due to the
presence of the ordered moment in $t_z$, and with coefficient ${\overline s}^2$
otherwise. It has the simple interpretation of scattering processes
involving magnon creation or destruction, and will be seen as a broadened
$\delta$-function line along $({\bf q}, \omega_{{\bf q} \alpha})$
corresponding to the magnon dispersion relations. $S^{R+} ({\bf q}, \omega)$
(\ref{edsfnms}) describes magnon-magnon scattering (first line) and pair
creation or destruction processes (second line), each involving magnons of a
different type, and so will have $(\sigma, \sigma)$ and $(\sigma, z)$
components which could in principle be distinguished by polarized neutron
scattering.
The dynamic structure factor $S^{R-} ({\bf q}, \omega)$ for magnon
creation ($ \omega > 0$) is shown in Fig. 11, and as stated above is found to
consist of a series of peaks at the dispersion relations of all magnon
branches. A small broadening is inserted by hand in the calculation. In the
disordered system (Fig. 11(a)) there are four triply degenerate branches in
the physical unit cell, and the lowest of these has a gap at the bottom of its
band ($q = 0$), as required for a spin liquid. In the ordered system
(Fig. 11(b)), this gap has vanished, and branches are split into their
$\sigma$ and $z$ components. In the lowest-lying branch, the intensity of the
spin-wave peak diverges as $q \rightarrow 0$ and the magnetic Bragg peak is
approached, while the corresponding $z$ mode appears at the energy $\omega =
0.29J$ at the chosen value of interladder coupling. This energy splitting is
expected to be resolvable, and so the amplitude mode should be detected as a
neighboring peak, or at least as a significant shoulder to the elastic peak,
and to each of the dispersion branches.
The component $S^{R+} ({\bf q}, \omega)$ for two-magnon scattering
and pair creation has an intensity between one and two orders of magnitude
lower than that from magnon creation, and so will be barely detectable.
Because inelastic neutron scattering couples directly to the one-magnon
process, it is the best technique to observe a clear signature of the
low-lying amplitude mode.
We have considered also the expected form of the intensity
$I(\omega)$ in a Raman light scattering experiment
on the magnetically ordered but nearly critical system. Unambiguous
spectroscopic evidence for the amplitude mode, as in the magnon creation
signal expected in the neutron scattering cross section, would be provided by
one-magnon processes with a significant weight. Following Fleury and Loudon,
\cite{rfl} it is clear that in this system neither the magnetic-dipole nor
electric-dipole first-order interactions detailed by these authors can
provide appreciable signals. Concluding for completeness with second-order
light scattering, we have computed the $\langle ({\bf S}_i {\bf .S}_j)
({\bf S}_i {\bf .S}_j) \rangle$ correlation function which gives the intensity
in a spin-only model. \cite{rsss} We find a ``3$J$'' peak due to scattering
of zone-boundary magnons in the ladder ($\pm k_z$) direction, and that this
peak is broadened in the presence of an amplitude mode split off from the spin
wave band in an AF system close to the QCP. However, mindful of the facts
that measured spectra are not fully explained by two-magnon scattering
considerations, and that these are found to be anomalously broad in other
cuprate materials, we conclude that Raman scattering is unlikely to be a
suitable probe of the dynamics of the amplitude mode.
\section{Conclusion}
We have presented a theoretical description of the three-dimensionally
coupled spin ladder system which is realized in the insulating compound
LaCuO$_{2.5}$. This spin configuration is such that for weak interladder
magnetic coupling the ground state is the spin-liquid phase characteristic
of the isolated ladder, while for larger coupling values an unfrustrated,
magnetically ordered state is expected. We employ a representation based on
singlet and triplet bond operators on each rung of the ladders to obtain a
uniform description of both phases within the same framework.
Within the mean-field approach to the bond-operator formalism,
solution at zero temperature gives the interladder coupling at the QCP as
$J^{\prime} = 0.121J$. The transition is found to
be very weakly first order, and the increase of the ordered moment to be
logarithmic in the coupling ratio $\lambda^{\prime} - \lambda_{c}^{\prime}$
in the vicinity of the transition, with the mean-field exponent recovered at
higher $\lambda^{\prime}$. Finite-temperature solutions are obtained by
incorporating the constraint on the triplet excited states into an effective
magnon statistics, and show the increase with temperature of the disordered
regime and the critical coupling, allowing the deduction of the N\'eel
temperature.
The real material has an ordering temperature extremely small on the
scale of the superexchange parameter $J$ within the ladders, implying that it
is located very close to the QCP on the ordered side. In
this case the magnon mode corresponding to amplitude fluctuations of the
ordered moment has only a small gap at the bottom of its band, and will
contribute to the low-energy dynamic and thermodynamic properties of the
system. The constant part of the static susceptibility is proportional to the
square of the ordered moment and is small, reconciling measured susceptibility
data with the fact that the system is ordered. The part of the static
susceptibility due to thermal excitation of modes of the system increases
quadratically with temperature for spin waves in three dimensions. The results
of the mean-field theory are supported by comparison with the detailed Quantum
Monte Carlo studies of Troyer {\it et al.} \cite{rtzu}. We have demonstrated
that the low-lying amplitude mode, which is
unique to this type of AF, contributes to the dynamical response
function. Inelastic neutron scattering is an especially suitable probe for
investigating this mode as it may access one-magnon excitation processes, and
we have presented an explicit calculation of its effects for comparison with
experiment.
In closing, for a system so close to the QCP we may
speculate on ways of controlling the tuning parameter represented by the
interladder coupling in order to pass through the transition. As discussed in
the preceding section, the superexchange processes contributing to the
interladder magnetic interaction are not well understood, but the coupling
is known to be a decreasing function of the bond angle away from 180$^0$. It
may be possible by application of hydrostatic
pressure, or better uniaxial pressure along the $x$- or $y$-axis of a single
crystal, to cause an alteration of this bond angle significant enough to
measure, at least as a raising or lowering of $T_N$ as deduced from
${\textstyle \frac{1}{T_1}}$ by NMR. As an alternative to physical pressure it
is possible
\cite{rhpc} also to apply chemical pressure by substituting other trivalent
atoms for La. In LaCuO$_{2.5}$ the bond angle is found to increase on
substitution of Y for La, while it is decreased in equal amount by
substitution of Nd. We await with interest more detailed results on the
evolution of the magnetic state, which may provide evidence of the system
being moved through the QCP. However, for a random distribution of
substituents there will be a random distribution of local distortions,
from which one may deduce only an average value for the bond angle and
interladder coupling, so that interpretation within the above framework will
be less transparent. Finally, as we have shown in the preceding sections, the
changes in many bulk quantities across the quantum critical regime remain
essentially smooth, and it will be necessary to adopt a criterion based on the
appearance of the staggered moment or the associated massive mode for
unambiguous identification of the transition point.
\section*{Acknowledgements}
We are grateful to Z. Hiroi, T.-K. Ng, S. Sachdev, M. Sigrist, M.
Troyer and M. E. Zhitomirsky for helpful discussions.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,209
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This article is an extension to the [getting started guide](gettingstarted.html). You will improve the same Calculator sample with a task for [FxCop](http://msdn2.microsoft.com/en-us/library/bb429476.aspx).
If you need more details please see the [API docs for the FxCop](apidocs/fake-fxcophelper.html) task.
## Setting up FxCop
Open *build.fsx* from your Calculator sample folder and add a new target *FxCop* to the targets section:
Target "FxCop" (fun () ->
!! (buildDir + @"\**\*.dll")
++ (buildDir + @"\**\*.exe")
|> FxCop
(fun p ->
{p with
// override default parameters
ReportFileName = testDir + "FXCopResults.xml"
ToolPath = "FxCopCmd.exe"})
)
In the dependencies section modify the build order to:
"Clean"
==> "BuildApp"
==> "FxCop"
==> "BuildTest"
==> "Test"
==> "Deploy"
==> "Default"
That's it. If you run your build script you will get new *.xml file in the *./test* folder:
## Letting the build fail
If you want to let the build fail in the case that FxCop reports any errors or warnings you can use the *FailOnError* parameter:
Target "FxCop" (fun () ->
!! (buildDir + @"\**\*.dll")
++ (buildDir + @"\**\*.exe")
|> FxCop
(fun p ->
{p with
// override default parameters
ReportFileName = testDir + "FXCopResults.xml"
FailOnError = FxCopErrorLevel.CriticalWarning
ToolPath = "FxCopCmd.exe"})
)
If you activate this option FxCop errors will cause your build to fail. Possible values are:
* FxCopErrorLevel.Warning
* FxCopErrorLevel.CriticalWarning
* FxCopErrorLevel.Error
* FxCopErrorLevel.CriticalError
* FxCopErrorLevel.ToolError
* FxCopErrorLevel.DontFailBuild
The values are cummulative. If you choose *FxCopErrorLevel.CriticalWarning* the build will fail for critical warnings, errors, critical errors and FxCop tool errors but not for simple warnings. The default is *FxCopErrorLevel.DontFailBuild*.
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,957
|
\section{Introduction}
If magnetic frustration is sufficiently strong, a spin system may evade spontaneous symmetry breaking at low temperatures and instead form a highly entangled state where the spins fluctuate in a cooperative manner. This so-called spin liquid state generally exists in two different flavors: the quantum~\cite{anderson73,balents10,savary16} and the classical spin liquid~\cite{ramirez99,bramwell01,bergmann07,gao16}. The first case preferably occurs for small quantum spins in combination with frustrated lattice geometries and/or anisotropic interactions where quantum fluctuations may reach the size of the local spin magnitude thus hindering the system from developing magnetic long-range order.
In the second case, spin liquid-like behavior even survives in the complete absence of quantum fluctuations such as for classical ($S\to\infty$) spins. The suppression of long-range magnetic order now relies on a macroscopic degeneracy of classical ground states through which the system fluctuates collectively, thus justifying the notion of a classical spin liquid. Paradigmatic examples are pyrochlore spin-ice systems~\cite{ramirez99,bramwell01}, where at zero temperature an ice rule (e.g., the famous two-in-two-out rule) imposes local constraints on possible spin states. Since these rules leave the ground-state spin configuration underdetermined, the system maintains a macroscopic (extensive) classical degeneracy~\cite{pauling35}.
Interestingly, for certain lattice geometries and special arrangements of frustrating interactions, classical spin liquids even exist without a local ice-rule constraint. This rare situation is realized on the three-dimensional diamond lattice [Fig.~\ref{fig1}(a)] with first ($J_1$) and second ($J_2$) neighbor Heisenberg interactions when $\frac{J_2}{|J_1|}>\frac{1}{8}$ and $J_2$ is antiferromagnetic~\cite{bergmann07,lee08,savary11,buessen18}. The competing interactions force the system into classical coplanar spin-spirals. Remarkably, the ground state is formed from a highly degenerate set of such spirals where the corresponding wave vectors $\mathbf{q}$ occupy a closed surface in reciprocal space (note that a similar scenario also occurs on the two-dimensional honeycomb lattice~\cite{fouet01,mulder10,baez17}). Due to the cooperative motion of spins through the degenerate manifold of spirals, this state has been dubbed a {\it spiral spin liquid}.
Spiral spin liquids are generally very fragile to perturbations of various different types. Any finite additional term in the Hamiltonian such as third neighbor couplings $J_3$ or dipolar interactions typically selects specific spirals out of the degenerate manifold and consequently generates long-range magnetic order. Even in the absence of such perturbations, a lifting of the degeneracy takes place due to thermal fluctuations, i.e., a finite temperature transition into a magnetically ordered state is induced by an entropic ``order-by-disorder'' selection~\cite{villain80} of spirals. As has been found in Ref.~\cite{bergmann07}, by varying $\frac{J_2}{|J_1|}>\frac{1}{8}$ the system goes through a sequence of different magnetic phases. While strictly speaking this effect destroys spiral spin liquids at any finite temperature, an approximate version of this state may still survive in a temperature range {\it above} the transition where the thermal selection is not yet active. Finally, quantum fluctuations at large but finite spin magnitudes have been found to induce an order-by-disorder effect similar to thermal fluctuations~\cite{buessen18}.
Currently, the most promising material to approximately realize a spiral spin liquid is the A-site spinel MnSc$_2$S$_4$~\cite{fritsch04,giri05,krimmel06,kalvius06,muecksch07,gao16} where spin\textendash5/2 Mn$^{2+}$ ions occupy the sites of a diamond lattice. At $\sim$$2.9$ K which is well below the Curie-Weiss temperature of $|\Theta_{\text{CW}}|=23$ K~\cite{fritsch04} but still inside the paramagnetic phase of this compound (which survives down to $\sim$$2.3$ K~\cite{fritsch04,krimmel06,kalvius06,muecksch07,gao16}) neutron scattering directly observes surface-like scattering profiles in momentum space, reminiscent of a spiral spin liquid~\cite{gao16}. From the radius of this surface a coupling ratio of $\frac{J_2}{|J_1|}=0.85$ has been determined~\cite{bergmann07} (where $J_1$ is ferromagnetic). The measured spin-structure factor is not evenly distributed on the spiral surface but shows higher intensities for spirals with wave vectors $\mathbf{q}\sim2\pi(0.75,0.75,0)$ and symmetry-related positions~\cite{krimmel06,muecksch07,gao16}. This spiral selection turns into real magnetic long-range order below $T_{\rm c}=2.3$ K~\cite{krimmel06,muecksch07} (other works report slightly smaller values of $T_{\rm c}\approx2.1$ K~\cite{fritsch04,kalvius06,gao16}). It is worth emphasizing that this peak position does not coincide with the thermal selection predicted in Ref.~\cite{bergmann07} but rather points towards the presence of longer-range $J_3$ interactions.
\begin{figure}[t]
\includegraphics[width=0.99\linewidth]{fig1.pdf}
\caption{(a) Cubic unit cell of the diamond lattice with first ($J_1$), second ($J_2$), and third ($J_3$) neighbor couplings. (b) Couplings $J_{1}$\textendash$J_{4}$ from DFT as a function of the Hubbard $U$ interaction. The vertical line indicates the exchange couplings investigated in the main text. \label{fig1}}
\end{figure}
\begin{figure*}
\includegraphics[width=0.99\linewidth]{fig2.pdf}
\caption{(a) Classical spin-spiral surface in the extended Brillouin zone formed by the wave-vectors of the degenerate spiral ground states of the model with $J_2/|J_1|=1.64$ and $J_3=0$, (b) The classical spin-spiral surface in the $q_x$\textendash$q_y$ plane for $J_2/|J_1|=1.64$ and $J_3=0$ (red line). The blue crosses indicate the Bragg peak position for an additional third neighbor coupling $J_3/|J_1|=0.57$. Black dots highlight the measured magnetic order at $\mathbf{q}\sim2\pi(0.75,0.75,0)$. (c) Red: Size of the spiral surface [given by the intersection with the line $(q,q,0)$] as a function of $J_2$ and for $J_3=0$. Blue: $(q,q,0)$ position of the ordering wave vector for $J_3/|J_1|=0.57$. Vertical full (dashed) lines indicate the coupling ratios $J_2/|J_1|=1.64$ ($J_2/|J_1|=0.85$~\cite{bergmann07}). The shaded area marks the position and width of the measured magnetic Bragg peak $\mathbf{q}\sim2\pi(0.75,0.75,0)$~\cite{gao16}.
\label{fig2}}
\end{figure*}
This article complements recent experimental works by theoretically investigating the fate of the spiral spin liquid when assuming a realistic model for MnSc$_2$S$_4$. To this end, we first employ {\it ab initio} density functional theory (DFT) calculations to determine the microscopic Hamiltonian of this compound. We then treat the resulting model within the pseudofermion functional-renormalization group (PFFRG) method~\cite{reuther10} which is capable of resolving the effects of thermal and quantum fluctuations, and we clarify the role of third neighbor $J_3$ interactions. In particular, we investigate to which degree the spiral spin liquid phase in MnSc$_2$S$_4$ remains stable under such perturbations and compare the $\mathbf{q}$\textendash space resolved magnetic susceptibility with neutron scattering experiments. Our main results are summarized as follows: (i) We find that the $J_2$ and $J_3$ interactions are both considerably larger than previously assumed~\cite{bergmann07}. (ii) Close to the magnetic phase transition but still inside the paramagnetic regime the spin correlations are dominated by $J_3$ couplings which induce a pronounced selection of spirals with wave vectors $\mathbf{q}\approx2\pi(0.72,0.72,0)$, in excellent agreement with experiments. (iii) We identify a temperature regime around $3T_{\rm c}$ to $5T_{\rm c}$ where the spiral selection due to $J_3$ couplings is suppressed such that the system realizes an approximate spiral spin liquid. (iv) PFFRG calculations for our model Hamiltonian reproduce the measured spin structure factor for MnSc$_2$S$_4$ with remarkable accuracy.
The paper is structured as follows: In Sec.~\ref{sec:methods}, we describe the DFT and PFFRG methods, and provide details of the calculations. In Sec.~\ref{sec:mm} we discuss the model Hamiltonian determined from DFT and compare our exchange couplings with those of the previously proposed model. We also discuss the physical implications of these new couplings for the corresponding classical model employing the Luttinger-Tisza method. Section~\ref{sec:PFFRG} contains the results obtained from the PFFRG calculations for the newly proposed Hamiltonian, which are also compared and contrasted with those obtained for the previously proposed model. Finally, in Sec.~\ref{sec:discussions} we summarize and discuss our findings, and give concluding remarks.
\begin{table}[t]
\setlength{\tabcolsep}{2.27pt}
\centering
\begin{tabular}{lllllc}
\hline \hline
\multicolumn{1}{c}{$U$ (eV)}
& \multicolumn{1}{c}{$J_{1}$ (K)}
& \multicolumn{1}{c}{$J_{2}$ (K)}
& \multicolumn{1}{c}{$J_{3}$ (K)}
& \multicolumn{1}{c}{$J_{4}$ (K)}
& \multicolumn{1}{c}{$\Theta_{\rm CW}$ (K)} \\ \hline
\multirow{1}{*}{3.0} & $-0.465(2)$ & 1.117(1) & 0.364(1) & 0.0039(6) & $-46$ \\
\multirow{1}{*}{3.5} & $-0.433(2)$ & 0.918(1) & 0.305(1) & 0.0029(5) & $-38$ \\% MnSc2S4, U=3.5, 6x6x6
\multirow{1}{*}{4.0} & $-0.404(1)$ & 0.755(1) & 0.257(1) & 0.0022(4) & $-31$ \\% MnSc2S4, U=4, 6x6x6
\multirow{1}{*}{$\mathbf{4.5}$} & $\mathbf{-0.378(1)}$ & $\mathbf{0.621(1)}$ & $\mathbf{0.217(1)}$ & $\mathbf{0.0015(3)}$ & $\mathbf{-25}$ \\% MnSc2S4, U=4.5, 6x6x6
\multirow{1}{*}{5.0} & $-0.356(1)$ & 0.509(1) & 0.184(1) & 0.0009(3) & $-20$ \\ \hline
\end{tabular}
\caption{Exchange couplings of MnSc$_2$S$_4$ calculated within GGA+$U$ at $J_{\rm H}=0.76$~eV and $6\times 6\times 6$ $q$-points. The parameters corresponding to $U = 4.5$ eV (marked in bold) are used for the PFFRG simulations [see also Fig.~\ref{fig1}(b)].}
\label{tab:DFT}
\end{table}
\section{Methods}\label{sec:methods}
We base our calculations on the cubic spinel structure determined by
neutron powder diffraction at $T=1.6$~K~~\cite{krimmel06}. The
Mn$^{2+}$ ions form a diamond lattice as shown in
Fig.~\ref{fig1}(a). We use an energy mapping technique to determine
the most important exchange interactions in
MnSc$_2$S$_4$~\cite{Jeschke2013,Guterding2016,Iqbal2017}. For
this purpose we construct a $2\times 2\times 1$ supercell of the
original primitive cell containing two Mn$^{2+}$ ions; in $P\,m$ space
group, this supercell has eight inequivalent Mn sites allowing for 20
distinct spin configurations. This allows us to determine the first
four exchange couplings, extending up to a Mn\textendash Mn distance of
10.6~{\AA}. We perform density functional theory calculations with the
all electron full potential local orbital (FPLO)~\cite{Koepernik1994}
basis set and generalized gradient approximation
(GGA)~\cite{Perdew1996} exchange correlation functional, accounting
for the strong correlations on the Mn $3d$ orbitals by a
GGA+$U$~\cite{Liechtenstein95} correction. The Hunds rule coupling for
Mn $3d$ was fixed at $J_{\rm H}=0.76$~eV~~\cite{Mizokawa1996}. The result of
fitting the DFT total energies against the Heisenberg Hamiltonian
\begin{equation}
\mathcal{\hat{H}}=\sum_{k=1}^4\sum_{\langle ij\rangle_k}J_k\mathbf{\hat{S}}_i\cdot\mathbf{\hat{S}}_j\,,\label{ham}
\end{equation}
where $\langle ij\rangle_k$ denotes pairs of $k$th neighbor sites on the diamond lattice, is shown in Fig.~\ref{fig1}(b) and Table~\ref{tab:DFT} for five values of the interaction strength $U$. Note that each pair of sites in the summation of Eq.~(\ref{ham}) is accounted for only once, i.e., we adopt the convention of single counting of bonds. As explained below, the value of $U$ is fixed by the experimentally observed Curie-Weiss temperature $\Theta_{\text{CW}}$.
The spin Hamiltonian from DFT is taken as an input for the PFFRG method~\cite{reuther10}. To treat this model within standard many-body techniques, the PFFRG first expresses the spin operators in terms of Abrikosov pseudofermions~\cite{abrikosov65}. The implementation of the local spin\textendash5/2 moments is performed as described in Ref.~\cite{baez17} where multiple copies of spin\textendash1/2 degrees of freedom effectively realize spins with larger magnitudes. The resulting fermionic Hamiltonian is then investigated using the well-developed FRG method~\cite{metzner12,platt13}, which calculates the evolution of $m$-particle vertices as a function of an RG parameter $\Lambda$. Effectively, the vertex flow takes into account leading diagrammatic contributions in $1/S$~\cite{baez17} and $1/N$~\cite{buessen17,roscher17,Rueck-2018}, such that classical spin correlations and quantum fluctuations (described in large $S$ and large $N$ approaches, respectively) are both faithfully captured. After its initial development in two dimensions~\cite{reuther10}, the PFFRG was further refined and applied to various models of frustrated magnetism including multilayer, and, eventually, three-dimensional magnets~\cite{Balz16,iqbal16,iqbal15,reuther11,reuther11_2,reuther11_3,singh12,reuther14,suttner14,iqbal16_2,iqbal16_3,buessen16,Iqbal2017,buessen17,roscher17,baez17,buessen18,keles18,Chillal-2017,Iqbal-2018,Hering-2018}. The finite-size approximation in the PFFRG amounts to limiting the real-space distance of spin correlations, which in our calculations extends over $12$ nearest-neighbor lattice spacings, corresponding to a correlation volume of $1963$ sites. Likewise, the continuous frequency arguments of the vertex functions are approximated by a discrete set of $64$ frequencies. The central physical quantity studied within the PFFRG is the static (zero-frequency) momentum-resolved susceptibility (or spin structure factor) which can be directly compared with experimental neutron scattering data.
\section{Model Hamiltonian and classical considerations}\label{sec:mm}
We first discuss the exchange couplings $J_k$ in Eq.~(\ref{ham}) determined
from DFT. As shown in Fig.~\ref{fig1}(b), DFT calculates these couplings as a function of the Hubbard onsite interaction $U$. Upon increasing $U$, all couplings decrease but their ratios remain relatively constant. The actual size of $U$ is determined via the known Curie-Weiss temperature $\Theta_{\text{CW}}=-\frac{S(S+1)}{3k_\text{B}}\sum_{k=1}^4 z_k J_k=-23$ K~\cite{fritsch04} (where $z_k$ is the coordination number of the $k$th neighbor bonds). This condition is best fulfilled for $U\approx4.5$ eV, yielding three significant couplings $J_1=-0.378$ K, $J_2=0.621$ K, $J_3=0.217$ K, and $J_4=0.0015$ K. Since $J_4$ is more than an order of magnitude smaller than all other couplings it will be neglected in the ensuing analysis. The small absolute values of the exchange couplings can be understood from the fact that in the diamond lattice of MnSc$_{2}$S$_{4}$ even the nearest-neighbor exchange couplings $J_{1}$ are mediated via rather long Mn-S-Sc-S-Mn superexchange paths. While the exchange couplings of $<1$~K are small, importantly the energy differences that need to be resolved within DFT are not: due to the spin\textendash$5/2$ moments, the energies for the different spin configurations vary in a window of $20$~meV, which is an energy scale that can be comfortably resolved by our highly converged all electron full potential DFT calculations.
\begin{figure*}
\includegraphics[width=1.0\linewidth]{fig3.pdf}
\caption{The evolution of the spin susceptibility profile in the $q_{x}$\textendash$q_{y}$ plane with temperature for the Heisenberg Hamiltonian of MnSc$_2$S$_4$ as determined from DFT. (a)\textendash(e) Evaluated for a $J_{1}$\textendash$J_{2}$ {\it only} model with $J_{2}/|J_{1}|=1.64$ and $J_{3}=0$; (f)\textendash(j) evaluated for the full $J_{1}$\textendash$J_{2}$\textendash$J_{3}$ Hamiltonian with $J_{2}/|J_{1}|=1.64$ and $J_{3}/|J_{1}|=0.57$. The temperatures are expressed in units of the critical (ordering) temperature $T_{\rm c}^{J_{3}}$ of the full model Hamiltonian with $J_{3}/|J_{1}|=0.57$. Note that in both models, at each temperature, we have rescaled the susceptibility so as to make the minimum and maximum plotted values lie between 0 and 1, which makes prominent the important features characterizing the spiral spin liquid. The absolute values of the maxima can be read off from the temperature evolution of the susceptibility shown in Fig.~\ref{fig10} [see black curve for system size L $=12$]. For each of the above profiles, the variation of the susceptibility along the radial $(q,q,0)$ direction is shown in Fig.~\ref{fig7}.} \label{fig3}
\end{figure*}
The DFT couplings might first appear unexpected because the ratios $\frac{J_2}{|J_1|}=1.64$ and $\frac{J_3}{|J_1|}=0.57$ are considerably larger compared to the values $\frac{J_2}{|J_1|}\approx0.85$ and $\frac{J_3}{|J_1|}\lesssim0.1$ proposed earlier (see Refs.~\cite{bergmann07,lee08}, respectively). These values were obtained from matching calculated and measured inelastic neutron scattering spectra under the assumption that $J_{3}$ is negligible~\cite{gao16}. However, in materials featuring a number of competing interactions, fitting methods are known to be ambiguous (see e.g., Refs.~\cite{Jeschke-2011,Janson-2016}), and thus, DFT based methods provide an important complementary path towards extraction of couplings allowing for an identification of the relevant Hamiltonian. Indeed, our DFT results reproduce the sign of the nearest- and next-nearest-neighbor exchange couplings of MnSc$_2$S$_4$ proposed earlier~\cite{bergmann07}, furthermore, they refine the previous picture by highlighting the presence of significant $J_{3}$ couplings, which considerably alters our understanding of the mechanism leading to the stabilization of a spiral spin liquid.
To shed further light on the physical implications of these new couplings, we first treat Eq.~(\ref{ham}) in the classical limit, employing the Luttinger-Tisza method~\cite{luttinger46,luttinger51}. This method aims at calculating the ground state of the corresponding classical Heisenberg Hamiltonian by minimizing the energy given by Eq.~(\ref{ham}), and does so by relaxing the spins' length constraint at each site, however, on the diamond-lattice geometry this soft-spin approach even becomes exact (see Appendix~\ref{sec:LT}). Ignoring $J_3$ for a moment, the $J_1$\textendash$J_2$ only model with $\frac{J_2}{|J_1|}=1.64$ exhibits a spiral surface in momentum space [see Fig.~\ref{fig2}(a)], which cuts through the first Brillouin-zone boundary [see Fig.~\ref{fig2}(b)]. This surface is slightly larger than the one for $\frac{J_2}{|J_1|}=0.85$, where the latter ratio has been determined in Ref.~\cite{bergmann07} to match the measured magnetic Bragg peak position $\mathbf{q}\approx2\pi(0.75,0.75,0)$ for $J_3=0$. Although the spiral surface only undergoes a moderate increase between $\frac{J_2}{|J_1|}=0.85$ and $\frac{J_2}{|J_1|}=1.64$, the DFT couplings first seem to overestimate the ordering wave vector even when the finite Bragg-peak width is taken into account [see Fig.~\ref{fig2}(c)]. The situation changes when $J_3$ couplings are considered. Already an infinitesimally small $J_3$ lifts the degeneracy and selects spirals with $\mathbf{q}=(q,q,0)$ along the surface. For larger (antiferromagnetic) $J_3$ this Bragg-peak position moves inwards in $\mathbf{q}$ space. As shown in Figs.~\ref{fig2}(b) and \ref{fig2}(c), the third neighbor coupling $\frac{J_3}{|J_1|}=0.57$ from DFT indeed shifts the Bragg peak back to $\mathbf{q}=2\pi(0.73,0.73,0)$, in very good agreement with the measured position. As discussed in Ref.~\cite{lee08}, small remaining discrepancies might disappear when incommensurate/commensurate ``lock-in'' transitions are considered.
\section{PFFRG results}\label{sec:PFFRG}
Having argued that our model parameters are generally compatible with the experimental findings, we next investigate to what extent the strong $J_3$ coupling together with thermal and quantum fluctuations destabilize the spiral spin liquid. To this end, we first calculate the spin-structure factor via PFFRG for $\frac{J_2}{|J_1|}=1.64$ and $J_3=0$, where only the effects of thermal and quantum fluctuations lift the spiral degeneracy [see Figs.~\ref{fig3}(a)\textendash\ref{fig3}(e)], and then compare with $\frac{J_3}{|J_1|}=0.57$, to study the influence of additional third neighbor couplings [see Figs.~\ref{fig3}(f)\textendash\ref{fig3}(j), and Figs.~\ref{fig8} and~\ref{fig9}]. In both cases, the spin-structure factor is investigated as a function of the RG parameter $\Lambda$ which has been argued to mimic finite temperatures $T$~\cite{reuther11,iqbal16,buessen16}. Indeed, the conversion factor between the RG scale $\Lambda$ and temperature $T$ evaluates to $\frac{T}{J}=\Big{(}\frac{2\pi S(S+1)}{3}\Big{)}\frac{\Lambda}{J}$. This is determined by comparing the limit of PFFRG where only the RPA diagrams contribute, i.e., a mean-field description, and the conventional spin mean-field theory formulated in terms of temperature $T$ instead of $\Lambda$~\cite{iqbal16,baez17}.
\begin{figure}[t]
\includegraphics[width=1\linewidth]{fig4.pdf}
\caption{The variation with temperature of the peak-position $q$ of the susceptibility maximum along the radial $(q,q,0)$ direction for the model Hamiltonian with $\frac{J_{2}}{|J_{1}|}=1.64$ and (a) $J_{3}=0$ and (b) $J_{3}=0.57|J_{1}|$.}\label{fig4}
\end{figure}
\begin{figure}[b]
\includegraphics[width=1.0\linewidth]{fig5.pdf}
\caption{The ratio of the susceptibility maxima along the $(q,q,0)$ and $(q,0,0)$ directions shown as a function of temperature $T$. Note that this ratio does not need to diverge when approaching criticality. (Inset) Temperature evolution of the width of the spiral surface along the $(q,q,0)$ direction. The width is defined as the difference of the $q$ values for the maxima and half-maxima of the susceptibility, respectively.}\label{fig5}
\end{figure}
For $J_3=0$ and at the critical RG scale (which corresponds to $\Lambda_{\rm c}^{0}=0.83(1)|J_1|$), the PFFRG detects a sharp spiral contour of strong intensities. Along finite segments centered around $(q,q,0)$ we find somewhat larger (and nearly constant) responses; however, this modulation quickly disappears with increasing temperature (i.e., RG scale $\Lambda$) such that an almost perfect spiral surface appears. Interestingly, the size and shape of the spiral surface remains nearly constant as a function of temperature [see Fig.~\ref{fig4}(a)] while its width increases considerably (see inset of Fig.~\ref{fig5}). Note that due to the missing $J_3$ coupling in Fig.~\ref{fig4}(a) the calculated maximum position $q$ is considerably larger than the experimentally measured wave vector $2\pi(0.75,0.75,0)$ [see also Figs.~\ref{fig2}(b) and \ref{fig2}(c)]; however, the inclusion of a third neighbor coupling $J_3=0.57|J_{1}|$ shifts the peaks to a position very close to the measured value, as discussed below. A more quantitative measure for the intactness of the spiral surface is shown in Fig.~\ref{fig5}, where the ratio of the intensity maximum along the $(q,q,0)$ and along the $(q,0,0)$ direction is plotted. For $J_{3}=0$, this quantity approaches unity, i.e., $\chi(q,q,0)/\chi(q,0,0)\approx1$ at around $\Lambda\approx2 \Lambda_{\rm c}^0$, indicating that the spiral surface quickly recovers. We also note that, compared to our classical Luttinger-Tisza analysis, the location of the spiral surface does not undergo any noticeable changes when including quantum fluctuations.
Switching on the third neighbor coupling $\frac{J_3}{|J_1|}=0.57$ induces a much stronger spiral-selection effect. At criticality, we observe pronounced peaks at $\mathbf{q}=2\pi(0.719,0.719,0)$ [see Fig.~\ref{fig3}(f)], which are found to be shifted slightly inwards compared to the classical wave-vector position $\mathbf{q}=2\pi(0.727,0.727,0)$. The critical RG scale $\Lambda_{\rm c}^{J_{3}}=0.99(1)|J_1|$ is slightly larger compared to the one for $J_3=0$, indicating that third neighbor interactions {\it reduce} the frustration (see Figs.~\ref{fig8} and~\ref{fig9} for a general trend with $J_{3}$). With increasing temperature, the response again becomes more evenly distributed along the spiral surface (see Fig.~\ref{fig5}); however, this intensity smearing occurs more slowly than for $J_3=0$ (see Fig.~\ref{fig8} for susceptibility plots corresponding to different values of $J_{3}$ for fixed $J_{2}/|J_{1}|=1.64$, and Fig.~\ref{fig9} for results with different $J_{3}$ with fixed $J_{2}/|J_{1}|=0.85$~\cite{bergmann07}).
We now highlight a number of features of our susceptibility data which enable us to establish the existence, stability, and extent of the spiral spin liquid. First, and foremost, a spiral spin liquid is expected to display a near uniform distribution of the susceptibility along a ring-like pattern. To this end, we plot in Fig.~\ref{fig5} the ratio of the susceptibility maxima along $(q,q,0)$ and $(q,0,0)$ directions as a function of temperature. We see that while the ratio starts with a large (diverging) value at $\Lambda_{\rm c}^{J_{3}}$, it slowly converges towards 1. Indeed, at around $\Lambda\approx3\Lambda_{\rm c}^{J_{3}}$ we observe the beginning of a temperature regime where the surface appears relatively intact (note that this temperature reflects a smooth crossover and not a sharp transition). Second, the width of the spiral surface is also seen to decrease upon inclusion of a $J_{3}$ coupling (see inset of Fig.~\ref{fig5}), implying that the response is concentrated within a narrower stripe around the spiral surface compared to the case with $J_{3}=0$, leading to a well-defined and ``intact'' spiral spin liquid. Third, we can obtain a rough estimate for the upper crossover temperature into the spiral spin liquid regime, defined as the temperature where the width of the peaks in the $(q,q,0)$ direction (as shown in the inset of Fig.~\ref{fig5}) equals their separation [where the separation refers to the two peaks which are approximately located at $2\pi(0.75,0.75,0)$ and $2\pi(1.25,1.25,0)$]. Below this temperature, individual spiral surfaces are clearly discernible, which is an important requirement for the realization of a stable spiral spin liquid. For $J_3=0.57|J_1|$ this crossover temperature is roughly given by $T_{\rm crossover} \approx 5 T_\text{c}^{J_3}$, while for $J_3=0$ we find $T_{\rm crossover} \approx 3 T_\text{c}^{J_3}$. These results, taken together, lead to the following approximate phase diagram: (i) Starting from the low-temperature regime, we have for $T/T_{\rm c}^{J_{3}}\leqslant 1$ long-range spiral magnetic order. (ii) For $1<T/T_{\rm c}^{J_{3}}\lesssim3$, we see fingerprints of a ``molten'' spiral order wherein the spectral weight remains concentrated around the ordering wave vectors of the parent spiral order, but the phase is not magnetically long-range ordered. (iii) In the interval $3\lesssim T/T_{\rm c}^{J_{3}}\lesssim5$ we find that not only is the spectral weight nearly uniformly distributed along a spiral surface but also the individual classical spiral spin surfaces are clearly discernible, and the system thus approximately realizes a stable spiral spin liquid. It is worth noting that the temperatures in this window are still much smaller compared to the absolute value of the Curie-Weiss temperature $|\Theta_{\rm CW}|=23$ K. (iv) At higher temperatures $T/T_{\rm c}^{J_{3}}\gtrsim5$, the different spiral surfaces start merging, being no longer individually distinguishable, and the spiral spin liquid becomes unstable towards a high-temperature paramagnet. Most importantly, our PFFRG results indicate that, in a temperature interval of around three to five times
the ordering temperature of $T_{\rm c}=2.3$ K, MnSc$_2$S$_4$ indeed realizes an approximate spiral spin liquid.
\begin{figure}[t]
\includegraphics[width=1.0\linewidth]{fig6.pdf}
\caption{Calculated (a) and measured (b) spin-structure factors in the $q_x$\textendash$q_y$ plane for $T/T_{\rm c}^{J_{3}}=1.33$ and $T=2.9$ K, respectively [(b) has been reproduced from Ref.~\cite{gao16}]. The calculated susceptibilities from PFFRG are given in units of $1/|J_{1}|$, while the experimental data are shown in arbitrary units.} \label{fig6}
\end{figure}
\begin{figure}[b]
\includegraphics[width=1\linewidth]{fig7.pdf}
\caption{The variation of the susceptibility along the radial $(q,q,0)$ direction at different temperatures for a model Hamiltonian with $J_{2}/|J_{1}|=1.64$ and (a) $J_{3}=0$ and (b) $J_{3}=0.57|J_{1}|$.}\label{fig7}
\end{figure}
Finally, to directly assess the quality of our simulations, we compare the measured spin structure-factor at $T=2.9$ K $=1.33T_{\rm c}$~\cite{gao16} with the PFFRG result for the full DFT model at the same RG-scale ratio $\Lambda=1.33\Lambda_{\rm c}^{J_{3}}$. For a proper comparison between theory and experiment, one has to take into consideration the extended orbital structure of the Mn$^{2+}$ magnetic moments as probed by neutron scattering wherein the measured spin structure factor is modulated by a $|\mathbf{q}|$-dependent function \textemdash the so called magnetic form factor~\cite{brown} \textemdash which describes the scattering from single moments (note that the susceptibility profiles in Fig.~\ref{fig3} assume point-like magnetic moments). The magnetic form factor is given by a sum of Gaussian curves with coefficients that can be found in Ref.~\cite{brown}. We have therefore multiplied our PFFRG result with the magnetic form factor of Mn$^{2+}$ ions which leads to a slight decrease of the intensity with increasing $|\mathbf{q}|$. The corresponding susceptibility profile is presented in Fig.~\ref{fig6}. As can be seen, the measured intensity modulation and, in particular, the spiral selection (which is still pronounced at these temperatures) is nicely reproduced by our calculations.
\section{Discussions and conclusions}\label{sec:discussions}
By combining {\it ab initio} DFT and PFFRG calculations we have shown that close to criticality the magnetic ordering process of MnSc$_2$S$_4$ is dominated by a pronounced $(q,q,0)$ spiral selection due to strong $J_3$ couplings, i.e., $J_{3}/|J_{1}|=0.57$, which are significantly larger than previously assumed~\cite{bergmann07}. Yet, as temperature increases, thermal fluctuations largely restore the spiral surface such that an approximate version of a spiral spin liquid is realized at around three to five times the ordering temperature. Interestingly, we find that the $J_3$ coupling is not entirely detrimental to a spiral spin liquid, since the selection induced by such interactions is accompanied by a reduction of the spiral surface's width.
While the Heisenberg couplings considered here determine the momentum structure of the spin correlations, they leave the plane of spiral rotation undetermined. This remaining degeneracy may be further lifted by anisotropic interactions such as dipolar couplings~\cite{lee08,gao16}. However, with a magnitude of a few percent of $J_1$ (Ref.~\cite{gao16} gives an estimate of $\sim0.026$ K on nearest-neighbor bonds) we expect dipolar interactions to become relevant only very close to the ordering transition. On the other hand, {\it below} criticality such couplings might be crucial for explaining the measured multistep ordering process involving sinusoidal collinear, incommensurate, and helical spin orders~\cite{gao16}. Since the PFFRG in its current formulation does not explicitly take into account spontaneous symmetry breaking, an analysis of such phases goes beyond the scope of the present work.
\begin{figure*}
\includegraphics[width=0.95\linewidth]{fig8.pdf}
\caption{The evolution of the spin susceptibility profile in the $q_{x}$\textendash$q_{y}$ plane with temperature for the $J_{1}$\textendash$J_{2}$\textendash$J_{3}$ Heisenberg Hamiltonian for different ratios of $J_{3}/|J_{1}|$ (different rows) keeping fixed the ratio $J_{2}/|J_{1}|=1.64$. The ratio of $J_{3}/|J_{1}|=0.57$ corresponds to the DFT model parameters of MnSc$_2$S$_4$. The temperatures are expressed in units of the critical (ordering) temperature $T_{\rm c}^{J_{3}}$ of the model Hamiltonian with $J_{3}/|J_{1}|=0.57$. Note that at each temperature we have rescaled the susceptibility so as to make the minimum and maximum plotted values lie between 0 and 1. Corresponding to the $J_{3}=0$ and $J_{3}=0.57|J_{1}|$ profiles, the variation of the susceptibility along the radial $(q,q,0)$ direction is shown in Fig.~\ref{fig7}.}\label{fig8}
\end{figure*}
\begin{figure*}
\includegraphics[width=0.95\linewidth]{fig9.pdf}
\caption{The evolution of the spin susceptibility profile in the $q_{x}$\textendash$q_{y}$ plane with temperature for the $J_{1}$\textendash$J_{2}$\textendash$J_{3}$ Heisenberg Hamiltonian for different ratios of $J_{3}/|J_{1}|$ (different rows) keeping fixed the ratio $J_{2}/|J_{1}|=0.85$. The value of $J_{3}=0$ (first row) corresponds to an estimation of the model parameters of MnSc$_2$S$_4$ as previously determined from the radius of the spiral surface in Ref.~\cite{bergmann07}. The temperatures are expressed in units of the critical (ordering) temperature $T_{\rm c}^{J_{3}}$ of the DFT model Hamiltonian of MnSc$_2$S$_4$ with $J_{2}/|J_{1}|=1.64$ and $J_{3}/|J_{1}|=0.57$. Note that at each temperature we have rescaled the susceptibility so as to make the minimum and maximum plotted values lie between 0 and 1.}\label{fig9}
\end{figure*}
\section{Acknowledgments}
We gratefully acknowledge the Gauss Centre for Supercomputing e.V. for funding this project by providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (LRZ). The work in W\"urzburg was supported by ERC-StG-Thomale-336012, DFG-SFB 1170, and DFG-SPP 1666. J.R. is supported by the Freie Universit\"at Berlin within the Excellence Initiative of the German Research Foundation. Y.I. acknowledges the kind hospitality of the Helmholtz-Zentrum f\"ur Materialien und Energie, Berlin, where part of this work was accomplished.
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Understanding How the Brain Processes Music through the Bach Trio Sonatas
The composer Varèse defined music as "organized sound," yet much remains to be learned about how sound is organized to make music. This project takes a "big data" approach to understanding the patterns and principles of music with a focus on the Bach Trio Sonatas for organ. Bach is the go-to composer for understanding the fundamentals of music. His Trio Sonatas have been important pedagogical tools from Bach's time to today in teaching performance, improvisation and composition. The organ is the ideal instrument for performance digitization since the performer only controls the attack and release of each note: once a note is started, pitch and timbre remain the same.
The team will develop a library of digitized performances of the Bach trio sonatas by the two Principal Investigators and students and faculty in the U-M Organ Department. From this dataset, the team will study harmony, counterpoint and music structure from a data science perspective. They will also mathematically compare different performances to determine features that make performances artistic, as well as common mistakes students and performers make with these pieces. Some of the research questions include: What mathematical representations reveal interesting patterns and features in these works? Can we identify patterns in these works that can be used to help guide future compositions? How can we use network theory to represent the chord progressions in these works? How can we quantify common features and differences between performances? Faculty members of the U-M Organ Department will review and validate the findings and suggest hypotheses to test by the analysis team, for example, particularly tricky sections of the trio sonatas, common ornaments, or other interesting facts about the trio sonatas. The digitized trio sonatas will be shared with U-M researchers and will enable research and pedagogy in many disciplines, including data science, music performance, mathematics and music psychology.
Interested readers can find audio recordings of the Trio Sonatas performed by Dr. Kibbie: http://www.blockmrecords.org/bach/kibbie.htm
Audio: The team's creation of Hail to the Victors with chords in the style of Bach's Trio Sonatas.
http://midas.umich.edu/wp-content/uploads/sites/3/2018/07/hail.m4a
Daniel Forger, co-Principal Investigator, Professor, Mathematics, College of Literature, Sciences and the Arts
James Kibbie, co-Principal Investigator, Professor and Chair, Organ, School of Music, Theatre and Dance
A network graph of chord progressions in Bach's Trio Sonatas
https://midas.umich.edu/wp-content/uploads/sites/3/2017/12/music-bg-light.png
Understanding and Mining Patterns of Audience Engagement and Creative Collaboration in Largescale Crowdsourced Music Performances
/* ----------------------------------------- */ /* Content Template: Template for MIDAS Funded Projects - start */ /* ----------------------------------------- */ .post-area { width: 100% !important; } .container-wrap { padding-top: 0 !important; } .header-image-data { display: none; } .heading-title.hentry { display: none; } .sidebox { border-radius: 1rem; overflow: hidden; } p { font-size: 14pt; line-height: 1.33; } /* ----------------------------------------- */ /* Content Template: Template for MIDAS Funded Projects - end */ /* ----------------------------------------- */
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The next EuroDIG will take place from 9-10 June 2016 in the heart of Brussels, at the Square-Meeting centre. EuroDIG 2016 is hosted by EURid in cooperation with the European Commission.
Submit your proposal now: Until the 31 December 2015 the EuroDIG Secretariat is looking forward to receive proposals on issues and topics for the EuroDIG 2016 programme. Each submission will only take 3 min, since we are not asking for session proposals but only for possible topics to be discussed next year. This is a joint call of SEEDIG* and EuroDIG 2016.
Planning meeting on 26th of January 2016 in Brussels. After the call for issues is finished, everyone is invited to participate in the public planning meeting at the Albert Borschette Conference Centre in Brussels. Please register and find all details here.
Messages from Sofia: During the IGF in João Pessoa we distributed the Messages from Sofia. If you like to get some copies for distribution among your networks please send us an email. You can also find all reports, pictures and (some) videos at the EuroDIG Wiki.
* Launched in June 2015 the South Eastern European Dialogue on Internet Governance (SEEDIG) moves forward as a sub-regional IGF initiative, and will be reconvened on 22 April 2016 for its second annual meeting in Belgrade. SEEDIG 2016 is being built as an integral part of the EuroDIG process. Thus, results of the SEEDIG discussions will be integrated into the EuroDIG session planning process and become part of the EuroDIG programme.
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\section{Observational results}
\label{section:obs_results}
\subsection{Features in PACS spectra}
\begin{figure*}
\centering
\includegraphics[width=0.98\linewidth]{specs1_2.pdf}
\caption{Continuum-subtracted PACS spectra of our sample. The vertical segments (at the top) and bars indicate the rotational transitions of $\rm{}^{12}CO$ (light gray, dotted lines), $\rm{}^{13}CO$ (dark gray), HCN (red), CS (yellow), OH (cyan), $\rm CH^{+}$ (green) and forbidden lines (blue) of [\ion{C}{II}] $157.7\,\rm \mu m$, [\ion{O}{I}] $63.2, 145.5\,\rm \mu m$, [\ion{N}{II}] $121.9\,\rm \mu m$. } \label{fig:specs}
\end{figure*}
\begin{figure*}
\centering
\ContinuedFloat
\includegraphics[width=0.98\linewidth]{specs2_2.pdf}
\caption{Continued. } \label{}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{irc10216_specs_2.pdf} \\
\includegraphics[width=\linewidth]{cont_var_2.pdf}
\caption{Line and continuum variability of IRC+10216 in the FIR. Top: Continuum subtracted spectra at different epochs (see also Fig.\,\ref{fig:irc10216_split}); the dotted lines sign the CO transitions. Bottom: Sine-wave fit to the continuum variability at two selected wavelengths for a fixed pulsation period of 630 days (solid lines) and free-period fit corresponding to 680 and 660 days for the blue and red dashed curves, respectively.} \label{fig:irc10216specs}
\end{figure*}
The PACS continuum-subtracted spectra are plotted in Fig.\,\ref{fig:specs} sorted in terms of stellar effective temperature (\teff\ increases from top to bottom) and arbitrarily scaled for better comparison of the CO spectra. The continuum was fitted using a non-parametric method after identifying the line-free regions of the spectrum for each target. A brief line-detection statistics summary is given in Table\,\ref{tab:lines}. Broad emission or absorption features due to instrumental artifacts are occasionally visible in some of the targets (e.g. near 62\,$\rm \mu m$). The spectra of IRC+10216 taken at seven different epochs are shown in the top panel of Fig.\,\ref{fig:irc10216specs}, and in more detail in Fig.\,\ref{fig:irc10216_split} in the Appendix.
Straightaway we see significant differences between the spectra of these targets such as the larger line density, resulting from a richer molecular content, in AGB stars compared to post-AGBs and yPNe as expected. In AGB stars the strongest spectral lines are due to CO emission,
while post-AGBs and yPNe show much more prominent forbidden lines, except for AFGL\,2688 and AFGL\,618. Additional molecular features with intense emission are attributed to rotational transitions of common C-bearing species such as $^{13}$CO, HCN, CS, etc.
Not so common in C-rich stars is the presence of OH, yet we identify several OH doublet lines (at 79, 84, 119 and 163\,$\mu$m) in all yPNe and in the post-AGB star IRAS\,16594-4656, with the highest \teff$\sim$10\,000K in its class and where the presence of shocks is plausible. Two additional OH doublets at 65 and 71\,$\mu$m, with high-excitation energy of about 500 and 600\,K, respectively, are also observed in IRAS\,16594-4656 and in the yPNe Hen\,2-113.
In those five (\teff$>$10\,000\,K) targets, we also identify pure
rotational transitions of CH$^+$, with the $J$=3-2 line at 119.86\,$\mu$m being one of the most clearly detected and best isolated lines\footnote{The lines of CH$^+$ covered by PACS are $J$=2-1, 3-2, 4-3, 5-4 and 6-5 at 179.61, 119.86, 90.02, 72.15 and 60.26\,$\rm \mu m$, respectively.}. The 179.61\,$\mu$m CH$^+$ line could be blended with the water 2$_{12}$-1$_{01}$
transition. The highest-$J$ lines of CH$^+$ are also present in IRAS\,16594-4656 and Hen\,2-113.
Interestingly, we do not see CH$^+$ in the spectrum of the Red Rectangle, where sharp emission features near 4225\,\AA\ were discovered and assigned to this ion by \cite{1992A&A...261L..25B}.
The detection of CH$^+$ in AFGL\,618 and the more evolved PNe NGC\,7027 had been previously reported by \citet[][{\it Herschel}/SPIRE]{2010A&A...518L.144W} and \citet[][{\sl ISO} data]{1538-4357-483-1-L65}. As we see below in this section, the targets with CH$^+$ and OH emission also show the strongest atomic/ionic fine-structure lines.
We identify lines due to $\rm H_{2}O$ (orto- and para-) transitions in all AGBs, in the post-AGB AFGL\,2688 and in the yPNe AFGL\,618. The study of water lines in the C-rich AGB stars of our sample (except for AFGL\,2513) was already conducted by \citet{2016A&A...588A.124L} who suggested that both shocks and UV photodissociation may play a role in warm $\rm H_{2}O$ formation. Water lines had also been previously identified in the SPIRE spectrum of AFGL\,618 and AFGL\,2688 \citet{2010A&A...518L.144W}.
The fine-structure lines [\ion{O}{I}] at 63.18\,$\rm \mu m$ and 145.53\,$\rm \mu m$ and [\ion{C}{II}] at 157.74\,$\rm \mu m$ are very prominent in the spectra of all post-AGBs and yPNe in our sample with the exception of AFGL\,2688. The post-AGB object IRAS\,16594-4656 demonstrates clear signs of ionization since it shows a [\ion{C}{II}] line even stronger than that of AFGL\,618, which has a hotter central star (Table B.1). Weak [\ion{N}{II}] 121.89\,$\rm \mu m$ emission is also detected in Hen\,2-113 and IRAS\,21282+5050, and, tentatively in IRAS\,16594-4656 and HD\,44179.
The 69\,$\rm \mu m$ band of forsterite (Mg$_2$SiO$_4$) is visible in \,CPD-$56^{\circ}8032$, Hen\,2-113, and HD\,44179,
as already reported by \citet{2002MNRAS.332..879C} and \citet{2014A&A...565A.109B}. The presence of O-rich dust in such C-rich objects corroborates a mixed chemistry nature.
\subsection{Line flux variability in IRC+10216}
Variability of the continuum at optical and IR wavelengths due to stellar pulsations is a common property of AGB stars. We present PACS spectra at multiple epochs of the Mira-type C-rich star IRC+10216 displaying periodic variations both in the continuum and in the lines (see Fig.\,\ref{fig:irc10216specs} and Fig.\,\ref{fig:irc10216_split}).
\citet{2041-8205-796-1-L21} first reported on the discovery of strong intensity variations in high-excitation lines of abundant molecular species towards IRC+10216 using {\it Herschel}/HIFI and IRAM 30m data. Line variability was attributed to periodic changes in the IR pumping rates and also possibly in the dust and gas temperatures in the innermost layers of the CSE. From the analysis of a 3 yr-long monitoring of the
molecular emission of IRC+10216 with {\it Herschel}~(including HIFI, SPIRE and PACS data), \citet{2015ASPC..497...43T} concluded that intensity changes of CO lines with rotational numbers up to $J$=18 are within the typical instrument calibration uncertainties, but in the higher PACS frequency range ($J\geq28$), line strength variations of a factor
$\ga$1.6, and scaling with $J$, were found. More recently, \cite{2017ApJ...845...38H} reported 5\%-30\% intensity variability of additional mm lines with periodicities in the range 450-1180 days.
In section \ref{section:irc10216_var} we study in greater detail what is the impact of CO line variability on the estimate of \mbox{$T_{\rm rot}$}\ and \mass\ from the rotational diagram analysis using PACS data for different epochs.
\subsection{CO line fluxes}
\label{section:fluxes}
We now focus on the purely rotational spectrum of CO in the ground vibrational state ($v$=0), which we use to study the physical properties of the warm regions of the molecular CSEs of our targets. In the PACS range one can potentially find very high CO rotational transitions, from $J=14-13$ ($E_{\rm u}$$\sim$581\,K) to $J=45-44$ ($E_{\rm u}$$\sim$5688\,K). The lack of data between 95 and 101 $\rm\mu m$ means we cannot detect the transitions $J$=26-25 and $J$=27-26. In the case of IRC\,+10216, the much larger gap in the PACS coverage ($\sim$95-140\,$\mu$m, Fig.\,\ref{fig:irc10216specs}) prevents detection of CO transitions with upper-level rotational number between $J_{\rm u}$=19 and $J_{\rm u}$=27.
The line fluxes are given in Table\,\ref{tab:fluxes}. The quoted uncertainties correspond to the propagated statistical errors and do not contain absolute flux calibration errors (typically of $\sim$15\%-20\%) or underlying continuum subtraction uncertainties.
As expected, the resolving power of PACS ($\sim$80-300\,\mbox{km~s$^{-1}$}) does not allow to spectrally resolve the CO profiles. This is true not only for AGB CSEs with full-widths-at-half-maximum (FWHM) similar to or smaller than the terminal expansion velocity of the envelopes (FWHM$\sim$10-25\,\mbox{km~s$^{-1}$}, Table \ref{tab:2}), but also for post-AGB objects and yPNe, even in targets that are known to have fast ($\approx$100\,\mbox{km~s$^{-1}$}) molecular outflows like AFGL\,618 \citep[e.g.][]{2010A&A...521L...3B}. For this reason we measured CO fluxes by simply fitting a Gaussian function to the PACS lines.
Due to insufficient spectral resolution some reported line fluxes are affected by line blend. These are identified with asterisks in the tables and figures. Some of the well-known line blends are CO\,$J$=30-29 with HCN\,$J$=39-38 and CO\,$J$=20-19 with HCN\,$J$=26-25 at 87.2 and 130.4\,$\mu$m, respectively. Also, the CO transitions $J$=21-20 and $J$=22-21 are blended with $^{13}$CO\ $J$=22-21 and $J$=23-22 at $124.2~\rm \mu m$ and $118.6~\rm \mu m$, respectively.
Figure\,\ref{fig:line_cont} compares the integrated flux of the CO\,($J$=15-14) line ($F_{\rm CO\,(15-14)}$) with the IRAS 100\,\micron\ flux (IRAS$_{\rm 100}$), the PACS continuum at 170\,\micron\ (PACS$_{\rm 170}$), i.e.\, near the CO\,($J$=15-14) line, and the [12]-[25] IRAS color. The $J$=15-14 transition is a strong, non-blended line, detected towards all of our targets, and it was also used by \citet{2016A&A...588A.124L} who analyzed PACS data of most of the AGBs here studied, which facilitates comparison.
As for the O-rich sample (paper I), there is a positive correlation between the CO line strength and the IRAS$_{100}$ and PACS$_{170}$ continuum fluxes. The correlation between the CO\,$J$=1-0 line and the IRAS fluxes of evolved stars of various chemical types had been previously reported \citep{1987A&A...183L..13O,1988A&A...196L...1O,1992A&A...257..701B}.
We also confirm the anticorrelation between the line-to-continuum ($F_{\rm CO\ 15-14}$/PACS$_{\rm 170}$) ratio and the IRAS [12]-[25] color (both distance-independent) for the AGB stars, which was noted in Paper I. For the more evolved targets the trend is not so obvious, but the sample size is small. We also see that, in general, the ratio between the molecular emission and the dust emission is higher in less evolved objects than in the most evolved ones, which could be partially attributed to more prominent CO photodissociation as the objects evolves along the AGB-to-PNe track. AFGL 618 and IRAS\,16594-4656 are two clear outliers in this relation since they show a line-to-continuum emission ratio as large as that of the AGB class. The Red Rectangle (HD\,44179) is well isolated in all the panels due to its comparatively weak CO emission and low CO-to-dust ratio. This surely reflects the different nature of this object with respect to the rest of post-AGB and yPNe in our sample, which is well known from previous works. The Red Rectangle belongs to a special class of post-AGB objects with relatively weak CO emission coming from large ($\sim$1000-2000\,AU) circumbinary rotating disks, with very prominent IR emission by warm/hot dust in the disk, but lacking massive molecular outflows found in many other evolved stars \citep[e.g.,][and references therein]{2016A&A...593A..92B}.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{cont2_4.pdf}
\caption{Correlation between CO line flux and continuum intensities. Top: CO $J=15-14$ vs IRAS 100\,$\rm \mu m$ flux. Middle: CO $J=15-14$ vs PACS 170\,$\rm \mu m$ continuum flux. Bottom: continuum normalized CO $J=15-14$ flux vs IRAS [12]-[25] color. The symbols and colors are the same as in Fig.\,\ref{fig:IRAS}.} \label{fig:line_cont}
\end{figure}
\section{Rotational diagram analysis}
\label{section:RD}
\begin{figure*}[!htp]
\begin{tabular}{ccc}
\includegraphics[valign=t]{plot1_4.pdf} & \hspace{-0.3cm} \includegraphics[valign=t]{plot2_4.pdf} & \hspace{-0.3cm} \includegraphics[valign=t]{plot3_4.pdf}\\
\includegraphics[valign=t]{plot4_4.pdf} & \hspace{-0.3cm} \includegraphics[valign=t]{plot6_4.pdf} & \hspace{-0.3cm} \includegraphics[valign=t]{plot5_4.pdf}\\
\includegraphics[valign=t]{plot7_4.pdf} & \hspace{-0.3cm} \includegraphics[valign=t]{plot8_4.pdf} & \hspace{-0.3cm} \includegraphics[valign=t]{plot9_4.pdf}\vspace{-0.8cm} \\
\includegraphics[valign=t]{plot10_4.pdf} & \hspace{-0.3cm} \includegraphics[valign=t]{plot11_4.pdf} &
\end{tabular}
\caption{Rotational diagrams of the CO molecule. The gray line correspond to a single least-squares fit to the full range of transitions from where a rotational temperature, \mbox{$T_{\rm rot}$}, and total gas mass, \mass, is computed. The characteristic radius of the CO-emitting volume ($r_{\rm CO}$) adopted is indicated. The red and blue lines correspond to a two-component model consisting of a "warm" and "hot" region, respectively. Asterisks mark line blends that were excluded in the fit.} \label{fig:rots}
\end{figure*}
\begin{figure*}[!htp]
\ContinuedFloat
\begin{tabular}{ccc}
\includegraphics[]{plot12_3.pdf} & \hspace{-0.2cm}\includegraphics[]{plot13_3.pdf} & \hspace{-0.2cm}\includegraphics[]{plot14_3.pdf}\\
\end{tabular}
\caption{Continued.}
\end{figure*}
\begin{figure}
\includegraphics[width=\linewidth]{irc10216_RDs_3.pdf}
\caption{Rotational diagram of the CO molecule in IRC+10216 at different epochs. Analogous to Fig.\,\ref{fig:rots}.} \label{fig:irc10216_rds}
\end{figure}
Following the same approach of Paper I, we have used the well-known and widely employed rotational diagram (RD) technique \citep[e.g.][]{1999ApJ...517..209G} to obtain a first estimate of the (average) excitation temperature (\mbox{$T_{\rm rot}$}) and total mass (\mass) of the warm inner layers of the molecular envelopes of our sample.
A canonical opacity correction factor, $C_\tau$, as defined by \citet{1999ApJ...517..209G}, has been included to take into account moderate optical depth effects (\S\,\ref{appendix:opCorrection}). We refer to Paper\,I for a more detailed description of the method.
\subsection{The characteristic size of the CO-emitting layers}
\label{section:rco}
To compute the optical depth of the line and to make the corresponding \ctau\ correction, we need an estimate of the column density of CO, which is computed in a simplified manner dividing the total number of CO molecules (\mbox{$N_{\rm CO}$}) by the projected area of the CO-emitting volume on the sky. The {\sl characteristic} size of the envelope regions where the CO PACS emission is produced (\mbox{$r_{\rm CO}$}) is one of the main sources of uncertainty since, except for a few targets (see below), these high-$J$ CO-emitting layers are unresolved by PACS. Therefore, the value of \mbox{$r_{\rm CO}$}\ needs to be adopted based on several criteria. These criteria are described in much detail in paper I, Appendix B. In the following, we provide a brief summary of the general method and provide additional arguments particularized to the C-rich targets here under study.
For AGB CSEs, a first estimate of the size of the CO-emitting volume can be derived from the envelope temperature structure, $T(r)$, estimated from detailed non-LTE molecular excitation and radiative transfer (nLTEexRT) calculations in the literature. These have been done for many AGB CSEs using low-$J$ CO transitions (typically $J_{\rm u}$$\la$6), and, for a few cases, also using certain CO transitions from higher $J$ levels \citep[see e.g.,][]{2010A&A...518L.143D,2014A&A...561A...5K,2014A&A...569A..76D,2016A&A...591A..44M,2018A&A...609A..63V}. As deduced from these studies, the gas temperature is approximately 1000-2000\,K close to the dust condensation radius ($\sim$5-15 \mbox{$\rm R_{\star}$}), and decreases gradually towards the outermost layers approximately following a power-law\footnote{See also the temperature profiles for the O-rich AGBs in Fig.\,B.1 of Paper I.} of the type $\sim$1/r$^{\alpha}$, with $\alpha \sim$0.5-1.0. Such models also exist in literature for three targets in our C-rich sample: IRAS\,15194-5115, CIT\,6 and IRC+10216 \citep{1999A&A...345..841R,2002A&A...391..577S,2012A&A...539A.108D}.
According to this, the high-excitation transitions observed with PACS require relative proximity to the central star and, in particular, we anticipate that regions with a few hundred kelvin, as deduced from our RDs (Figs.\,\ref{fig:rots} and \ref{fig:irc10216_rds}), should not extend much farther than a few $10^{15}$\,cm ($\rm \la10^2\,R_{\star}$) in AGB stars.
An additional constraint on the radius can be imposed from the fact that the deepest layer traced by the observed CO emission must be such that $\tau<1$ because of the almost null escape probability from deeper, very optically thick regions (\S\,\ref{appendix:opCorrection}). For all our AGB CSEs, we have explored a range of radii around 1\ex{15}\,cm (Fig.\,\ref{fig:opacity}). We found that values around \mbox{$r_{\rm CO}$}$\sim$[1-4]\ex{15}\,cm result in line optical depths close to, but smaller than unity (typically $\tau_{J=14\raw13}\sim$0.5-0.9) that yield moderate $C_\tau$ opacity correction factors for lines with $J_{\rm u}$$<$19 and negligible for higher-$J$ transitions. For \mbox{$r_{\rm CO}$}$<$1\ex{15}\,cm, the opacity of the CO\,$J$=14-13 line, which is the optically thickest transition in our sample, becomes larger than 1 in all our targets (also including post-AGBs and yPNes).
We have checked that the range of plausible radii found for the AGB CSEs in our sample, \mbox{$r_{\rm CO}$}$\sim$[1-4]\ex{15}\,cm, is consistent with the upper limits to the size of their envelopes deduced from the PACS spectral cubes and/or photometric maps by \citet{2016MsT..........1D} and with any other information on the molecular envelope extent from the literature. IRC+10216 is the only AGB CSE that is partially resolved in the PACS cubes, and it is also the closest in our sample. The PACS cubes show a slightly extended source (with a half-intensity size 4$\sigma$ above the instrument PSF in both bands) that implies a deconvolved Gaussian radius of about 2\arcsec, that is $\sim4\times10^{15}$\,cm at $d$=150\,pc. This value is in good agreement with the lower limit to \mbox{$r_{\rm CO}$}\ needed to satisfy the $\tau_{J=14\raw13}<$1 criteria in this target, \mbox{$r_{\rm CO}$}$\sim$2\ex{15}\,cm. In this case, we then rather confidently use an intermediate value of \mbox{$r_{\rm CO}$}=3\ex{15}\,cm.
Contrary to AGBs, for post-AGBs and yPNe there are no model temperature profiles in the literature of the molecular gas in the CSEs (except for the rotating, circumbinary disk of the Red Rectangle, \S\,\ref{Section:individual}).
The range of representative radius adopted for post-AGBs and yPNe is \mbox{$r_{\rm CO}$}=[0.4-4]\ex{16}\,cm (Table\,\ref{tab:rot}) based on a moderate opacity criteria, the extent of the emission in the PACS cubes and photometric maps, and on additional information on the extent of the intermediate-to-outer molecular envelope from the literature. In particular, in AFGL\,2688, a deconvolved diameter of $\sim$4\arcsec\ in the PACS blue band suggests that \mbox{$r_{\rm CO}$}$\la1\times10^{16}$\,cm (at $d$=340\,pc). Previous CO\,$J$=2-1 mapping observations identified a compact shell of radius $\sim$2\arcsec\ ($\sim$5\ex{15}cm) around the center of the nebula \citep{2000A&A...353L..25C}. Since the CO\,$J$=14-13 emission is optically thick at \mbox{$r_{\rm CO}$}$\la$6\ex{15}\,cm (as shown in Fig.\,\ref{fig:opacity}), we adopt as representative radius an intermediate value of \mbox{$r_{\rm CO}$}=8\ex{15}\,cm. HD\,44179 (The Red Rectangle) is a point-source in the PACS photometric maps, therefore, for a distance of 710 pc, $r_{\rm CO}$ should be of the same order as in AFGL\,2688, which is roughly consistent with interferometric observations \citep[e.g.,][]{2016A&A...593A..92B}. For IRAS\,16594-4656, only a very loose upper limit to the radius of \mbox{$r_{\rm CO}$}$<$3\ex{16}\,cm is inferred from optical images and H$_2$ emission maps in this object \citep[e.g.,][]{2008ApJ...688..327H}. We explored the range, $r_{\rm CO}\sim0.8-2\times10^{16}$, similar to AFGL\,2688, and adopted as a reference value the midpoint value where $\tau_{J=14\raw13}\sim$0.7.
All yPNe in our sample are point-sources in the PACS spectral cubes, which means that the upper limit to the radius is of about
\mbox{$r_{\rm CO}$}$\la$1\arcsec$\sim$[1-2]\ex{16}\,cm for AFGL\,618, and $\sim$[4-6]\ex{16}\,cm for the rest. They are known to have central \ion{H}{ii} regions that have recently formed as the star has become progressively hotter along the PNe evolution. Because the CO envelope surrounds the ionized nebula, a lower limit to \mbox{$r_{\rm CO}$}\ can be established from the extent of the latter. Taking this into account, we set a representative radius to \mbox{$r_{\rm CO}$}=1\ex{16}\,cm for AFGL\,618
\citep{2017arXiv170401773S,2013ApJ...777...37L}, and \mbox{$r_{\rm CO}$}=4\ex{16}\,cm for the rest \citep[see e.g.,][]{2015MNRAS.449L..56D,2010A&A...523A..59C}.
Given the uncertainty in \mbox{$r_{\rm CO}$}, we have systematically explored a range of radii around optimal/plausible values of \mbox{$r_{\rm CO}$}\ to asses the impact of this parameter in our results (Fig.\,\ref{fig:opacity}). The opacity correction increases the smaller \mbox{$r_{\rm CO}$}~is, therefore the slope and y-intercept of the RD increase as a direct result of the frequency-dependence of C$_\tau$ (\S\ \ref{appendix:opCorrection}), which results in lower values of \mbox{$T_{\rm rot}$}\ and larger values of $N_{\rm CO}$, thus \mass. This also means that, in practice, only the lowest-frequency-points are affected by C$_\tau$, while the highest-frequency transitions (i.e., highest excitation energies) are unaltered regardless of the radius that we chose within the reasonable constraints that we have put.
\subsection{Non-LTE effects}
In Paper I we examine and discuss extensively the impact of non-LTE excitation effects (if present) on the values of \mbox{$T_{\rm rot}$}\ and \mass\ derived from the RDs in a sample of 26 non C-rich evolved stars with mass-loss rates in the range \mbox{$\dot{M}$}$\sim$2\ex{-7}-1\ex{-4}\,$\rm M_{\sun}\,yr^{-1}$. The C-rich targets studied here have on average larger mass-loss rates than those in Paper I, therefore, using a similar reasoning, the CO population levels are also most likely close to thermalization in the inner dense regions of the CSEs under study.
This is further supported by nLTEexRT computations of a selection of high-$J$ CO transitions observed with PACS (from $J_{\rm u}$=14 to 38) by \cite{2016A&A...588A.124L}. Their sample included all our targets except for AFGL\,2513 and IRC+10216. These authors conclude that over
a broad range of mass-loss rates (\mbox{$\dot{M}$}$\sim$10$^{-7}$-2\ex{-5}\,$\rm M_{\sun}\,yr^{-1}$) the CO molecule is predominantly excited through collisions with H$_2$, with a minor effect of FIR radiative pumping due to the dust radiation field. The role of dust-excitation on FIR CO lines was also investigated by \citet{2002A&A...391..577S} and found to be of minor importance for AGBs with typical mass-loss rates of $\sim10^{-5}\rm \,M_{\sun}\,yr^{-1}$. We assess the FIR pumping effect further in Section\,\ref{section:irc10216_var},
where we study multi-epoch RDs of IRC+10216, which is a source with well-known CO line variability.
We stress that even under non-LTE conditions, for a simple diatomic molecule like CO, the RD method provides a reliable measure of the total mass within the emitting volume. Although \mbox{$T_{\rm rot}$}\ may deviate from the kinetic temperature in regions where the local density is lower than the critical densities of the transitions considered (\nc$\sim$5\ex{5}-3\ex{6}\,\cm3, for $J_{\rm u}$=14 and 27, respectively, and \nc$\approx$10$^6$-10$^7$\,\cm3 for $J_{\rm u}$$>$27), it does describe quite precisely the molecular excitation, i.e.\, the real level population. Therefore, the total number of emitting molecules (and, thus, the mass) is quite robustly computed by adding up the populations of all levels. This is also supported by the good agreement (within uncertainties) between the mass-loss rates derived from this (and other works) using similar LTE approximations, and those obtained from nLTEexRT models (including PACS lines for a few targets) -- see paper I and Section\,\ref{section:Mloss}.
As discussed in paper I, conceivable LTE deviations would have its largest impact on the excitation temperature of the hot component, since high-$J$ levels have the lowest critical densities (see above). If this is the case, in low mass-loss rate objects, the value of \mbox{$T_{\rm rot}$}\ deduced for the hot component could deviate from the temperature of the gas and approach to that of the dust within the CO-emitting volume. We note that, in any case, the gas and dust temperatures, although not equal, are not excessively divergent in the warm envelope regions around $\sim$10$^{15}$\,cm under study \citep[e.g.,][]{2014A&A...569A..76D,2002A&A...391..577S}.
\section{Results}
\label{section:results}
\subsection{Gas temperatures and masses}
\begin{figure}[t]
\centering
\includegraphics[width=0.88\linewidth]{T_vs_M.pdf} \\ \hspace{-1.15cm}
\includegraphics[width=0.70\linewidth]{extra5.pdf}
\caption{Summary of the RD results. Top: The colored symbols correspond to the opacity corrected values (single fit) which are connected by dotted lines to the corresponding uncorrected values in gray; on the sides we show the histograms of temperature and mass of the single fits; in the case of IRC+10216 we show weighted averages of four observations. Bottom: temperature and mass of the warm and hot components connected by dashed lines for the same target (if applicable). } \label{fig:props}
\end{figure}
The opacity-corrected RDs of the CO molecule are plotted in Fig.\,\ref{fig:rots} and in Fig.\,\ref{fig:irc10216_rds}, where multi-epoch RDs for IRC+10216 are separately shown, together with the best-fit parameters using a single-temperature component (gray line) and double-temperature component (red and blue lines). For 8 out of 14 sources, the RDs include CO transitions with upper-level energies that range from $E_{\rm u}$$\sim$580\,K to $E_{\rm u}$$\la$2000-3000\,K. For 6 targets, CO transitions with upper-level energies of up to $E_{\rm u}$$\sim$5000\,K are also detected. Our temperatures for the group of AGBs are consistent with the ones reported in \citet{2018arXiv180803467N} within the uncertainties, although for many AGBs we find lower values because of the opacity correction term that we have included.
There are cases where it is clear by the analysis of residuals that a single straight line does not fit the entire range of excitation energies. That is the case of V\,Hya, IRAS\,16594-4656, AFGL\,3116, CIT\,6 and AFGL\,618.
Because it is not always clear by eye whether the slope changes and at which point it occurs, we used the Bayesian information criterion (BIC) to help us deciding where to split the diagram and to quantify significance. This is explained in the appendix \ref{appendix:multiFit} and illustrated in the supplementary Fig.\,\ref{fig:stats2}.
We provide the fitting parameters in Table \ref{tab:rot} corresponding to a single-fit and a double-component fit in targets where a single line does not equally fit all data points. As in Paper I, we call these two components "warm" and "hot" with $T_{\rm w}<T_{\rm single}<T_{\rm h}$. Their mean temperatures are $\overline{T}_{\rm w}\sim$ 400\,K and $\overline{T}_{\rm h}\sim$ 820 K respectively. The corresponding masses are $M_{\rm w}$ and $M_{\rm h}$ with the former being 4-10 times larger than the latter.
We find single-fit rotational temperatures in the range \mbox{$T_{\rm rot}$}$\sim$\,200-700\,K with some of the post-AGBs and yPNe being the targets with the coolest gas, except for the yPNe AFGL\,618 which has the largest rotational temperature in our sample similar to that of the warmest AGB CSEs.
The total number of CO molecules is in the range \mbox{$N_{\rm CO}$}$\sim10^{49}-10^{51}$, resulting in column densities of \ncol$\sim10^{16}-10^{19}$\,$\rm cm^{-2}$ for the adopted radii (Table \ref{tab:rot}). To estimate the total gas mass from CO we assumed the same fractional abundance $X_{\rm CO} = 8\times10^{-4}$ \citep[e.g.][]{2006A&A...450..167T} with respect to $\rm H_2$ for all targets. The single-fit values of the total mass of the CO-emitting volume range between \mass$\sim$3$\times$10$^{-5}$\,\mbox{$\rm M_{\sun}$}\ (V\,Cyg and HD\,44179) and $\sim$8$\times$10$^{-3}$\,\mbox{$\rm M_{\sun}$}\ (IRAS\,21282+5050), with a median value of \mass$\sim$4$\times$10$^{-4}$\,\mbox{$\rm M_{\sun}$}.
Figure\,\ref{fig:props} shows the single-fit temperature versus mass for the opacity corrected diagram (colored symbols) and uncorrected (gray symbols). The opacity-correction results in changes in \mbox{$T_{\rm rot}$}\ of 10-15\% in AGBs, and lower than 5\% in post-AGBs and yPNe.
In mass this typically corresponds to 60\% in AGBs and lower in the post-AGBs and AFGL\,618 ($<10$\% in $M_{\rm H_2}$), and negligible in the other three yPNes. Figure \ref{fig:opacity} shows that $\tau_{J=14-13}$ is close to unity but it quickly falls off with increasing $J$ meaning that $C_\tau \rightarrow 0$ for $E_{\rm u}/k > 2000$\,K, and that $M_{\rm h}$, in particular, is not underestimated by opacity effects.
As in Paper I, we find an anti-correlation between \mass\ and \mbox{$T_{\rm rot}$}, especially if we consider only the group of AGBs (Fig. \ref{fig:props}). In general, post-AGBs and yPNe have the the highest masses. One clear exception to trend is The Red Rectangle, which is one of the least massive targets (with only a few $\sim$10$^{-5}$\,\rm \mbox{$\rm M_{\odot}$~yr$^{-1}$}) in contrast to the rest of post-AGBs and yPNe. This is not surprising given the nature of this object, which is the prototype of a special class of post-AGB objects with hot rotating disks and tenuous winds very different from the massive and fast (high-momentum) outflows of standard pre-PNe \citep[][and references therein]{2016A&A...593A..92B}.
We also investigated the correlation between the CO $J=15-14$ flux and the gas mass and find that the strongest CO emitters have tendentially more massive envelopes, although there is a significant scatter (Fig.\,\ref{fig:orichcomp}).
Also, the targets with the highest temperatures have the highest line-to-continuum (F$_{\rm CO\,15-14}$/PACS$_{\rm 170}$) ratios. IRAS\,21282+5050, which has the most massive warm envelope in our sample, defies this trend since it shows relatively strong CO $J=15-14$ emission, although its \mbox{$T_{\rm rot}$}$\sim170$\,K is the lowest in the sample. The Red Rectangle appears isolated in a region of rather weak CO emission in spite of relatively high temperatures ($\sim$400-500\,K). We compare these results to Paper I in Section\,\ref{Section:comparison}.
\subsection{Mass-loss rates}
\begin{figure}[]
\centering \hspace{+0.1cm}
\includegraphics[width=0.9\linewidth]{extra8.pdf}
\caption{Logarithm of the mass-loss rate versus total CO $J=$15-14 flux.} \label{fig:mloss_results}
\end{figure}
The mass-loss rates have been estimated by simply dividing the total mass by the crossing time of the CO-emitting layers, that is:
\begin{equation}\label{MassLossEquation}
\mbox{$\dot{M}$} = \frac{\mtot\,\vexp}{\mbox{$r_{\rm CO}$}}
\end{equation}
\noindent where \vexp\ is the expansion velocity of the gas, which has been taken from literature (Table\,\ref{tab:2}). For the characteristic radius of the CO-emitting region we use the same value (\mbox{$r_{\rm CO}$}) adopted for the opacity correction. We note that this estimate represents a mean or "equivalent" mass-loss rate assuming constant-velocity spherically-symmetric mass-loss during the time when the warm-inner envelope layers where the CO PACS lines arise were ejected, that is, during the last $\sim$20-50\,yr and $\la$300\,yr for AGBs and post-AGBs/yPNe, respectively, given the values of \vexp\ and \mbox{$r_{\rm CO}$}.
The mass-loss rates are listed in Table\,\ref{tab:rot}. We find a range of values of $\dot{M}\sim10^{-7}-10^{-5}\rm~M_{\sun}~yr^{-1}$, with a median value in our AGB stars of \mbox{$\dot{M}$}$\sim6\times10^{-6} \rm~M_{\sun}~yr^{-1}$. These values are not to be taken as representative of the whole class of C-rich evolved stars, since our sample is small and not unbiased. This is because the objects in the THROES catalogue were originally selected for {\it Herschel}\ observations due to various reasons, probably including their strong CO emission.
As in Paper I, we investigated a possible correlation between $\dot{M}$, and \mbox{$T_{\rm rot}$}, \vexp. We see little evidence of an anti-correlation between \mbox{$\dot{M}$}\ and \mbox{$T_{\rm rot}$}, although the relation is strongly influenced by AFGL\,618, which is a strong outlier in this parameter space (Fig.\,\ref{fig:orichcomp}). In this, and maybe other objects (mainly post-AGB/yPNe), we expect departures from the simple (constant mass-loss rate, spherically symmetric) model adopted to estimate the "equivalent" \mbox{$\dot{M}$}.
We compare the results obtained here and in Paper I in Section \ref{Section:comparison}.
Figure\,\ref{fig:mloss_results} shows the logarithm of the integrated flux of the CO $J=15-14$ line versus the logarithm of the mass-loss rate of the single component fit. The upper and lower limit of the errorbars in $\dot{M}$ correspond to a range of radii around the representative one (see Fig.\,\ref{fig:opacity}). We find a positive trend which is consistent with a power-law relation similar to that found by \citet{2016A&A...588A.124L} in their sample of C-rich AGB CSEs with \water\ FIR emission lines.
Separate values of $\dot{M}$ for the hot and warm components are computed for completeness, but the difference found ($M_{\rm h}<M_{\rm w}$) should not be overinterpreted as a recent decrease of the mass-loss rate. The hot and warm components most likely trace adjacent layers of the inner-winds of our targets, with the hot component presumably best sampling regions closer to the center. However, for simplicity and since we ignore the true CO excitation structure, we use the same radius to formally compute $\dot{M}$ for both components. We note that due to the $1/r_{\rm CO}$ dependence, the values of $M_{\rm h}$ and $M_{\rm w}$ can be brought closer to the single-fit value if the warm and hot correspond to different $r_{\rm CO}$. We refrain from discussing $M_{\rm h}$ and $M_{\rm w}$ separately since a more sophisticated analysis, including nLTEexRT modeling, is needed in order to assess mass-loss time variability. For this reason we only compare our single-component mass-loss rates to the literature (\S\,\ref{section:Mloss}).
\subsection{The influence of line variability on \mbox{$T_{\rm rot}$}\ and \mass}
\label{section:irc10216_var}
We have shown in Fig.\,\ref{fig:irc10216specs} the temporal variability of the continuum and line fluxes in the case of the Mira-type variable AGB star IRC+10216. The higher-$J$ CO lines ($E_{\rm u}$$>$2000\,K) are the ones that show the strongest variations with time. Here, we are interested in studying how CO line variability affects the values of \mass\ and \mbox{$T_{\rm rot}$}\ derived using the RD method but using data acquired at different epochs.
We use the same seven available OBSIDS\footnote{Which have been reprocessed and are part of the THROES catalogue.} of IRC+10216 as \citet{2015ASPC..497...43T}, which span a time period of 551 days (Table \ref{tab:1}).
The RDs of IRC+10216 for these seven different observing epochs are shown in Fig.\,\ref{fig:irc10216_rds}, and the results of the RD analysis are tabulated in Table \ref{tab:rot}, together with the remaining targets. Since three of the OBSIDs corresponding to the ODs 894, 1133 and 1288 have a more restricted wavelength coverage, the fits to the RDs have an inherently larger uncertainty because the fit is more sensitive to the low number statistics. The error-weighted mean (single-fit) rotational temperature and mass are \mbox{$T_{\rm rot}$}$\sim$520\,K and \mass$\sim$4$\times10^{-4}$\,\mbox{$\rm M_{\sun}$}, respectively.
In Fig.\,\ref{fig:irc10216_var}, we plot the temperature and
the mass, for single- and double-temperature components, versus the
operational day of the observations. The bottom panel shows that
the total gas mass deduced from the single-fit of the RD or
for the warm and hot components does not reflect the line flux
variability since it stays essentially constant with time about the
average value (dotted lines), well within the estimated
uncertainties.
In a similar manner, the temperature of the warm component does not clearly reflect the CO line flux variations, since all multi-epoch values are in good agreement within uncertainties. The hot component is the one that shows the largest variations (perhaps periodic) of the temperature, with $T_{\rm h}$ going $\sim$100\,K ($\sim$16\%) above the average ($\sim$\,640\,K) at OD\,1087, and then relaxing back to normal values in the remaining epochs. This variation is echoed in the single-fit value of \mbox{$T_{\rm rot}$}\, ($\sim$12\%).
Temperature variations are not necessarily expected to be periodic, but if we assume that to be true by fitting a sinusoidal function with the known pulsation period $P=630$ days to the RD parameters, we find that such model accommodates reasonably well the data points corresponding to $T_{\rm h}$ (and also the single-fit \mbox{$T_{\rm rot}$}).
It is also possible that IRC+10216 underwent an abrupt change of the physical conditions in its inner wind layers at epoch OD\,1087, since the remaining data points by themselves will not justify/indicate periodic variability.
In summary, probably due to a compensation between the changes in the
y-intercept and the slope of the RD due to CO line flux variations,
which are largest for transitions with the highest $J$, the total gas mass \mass\ appears to be quite robust to FIR pumping (non-LTE) effects. These, however, could have a measurable, yet moderate impact on the rotational temperatures.
\begin{figure}[t]
\centering
\includegraphics[width=0.85\linewidth]{irc10216_var_2.pdf}
\caption{Rotational temperature and mass versus operational day in IRC+10216. Top: sinusoidal fit to the \mbox{$T_{\rm rot}$} variation with fixed period of 630 days (solid line). Bottom: total gas mass over time. In each panel, the dotted lines are the average of each component and the asterisks mark unused data points in the fit (see text). The color code is the same as in Fig.\,\ref{fig:irc10216_rds}.}
\label{fig:irc10216_var}
\end{figure}
\section{Discussion}
\label{section:DIS}
\subsection{Gas temperatures}
\label{section:gasmass}
The detection of high-$J$ CO rotational lines is an indication of a significant amount of molecular gas under relatively high temperature conditions. From our simple RD analysis we inferred that the average gas temperatures of the layers sampled by FIR CO lines are much larger ($T_{\rm rot}\sim200-900$\,K) than those typically derived from mm/sub-mm observations, which are sensitive to $\la$\,100\,K gas from the intermediate-to-outer layers of the envelopes of evolved stars \citep[at $\approx$10$^{16}$-10$^{17}$\,cm, see e.g.][]{2010A&A...523A..18D,2011A&A...530A..83S}.
In a number of targets we identified a double-temperature ("warm" and "hot") component. Deviations from a single straight line fit to the RD have been found also in some of the O-rich objects in Paper I and in other previous works using, for example, ISO and/or \textit{Herschel}/SPIRE CO spectra in a number of AGB and post-AGB CSEs \citep[e.g.,][]{2000A&A...360.1117J,2010A&A...518L.144W,2014MNRAS.437..532M,2015ApJ...806L...3C,2016ApJ...828...51C}.
AFGL\,618 and IRAS\,16594-4656 are the targets whose RDs show the most obvious departure from linearity. IRAS\,16594-4656 is particularly interesting since we found a much cooler warm-component of just $T_{\rm w}\sim220$\,K, and a breakpoint at lower energies ($E_{\rm u}/k\approx1049.9$\,K) compared to other targets typically with $T_{\rm w}\sim450$\,K up to $E_{\rm u}/k\approx1794.3$\,K.
As in Paper I, in order to investigate if the
double-temperature component is consistent with resulting from the temperature
stratification within the inner layers of the CSEs, we compared the
hot-to-warm \mass\ and \mbox{$T_{\rm rot}$}\ ratios (Fig.\,\ref{fig:2comps}), and
find that they are correlated. If the temperature profiles in the
envelope follow a power-law of the type $T(r)\propto r^{-\alpha}$,
with $\alpha$ being a constant, then the trend in
Fig.\,\ref{fig:2comps} should also follow approximately a power-law
function. We find $\alpha\sim0.4$, which is similar to the value
found for the O-rich targets studied in Paper I. The value of
$\alpha$ is in agreement with past works that suggested that the
kinetic temperature distribution is shallower, with values of
$\alpha$ down to $\sim$0.4-0.5, for the inner
($\sim$5\ex{14}-3\ex{15}\,cm) CSE layers
\cite[][]{2012A&A...539A.108D,2016A&A...588A.124L,2014MNRAS.437..532M} than for the
outer regions, where the steepest temperature variations ($\alpha
\sim$1-1.2) are found
\cite[$\ga$10$^{16}$\,cm;][]{2006A&A...450..167T}.
In addition to the average power-law exponent obtained by fitting all the targets simultaneously, we can also derive a value for each individual case applying:
\begin{equation}
\alpha = -\frac{\log(T_{\rm h}/T_{\rm w})}{\log(M_{\rm h}/M_{\rm w})}
\end{equation}
\noindent as shown in the bottom panel of Fig.\,\ref{fig:2comps}. In the case of IRC+10216, we find $\alpha=0.45\pm0.06$ which is in good agreement with the power-law exponent in the inner CSE between 9 and 65 stellar radii (i.e., up to $\sim3\times10^{15}\rm\,cm$) deduced from detailed non-LTE excitation and radiative transfer models \citep[][]{2012A&A...539A.108D}.
Therefore, the empirical relation found between the
hot-to-warm ratio of \mass\ and \mbox{$T_{\rm rot}$}\ is consistent with the
double-\mbox{$T_{\rm rot}$}\ component in some of our targets stemming (at least
partially) from the temperature stratification across the inner
envelope layers. The two components in the RD do not necessarily
imply two distinct/detached shells of gas at different temperatures,
but they most likely reflect the temperature decay laws. As
explained in Paper I, in case of LTE deviations (not impossible in
the lowest mass-loss rate objects), the value of $\alpha$ obtained
from this simple approach would more closely represent the dust
(rather than the gas) temperature distribution. This needs
confirmation by detailed nLTEexRT models to the individual targets,
which will be done in a future publication.
\begin{figure}[t]
\centering
\includegraphics[width=0.85\linewidth]{2comps_2.pdf}\\
\includegraphics[width=0.85\linewidth]{alpha_vs_Mloss_crich.pdf}
\caption{Hot/warm temperature ratio versus hot/warm mass ratio for AGBs and post-AGBs and the power-law index. Top: the lines correspond to power-law fits with index $\alpha=0.4$ for the opacity corrected RDs (solid) and $\alpha=0.44$ for the uncorrected RDs (dashed); the open symbols correspond to the O-rich AGB stars in Paper I. Bottom: power law index for each individual target versus mass-loss rate (single fit).} \label{fig:2comps}
\end{figure}
\subsubsection{Individual targets: comparison with previous works} \label{Section:individual}
It is not possible to directly compare most of our results with literature because past studies have been focusing on the cold, outer components of CSEs.
Prior to \textit{Herschel} there was a study based on ISO LWS data in roughly the same wavelength range by \citet{2000A&A...360.1117J} who also performed RD analysis (without opacity correction). They found $T_{\rm rot}\sim700(\pm 90)$\,K and $T_{\rm rot}\sim380(\pm 30)$\,K for AFGL\,618 and AFGL\,2688 respectively. We obtained similar results without opacity correction, but introducing this effect lowered these values to $T_{\rm rot}\sim690(\pm30)$\,K and $T_{\rm rot}\sim300(\pm20)$\,K. In AFGL\,618 the central star is hot enough ($T_{\rm eff}\sim 33000$\,K) to produce FUV photons that heat the gas. In AGFL\,2688 this is probably not the case since the central star is much cooler ($T_{\rm eff}\sim7250$\,K), so low-velocity shocks are the most likely heating mechanism.
Also partially based on ISO data, \citet{1999A&A...345..841R}
used a number of spectral lines of CO between $J=1-0$ and $J=21-20$
to infer the kinetic temperature profile across the CSE of
IRAS\,15194-5115. According to their model, $T_{\rm kin}\sim400$\,K
at $r_{\rm CO}\sim$\,(1-2)$\times10^{15}\rm\,cm$, which is in
excellent agreement with our opacity-corrected single component
$T_{\rm rot}\sim410$\,K for $r_{\rm CO}=2\times10^{15}\rm\,cm$. This
is because, as already pointed out by \citet{1999A&A...345..841R},
these high-$J$ levels are mainly populated by collisions, therefore
they are proxies for the kinetic temperature, at least out to a few
$\sim10^{15}$\,cm. Also using {\sl ISO} data,
\citet{2002A&A...391..577S} presented a kinetic temperature model
for CIT\,6 that shows that $T_{\rm kin}\sim400-500$\,K at
approximately $r_{\rm CO}\sim$\,(1-2)$\times10^{15}\rm\,cm$, which
is consistent with the warm component ($T_{\rm w}\sim460$\,K) that
we infer and the representative radius $r_{\rm
CO}=2\times10^{15}\rm\,cm$ adopted. The hot component ($T_{\rm
w}\sim830$\,K) found by us would imply that regions closer to the
star have an important contribution to the emission of the
highest-$J$ lines.
The only star whose inner/warmer (gaseous) CSE had been studied in detail before using \textit{Herschel}/PACS data is IRC+10216. This was done by \citet{2010A&A...518L.143D} using high-$J$ CO spectral lines to infer the kinetic temperature profile as a function of radial distance. They find $T_{\rm kin}\sim500-600$\,K at $r_{\rm CO}\sim$ (1-2)$\times10^{15}\rm\,cm$. Here we applied the opacity correction for $r_{\rm CO}=3\times10^{15}$\,cm which seems to be the layer at which $T_{\rm kin}\sim300$\,K in their model. This is also the average value of $T_{\rm w}$ that we found by fitting only the lowest $J$ transitions, wich does not change appreciably with time despite strong line flux variability (Fig.\,\ref{fig:irc10216_var}). The hot component ($\sim$600-700\,K) may correspond to $r_{\rm CO}\sim1\times10^{15}\rm\,cm$ according to their model.
Using \textit{Herschel}/SPIRE data, \citet{2010A&A...518L.144W} also performed RD analysis of CO spectra and derived $T_{\rm rot}\sim$70-230\,K and $T_{\rm rot}\sim$100-200\,K for AFGL\,618 and AFGL\,2688, respectively. The rotational temperatures we find are thus higher than the ones obtained in \citet{2010A&A...518L.144W} as expected.
Further Large-velocity gradient (LVG) calculations by those authors suggested that hot material at approximately $1000$\,K might exist at the high-velocity wind region of AFGL\,618, which could be the one traced by PACS since we found $T_{\rm h}\sim 900$\,K.
For HD\,44179 (the Red Rectangle) we obtained $T_{\rm rot}\sim440$\,K, which is about 2-3 times larger than the range of values inferred by \citet{2013A&A...552A.116B} from \textit{Herschel}/HIFI lower-$J$ CO observations. It is not surprising that we find a larger value since the higher $J$ lines probed by PACS are probably formed deeper inside the rotating circum-binary disk at the core of this object. Meanwhile follow up analysis has shown more clearly an outflow with $\sim500$\,K
\citep{2016A&A...593A..92B}, but also seems that such temperature conditions could exist in a region of the inner disk with a radius of about $r_{\rm CO}\sim2\times10^{15}$\,cm, unresolved by PACS. The opacity correction would still be moderate for this value ($\tau_{J=14-13}\sim 0.67$), and it lowers the rotational temperature to $T_{\rm rot}\approx410$\,K which is still within the uncertainties.
For the remaining yPNes (IRAS\,21282+5050, CPD-568032, Hen\,2-113) there are no kinetic temperature models in the literature that we could compare our results to.
\input{sec6p2.tex}
\subsection{Comparison between the sample of O-rich and C-rich stars}
\label{Section:comparison}
In the Appendix we provide Fig.\,\ref{fig:orichcomp} where we plot together the results presented here to the ones obtained for the sample of O-rich and S-type stars in Paper I.
We find that the range of \mbox{$T_{\rm rot}$}\ is approximately the same among C-rich and O-rich stars, but the O-rich AGBs have less massive CSEs by typically one order of magnitude, and lower expansion velocities. This in turn reflects on lower mass-loss rates on average. The scenario is reversed in the groups of PNes with the few O-rich PNes having more warm gas than the carbon counterparts.
Probably the O-rich AGB stars studied in Paper I are typically low massive stars with very large evolution times, while the O-rich post-AGBs and PNe are very massive objects that have undergone through the Hot Bottom Burning stage.
We stress that despite these trends being indicative of clear differences in the properties of the CSEs of the targets in the THROES catalogue, they may not be a general property of C-rich versus non C-rich targets, since our samples are not necessarily unbiased and they are definitely not statistically significant.
\section{Summary}
\label{section:conclusion}
In this paper (Paper II), we use \textit{Herschel}/PACS FIR spectra of a sample of 15 C-rich evolved stars, including AGBs, post-AGBs and yPNe, from the THROES catalogue \citep{2017arXiv171105992R}.
These data contain valuable information about the physical-chemical properties of evolved stars as shown, for instance, by the striking differences of spectral features (molecular, atomic/ionized and solid state) as a function of evolutionary stage. In this work, we focus on the rotational spectrum of CO (up to $J=45-44$) which was used as a proxy for the molecular component of the gas in the warm regions of the CSEs. Our findings can be summarized as follows:
\begin{itemize}
\item Due to \textit{Herschel's} higher sensitivity compared to ISO, the range of detected $^{12}$CO\, transitions has been extended to high rotational levels of up to $J_{\rm u}$=45 in low-to-intermediate mass evolved stars. Rotational diagrams using high-excitation CO\,($v$=0) rotational emission lines, with upper-level energies $E_{\rm u}$$\sim$580 to 5000 K, have been plotted to estimate rotational temperatures (\mbox{$T_{\rm rot}$}), total molecular mass in the CO-emitting layers (\mtot) and average mass-loss rates during the ejection of these layers (\mbox{$\dot{M}$}).
\item The range of temperatures found in our sample, \mbox{$T_{\rm rot}$}$\sim$200-700\,K, is larger than what had been deduced from mm/sub-mm observations, and even {\it Herschel}/HIFI and SPIRE observations, confirming that PACS CO lines probe deeper layers yet poorly studied to date (typically, $\approx$10$^{15}$ cm for AGBs and $\approx$10$^{16}$ cm for post-AGBs and yPNes).)
\item The total gas mass of the warm envelope layers sampled by PACS data are between $M\sim10^{-5}-10^{-3}~\rm M_{\sun}$, with post-AGBs and yPNe being overall more massive.
\item We find clearly different temperature distributions for the different classes with AGBs having typically hotter gas (up to $T_{\rm rot}\sim1000$\,K) than post-AGBs ($T_{\rm rot}\la500$\,K) and yPNes ($T_{\rm rot}\la400$\,K). The yPN AFGL\,618 is a clear outlier with a very high amount (\mass$\sim$2\ex{-3}\,\mbox{$\rm M_{\sun}$}) of rather hot (up to T$_{\rm h}$$\sim$900\,K) gas, similar to the most massive AGBs in the sample.
\item For AFGL\,3116, CIT\,6, AFGL\,2513, V\,Hya, IRAS\,16594-4656 and AFGL\,618 a double temperature (hot and warm) component is inferred from the RDs. The mean temperatures of the warm and hot components are $\sim$400\,K and $\sim$820\,K, respectively. The mass of the warm component
($\sim$10$^{-5}$-8$\times$10$^{-3}$\mbox{$\rm M_{\sun}$}) is always larger than that of the hot component, by a factor $\sim$4-10.
\item The warm-to-hot \mass\ and \mbox{$T_{\rm rot}$}\ ratios in our sample are correlated and are consistent with an average temperature radial profile of $T\propto$\,$r^{-0.4}$, that is, slightly shallower than in the outer envelope layers, in agreement with recent studies.
\item The mass-loss rates estimated are in the range
\mbox{$\dot{M}$}$\approx$10$^{-7}$-10$^{-4}$\mbox{$\rm M_{\odot}$~yr$^{-1}$}, in agreement (within the uncertainties) with values found in the literature for our targets.
\item We investigated the impact of CO line flux variability on the values of \mass\ and \mbox{$T_{\rm rot}$}\ derived from the simple RD analysis. We studied in detail the case of the Mira-variable AGB star IRC+10216, for which multi-epoch PACS data exist. In spite of strong line flux variability we find that the total gas mass and the average temperature derived from the RDs at different epochs are minimally affected. Only the hot component does show the sign of line variability ($\delta T/T\sim16$\%), roughly in-phase with the continuum periodicity.
\item Similarly to Paper I, we find an anti-correlation between \mbox{$T_{\rm rot}$}\ and \mass, which may result from a combination of CO line cooling and opacity effects, and we find a correlation between \mbox{$\dot{M}$}\ and \vexp, which is consistent with the wind acceleration mechanism being more efficient the more luminous/massive the star is. These trends had been reported in previous studies using low-$J$ CO transitions.
\end{itemize}
We show that high-$J$ CO emission lines probed by \textit{Herschel}/PACS are good tracers of the warm gas ($T\sim200-900$\,K) surrounding evolved carbon stars. Using the simple RD technique, we have provided systematic and homogeneous insight into the deepest layers of these CSEs, though it relies on several approximations. Detailed non-LTE excitation and radiative transfer calculations are needed to determine the temperature stratification of the CSEs, to infer mass-loss rates and to address their time-variability.
\input{tab_2.tex}
\begin{acknowledgements}
We thank the referee for the useful comments and remarks. PACS has been developed by a consortium of institutes led by MPE (Germany) and including UVIE (Austria); KU Leuven, CSL, IMEC (Belgium); CEA, LAM (France); MPIA (Germany); INAF-IFSI/OAA/OAP/OAT, LENS, SISSA (Italy); IAC (Spain). This development has been supported by the funding agencies BMVIT (Austria), ESA-PRODEX (Belgium), CEA/CNES (France), DLR (Germany), ASI/INAF (Italy), and CICYT/MCYT (Spain). This publication makes use of data products from the THROES catalog, which is a project of the Centro de Astrobiología (CAB-CSIC) with the collaboration of the Spanish Virtual Observatory (SVO), funded by the European Space Agency (ESA).
J.M.S.S.\, acknowledges financial support from the ESAC Faculty and the ESA Education Office under the ESAC trainee program. The Institute for Solar Physics is supported by a grant for research infrastructures of national importance from the Swedish Research Council (registration number 2017-00625). C.S.C.\,acknowledges financial support by the Spanish MINECO through grants AYA2016-75066-C2-1-P and by the European Research Council through ERC grant 610256: NANOCOSMOS.
\end{acknowledgements}
\bibliographystyle{aa}
\section{Introduction}
The asymptotic giant branch (AGB) is a late evolutionary stage of
low-to-intermediate mass stars
(1\,\mbox{$\rm M_{\sun}$}$\la$M$_\star$$\la$8\mbox{$\rm M_{\sun}$}) which is largely dominated
by mass-loss processes. AGB stars can shed significant portions of
their outer atmospheric layers in a dust-driven wind, with mass-loss
rates of up to {\mbox{$\dot{M}$}$\sim$10$^{-4}\rm\,M_{\sun}\,yr^{-1}$}
\citep[e.g.][]{Habing1996,Hofner2018}. The material expelled by the
central star (with very low effective temperatures of
\teff$\sim$2000-3000\,K) forms a cool, dense circumstellar envelope
(CSE) that is rich in dust grains and a large variety of molecules.
After a significant decrease of the mass-loss
rate, the AGB phase ends and the `star+CSE' system begins to evolve to the Planetary Nebula
(PN) phase, at which the CSE is fully or almost fully ionized due to
the much higher central star temperatures
(\teff$\approx$10$^4$-10$^5$\,K) and more diluted envelopes. During the AGB-to-PN transition, shocks -- resulting from the
interaction between slow and fast winds at the end of the AGB phase
(or early post-AGB) -- also play an important role changing the
morphology, dynamics and chemistry of the CSEs \citep[e.g.][]{2000oepn.book.....K,2003ARA&A..41..391V,2006IAUS..234..193B}.
The CO\footnote{We abbreviate $\rm ^{12}C^{16}O$ as simply CO
throughout the paper.} molecule is an excellent tracer of the CSEs of AGB stars, post-AGB objects and PNe \cite[e.g][]{1996A&A...306..241G,2001A&A...368..969S,2002A&A...391..577S,2006A&A...450..167T}. The rotational transitions of the ground vibrational level over a wide range of excitation energies sample from cold ($\sim$10 K) to hot gas ($\sim$1000 K). Literature on observations of the cold, extended CO $J = 1-0$ to $J=6-5$ emission around many evolved stars is abundant at mm/sub-mm wavelengths
\citep[e.g.][]{1982ApJ...252..616K,1985ApJ...292..640K,1985ApJ...293..281K,1993ApJS...87..267O,1989A&A...219..256B,1998A&AS..129..363J,2010A&A...523A..59C,2012ApJS..203...16S,2014A&A...566A.145R}. Studies looking at far-infrared (FIR) observations of even higher $J$ CO transitions probing the warmest gas ($\sim$200-1000 K) in deeper layers of CSEs are much more scarce. Pioneering works based on observations with the Infrared Space Observatory ($ISO$) \citep[e.g.][]{2000A&A...360.1117J,2002A&A...391..577S} have continued more recently with \textit{Herschel} \citep[e.g.][]{2011A&A...526A.162G,2012A&A.537A.8B,2014A&A...561A...5K,2015A&A...581A..60D,2018arXiv180803467N}.
This is the second of a series of papers \citep[Paper I]{PaperI} where we analyze in an uniform and systematic way \textit{Herschel}/PACS spectra of a large sample of evolved stars from the THROES catalogue \citep{2017arXiv171105992R} to study their warm inner envelope regions
using high-$J$ CO transitions at FIR wavelengths. As in other previous studies, we divide our sample in O-rich and C-rich targets (papers I and II, respectively) since these two major chemistry classes correspond to progenitor stars with different masses, which follow somewhat different evolutionary paths, and also have a dissimilar dust composition, both facts potentially affecting the mass-loss process. We include targets with different evolutionary
stages: AGB, post-AGB (or pre-PNe) and young planetary nebulae (yPNe).
The goal of our study (papers I and II) is to obtain a first
estimate of the average excitation temperature (\mbox{$T_{\rm rot}$}) and mass
(\mass) of the warm envelope layers traced by the PACS CO lines in a
uniform way using a simple analysis technique, the well-known
rotational diagram (RD) method. The RD technique is useful to
rapidly analyze large data sets (large number of lines and/or large
samples) and to provide some constraints on these fundamental
parameters. With the aim of benchmarking the results of the RD
approximation, we obtain rough estimates of mass-loss rates
(\mbox{$\dot{M}$}) and compare them to values in the literature, paying particular attention to studies including at least a few high-$J$ CO (FIR) transitions.
The impact of possible non-LTE effects on the results from the simple RD analysis was investigated in paper\,I. It was concluded that, though they are expected to be minor in general and probably only affecting the highest-$J$ CO transitions studied here ($J$$\ga$27) at most, their existence cannot be ruled out in the lowest mass-loss rate stars and/or the outermost layers of the PACS CO-emitting volume. We also showed that even under non-LTE conditions, the masses derived from the RDs are approximately correct (or, at the very least, not affected by unusually large uncertainties) since the average excitation temperature describes rather precisely the molecular excitation (i.e., the real level population). This is also corroborated by the good agreement found between our estimates of \mbox{$\dot{M}$}\ and those from detailed non-LTE excitation and radiative transfer (nLTEexRT) studies that exist for a number of targets. We stress that the RD method enables a characterization of the warm CSEs of evolved stars in a first approximation and that for a more robust and detailed study of the radial structure of the density, temperature and velocity in the CSEs, as well as for establishing potential mass-loss rate variations with time, more sophisticated analysis is needed \citep[e.g.,][]{1999A&A...345..841R,2002A&A...391..577S,2012A&A...539A.108D}.
Upon submission of this manuscript (and after our paper I was accepted for publication) we became aware of a recent work by \citet{2018arXiv180803467N} who independently presented \textit{Herchel}/PACS (and SPIRE) range spectroscopy of a sample of 37 AGB stars. These authors perform a similar RD analysis of the CO spectra and focus on deriving excitation temperatures (in contrast to our study, estimates of the envelope mass or mass-loss rates are not reported). Other differences with respect to the work by \citet{2018arXiv180803467N} is that we introduce a canonical opacity correction in the RDs and that we include post-AGBs and PNe.
\section{Observations}
\label{section:Observations}
\subsection{Observations and data reduction}
PACS is a photometer and medium resolution grating spectrometer \citep{2010A&A...518L...2P} onboard the Herschel Space Telescope \citep{2010A&A...518L...1P} probing the FIR wavelength range. The PACS spectrometer covers the wavelength range from 51 to 210 $\mu$m in two different channels that operate simultaneously in the blue (51-105 $\mu$m) and red (102-220 $\mu$m) bands. The Field of View (FoV) covers a 47\arcsec$\times$47\arcsec\ region in the sky, structured in an array of 5$\times$5 spatial pixels ("spaxels") with 9.4\arcsec$\times$9.4\arcsec. PACS provides a resolving power between 5500 and 940, i.e. a spectral resolution of approximately 55-320 $\rm km~s^{-1}$, at short and long wavelengths, respectively, and the PSF of the PACS spectrometer ranges from $\sim$\,9\arcsec\ in the blue band to $\sim$13\arcsec\ in the red band. The PSF is described in \cite{2016MsT..........1D} and \cite{2016A&A...591A.117B}. The technical details of the instrument can be found in the \textit{PACS Observer's Manual}\footnote{\url{herschel.esac.esa.int/Docs/PACS/html/pacs_om.html}}.
The PACS (1D) spectral data were taken from the THROES (caTalogue of HeRschel Observations of Evolved Stars) website\footnote{\url{https://throes.cab.inta-csic.es}} which contains fully reduced PACS spectra of a collection of 114 stars, mostly low-to-intermediate mass AGB stars, post-AGB and PNe. The data reduction is explained in \citet{2017arXiv171105992R} in full detail. The catalogue also consists of a compilation of previous photometry measurements at 12, 25, 60, 100 $\rm \mu m$ with the Infrared Astronomical Satellite (IRAS).
Table \ref{tab:1} offers a description of the observations used here where we
provide the target name as listed in the header of the PACS FITS files
and an alternative name for which some of the stars are more well
known in the literature. We only analyzed the spectra between [55-95]
and [101-190] $\rm \mu m$ because the flux densities are unreliable
above 190 $\rm\mu m$, below 55 $\rm \mu m$ and between 95-101 $\rm\mu
m$ due to spectral leakage. Two observation identifiers (OBSIDs) per
target, corresponding to the bands B2A, B2B and R1, are necessary to
cover the full PACS wavelength range. In the case of IRC+10216,
band B2A data exists \citep[see][]{2010A&A...518L.143D}, but these
were acquired in a non-standard spectroscopy mode and have
restricted access in the {\it Herschel}\ Science Archive and, thus, are not
included in the THROES catalogue. Instead, we used 4+3 OBSIDs
corresponding to seven different operational days: OD = [745, 1087,
1257 and 1296] covering the spectral range [69-95, 140-190] $\rm \mu
m$ and OD = [894, 1113 and 1288] covering a narrower interval [77-95,
155-190] $\rm \mu m$.
\begin{figure*}[t]
\centering
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{IRAS1.png}
\end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{IRAS100vsIRAS12_25Crich.png}
\end{minipage}
\caption{IRAS color diagrams for the stars in the THROES catalogue. The colors are defined from the infrared fluxes at 12, 25 and 60 $\rm \mu m$. The boxes on the left panel are the ones defined in \citet{1988A&A...194..125V}, and the highlighted points correspond to the stars studied in this paper.} \label{fig:IRAS}
\end{figure*}
\subsection{Sample overview}
We searched for CO emission amongst the entire THROES catalog, but in
this paper we focus on C-rich CSEs (29\% of the entries) observed in
PACS range mode. We found 15 evolved stars with at
least three CO emission lines with signal-to-noise ratio above 3.
This sample contains bright infrared targets spanning a range of
evolutionary stages from the AGB to the PN phase, but sharing similar
carbon chemistry with strong CO emission at high excitation
temperatures (up to $E_{\rm u}/k\approx5688$ K). The AGBs, which are
mostly Mira variables, are known for their high mass-loss rates
compared to the mean value of
$\dot{M}\sim1.5\times10^{-7}\rm~M_{\sun}~yr^{-1}$ derived from studies
with large samples of carbon stars \citep{1993ApJS...87..267O}. We
also include two mixed-chemistry post-AGBs \citep[Red Rectangle and
IRAS\,16594-4656, ][]{1998Natur.391..868W,2005A&A...429..977W} and
two mixed-chemistry yPNes \citep[Hen 2-113 and CPD-56\degr8032,][and
references therein]{1998MNRAS.296..419D,2015MNRAS.449L..56D}, that
is, they show both C-rich and O-rich dust grains. In the Appendix we
provide Table \ref{tab:2} with a summary of some relevant properties
such as distance, effective temperature and gas mass-loss rate, along
with additional references.
Figure \ref{fig:IRAS} shows the classic IRAS color-color diagram
\citep{1988A&A...194..125V} featuring the colors $[25] - [60] =
-2.5\log\frac{IRAS_{25}}{IRAS_{60}}$ and $[12] - [25] =
-2.5\log\frac{IRAS_{12}}{IRAS_{25}}$ of all the stars in THROES. The
stars here studied are highlighted with colored filled symbols which
we will use consistently throughout the paper. This diagram is known
to be a good indicator of the evolutionary stage of
low-to-intermediate evolved stars, with AGBs populating the lower left
corner and more advanced stages being located on the diametrically
opposed, so-called cold, side of the diagram. It also shows an
evolution in terms of the mass-loss rate and/or progressively
increasing optical depths \citep{1987A&A...186..136B}. C-rich AGBs
clearly constellate in a different box compared to the O-rich stars in
Paper I, which has been interpreted as a consequence of different
grains' emissivities \citep{1986ApJ...311..345Z}. The AGB star that
falls outside the expected box with a clear 25 $\rm \mu m$ excess
($[12]-[25]=0.1$) is AFGL 3068 (LL Peg), which is an "extreme carbon
star": very dust-obscured by optically thick shells due to high
mass-loss rate \citep{1992ApJ...391..285V,1997A&A...326..305W}. The
post-AGB object HD\,44179, best known as The Red Rectangle, is also
outlying in this same box with respect to objects in a similar
evolutionary status beyond the AGB, which typically show much larger
25 $\rm \mu m$ excess indicative of detached cold dust (and gas)
envelopes.
\subsection{Mass-loss rate}
\label{section:Mloss}
As in Paper I, we have compared the values of the mass-loss rates derived from our simple RD analysis with other values found in the literature mostly from low-$J$ observations, paying special attention to a few targets with detailed non-LTE excitation and radiative transfer analysis of CO data including at least some high-$J$ transitions observed with {\it Herschel}. This is also a way of ascertaining the robustness of the RD method.
Figure\,\ref{fig:mLoss2} shows our estimate of the mass-loss rate versus that found in the literature (see Table\,\ref{tab:2}). For each target the markers correspond to the single temperature component for the radius mentioned in Table\,\ref{tab:rot}, and the error bars represent the uncertainty in the radius for the adopted $v_{\rm exp}$. We show the computed opacities for the considered radii in each target in the supplementary Fig. \ref{fig:opacity}. The range of values found in literature are shown by the gray shaded area whose bounds are set by the maximum and minimum $\dot{M}$ plus uncertainties when reported \citep[factor of $\sim$3 in AGBs, ][]{2008A&A...487..645R,2010A&A...523A..18D}. Those values have been rescaled to the same distance, $v_{\rm exp}$ and $X_{\rm CO}$ here adopted.
Similarly to the non C-rich THROES targets in Paper I, our mass-loss rates are in good agreement with values in the literature within the large uncertainties. In our case, these are dominated by the uncertainty in \mbox{$r_{\rm CO}$}. We see that the range of radii we explored yields a $\dot{M}$ that fits within the shaded area. In many cases, the error bars are truncated at the upper limit above which the line opacities would be too large to allow reliable estimates of the masses and mass-loss rates (see Fig.\,\ref{fig:opacity}). For example, in the case of AFGL\,3068 this seems to imply smaller radius than what we have adopted to better match the values in literature.
In the case of IRC+10216, the opacity correction for $r_{\rm CO}=2\times10^{15}$\,cm (instead of \mbox{$r_{\rm CO}$}=3\ex{15}\,cm) would result in a mass-loss of $\dot{M}\approx1.4\times10^{-5}~\rm M_{\sun}~yr^{-1}$ (not displayed) that would best match the estimate from mm observations. However in these circumstances the expected opacities in the lowest $J$ lines would be too large ($\tau \gg1$). Nonetheless, the derived $T_{\rm w}$ and $T_{\rm h}$ are consistent with kinetic temperature profile models as explained in Section\,\ref{section:gasmass}.
We find an average value of four OBSIDs of $\dot{M}\sim5\times10^{-6}~\rm M_{\sun}~yr^{-1}$, which is lower than the range $\dot{M}\sim$(1-3)$\times10^{-5}~\rm M_{\sun}~yr^{-1}$ scaled from \cite{2006A&A...450..167T, 2010A&A...523A..18D,2012A&A...539A.108D,2017arXiv170904738G}. However the value obtained for the warm component agrees with the lower limit of this range (because of the larger $M_{\rm H_2}$). We obtain $T_{\rm w}\sim300$ K and $\dot{M_{\rm w}}\sim$1$\times10^{-5}~\rm M_{\sun}~yr^{-1}$ for a radius of $r_{\rm CO}=3\times10^{15}$ cm, in good agreement with the results of radiative transfer modeling of the same high $J$ CO lines by \citet{2010A&A...518L.143D}. Scaling their $\dot{M}$ to the same $d$ and $X_{\rm CO}$ gives
$\dot{M}\sim$1.2$\times10^{-5}$\,\mbox{$\rm M_{\odot}$~yr$^{-1}$}\ with an uncertainty of a factor 2.
For V\,Hya we obtained $\dot{M}\sim8\times10^{-6}~\rm
M_{\sun}~yr^{-1}$ which is lower than $\dot{M}\sim3\times10^{-5}~\rm
M_{\sun}~yr^{-1}$ from \citet{thesis-camps} who performed radiative transfer calculations using the same PACS spectrum. In this case, however, the spatio-kinematic structure of the molecular outflow is more complex than assumed here. In particular, multiple kinematic (fast and slow) components seem to be present \citet{2004ApJ...616L..43H,0004-637X-699-2-1015}, which not only translates into a larger uncertainty in the characteristic value of \vexp\ in the PACS CO-emitting layers, but also implies that the "equivalent" mass-loss rate is particularly questionable.
As for V\,Hya, for pPNe and yPNe, the assumption of constant-velocity spherically- symmetric mass loss may also not hold. For completeness, the mass-loss rates estimates for these targets are shown Table\,\ref{tab:2}, but they are subject to larger uncertainties and have to be interpreted with caution. We have checked that, even in these cases, our values are in good agreement with previous estimates (making similar simplifying assumptions) in the literature. For example, for the Red Rectangle, we obtained
$\dot{M}\sim2\times10^{-7}$\,\mbox{$\rm M_{\odot}$~yr$^{-1}$}\ which is within the (scaled) range of $\dot{M}\sim$(0.2-1.4)$\times10^{-7}$\,\mbox{$\rm M_{\odot}$~yr$^{-1}$}\ reported by \cite{2010A&A...523A..18D}.
\begin{figure}[t]
\centering
\includegraphics[width=0.85\linewidth]{MlossComp2.pdf}
\caption{Comparison between the mass-loss rate in this work and the literature. The markers correspond to the representative radii listed in Table \ref{tab:rot} for each target and the error bars are mass-loss rates for a given range of radii. The shaded area encloses the range of values found in literature scaled to the same parameters here assumed (see text).} \label{fig:mLoss2}
\end{figure}
In summary, our results are consistent with the literature within the
typical uncertainties, but it is hard to tell what is the exact cause
of the slight discrepancies from case to case. One obvious
reason is that the simple RD method and our assumption of a
characteristic value of \mbox{$r_{\rm CO}$}\ (unknown, but crudely constrained
from first principles and observations) only provides a rough
estimate of the mass-loss. We also note, that the bulk of the CO
emission under study is produced in the warm inner layers of the
CSEs of our targets down to a location where $\tau$$\sim$1. For
very optically thick CSEs, there may be an additional amount of gas
that is not fully recovered after the moderate opacity correction
applied. Another reason for \mbox{$\dot{M}$}\ discrepancies is the different
number of transitions and range of $E_{\rm u}$\ covered by different
studies. Non-LTE excitation and radiative transfer models of the CO
emission including a wide range of $J$- transitions is needed to
obtain accurate estimates of the mass-loss rates and, in particular,
to address \mbox{$\dot{M}$}\ time modulations.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Two-photon probe of the Jaynes-Cummings model and controlled symmetry breaking in circuit QED — Frank Deppe1, Matteo Mariantoni1, Edwin P. Menzel1, •Achim Marx1, Rudolf Gross1, S. Saito2, K. Kakuyanagi2, H. Tanaka2, K. Semba2, T Meno3, H. Takayanagi4, and E. Solano5 — 1Walther-Meissner-Institut and TU München, Germany — 2NTT Basic Research Laboratories, NTT Corp., Japan — 3NTT Advanced Technology, NTT Corp., Japan — 4Tokyo University of Science and International Center for Materials Nanoarchitectronics, Japan — 5Universidad del País Vasco - Euskal Herriko Unibertsitatea, Spain
Superconducting qubits behave as artificial two-level atoms. Coupling them to on-chip microwave resonators has given rise to the field of circuit quantum electrodynamics. In this work, we report on the observation of key signatures of a two-photon driven Jaynes-Cummings model, which unveils the upconversion dynamics of a superconducting flux qubit coupled to an on-chip resonator. Our experiment and theoretical analysis show clear evidence for the coexistence of one- and two-photon driven level anticrossings of the qubit-resonator system. This results from the controlled symmetry breaking of the system Hamiltonian, causing parity to become a not well-defined property. Our study provides deep insight into the interplay of multiphoton processes and symmetries in a qubit-resonator system. We ackowledge support from SFB 631, NIM, CREST-JST, JSPS-KAKENHI(18201018) and MEXT-KAKENHI(18001002), EuroSQIP, and the Ikerbasque Foundation.
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«Пилот» () — первый эпизод психологического триллера «Родина». Премьера состоялась на канале Showtime 2 октября 2011 года.
Эпизод фокусируется на возвращении домой сержанта морской пехоты Николаса Броуди (Дэмиэн Льюис), спасённого после восьми лет, находившегося в качестве военнопленного в Афганистане. В то время как Броуди отмечается как герой, офицер ЦРУ Кэрри Мэтисон (Клэр Дэйнс) считает, что Броуди действует в качестве спящего агента для «Аль-Каиды».
«Пилот» получил широкую похвалу от критиков и стал самой высокорейтинговой драматической премьерой на Showtime с 2003 года.
Сюжет
Во флэшбеке, Кэрри Мэтисон (Клэр Дэйнс) появляется в Ираке, где она работает сотрудником ЦРУ. Она даёт взятку, чтобы пройти в тюрьму, где держат одного из её информаторов — создателя бомб, которого в скором времени казнят. Когда охранники засекают Кэрри и уводят ее, информатор что-то шепчет ей на ухо.
Настоящее время, Кэрри с опозданием прибывает на собрание в антитеррористический центр ЦРУ, куда она распределена после случая в иракской тюрьме. Директор по борьбе с терроризмом Дэвид Эстес (Дэвид Хэрвуд) объявляет, что во время рейда на лагерь Аль-Каиды был спасён находившийся в плену сержант морской пехоты Николас Броуди (Дэмиэн Льюис), которого все считали погибшим. Кэрри позже поведает своему коллеге и наставнику, Солу Беренсону (Мэнди Патинкин), что ей сказал её информатор в последний момент: «Американский военнопленный сменил хозяина». Она приходит к выводу, что этот военнопленный и есть Броуди. Сол категорически отвергает возможность того, чтобы ЦРУ проводило какое-либо расследование в отношение Броуди, который теперь всеми любимый герой войны.
Джессика Броуди (Морена Баккарин), жена Николаса, занимается сексом с Майком Фабером (Диего Клаттенхофф). Позже выясняется, что Майк лучший друг Ника, морпех, с которым служил Броуди до того, как был захвачен в плен. Джессика потрясена раздавшимся телефонным звонком от Броуди, объявляющего о своем возвращении, и она отправляется в аэропорт со своими детьми, 16-летней Даной (Морган Сэйлор) и 12-летним Крисом (Джексон Пейс), чтобы встретить его. В то время как Броуди возвращается домой, Кэрри готовит несанкционированную (и незаконную) операцию по наблюдению и слежке за ним. Она вербует своего друга Вёрджила, независимого подрядчика, для установки скрытых камер и микрофонов по всему дому Броуди, за которым она сможет следить из своего дома. Они успешно устанавливают скрытые камеры и микрофоны, прежде чем Броуди возвращается домой. Кэрри начинает наблюдать за каждым движением Броуди.
На следующий день, Броуди является на дебрифинг в ЦРУ, где присутствуют Кэрри, Дэвид и другие сотрудники ЦРУ. Броуди расспрашивают о том, что с ним происходило, когда он был пленником Аль-Каиды. Кэрри спрашивает его, имел ли он когда-либо любой контакт с Абу Назиром, лидером Аль-Каиды. Он говорит нет, но мы понимаем, что он лжёт, ведь в воспоминаниях память показывает самого Броуди с Абу Назиром. Кэрри скептична и неоднократно расспрашивает его снова и снова, прежде чем Дэвид останавливает ее.
Позже, Броуди идёт с кем-то на встречу в парке. Полагая, что он может встречается со связным Аль-Каиды, Кэрри, Вёрджил и Макс (брат Вёрджила) следуют за ним. Но оказывается, что Броуди встречается с Хелен Уокер (Афтон Уильямсон), женой Тома Уокера, тоже морпеха, который восемь лет назад попал в плен вместе с Броуди. Уокер также считается пропавшим без вести, и Броуди рассказывает Хелен, что её муж, находясь в плену был забит до смерти. Хелен спрашивает Броуди, был ли он рядом с Томом, когда он умер, и он говорит «нет». Но Броуди снова лжет, так как его воспоминания показывают сцену избиения и мы видим, что Броуди находится рядом с Томом. Кэрри возвращается домой, где застает разъяренного Сола. Он обнаружил, что она организовала незаконную слежку и говорит Кэрри, что она должна доложить обо всем генералу-инспектору и чтобы «нашла адвоката, он ей понадобится». Кэрри, в отчаянии и резко разговаривает с Солом, который уходит от нее в гневе. Кэрри расстроена и на грани срыва, но в конце концов берёт себя в руки и идёт в бар. Кэрри знакомится в баре с мужчиной, во время разговора с ним наблюдает за музыкантами, играющими в баре, параллельно по телевидению идут новости о возвращении Броуди, и у неё вдруг появляется мысль о разоблачении Броуди. Она бросается к дому Сола и показывает ему различные новостные сюжеты с Броуди в этот день. Она демонстрирует Солу, что каждый раз, когда Броуди снимает камера, он двигает пальцами в какой-то последовательности. Кэрри предполагает, что это какое-то зашифрованное послание, возможно предназначенное для его связного или тайной ячейки. Сол соглашается, что это нуждается в дальнейших расследовании.
В финальной сцене, Броуди бежит трусцой через парк Вашингтона. Во время пробежки, мы видим его воспоминания с избиении Тома Уокера. На этот раз становится понятно, что на самом деле Броуди, по приказу Абу Назира, избивает Уокера до смерти. Во время пробежки Броуди останавливается и смотрит на здание Капитолия.
Производство
Сценарий к эпизоду был написан исполнительными продюсерами Алексом Гансой, Гидеоном Раффом и Говардом Гордоном, в то время как исполнительный продюсер Майкл Куэста стал режиссёром.
Реакция
Рейтинги
Оригинальная трансляция пилотного эпизода 2 октября 2011 года в 10:00 получил 1.08 миллионов зрителей, став самой высокорейтинговой драматической премьерой на Showtime за восемь лет (с момента выхода «Мёртвые, как я», где также играл Патинкин, с 1.11 миллионом зрителей). Эпизод получил в целом 2.78 миллионов зрителей с дополнительными трансляциями, по запросу и по онлайн-просмотрам.
Рецензии
Пилотный эпизод получил всеобщее признание, получив рейтинг 91/100 на основе 28 отзывов на Metacritic. Хэнк Стёвер из Washington Post дал эпизоду A-, сказав, что «это заставляет "Родину" возвыситься над другими, это звёздная работа Дэйнс в роли Кэрри» и что «вторая половина первого эпизода будоражит. Я подсел.» Мэттью Гилберт из The Boston Globe сказал, что это его любимый драматический пилот сезона, дав ему оценку A. Кен Такер из Entertainment Weekly дал эпизоду оценку A-, заявив: «Это самая интригующая, напряжённая головоломка осени.»
Награды и номинации
Майкл Куэста получил номинацию на премию Гильдии режиссёров США за лучшую режиссуру драматического сериала, проиграв Пэтти Дженкинс за пилот «Убийства».
Эпизод получил номинации за лучшую режиссуру драматического сериала и лучший сценарий драматического сериала на церемонии вручения премии в 2012 году; Алекс Ганса, Говард Гордон и Гидеон Рафф выиграли премию за лучший сценарий драматического сериала.
Джордан Голдман и Дэвид Лэтэм выиграли премию «Эмми» за лучший монтаж драматического сериала за их работу над «Пилотом».
Эпизод выиграл премию Эдгара Аллана По за лучший телесценарий к эпизоду в 2012 году.
Примечания
Ссылки
«Пилот» на Showtime
Пилотные серии телесериалов
Эпизоды телесериала «Родина»
Эпизоды телесериалов — лауреаты премии «Эмми»
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\section{Introduction}
Early measurements of atmospheric densities and chemical composition close to the surface of Mars were carried out using earth-based remote sensing techniques of high resolution optical spectrometers incorporating the Doppler shift method. These measurements in visible and IR bands indicated $CO_2$ as the dominant constituent with the total atmospheric pressure of about $\approx$7 mb which were later found to be corroborated with the Viking 1 and 2 lander spacecraft results (Moroz et al., 1998 and references there in). Similarly from earth-based spectral line measurements, the Martian surface-air concentrations of water vapour ($H_2O$), molecular oxygen ($O_2$), carbon monoxide ($CO$) and isotope ratios of oxygen ($O$) and carbon ($C$) were determined (Kaplan et al.,1969; Barker et al., 1972; Carleton and Traub, 1972; Young and Young, 1977). During the initial years of space exploration, optical spectrometers in UV and IR bands were used in e.g., Mariner 9 orbiter mission resulting in the detection of hydrogen corona and the confirmation of small densities of ozone ($O_3$) in the polar region of Mars (Lane et al., 1973; Barth and Dick, 1974). There were similar observations of lower atmospheric constituents from other spacecraft missions including Mars 3 \& 5 and Mariner 6 \& 7. While the efforts to measure the atmospheric parameters including the densities of the gaseous constituents near the surface of Mars in the pre-Viking period i.e., before 1976, have provided some basic information, there was a need to carry out further observations particularly related to the spatial distribution of the atmospheric constituents in the thermosphere and exosphere of Mars. Such an opportunity was provided by the launch of two Viking spacecraft missions during September~1976 to carry out the observations using mass spectrometers both on the entry probe as well as on the lander. The first in-situ data of the Martian upper atmospheric composition, densities and temperatures were obtained from this twin Viking mission (Hanson et al., 1977; Nier and McElroy, 1977; Owen et al., 1977; Hanson and Mantas, 1988). This was followed by Phobos 2 (1989), Mars Global Surveyer (MGS, 1997) and the Mars Express (MEX, 2004) missions with main emphasis to study the interaction of solar wind charged particles with Mars, global mapping of the magnetic field and estimating escape rates of ions from Mars (Sagdeev and Zakharov, 1989; Acuna et al., 1999; Barabash et al., 2007). The Mars Pathfinder mission (1997) was the first successful lander-rover mission with similar findings as that of Viking (Magalhaes et al., 1999). Mars Odyssey (MO, 2002) and Mars Exploration Rovers Spirit and Opportunity (MER, 2004) searched for the surface features affected by flow of water and the atmosphere close to the ground (Mangold et al., 2004; Squyres et al., 2006). More recent missions like Mars Reconnaissance Orbiter (MRO, 2006), Phoenix lander (2008) and Mars Science Laboratory (MSL, 2012) revealed hemispherical differences in the variation of precipitable water (Smith et al., 2009), role of recent degassing due to volcanic activity in the evolution of modern atmosphere of Mars (Niles et al., 2010) and enhanced $D/H$ (Deuterium/Hydrogen) ratio in the clay minerals pointing to a longer history of hydrogen escape and hence water (Mahaffy et al., 2015). The quadrupole mass spectrometer (QMS) as part of the instrument suite on the MSL's Curiosity rover measured the chemical concentrations and isotopic fractions of volatile compounds. The surface atmospheric composition of Mars consists of about 96\% carbon dioxide ($CO_2$) and small concentrations of nitrogen ($N_2$), argon ($Ar$), and trace amounts of oxygen ($O_2$), Water vapour ($H_2O$) and ozone ($O_3$) (Mahaffy et al., 2013). The complete set of near surface meteorological data obtained from Viking landers (1976) to the Curiosity rover (2012) have been analysed and results have been consolidated in terms of diurnal, seasonal and inter-annual variations of meteorological parameters including dust storms over a span of more than 20 Martian years (Martinez et al., 2017). A similar analysis to characterise the thermosphere-exosphere system has not been possible due to paucity of observational data building up time sequences of the altitude and latitudinal profiles of the meteorological parameters and neutral/ionised gas concentrations (Bougher et al., 2014).
Recently the Indian Mars Orbiter Mission (MOM, 2014) and the Mars Atmosphere and Volatile Evolution (MAVEN, 2014) from NASA have been launched for the study pertaining to the evolution and escape of atmospheric/ionospheric constituents. MAVEN makes additional observations of the constituents in the fringe of the mesosphere. It is an important point to note that till September~2014 when MOM and MAVEN arrived at Mars, the data from the Viking mission remained the only in-situ measurements of neutral and ion composition of the thermosphere and its fringe region just touching the exosphere of Mars. The main purpose of this paper is to study the spatial and temporal distribution of atmospheric composition of the thermosphere-exosphere region of Mars which is essential to understand the diurnal, latitudinal, seasonal and solar activity driven variations and the escape of neutral/ion atmospheric species from Mars. The in-situ observations carried out by MENCA (Mars Exospheric Neutral Composition Analyser) and NGIMS (Neutral Gas and Ion Mass Spectrometer) on board MOM and MAVEN orbiter missions have opened up new vistas to further characterise the thermosphere-exosphere of Mars. The primary objective is therefore to reduce the gap in knowledge about the variations of atmospheric constituents in the exobase to exosphere of Mars using recent data from MENCA.
\begin{figure}[h]
\vspace*{0cm}
\centering
\makebox[0pt]{%
\includegraphics[width=0.8\paperwidth]{01.pdf}}
\caption{Neutral composition number density profiles in the thermosphere and exobase regions of Mars as measured by Viking 1 lander (Nier and McElroy, 1977). The lander covered SZA of 41$^\circ$-44$^\circ$ and ground co-ordinates in latitude and longitude of 14$^\circ$-17$^\circ$ and 302$^\circ$-306$^\circ$ respectively.\label{overflow}}
\end{figure}
\section{Early measurements of the upper atmosphere of Mars}
The first in situ observations providing the vertical profiles of neutral gases, ion, electron densities and temperatures of the atmosphere of Mars were carried out by the Viking 1 and 2 landers when these descended down to the surface of Mars.
The neutral mass spectrometric data on the composition of the upper atmosphere of Mars indicates that the main constituents of the Martian atmosphere are carbon dioxide ($CO_2$), Argon ($Ar$), molecular nitrogen ($N_2$), carbon monoxide ($CO$), and with photo-chemically produced atomic oxygen ($O$) (Nier and McElroy, 1977). The surface level volume mixing ratios of $CO_2$, $Ar$ and $N_2$ are 0.960, 0.0193 and 0.0189 respectively (Mahaffy et al., 2013). The altitude variations of some important atmospheric constituents above the well mixed atmosphere, i.e., above the mesosphere and into the thermosphere, where the diffusion of individual atmospheric species dominate in comparison to turbulent mixing (Haberle et al., 2002), are shown in Figure~1 as an example taken from Viking results (Nier and McElroy, 1977).
During the period of atmospheric observation while in descent from 200 km, the lander covered SZA of 41$^\circ$-44$^\circ$ and ground coordinates in latitude and longitude of 14$^\circ$-17$^\circ$ and 302$^\circ$-306$^\circ$ respectively (Withers and Lorenz, 2002). So the data on altitude variation of the gaseous constituents shown in Figure 1 are taken under similar solar radiation condition and within a marginal variation of 4 degrees of latitude and longitude. These results were later extended from 200 to 300 km through scale height extrapolations (Hanson et al., 1977). After the operation of Viking, a number of orbiter, lander and rover missions have enriched the information about Mars's atmospheric phenomena mainly in its meteorological context. Based on theoretical considerations, Mars atmospheric general circulation models such as Mars Climate Database (MCD) and Mars Thermospheric General Circulation Model (MT-GCM) have been developed extending into the exosphere but validated mainly using the in-situ observations from Viking (Lewis et al., 1999; Bougher et al., 2012). Another advantage of MOM/MAVEN data is that it helps determining the solar activity effects as the observations have been taken during the period of moderately high solar activity condition as compared to Viking probe measurements.
The present work therefore involves conversion of the available near-raw data of MENCA experiment into a calibrated data set with associated tags of time, altitude, latitude, longitude and solar zenith angles for detailed analysis to derive the exospheric composition and its variability.
\section{Observation and Data Analysis}
The Indian Mars Orbiter Mission (MOM) was launched on 5~November~2013. The probe arrived in Mars on 24~September~2014 into an highly eccentric orbit of 422~km x 76,993~km with an orbital period of $\approx$~67~hours. Later manoeuvres during December~2014 brought down the periareion altitude to around 262~km (Bhardwaj et al., 2016). One of the payloads of MOM is MENCA for the measurement of total atmospheric pressure and partial pressures of various atmospheric constituents covering 1-300 amu (Bhardwaj et al., 2015). The data sets used for this paper are obtained when MENCA was operated in the mass range of 1-100 amu. Raw data of MENCA payload were made to public for the period 24~September~2014 to 23~September~2017 through the ISRO Space Science Data Center (ISSDC). This archived data consists of total and partial pressure values in the units of Torr with time resolution of 12 to 30 seconds near periareion. Before this data can be used for scientific studies, it needs further processing in terms of calibration factors and ancillary data like latitude, longitude, altitude and solar zenith angle.
\begin{table}[h]
\vspace*{0cm}
\caption{Data availability sample; giving MOM orbit number and number of data files along with their time coverage}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
Orbit \# & Observation start time & Observation end time & Total files \\
\hline
0031 & 2014-10-17 11:01:17 & 2014-10-19 18:43:23 & 28 \\
0032 & 2014-10-20 02:01:18 & 2014-10-21 06:38:39 & 10 \\
0034 & 2014-10-28 05:41:41 & 2014-10-30 20:32:55 & 16 \\
0035 & 2014-11-01 23:16:21 & 2014-11-02 14:01:41 & 04 \\
0036 & 2014-11-02 16:26:04 & 2014-11-05 07:11:51 & 16 \\
0038 & 2014-11-05 10:13:05 & 2014-11-08 00:40:58 & 10 \\
0039 & 2014-11-08 03:34:50 & 2014-11-10 18:08:43 & 12 \\
0040 & 2014-11-11 19:03:42 & 2014-11-13 11:57:10 & 10 \\
\hline
\end{tabular}
\end{center}
\end{table}
The data is identified with respect to different orbit numbers of MOM for aforementioned period covering 88 orbits. The raw data of each orbit is included in a number of file pairs, each pair containing: (a) total atmospheric pressure (in Torr) and (b) partial pressures (in Torr) of the constituents for different time intervals. The individual files for partial pressures contain the data in a continuous time sequenced array form, which is converted into tabulated columnar format by marking each column for: (i) time of observation in UTC, (ii) corresponding 9-values of partial pressures for each constituent with amu between 1-100 and (iii) total pressures time synchronised with partial pressure measurement times using the second file in the pair. Table 1 shows an example of information on number of files containing data in a streaming time sequence mode for different orbits. ISSDC has also provided the specific SPICE (Spacecraft Planet Instrument C-matrix Events) kernel files for one year period of observations consistent with the internationally standardized service modules (Acton C.H. et al., 1996; 2017) required to compute the altitude, latitude, longitude and solar zenith angle values tagged to each time of observation so as to characterize and tabulate the pressure data with respect to these localization parameters.
Further data processing involves: (a) combining each pair of file to convert into one file having the partial pressures (maximum pressure value out of 9-values of observation for each of the 100 amu bins) as well as the time synchronised total pressure. Thus 100 maximum pressure values are selected for each amu from 1-100 for the same time epoch constituting a single record for further analysis, (b) computing the altitude, latitude, longitude and solar zenith angle for each time epoch of observation using the kernels of SPICE system mentioned earlier and (c) conversion of raw partial pressure data using calibration information provided along with the science data for different amu values. The final calibrated partial pressure values along with the associated ephemeral and spatial information for each orbit is used for further analysis and scientific studies of the atmospheric constituents of Mars. After the above treatment of data, the MOM orbit-wise analysis is carried out to select the useful partial pressure values from the lowest altitude to about 500 km. This height limit is decided based on the requirement of our present study. The sum of the partial pressures of major gas constituents is also used to compute the total pressures for studying relative variation with time, altitude, etc. This method is found useful when the concentrations of certain constituents like water vapour, molecular and atomic hydrogen etc. show abnormally high values due to out-gassing. Such contamination due to out-gassing has been reported by the payload team (Bhardwaj et al., 2017). Statistical analysis is carried out for deriving the monthly mean values and standard errors at 95\% confidence interval.
\section{Results}
The MENCA in-orbit payload operation plan was governed by the science objective to study the atmospheric composition in the exosphere which has been constrained by the variation of periareion (260-430 km) for the first year (September~2014 to October~2015) of observation. However this periareion coverage of lowest altitudes is best suited to meet the scientific objectives compared to the situation of subsequent observations.
\begin{figure}[h]
\vspace*{0cm}
\centering
\makebox[0pt]{%
\includegraphics[height=8.5cm,width=0.8\paperwidth]{02.pdf}}
\caption{Periareion altitude coverage of MENCA observations during October~2014 to October~2015\label{overflow}}
\end{figure}
\begin{figure}[h]
\vspace*{0cm}
\centering
\makebox[0pt]{%
\includegraphics[height=8.5cm,width=0.8\paperwidth]{03.pdf}}
\caption{MOM trajectory (dotted) and MENCA observations (coloured) in each orbit during Dec~2014 \label{overflow}}
\end{figure}
Figure 2 shows this variation of the altitude of the periareion of MOM spacecraft during the first year of data collection. From 27~May~2015 to 01~July~2015, MOM was behind the Sun as viewed from the Earth and hence no observations were made during the communication black-out period. During each MOM orbit, the MENCA payload has been operated to cover the descending and ascending tracks near the periareion as well as at locations away from the periareion, mainly to serve as background information.
Figure 3 shows the MOM tracks for the 11-orbits during December~2014 with the MENCA observation periods indicated by purple colour along these tracks. It can be seen that the observations have been taken at different sections of the track ensuring the altitudes near the periareion of MOM for observing the exosphere of Mars starting from about 260 km altitude. This height coverage of composition data is unique as there have been hardly any such in-situ measurements of Martian upper atmosphere deep into the exosphere.
The in-orbit mass spectrometric observations of MENCA payload are linked to the planetary co-ordinates in terms of altitude, latitude, longitude and SZA. These are determined using the auxiliary data files provided along with the pressure data of the atmospheric constituents.
\begin{figure}[h]
\vspace*{0cm}
\centering
\makebox[0pt]{%
\includegraphics[width=0.8\paperwidth]{04.pdf}}
\caption{Partial pressures of exospheric constituents from MENCA observations related to MOM orbit~$\#$55 on 24~December~2014, shown for both inbound and outbound separately along with the coverage of solar zenith angle, latitude and longitude.\label{overflow}}
\end{figure}
The MENCA dataset obtained following the above procedure has been analysed mainly to study the variation of partial pressures of gaseous composition. The exospheric profiles of these parameters are obtained for individual orbits specifically covering the region around periareion of MOM. A typical example of the altitude profiles of exospheric constituents derived from MENCA observations of the orbit $\#$54 for descending and ascending tracks of MOM is shown in Figure 4. Here we notice that pressure values drop with altitude by about 2-orders of magnitude between 260 and 500~km for descending track. However for the ascending part of orbit (a) the gradient is slower particularly between 260-350~km and (b) the absolute magnitude of this reduction in pressure is lower between 260 and 350 km during sunset as evident from the variation of SZA shown in the same figure. Combined together this decrease of pressure is slower with altitude compared to the Viking results below ~200 km due to the difference that above the thermosphere the gaseous elements follow diffusion equilibrium paths which are different for different species depending on their mass and temperature. It can also be seen that in the region of exosphere above 300~km, $CO_2$, $N_2$, $O$ and $O_2$ are the dominant atmospheric constituents. The figure also shows considerable reduction in atomic oxygen ($O$) density after sunset. This is because $O$ is mainly a product of photo-dissociation of $O_2$ and $CO_2$. As it is not possible to separate $N_2$ and $CO$ having the same amu of 28, the effective value of $N_2$ would be less due to the contribution of $CO$ which is a photo-dissociation product of $CO_2$ (Bhardwaj et al., 2017).
\newpage
\begin{figure}[h]
\vspace*{0cm}
\centering
\makebox[0pt]{%
\includegraphics[width=0.8\paperwidth]{05.pdf}}
\caption{Total pressure values estimated by summing partial pressures of a number of constituents for the minimum altitude of each MOM orbit ($\#$28-107) during the period October~2014 to May~2015.\label{overflow}}
\end{figure}
\begin{figure}[h]
\vspace*{0cm}
\centering
\makebox[0pt]{%
\includegraphics[width=0.8\paperwidth]{06.pdf}}
\caption{Variation of smoothed daily mean sunspot numbers (SSN) and solar zenith angle for the same period as in Figure 5\label{overflow}}
\end{figure}
\begin{figure}[h]
\vspace*{0cm}
\centering
\makebox[0pt]{%
\includegraphics[height=13cm, width=0.8\paperwidth]{07.pdf}}
\caption{Vertical profiles of total pressure for the inbound and outbound of MOM orbits during December~2014\label{overflow}}
\end{figure}
Hence, for the rest of the paper we would consider $CO_2$, $O$, $N_2$ and $Ar$ as the main elements to study the variation of exospheric composition. Also the total pressure is computed by summing up the partial pressures of all the major constituents as $CO_2$, $O$, $N_2$, $CO$, $NO$, $O_2$, $Ar$, $N$ and $He$. The contribution due to water vapour ($H_2O$), $H_2$ and $H$ are excluded from the sum as these constituents may be modulated by the degassing of the spacecraft.
The total pressure profiles for exospheric altitudes have been estimated from the available data per orbit and the results for both descending and ascending tracks are shown for the month of December, 2014 in Figure 7. It can be seen that, except for the anomalous pressure values on 24 December 2014 (reasons explained later in this paper), there is a progressive decrease in total pressure at all altitudes from beginning to end of the month and this decrease correlates well with the decrease in the smoothed daily mean SSN mentioned earlier. The large reduction of the total pressure within a month is difficult to explain without invoking the solar activity effects. By checking the variations of solar zenith angles, latitude and longitude, it is found that the effect of these is at best very marginal and not effective in bringing a change of this magnitude within a period of one month for these altitudes.
\begin{figure}[h]
\vspace*{0cm}
\centering
\makebox[0pt]{%
\includegraphics[height=11cm,width=0.8\paperwidth]{08.pdf}}
\caption{Mean total pressure profiles with standard errors derived from all the measurements during December 2014 both for inbound and outbound.\label{overflow}}
\end{figure}
The changes in solar UV/X-ray fluxes arriving at Mars due to variations of solar activity have a strong modulating effect on exospheric pressures. These variations due to solar activity are found to be larger than the standard errors at 95\% confidence interval for both descending and ascending orbital paths as shown in Figure 8. Where the standard error of the means at 95\% confidence interval is given by m$\pm$(1.96 $\sigma/\sqrt{n}$) (m is the sample mean, $\sigma$ the standard deviation of mean and $n$ is the number of samples). These error bars are shown in the figure.
Only at two points between 250-300 km altitude for the descending track of the orbit the error bars are relatively large due to the anomalous pressures profile of 24 December, 2014. This point is addressed again later in this paper. Similar analysis is carried out separately for the partial pressures of $CO_2$ and $O$ by taking the mean pressure and altitude values of descending and ascending part of the orbital tracks. Figure 9 shows the individual days profiles of $CO_2$ and $O$ pressures for different days (pertaining to different orbit numbers) of December, 2014. From this figure the following points can be noted: (a) there is a considerable day to day variability of the altitude profiles of both $CO_2$ and $O$ with a consistent decrease of pressure at all atitudes with respect to advancing days in the period, (b) the $O$ pressure is generally higher than that of $CO_2$ before the anomalous event of 24 December, (c) On 24 December there is a large enhancement of $CO_2$ and $O$ pressures between 260-350 km when the solar proton fluxes are enhanced due to a moderate Coronal Mass Ejection (CME) event in progress during 20-25 December, 2014, (d) the anomalous pressure values of $CO_2$ are higher than that of $O$ at all altitudes and the $O$ pressures are lower compared to $CO_2$ pressures while recovering from the event after 24 December 2014.
In order to explore the anomalous pressure enhancements of $CO_2$ and $O$, contours of these pressures with respect to time and altitude are plotted for periods between 21-27 December and 18-27 December respectively. These figures depict the following (a) $CO_2$ pressures started increasing anomalously from 22 December and reached 2 peaks by 24 December one around 275 km and the other around 295 km, (b) atomic oxygen ($O$) started increasing from 19 December and reached one strong peak around 275 km between 22-23 December, (c) the height range of the effect of pressure increase for $CO_2$ continued above 360 km but for $O$ it tapered around 330 km.
\begin{figure}[h]
\centering
\begin{minipage}[b]{0.4\paperwidth}
\includegraphics[height=6cm,width=\textwidth]{09a.pdf}
\end{minipage}
\hfill
\begin{minipage}[b]{0.4\paperwidth}
\includegraphics[height=6cm,width=\textwidth]{09b.pdf}
\end{minipage}
\caption{MOM orbit-wise vertical profiles of partial pressure measured during December~2014\label{overflow}}
\end{figure}
\newpage
\begin{figure}[t]
\centering
\begin{minipage}[b]{0.4\paperwidth}
\includegraphics[height=7cm,width=\textwidth]{10a.pdf}
\end{minipage}
\hfill
\begin{minipage}[b]{0.4\paperwidth}
\includegraphics[height=7cm,width=\textwidth]{10b.pdf}
\end{minipage}
\caption{Partial pressure contours of $CO_2$ during 21-27~December~2014 (left) and partial pressure contours of $O$ during 18-27~December~2014 from MENCA observations (right).}
\end{figure}
As can be inferred from the results presented in Figure 5 and 6, these anomalous effects are to be examined in terms of any eruptive events of solar high energy electromagnetic and charged particle radiations during December~2014 as both these radiations would arrive at Mars and interact with its atmosphere and surface. Since Mars does not have a magnetic field like that of Earth, in addition to the UV/X-rays, the charged particles or solar plasma components like energetic protons would directly interact with the neutral composition and initiate photodissociation and photoionisation. The absorption of particle energy would also result in modified densities of atmospheric species in certain height ranges due to increase in temperatures. To check this phenomenon, the solar activity as revealed by the smoothed daily mean SSN mentioned earlier, and the solar proton densities measured by Advanced Composition Explorer (ACE) spacecraft (srl.caltech.edu, 2018) are plotted for December~2014 and shown in Figure 11.
\begin{figure}[h]
\begin{center}
\includegraphics[height=9cm,width=0.8\paperwidth]{11.pdf}
\caption{Variation of smoothed daily mean sunspot numbers and solar proton flux along with moving averages during December~2014}
\end{center}
\end{figure}
The UV/X-rays fluxes for the period did not show any occurrence of solar flare during this period and hence are not shown in the figure. While it is known that CMEs, in large number of cases, are preceded by solar flares, it is also shown that this may not always be the case and occurrence of moderately strong CMEs during a very low solar activity year of 2009 has occurred without any associated solar flare (Nagaraja et al., 2018). Hence the variation of solar proton fluxes along with their moving averages in Figure 11 indicates that there is a large increase of energetic proton fluxes during the period 21-26~December~2014 with peak fluxes around 23~December~2014. The exact date of the peak of the corresponding $CO_2$ and $O$ pressures is not accurately discernable due to the fact that the observation periodicity is restricted to about 3-days orbital period of MOM. However it is clear the peaks of enhanced pressures of $CO_2$ has occurred with a delay of about 1 to 1.5 days. The $O$ pressures on the other hand show a broad maximum. The density of $O$ depends, apart ftom temperature on the photolysis and photo-ionisation effects and hence its enhancement and persistence of its peak values is primarily a cumulative effect compared to that of $CO_2$.
\section*{Summary and Conclusion}
\begin{itemize}
\item MENCA data available from ISSDC in its nearly raw form has been processed to create a new data set with orbit-wise data assimilation particularly between 250 and 500 km for the period from September~2014 to September~2015. The ancillary information on the altitude, latitude, longitude and solar zenith angles obtained using the SPICE kernels have been tagged to each epoch of time and measurement. Also the partial pressures of the exospheric constituents of Mars have been converted to actual values using the calibration information and normalisation procedure.
\item As a sample of altitude-pressure profiles of exospheric constituents $CO_2$, $O$, ($N_2$+$CO$), $O_2$,$NO$, $He$ \& $Ar$ obtained from MOM orbit $\#$55 near the periareion (covering $\approx$ 262-500 km in the exosphere of Mars) are generated and presented using this data set. The variation of profiles generally follows the exponential decrease with altitude with atomic Oxygen ($O$) concentrations being larger than that of $CO_2$ during daylight hours.
\item The total pressure is computed by summing up the partial pressures of all the major constituents i.e., $CO_2$, $O$, $N_2$, $CO$, $NO$, $O_2$, $Ar$, $N$ and $He$ to ensure removing those constituents affected by the degassing problem. The variation of total pressure at the periareion altitudes during October~2014 to May~2015 is well correlated with the smoothed daily sunspot numbers.
\item A time series plot of partial pressure-altitude profiles of $CO_2$ and $O$ during December~2014 shows gradual decrease in partial pressure values (at all altitudes) from beginning to end of December which is again consistent with the decrease in the smoothed daily sunspot numbers.
\item Superimposed on the December~2014 profiles of partial pressures of $CO_2$ and $O$ , there are anomalously high pressure profiles of both $CO_2$ and $O$ peaking on 24~December~2014. The contour plots show more details of the temporal and altitude build up, maximum and recovery phases of this anomaly. It is found that this anomaly is due to a moderately strong CME event of sun which peaked around 23~December~2014.
\end{itemize}
\section*{Acknowledgement}
The authors are grateful to Indian Space Research Organisation (ISRO) for providing necessary funds to carry out this work under a research project vide reference ISRO:SPL:01.01.33/16. We acknowledge the use of data from the Mars Orbiter Mission (MOM), first inter-planetary mission of the Indian Space Research Organisation (ISRO), archived at the Indian Space Science Data Centre (ISSDC) and, NASA's Navigation and Ancillary Information Facility (NAIF) for the necessary SPICE kernels of Mars. Thanks are also due to Dr. Anil Bhardwaj, MENCA Principal Investigator and the payload team members notably, Dr. Smitha V. Thampi, Dr. T. P. Das, Dr. M. B. Dhanya, Space Physics Laboratory (SPL), Vikram Sarabhai Space Center for their valuable inputs and discussions.
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\end{description}
\end{document}
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"redpajama_set_name": "RedPajamaArXiv"
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YourFireShop's collection of indoor fully vented Gas Log and Burner Systems. The Master Flame gas logs and burners are lifelike and offer a very natural looking flame. The Master Flame Triple Burner Gas Log Fireplace System or Single Burner Gas Log Fireplace System includes choice of log style: Red Oak, Aged Oak and Charred Split Oak.
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{
"redpajama_set_name": "RedPajamaC4"
}
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Optic nerve sheath meningiomas (ONSM) are rare benign tumors of the optic nerve. 60–70% of cases occur in middle age females, and is more common in older adults (mean age 44.7 years). It is also seen in children, but this is rare. The tumors grow from cells that surround the optic nerve, and as the tumor grows, it compresses the optic nerve. This causes loss of vision in the affected eye. Rarely, it may affect both eyes at the same time.
It is typically a slow growing tumor, and has never been reported to cause death. However, there is concern that the tumor can grow into the brain and cause other types of neurological damage. In some patients, the tumor grows so slowly that treatment is not necessary. Standard treatments are observation, surgery, radiation therapy, and combinations of the above.
Symptoms and signs
The most common symptom of ONSM is a gradual loss of vision in one eye. In a minority of patients this may be intermittent, at least to begin with. Less common symptoms include pain in the affected eye, protrusion of the eye, or double vision.
Natural history
ONSM does not improve without treatment. In many cases, there is gradual progression until vision is lost in the affected eye. However, this takes at least several months to occur, and a minority of patients remain stable for a number of years.
Diagnosis
Clinical examination will show an abnormal optic disc, either swollen or atrophic. Optociliary shunt vessels may be seen; the combination of these with progressive visual loss and optic disc atrophy is known as the Hoyt-Spencer triad. Visual acuity is usually but not always reduced.
When ONSM is suspected, MRI of the brain or orbits should be performed. This will usually show characteristic findings and confirm the diagnosis.
Treatment
Most ophthalmologists will not advocate any treatment unless visual loss is present and ongoing. Reports of patients with ONSM having no change in their vision for multiple years are not uncommon. If loss of vision occurs, radiation therapy will improve vision in about ⅓ of cases, and preserve vision in about ⅓ of cases. Surgery has traditionally been associated with rapid deteroriation of vision. However, newer surgical techniques may prove better for the treatment of ONSM.
Incidence
About 1–2% of all meningiomas are optic nerve sheath meningiomas. Meningiomas have an incidence of ~4.18/100,000 persons each year. Thus, ~10,000 meningiomas are diagnosed in the US each year; corresponding to ~100 cases of ONSM each year in the US. The actual number of meningiomas is likely much higher as it is very common in elderly women. ONSM comprises about 2% of orbital tumors, and about 10% of optic nerve lesions.
Neurofibromatosis type II (NF-2) affects around 9% of ONSM patients, where the incidence in the general population is around 0.03–0.05%. Thus NF-2 is felt to be a risk factor for the development of ONSM.
References
Ocular neoplasia
Brain tumor
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"redpajama_set_name": "RedPajamaWikipedia"
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Q: Creating Tables with MYsql in yii 1.1 Here is the scenario: I have been following the "yii" book by Larry Ullman in which he gave his MYsql for CMS but he didn't described any tool , how to create these tables of sql. The only way I know is through migrations but the sql written in the book is not working in migrations.
The sample sql is given for a table from the book :
CREATE TABLE IF NOT EXISTS yii_cms.user ( id INT UNSIGNED NOT NULL AUTO_INCREMENT,
username VARCHAR(45) NOT NULL,
email VARCHAR(60) NOT NULL,
pass CHAR(64) NOT NULL,
type ENUM('public','author','admin') NOT NULL,
date_entered TIMESTAMP NOT NULL DEFAULT CURRENT_TIMESTAMP,
PRIMARY KEY (id),
UNIQUE INDEX username_UNIQUE (username ASC),
UNIQUE INDEX email_UNIQUE (email ASC) )
ENGINE = InnoDB DEFAULT
CHARACTER SET = utf8
A: If you don't mind, you can use straight-forward way to execute sql so:
$sql = 'your sql here';
Yii::app()->db->createCommand($sql)->execute();
The other way is to use Yii QueryBuilder and the createTable() command. More information for QueryBuilder is here and for createTable() here.
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"redpajama_set_name": "RedPajamaStackExchange"
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Dan, here is a forum on care and storing helmets. Maybe you can find good information from someone's posting. Heat and cold are bad for helmets.
My grand kids hang their helmets on the handlebars - a good reminder to use them when going bike riding.
You can buy bins just about the right size at the Dollar Store. Pick up one and drop your helmet, riding gloves, etc. right into the bin and store it away.
Try Command hooks. They come in all sizes and for lots of different surfaces. And if it doesn't suit you, you haven't put a hole in your wall.
Well you could store it in sports bags and hang so they're easy to grab when you need them but closed in at least they won't get dusty.
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{
"redpajama_set_name": "RedPajamaC4"
}
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{"url":"https:\/\/yoursageinformation.com\/is-inorganic-benzene-polar-or-nonpolar\/","text":"# Is inorganic benzene polar or nonpolar?\n\n## Is inorganic benzene polar or nonpolar?\n\nInorganic benzene is completely non-polar.\n\n## Is organic benzene polar?\n\nBenzene is an organic chemical compound composed of six carbon atoms joined to each other forming a planar ring, and one hydrogen atom attached to each. The molecular formula for benzene is C6H6. Although C-H bonds are slightly polar, benzene is a nonpolar compound.\n\nHow is Borazine polar?\n\nWell as a whole molecule it should be polar. As dipole moment is a vector quantity and there are poles in inorganic benzene. As lone pair of electron on nitrogen are shared with boron there exist a formal positive charge on nitrogen and negative charge on boron.\n\n### Is Borazole polar compound?\n\nBorazine, also known as borazole, is a polar inorganic compound with the chemical formula B3H6N3. In this cyclic compound, the three BH units and three NH units alternate.\n\nREAD ALSO: \u00a0 Is it OK to put eye drops while sleeping?\n\n### Is BH3 polar?\n\nOriginally Answered: Is BH3 polar or nonpolar? Each B-H bond in BH3 is polar \/ forms a dipole because the B and H atoms have different electronegativities. The shape of the molecule is trigonal planar which is symmetrical, so the dipoles \/ bond polarities cancel. The resultingBH3 molecule is non-polar.\n\nWhy C6H6 is non polar?\n\nBenzene is nonpolar because the carbon C is slightly electronegative than the H (dipole moment is $\\text{ 0}\\text{. 35 }$ ) so $\\text{ C}-\\text{H }$ the bond is very slightly polar. However, the benzene has six such bonds which are directed in the opposite direction. Thus benzene has a zero dipole moment.\n\n#### Which benzene is most polar?\n\nHence the product of both is less than that of benzene chloride. Nd so benzene chloride is more polar than benzene flouride. This case of polarity reverse is there fr halogen series.\n\n#### Why is borazole more reactive than benzene?\n\nBorazole is more reactive than benzene towards EAS reaction. Borazine is a highly polar molecule due to high Electronegativity difference between Boron and Nitrogen. The pi bonds in borazine are highly polarized than pi bonds in benzene. Thus borazine is more nucleophillic hence more reactive than benzene.\n\nREAD ALSO: \u00a0 What is an example of electrical energy being converted to chemical energy?\n\nIs BrF3 polar or nonpolar?\n\nBrF3 has a pungent odor and appearance as straw colored liquid. Is BrF3 a polar or nonpolar molecule in nature? Because the presence of two lone pairs on the bromine atom causes the molecule\u2019s form to be deformed or bent, BrF3 (bromine trifluoride) is a polar molecule.","date":"2023-03-26 09:23:57","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6268473267555237, \"perplexity\": 6122.2872666444155}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296945440.67\/warc\/CC-MAIN-20230326075911-20230326105911-00366.warc.gz\"}"}
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Las termas romanas situadas en la localidad de Bath (Somerset) son un edificio de interés histórico, uno de los más importantes a nivel turístico de Inglaterra. El complejo está muy bien conservado, gracias a lo cual se pueden apreciar muy bien los elementos arquitectónicos presentes en el edificio. Las termas propiamente dichas se sitúan por debajo del nivel de la calle y los edificios construidos a raíz de su descubrimiento se pueden dividir en cuatro grupos entre los que están el «Manantial Sagrado», el Templo Romano, el Baño Romano y la Casa Museo. Estas estructuras, que se encuentran a nivel de la calle, datan del .
Los baños suponen una gran atracción turística y pueden llegar a recibir un millón de visitantes al año. En 2005 se los presentó en el programa de televisión del mismo nombre como una de las «Siete Maravillas Naturales» del West Country. Una vez en el complejo, los visitantes pueden ver los baños y el museo, aunque no pueden acceder al agua. Está disponible una guía de audio en varios idiomas.
Cómo se forman las aguas termales en Bath
El agua que finalmente constituye el núcleo de las aguas termales de los baños de Bath proviene originalmente de las lluvias que caen sobre Mendip Hills. Esta se filtra a través de los acuíferos de piedra caliza situados a una profundidad entre los 2700-4300 metros, donde la energía geotérmica eleva la temperatura del agua hasta los 64 °C (147,2 °F) y 96 °C (204,8 °F). Bajo dicha presión, el agua caliente sube a la superficie a lo largo de fisuras y fallas localizadas en la piedra caliza. Este proceso es similar al artificial conocido como Enhanced Geothermal System que también hace uso de las altas presiones y temperaturas por debajo de la corteza terrestre. El agua caliente a una temperatura de 46 °C (114,8 °F) se eleva aquí todos los días una tasa de 1 170 000 litros (257 364 galones imp), de una falla geológica (la Falla de Pennyquick). En 1983 apareció un nuevo hoyo en el interior del complejo que aseguraba un suministro continuo y limpio de agua a las instalaciones.
Historia
Imperio romano
El primer santuario de aguas termales erigido en este lugar fue construido por los celtas, que lo dedicaron a la diosa Sulis, cuyo equivalente romano sería Minerva. No obstante, el nombre de Sulis siguió usándose tras la conquista romana de Britania, dato probado debido al nombre de la población de Aquae Sulis (literalmente, «las aguas de Sulis»). El templo romano fue construido entre los años 60-70 y el complejo termal durante los siguientes 300 años. Durante la ocupación romana de la isla bajo el reinado del emperador Claudio, este ordenó a sus ingenieros que trajeran postes de roble con el fin de proporcionar al complejo una base sólida y que rodearan la fuente de la que brotaban las aguas termales con un cámara de piedra irregular recubierta de plomo. El complejo incluía un caldarium (baño caliente), un tepidarium (baño templado) y un frigidarium (baño frío). Tras la retirada de los romanos de Britania durante el el edificio cayó en desuso y finalmente quedó enterrado bajo un constante proceso de sedimentación. La Crónica anglosajona sugiere que los baños originales fueron destruidos durante el .
Reurbanización
Los baños se han visto sometidos a diversas modificaciones entre las que se incluyen las del , cuando Juan de Tours construyó un edificio de aguas curativas en la misma fuente del manantial que provee de agua a las termas y las del , cuando el gobierno de la ciudad construyó unos nuevos baños (Queen's Bath) ubicados al sur del manantial. El manantial está actualmente localizado en el interior de un complejo construido en el por los arquitectos John Wood (padre e hijo). Los visitantes podían beber el agua del manantial situada en una habitación llamada Pump Room, un salón de estilo neoclásico que actualmente permanece en funcionamiento, tanto para recoger las aguas del manantial como para albergar a los visitantes. La ampliación victoriana siguió la tradición neoclásica establecida por los Woods. En 1810 William Smith abrió un nuevo edificio llamado Bath Hot Spring en la parte inferior del complejo, donde se encontró con que el fracaso económico que suponían los baños no se debía a la sequía del manantial, sino a que este circulaba por un nuevo canal. Smith restauró el rumbo del agua a su curso original y los baños se llenaron sin ningún problema.
La entrada de los visitantes se realiza a través de una sala construida en 1897 por J. M. Brydon. Constituye una continuación hacia el este de la Great Pump Room con una cúpula de cristal en el techo. La construcción de la Great Pump Room se inició en 1789 por Thomas Baldwin. Baldwin dimitió en 1791 y John Palmer se hizo cargo del proyecto hasta su culminación en 1799. La elevación de la Abbey Church Yard tiene un centro constituido por cuatro columnas corintias con entablamentos y frontón. Ha sido designada por el English Heritage como un edificio de grado I. La columnata norte fue también diseñada por Thomas Baldwin, similar a la columnata sur a excepción del añadido en el segundo caso de un piso superior erigido a finales del . El museo y el Queen's Bath incluían un «puente» construido en 1889 por C. E. Davis que abarcaba el espacio situado entre la calle York y la lavandería de la ciudad.
Museo
El museo que alberga el complejo termal exhibe utensilios de la época romana entre los que se incluyen los que fueron arrojados al manantial sagrado, seguramente como ofrendas a la diosa Sulis. Entre los distintos descubrimientos realizados en la zona, se han encontrado 12 000 monedas romanas, que suponen la mayor ofrenda votiva de Gran Bretaña. También se puede ver en el museo una cabeza de bronce dorado de la diosa Sulis Minerva encontrada en 1727.
El templo del baño se alzaba sobre un podio de más de dos metros de alto y se accedía al templo subiendo unos escalones. En la entrada había cuatro grandes columnas acanaladas de estilo corintio que sostenían un friso y un frontón triangular decorado. En el museo se exhiben algunas partes del frontón, que medía 7,9 metros de ancho y 2,4 metros de alto. Destacaba la poderosa imagen central de la cabeza de Gorgona en el frontón, que miraba desde una altura de 15 metros a los que se acercaban al templo.
En las esquinas del frontón hay una pareja de tritones, criatura mitológica mitad hombre mitad pez y sirviente del dios de las aguas, Neptuno. El centro de la parte inferior izquierda está decorado por un delfín, mientras que la parte inferior derecha está protagonizada por un búho escondido. La parte central está decorada con grabados de mujeres que portan un escudo de hojas de roble, simbolizando con ello la Victoria. Por encima de todo destaca una gran estrella situada en lo que sería la parte más alta del edificio. Subyugada a la estrella se halla la cabeza de la gorgona con serpientes entrelazadas entre sus barbas, alas por encima de las orejas y un gran bigote. No obstante, existe un debate en relación con si este relieve representa una gorgona, ya que esta criatura es normalmente del sexo femenino. Existen interpretaciones alternativas que ven a la cabeza como la representación del dios del mar, Océano o como el dios del Sol de los celtas.
También se exponen los restos del sistema de calefacción de las saunas, el hipocausto.
Conservación
Las estatuas de finales del de emperadores romanos y gobernadores de la provincia de Britania son vulnerables a los efectos de la lluvia ácida, por lo que se han tenido que proteger mediante la aplicación cada pocos años de una capa de barniz. Las piezas dentro del templo son vulnerables al aire caliente que tenía el efecto de depositar sales corrosivas. Para intentar reducir esta erosión, se instaló en 2006 un nuevo sistema de ventilación.
Seguridad del agua
La ciudad de Bath fue investida con la responsabilidad sobre el Hot Springs en el Estatuto Real de 1591 otorgado por la reina Isabel I. Esta obligación ha pasado ahora al organismo conocido como Bath y North East Somerset Council, que lleva a cabo la vigilancia de la presión, temperatura y caudal de estas aguas. Los análisis realizados a estas aguas muestran que contienen sodio, calcio e iones de cloruro y sulfato en altas concentraciones.
El agua que fluye a través de los baños no se considera segura como para bañarse, en parte debido a su aún actual uso de lugar de paso de diversas tuberías y al descubrimiento durante la Segunda Guerra Mundial de la radioactividad que contenía. No obstante, el mayor peligro de todos reside en su condición de lugar de propagación de enfermedades infecciosas. En 1979 una niña bebió accidentalmente un poco del agua de los baños mientras nadaba en ellos y murió al cabo de cinco días a causa de una meningitis amébica. Las pruebas demostraron que el origen de la meningitis era un bacilo de Naegleria fowlerii que la niña había cogido en la piscina. Tras esta muerte, la piscina fue cerrada al público, estado en el que permanece hoy en día. En las cercanías de los baños romanos se construyó un edificio conocido como Thermae Bath Spa, diseñado por Nicholas Grimshaw & Partners. Este edificio permite a sus visitantes bañarse en aguas procedentes de unos pozos practicados inmediatamente antes del término de su construcción.
Galería
Referencias
Enlaces externos
Sitio oficial de las Termas romanas de Bath (en inglés)
Termas romanas Quicktime VR (en inglés)
Las termas romanas (video) (en inglés)
BBC vista de 360º (en inglés)
Bath
Bath
Arquitectura de la Antigua Roma en Inglaterra
Arquitectura de Inglaterra del siglo I
Arquitectura de Inglaterra del siglo XII
Arquitectura de Inglaterra del siglo XVIII
Museos de Inglaterra
Monumentos de Inglaterra
Patrimonio de la Humanidad en Inglaterra
Edificios listados de Grado I de Inglaterra
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The Valdez Star - Serving Prince William Sound and Copper River Basin
By Ned Rozell
UAF Science Writer
Volcano study brings risk of fire and ice
Scientists endure nights on mountaintop perch
Photo courtesy Taryn Lopez
Volcano researcher John Paskievitch covered in ice after three hours of creating and securing anchor points to tie off a Jet Ranger helicopter stranded by ice on the summit of Mount Mageik in the Valley of 10,000 Smokes.
Leaning against her Thermarest pad in a helicopter coated with ice, Taryn Lopez imagined herself as the little girl rocking to sleep in her parent's boat. Just before she drifted off on that early September night, the volcano researcher wondered if the climbing ropes would hold the Jet Ranger to the wind-pounded volcano on the spine of the Alaska Peninsula.
"We weren't sure if we'd wake up the next morning having moved a couple feet," she said.
In the back seat of the stranded helicopter, John Paskievitch was confident in his improvised anchors, but had a harder time falling asleep. He couldn't help thinking of the flying-rock windstorms he had experienced in 25 years of fieldwork in the Valley of 10,000 Smokes. And how most of that extreme weather occurred in places not nearly as exposed as this.
Sleep also eluded pilot Sam Egli of King Salmon as he shifted in his seat at the unfamiliar sensation of being wrapped in a sleeping bag. Egli made the call to stay on top of Mount Mageik when ice formed on blades of his helicopter during what was meant to be a short trip.
Overnighting near a steaming volcanic crater in a vessel weighing less than a compact car was not what any of the trio wanted, but it was a circumstance each had thought about before it occurred. Their foresight, experience and calm allowed them to survive 48 hours on top of Mount Mageik. Theirs is a story of a rare circumstance but one that is always possible when scientists perform fieldwork in remote spots.
The adventure started in a routine fashion. Lopez, who had flown down from Fairbanks, and Paskievitch, who lives near Anchorage, met at the airport in King Salmon. There, Egli operates Egli Air Haul with his family. Lopez, a postdoctoral student at the University of Alaska Fairbanks Geophysical Institute, is studying the relationship of volcanic gases to seismicity on Mounts Mageik and Martin and Trident Volcano. Paskievitch installs and fixes scientific equipment all over the Alaska Peninsula.
The next afternoon, with the weather clearing, Egli flew them into the Valley of 10,000 Smokes. First on Paskievitch's list was to fix a radio repeater. He made the repair quickly and Egli flew them deep into the valley, where he landed near the Baked Mountain huts, built by researchers a few decades ago and the only shelter for miles. To the south, they could see the blue-white summit of 7,103-foot Mount Mageik. Their next stop was to retrieve Lopez's equipment from near the steaming crater on top.
At the huts, the scientists dropped off excess gear, such as a computer and test equipment Paskievitch used at the repeater site. Lopez pulled on heavy long johns, quick-drying field pants, rain pants, wool socks, two wool shirts, a fleece sweater and a rain-shell. Paskievitch stepped into insulated coveralls and pulled on his climbing boots.
They boarded the helicopter with the summit of Mount Mageik visible seven miles away. Egli floated them up, and they were soon on the rim of the summit crater with a volcanic lake on one side and crevassed glacier on the other.
"We landed in excellent conditions," Paskievitch, who works at the USGS Volcano Science Center, said over the phone from Anchorage a few weeks after his adventure. "We go into places a lot where weather is an obvious factor you have to consider and you're on high guard. This wasn't one of those times. Nothing was threatening."
While Egli sat at the controls of the helicopter, Lopez and Paskievitch disassembled her monitoring equipment at the site, which included an ice-caked antenna attached to an aluminum pole. This was the last trip of the season — they would remove the instruments that had provided Lopez data on what types and amounts of gases the volcano emitted.
"We had worked for 28 minutes when Sam called out to us," Paskievitch said. "He said ice was forming on the helicopter blades and it was time to go."
Their job unfinished, the scientists gathered up loose gear and headed back to the helicopter. As they buckled in, Egli started rotating the blades.
"At that point, the weather did creep in," Paskievitch said. "(Sam) sat there at full throttle waiting for an opening in the weather. While he was waiting there, we watched his torque meter go from 27 to 35 to 40 percent without him doing anything."
The torque meter measures the strain on the rotor shaft. The increasing numbers showed that more ice was forming on the blades. Egli shut down the engine. He asked Paskievitch to clear the ice.
Paskievitch stepped out, leaned into the wind, and tapped the leading edges of each blade with the synthetic handle of a pick. Lopez got out and held the blades down while Paskievitch zipped his climbing rope over them to remove more ice.
After the 15-minute task was complete, Paskievitch and Lopez slipped back into the helicopter. Egli again cranked the engine to life. As the three waited for a hole in the clouds, they again noticed the numbers climbing on the torque meter. Egli shut down the engine. Paskievitch left the helicopter once more to manually de-ice the blades.
"After I cleared one off and was working on the other, I looked over and saw the clean one was picking up ice again," he said. "It was obvious it was a futile effort."
That was the moment all three realized they were not leaving the volcano any time soon. With the wind gusting at 70 mph, they made a mental switch. The helicopter was no longer transportation; it was a shelter that was far better than their second option, a blue plastic tarp.
Egli remained in the pilot's seat to help keep the helicopter pinned to the ground. Paskievitch pulled on his coat and squeezed out the door. Knowing the Jet Ranger has on its underside three metal tie-in points, he had a plan.
Using a Sawzall, which he carries to hack through corroded fasteners during equipment removal, Paskievitch sawed through the 8-foot long, 2-inch diameter aluminum pipe that had been Lopez's antenna mast. He cut it into three pieces, each a little more than two feet long.
Using a pick and shovel, he dug three trenches that would hold the buried pipes at right angles to the helicopter. Each trench was about three feet deep in the thawed rocks around the crater vent.
Paskievitch also had stout climbing rope, a good length of it, because he thought they might need to sling parts off the mountain back to Baked Mountain huts. He tied a clove hitch at the center of each pipe, dropped it in its trench and cut a thin channel for the rope in the direction of the helicopter. Then he backfilled the trenches with rocks and stomped on top. To finish, he tensioned the three ropes with a trucker's hitch.
"(The anchors) were set pretty good as these things go," he said. "It took about three hours. Whenever I got hot and started to sweat, I laid down and paced myself."
The weather worsened during the time Paskievitch secured the ship, so much that he found himself wearing a suit of ice armor as he slipped back into the helicopter. He took off his outerwear and stowed it in a trash bag, which he stored at his feet in the back seat, hoping his clothes would stay frozen and would retain some of their insulating value.
With their shelter secured, the three slowed down for a long wait. They each wiggled into their own sleeping bags, with Lopez and Paskievitch also slipping bivvy sacks over the top of their bags. Lopez had two liters of extra water; they decided to stash that under the seat for future use, while adding snow to their water bottles each time they exited the helicopter. They had a decent amount of food, even a slice of leftover pizza from a Naknek restaurant. Lopez also had a quart-size bag of trail mix and bars, Egli had a supply of survival food in the helicopter and Paskievitch had a large bag of granola and other edibles.
"I don't go anywhere without cheese," he said.
Equipped with three satellite phones and several radio systems including hand-helds, they started a routine of regular calls. Their early communications were to Egli's base of operations in King Salmon; to Michelle Coombs, Duty Scientist of the Alaska Volcano Observatory in Anchorage; and to Lopez's boyfriend David Fee, the acting Coordinating Scientist at the Alaska Volcano Observatory's branch in Fairbanks.
"I said, 'We're probably stuck for the night, please call my family so they can pray for us,'" Lopez said.
The helicopter grew heavier by the hour. Ice formed a shell several inches thick, with up to eight inches growing on the helicopter's windward side. They opened the doors as few times as possible, only leaving the helicopter to relieve themselves.
"When I went out one time, I kept getting knocked down by the wind," said Lopez, the petit member of the group.
After 24 hours with no signs of things getting better, and realizing that worsening weather or the loss of a helicopter door could turn their situation into life or death, the three decided to request a rescue. Their only other option, walking down the heavily crevassed mountain with a few dozen feet of visibility, was not a viable one.
Egli activated his Emergency Locator Transmitter. Detecting his signal, Alaska Air National Guardsmen at the Alaska Rescue Coordination Center, operating out of Joint Base Elmendorf-Richardson in Anchorage, started to move. Within six hours, an HC-130 was circling Mount Mageik and the pilot of a Pave Hawk helicopter was touching down in the Valley of 10,000 Smokes waiting for a clearing in weather. Also in the area was Bob Egli, Sam's son, who piloted another Egli Air Haul helicopter.
After their second night on the mountain, Paskievitch noticed a clear patch of sky overhead. He radioed that information down to the pilot of the Pave Hawk, ready in the valley below.
The large helicopter did not find that hole, but made overpasses for about three hours until another emerged. When the pilot radioed that he could finally see them from above, Egli, Paskievitch and Lopez exited the Jet Ranger and secured the doors.
The Pave Hawk landed on a flat spot on a summit glacier a few hundred yards from the Jet Ranger, which then resembled an ice sculpture. Two climbers on ropes attached to the Pave Hawk hiked over to the stranded trio. The rescuers told Egli, Paskievitch and Lopez to grab the rope and follow them back to the Pave Hawk.
"We were inside the helicopter in two minutes when I thought it would take forever," Lopez said. "The glacier looked bigger to me."
Shortly after they were inside the Pave Hawk and the rescuers closed the door, the rescue helicopter was on its way to King Salmon. Their ordeal was over, and just in time.
"They took advantage of that very brief window (of clear weather)," Paskievitch said. "The first time the site was accessible in a casual, routine way was six days after we came off."
Back home and safe in Fairbanks, Lopez remembers a "pretty mellow" wait in the plastic bubble clinging to a mountain.
"Because I was with really experienced people and we had shelter I was OK," Lopez said a few weeks after the incident. "Sam and John were really calm and collected and they liked to joke around. I never felt scared."
She learned a lot by watching Paskievitch install improvised deadman anchors in the mountain. She also appreciated how he would exit his side door of the helicopter and walk over to chip ice from her door when she needed to get out.
Ice buildup on a Jet Ranger helicopter that became stranded (by ice buildup) on the summit of Mount Mageik near the Valley of 10,000 Smokes in early September 2013. Sam Egli, the owner of the helicopter, later flew up with another helicopter and a mechanic who thawed this helicopter. Egli eventually flew this helicopter back to his base in King Salmon.
"I was very lucky to be stuck with him," Lopez said.
As for Paskievitch, he appreciated Egli's seasoned decision to remain on the mountain in poor conditions, the professionalism of the rescue team, and that he got a chance to see the Pave Hawk refueled by an HC-130 on the way to King Salmon.
"I was very impressed and thankful," he said.
At the end of the adventure, after arriving safely at Sam Egli's hanger in King Salmon, Paskievitch reached into his pack and felt the plastic bag that contained the slice of pizza he saved for when they might really need it. He pulled out the Ziploc, opened it, and ate the pizza for lunch.
(Since the late 1970s, the University of Alaska Fairbanks' Geophysical Institute has provided this column free in cooperation with the UAF research community. Ned Rozell is a science writer for the Geophysical Institute.)
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Monitoring a mystery b...NED ROZELL
The Valdez Star
310 Pioneer Street, box 2949
Valdez, Alaska, 99686
info@valdezstar.net
© 2019 Far North Media Inc.
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{"url":"https:\/\/discuss.codechef.com\/t\/mancbst-editorial\/17708","text":"# MANCBST - Editorial\n\nTester: Misha Chorniy\n\nEditorialist: Animesh Fatehpuria\n\n### PROBLEM\n\nCount number of semi-BSTs having exactly n nodes. A semi-BST is a tree with n nodes which satisfies the following conditions:\n\n\u2022 Each node contains a distinct value from \\{1, 2, 3, \\dots, n\\}.\n\u2022 The root has at most two children.\n\u2022 The right subtree of the root (if it exists) is a valid semi-BST.\n\u2022 The left subtree of the root (if it exists) is any unordered rooted tree. That is, the children of a vertex are not ordered, and hence no concept of left or right child.\n\u2022 The BST property for the root is maintained, i.e. all the values in the left subtree of the root are lesser than that of the root and all the values in the right subtree of the root are greater than that of the root.\n\nConstraints: n \\le 10^5, ext{test cases} \\le 10^5.\n\nPrint the answer modulo 663224321 i.e. 2^{19} imes 5 imes 11 imes 23 + 1.\n\n### PREREQUISITES\n\nKnowledge of \u201conline\u201d FFT trick.\n\n### EXPLANATION\n\nWe first solve the problem in O(n^2), and then optimize it using a trick.\n\n## An O(n^2) Solution\n\nWe use a dynamic program similar to \u201ccount number of BSTs having n nodes\u201d. Define T(n) to be number of semi-BSTs of size n. To find a recurrence for T(n), we will iterate over the number of nodes in the left subtree. Note that fixing the number of nodes in the left subtree determines the value of the root. Suppose we say that the left subtree has i\nnodes. Then, the number of labeled trees with i nodes is i^{i - 2} by Cayley\u2019s Formula. Thus, the number of ways for us to choose a left subtree is i imes i^{i - 2} = i^{i - 1} since we have i choices for the root of the left subtree as well. By definition, the right subtree can be chosen in T(n - 1 - i) ways. Note that these two are independent, making our recurrence:\n\nT(n) = \\sum_{i = 0}^{n - 1} (i^{i - 1} imes T(n - 1 - i)) with base cases T(0) = T(1) = 1.\n\nClearly, there are O(n) states, and it takes O(n) to compute each state. Therefore, this solution is O(n^2).\n\n## Speedup using Online FFT\n\nIt turns out that we can speed up the above dynamic program using a trick known as online FFT. The special modulo would have given this hint as well! For reference, please solve this Codeforces problem. The editorial for this problem explains this trick!\n\nAdditionally, you can check out these brilliant slides written by Tanuj Khattar! Interestingly, his team TooWeakTooSlow were the only team to solve this problem in the onsite regional.\n\nThe final complexity would be O(n \\log^2{n}).\n\n### AUTHOR\u2019S AND TESTER\u2019S SOLUTIONS:\n\nAuthor\u2019s solution can be found here.\nTester\u2019s solution can be found here.\n\nAlternate (much simpler?) solution using generating functions: https:\/\/www.codechef.com\/viewsolution\/18002620\n\nLet s* = i * (i -1),\nand t[n] be the answer for n nodes.\n\nS(z) = s[0] * x^0 + s[1] * x^1 + \u2026\nT(z) = t[0] * x^0 + t[1] * x^1 + \u2026\n\nT(z) = z * S(z) * T(z) + 1\n\nT(z) = 1 \/ (1 - z * S(z))\n\nUse polynomial inverse to get AC. (Refer http:\/\/codeforces.com\/problemset\/problem\/438\/E for more details)\n\n2 Likes","date":"2019-07-19 03:43:29","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8257933855056763, \"perplexity\": 1306.8747180733506}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-30\/segments\/1563195525974.74\/warc\/CC-MAIN-20190719032721-20190719054721-00440.warc.gz\"}"}
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\section{}
\section{Introduction}
Let $E$ be an elliptic curve over the rational numbers $\Q$. The $\Q$-rational points form a finitely generated abelian group $E(\Q)=\Z^r\oplus E(\Q)_{tors}$, and the classical Birch and Swinnerton-Dyer conjectures predict that the order of vanishing $r^{an}_\C$ of the Hasse-Weil $L$-function $L(E,s)$ at $s=1$, an analytic quantity, should be equal to $r$, an algebraic quantity. The second part of their conjecture says that the leading Taylor coefficient of $L(E,s)$ should encode $E(\Q)_{tors}$ and the size of the Tate-Shafarevich group $\Sha(E/\Q)$, among other algebraic quantities. More precisely, this conjecture says the following.
Denote by $\omega=\omega_E$ the N\'{e}ron differential and by $\Omega_E=\int_{E(\R)}\omega\in\R^{>0}$ the N\'{e}ron period of $E$.
We denote by $L^*(E)$ the leading coefficient of the Taylor expansion at $s=1$.
\begin{conjecture}[BSD]\quad
\begin{enumerate}
\item We have $r^{an}_\C=r$.
\item $\frac{L^*(E)}{\Omega_E}=\frac{\prod_v c_v \cdot \#\Sha(E/\Q)\cdot \Reg_\C(E/\Q)}{(\#E(\Q)_{tors})^2}$.
\end{enumerate}
Here, $c_v$ denotes the Tamagawa number for a place $v$, and the regulator $\Reg_\C(E/\Q)$ is the discriminant of the N\'{e}ron-Tate canonical height pairing on $E(\Q)$
\end{conjecture}
To construct a $p$-adic analogue, we identify algebraic numbers with $p$-adic numbers by fixing an embedding $\overline{\Q}\hookrightarrow \C_p$. A $p$-adic analogue of this conjecture should look as follows. There should be a $p$-adic $L$-function $L_p(E,T)$, whose order of vanishing $r^{an}$ at $T=0$ should equal $r$, and whose leading Taylor coefficient should again encode algebraic quantities of $E$ including $E(\Q)_{tors}$ and $\#\Sha(E/\Q)$ as in the above formula, with the regulator $\Reg_\C$ replaced by a $p$-adic avatar. There are two types of such $p$-adic analogues.
The first type concerns the case when $p$ is a prime of ordinary reduction (meaning that $p$ is coprime to $a_p:=p+1-\#E(\Fp)$). Here, Mazur, Tate, and Teitelbaum formulated a $p$-adic version of these conjectures. The $p$-adic $L$-function $L_p(E,T)$ they employed gave rise to the expected version of $p$-adic BSD when $p$ was of good ordinary reduction, but in the split multiplicative case $r^{an}$ corresponded to $r+1$ in view of an extra zero.
The second type is the supersingular case (the case when $p|a_p$), which is more complex. There are \textit{two} classical $p$-adic $L$-functions, denoted $L_\alpha(E,T)$ and $L_\beta(E,T)$, constructed independently by Amice and V\'{e}lu, and Vishik. The subscripts $\alpha$ and $\beta$ denote the roots of the Hecke polynomial $Y^2-a_pY+p$. The formulation of a $p$-adic analogue of the Birch and Swinnerton-Dyer conjectures in terms of these $L_\alpha(E,T)$ and $L_\beta(E,T)$ when $p$ is of good reduction is due to Bernardi and Perrin-Riou. For questions of formulating Birch and Swinnerton-Dyer conjectures, this seems to suggest that the picture is complete except for some primes of bad reduction. The goal of this paper is to indicate that this is not the case.
We do this by formulating $p$-adic versions of BSD for supersingular primes $p$ using a more natural pair of $p$-adic $L$-functions $L_\sharp(E,T)$ and $L_\flat(E,T)$. This hints at a formulation of $p$-adic BSD in the ordinary case in terms of such a pair as well. These $p$-adic $L$-functions had been constructed by Pollack, Kobayashi, and the author in the supersingular case and are functions living in the power series ring $\Z_p[[T]]$, unlike $L_\alpha(E,T)$ and $L_\beta(E,T)$ which have more complicated growth properties. Apart from being an ingredient for a natural formulation of the Iwasawa Main Conjecture, the appearance of the Iwasawa invariants of this pair of $p$-adic $L$-functions indicates that this pair is the natural choice: The Iwasawa invariants appear in analytic estimates for the sizes of the $p$-primary parts of the Tate-Shafarevich group along the cyclotomic $\Z_p$-extensions. The term ``analytic estimates" refers to the corresponding special values of the Hasse-Weil $L$-functions twisted by characters of $p$-power conductor, which should encode these sizes. These \textit{analytic} estimates can be found in \cite{pollack} and \cite
{surprisesha}. There are algebraic counterparts of $L_\sharp(E,T)$ and $L_\flat(E,T)$, which are modified Selmer groups $\Sel^\sharp(E)$ and $\Sel^\flat(E)$. Using their Iwasawa invariants one can estimate these sizes directly, see e.g. \cite{kobayashi} and \cite{sha3}. In addition to these estimates, the Iwasawa invariants appear in upper bounds for the rank of the elliptic curve in the cyclotomic $\Z_p$-extension of $\Q$. For analytic estimates, see \cite{pollack} and \cite{surprisesha}, and for their algebraic counterparts, see \cite{kobayashi} and \cite{rank}.
Concretely, our conjecture says the following.
\begin{conjecture}[Tandem $p$-adic BSD]Let $E$ be an elliptic curve and $p$ a prime of good supersingular reduction.
Denote by $\overrightarrow{L}_p^*$ the first non-zero Taylor coefficient around $T=0$ of the vector of $p$-adic $L$-functions $(L_\sharp(E,T),L_\flat(E,T))$.
\begin{enumerate}
\item The minimum of the orders of vanishing of $L_\sharp(E,T)$ and $L_\flat(E,T)$ at $T=0$ is equal to $r$.
\item $\overrightarrow{L}_p^*=\frac{\prod_v c_v \cdot \#\Sha(E/\Q)}{(\#E(\Q)_{tors})^2} \Reg_p^\natural(E/\Q).$
\end{enumerate}\end{conjecture}
Our regulator $\Reg_p^\natural(E/\Q)$ is constructed explicitly from height functions that mirror the construction of $L_\sharp(E,T)$ and $L_\flat(E,T)$.
\begin{theorem}This conjecture is equivalent to the conjectures of Bernardi and Perrin-Riou. \end{theorem}
Since we have two functions at hand, we may consider their quotient, and give the following criterion for detecting non-zero rank:
\begin{theorem} Assume property $(*)$ below holds and that $\Sha(E/\Q)[p^\infty]<\infty$. Then
$$ \rank{E(\Q)}>0 \iff \left. \frac{L_\sharp(E,T)}{L_\flat(E,T)} \right|_{T=0}\neq \frac{-a_p^2+2a_p+p-1}{2-a_p} \text{ for odd $p$, and } $$
$$ \rank{E(\Q)}>0 \iff \left. \frac{L_\sharp(E,T)}{L_\flat(E,T)} \right|_{T=0}\neq \frac{-a_2^3+2a_2^2+3a_2-4}{-a_2^2+2a_2+1} \text{ for $p=2$}. $$\end{theorem}
This theorem is a generalization of \cite[Corollary 0.5]{kuriharapollack} who assumed $p$ to be odd and $a_p=0$, which is automatically satisfied when $p\geq5$. We remark that the proof of Kurihara and Pollack almost immediately generalizes once the $L_\sharp(E,T)$ and $L_\flat(E,T)$ have been defined in complete generality, as done in \cite{shuron}. There is only one proposition in their tools that assumes $a_p=0$, which is fixed in this paper by adhering to a generalization found in \cite{sha3}.
As a corollary to this, we get:
\begin{corollary}Let $\rank E(\Q)=0$. Then both $L_\sharp(E,T)$ and $L_\flat(E,T)$ are non-zero functions, confirming \cite[Conjecture 6.15]{shuron} in this case.
\end{corollary}
Another object to consider given two quantities is their greatest common divisor. Denote by $d_n$ the normalized jump in ranks $\frac{1}{p^n-p^{n-1}}\left(\rank E(\Q_n)- \rank E(\Q_{n-1})\right)$, where $\Q_n$ is the $n$th layer in the cyclotomic $\Z_p$-extension numbered so that $\Q_0=\Q$. We also let $\Phi_{p^n}$ be the $p^n$th cyclotomic polynomial. Kurihara and Pollack have formulated the following conjecture in the case $a_p=0$ and $p$ odd:
\begin{conjecture}Let $E/\Q$ be an elliptic curve, and $p$ a good supersingular prime. Then
$$\gcd(L_\sharp(E,T),L_\flat(E,T))=\left(T^{r}\prod_{ d_n \geq1 \text{ and } n\geq 1}\Phi_{p^n}^{d_n-1}(1+T)\right).$$
\end{conjecture}
Note that we excluded their assumptions. This is because we give some evidence towards their conjecture by proving the following proposition which works for general supersingular $p$. Denote by $r^{an}$ the order of vanishing of $L(\alpha,T)$ (or $L(\beta,T)$) at $T=0$:
\begin{proposition}\label{kuriharapollack}Let $E/\Q$ be an elliptic curve and $p$ a prime of good reduction. For some polynomial $P_\Sha(E,T)$ with $P_\Sha(E,\zeta_{p^n}-1)\neq0$ for $n\geq0,$
$$\gcd\left(L_\sharp(E,T),L_\flat(E,T)\right)=\left(P_\Sha(E,T)\cdot T^{r^{an}}\prod_{ d_n \geq1 \text{ and } n\geq 1}\Phi_{p^n}^{\epsilon_n^{an}-1}(1+T)\right),$$
where $\epsilon_n^{an}=d_n^{an}$ or $\epsilon_n^{an}=d_n^{an}+1.$
\end{proposition}
Finally, we ask a question on a possible generalization of the situation to the ordinary case. When $p$ is ordinary, the functions $L_\sharp(E,T)$ and $L_\flat(E,T)$ are not uniquely defined, but their values at $T=0$ are. We find that when $p$ is odd and $a_p=2$, $L_\flat(E,0)=0$.
\begin{question} Where does this `extra zero phenomenon' in the ordinary case come from?
\end{question}
In \cite{loefflerzerbes}, Loeffler and Zerbes find a pair of Iwasawa functions in the ordinary case using the theory of Wach modules. The above question suggests that an extra zero phenomenon should occur in terms of their pair as well.
\section{Review of $p$-adic $L$-functions for elliptic curves}
Let $p$ be a prime of good reduction for our elliptic curve $E$. The work of Mazur, Swinnerton-Dyer, Amice and V\'{e}lu, and Vi\v{s}ik gives constructions of $p$-adic $L$-functions that should encode the behavior of the $\Q_n$-rational points $E(\Q_n)$ of the elliptic curve $E$. Concretely, denote by $\alpha$ and $\beta$ the roots of the Hecke polynomial $Y^2-a_pY+p$ ordered so that $\ord_p(\alpha)\leq\ord_p(\beta)$ for any $p$-adic valuation $\ord_p$. We say that $\alpha$ resp. $\beta$ is an allowable root if $\ord_p(\alpha)<\ord_p(p)$ resp. $\ord_p(\beta)<\ord_p(p)$. Notice that $\alpha$ is always allowable by convention, while $\beta$ is only when $p$ is supersingular.
\notation We denote by $\zeta_{p^n}$ a primitive $p^n$th root of unity, and we let $N=n+1$ when $p$ is odd and $N=n+2$ when $p=2$. We also let $\chi_u$ be a group morphism from $1+2p\Z_p$ into $\C_p^\times$ sending a topological generator $1+2p$ to some $u\in\C_p$ so that $|u-1|_p<1$, where $|\quad |_p$ is the normalized $p$-adic absolute value (i.e. $|\frac{1}{p}|_p=1$).
\begin{theorem}[Mazur and Swinnerton-Dyer, Amice and V\'{e}lu, Vi\v{s}ik]\cite[Proposition in Section 14]{mtt} Let $E$ be an elliptic curve over $\Q$ and $p$ a prime of good reduction. Regard $\chi_{\zeta_{p^n}}$ and $\chi_{\zeta_{p^n}^{-1}}$ as characters of $\Z_p/p^n\Z_p.$ There is a $p$-adic analytic function $L_\alpha(E,T)$ converging on the open unit disk with the following interpolation properties:
$$L_\alpha(E,\zeta_{p^n}-1)=\frac{p^N}{\alpha^N\tau(\chi_{\zeta_{p^n}^{-1}})}\frac{L(E,\chi_{\zeta_{p^n}^{-1}},1)}{\Omega_E} , and$$
$$ L_\alpha(E,0)=\left(1-\frac{1}{\alpha}\right)^2\frac{L(E,1)}{\Omega_E}.$$
When $\beta$ is allowable (i.e. $p$ is supersingular), there is a companion $p$-adic analytic function $L_\beta(E,T)$ with the same interpolation properties with $\alpha$ replaced by $\beta$.
\end{theorem}
While these classical $L_\alpha(E,T)$ have bounded coefficients when $p$ is ordinary, they are not elements of $\Z_p[[T]]$ (or even of $\Z_p[[T]]\otimes \overline{\Q}_p$) when $p$ is supersingular. The following theorem fixes this situation:
\begin{theorem}\label{maintheorem}\cite[Theorem 5.6 for $a_p=0$]{pollack}\cite[Theorem 6.12 for general $p|a_p$]{shuron} Let $p$ be a prime of good supersingular reduction. Then there are two $p$-adic $L$-functions $L_\sharp(E,T)$ and $L_\flat(E,T)$ which are elements of $\Z_p[[T]]$ so that we may write
$$(L_\alpha(E,T),L_\beta(E,T))=(L_\sharp(E,T), L_\flat(E,T))\Log_{\alpha,\beta}(1+T),$$
where $$\Log_{\alpha,\beta}(1+T):=\lim_{n \rightarrow \infty}\smat {a_p & 1 \\ \Phi_{p^1}(1+T) & 0 }\smat {a_p & 1 \\ \Phi_{p^2}(1+T) & 0 }\cdots \smat {a_p & 1 \\ \Phi_{p^n}(1+T) & 0 }\smat{ a_p & 1 \\ p & 0}^{-(N+1)}\smat{-1 & -1\\ \beta & \alpha}$$
is a matrix of $p$-adic analytic functions converging on the open $p$-adic unit disk.
\end{theorem}
\begin{theorem}\cite[Theorem 2.14]{surprisesha} Let $p$ be an ordinary good prime. Then the first column $\log_\alpha$ of $\Log_{\alpha,\beta}(1+T)$ converges and we have $$L_\alpha(E,T)=(L_\sharp(E,T), L_\flat(E,T))\log_\alpha, $$ for two $p$-adic analytic functions converging on the closed $p$-adic unit disk $L_\sharp(E,T)$ and $L_\flat(E,T)$.
\end{theorem}
We remark that these $L_\sharp(E,T)$ and $L_\flat(E,T)$ are not uniquely defined in the ordinary case, but their values at $T=0$ are unique:
\begin{lemma}The value of the vector $\left(L_\sharp(E,0),L_\flat(E,0)\right)$ equals\begin{linenomath}$$\begin{cases}(-a_p^2+2a_p+p-1,-a_p+2)\cdot\frac{L(E,1)}{\Omega_E}& \text{ when $p$ is odd,}\\(-a_p^3+2a_p^2+2pa_p-a_p-2p,-a_p^2+2a_p+p-1)\cdot\frac{L(E,1)}{\Omega_E}& \text{ when $p$ is even.}
\end{cases}$$\end{linenomath}
\end{lemma}
\begin{proof}This follows from the table before Proposition 6.14. in \cite{shuron} in the supersingular case and the one before Conjecture 5.18 in \cite{surprisesha} for the general case.
\end{proof}
\section{The behavior at $T=0$: A criterion for non-zero rank}
The goal of this short section is to prove:
\begin{theorem} Assume property $(*)$ below holds and that $\Sha(E/\Q)[p^\infty]<\infty$. Then
$$ \rank{E(\Q)}>0 \iff \left. \frac{L_\sharp(E,T)}{L_\flat(E,T)} \right|_{T=0}\neq \frac{-a_p^2+2a_p+p-1}{2-a_p} \text{ for odd $p$, and } $$
$$ \rank{E(\Q)}>0 \iff \left. \frac{L_\sharp(E,T)}{L_\flat(E,T)} \right|_{T=0}\neq \frac{-a_2^3+2a_2^2+3a_2-4}{-a_2^2+2a_2+1} \text{ for $p=2$}. $$
\end{theorem}
Let $T_p(E)$ be the Tate module for $E$.
Here is property $(*)$: The composite of natural maps
$$ \mathbf{H}^1_{\text{glob}}=\varprojlim H^1_{\text{\'{e}t}}(\O_{\Q_n}[1/S],T_p(E))\rightarrow H^1(\Q,T_p(E))\rightarrow H^1(\Q_p,T_p(E))$$
is not zero. Here, the limit is taken with respect to corestriction, and $S$ is the product of the bad reduction primes and $p$.
We remark that property $(*)$ should always be true.
\begin{proof}[Proof of Theorem]The arguments of Kurihara and Pollack of \cite[Section 1.4]{kuriharapollack} almost work with the appropriate modifications. In terms of notation, we write $Col(z)=(Col^\sharp(z),Col^\flat(z))$ with the Coleman maps from \cite{shuron} instead of the functions $h_z(T)$ and $k_z(T)$. We note that their arguments work in spite of having to take into account the possibility of one of $L_\sharp(E,T)$ or $L_\flat(E,T)$ to be zero, cf. \cite[6.15]{shuron} and the surrounding discussions! Finally, \cite[Proposition 1.2]{kuriharapollack} is only proved in the case $a_p=0$. To remedy this, we refer to the isotypical component of the trivial tame character of the inverse limit of \cite[Proposition 4.7]{sha3} and remark that the arguments found therein all apply when $p=2$ as well.\end{proof}
\begin{corollary} \cite[Conjecture 6.15]{shuron} said that both $L_\sharp(E,T)$ or $L_\flat(E,T)$ are non-zero. This conjecture holds in the case where $\rank E(\Q)=0$.
\end{corollary}
\begin{proof}We know that at least one of $L_\sharp(E,T)$ or $L_\flat(E,T)$ is non-zero by \cite[Proposition 6.14]{shuron}, so the theorem tells us they both have to be.
\end{proof}
\section{The behavior at $T=0$: $p$-adic versions of the BSD conjectures}
The goal of this section is to formulate $p$-adic versions of BSD in terms of the vector $(L_\sharp(E,T),L_\flat(E,T))$ in the supersingular case. We thus assume that $p$ is supersingular for this section.
\subsection{Dieudonn\'{e} modules and $p$-adic heights}
The Dieudonn\'{e} module is the following two-dimensional $\Q_p$-vector space:
\begin{linenomath}$$D_p(E):=\Q_p\otimes H^1_{dR}(E/\Q)$$\end{linenomath}
There is a Frobenius endomorphism $\phi$ which acts on $D_p(E)$ linearly. We refer the reader to \cite[Paragraph 2]{bpr} for a concrete definition, but let us record that its characteristic polynomial is $Y^2-\frac{a_p}{p}Y+\frac{1}{p}$, as opposed to the definition of \cite{mazursteintate} or \cite{kedlaya} (where it is $Y^2-a_pY+p$). This vector space admits a basis $\omega$ and $\phi(\omega)$, where $\omega$ is the invariant/N\'{e}ron differential of $E$.
We want to define eigenvectors $\nuek:=\nu_{\frac{1}{\alpha}}$ and $\nudo:=\nu_{\frac{1}{\beta}}$ of $\phi$ with eigenvalues $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ which live in the $\Q_p(\alpha)$-vector space
\begin{linenomath}$$D_p(E)(\alpha):=\Q_p(\alpha)\otimes H^1_{dR}(E/\Q).$$\end{linenomath}
\begin{definition}We define (scale) both eigenvectors as follows:
\begin{linenomath}$$\hidari \nuek \\\nudo \migi :=\links -\alpha & p \\ \beta & -p \rechts \frac{1}{\beta-\alpha}\hidari \omega \\ \phi(\omega) \migi$$\end{linenomath}
\end{definition}
\begin{definition}Perrin-Riou's $p$-adic $L$-function can be defined via the classical $p$-adic $L$-functions $L_\alpha:=L_\alpha(E,\alpha,T)$ and $L_\beta:=L_p(E,\beta,T)$:
\begin{linenomath}$$L_p^{PR}(E,T):=(L_\alpha,L_{\beta})\hidari \nuek \\ \nudo \migi$$\end{linenomath}
\end{definition}
This is equivalent via the arguments in \cite[Section 3.5]{steinwuthrich} to Perrin-Riou's construction in \cite[Section 2.2]{perrinriou}.
\begin{lemma}[$D_p(E)$-rationality of coefficients] We have $L_p^{PR}(E,T) \in D_p(E)[[T]]$.
\end{lemma}
\begin{proof}By Theorem \ref{maintheorem}, we can write
\begin{linenomath}$$L_p^{PR}(E,T)=(L_\sharp, L_\flat)\Log_{\alpha,\beta}\smat{1&\alpha\\1&\beta}^{-1}\smat{1 & 0\\a_p & -p}\smat{\omega \\ \phi(\omega)}.$$\end{linenomath}
From the definition of $\Log_{\alpha,\beta}$, we then see that $L_p^{PR}(E,T)\in D_p(E)[[T]]$, as desired.
\end{proof}
Given a globally minimal Weierstrass equation over $\Z$
\begin{linenomath}$$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$\end{linenomath} for $E$, recall that the associated N\'{e}ron/invariant differential is $\omega=\frac{dx}{2y+a_1x+a_3}$ (see e.g. \cite[Chapter III.1]{silverman}). The $\Q$-vector space $H^1_{dR}(E/\Q)$ admits a basis $\{\omega,x\omega\}$ and is equipped with a canonical alternating bilinear form $[\cdot,\cdot]$ so that ${[\omega,x\omega]=1}$. We extend it linearly to the Dieudonn\'{e} modules above and denote these extensions by $[\cdot,\cdot]$ as well.
Fix $\omega$. Then for each $\nu\in D_p(E)$ (resp. $\nu\in D_p(E)(\alpha)$), one can associate a quadratic form $h_\nu$ mapping $E(\Q)$ to $\Q_p$ (resp. to $\Q_p(\alpha))$. One can do this (see \cite{bpr}, or \cite{steinwuthrich}) by defining preliminary height functions $h'_{\omega}$ and $h'_{x\omega}$, and then extending linearly, i.e. given $\nu=a\omega+bx\omega$, let $h'_\nu=ah'_\omega+bh'_{x\omega}$. Explicitly, we have $h'_\omega(P)=-\log_{\omega}(P)^2$, where $\log_{\omega}$ is the logarithm associated to $\omega$. The definition of $h'_{x\omega}$ involves the $\sigma$-functions of either Mazur and Tate or of Bernardi. We refer to \cite[Section 4]{steinwuthrich} for an explicit definition, since it won't be needed in this paper. We then normalize and put $h_\nu:=\frac{h'_\nu}{\log_p(\gamma)}$.
\remark The reason for this normalization is that $p$-adic $L$-functions $L_p(T)$ are classically thought of as functions of a variable $s$ via the substitution $T=\gamma^{s-1}-1$, cf. \cite[\S II.13]{mtt}. The original formulation of $p$-adic BSD then investigated the behavior of a particular $L_p(\gamma^{s-1}-1)$ at $s=1$. Note that \begin{linenomath}$$\frac{d^r}{ds^r}L_p(\gamma^{s-1}-1)|_{s=1}=\left.\frac{d^r}{dT^r}L_p(T)\right|_{T=0}\cdot\log_p(\gamma)^r.$$\end{linenomath}
The bilinear form associated to this height function has values in $\Q_p$ (resp.$\Q_p(\alpha)$):
\begin{linenomath}$$\langle P,Q\rangle_\nu=\frac{1}{2}\left(h_\nu(P+Q)-h_\nu(P)-h_\nu(Q)\right)$$\end{linenomath}
\definition Let $\Reg_\nu$ be the discriminant of this height pairing on $E(\Q)/E(\Q)_{tors}$.
\begin{definition}Let $\nuek$ and $\nudo$ be as above. We define normalized height functions
\begin{linenomath}$$\hat{h}_\nuek:=\frac{h_\nuek}{\left[\nudo,\nuek\right]}=\frac{h_\nuek}{\left[\omega,\nuek\right]} \text{ and } \hat{h}_\nudo:=\frac{h_\nudo}{\left[\nuek,\nudo\right]}=\frac{h_\nudo}{\left[\omega,\nudo\right]},$$\end{linenomath}
and denote their corresponding regulators by $\Reg_{\frac{1}{\alpha}}$ and $\Reg_{\frac{1}{\beta}}$. Note that the
height functions (and thus the regulators) are independent of the choice of our Weierstra\ss\text{ }equation.
\end{definition}
Denote by $r(E)$ the rank of $E(\Q)$. In the supersingular case, Perrin-Riou defines the regulator $\Reg_p^{BPR}(E/\Q)$ as the unique element in $D_p(E)$ so that for any $\nu\in D_p(E)$ with $\nu\not\in\Q_p\omega,$ we have \footnote{This characterization is that of \cite[Lemma 4.2]{steinwuthrich}, which is a corrected version of Perrin-Riou's original lemma, \cite[Lemme 2.6]{perrinriou}.}
\begin{equation}\label{universal}
\left[\Reg_p^{BPR}(E/\Q),\nu\right]=\frac{\Reg_\nu}{[\omega,\nu]^{r-1}}\text{ where $r=r(E)>0$}.
\end{equation} For $r(E)=0$, she puts $\Reg_p^{BPR}(E/\Q)=\omega.$
\begin{definition}As an element of $D_p(E)(\alpha)$, define \begin{linenomath}$$\Reg_p(E/\Q):=( \Reg_{\frac{1}{\beta}},\Reg_{\frac{1}{\alpha}})\hidari \nuek \\ \nudo \migi.$$\end{linenomath}
\end{definition}
\begin{proposition}\label{computationofregulator}We have $\Reg_p^{BPR}(E/\Q)=\Reg_p(E/\Q)\in D_p(E)$.
\end{proposition}
\begin{proof
When $r(E)=0$, this follows from the fact that $\Reg_{\frac{1}{\alpha}}=1$ and $\Reg_{\frac{1}{\beta}}=1$.
When $r(E)>0$, note that
\begin{linenomath}$$\left[\Reg_p(E/\Q),\nuek\right]=\Reg_{\frac{1}{\alpha}}\cdot \left[\nudo,\nuek\right]=\frac{\Reg_\nuek}{\left[\nudo,\nuek\right]^{r-1}}=\frac{\Reg_\nuek}{\left[\omega,\nuek\right]^{r-1}},$$\end{linenomath}
and similarly, $\left[\Reg_p(E/\Q),\nudo\right]=\frac{\Reg_\nudo}{\left[\omega,\nudo\right]^{r-1}}$. Since $\nuek$ and $\nudo$ form a basis for $D_p(E)(\alpha)$, linearity tells us that the property described in equation (\ref{universal}) holds for any $\nu\in D_p(E)(\alpha)$.
Now suppose that $\Delta:=\Reg_p(E/\Q)- \Reg_p^{BPR}(E/\Q)\neq 0$. Then
\begin{linenomath}$$\left[\Delta,\nu\right]=0 \text{ for any }\nu\in D_p(E),$$\end{linenomath}
so this would in particular hold for $\nu=\omega$ or $\nu=x\omega$, from which we conclude by linearity that
\begin{linenomath}$$\left[\Delta,\nu\right]=0 \text{ for any }\nu\in D_p(E)(\alpha).$$\end{linenomath}
But this would imply $\Delta=0$. \renewcommand{\qedsymbol}{Q.E.A.}
\end{proof}
\renewcommand{\qedsymbol}{Q.E.D.}
\subsection{Statement of the conjectures}
The following is a $p$-adic analogue of the Birch and Swinnerton-Dyer conjectures when $p$ is supersingular (cf. \cite[Conjecture on page 229]{bpr} and \cite[Conjecture 2.5]{perrinriou}):
\begin{conjecture}[Bernardi and Perrin-Riou]\label{bernardiandperrinriou} Let $p$ be a good supersingular prime, and denote by $r^{an}$ the order of vanishing of $L_p^{PR}(E,T)$ at $0$, and by $L_p^{PR*}(E)$ its leading coefficient (with value in the Dieudonn\'{e} module) of its Taylor expansion around $0$.
\begin{enumerate}
\item We have $r^{an}=r(E)$.
\item $L_p^{PR*}(E)=\left(1-\phi\right)^2\frac{\prod_v c_v \cdot \#\Sha(E/\Q)}{(\#E(\Q)_{tors})^2}\Reg_p(E/\Q)$.
\end{enumerate}
\end{conjecture}
\remark Note that while the objects in the second part of this conjecture depend on the choice of Weierstra\ss{ }equation, their coordinates with respect to the basis $\{\nu_\alpha,\nu_\beta\}$ don't. In fact, one can formulate Bernardi's and Perrin-Riou's conjecture in a form that resembles more closely that of the one given by Mazur, Tate, and Teitelbaum:
\begin{conjecture}[Equivalent formulation of above]\label{equivalentformulationofabove} Let $r^{an}$ be as in Lemma \ref{ordersofvanishingagree} below, and $L_\alpha^*$ and $L_\beta^*$ be the leading coefficients in the Taylor expansion. Then
\begin{enumerate}
\item $r^{an}=r(E)$.
\item $L_\alpha^*=(1-\frac{1}{\alpha})^2\frac{\prod_v c_v \cdot \#\Sha(E/\Q)\cdot \Reg_{\frac{1}{\beta}}}{(\#E(\Q)_{tors})^2}$, and $L_\beta^*=(1-\frac{1}{\beta})^2\frac{\prod_v c_v \cdot \#\Sha(E/\Q)\cdot \Reg_{\frac{1}{\alpha}}}{(\#E(\Q)_{tors})^2}$.
\end{enumerate}
\end{conjecture}
This version can be found in \cite[Conjecture 0.12]{colmez}, where it is attributed to Mazur, Tate and Teitelbaum.
\begin{lemma}\label{ordersofvanishingagree} Since $p$ is supersingular, $r^{an}:=\ord_{T=0} L_p(E,\alpha,T)=\ord_{T=0} L_p(E,\beta,T)$.
\end{lemma}
\proof The same proof as in \cite[Lemma 6.6]{pollack} works.
\subsection{A version of the conjectures via $L_\sharp$ and $L_\flat$ in the supersingular case}
We now reformulate Conjecture \ref{bernardiandperrinriou} using $L_\sharp$ and $L_\flat$. \begin{definition}\label{lvector} We call $\overrightarrow{L}_p(E,T):=\overrightarrow{L}_p:={(L_\sharp,L_\flat)}$ the \textit{$p$-adic $L$-vector of $E$}, and denote by $r_p^\natural$ the minimum of the orders of vanishing of $L_\sharp$ and $L_\flat$. \end{definition}
We would now like to find a pair of elements $\nu_\sharp$ and $\nu_\flat$ in $D_p(E)$ that give rise to regulators corresponding to our $p$-adic $L$-functions. Recall that $L_p^{PR}(E,T) = (L_\sharp, L_\flat) \Log_{\alpha,\beta}\hidari\nuek \\ \nudo \migi $.
\begin{definition} Let $Z:=\left.\Log_{\alpha,\beta}(1+T)\right|_{T=0}=\Log_{\alpha,\beta}(1)$.
We define
\begin{linenomath}$$\hidari \nu_\sharp\\ \nu_\flat \migi :=Z\hidari\nuek \\ \nudo \migi,$$\end{linenomath}
\begin{linenomath}$$(N_\sharp, N_\flat) :=(\nu_B,-\nu_A)\smat{(1-\frac{1}{\alpha})^2 & 0 \\ 0 & (1-\frac{1}{\beta})^2}Z^{-1}\times \det Z.$$\end{linenomath}
\end{definition}
\begin{lemma}\label{nonlinearity}$\nu_\sharp, \nu_\flat, N_\sharp, N_\flat$ are in $D_p(E)$ and are not $\Q_p$-multiples of $\omega$.
\end{lemma}
\proof Calculation.
\begin{definition} We let $\Reg_\sharp:=\Reg_{\frac{N_\sharp}{[\omega,N_\sharp]}}$ and $\Reg_\flat:=\Reg_{\frac{N_\flat}{[\omega,N_\flat]}}$ be the regulators for the normalized heights associated to $N_\sharp$ and $N_\flat$. Also, we let
\begin{linenomath}$$\Reg_p^\natural:=
\begin{cases}
\left(
\begin{array}{cc} (-a_p^2+2a_p+p-1)\Reg_\sharp,& (-a_p+2)\Reg_\flat \end{array}\right)
& \text{ for odd $p$,}\\
\left(\begin{array}{cc}(-a_p^3+2a_p^2+2pa_p-a_p-2p)\Reg_\sharp,& (-a_p^2+2a_p+p-1)\Reg_\flat\end{array}\right)
& \text{ for even $p$.}
\end{cases}
$$\end{linenomath}
\end{definition}
We are now ready to give our $p$-adic version of BSD:
\begin{conjecture}[Tandem $p$-adic BSD]\label{tandempadicbsd}Let $E$ be an elliptic curve and $p$ a prime of good supersingular reduction.
Denote by $\overrightarrow{L}_p^*$ the first non-zero leading Taylor coefficient around $T=0$ of $\overrightarrow{L}_p=\overrightarrow{L}_p(E,T)$.
\begin{enumerate}
\item We have $r_p^\natural=r(E)$.
\item $\overrightarrow{L}_p^*=\frac{\prod_v c_v \cdot \#\Sha(E/\Q)}{(\#E(\Q)_{tors})^2} \Reg_p^\natural(E/\Q)$
\end{enumerate}\end{conjecture}
\remark The term $\Reg_p^\natural(E/\Q)$ is independent from the choice of Weierstra\ss{ }equation. This follows from the proof of part 2 of Theorem \ref{belowthisone}, which only compares \textit{coordinates} with respect to the basis $\nu_\alpha,\nu_\beta$.
\begin{theorem}\label{belowthisone}This conjecture is equivalent to that of Bernardi and Perrin-Riou (i.e. Conjecture \ref{bernardiandperrinriou}).
\end{theorem}
\begin{definition}Let $r=r(E)>0$. Given a vector $\nu\in D_p(E)$ (or in $D_p(E)(\alpha))$ that is not a linear multiple of $\omega$, we put \begin{linenomath}$$\widetilde{\Reg}_{\nu}:=\frac{\Reg_\nu}{[\omega,\nu]^{r-1}}.$$\end{linenomath}
\end{definition}
\remark We know that $\widetilde{\Reg}_{\nu}$ is linear in $\nu$. See e.g. \cite[proof of Lemma 4.2]{steinwuthrich}.
\begin{proof}[Proof of equivalence for part $1$] This follows from Lemma \ref{ordersofvanishingagree} and the product rule.
\end{proof}
\begin{proof}[Proof for part $2$]From the equivalence of part $1$ and the product rule, we have \begin{linenomath}$$\overrightarrow{L}_p^*\Log_{\alpha,\beta}(1)=\overrightarrow{L}_p^*Z=(L_{\alpha}^*,{L_{\beta}^*}).$$\end{linenomath} But we also have, for $r>0$, \begin{linenomath}$$(1-\phi)^2(\Reg_{\frac{1}{\beta}},\Reg_{\frac{1}{\alpha}})\hidari\nu_A \\ \nu_B\migi=\left(\frac{\widetilde{\Reg}_{\nu_B}}{[\omega,\nu_B]},\frac{\widetilde{\Reg}_{\nu_A}}{[\omega,\nu_A]}\right)\smat{ (1-\frac{1}{\alpha})^2 & 0 \\ 0 & (1-\frac{1}{\beta})^2}Z^{-1}\hidari \nu_\sharp \\ \nu_\flat \migi.$$\end{linenomath}
Since $[\omega,\nu_B]=[\nu_A,\nu_B]=-[\omega,\nu_A]$ and $\widetilde{\Reg}_{\nu}$ is linear in $\nu$, and by Lemma \ref{nonlinearity}, this is equal to
\begin{linenomath}$$\frac{1}{\det Z} \left(\frac{1}{[\nu_A,\nu_B]}\widetilde{\Reg}_{N_\sharp},
\frac{-1}{[\nu_A,\nu_B]}\widetilde{\Reg}_{N_\flat}\right)\hidari \nu_\sharp \\ \nu_\flat \migi.$$
But $\widetilde{\Reg}_{N_\sharp}=[\omega,N_\sharp]\Reg_\sharp$, so this is equal to
$$
\left(\frac{[\omega,N_\sharp]}{[\nu_\sharp,\nu_\flat]}\Reg_\sharp, \frac{-[\omega,N_\flat]}{[\nu_\sharp,\nu_\flat]}\Reg_\flat\right)\hidari \nu_\sharp \\ \nu_\flat \migi.$$\end{linenomath} The rest follows from explicit calculation of the factors preceding the regulators.
\end{proof}
What is know so far is the following theorem of Kato:
\theorem[{\cite{kato}, cf. \cite[Theorem 9.4]{kobayashi} when $a_p=0$}]
In Conjectures \ref{tandempadicbsd}, \ref{equivalentformulationofabove}, \ref{bernardiandperrinriou}, and \ref{mazurtateteitelbaum}, the orders of vanishing of the $p$-adic $L$-functions are all $\geq r(E)$.
\subsection{A remark in the ordinary case}\label{ordinarybsd}
When $p$ is ordinary, there is the following conjecture of Mazur, Tate, and Teitelbaum.
\begin{conjecture}[Mazur, Tate, and Teitelbaum]\label{mazurtateteitelbaum} Let $p$ be a good ordinary prime, and denote by $r^{an}$ the order of vanishing of $L_p(E,\alpha,T)$ at $0$, and by $L_p^*(E)$ the leading coefficient of the Taylor expansion at $0$.
\begin{enumerate}
\item We have $r^{an}=r(E)$.
\item $L_p^*(E)=\left(1-\frac{1}{\alpha}\right)^2\frac{\prod_v c_v \cdot \#\Sha(E/\Q)\cdot \Reg_{\frac{1}{\beta}}(E/\Q)}{(\#E(\Q)_{tors})^2}$.
\end{enumerate}
\end{conjecture}
\remark These conjectures are a combination of \cite[\S II.10, Conjecture (BSD($p$))]{mtt}, which asserts that $r^{an}\geq r(E)$, and the remark thereafter, which predicts the non-vanishing of $\Reg_{\frac{1}{\beta}}(E/\Q)$.
\remark We encounter the term $\Reg_{\frac{1}{\beta}}$ (rather than $\Reg_{\frac{1}{\alpha}}$) because of our choice of Frobenius $\phi=\frac{F}{p}$, where $F$ is the Frobenius as chosen in \cite{mazursteintate} or \cite{kedlaya}. The regulator comes from the normalized height associated to the unit-eigenvector $\alpha$ of $F$ on $D_p(E)$, so that the eigenvalue for $\phi$ becomes $\frac{\alpha}{p}=\frac{1}{\beta}$.\footnote{In \cite[Section 4.1]{steinwuthrich}, the regulator was accidentally constructed from the height coming from the normalized eigenvector of $\phi$ with eigenvalue $\frac{1}{\alpha}$. Everything works in that section if one replaces $\alpha$ by $\beta$.}
In the ordinary case, recall that $L_\sharp$ and $L_\flat$ are not well-defined, but their values at $T=0$ are. In particular this means that when $a_p=2$ and $p$ is odd, we may have $L_\flat(E,0)=0$ while $L(E,1)\neq0$, reminiscent of an extra zero phenomenon. This leads us to ask:
\begin{question} \textit{Where does this extra-zero phenomenon come from?}
\end{question}
\rm
\section{The greatest common divisor}
We now generalize and give some evidence for the following conjecture found in \cite[Problem 3.2]{kuriharapollack}.
Recall that $d_n$ denoted the normalized jump in the ranks at the $n$th level of the cyclotomic tower:
$$ d_n=\frac{1}{p^n-p^{n-1}}(\rank E(\Q_n)- \rank E(\Q_{n-1}))$$
\begin{conjecture}[The problem of Kurihara and Pollack]\label{kuripola}
Let $E/\Q$ be an elliptic curve so $p$ is an odd prime of good supersingular reduction and $a_p=0$. Then
$$\gcd(L_\sharp(E,T),L_\flat(E,T))=\left(T^{r}\prod_{ d_n \geq1 \text{ and } n\geq 1}\Phi_{p^n}^{d_n-1}(1+T)\right).$$
\end{conjecture}
\rm Note that this is an equality of {\it ideals}, since the greatest common divisor of two functions of ${\Q\otimes \Z_p[[T]]}$ is only well-defined as a $\Z_p[[T]]$-ideal. We can give the following proposition:
\begin{proposition}\label{kuriharapollack}Let $E/\Q$ be an elliptic curve and $p$ a prime of good supersingular reduction. For some polynomial $P_\Sha(E,T)$ with $P_\Sha(E,\zeta_{p^n}-1)\neq0$ for $n\geq0,$
$$\gcd\left(L_\sharp(E,T),L_\flat(E,T)\right)=\left(P_\Sha(E,T)\cdot T^{r^{an}}\prod_{ d_n \geq1 \text{ and } n\geq 1}\Phi_{p^n}^{\epsilon_n^{an}-1}(1+T)\right),$$
where $\epsilon_n^{an}-1=d_n^{an}-1$ or $\epsilon_n^{an}-1=d_n^{an}.$
\end{proposition}
\convention Given a vector $(f(T),g(T))$ of $p$-adic analytic functions, we define its order of vanishing at $s$ by $\ord_{T=s}(f(T),g(T)):=\min(\ord_{T=s}f(T),\ord_{T=s}g(T))$.
\begin{lemma}\label{lemma1}
Denote by $\iota$ (any) complex conjugation. Let $f(T), g_1(T)$, $g_2(T)$, and the entries of a $2\times2$ matrix $M(T)$ be $p$-adic analytic functions on the open unit disc and $e=\ord_{T=s}f(T)$ so that
$$\left(f(T),\iota\left(f(T)\right)\right)=(g_1(T),g_2(T))M(T)$$
and $\det M(s) \neq 0$. Then we have $\ord_{T=s}(g_1(T),g_2(T))=e$.
\end{lemma}
\begin{proof}By calculus, $\left(f^{(m)}(s),\iota\left({f^{(m)}}(s)\right)\right)=(0,0)$ if and only if $(g_1^{(m)}(s),g_2^{(m)}(s))=(0,0)$ for ${m \geq 0}$.
\end{proof}
\rm
\begin{corollary}\label{lemma3}
The exact power of $T$ dividing $\gcd\left(L_\sharp(E,T),L_\flat(E,T)\right)$ is $T^{r^{an}}$.
\end{corollary}
\begin{lemma}\label{lemma2}
Let $\overrightarrow{f}(T)=(f_1(T),f_2(T)) \text{ and }\overrightarrow{g}(T)=(g_1(T),g_2(T))$ be vectors of analytic functions on the open unit disc satisfying
$$\overrightarrow{f}(T)=\overrightarrow{g}(T)\\\CCC_n,\text{ where } \CCC_n= \links a_p & p \\ -\Phi_{p^n}(1+T) & 0 \rechts.$$
Let $s=\zeta_{p^n}-1$. Then $\ord_{T=s}\overrightarrow{g}(T)=\ord_{T=s}\overrightarrow{f}(T)$ or $\ord_{T=s}\overrightarrow{g}(T)=\ord_{T=s}\overrightarrow{f}(T)-1$.
\end{lemma}
\begin{proof}Since $\ord_{T=s} g_1(T)=\ord_{T=s} f_2(T)$ and {$a_pg_1(T)-f_1(T)=-\Phi_{p^n}(1+T)g_2(T)$},
$$\begin{cases}\ord_{T=s} f_1(T)< \ord_{T=s} f_2(T)& \text{ implies } \ord_{T=s} g_2(T)=\ord_{T=s} f_1(T)-1,\\
\ord_{T=s} f_1(T)= \ord_{T=s} f_2(T)\text{ and $a_p\neq0$} &\text{ implies } \ord_{T=s} g_2(T)\geq\ord_{T=s} f_1(T)-1,\\
\ord_{T=s} f_1(T)> \ord_{T=s} f_2(T)\text{ and $a_p\neq0$} &\text{ implies }\ord_{T=s} g_2(T)=\ord_{T=s} f_2(T)-1,\\
\ord_{T=s} f_1(T)\geq \ord_{T=s} f_2(T)\text{ and $a_p=0$} &\text{ implies } \ord_{T=s}g_2(T)=\ord_{T=s} f_1(T)-1.\end{cases}$$
\end{proof}
\begin{proof}[Proof of Proposition \ref{kuriharapollack}] We use Corollary $\ref{lemma3}$ and the following argument: Let $\mathcal{M}=I$ when $n=1$ and $\mathcal{M}=\CCC_1\cdots\CCC_{n-1}$ when $n>1$. Recall that $\overrightarrow{L}_p=\left(L_\sharp(E,T),L_\flat(E,T)\right)$, so that
$$\left(L_p(E,\alpha,T),L_p(E,\beta,T)\right)=\overrightarrow{L}_p\mathcal{M}\CCC_n\Xi_n$$
for some $2\times2$ matrix $\Xi_n$ so that $\det \Xi_n(\zeta_{p^n}-1)\neq0$. From Lemma \ref{lemma1}, ${\ord_{T={\zeta_{p^n}-1}}\left(\overrightarrow{L}_p\mathcal{M}\CCC_n\right)=d_n^{an}}$. Lemma \ref{lemma2} then implies $${\ord_{T={\zeta_{p^n}-1}}\left(\overrightarrow{L}_p\mathcal{M}\right)=d_n^{an}-1}\text{ or }{\ord_{T={\zeta_{p^n}-1}}\left(\overrightarrow{L}_p\mathcal{M}\right)=d_n^{an}}.$$ From $\det \mathcal{M} (\zeta_{p^n}-1)\neq0$ and Lemma \ref{lemma1} again, ${\ord_{T={\zeta_{p^n}-1}}\overrightarrow{L}_p=d_n^{an}-1}$ or ${\ord_{T={\zeta_{p^n}-1}}\overrightarrow{L}_p=d_n^{an}}$.
\end{proof}
In view of the problem of Kurihara and Pollack (Conjecture \ref{kuripola}), we make the following conjecture:
\begin{conjecture}Let $E/\Q$ be an elliptic curve, and $p$ a good supersingular prime. Then
$$\gcd(L_\sharp(E,T),L_\flat(E,T))=\left(T^{r}\prod_{ d_n \geq1 \text{ and } n\geq 1}\Phi_{p^n}^{d_n-1}(1+T)\right).$$
\end{conjecture}
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Newly Acquired Echo Lake Family Campground Sees Expansion
Previously county-owned Echo Lake Family Campground (Alberta, Canada) was sold for CA$255,000 on April 1.
According to a report, the 55-acre property, once owned by Westlock County, was acquired by Diverse Bridges, a local construction company co-owned by Adam Esch.
"The county has owned the campground for a number of years, somewhere during the budget process, it was suggested that the county might want to look at disposing of the property," said county interim CAO Pat Vincent.
Last fall, the staff promoted the property and received a couple of inquiries and an offer. The original offer was rejected, and while the property was on the market for several months, the process progressed quickly.
"We received a second offer for it, which met all of the conditions that had been set out by council, so we entered into an agreement to sell the property to the party that expressed interest," Vincent added.
Echo Lake Family Campground is situated on the county's east side, just east of Vimy. Vincent identified various reasons for selling the property, including the cost of running the smaller campground.
"I think the time and effort that was being extended by the county staff in looking after the operation and maintenance of it, was part of the contributing factors for the decision to look at selling the property," he said.
Esch was the county's previous recreation coordinator and is familiar with the campground and the surrounding land. The campground was purchased believing it was a good investment, and they plan to maintain it as a place for recreation for families to enjoy.
They have already started planning some changes and improvements to the campground.
One of the changes involves increasing the number of available RV sites from 12 to 40 this year.
"We do plan expanding that number up to 40 this year. There's more room to expand, and we'll probably expand next year as well," said Esch. "We've hit the ground running — we are bringing our equipment out there, clearing some snow, and Friday (April 8), we are finalizing where we are putting the extra 28 sites."
The expansion will include a variety of new amenities and features, including a two-acre off-leash dog park, extra-large RV sites that have added space between RV spots, numerous camping units per site, a fire pit, picnic table and power to each site, which includes multi-plugged powered sites. There would also be wildlife watching areas as well as other spots for recreation for campers.
"We plan on putting in about a kilometer and a half of walking trails on the site," said Esch. "We're going to try and put a beach on it as well, we're just working on approvals with the province for that."
Vincent said that the county is happy to see the campground sold and remain a local attraction.
"Council and administration are very pleased that we were able to sell the property and that the party that purchased it is familiar with it and that they're going to continue to operate it as a campground," said Vincent. "I think it continues to provide a service that hadn't been provided for at least the last year."
This story originally appeared on Town and Country Today.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Q: Allow to add all possible options to the site menu for iOS app In iOS app you are allowed to select up to 4 items as site options:
This limitation happening due to screen width when app is used on the phone or in split view mode for iPad:
My request is to allow to select all of items simultaneously for the site submenu and show all selected items if the space is enough (like on the first picture when items aligned vertically in the main toolbar, there is no limit in space due to scrolling). Items that are not selected will be available via ... as well as early. Alternative solution would be is to scroll items horizontally if they don't fit in width, but I think it's harder to implement.
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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To unlock certain features offered in the Lydia app, you can upgrade your account by verifying it for free.
Verified users can be identified by a blue dot next to their profile photo.
Tap here if you already have Lydia on this device.
You can take a photo of the document from the app itself or simply import it.
Once you have provided all the necessary documents, your account should be verified within 3 working days. If the quality of the photos is poor or if they are blurred, this could take longer.
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{
"redpajama_set_name": "RedPajamaC4"
}
| 683
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{"url":"http:\/\/mathematica.stackexchange.com\/questions\/16867\/what-is-causing-a-type-error-when-trying-to-modify-mathematica-state-from-a-func\/16983","text":"# What is causing a type error when trying to modify Mathematica state from a function linked with LibraryLink\n\nI've managed to reduce my problem to a very simple C program:\n\n#include \"mathlink.h\"\n#include \"WolframLibrary.h\"\n\nint SetX( WolframLibraryData libData )\n{\nint errorCode;\nint nextPacket;\nif ( errorCode != 0 )\nreturn errorCode;\nif ( nextPacket == RETURNPKT )\nreturn 0;\n}\n\nDLLEXPORT int WolframLibrary_getVersion()\n{\nreturn WolframLibraryVersion;\n}\n\nDLLEXPORT int WolframLibrary_initialize(WolframLibraryData libData)\n{\nreturn 0;\n}\n\nDLLEXPORT int testIt(WolframLibraryData libData, mint argc, MArgument*\nmargs, MArgument result)\n{\nint error = SetX( libData );\nif ( error != 0 ) {\nlibData->Message( \"error setting x\" );\n}\nreturn error;\n}\n\n\nI use AppendTo to add the directory containing the compiled shared object\/Dll to $LibraryPath, then do: In[4]:= LibraryFunctionLoad[ \"CB\", \"testIt\", {}, Void ] Out[4]= LibraryFunction[ <>, testIt, {}, {}] In[5]:= %[] During evaluation of In[5]:= LibraryFunction::error setting x: -- Message text not found -- >> During evaluation of In[5]:= LibraryFunction::typerr: An error caused by inconsistent types was encountered evaluating the function testIt. >> Out[5]= LibraryFunctionError[LIBRARY_TYPE_ERROR,1] Regretfully, the error message doesn't tell me what type is wrong, and what type I should be using instead. (I've tried using MLPutString, insteamd of MLPutSymbol, in the above, with the same results.) I've tried this with the same results under Windows and Linux, so I don't think it's a problem with the build procedure I'm using; but FWIW, here is the command line used to build the shared object under Linux: Gabi (41): echo$MATHINCL\nGabi (42): echo $MATHLIB \/usr\/local\/Wolfram\/Mathematica\/8.0\/SystemFiles\/Links\/MathLink\/DeveloperKit\/Linux-x86-64\/CompilerAdditions\/libML64i3.so Gabi (43): gcc -shared -fPIC$MATHINCL CB.c \\$MATHLIB -o CB.so\n\n\n(I use something similar from CygWin bash under Windows, with options \/MDd and \/Tc.)\n\n-\nLooking into this, I can see that the docs are simply wrong. The processMathLink() function returns TRUE on success, FALSE otherwise, not what the docs describe. We'll make sure this gets fixed. Your program looks to be correct in all other ways.\nThanks. Not only does my small test work now, but I was able to get the actual code working. In the actual code, I had another small error: I'd spelled the first function \"EvalutatePacket\". Is there anyway to get better error detection---as far as I can see, the spelling error only made the packet a no-op. (Maybe by processing the response packet? In the actual code, I know that the head of what I'm setting should be a LibraryFunction; just checking for that would probably suffice.) \u2013\u00a0 James Kanze Dec 27 '12 at 12:52","date":"2014-03-11 16:53:39","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.1757199615240097, \"perplexity\": 8235.87432487177}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-10\/segments\/1394011232483\/warc\/CC-MAIN-20140305092032-00084-ip-10-183-142-35.ec2.internal.warc.gz\"}"}
| null | null |
About ISOTC 223
Published standards
Major milestones
The sinking of the Russian submarine Kursk to the bottom of the Barents Sea in 2000 can be cited as a major impetus for the formation of ISO/TC 223. The salvage operation that followed the accident provided painful evidence that the international community lacked the tools necessary to cooperate effectively in emergency situations, resulting in an initiative from the Russian standards organization, GOST, to establish ISO/TC 223. Originally titled "Civil defense", the committee was created to standardize international emergency procedures. Unfortunately, the initiative lay dormant for some time. However, terrorist actions, including the 9/11 attacks on New York and Washington, as well as a surge in natural disasters at the time, led ISO to conduct a large-scale assessment of the role of standardization in the security field. One important decision was to restart the work originally started by ISO/TC 223.
In 2005 the chairmanship of the committee was taken over by the SIS, Swedish Standards Institute and Ambassador Krister Kumlin was appointed as Chair. To better reflect its ambition of taking a broader approach toward disruptive incidents that threaten civil society, ISO/TC 223 was renamed Societal security. The scope of the committee is very broad, covering all phases of man-made or naturally-caused disaster situations. Since its first meeting in Stockholm in May 2006 the membership of ISO/TC 223 has grown steadily and today consists of 47 participating (P) members, 20 observer (O) members and several liaisons.
In 2012, Ambassador Krister Kumlin retired after six years of service and Mrs. Åsa Kyrk Gere was appointed as new Chair of the committee.
Previous chairs and secretaries
Krister Kumlin Ambassador Kumlin served as the first Chair of ISO/TC 223 for a six years period between 2006-2012.
Per Forsgren Mr Forsgren served as the first Secretary of ISO/TC 223 and supported the committee during its establishment and at its first meeting.
Stefan Tangen Dr Tangen served as the Secretary of ISO/TC 223 between 2006-2013. He is still part of the committee but has taken on other roles including the convenorship of the ISO/TC 223 communication group.
Sanna Edlund Mrs Edlund served as the Secretary of ISO/TC 223 in a short period of 2013
20-24 January in Bogota, Colombia
Working Group 9 meeting
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Communication Group meeting
[TBD] Mars in London, UK
Working Group 6 and Working Group 8 meeting
21-26 June, 2020, in Berlin, Germany
8th ISO/TC 292 plenary meeting
Authenticity, integrity and trust for products and documents
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Mass evacuation
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 4,001
|
\section{Introduction}
Symmetry is a fundamental concept in Physics. In the Lagrangian formalism, Noether's symmetries play a central role, since Noether's theorem shows a direct connection between symmetries and conservation laws \cite{Noether}. In particular, the starting point of modern gauge theories \cite{Kleinert, Ryder} is a Lagrangian density admitting a prescribed symmetry group. The associated conservation laws should then reflect experimental observations. Examples of conserved quantities include energy, linear momentum, angular momentum, and electric charge.
A natural requirement in relativistic theories is that the Lagrangian density must be a Lorentz scalar. This assures the validity of the Poincar\'e group, which is the fundamental transformation group on Minkowski space. In another context, in gauge theories, interactions are obtained from local gauge invariance principles. Usually, the local gauge symmetry is postulated in advance. Afterward, the action invariance is explicitly verified \cite{Kleinert, Ryder}. The question, regarding an opposite point of view, is how to systematically derive Noether symmetries uniquely from the Lagrangian density, without additional hypothesis. The advantage of the deductive approach is the possibility of unveiling symmetries not apparent from the very beginning. In addition, given a broad class of Lagrangians, one might be interested in the identification of the particular subclasses admitting Noether symmetries. In Section 7, this approach is put forward in the case of the charged scalar particle under prescribed external electromagnetic fields. In this case, not necessarily the external field will be compatible with some symmetry transformation. It should be mentioned that the traditional approach, put forward e.g. in authoritative textbooks
\cite{Kleinert, Ryder, Schweber, Weinberg} is of course perfectly well justified, as long as the overall symmetry structure of space-time and internal space is concerned. However, sometimes (as in the aforementioned case of the charged scalar particle under external fields) one might be interested in subclasses admitting Noether point symmetry, as a toll in the search for sufficiently simple benchmark systems.
The present article shows in detail the procedure for the systematic derivation of Noether point (or geometric) symmetries, applied to special relativistic field theories. In fact, in comparison with systems of a discrete number of degrees of freedom, there are fewer examples of
step-by-step calculation of Noether invariance results for continuous systems, in relativity. For instance, Ref. \cite{Kobussen} considers the derivation of integrals of motion for the N-body problem. With few exceptions \cite{Havas, Schoeller}, most works in this direction assume non-relativistic and discrete systems.
The work considers some of the basic relativistic field theories, going from the simplest to the more elaborate models. Namely, in a sequence, the real scalar, complex scalar, vacuum electromagnetic and coupled complex scalar and electromagnetic field theories will be treated, as case examples for the systematic application of Noether's theorem in the search for geometric transformations. In this way, an hierarchy of models will be described from the symmetry analysis point of view. Although in all studied cases the external symmetries are given by the Poincar\'e group, as expected, the internal symmetries have distinctive features, including global or local gauge symmetries, together with an additional internal symmetry due to linearity, in the non-interacting field cases - to be detailed in the next sections. It is assumed that the present communication has essentially a methodological character. Nevertheless, it is precisely the systematic (non {\it ad hoc}) procedure that allows the identification of the aforementioned extra internal symmetry. Moreover, it will be addressed the case of a charged scalar particle under external electromagnetic fields. The class of external fields so that Noether point symmetries are admissible will be so determined, for the first time, as well as the accompanying conservation laws. The problem has interest for strong laser-plasma interactions, where test-particle dynamics has both relativistic and quantum aspects \cite{Marklund}.
Evidently, nowadays a large number of packages is available for the calculation of Noether point symmetries using computer algebra software. Nevertheless, the interest still remains for some people at least, to not blindly follow the computer's advice and to personally understand all the steps in the symmetry procedure. Moreover, in the same trend, the version of the Noether theorem presented below, is certainly not the most abstract, general or rigorous possible. Nevertheless, for practical applications, it might be of some interest to present the subject in a modest, more readable fashion for non-mathematicians. Finally, it should be noted that the manuscript is not intended to be a review. Therefore a detailed, encyclopaedic account of updated references on symmetries and conservation laws should be found elsewhere.
This work is organized as follows. In Section 2, the Noether theorem is reviewed. In Section 3, the Noether point symmetries and conservation laws are deduced in the case of the real scalar field. The same procedure is repeated in the remaining sections, for increasing complexity of the models. Section 4 is dedicated to the complex scalar field. Section 5 is dedicated to the vacuum electromagnetic field. Section 6 considers the coupled complex scalar and electromagnetic fields. Section 7 is devoted to the charged scalar particle under external electromagnetic fields. Section 8 presents some conclusions.
\section{Noether's Theorem}
\label{Noethertheorem}
In this Section, the Noether's theorem is enunciated in the case of a single field, the generalization to the multi-field case being straightforward. The starting point is the action functional,
\begin{equation}
\nonumber
S = \int\,{\cal{L}}(\phi,\partial_{\mu}\phi,x)\,d^{4}x \,,
\end{equation}
where $\phi = \phi(x)$ is the pertinent field, ${\cal L}$ is the Lagrangian density of the indicated arguments, and $x$ is a 4-vector with covariant components $x_\mu = (t, {\bf r}) = (x_0, x_1, x_2, x_3)$. The metric tensor will be taken as
$g^{\mu\nu} = {\rm diag}(1,-1,-1,-1)$. Natural units $c = 1, \hbar = 1$ and the Einstein summation convention will be employed. Greek indexes run from 0 to 3, and Latin indexes from 1 to 3.
The infinitesimal point transformations given by
\begin{eqnarray}
x^\mu &\rightarrow& x^{\mu} + \varepsilon\,\eta^{\mu}(\phi,x) \,, \nonumber \\
\phi &\rightarrow& \phi + \varepsilon\,\psi(\phi,x) \,, \nonumber
\end{eqnarray}
will be considered, where $\varepsilon$ is a real infinitesimal parameter and where
$\eta^{\mu}(\phi,x)$, $\psi(\phi,x)$ are smooth functions not dependent on the field derivatives. Dynamical symmetries, where the transformation law involves the field derivatives, are fundamental in many cases \cite{Bluman}.
For instance, the derivation of the infinite hierarchy of conserved functionals for the Korteweg-de Vries equation \cite{Olver} needs the application of dynamical symmetries. Nevertheless, for simplicity this work is restricted to point transformations only.
The (quasi) invariance condition for the action is given \cite{Hill, Sarlet} by
\begin{equation}
\label{eq1} \frac{\partial {\cal{L}}}{\partial\phi}\,\psi +
\frac{\partial
{\cal{L}}}{\partial(\partial_{\mu}\phi)}\,(d_{\mu}\psi
-
d_{\nu}\phi\,\partial_{\mu}\eta^{\nu}) +
\partial_\mu{\cal{L}}\,\,\eta^\mu +
{\cal{L}}\,\,d_{\mu}\eta^\mu = d^{\mu}\sigma_\mu \,,
\end{equation}
where $\sigma_\mu = \sigma_{\mu}(\phi,x)$ is at this stage an arbitrary 4-vector of the indicated arguments. Actually, the condition
(\ref{eq1}) does not assures the strict invariance of the action, which can be modified by the addition of a constant surface term, namely, the surface integral of $\sigma_\mu$ at infinity. However, a numerical constant added to the action, has not any effect on the form of the Euler-Lagrange equations.
For the sake of notation, we denote total derivatives as $d_\mu$ and partial derivatives (maintaining constant fields and derivatives of the fields) as
$\partial/\partial\,x^\mu = \partial_\mu$. For instance,
\begin{equation}
d_{\mu}{\cal{L}} = \partial_\mu{\cal{L}} +
\frac{\partial {\cal{L}}}{\partial\phi}\,\partial_{\mu}\phi +
\frac{\partial
{\cal{L}}}{\partial(\partial_{\nu}\phi)}\,\partial_{\mu}\partial_{\nu}\phi
\,. \nonumber
\end{equation}
Noether's theorem assures that whenever the symmetry condition (\ref{eq1}) is satisfied, there is a conserved current given by
\begin{equation}
\label{eqq1} J^\mu = \theta^{\mu}_{\,\nu}\,\eta^\nu - \frac{\partial
{\cal{L}}}{\partial(\partial_{\mu}\phi)}\,\psi + \sigma^\mu \,,
\end{equation}
where the energy-momentum tensor $\theta^{\mu}_\nu$ is defined by
\begin{equation}
\nonumber
\theta^{\mu}_{\,\nu} = \frac{\partial
{\cal{L}}}{\partial(\partial_{\mu}\phi)}\,\partial_{\nu}\phi -
\delta^{\mu}_{\,\nu}\,{\cal{L}} \,.
\end{equation}
The conservation law reads
\begin{equation}
\nonumber
d_{\mu}J^\mu = 0 \,,
\end{equation}
where $\phi$ solves the Euler-Lagrange equation,
\begin{equation}
\label{el}
\frac{\partial{\cal L}}{\partial\phi} - d_{\mu}\left(\frac{\partial{\cal L}}{\partial(\partial_{\mu}\phi)}\right) = 0 \,.
\end{equation}
The basic question to be addressed here is: given the Lagrangian density, how to systematically derive the infinitesimal symmetry transformations leaving the action functional invariant up to the addition of a surface term? The answer, to be developed in the examples in the next sections, is as follows. Inserting the Lagrangian density into the invariance condition (\ref{eq1}), typically we obtain a polynomial equation on the field derivatives. The coefficient of each different field derivative must be zero. Otherwise, one would impose additional constraints on the field, which should be ideally leaved free as much as possible. Therefore, a set of partial differential equations for the symmetry functions will be derived, not involving the derivatives of the field. Solving the determining partial differential equations in all generality, we obtain the full set of Noether point symmetries, without {\it ad hoc} postulates. The procedure will be worked out in the following examples, starting with one of the simplest relativistic models, namely, the Klein-Gordon field.
\section{Real Scalar Field}
\label{realscalar}
\subsection{Noether Symmetries for the Real Scalar Field}
The Lagrangian density for the real scalar field is
\begin{equation}
\label{lsf}
{\cal{L}} = \frac{1}{2}\,\partial^{\mu}\phi\,
\partial_{\mu}\phi - \frac{1}{2}\,
m^2\,\phi^2 \,,
\end{equation}
where $m$ is the particle mass. Using the Euler-Lagrange equation (\ref{el}), we derive Klein-Gordon's equation,
\begin{equation}
\nonumber
\partial^{\mu}\partial_{\mu}\phi + m^{2}\phi = 0 \,.
\end{equation}
Inserting ${\cal{L}}$ from Eq. (\ref{lsf}) into the symmetry condition (\ref{eq1}),
it follows that
\begin{equation}
\label{equa2} - m^{2}\phi\,\psi +
\partial^{\mu}\phi\,(d_{\mu}\psi -
\partial_{\nu}\phi\,d_{\mu}\eta^{\nu}) +
\frac{1}{2}\,(\partial^{\mu}\phi\,\partial_{\mu}\phi -
m^{2}\phi^2)d_{\nu}\eta^\nu = d^{\mu}\sigma_\mu \,.
\end{equation}
The quantities $\eta^\mu$ and $\psi$ should be managed so that the left-hand side of Eq. (\ref{equa2}) becomes the divergence of some appropriate 4-vector $\sigma_\mu$.
The following total derivatives
\begin{eqnarray} \nonumber
d_{\mu}\psi &=& \partial_\mu\psi +
\frac{\partial\psi}{\partial\phi}\,\partial_{\mu}\phi \,,\\
\nonumber
d_{\mu}\eta^\nu &=& \partial_\mu\eta^\nu +
\frac{\partial\eta^\nu}{\partial\phi}\,\partial_{\mu}\phi \,,\\
\nonumber
d^{\mu}\sigma_\mu &=& \partial^\mu\sigma_\mu +
\frac{\partial\sigma_\mu}{\partial\phi}\,\partial^{\mu}\phi \,,
\end{eqnarray}
when inserted on Eq. (\ref{equa2}), give the expression
\begin{eqnarray}
-
\frac{1}{2}\,\partial^{\mu}\phi\,\partial_{\mu}\phi\,\partial_{\nu}
\phi\,\frac{\partial\eta^\nu}{\partial\phi}
+ \partial^{\mu}\phi\,\partial_{\mu}\phi\,\left(\frac{\partial\psi}{\partial\phi}
+ \frac{1}{2}\,\partial_\nu\eta^\nu\right) -
\partial^{\mu}\phi\,\partial_{\nu}\phi\,\partial_\mu\eta^\nu + \nonumber
\\
\label{equa3}
\partial^{\mu}\phi\,\left(\partial_\mu\psi -
\frac{1}{2}\,m^{2}\phi^{2}\,\frac{\partial\eta_\mu}{\partial\phi}\right) -
m^{2}\phi\,\psi -
\frac{1}{2}\,m^{2}\phi^{2}\,\partial_\mu\eta^\mu =
\partial^{\mu}\phi\,\frac{\partial\sigma_\mu}{\partial\phi} +
\partial^\mu\sigma_\mu \,.
\end{eqnarray}
Equation (\ref{equa3}), to be identically satisfied, is a polynomial expression on the field derivatives.
Therefore, the coefficient of each monomial (term with equal derivative power) should vanish. The third order terms give
\begin{equation}
\nonumber
\partial^{\mu}\phi\,\partial_{\mu}\phi\,\partial_{\nu}
\phi\,\frac{\partial\eta^\nu}{\partial\phi}
= 0 \quad \Rightarrow \quad \eta^{\mu} = \eta^{\mu}(x) \,.
\end{equation}
In another words, the external transformations (affecting the space-time coordinates only) are
not dependent on $\phi$.
The second order terms in Eq. (\ref{equa3}) imply
\begin{equation}
\partial^{\mu}\phi\,\partial_{\mu}\phi\,\left(\frac{\partial\psi}{\partial\phi}
+ \frac{1}{2}\,\partial_\nu\eta^\nu\right) -
\partial^{\mu}\phi\,\partial_{\nu}\phi\,\partial_\mu\eta^\nu = 0 \,.
\nonumber
\end{equation}
The last equation decomposes itself into a set of equations, corresponding to terms proportional to
$\partial_{0}\phi\,\partial_{0}\phi$, $\partial_{0}\phi\,\partial_{i}\phi$ and
$\partial_{i}\phi\,\partial_{j}\phi$. A detailed examination shows that the resulting equations are reducible to
\begin{eqnarray}
\label{equa4} - \partial_{0}\eta_0 &=& \partial_{1}\eta_1 =
\partial_{2}\eta_2 = \partial_{3}\eta_3 \,,\\ \label{equa5}
\partial_{\mu}\eta_\nu &+& \partial_{\nu}\eta_\mu = 0 \,,\quad \mu \neq \nu
\,,
\end{eqnarray}
together with
\begin{equation}
\label{equa6}
\frac{\partial\psi}{\partial\phi} = - \partial_{0}\eta_0 \,,
\end{equation}
where Eq. (\ref{equa4}) was taken into account for Eq. (\ref{equa6}).
We left the system (\ref{equa4})-(\ref{equa5}) untouched by now. Equation (\ref{equa6}) gives
\begin{equation}
\label{equa7} \psi = - \phi\,\partial_{0}\eta_0 + \tilde\phi(x)
\,,
\end{equation}
where $\tilde\phi(x)$ is an arbitrary function of $x$.
The invariance condition (\ref{equa3}), for the terms which are of first order in the derivatives, leaves us with
\begin{equation}
\frac{\partial\sigma_\mu}{\partial\phi} = \partial_\mu\psi = -
\phi\,\partial_{\mu}\partial_{0}\eta_0 + \partial_{\mu}\tilde\phi \,,
\nonumber
\end{equation}
with the solution
\begin{equation}
\label{eq8}
\sigma_\mu = - \frac{\phi^2}{2}\,\partial_{\mu}\partial_{0}\eta_0
+ \phi\,\partial_{\mu}\tilde\phi + \tilde\sigma_{\mu}(x) \,,
\end{equation}
where $\tilde\sigma_{\mu}(x)$ is an arbitrary 4-vector depending only on $x$.
The zeroth-order term on Eq. (\ref{equa2}) implies
\begin{equation}
\label{zo}
- m^{2}\phi\,\psi - \frac{1}{2}\,m^{2}\phi^{2}\,\partial_\mu\eta^\mu =
\partial^\mu\sigma_\mu \,.
\end{equation}
Inserting the results from Eqs. (\ref{equa4}), (\ref{equa7}) and (\ref{eq8}) into Eq. (\ref{zo}), one get
\begin{equation}
\label{ee}
\left(- \frac{1}{2}\,\partial^{\mu}\partial_{\mu}\partial_{0}\eta_0
+ m^{2}\partial_{0}\eta_0\right)\,\phi^2 +
\left(\partial^{\mu}\partial_{\mu}\tilde\phi +
m^{2}\tilde\phi\right)\,\phi + \partial^{\mu}\tilde\sigma_\mu = 0
\,.
\end{equation}
Equation (\ref{ee}), being identically satisfied for any $\phi$, implies that the coefficients of different powers of the field vanish, or,
\begin{eqnarray}
\label{eq9} -
\frac{1}{2}\,\partial^{\mu}\partial_{\mu}\partial_{0}\eta_0 +
m^{2}\partial_{0}\eta_0 = 0 \,,\\ \label{eq10}
\partial^{\mu}\partial_{\mu}\tilde\phi + m^{2}\tilde\phi = 0 \,,\\
\label{eq11}
\partial^{\mu}\tilde\sigma_\mu = 0 \,.
\end{eqnarray}
Equation (\ref{eq9}) can be rewritten as
\begin{equation}
\label{eq12}
\partial_{0}\left(\partial^{\mu}\partial_{\mu}\eta_0 - 2\,m^{2}\eta_0\right) = 0 \,,
\end{equation}
or,
\begin{equation}
\label{eqq300} \partial^{\mu}\partial_{\mu}\eta_0 - 2\,m^{2}\eta_0 =
- 2\,m^{2}\tilde\eta_{0}({\bf r}) \,,
\end{equation}
where $\tilde\eta_{0}({\bf r})$ is a function of space coordinates only.
The next information, comes from Eq. (\ref{eq10}) showing that
$\tilde\phi$ solves the Klein-Gordon equation. Therefore, adding to $\phi$ a particular solution of the Klein-Gordon equation is a Noether symmetry, reflecting the linearity of the equation. In addition, Eq. (\ref{eq11}) shows that
\begin{equation}
\nonumber
\tilde\sigma_\mu = 0 \,,
\end{equation}
without loss of generality.
While Eqs. (\ref{eq10}) and (\ref{eq11}) have already been fully examined, there remains Eq. (\ref{eq9}), which is equivalent to Eq.
(\ref{eqq300}). To analyze the last one, we take into account Eqs. (\ref{equa4}) and (\ref{equa5}). For instance, considering the first line in Eq. (\ref{equa4}), for $\mu = 0$, $\nu = 1$ in Eq. (\ref{equa5}), it results
\begin{equation}
\nonumber
\partial_{1}\eta_1 = - \partial_{0}\eta_0 \,,\quad \partial_{0}\eta_1 = -
\partial_{1}\eta_0 \,.
\end{equation}
To satisfy Cauchy's condition $\partial_{0}\partial_{1}\eta_1 =
\partial_{1}\partial_{0}\eta_1$, necessarily
\begin{equation}
\label{eq13}
\partial_{1}\partial_{1}\eta_0 = \partial_{0}\partial_{0}\eta_0 \,.
\end{equation}
Similarly, using again Eqs. (\ref{equa4}) and (\ref{equa5}),
we conclude that
\begin{equation}
\label{eq14}
\partial_{2}\partial_{2}\eta_0 =
\partial_{3}\partial_{3}\eta_0 =
\partial_{0}\partial_{0}\eta_0 \,.
\end{equation}
Equations (\ref{eq13}) and (\ref{eq14}) allow to write
\begin{equation}
\nonumber
\partial^{\mu}\partial_{\mu}\eta_0 = - 2\,\partial_{0}\partial_{0}\eta_0 \,.
\end{equation}
Inserting the last into Eq. (\ref{eq12}) gives
\begin{equation}
\label{eq15}
\partial_{0}\partial_{0}\eta_0 + m^{2}\eta_0 = m^{2}\tilde\eta_{0}({\bf r}) \,.
\end{equation}
Only time-derivatives appear in the differential equation (\ref{eq15}). Therefore, in this context,
$\tilde\eta_{0}({\bf r})$ is a constant, and the general solution obviously is
\begin{equation}
\label{eq16} \eta_0 = \tilde\eta_{0}({\bf r}) + F({\bf r})e^{imt} +
G({\bf r})e^{- imt} \,,
\end{equation}
where $F({\bf r})$ and $G({\bf r})$ are arbitrary functions of the space coordinates only.
From now on, sometimes we denote ${\bf r} = (x,y,z)$ whenever convenient, as long as there is no risk of confusion between the space-time 4-vector $x = (t, {\bf r})$ and the coordinate $x$. Following this definition, and inserting Eq. (\ref{eq16}) into (\ref{equa4}), the result is
\begin{eqnarray}
\label{eq301} \eta_1 &=& \tilde\eta_{1}(y,z,t) -
im\,e^{imt}\int\,dx\,F + im\,e^{-imt}\int\,dx\,G \,,\\
\label{eq302} \eta_2 &=& \tilde\eta_{2}(x,z,t) -
im\,e^{imt}\int\,dy\,F + im\,e^{-imt}\int\,dy\,G \,,\\
\label{eq303} \eta_3 &=& \tilde\eta_{3}(x,y,t) -
im\,e^{imt}\int\,dz\,F + im\,e^{-imt}\int\,dz\,G \,,
\end{eqnarray}
where $\tilde\eta_1, \tilde\eta_2$ and $\tilde\eta_3$ are arbitrary functions of the indicated arguments.
The solution presented in Eqs. (\ref{eq16})-(\ref{eq303}) must be compatible with Eq. (\ref{equa5}).
For instance, for $\mu = 0, \nu = 1$ in Eq. (\ref{equa5}), we obtain
\begin{equation}
\label{eqq18}
\frac{\partial}{\partial\,t}\tilde\eta_{1} +
\frac{\partial}{\partial\,x}\tilde\eta_{0} +
e^{imt}\,\left(m^{2}\int\,dx\,F +
\frac{\partial\,F}{\partial\,x}\right) + e^{- imt}\,\left(m^{2}\int\,dx\,G +
\frac{\partial
G}{\partial\,x}\right) = 0 \,. \nonumber
\end{equation}
The derivative of the last equation with respect to $x$ implies
\begin{equation}
\label{eq19}
\frac{\partial^{2}\tilde\eta_{0}}{\partial\,x^2} +
e^{imt}\left(\frac{\partial^{2}F}{\partial\,x^2} + m^{2}F\right) +
e^{- imt}\left(\frac{\partial^{2}G}{\partial\,x^2} + m^{2}G\right) \nonumber
= 0 \,.
\end{equation}
Since neither $\tilde\eta_{0}$, nor $F, G$ have a dependence on time, it follows that
\begin{eqnarray}
\label{eq400}
\frac{\partial^{2}F}{\partial\,x^2} + m^{2}F &=&
\frac{\partial^{2}G}{\partial\,x^2} +
m^{2}G = 0 \,,\\ \label{eq401}
\frac{\partial^{2}\tilde\eta_{0}}{\partial\,x^2} &=& 0 \,.
\end{eqnarray}
Equation (\ref{eq401}) will not have immediate consequences. On the other hand, following a procedure similar to the derivation of Eq.
(\ref{eq400}), we obtain
\begin{eqnarray}
\label{eq402}
\frac{\partial^{2}F}{\partial\,y^2} + m^{2}F &=& \frac{\partial^{2}F}{\partial\,z^2} +
m^{2}F = 0 \,,\\ \label{eq403}
\frac{\partial^{2}G}{\partial\,y^2} + m^{2}G &=& \frac{\partial^{2}G}{\partial\,z^2} +
m^{2}G = 0 \,.
\end{eqnarray}
The (unique) solution for the system composed by Eqs. (\ref{eq400}),
(\ref{eq402})-(\ref{eq403}) is
\begin{eqnarray}
F &=& c_{1}\exp[im(x+y+z)] + c_{2}\exp[im(x+y-z)] \nonumber \\
&+& c_{3}\exp[im(x-y+z)] + c_{4}\exp[im(-x+y+z)] \nonumber \\
&+& c_{5}\exp[im(x-y-z)] + c_{6}\exp[im(-x+y+z)] \nonumber \\ &+&
c_{7}\exp[im(-x-y+z)] + c_{8}\exp[-im(x+y+z)] \,, \nonumber \\ G &=&
c_{9}\exp[im(x+y+z)] + c_{10}\exp[im(x+y-z)] \nonumber \\
&+& c_{11}\exp[im(x-y+z)] + c_{12}\exp[im(-x+y+z)] \nonumber \\
&+& c_{13}\exp[im(x-y-z)] + c_{14}\exp[im(-x+y+z)] \nonumber \\ &+&
c_{15}\exp[im(-x-y+z)] + c_{16}\exp[-im(x+y+z)] \,, \nonumber
\end{eqnarray}
where $c_1,\dots\,c_{16}$ are numerical constants. Hence,
\begin{eqnarray}
\label{equa500} \eta_0 &=& \tilde\eta_{0}(x,y,z) +
c_{1}\exp[im(x+y+z+t)] \nonumber \\ &+& \dots +
c_{16}\exp[-im(x+y+z+t)] \,,\\ \label{eq502} \eta_1 &=&
\tilde\eta_{1}(y,z,t) - c_{1}\exp[im(x+y+z+t)] \nonumber
\\ &+& \dots - c_{16}\exp[-im(x+y+z+t)] \,,\\ \label{eq503} \eta_2 &=&
\tilde\eta_{2}(x,z,t) - c_{1}\exp[im(x+y+z+t)] \nonumber
\\ &+& \dots - c_{16}\exp[-im(x+y+z+t)] \,,\\ \label{eq501} \eta_3 &=&
\tilde\eta_{3}(x,y,t) - c_{1}\exp[im(x+y+z+t)] \nonumber
\\ &+& \dots - c_{16}\exp[-im(x+y+z+t)] \,,
\end{eqnarray}
where the terms depending on $c_2,
\dots, c_{15}$ were omitted, for brevity.
It remains the constraint (\ref{equa5}). Taking $\mu = 1, \nu = 2$ in Eq.
(\ref{equa5}) and inserting Eqs. (\ref{eq502})-(\ref{eq503}), results in
\begin{eqnarray}
\partial_{1}\tilde\eta_2 + \partial_{2}\tilde\eta_1 &-&
2\,imc_{1}\exp[im(x+y+z+t)] \nonumber \\ &+& \dots +
2\,imc_{16}\exp[- im(x+y+z+t)] = 0 \nonumber \,.
\end{eqnarray}
Derivation of the last with respect to $x$ and
$y$, shows that
\begin{equation}
2\,im^{3}\left\{c_{1}\exp[im(x+y+z+t)] + \dots + c_{16}\exp[-
im(x+y+z+t)]\right\} = 0 \,, \nonumber
\end{equation}
so that
\begin{equation}
c_1 = \dots = c_{16} = 0 \,. \nonumber
\end{equation}
Since all numerical constants $c_1,\dots,c_{16}$
vanish, and using Eqs. (\ref{equa5}), (\ref{equa500})-(\ref{eq501}),
we get the compact expressions
\begin{equation}
\label{eq600}
\partial_{\mu}\eta_\nu + \partial_{\nu}\eta_\mu = 0 \,,
\end{equation}
generalizing Eq. (\ref{equa5}) to arbitrary indexes $\mu, \nu$.
We are almost done. However, we should still take into account Eq. (\ref{eq600}), applying appropriate derivatives to it. For instance, for
$\mu = 0, \nu = 1$, differentiation with respect to $x$ recalling that $\tilde\eta_0$ does not depend on time, gives
\begin{equation}
\partial_{1}\partial_{1}\eta_{0} = 0 \,. \nonumber
\end{equation}
Similar calculations, involving appropriate indexes and derivatives of Eq. (\ref{eq600}), shows that the functions $\eta^\mu$ are at most linear functions of space-time coordinates. In other words,
\begin{equation}
\tilde\eta^\mu = a^\mu + R^{\mu}_{\,\nu}\,x^\nu \,, \nonumber
\end{equation}
where $a^\mu$ is an arbitrary constant 4-vector and $R^{\mu}_{\,\nu}$ is a constant second-rank tensor. However, the components
$R^{\mu}_{\,\nu}$ are not entirely free. Indeed, Eq. (\ref{eq600}) implies
\begin{equation}
\nonumber
R_{\mu\nu} + R_{\nu\mu} = 0 \,,
\end{equation}
so that $R^{\mu}_{\,\nu}$ is an anti-symmetric tensor. This exhausts the information contained in the
Noether symmetry condition (\ref{eq1}), applied to the real scalar field.
We are at a convenient point to enumerate the results until now. The Noether symmetries are completely specified by
\begin{eqnarray}
\label{poi}
\eta^\mu &=& a^\mu + R^{\mu}_{\,\nu}\,x^\nu \,, \nonumber \\ \psi &=&
\tilde\phi(x) \,, \nonumber
\end{eqnarray}
where $a^\mu$ is a constant 4-vector and $R^{\mu}_{\,\nu}$ a constant anti-symmetric tensor, while $\tilde\phi$ is any solution of the Klein-Gordon equation. Therefore, we are left with a 10-parameter external symmetry group for space-time coordinates, plus the internal symmetry transformation due to linearity. Naturally, the external symmetry group is Poincar\'e's group, where $a^\mu$ relates to space-time translations, $R^{ij}$ relates to spatial rotations, and $R^{0i}$ corresponds to Lorentz boosts.
\subsection{Conserved Currents for the Real Scalar Field}
\label{ccrsf}
To obtain the conserved current defined in Eq. (\ref{eqq1}), the 4-vector $\sigma^\mu$ is needed. From Eq. (\ref{eq8}), we obtain
\begin{equation}
\nonumber
\sigma^\mu = \phi\,\partial^{\mu}\tilde\phi \,,
\end{equation}
so that
\begin{eqnarray}
J^\mu &=& \partial^{\mu}\phi\,a^{\nu}\partial_{\nu}\phi -
a^{\mu}{\cal{L}} +
\partial^{\mu}\phi\,\partial_{\nu}\phi\,R^{\nu}_{\,\alpha}x^\alpha \nonumber \\
\label{eq22} &-& {\cal{L}}\,R^{\mu}_{\,\nu}x^\nu +
\phi\,\partial^{\mu}\tilde\phi - \tilde\phi\,\partial^{\mu}\phi
\,,
\end{eqnarray}
where ${\cal{L}}$ is given by Eq. (\ref{lsf}).
It is interesting to examine the conservation laws associated with different symmetries. For time translations, we set
$a^0 = 1$, together with the remaining parameters and $\tilde\phi$ vanishing. From expression (\ref{eq22}), we get
\begin{eqnarray}
J^0 &=& \frac{1}{2}\,\left[(\partial_{0}\phi)^2 + (\nabla\phi)^2 +
m^{2}\phi^2\right] \,, \nonumber \\
J^i &=& \partial^{i}\phi\,\partial_{0}\phi \,, \nonumber
\end{eqnarray}
corresponding to the energy conservation law,
\begin{equation}
\frac{d}{dt} \left\{\frac{1}{2}\,\int \left[(\partial_{0}\phi)^2 + (\nabla\phi)^2 +
m^{2}\phi^2\right] d{\bf r}\right\} = 0 \,. \nonumber
\end{equation}
For space translations, we set $a^\mu = (0,a^{1},a^{2},a^{3})$, together with vanishing remaining parameters and $\tilde\phi$. The conserved currents (one for each component of the translation vector ${\bf a}$) can be expressed as
\begin{eqnarray}
J^0 &=& a^{j}\,\partial^{0}\phi\,\partial_{j}\phi \,,
\label{mom}
\\
J^i &=& a^{j}\,\left[\partial^{i}\phi\,\partial_{j}\phi -
\frac{\delta^{i}_{\,j}}{2}\Bigl((\partial_{0}\phi)^2 - (\nabla\phi)^2 - m^{2}\phi^2\Bigr)\right] \,, \nonumber
\end{eqnarray}
associated with linear momentum conservation,
\begin{equation}
\frac{d}{dt} \int \partial^0\phi\,\nabla\phi\,d{\bf r} = 0 \,. \nonumber
\end{equation}
Spatial rotations are associated with $R^{i}_{\,j} \neq 0$, which gives
\begin{eqnarray}
J^0 &=& R^{j}_{\,k}\,x^k\,\partial^{0}\phi\,\partial_{j}\phi\,, \nonumber \\
J^i &=& R^{j}_{\,k}\,x^k\,\left(\partial^{i}\phi\,\partial_{j}\phi - \delta^{i}_{\,j}\,{\cal{L}}\right) \,, \nonumber
\end{eqnarray}
associated with the angular momentum conservation,
\begin{equation}
\frac{d}{dt} \int \partial^{0}\phi \,\,{\bf r}\times\nabla\phi\,\,d{\bf r} = 0 \,.
\nonumber
\end{equation}
For Lorentz boosts, we take $R^{0}_{\,j} \neq 0$, leaving us with
\begin{eqnarray}
J^0 &=& R^{0}_{\,j}\left[x^{j}\left(\partial^{0}\phi\,\partial_{0}\phi - {\cal L}\right) - x^0 \,\partial_{0}\phi\,\partial^{j}\phi\right] \,, \nonumber \\
J^i &=& R^{0}_{\,j}\left[x^{j}\,\partial^{i}\phi\,\partial_{0}\phi - x^{0}\left(\partial^{i}\phi\,\partial^{j}\phi + {\cal L}\,\delta^{i}_{\,j}\right)\right] \,. \nonumber
\end{eqnarray}
In consequence,
\begin{equation}
\frac{d}{dt} \int \left[x^0 \,\partial_{0}\phi\,\partial_{i}\phi - x_{i}\,\left(\partial^{0}\phi\,\partial_{0}\phi
- {\cal L}\right)\right]\,d{\bf r} = 0 \,.
\label{boost}
\end{equation}
Taking into account the momentum density $\pi_i = \partial_{0}\phi\,\partial_{i}\phi$ which follows from Eq. (\ref{mom}) and the energy density ${\cal H} = \partial^{0}\phi\,
\partial_{0}\phi - {\cal L}$, Eq. (\ref{boost}) reads
\begin{equation}
\frac{d}{dt}\,\int\left(x^{0}\,\boldsymbol{\pi} - {\cal H}\,{\bf r}\right)\,d{\bf r} = 0 \,, \nonumber
\end{equation}
showing, in a more transparent way, that the linear momentum relative to the center of mass is constant. Notice that the Noether momentum density, a quantity arising from space translation symmetries, in this case is not the same as the canonical momentum density $\partial{\cal L}/\partial(\partial_{0}\phi) = \partial^{0}\phi$.
Similarly, for a free relativistic particle with momentum ${\bf p}$ and energy $H$, one has $(d/dt)({\bf p}\,t - H\,{\bf r}) = 0$.
Finally, for the internal symmetry, only $\tilde\phi$ is non-vanishing. In this case,
\begin{equation}
J^\mu = \phi\,\partial^{\mu}\tilde\phi - \tilde\phi\,\partial^{\mu}\phi \quad \Rightarrow \quad \frac{d}{dt}\int \left(\phi\,\partial^{0}\tilde\phi - \tilde\phi\,\partial^{0}\phi\right)\,d{\bf r} = 0 \,. \nonumber
\end{equation}
This conserved current is analogous to the constant wronskian of two par\-ti\-cu\-lar solutions $y_{1,2}(t)$ for a
linear harmonic oscillator equation, namely,
\begin{equation}
\frac{d^2 y_{1,2}}{dt^2} + y_{1,2} = 0 \quad \Rightarrow \quad \frac{d}{dt}\left(y_1 \frac{dy_2}{dt} - y_2 \frac{dy_1}{dt}\right) = 0 \,. \nonumber
\end{equation}
Recapitulating, we have derived, in a systematic way, the Noether point symmetries group for the real scalar field, obtaining the Poincar\'e group (external symmetries) plus an infinite dimensional internal symmetry group, reflecting the linearity of the Klein-Gordon equation. It can be mentioned that such internal Noether symmetry transformation
is not recognized in the literature, as far as we know.
\section{Complex Scalar Field}
\label{csf}
In this Section, we consider the case of the complex scalar field, so that a global gauge symmetry is expected. Let us verify this, proceeding in a systematic manner, as done in the case of the real scalar field.
\subsection{Noether Symmetries for the Complex Scalar Field}
The Lagrangian density for the complex scalar field is
\begin{equation}
{\cal L} = \partial^{\mu}\phi^{*}\,\partial_{\mu}\phi -
m^{2}\,\phi^{*}\,\phi \,,
\label{ldcsf}
\end{equation}
implying two separate equations for the independent fields $\phi, \phi^*$,
\begin{eqnarray}
\partial^{\mu}\partial_{\mu}\phi + m^{2}\phi &=& 0 \,, \nonumber \\ \partial^{\mu}\partial_{\mu}\phi^* + m^{2}\phi^*
&=& 0 \, \nonumber
\end{eqnarray}
Due to the linearity of the equations, it is reasonable to suppose the existence of internal Noether symmetries corresponding to the addition of particular solutions.
Consider the infinitesimal point transformations,
\begin{eqnarray}
x^\mu &\rightarrow& x^{\mu} +
\varepsilon\,\eta^{\mu}(\phi,\phi^{*},x) \,, \nonumber
\\ \phi &\rightarrow& \phi + \varepsilon\,\psi(\phi,\phi^{*},x)
\,, \nonumber \\ \phi^* &\rightarrow& \phi^* +
\varepsilon\,\psi^{*}(\phi,\phi^{*},x)\,, \nonumber
\end{eqnarray}
where $\varepsilon$ is an infinitesimal parameter, and where
$\eta^{\mu}(\phi,\phi^{*},x)$, $\psi(\phi,\phi^{*},x)$, $\psi^{*}(\phi,\phi^{*},x)$
are smooth functions to be determined and not depending on field derivatives. The Noether invariance condition (\ref{eq1}), generalized to more than a single field, yields
\begin{eqnarray}
\frac{\partial{\cal{L}}}{\partial\phi}\,\psi +
\frac{\partial{\cal{L}}}{\partial\phi^*}\,\psi^* + \frac{\partial
{\cal{L}}}{\partial(\partial_{\mu}\phi)}\,(d_{\mu}\psi -
\partial_{\nu}\phi\,d_{\mu}\eta^{\nu}) &+& \nonumber \\ \label{equa1} +
\frac{\partial
{\cal{L}}}{\partial(\partial_{\mu}\phi^*)}\,(d_{\mu}\psi^*
-
\partial_{\nu}\phi^{*}\,d_{\mu}\eta^{\nu}) +
\partial_\mu{\cal{L}}\,\eta^\mu +
{\cal{L}}\,d_{\mu}\eta^\mu &=& d^{\mu}\sigma_\mu \,,
\nonumber
\end{eqnarray}
where $\sigma_\mu = \sigma_{\mu}(\phi,\phi^{*},x)$ is, at this stage, an arbitrary 4-vector. The corresponding conserved current reads
\begin{equation}
\label{eq300} J^\mu = \theta^{\mu}_{\,\nu}\,\eta^\nu -
\frac{\partial{\cal L}}{\partial(\partial_{\mu}\phi)}\,\psi -
\frac{\partial{\cal L}}{\partial(\partial_{\mu}\phi^{*})}\,\psi^* +
\sigma^\mu \,,
\end{equation}
where the energy-momentum tensor
\begin{equation}
\theta^{\mu}_{\,\nu} = \frac{\partial{\cal
L}}{\partial(\partial_{\mu}\phi)}\,\partial_{\nu}\phi +
\frac{\partial{\cal
L}}{\partial(\partial_{\mu}\phi^*)}\,\partial_{\nu}\phi^* -
\delta^{\mu}_{\,\nu}\,{\cal L} \,. \nonumber
\end{equation}
was used.
Inserting ${\cal{L}}$ from Eq. (\ref{ldcsf}) into the symmetry condition (\ref{eq1}),
it results a polynomial on the derivatives of $\phi, \phi^*$, to be identically set to zero. The coefficient of each monomial (term with equal power of the derivatives of the fields) should then vanish. For brevity, we will not show
the full calculations, which are analogous to the case of the real scalar field.
The terms of degree three imply
\begin{equation}
\eta^{\mu} = \eta^{\mu}(x) \,,
\nonumber
\end{equation}
so that the external transformations are field-independent.
The terms of degree three and two imply
\begin{eqnarray}
\label{eq2} \frac{\partial\psi}{\partial\phi^*} =
\frac{\partial\psi^*}{\partial\phi} &=& 0 \,,\\ \label{eq3}
\delta_{\mu\nu}\Bigl(\frac{\partial\psi}{\partial\phi} +
\frac{\partial\psi^*}{\partial\phi^*} &+&
\partial_{\alpha}\eta^\alpha\Bigr) - \partial_{\mu}\eta_\nu -
\partial_{\nu}\eta_\mu = 0 \,.
\end{eqnarray}
A tedious analysis shows that Eqs. (\ref{eq2}) and (\ref{eq3}) satisfy the appropriate Cauchy conditions if and only if
\begin{eqnarray}
\label{x1}
\partial_{1}\eta_1 &=& \partial_{2}\eta_2 = \partial_{3}\eta_3 = -
\partial_{0}\eta_0 \,, \nonumber \\
\partial_{\mu}\eta_\nu &+& \partial_{\nu}\eta_\mu = 0 \,,\quad \mu \neq
\nu \,. \nonumber
\label{xx1}
\end{eqnarray}
In this case, the solution for Eqs. (\ref{eq2})-(\ref{eq3}) is
\begin{eqnarray}
\psi &=& - \phi\,\partial_{0}\eta_0 - i\,\lambda(x)\phi +
\tilde\phi(x) \,, \label{p1} \\ \psi^* &=& - \phi^{*}\,\partial_{0}\eta_0 +
i\,\lambda(x)\phi^* + \tilde\phi^{*}(x) \,, \label{p2}
\end{eqnarray}
where $\lambda(x)$, $\tilde\phi(x)$ and $\tilde\phi^{*}(x)$ are arbitrary functions not depending on the fields. For consistency,
\begin{equation}
\label{x2}
\lambda(x) = \lambda^{*}(x) \,.
\end{equation}
The terms depending on $\lambda(x)$ are associated with local gauge transformations.
The invariance condition, regarding terms of first order in the derivatives of the fields, gives
\begin{eqnarray}
\frac{\partial\sigma_\mu}{\partial\phi} &=&
\partial_\mu\psi^* \,, \nonumber \\
\frac{\partial\sigma_\mu}{\partial\phi^*} &=&
\partial_\mu\psi \,. \nonumber
\end{eqnarray}
Taking into account $\psi$, $\psi^*$ from Eqs. (\ref{p1}) and (\ref{p2}), it can be verified that the Cauchy condition
\begin{equation}
\frac{\partial^{2}\sigma_\mu}{\partial\phi\partial\phi^*} =
\frac{\partial^{2}\sigma_\mu}{\partial\phi^{*}\partial\phi} \label{cau}
\end{equation}
yields
\begin{equation}
\partial_{\mu}\lambda(x) = 0 \,.
\nonumber
\end{equation}
In other words, $\lambda$ does not depend on $x$ and gauge transformations are global. Besides, in this case we can solve for
\begin{equation}
\label{eq500} \sigma_\mu = -
\phi^{*}\phi\,\partial_{0}\partial_{\mu}\eta_0 +
\phi^{*}\partial_{\mu}\tilde\phi + \phi\,\partial_{\mu}\tilde\phi^*
+ \tilde\sigma_{\mu}(x) \,,
\end{equation}
where $\tilde\sigma_{\mu}(x)$ is an arbitrary 4-vector, depending on $x$ only.
Finally, the term not containing field derivatives in the invariance condition (\ref{eq1}) gives
\begin{eqnarray}
\phi^{*}\phi\left(\partial^{\mu}\partial_{\mu}\partial_{0}\eta_0 -
2m^{2}\partial_{0}\eta_0\right) -
\phi^{*}\left(\partial^{\mu}\partial_{\mu}\tilde\phi +
m^{2}\tilde\phi\right) \nonumber \\ -
\phi\left(\partial^{\mu}\partial_{\mu}\tilde\phi^* +
m^{2}\tilde\phi^*\right) - \partial^{\mu}\tilde\sigma_\mu = 0 \,. \nonumber
\end{eqnarray}
The terms proportional to $\phi^{*}\phi$, $\phi^*$, $\phi$ and the remaining contribution should identically vanish. Hence,
\begin{eqnarray}
\label{e4}
\partial_{0}(\partial^{\mu}\partial_{\mu}\eta_0 - 2m^{2}\eta_0) &=& 0 \,,\\
\label{eq5} \partial^{\mu}\partial_{\mu}\tilde\phi + m^{2}\tilde\phi &=& 0 \,,\\
\label{eq6} \partial^{\mu}\partial_{\mu}\tilde\phi^* + m^{2}\tilde\phi^* &=& 0 \,,\\
\label{eq7}
\partial^{\mu}\tilde\sigma_\mu &=& 0 \,.
\end{eqnarray}
Equations (\ref{eq5}) and (\ref{eq6}) show that $\tilde\phi$
and $\tilde\phi^*$ are particular solutions of the Klein-Gordon equation, while Eq. (\ref{eq7})
shows that
\begin{equation}
\tilde\sigma_\mu = 0 \,,
\nonumber
\end{equation}
without loss of generality. Finally, Eq. (\ref{e4}) reveals that
\begin{equation}
\label{ult}
\partial^{\mu}\partial_{\mu}\eta_0 - 2m^{2}\eta_0 = - 2m^{2}\tilde\eta_{0}({\bf r}) \,,
\end{equation}
where $\tilde\eta_{0}({\bf r})$ is an arbitrary time-independent function. Equation (\ref{ult}) coincides with
Eq. (\ref{eqq300}). In this way, we realize that there is no need to repeat the previous calculations, with the conclusion that the external Noether symmetries are given by the Poincar\'e group. Therefore,
\begin{eqnarray}
\eta^\mu &=& = a^\mu + R^{\mu}_{\,\nu}x^\nu \,, \nonumber \\ \psi &=& -
i\lambda\,\phi + \tilde\phi(x) \,, \nonumber \\ \psi^* &=& i\lambda\,\phi^* +
\tilde\phi^{*}(x) \,, \nonumber
\end{eqnarray}
where $a^\mu$ is a constant 4-vector, $R^{\mu}_{\,\nu}$ is a second-rank antisymmetric tensor, $\lambda$ is a real constant and
$\tilde\phi, \tilde\phi^*$ are particular solutions of the Klein-Gordon equation. To conclude, besides the Poincar\'e group, one has the internal symmetries, composed by a global gauge transformation and a symmetry due to the linearity of the model.
\subsection{Conserved Currents for the Complex Scalar Field}
To compute the conserved currents, there is the need of the 4-vector
$\sigma_\mu$, obtained from Eq. (\ref{eq500}),
\begin{equation}
\sigma^\mu = \phi^{*}\partial^{\mu}\tilde\phi +
\phi\,\partial^{\mu}\tilde\phi^* \,. \nonumber
\end{equation}
Inserting this 4-vector into $J^\mu$ in Eq.
(\ref{eq300}), we get a somewhat long expression. In comparison to the case of the real scalar field, the distinctive feature comes from the global gauge symmetry. Setting $\lambda = 1$ and the remaining contributions to zero, we derive the Noether's current,
\begin{equation}
J^{\mu}_{(\lambda)} = i\,(\phi\,\partial^{\mu}\phi^* -
\phi^{*}\partial^{\mu}\phi) \,, \nonumber
\end{equation}
corresponding to global electric charge conservation,
\begin{equation}
\frac{d}{dt}\left[\,i\,\int (\phi\,\partial^{0}\phi^* -
\phi^{*}\partial^{0}\phi)\,d{\bf r}\right] = 0 \,. \nonumber
\end{equation}
The remaining Noether currents are analogous to those derived in Section \ref{ccrsf}.
\section{Vacuum Electromagnetic Field}
\label{ef}
\subsection{Noether Symmetries for the Electromagnetic Field in Vacuum}
Following our schedule, the Noether point symmetries for the vacuum electromagnetic field will be studied, without {\it ad hoc} claims. The Lagrangian density is
\begin{equation}
{\cal L} = - \frac{1}{4}\,F^{\mu\nu}\,F_{\mu\nu} \,,
\label{vl}
\end{equation}
where
\begin{equation}
F_{\mu\nu} = \partial_{\mu}A_\nu - \partial_{\nu}A_\mu \nonumber
\end{equation}
is the electromagnetic tensor, while $A_\mu = (A_{0},{\bf A})$ denotes the electromagnetic 4-potential. The
Euler-Lagrange equations are the vacuum Maxwell's equations,
\begin{equation}
\partial^{\nu}\partial_{\nu}\,A^\mu - \partial^{\mu}\partial_{\nu}A^\nu = 0 \,.
\nonumber
\end{equation}
We can anticipate the existence of internal Noether symmetries, associated with the linearity of the equations.
Suppose the infinitesimal point transformations,
\begin{eqnarray}
x^\mu &\rightarrow& x^{\mu} + \varepsilon\,\eta^{\mu}(A,x) \,, \nonumber
\\ A^\mu &\rightarrow& A^\mu + \varepsilon\,\Gamma^{\mu}(A,x) \nonumber
\,,
\end{eqnarray}
where $A$ is a shorthand for the 4-potential, $\varepsilon$ an infinitesimal parameter, and $\eta^{\mu}(A,x)$, $\Gamma^{\mu}(A,x)$
functions to be determined, not depending on field derivatives.
Since the Lagrangian density contains only field derivatives, the Noether symmetry condition simplifies to
\begin{equation}
\frac{\partial{\cal
L}}{\partial(\partial_{\mu}A_\nu)}\,\left(d_{\mu}\Gamma_\nu -
\partial_{\alpha}A_{\nu}\,d_{\mu}\eta^\alpha\right) +
{\cal{L}}\,d_{\mu}\eta^\mu =
d^{\mu}\sigma_\mu \,,
\label{cn}
\end{equation}
where, at this stage, $\sigma_\mu = \sigma_{\mu}(A,x)$ is an arbitrary 4-vector with the indicated functional dependence.
The procedure must be clear now. Inserting ${\cal L}$ from Eq. (\ref{vl}) into Eq. (\ref{cn}) and considering field derivatives of third degree, it follows that
\begin{equation}
\eta^{\mu} = \eta^{\mu}(x) \,. \nonumber
\end{equation}
Once again, the space-time transformation rules do not depend on the fields.
After several elementary calculations, the second-order field derivative terms give
\begin{eqnarray}
\label{f1} \Gamma^\mu = - \partial^{\mu}\eta_{\nu}\,\,A^\nu &+&
\tilde{A^{\mu}}(x) \,,\\ \label{ff1}
\partial_{\mu}\eta_\nu + \partial_{\nu}\eta_\mu &=& 0 \,,
\end{eqnarray}
where the 4-vector $\tilde{A^\mu}$ is field-independent.
As seen before, the Poincar\'e group is obtained from Eq. (\ref{ff1}).
The first-order in the field derivative terms give
\begin{equation}
\partial_\mu\Gamma_\nu -
\partial_\nu\Gamma_\mu =
\frac{\partial\sigma_\nu}{\partial A^\mu} \,, \nonumber
\end{equation}
or, taking into account Eq. (\ref{f1}),
\begin{equation}
\partial_\mu{\tilde A}_\nu -
\partial_\nu{\tilde A}_\mu =
\frac{\partial\sigma_\nu}{\partial A^\mu} \,, \nonumber
\end{equation}
with the solution
\begin{equation}
\sigma_\mu = (\partial_{\nu}\tilde{A_\mu} -
\partial_{\mu}\tilde{A_{\nu}})\,A^\nu + \tilde\sigma_{\mu}(x) \nonumber
\,,
\end{equation}
where $\tilde\sigma_{\mu}(x)$ is an arbitrary field-independent 4-vector.
From the remaining term in the invariance condition, we get
\begin{eqnarray}
\label{e2} \partial^{\mu}\tilde\sigma_\mu &=& 0 \,, \\ \label{e3}
\partial^{\nu}\partial_{\nu}\tilde{A^\mu}
-
\partial^{\mu}(\partial_{\nu}\tilde{A^\nu}) &=& 0 \,.
\end{eqnarray}
While Eq. (\ref{e2}) shows that $\tilde\sigma_\mu$ is superfluous, Eq.
(\ref{e3}) implies that $\tilde{A^\mu}$ is a particular solution for vacuum Maxwell's equations.
Summing up results, and using Eq. (\ref{f1}), we derive the following symmetry transformation functions,
\begin{eqnarray}
\eta^\mu &=& a^\mu + R^{\mu}_{\,\nu}x^\nu \,,\quad
R_{\mu\nu} + R_{\nu\mu} = 0 \,, \nonumber \\ \label{eq4} \Gamma^\mu &=& \tilde{A^\mu}(x) +
R^{\mu}_{\,\nu}A^\nu \,, \nonumber
\end{eqnarray}
where $\tilde{A^\mu}(x)$ solves Maxwell's vacuum equations.
A notorious particular vacuum solution is specified by local gauge transformations,
\begin{equation}
\tilde{A^\mu} = \partial^{\mu}\lambda(x) \,, \nonumber
\end{equation}
where $\lambda(x)$ is an arbitrary space-time function. In this context there is a link between
the linearity of vacuum Maxwell's equations and local gauge transformations.
\subsection{Conserved Currents for the Electromagnetic Field in Vacuum}
The conserved current in all generality reads
\begin{eqnarray}
J^\mu &=& a^\nu\,\left(- F^{\mu\alpha}\,\partial_{\nu}A_\alpha + \frac{1}{4}\,\delta^{\mu}_{\,\nu}\,F^{\alpha\beta}F_{\alpha\beta}\right) \nonumber \\
&+& R^{\alpha}_{\,\beta}\left(-x^\beta\,F^{\mu\nu}\,\partial_\alpha A_\nu + \frac{1}{4}\,x^\beta \delta^{\mu}_{\,\alpha}\,F^{\nu\gamma}\,F_{\nu\gamma} + A^\beta\,F^{\mu}_{\,\,\alpha}\right) \nonumber \\
&+&
F^{\mu\nu}\tilde{A}_\nu - \tilde{F}^{\mu\nu}\,A_\nu \,, \label{eff}
\end{eqnarray}
where it was defined $\tilde{F}^{\mu\nu} = \partial^{\mu}\tilde{A}^\nu - \partial^{\nu}\tilde{A}^\mu$.
The usual symmetries (space-time translations and spatial rotations) are fairly well discussed in the literature, including additional steps such as the symmetrization of the energy-momentum tensor \cite{Kleinert}. For instance, in terms of the electric field $E_i = - F_{0i}$ and the magnetic field $B_i = - (1/2)\,\varepsilon_{ijk}\,F^{jk}$, where $\varepsilon_{ijk}$ is the 3-dimensional Kronecker symbol, for time-translations (only $a^0 = 1$ is non-zero) one finds
\begin{equation}
J^0 = \frac{1}{2}\,(E^2 + B^2) + \nabla\cdot(A_0\,{\bf E}) - A_0\,\nabla\cdot{\bf E} \,. \nonumber
\end{equation}
The third term is a surface term and so does not contribute to the associated conserved quantity. The last term vanishes since $\nabla\cdot{\bf E} = 0$ in vacuum. The only effective contribution is
\begin{equation}
\int\,J^{0}\,d{\bf r} = \frac{1}{2}\,\int\,(E^2 + B^2)\,d{\bf r} \,, \nonumber
\end{equation}
which is the well-known electromagnetic energy density.
Actually the conserved current (\ref{eff}) can be put into a more traditional and gauge-invariant form, as follows. One has
\begin{eqnarray}
J^\mu &=& a^\nu\,\left(- F^{\mu\alpha}\,F_{\nu\alpha} + \frac{1}{4}\,\delta^{\mu}_{\,\nu}\,F^{\alpha\beta}F_{\alpha\beta} - \partial_\alpha(F^{\mu\alpha}\,A_\nu) + (\partial_\alpha F^{\mu\alpha})\,A_\nu\right) \nonumber \\
&+& \frac{1}{2}\,R^{\alpha\beta}\Bigl[x^\beta\,\left(- F^{\mu\nu}\,F_{\alpha\nu} + \frac{1}{4}\,\delta^{\mu}_{\,\alpha}\,F^{\nu\gamma}\,F_{\nu\gamma}\right) - x^\alpha\,\left(- F^{\mu\nu}\,F_{\beta\nu} + \frac{1}{4}\,\delta^{\mu}_{\,\beta}\,F^{\nu\gamma}\,F_{\nu\gamma}\right)
\nonumber \\ &+& \partial_\nu\left(x_\alpha F^{\mu\nu}\,A_\beta - x_\beta F^{\mu\nu}\,A_\alpha\right) - (x_\alpha\,A_\beta - x_\beta\,A_\alpha)\,\partial_\nu\,F^{\mu\nu} \Bigr] \nonumber \\
&+&
F^{\mu\nu}\tilde{A}_\nu - \tilde{F}^{\mu\nu}\,A_\nu \,, \label{efff}
\end{eqnarray}
Most terms in (\ref{efff}) are total divergence terms, integrating to zero, or vanish due to Maxwell's equations in vacuum. Therefore essentially we have the well-known \cite{Hill, Kleinert} results,
\begin{equation}
\label{rec}
J^\mu = a^\nu P^{\mu}_{\,\nu} + \frac{1}{2}\,R^{\alpha\beta}\,M_{\beta\alpha}^\mu + F^{\mu\nu}\tilde{A}_\nu - \tilde{F}^{\mu\nu}\,A_\nu \,,
\end{equation}
where
\begin{equation}
P^{\mu}_{\,\nu} = - F^{\mu\alpha}\,F_{\nu\alpha} + \frac{1}{4}\,\delta^{\mu}_{\,\nu}\,F^{\alpha\beta}F_{\alpha\beta} \nonumber
\end{equation}
is associated to momentum conservation and
\begin{equation}
M_{\alpha\beta}^\mu = x_\alpha\,P^{\mu}_{\,\beta} - x_\beta\,P^{\mu}_{\,\alpha} \nonumber
\end{equation}
is related to angular momentum conservation. As can be checked, $\partial_\mu\,J^\mu = 0$ is maintained in the reshaped form (\ref{rec}).
The internal symmetry due to linearity is basically ignored in the usual treatments. As apparent from Eq. (\ref{eff}), one has the conservation law
\begin{equation}
\nonumber
\frac{d}{dt}\,\int (F^{0\mu}\tilde{A}_\mu - \tilde{F}^{0\mu}\,A_\mu)\,d{\bf r} = 0 \,,
\end{equation}
where both $A_\mu, \tilde{A}_\mu$ solve Maxwell's equations.
The case of local gauge symmetries with $\tilde{A}_\mu = \partial_{\mu}\lambda$ yields $\partial_{\mu}J^{\mu}_{(\lambda)} = 0$, where
\begin{equation}
J^{\mu}_{(\lambda)} =
F^{\mu\nu}\partial_{\nu}\lambda = \partial_\nu(F^{\mu\nu}\lambda) - \lambda\,\partial_\nu\,F^{\mu\nu}\,. \label{rhs}
\end{equation}
The subscript $\lambda$ refers to the particular gauge function employed. At least for $\lambda$ bounded at infinity, the first term in the right-hand side of Eq. (\ref{rhs}) contribute a vanishing surface term, while the last term vanishes due to Maxwell's equations in vacuum. Hence in this case ($\tilde{A}_\mu = \partial_\mu\lambda$ for some function $\lambda$) the related conserved charge vanishes. Otherwise it can happens that the conservation law yields a non-trivial result.
\section{Coupled Complex Scalar and Electromagnetic Fields}
\label{csem}
\subsection{Noether Symmetries for the Coupled Complex Scalar and Electromagnetic Fields}
The Lagrangian density follows from the minimal coupling assumption and is given by
\begin{equation}
{\cal L} = (\partial^{\mu}\phi^* -
i\,e\,A^{\mu}\,\phi^{*})(\partial_{\mu}\phi + i\,e\,A_{\mu}\,\phi) -
m^{2}\phi^{*}\phi - \frac{1}{4}\,F^{\mu\nu}F_{\mu\nu} \,, \label{laga}
\end{equation}
where $e$ is the particle's charge and $F^{\mu\nu}$ the electromagnetic tensor, as before.
The Euler-Lagrange equations are
\begin{eqnarray}
D^{\mu}D_{\mu}\phi + m^{2}\phi &=& 0 \,, \nonumber \\ (D^{\mu}D_{\mu}\phi)^* +
m^{2}\phi^* &=& 0 \,, \nonumber \\
\partial_\nu F^{\mu\nu} &=& i\,e\, \Bigl(\phi\,(D^{\mu}\phi)^* -
\phi^*\,(D^{\mu}\phi)\Bigr) \nonumber
\,,
\end{eqnarray}
where
\begin{eqnarray}
D^{\mu}\phi &=& (\partial^\mu + i\,e\,A^{\mu})\,\phi \,, \nonumber \\
(D^{\mu}\phi)^* &=& (\partial^\mu - i\,e\,A^{\mu})\,\phi^* \nonumber
\end{eqnarray}
denote covariant derivatives. Due to presence of matter and the corresponding nonlinearity, we expect a broken internal symmetry, previously corresponding to the linear superposition law.
Consider infinitesimal point transformations of the form
\begin{eqnarray}
x^\mu &\rightarrow& x^{\mu} +
\varepsilon\,\eta^{\mu}(A,\phi,\phi^{*},x) \,, \nonumber
\\
A^\mu &\rightarrow& A^\mu + \varepsilon\,\Gamma(A,\phi,\phi^{*},x)
\,, \nonumber
\\ \phi &\rightarrow& \phi + \varepsilon\,\psi(A,\phi,\phi^{*},x)
\,, \nonumber \\ \phi^* &\rightarrow& \phi^* +
\varepsilon\,\psi^{*}(A,\phi,\phi^{*},x) \,. \nonumber
\end{eqnarray}
The Noether symmetry condition is
\begin{eqnarray}
\frac{\partial{\cal L}}{\partial\,A^\mu}\,\Gamma^\mu
&+& \frac{\partial{\cal L}}{\partial\phi}\,\psi +
\frac{\partial{\cal L}}{\partial\phi^*}\,\psi^*
+
\frac{\partial{\cal L}}{\partial
(\partial_{\mu}A_\nu)}\,(d_{\mu}\Gamma_\nu
-
\partial_{\rho}A_{\nu}\,d_{\mu}\eta^\rho)
+ \nonumber \\ &+&
\frac{\partial{\cal{L}}}{\partial(\partial_{\mu}\phi)}\,(d_{\mu}\psi
- \partial_{\nu}\phi\,d_{\mu}\eta^{\nu}) +
\frac{\partial{\cal{L}}}{\partial(\partial_{\mu}\phi^*)}\,
(d_{\mu}\psi^*
- \partial_{\nu}\phi^{*}\,d_{\mu}\eta^{\nu}) + \nonumber
\\ &+& {\cal{L}}\,d_{\mu}\eta^\mu =
d^{\mu}\sigma_\mu \,, \nonumber
\end{eqnarray}
where $\sigma_\mu = \sigma_{\mu}(A,\phi,\phi^{*},x)$ is an arbitrary 4-vector.
From the cubic in the derivatives terms, it can be shown that
\begin{equation}
\eta^{\mu} = \eta^{\mu}(x) \,. \nonumber
\end{equation}
The quadratic terms give
\begin{eqnarray}
\label{e1} \Gamma^\mu \nonumber &=& - \partial^{\mu}\eta_{\nu}\,\,A^\nu +
\tilde{A^{\mu}}(x) \,,\\ \psi &=& - i\,e\,\lambda(x)\,\phi +
\tilde\phi(x) \,, \nonumber \\ \psi^* &=& i\,e\,\lambda(x)\,\phi^* +
\tilde\phi^{*}(x) \,, \nonumber \\ \label{eqqq1}
\partial_{\mu}\eta_\nu &+& \partial_{\nu}\eta_\mu = 0 \,,
\end{eqnarray}
where $\tilde{A^\mu}$, $\tilde\phi$, $\tilde\phi^*$ and
$\lambda$ are field independent. The factor $e$ was included for convenience. Moreover, $\lambda$ is a real function. Proceeding as in the vacuum case, from Eq. (\ref{eqqq1}) it follows that the external symmetries are associated with the Poincar\'e group.
The linear in the derivative terms imply
\begin{eqnarray}
\label{eq30}
\frac{\partial\sigma_\nu}{\partial\,A^\mu} &=&
\partial_\mu\tilde{A_\nu} -
\partial_\nu\tilde{A_\mu} \,,\\
\label{eq31}
\frac{\partial\sigma_\mu}{\partial\phi} &=& -
i\,e\,A_{\mu}\,\tilde\phi^* - i\,e\,\phi^{*}\,\tilde{A_\mu} +
i\,e\,(\partial_{\mu}\lambda)\,\phi^* + \partial_{\mu}\tilde\phi^* \,,\\
\label{eq32}
\frac{\partial\sigma_\mu}{\partial\phi^*} &=&
i\,e\,A_{\mu}\,\tilde\phi + i\,e\,\phi\,\tilde{A_\mu} -
i\,e\,(\partial_{\mu}\lambda)\,\phi + \partial_{\mu}\tilde\phi \,.
\end{eqnarray}
The integrability condition
\begin{equation}
\frac{\partial^{2}\sigma_\mu}{\partial\phi\,\partial\phi^*} =
\frac{\partial^{2}\sigma_\mu}{\partial\phi^{*}\,\partial\phi} \nonumber
\end{equation}
implies
\begin{equation}
\label{eq40} \tilde{A_\mu} = \,\partial_{\mu}\lambda
\,.
\end{equation}
From Eq. (\ref{eq40}) it can be seen that in the presence of matter only (local) gauge symmetries are allowed.
Once Eq. (\ref{eq40}) is satisfied, it is possible to solve Eqs. (\ref{eq30})-(\ref{eq32}), with the result
\begin{eqnarray}
\tilde\phi &=& \tilde\phi^* = 0 \,,\nonumber \\ \sigma_\mu &=&
\tilde\sigma_{\mu}(x) \,. \nonumber
\end{eqnarray}
As verified, the symmetries due to the linearity are completely eliminated.
The remaining terms in the invariance condition give
\begin{equation}
\partial^{\mu}\tilde\sigma_\mu = 0 \,, \nonumber
\end{equation}
so that there is no loss of generality to define
\begin{equation}
\tilde\sigma_\mu = 0 \,.
\nonumber
\end{equation}
To conclude, the external symmetries are composed by the Poincar\'e transformations, while the internal symmetries are specified by
\begin{eqnarray}
\Gamma^\mu &=& R^{\mu}_{\,\nu}A^\nu +
\,\partial^{\mu}\lambda(x) \,, \nonumber \\
\psi &=& - i\,e\,\lambda(x)\,\phi \,, \nonumber \\ \psi^* &=& i\,e\,\lambda(x)\,\phi^* \,. \nonumber
\end{eqnarray}
\subsection{Conserved Currents for the Coupled Complex Scalar and Electromagnetic Fields}
The general conserved current is
\begin{eqnarray}
J^{\mu} &=& a^{\nu}\,\Bigl(\partial_\nu\phi\,(D^\mu\phi)^* + \partial_\nu\phi^*\,(D^\mu\phi) - F^{\mu\alpha}\,\partial_\nu A_\alpha - \delta^{\mu}_{\,\nu}{\cal L} \Bigr) \nonumber \\
&+& R^{\alpha}_{\,\beta} \Bigl[x^\beta \Bigl(\partial_\alpha\phi\,(D^\mu\phi)^* + \partial_\alpha\phi^*\,(D^\mu\phi) - F^{\mu\nu}\,\partial_\alpha A_\nu - \delta^{\mu}_{\,\alpha}{\cal L}\Bigr) + A^\beta\,F^{\mu}_{\alpha}\Bigr] \nonumber \\
&+& F^{\mu\nu}\,\partial_\nu\lambda + i\,e\,\lambda\,\phi\,(D^\mu\phi)^* - i\,e\,\lambda\,\phi^*\,(D^\mu\phi) \,. \label{pen} \nonumber
\end{eqnarray}
The usual conservation laws (energy, linear momentum, angular momentum) are verified.
Besides, we have the gauge symmetry, with the conserved current
\begin{equation}
J^{\mu}_{(\lambda)} = F^{\mu\nu}\,\partial_\nu\lambda + i\,e\,\lambda\,\phi\,(D^\mu\phi)^* - i\,e\,\lambda\,\phi^*\,(D^\mu\phi) \,.
\nonumber
\end{equation}
In particular,
\begin{equation}
J^{0}_{(\lambda)} = \partial_{i}(F^{0i}\lambda) - \lambda\,\left[\partial_{i}F^{0i} + i\,e\,\phi^{*}\,(D^\mu\phi) - i\,e\,\phi\,(D^\mu\phi)^*\right] = \partial_{i}(F^{0i}\lambda) \,,
\nonumber
\end{equation}
the last equality coming from Maxwell's equations.
Therefore, $dQ_{(\lambda)}/dt = 0$, where
\begin{equation}
Q_{(\lambda)} = \int\,J^{0}_{(\lambda)}\,d{\bf r} = - \int\,\nabla\cdot(\lambda\,{\bf E})\,d{\bf r} \,,
\nonumber
\end{equation}
which is the electric charge for $\lambda = - 1$, where $E_i = - F_{0i}$ is the electric field.
\section{Charged Scalar Particle under External Electro\-magnetic Fields}
The Lagrangian density is the same as in Eq. (\ref{laga}), omitting the free electromagnetic field contribution, but retaining the interaction terms,
\begin{equation}
{\cal L} = (\partial^{\mu}\phi^* -
i\,e\,A^{\mu}\,\phi^{*})(\partial_{\mu}\phi + i\,e\,A_{\mu}\,\phi) -
m^{2}\phi^{*}\phi \,. \label{lagar} \nonumber
\end{equation}
The interpretation is that $A_\mu$ is a given external field, acting on a test charge. Such problem has implications in laser-plasma interactions in the quantum relativistic regime \cite{Marklund}.
The Noether symmetry condition is given by Eq. (\ref{eq1}), since $A_\mu$ is not subject to any transformation. The treatment of the third and second-order in the field derivative terms yields the results already shown in Eqs. (\ref{x1})-(\ref{x2}), with the replacement $\lambda \rightarrow e\,\lambda$ since now there is the test charge $e$.
From the first and zeroth order in the field derivatives terms, the following equations are derived,
\begin{eqnarray}
\frac{\partial\sigma_\mu}{\partial\phi} &=& \Bigl(i\,e\,A_\nu\,\partial^{\nu}\eta_\mu - \partial_{\mu}\partial_{0}\eta_0 + i\,e\,\partial_\mu\lambda - i\,e\,\eta_\nu\partial^{\nu}A_\mu - 2\,i\,e\,(\partial_0\eta_0)\,A_\mu\Bigr)\,\phi^* \nonumber \\ &+& \partial_\mu\tilde{\phi}^* - i\,e\,A_\mu\,\tilde{\phi}^* \,, \label{fu1}\\
\frac{\partial\sigma_\mu}{\partial\phi^*} &=& \Bigl(- i\,e\,A_\nu\,\partial^{\nu}\eta_\mu - \partial_{\mu}\partial_{0}\eta_0 - i\,e\,\partial_\mu\lambda + i\,e\,\eta_\nu\partial^{\nu}A_\mu + 2\,i\,e\,(\partial_0\eta_0)\,A_\mu\Bigr)\,\phi \nonumber \\ &+& \partial_\mu\tilde{\phi} + i\,e\,A_\mu\,\tilde{\phi} \,, \label{fu2}\\
\partial^\mu\sigma_\mu &=& 2\,\Bigl(-\,e^2\,A^\mu\partial_\mu\lambda + \,e^2\eta_\nu\,(\partial^\nu A_\mu)\,A^\mu + (\partial_{0}\eta_0)\,(e^2\,A^\mu A_\mu - m^2)\Bigr)\,\phi^*\,\phi \nonumber \\
&+& \Bigl(i\,e\,A^\mu\partial_\mu\tilde{\phi}^* + (e^2\,A^\mu A_\mu - m^2)\,\tilde{\phi}^*\Bigr)\,\phi \nonumber \\ &+& \Bigl(- i\,e\,A^\mu\partial_\mu\tilde{\phi} + (e^2\,A^\mu A_\mu - m^2)\,\tilde{\phi}\Bigr)\,\phi^* \label{fu3} \,.
\end{eqnarray}
From the equality of mixed partial derivatives as shown in Eq. (\ref{cau}), it follows that
\begin{equation}
\eta_\nu\partial^{\nu}A_\mu + 2\, (\partial_0\eta_0)\,A_\mu
- A_\nu\,\partial^{\nu}\eta_\mu = \partial_\mu\lambda \,, \label{tit}
\end{equation}
allowing to re-express Eqs. (\ref{fu1}) and (\ref{fu2}) as
\begin{eqnarray}
\frac{\partial\sigma_\mu}{\partial\phi} &=& -\tilde{\phi}^*\,\partial_\mu\partial_{0}\eta_0 + \partial_\mu\tilde{\phi}^*
- i\,e\,A_\mu\,\tilde{\phi}^* \,, \nonumber \\
\frac{\partial\sigma_\mu}{\partial\phi^*} &=& -\tilde{\phi}\,\partial_\mu\partial_{0}\eta_0 + \partial_\mu\tilde{\phi} + i\,e\,A_\mu\,\tilde{\phi} \,, \nonumber
\end{eqnarray}
which has the solution
\begin{equation}
\sigma_\mu = - (\partial_\mu\partial_{0}\eta_0)\,\phi^*\,\phi + (\partial_\mu\tilde{\phi}^*
- i\,e\,A_\mu\,\tilde{\phi}^*)\,\phi + (\partial_\mu\tilde{\phi}
+ i\,e\,A_\mu\,\tilde{\phi})\,\phi^* + \tilde{\sigma}_{\mu}(x) \,, \label{sig}
\end{equation}
where $\tilde{\sigma}_{\mu}(x)$ does not depend on $\phi, \phi^*$.
Plugging $\sigma_\mu$ from Eq. (\ref{sig}) into Eq. (\ref{fu3}) and considering the terms linear in $\phi^*$ and $\phi$, the results are
\begin{eqnarray}
\Bigl(\partial^\mu\partial_\mu + 2\,i\,e\,A^\mu\partial_\mu + i\,e\,\partial^\mu A_\mu - (e^2\,A^\mu A_\mu - m^2)\Bigr)\,\tilde{\phi} &=& 0 \,, \nonumber\\
\Bigl(\partial^\mu\partial_\mu - 2\,i\,e\,A^\mu\partial_\mu - i\,e\,\partial^\mu A_\mu - (e^2\,A^\mu A_\mu - m^2)\Bigr)\,\tilde{\phi}^* &=& 0 \,, \nonumber
\end{eqnarray}
which are the Klein-Gordon equations obeyed by $\tilde{\phi}, \tilde{\phi}^*$. Therefore, the addition of particular solutions is an allowable Noether symmetry, reflecting the linearity of the problem. On the same trend, the independent term (not containing $\phi$ or $\phi^*$) gives $\partial^\mu\tilde\sigma_\mu = 0$, so that $\tilde\sigma_\mu = 0$ without loss of generality.
In the continuation, the term proportional to $\phi^*\phi$ in Eq. (\ref{fu3}), taking into account Eq. (\ref{sig}), gives
\begin{equation}
\Bigl(\partial^\mu\partial_\mu + 2\,(e^2\,A^\mu A_\mu - m^2)\Bigr)\,\partial_{0}\eta_0 + 2\,e^2\eta_\nu\,(\partial^\nu A_\mu)\,A^\mu - 2\,e^2\,A^\mu\partial_\mu\lambda = 0 \,.
\nonumber
\end{equation}
Using Eq. (\ref{tit}), the last equation can be reshaped as
\begin{equation}
\Bigl(\partial^\mu\partial_\mu - 2\,(e^2\,A^\mu A_\mu + m^2)\Bigr)\,\partial_{0}\eta_0 + 2\,e^2\,A^\mu A_\nu\,\partial^\nu\eta_\mu = 0 \,,
\nonumber
\end{equation}
where the gauge function $\lambda$ was eliminated. From the identities shown in Eqs. (\ref{x1}) and (\ref{xx1}), it is possible to prove that $A^\mu A_\mu\,\partial_0\eta_0 = A^\mu A_\nu\,\partial^\nu\eta_\mu = 0$, so that
\begin{equation}
\Bigl(\partial^\mu\partial_\mu - 2\,m^2\Bigr)\,\partial_{0}\eta_0 = 0 \,.
\nonumber
\end{equation}
We have already meet the same equation, see Eq. (\ref{eq4}). Following the same procedure as before, we obtain $\partial_{0}\eta_0 = 0$, which in turn imply $\partial_\mu\eta_\nu + \partial_\nu\eta_\mu = 0$, showing that the symmetry transformations constitute the Poincar\'e group. In comparison, the symmetry treatment of non-relativistic charged particle motion under external electromagnetic fields, including magnetic monopoles, is compatible with much more general transformations of the time variable \cite{Haas1, Haas2, Haas3}. In the relativistic case, the space and time variables entanglement allows only linear coordinate transformations as determined by the Poincar\'e generators.
The external fields are not arbitrary. According to Eq. (\ref{tit}), they satisfy
\begin{equation}
\eta_\nu\partial^{\nu}A_\mu - A_\nu\,\partial^{\nu}\eta_\mu = \partial_\mu\lambda \,, \label{tiit} \nonumber
\end{equation}
for some function $\lambda$.
In other words,
\begin{equation}
\eta_\nu (\partial^\nu A_\mu - \partial_\mu A^\nu) = \partial_\mu (\lambda - \eta^\nu A_\nu) + (\partial_\mu \eta_\nu + \partial_\nu \eta_\mu)\,A^\nu \,,
\nonumber
\end{equation}
which is the same as
\begin{equation}
\label{grad}
\eta^\nu F_{\nu\mu} = \partial_\mu\tilde{\lambda} \,,
\end{equation}
where
\begin{equation}
\tilde{\lambda} = \lambda - \eta^\mu A_\mu \nonumber
\end{equation}
is a redefined arbitrary function. Equation (\ref{grad}) is in a manifestly covariant form. The last requirement is the verification of the homogeneous Maxwell equations,
\begin{equation}
\label{homo}
\partial_\mu F_{\nu\alpha} + \partial_\nu F_{\alpha\mu} + \partial_\alpha F_{\mu\nu} = 0 \,. \nonumber
\end{equation}
\subsection{Examples}
It is worthwhile to consider illustrations of the compact equation (\ref{grad}).
\subsubsection{Time-translations}
Supposing time-translations, one has $\lambda_0 = 1, \lambda_i = 0$. In this case, it is a simple matter to verify that Eq. (\ref{grad}) implies an electric field ${\bf E} = \nabla\tilde{\lambda}$, where $\tilde{\lambda} = \tilde{\lambda}({\bf r})$ is time-independent. Finally, Faraday's law shows that the magnetic-field is also time-independent.
\subsubsection{Space-translations}
For translations along a particular axis (say, the $z-$direction), one has $\eta_0 = \eta_1 = \eta_2 = 0, \eta_3 = 1$. In this case, applying the symmetry condition (\ref{grad}), one has the electromagnetic field components
\begin{equation}
B_1 = F_{32} = \partial^{2}\tilde{\lambda} \,, \quad B_2 = F_{13} = - \partial^{1}\tilde{\lambda} \,, \quad E_3 = F_{30} = - \partial_{0}\tilde{\lambda} \,, \nonumber
\end{equation}
where $\partial^{3}\tilde{\lambda} = 0$. Moreover, from the homogeneous Maxwell's equations it is found that $\partial^{3}E_1 = \partial^{3}E_2 = \partial^{3}B_3 = 0$.
\subsubsection{Circularly Polarized Electromagnetic Field}
In an inverse approach, one might consider first the electromagnetic configuration, and then ask about Noether symmetries plugging the field into Eq. (\ref{grad}). For instance, for the right circularly polarized wave one has
\begin{equation}
\label{a3}
{\bf A} = \frac{C_0}{\sqrt{2}} ({\boldsymbol \epsilon}\, e^{i\theta} + {\boldsymbol \epsilon}^* e^{-i\theta}) \,, \quad A_0 = 0 \,, \nonumber
\end{equation}
where $\omega, k$ are constants and $C_0$ is a slowly varying function of the phase
\begin{equation}
\label{phase}
\theta = k z - \omega t \,, \nonumber
\end{equation}
while ${\boldsymbol \epsilon} = (\hat{x} - i \hat{y})/\sqrt{2}$ denotes the polarization vector, with the unit vectors $\hat{x}, \hat{y}$ perpendicular to the direction of light propagation. It is a simple matter to verify that the circularly polarized field admits Noether symmetries, satisfying Eq. (\ref{grad}) with $\eta_0 = k, \eta_1 = \eta_2 = 0, \eta_3 = \omega$, together with $\tilde{\lambda} = 0$. In this case, it should be noticed that this is not the unique solution to the system provided by Eq. (\ref{grad}).
\subsubsection{Homogenous Static Magnetic Field}
As a final example, let us suppose an homogeneous static magnetic field
\begin{equation}
{\bf B} = B_0 \hat{z} \,, \nonumber
\end{equation}
where $B_0$ is a non-zero constant. What are the allowable electric fields, so that Noether symmetry exist?
From Faraday's law, one has $\nabla\times{\bf E} = 0$, so that ${\bf E} = - \nabla\,A_0$, where $A_0$ is the scalar potential. Therefore,
$F_{i0} = - \partial^{i}A_0$. Moreover, for our magnetic field we have $F_{12} = - B_0, F_{23} = F_{31} = 0$. The Noether symmetry
condition (\ref{grad}) reduces to four equations. For simplicity, limiting ourselves to the case $\eta_0 = 0$, these equations are given by
\begin{eqnarray}
\eta_{i}\partial^{i}A_0 &=& \partial\tilde{\lambda}/\partial t \,, \label{fffu1} \\
\eta_2 B_0 &=& \partial\tilde{\lambda}/\partial x \,, \label{fffu2} \\
- \eta_1 B_0 &=& \partial\tilde{\lambda}/\partial y \,, \label{fffu3} \\
0 &=& \partial\tilde{\lambda}/\partial z \,. \label{fffu4}
\end{eqnarray}
Since $\eta_0 = 0$ implies $a^0 = 0, R^{0i} = 0$, the general solution for Eqs. (\ref{fffu2}) and (\ref{fffu3}) is
\begin{equation}
\label{fffu5}
\tilde{\lambda} = B_0 (a^1 y - a^2 x) + \frac{1}{2}\,B_0 R^{12} (x^2 + y^2) - B_{0} z (R^{23} x + R^{31} y) + g(z,t) \,,
\end{equation}
for some function $g(z,t)$.
In view of Eq. (\ref{fffu4}), from Eq. (\ref{fffu5}) we conclude that $R^{23} = R^{31} = 0$, and that $g = g(t)$. Finally, Eq. (\ref{fffu1}) gives
\begin{equation}
\label{fffu6}
(a^1 + R^{12} y)\,\frac{\partial A_0}{\partial x} + (a^2 - R^{12} x)\,\frac{\partial A_0}{\partial y} + a^3\,\frac{\partial A_0}{\partial z} = - \dot{g}(t) \,.
\end{equation}
Just for simplicity, we will assume $a^3 = 0$. The results with $a^3 \neq 0$ are easily reachable, although of a less readable form.
There are two classes of solutions for Eq. (\ref{fffu6}) with $a^3 = 0$. The first, for $R^{12} \neq 0$, can be more simply written taking $R^{12} = 1$ and $a^{1} = a^2 = 0$ without loss of generality, after a rescaling of $g$ and appropriate space translations, if necessary. In this case one has
\begin{equation}
A_0 = \dot{g} \,\,{\rm arctan}\left(\frac{y}{x}\right) + \tilde{A}_{0}\left(\sqrt{x^2 + y^2}, z, t\right) \,, \nonumber
\end{equation}
where $\tilde{A}_0$ is an arbitrary function of the indicated arguments. Notice that the electric field found from the scalar potential is a single valued function.
The second class of solutions for Eq. (\ref{fffu6}) is found for $R^{12} = 0$ and is given by
\begin{equation}
A_0 = - \frac{1}{2}\,\dot{g}\left(\frac{x}{a^1} + \frac{y}{a^2}\right) + \tilde{A}_{0}(a^1 y - a^2 x, z, t) \,, \nonumber
\end{equation}
where $\tilde{A}_0$ is an arbitrary function of the indicated arguments and where $a^1 a^2 \neq 0$ was assumed, for simplicity.
\subsection{Conserved Currents for the Charged Scalar Particle under External Electromagnetic Fields}
Our conserved current reads
\begin{eqnarray}
J^\mu &=& \Bigl[\Bigl(D^\mu\phi\Bigr)\,\partial_\nu\phi^* + \Bigl(D^\mu\phi\Bigr)^{*}\,\partial_\nu\phi - \delta^{\mu}_{\,\nu}\,\cal{L}\Bigr]\,\eta^\nu \nonumber \\
&+& i\,e\,\lambda\,\Bigl[\phi\,\Bigl(D^\mu\phi\Bigr)^* - \phi^{*}\,\Bigl(D^\mu\phi\Bigr)\Bigr] \nonumber \\
&+& \phi\,\Bigl(D^\mu\tilde{\phi}\Bigr)^* + \phi^{*}\,\Bigl(D^\mu\tilde{\phi}\Bigr) - \tilde{\phi}\,\Bigl(D^\mu\phi\Bigr)^* - \tilde{\phi}^{*}\,\Bigl(D^\mu\phi\Bigr) \,. \nonumber
\end{eqnarray}
There are three components in the current: the external symmetries contribution, associated to $\eta^\mu$; the internal gauge symmetries associated to $\lambda$; the internal symmetries due to linearity, due to the superposition law for particular solutions
$\tilde{\phi}, \tilde{\phi}^*$.
\section{Conclusion}
\label{conclusion}
Adopting a systematic procedure, it was seen that the incorporation of symmetries requires the addition of adequate terms to the Lagrangian density. Already in the case of the complex scalar field, there is a global gauge symmetry, which becomes local when coupled to the electromagnetic field. In this context, gauge symmetries can be viewed as rotations in the internal space parametrized by the real and imaginary parts of the field \cite{Ryder}. For the vacuum electromagnetic field, we have a local gauge symmetry, which is in accordance with the causality principle. The accompanying linearity of the equations, is associated with a Noether conserved current not always emphasized in the literature. It was the main purpose of the present study, to pursue the calculations of Noether symmetries without {\it ab initio} assumptions. Such a procedure was shown to be useful for the clear identification of the internal symmetries due to linearity, in the case of the free theories (real and complex scalar field, vacuum Maxwell's equations). The conservation laws can be used to check the accuracy of numerical methods. It is expected, that the present systematic procedure could be more frequently applied, in both relativistic and non-relativistic studies. An example was applied, for the first time, to the case of a charged scalar particle under a general external electromagnetic field. The Noether symmetry condition, in this case, was reduced to the compact system provided by (\ref{grad}), which is manifestly gauge-invariant.
\acknowledgments
Work partially supported by Con\-se\-lho Na\-cio\-nal de De\-sen\-vol\-vi\-men\-to Cien\-t\'{\i}\-fi\-co e Tec\-no\-l\'o\-gi\-co
(CNPq).
|
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{"url":"http:\/\/mathoverflow.net\/questions\/142587\/how-to-detect-a-simple-closed-curve-from-the-element-in-the-fundamental-group","text":"How to detect a simple closed curve from the element in the fundamental group?\n\n(1) Given a fundamental group representation of a hyperbolic surface, i.e. $<a_j,b_j|\\prod[a_j,b_j]=1>$, and given an element in this group, can we determine whether this element can be represented by a simple closed curve?\n\n(2) More specifically, if we consider only a commutative element, whose abelizaions is trivial in the homology group, is there some method to determine whether this element can be represent by a simple closed separating curve on the surface?\n\nFor example, $a_1a_2a_1^{-1}a_2^{-1}$ cannot be represented by a simple closed separating curve. I guess the geometric interpretation of a simple closed separating curve is that it is the boundary of the neibourhood of a group of chained curves. Here a group of chained curves means a set of curves $\\{c_1, \\dots ,c_{2h}\\}$ satisfying the following conditions: the geometric intersection numbers are $i(c_j,c_{j+1})=1$ and $i(c_j,c_{k})=0$ for $k-j>1$. But even this criterion is true, the choice of $c_j$'s in the fundamental group are diverse. From an element in the fundamental group, we may not easily see that whether it is the boundary of a group of chained curves. Are there some easy way or algorithm to determine it?\n\nThe previous discussion is for question (2). What if we consider the elements in question (1)? I learn from Farb and Margalit's book that there is a neccesary condition on the homology group that the abelization element of a simple closed curve in the homology group must be primitive. What about the representation in the fundamental group?\n\n-\n\nThere is no simple necessary and sufficient condition for whether an element of the fundamental group can be realized by a simple closed curve. However, there are a variety of algorithms known. I believe that the first such algorithm is given in\n\nReinhart, Bruce L. Algorithms for Jordan curves on compact surfaces. Ann. of Math. (2) 75 1962 209\u2013222.\n\nBut probably the most used one is the one in\n\nBirman, Joan S.; Series, Caroline, An algorithm for simple curves on surfaces. J. London Math. Soc. (2) 29 (1984), no. 2, 331\u2013342.\n\nIt uses hyperbolic geometry.\n\nI should remark that both of these papers concern unbased curves; however, if $\\Sigma_g$ is a closed genus $g$ surface, then an element $\\gamma \\in \\pi_1(\\Sigma_g,\\ast)$ can be realized by a based simple closed curve if and only if it is freely homotopic to a simple closed curve. Indeed, if $\\gamma$ is freely homotopic to a simple closed curve, then there is some $x \\in \\pi_1(\\Sigma_g,\\ast)$ such that $x \\gamma x^{-1}$ can be realized by a based simple closed curve. But it follows from the Dehn-Nielsen-Baer theorem that inner automorphisms of $\\pi_1(\\Sigma_g,\\ast)$ can be realized by based homeomorphisms of the surface, so thus $\\gamma$ can also be realized by a based simple closed curve.\n\n-\n(it looks like HJRW posted some of this info while I was typing, but I'll leave this up since it does contain some other stuff). \u2013\u00a0Andy Putman Sep 19 '13 at 15:04\nSnap!${}{}{}{}$ \u2013\u00a0HJRW Sep 19 '13 at 15:34\n@HJRW : Wait, was my comment inadvertently mean? I was just trying to acknowledge your priority. Certainly no offense was intended. \u2013\u00a0Andy Putman Sep 19 '13 at 15:46\nI meant 'snap' in the English sense: en.wikipedia.org\/wiki\/Snap_%28card_game%29#Snap Sorry, I seem to have caused confusion. \u2013\u00a0HJRW Sep 19 '13 at 15:49\n@Andy - I think that \"snap\" in English translates to \"jinx\" in American. separatedbyacommonlanguage.blogspot.co.uk\/2006\/10\/\u2026 \u2013\u00a0Sam Nead Jan 2 '14 at 21:43\n\nVarious algorithms for determining whether a given conjugacy class contains a simple representative are given in the following papers.\n\nReinhart, Bruce L., 'Algorithms for Jordan curves on compact surfaces', Ann. of Math. (2) 75 1962 209\u2013222.\n\nZieschang, Heiner, 'Algorithmen f\u00fcr einfache Kurven auf Fl\u00e4chen', (German) Math. Scand. 17 1965 17\u201340.\n\nBirman, Joan S; Series, Caroline, 'An algorithm for simple curves on surfaces', J. London Math. Soc. (2) 29 (1984), no. 2, 331\u2013342.\n\n-\n\nA recent paper by Patricia Cahn, A Generalization of the Turaev Cobracket and the Minimal Self-Intersection Number of a Curve on a Surface gives a strengthening of Turaev's bracket. She shows that a related invariant of a free homotopy class is 0 if and only if the class is a power of a simple loop. This answers your question, modulo determining if your class is a proper power of some other element; I have no idea how difficult this latter is, or indeed how calculable her invariant is. There are some worked examples in the paper.\n\n-\n\nA necessary condition is that Turaev's cobracket is zero. See the paper of Moira Chas on combinatorial Lie bialgebras. There is also a later paper by Chas and Krongold, which promises an answer to your question, but I have not read it.\n\n-","date":"2016-02-09 18:17:26","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8639538884162903, \"perplexity\": 314.92862852421706}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-07\/segments\/1454701157443.43\/warc\/CC-MAIN-20160205193917-00130-ip-10-236-182-209.ec2.internal.warc.gz\"}"}
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Саут Вибер () град је у америчкој савезној држави Јута.
Демографија
По попису из 2010. године број становника је 6.051, што је 1.791 (42,0%) становника више него 2000. године.
Референце
Литература
Види још
Списак градова у САД по броју становника
Највећи градови у САД по деценијама
Спољашње везе
-{United States Census Bureau}-
Градови у Јути
Википројект географија/Насеља у САД
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You may be considering a fast cash loans Madisonville Texas. When you are looking into anything new it's good to know each step in Madisonville that is expected so you don't miss anything expected and make your experience in Madisonville TX any less than it should be. online cash advance lending usually break down to three steps that we will cover here in Madisonville Texas. The only thing that may be different is as most quick cash lenders businesses in the financing sector the process may have new added benefits to the Madisonville fast cash loans borrower as technology and ease of borrowing online cash advances.
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The way it can work is the loan receiver goes to the fast money loans lender to get the fast cash loan instead of online. The quick money loan receiver's part of the transaction is to leave a postdated check in Madisonville. A postdated check in Madisonville is a check that is written with a date in Madisonville in the future which can't be cashed until the Madisonville account holder expects to have adequate money in Madisonville to cover the amount the check is written for. In the quick money loans situation the loan receiver writes the check for the amount in Madisonville Texas needed plus fees which keep the loan store running in Madisonville. The agreement goes when the date written on the post dated check comes the loans receiver usually returns to the location to settle in Madisonville the amount on the check. The motivation for the transaction is if the online loan receiver doesn't show up the quick money loans provider will cash the check. In the case that there is not enough cash in the account to cash the check the receiver of the online cash lending will face not only the loan plus fee repayment but also any fees in Madisonville from the bank for what is called a bounced check in Madisonville Texas. This is of course very undesirable.
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Rick Ross Explains MMG Tour's Cancellation, Says Choking Young Jeezy Is An "Option" [VIDEO]
Say what you will about Rick Ross and his larger-than-life image, but the Maybach Music honcho knows how to stir up controversy. With his nationwide tour cancelled allegedly due to threats from the notorious Gangster Disciples gang, Ricky Rozay took a time to address the situation while returning to his home base of Miami in an on-air interview with 99 Jamz Felisha Monet.
The bombshell of the interview, however, was the Bawse saying he and rival Young Jeezy's run-in at the BET Hip Hop Awards was a minor dustup that has the potential to get uglier.
Rick Ross on the Young Jeezy confrontation:
"This is nothing personal to me. At the BET Awards it basically just boiled down to me running across Young Jeezy, he had five security guards in front of him, five police officers behind him and we crossed paths I said 'what's up?' soon as he said 'what's up?' I tried to choke him. His security guards hemmed me up; whoever the big black dude with the bumps on his face, he should give that dude something special for Christmas."
The Rich Forever rapper continued to lay out what may happen next in the ongoing saga, and even threw some darts in the direction of another nemesis of his in 50 Cent.
"I'm a hands on type dude. I'm better with my hands than I am at making music," said a confident Ross. "So it's not personal but if I see Young Jeezy again will I try to choke him? I don't know that's an option. If I see 50 Cent, is it personal? Not at all but it's an option."
Rick Ross gave a salute to Fif, who now added the title of boxing promoter to his long list of achievements outside the recording booth. Rozay promised in the interview to hop into the fight game as well, saying that they should "beef on the boxing level too."
This probably won't end well, nor anytime soon.
Check out Rozay speak on Jeezy below, and explain why he's feels he's good in any hood on the next page.
http://www.springboardplatform.com/js/overlay
50 Cent , 99 Jamz , beef , bet hip hop awards , boxing , Cancelled , Fight , Gangster Disciples , Gunplay , maybach music group , rozay , young jeezy
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San Diego State University Transit Center is a station on San Diego Trolley's Green Line. The station is underground (the only such station in the system) and has side platforms. The station is located in the Aztec Green on the south portion of the campus of San Diego State University. The station entrances are between College Avenue and Campanile Drive. The station is very popular with students and staff who commute to the university because of the high cost and low availability of parking around campus.
History
In 1989, the San Diego Metropolitan Transit Development Board began developing plans for an additional trolley line connecting its Blue and Orange lines. The new line was initially expected to cost $506 million and cover 5.6 miles of track. After initially looking at over ten different routes to connect the two lines, the Board initially recommended connecting San Diego State University at the north end of its campus, near Interstate 8. However SDSU officials wanted the trolley station to go through the center of the campus, which would require tunneling underneath the campus. The Board initially balked at this concept as it would increase construction costs by $40–50 million. Once the Board learned that tunneling would be cheaper than purchasing adjacent land they agreed to alter the route. Final construction costs for the new trolley station were $103 million. Construction for the new trolley line first began in 1999.
Construction
Several methods were used to excavate the tunnel that would contain the station and the tracks to be used for the trolleys to pass underneath the campus. 2,915 feet (of the total 4,000 feet of track) were excavated using the cut-and-cover method where a current road above the tunnel would have its pavement removed, the underlying ground entrenched, and the pavement for the road re-added. For the remaining 1,085 feet, the New Austrian Tunnelling method was used, which required the use of the geological stress of the surrounding rock mass to stabilize the tunnel.
Construction of the tunnel and station was designed (Began in September 1998) to minimally disrupt the campus and, as a result, it was halted during the opening weeks of each semester as well as finals. The station opened in September 2005 with sold out tickets purchased by students.
Design
The trolley station was designed by the architect firm ZGF Architects. The station was initially designed to limit the noise of the passing trolleys, so as to not disturb classes on campus. In addition to blue cold cathode lighting, natural light enters the station from the street level above through 20 openings.
Dedication
On June 29, 2011, the station was dedicated to Leon Williams, an SDSU alumnus and a former MTS board chair who was instrumental in bringing the light rail line to the heart of the SDSU campus.
Station layout
There are two tracks, each served by a side platform.
Bus connections
San Diego Metropolitan Transit System: 11, 14, 15, 115, Rapid 215, 856, 936, 955
See also
List of San Diego Trolley stations
References
Green Line (San Diego Trolley)
San Diego Trolley stations in San Diego
Transit Center
Railway stations in California at university and college campuses
Railway stations in the United States opened in 2005
Railway stations located underground in California
Underground rapid transit in the United States
2005 establishments in California
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\section{Introduction}
In four space-time dimensions, string solutions
with $N \le 4$ supersymmetries can be constructed,
through appropriate compactifications of the
six-dimensional internal manifold, from
any of the perturbative ten-dimensional strings:
heterotic, type I, type IIA or type IIB. Although these
constructions
appear different at the string perturbative level, they might be
non-perturbatively equivalent, provided the massless
spectrum and the number of supersymmetries is the
same \cite{ht}--\cite{pw}.
As far as $N=4$ supersymmetry is concerned, several tests in favour of
the non-perturbative
duality equivalence have been presented in the literature, not only in the
case of the heterotic string compactified on $T^6$ versus the type IIA, B
compactified on $K3\times T^2$\cite{ht},
but also for theories with a lower number of massless vector
multiplets \cite{6auth}, including the type IIA, B $N=4$ asymmetric,
freely acting, orbifold constructions \cite{sv}.
In the $N=4$ theories the space of the moduli fields is restricted
by supersymmetry to be the coset \cite{ht,fk}:
\begin{equation}
\left({SL(2,R)\over U(1)}\right)_S
\times\left({SO(6,6+r)\over SO(6)\times
SO(6+r)}\right)_{T}\, ,
\end{equation}
where $r=16$ in the heterotic and type IIA, B
compactified on $T^6$ and $K3 \times T^2$, respectively.
Models with lower rank, $r<16$, can
be constructed, via freely acting (asymmetric) orbifold
compactification,
from any of the perturbative superstring theories in ten dimensions
\cite{6auth, fk,sv}.
On the heterotic side, the dilaton $S_{\rm Het}$ is always in the
gravitational multiplet, while on the type II side it is either
one of the moduli of the vector multiplets ($S_{\rm II}=T^1$),
when the compactification is left--right-symmetric, with $N=(2,2)$
supersymmetry, or in the gravitational
multiplet in the asymmetric compactification with $N=(4,0)$
supersymmetry.
The non-perturbative string duality therefore implies interchanges
between the moduli fields $S_{\rm Het}$ and $T^{i}$ of the scalar
manifold \cite{ht, 6auth, sv}, with the perturbative states of
one string theory mapped to non-perturbative states of its
dual equivalent and vice versa \cite{6auth, sv, s, hmn=4}.
The non-perturbative equivalence of dual strings has been verified
on several occasions: for instance, the anomaly cancellation of the
six-dimensional heterotic string implies that there should be a
one-loop correction to the gravitational $R^2$ term in the type II
theory.
Such a term was found by direct calculation in \cite{6auth, vw2}. Its
one-loop threshold correction, upon compactification to four
dimensions,
implies instanton corrections on the heterotic side, due to
five-branes being wrapped around the six-torus.
Several other indications
are given for dual $N=4$ theories with rank $r<16$ \cite{6auth}.
Heterotic/type II dual pairs with lower supersymmetry, $N=2$,
share properties similar to those of $N=4$. In general
(non-freely-acting) symmetric orbifolds still give rise to $N=2$
heterotic/type II dual pairs in four dimensions \cite{kv}--\cite{re}. On
the heterotic side they can be interpreted as $K3$ plus gauge-bundle
compactifications, while on the type II side
they are Calabi--Yau compactifications of the ten-dimensional type
IIA theory. The heterotic dilaton is in a vector-tensor multiplet, dual
to a vector, and the
vector moduli space receives both perturbative and non-perturbative
corrections. The hypermultiplet moduli space, however, does
not receive perturbative corrections; if $N=2$ is assumed to be unbroken,
it does not receive non-perturbative corrections either.
The dilaton in the type II (symmetric) constructions is in a
tensor multiplet, dual to a
hypermultiplet, and the prepotential for the vector multiplets
receives only tree-level contributions. The tree-level type II
prepotential was computed and shown to give the correct one-loop
heterotic result. This provides a quantitative test of the duality
\cite{kv,re} and allows us to reach the non-perturbative corrections
of the heterotic side.
In the quantitative tests of non-perturbative
dualities, extended supersymmetry plays an essential role, since it
allows for the existence of BPS states (states in short representations
of the supersymmetry algebra). These states are (generically)
non-perturbatively stable and provide a reliable window into
non-perturbative corrections to some terms of the effective action
that receive contributions only from those states. The relevant
structures for this analysis are the helicity supertrace formulae,
which
distinguish between various BPS and non-BPS states
\cite{6auth, bk}--\cite{n=6}. For $N \ge 2$,
these supertraces appear in particular
in the $F^2$ and $R^2$ (two-derivative) terms or in a special class
of higher-order terms constructed out of the Riemann tensor and the
graviphoton field strength \cite{fg}. In the four-dimensional
heterotic string, these terms are anomaly-related and it can be shown
that they receive only tree- and one-loop corrections. In higher
dimensions, they do not receive non-perturbative corrections
either \cite{bk2}.
The non-perturbative equivalence of some
heterotic/type II dual pairs with $N=2$ supersymmetry has been
investigated
in Refs. \cite{sv,fhsv,hmn=2,gkp}. In this class of $N=2$ models the
scalar manifolds are coset spaces:
\begin{equation}
{SU(1,1)\over U(1)}\times {SO(2,2+N_V)\over SO(2)\times SO(2+N_V)}
\ \ {\rm and}
\ \ {SO(4,4+N_H)\over SO(4)\times SO(4+N_H)},
\end{equation}
describing the moduli space of the $N_V+3$ moduli in vector
multiplets and the $N_H+4$ in the hypermultiplets, respectively.
The type II symmetric duals correspond to
self-mirror Calabi--Yau threefold compactifications with Hodge
numbers $h^{1,1}= N_V +3=h^{2,1}=N_H + 3$, which are $K3$ fibrations,
necessary condition for the existence of heterotic duals
\cite{klm,al}. The equivalence of heterotic/type II
model(s) with $N_V=N_H=8$ was studied in
Refs. \cite{fhsv,hmn=2,gkp}; recently, in Ref. \cite{gkp}, this
analysis has been
extended to type II and heterotic duals with $N_{V}=N_{H}=4$ and
2.
The purpose of this work is to extend the analysis of Ref. \cite{gkp}
and establish the non-perturbative equivalence of three different
$N=2$ constructions with $N_V=N_H=0$: the heterotic construction with
supersymmetry
$N=(2,0)$, the symmetric type II construction with $N=(1,1)$ and
the asymmetric type II with $N=(2,0)$ supersymmetry.
All these constructions are based on six-dimensional
freely acting (asymmetric) orbifold compactifications and thus the
initial $N=8$ (in type II) or $N=4$ (in heterotic) supersymmetry is
spontaneously broken to $N=2$ \cite{kk}.
The heterotic scalar manifold of this $N=2$ construction is
described by the vector scalar manifold $\big(SU(1,1) \big/
U(1)\big)^3$
associated to the three moduli $S,T$ and $U$, and by the
hypermultiplet
quaternionic one, $SO(4,4)\big/ \big(SO(4) \times SO(4)\big)$.
The type IIA symmetric construction corresponds to a self-mirror
Calabi--Yau threefold compactification with Hodge numbers
$h^{1,1}=h^{2,1}=3$ \cite{sv}. The type II asymmetric construction
\cite{sv}
corresponds to
a spontaneous breaking of $N=(4,4)$ to $N=(2,0)$
supersymmetry \cite{kk}.
In Section 2 we construct the symmetric type II model with $N=(1,1)$
supersymmetry and calculate the one-loop gravitational corrections
associated to the $R^2$ term. The asymmetric type II
$N=(2,0)$ construction
is presented in Section 3; in the same section we show that the
$R^2$ corrections in the two type II dual theories
are in agreement with their non-perturbative equivalence.
The heterotic construction, as well as the corresponding corrections to
the gauge and gravitational couplings, are presented in Section
4. In Section 5 we compare the heterotic, the type II symmetric and the
type II asymmetric corrections, and show that the non-perturbative
equivalence of the three models is verified; we
furthermore show that due to this triality equivalence, there
exists a weak--strong coupling relation ($S$-duality) between the
heterotic and the asymmetric type II theory
($4 \pi S_{\rm Het}=-(4 \pi S_{\rm As})^{-1}$). We also claim
that the $N=8$ supersymmetry is restored in the heterotic strong
coupling regime. Finally, in Section 6 and in the appendix,
we derive the perturbative
prepotential, as well as part of its non-perturbative corrections, which
are argued to be valid for all three dual theories.
Our conclusion and comments are given in Section 7.
\boldmath
\section{The $N=(1,1)$ type II symmetric
construction}
\unboldmath
We start by considering a type II symmetric construction with
$N=(1,1)$ supersymmetry and no vector multiplets or hypermultiplets in
the twisted sector.
This model is obtained by compactification of the
ten-dimensional type II string on a Calabi--Yau manifold
${\rm CY}^{(3,3)}$,
with Hodge numbers $h^{1,1}=h^{2,1}=3$. This compactification
reduces the $N=(4,4)$ supersymmetry
to the desired $N=(1,1)$.
In what follows we will always work at the $Z_2^{(1)}\times
Z_2^{(2)}$
(freely acting) orbifold limit of this manifold, where the partition
function and the one-loop gravitational and gauge corrections
can be computed analytically.
At the orbifold point, where
${\rm CY}^{(3,3)}\equiv T^6\bigg / \left(Z_2^{(1)}\times
Z_2^{(2)}\right)$,
the partition function of the model can be
written easily in terms of the characters of the twisted and shifted
compactified left and right coordinates $X^I, \bar X^I,I=1,\ldots,6$,
and in terms of twisted fermionic superpartners $\Psi^I$ and
${\bar\Psi}^I$.
The remaining contribution to the partition function comes from the
left- and right-moving non-compact supercoordinates
$X^{\mu}, \Psi^{\mu}, {\bar X}^{\mu}, {\bar \Psi}^{\mu}$
and the super-reparametrization ghosts
$b,c,\beta,\gamma$ and ${\bar b},{\bar c},{\bar \beta},{\bar
\gamma}$.
The resulting partition function reads:
\begin{eqnarray}
Z_{\rm II}^{(1,1)} & = &
{1 \over \,{\rm Im}\, \tau \vert \eta \vert^{24} } {1 \over 4}
\sum_{H^1,G^1} \sum_{H^2,G^2}
\Gamma_{6,6} \ar{H^1,H^2}{G^1,G^2} \nonumber \\
&&\nonumber \\
&& \times {1 \over 2} \sum_{a,b} (-)^{a+b+ab} \vartheta \ar{a}{b}
\vartheta \ar{a+H^2}{b+G^2}
\vartheta \ar{a+H^1}{b+G^1}
\vartheta \ar{a-H^1-H^2}{b-G^1-G^2} \nonumber \\
&&\nonumber \\
&& \times {1 \over 2} \sum_{\bar{a},\bar{b}}
(-)^{\bar{a}+\bar{b}+\bar{a}\bar{b}} \bar{\vartheta}
\ar{\bar{a}}{\bar{b}}
\bar{\vartheta} \ar{\bar{a}+H^2}{\bar{b}+G^2}
\bar{\vartheta} \ar{\bar{a}+H^1}{\bar{b}+G^1}
\bar{\vartheta} \ar{\bar{a}-H^1-H^2}{\bar{b}-G^1-G^2}\, ,
\label{zII}
\end{eqnarray}
where the contribution of $\beta,\gamma, \Psi^{\mu},\Psi^{I}$ and
${\bar \beta}, {\bar \gamma}, {\bar \Psi}^{\mu},{\bar \Psi}^{I}$
gives rise to the functions $\vartheta$ and $\bar{\vartheta}$, while
$\Gamma_{6,6} \ar{H^1,H^2}{G^1,G^2}$ denotes
the contribution of $X^I$ and $\bar X^I$; $(H^{1},G^{1})$ refer to
the boundary conditions
introduced by the
projection $Z_2^{(1)}$ and $(H^{2},G^{2})$ to the
projection $Z_2^{(2)}$:
\begin{equation}
\Gamma_{6,6} \ar{H^1,H^2}{G^1,G^2}=
\Gamma_{2,2}^{(1)} \ar{H^2 \vert H^1}{G^2 \vert G^1}\,
\Gamma_{2,2}^{(2)} \ar{H^1 \vert H^1+H^2}{G^1 \vert G^1+G^2}\,
\Gamma_{2,2}^{(3)} \ar{H^1+H^2 \vert H^2}
{G^1+G^2 \vert G^2}\, .
\end{equation}
Here we have introduced the twisted and shifted
characters of a $c=(2,2)$ block,
$\Gamma_{2,2} \ar{h\vert h'}{g\vert g'}$; the first column refers to
the twist, the second to the shift. The non-vanishing components
are the following:
\begin{eqnarray}
\Gamma_{2,2} \ar{h\vert h'}{g\vert g'}
&=&
{4\, \vert\eta \vert^6\over
\left\vert
\vartheta{1+h\atopwithdelims[] 1+g}\,
\vartheta{1-h\atopwithdelims[] 1-g}
\right\vert} \ , \ \
{\rm for\ } (h',g')=(0,0) \
{\rm or\ } (h',g')=(h,g) \nonumber \\
&=&
\Gamma_{2,2} \ar{h'}{g'} \ , \ \
{\rm for\ } (h,g)=(0,0)\, ,
\label{g22ts}
\end{eqnarray}
where $\Gamma_{2,2} \ar{h'}{g'}$ is the $Z_2$-shifted $(2,2)$ lattice
sum. As usual, the shift has to be specified by
the way it acts on the momenta and windings (our conventions are
those of Refs. \cite{6auth,gkp,kkprn}).
Since the three complex planes of $T^6$ are translated,
there are no fixed points. Therefore there are no
extra massless states coming from the twisted sectors:
the massless spectrum of this model contains
the $N=2$ supergravity multiplet, 3 vector multiplets,
1 tensor multiplet and 3 hypermultiplets.
The tensor multiplet is the type II dilaton multiplet
and is equivalent to an extra hypermultiplet.
By using the techniques developed in Refs. \cite{kk, solving, kkprn},
it can be shown that this model possesses an $N=8$ supersymmetry
spontaneously broken through a super-Higgs phenomenon,
due to the free actions of $Z_2^{(1)}\times Z_2^{(2)}$.
There exist
appropriate limits of the moduli, which depend on the precise shifts
in the lattices, in which there
is an approximate restoration of 16 or 32 supercharges.
In such limits, the supersymmetry restoration
is accompanied by a logarithmic instead of a linear blow-up of
the various thresholds.
The logarithmic blow-up is an infrared artefact, which can be lifted
by switching on an infrared cut-off
$\mu$ larger than the mass of the extra massive gravitinos; the
thresholds thus vanish, as expected, in the limit ${m_{3/2}/ \mu} \to
0$ in which supersymmetry is extended to $N=8$.
The relevant quantities for the computation of the
string correction to the $R^2$ term are the helicity
supertraces $B_{2n}$.
These are defined as the vacuum expectation value
of $\left( Q+\overline{Q} \right)^{2n}$,
where $Q$, $\overline{Q}$ stand for the left- and
right-helicity contributions to the four-dimensional physical helicity.
For the details of the computation
of such quantities in the framework of the above models
we refer to previous publications \cite{gkp, kkprn}. Here we simply
quote the results.
A straightforward computation shows that $B_2 = 0$, as expected in
the models with $N_V=N_H$ \cite{gkp}.
This feature is common to all the
$N=2$ type II $Z_2 \times Z_2$ symmetric orbifolds \cite{gkr}, in which
$B_2$ can receive a non-zero contribution only from the
$N=(1,1)$ sectors of the orbifold. The
internal coordinates in these sectors are twisted; all corrections
are therefore moduli-independent and come from the massless
states only. One finds $B_2=B_2\vert_{\rm massless}$, which vanishes
in the model we are considering here.
On the
other hand, $B_4$ receives non-zero contributions from the $N=(2,2)$
sectors of the orbifold, and we find\footnote{The prime summation
over $(h,g)$ stands for $(h,g) =
\{(0,1),(1,0),(1,1)\} $.}:
\begin{equation}
B_4=6 \sum_{i=1,2,3} \sump
\Gamma_{2,2}^{(i)}\ar{0|h}{0|g}\, .
\label{B411}
\end{equation}
From (\ref{B411}),
by applying the techniques developed in \cite{kkprn},
one can see that there is a limit
in moduli space in which $B_4$ vanishes; this is the signal of
the restoration of the $N=8$ supersymmetry.
The four-derivative gravitational correction we consider here
is similar to those that were analysed in Refs. \cite{6auth} and
\cite{gkp};
in order to obtain it we proceed as in \cite{gkp}.
There is no tree-level contribution to this operator, and the $R^2$
correction appears at one loop; it is related to the contribution of
the
$h^{1,1}$ moduli, obtained through the insertion, in the
one-loop partition function,
of the two-dimensional operator $2Q^2 \overline{Q}^2$.
In this class of models the
contribution of the $N=(1,1)$ sectors to $B_4$ vanishes, and
therefore
$\left\langle2Q^2 \overline{Q}^2 \right\rangle$ is $B_4 / 3$.
The massless contributions of the latter
give rise to an infrared logarithmic behaviour
${B_4 \vert_{\rm massless}\over 3}
\log \left( M^{(\rm IIA)\,2} \Big/
\mu^{(\rm IIA)\,2}\right)$
\cite{infra,delgrav}, where
$M^{(\rm IIA)}\equiv 1/\sqrt{\alpha'_{\rm IIA}}$ is
the type IIA string scale and $\mu^{(\rm IIA)}$ is the type IIA
infrared cut-off.
Besides this running, the one-loop correction contains, as usual, the
thresholds $\Delta_{\rm IIA}$, which account for the infinite tower of
massive string modes.
The threshold corrections to the $R^2$ term are related to the
infrared-regularized genus-one integral of $B_4 / 3$. This
relationship can be made more precise by noting that the amplitude
$\left\langle2Q^2 \overline{Q}^2 \right\rangle$ contains more than
the $R^2$-term corrections: it also accounts for terms
such as $F^2$ or $H^2$. However, in the type IIA
string, the $R^2$ corrections depend on the K\"ahler moduli
$T^1, T^2$ and $T^3$
(spanning the vector manifold), and are independent of the
complex-structure moduli $U^1, U^2$
and $U^3$ (spanning the scalar manifold) \cite{6auth, gkp}. We
thus have
\begin{equation}
\partial_{T^i}\Delta_{\rm IIA}=\frac{1}{3}\int_{\cal F}\frac{{\rm d}^2\tau}{\im}
\partial_{T^i} B_4
\ , \ \
\partial_{U^i}\Delta_{\rm IIA}=0\, .
\label{IIAthr}
\end{equation}
For definiteness
we choose the half-unit shifts for $Z_2^{(1)}$ and $Z_2^{(2)}$
as defined by the following insertions:
$(-)^{n_2G^1}$, $(-)^{m_1(G^1+G^2)}$, $(-)^{n_2G^2}$ shifting
the lattices of the first, second and third plane, respectively.
With this choice of lattice shifts the one-loop-corrected
gravitational coupling reads (up to a constant):
\begin{equation}
{16 \, \pi^2\over g^2_{\rm grav}(\mu^{(\rm IIA)})} =
- 2 \sum_{i=1,3} \log \,{\rm Im}\, T^i
\left\vert \vartheta_2\left(T^i\right)
\right \vert^4
- 2\log \,{\rm Im}\, T^2
\left\vert \vartheta_4\left(T^2\right)
\right \vert^4 + 6 \log {M^{(\rm IIA)}\over \mu^{(\rm IIA)}}
\, . \label{thrint}
\end{equation}
The shifts on the $\Gamma^{(i)}_{2,2}$ lattices break
the $SL(2,Z)_{T^i}$ duality groups, and the actual subgroup
left unbroken depends on the kind of shifts performed
(see Refs. \cite{6auth, gkp, solving, kkprn}).
Furthermore, there are three $N=4$ restoration limits,
corresponding to $\left(\,{\rm Im}\, T^1, 1 / \,{\rm Im}\, T^2\right) \to 0$,
$\left(\,{\rm Im}\, T^1, \,{\rm Im}\, T^3\right) \to 0$ or
$\left(1/ \,{\rm Im}\, T^2, \,{\rm Im}\, T^3\right) \to 0$.
The masses of the three extra pairs of gravitinos are in fact given by
\begin{eqnarray}
m^2_{3/2}(1) & = & { 1 \over 4} \,{\rm Im}\, T^1 \,{\rm Im}\, U^1+
{ 1 \over 4} {\,{\rm Im}\, U^2 \over \,{\rm Im}\, T^2} \nonumber\\
m^2_{3/2}(2) & = & { 1 \over 4} \,{\rm Im}\, T^1 \,{\rm Im}\, U^1+
{ 1 \over 4} \,{\rm Im}\, T^3 \,{\rm Im}\, U^3 \\
m^2_{3/2}(3) & = & { 1 \over 4} {\,{\rm Im}\, U^2 \over \,{\rm Im}\, T^2}+
{ 1 \over 4} \,{\rm Im}\, T^3 \,{\rm Im}\, U^3 \, ,\nonumber
\end{eqnarray}
and each of them vanishes in one of the above limits.
Owing to the effective restoration of the $N=4$ supersymmetry,
there is no linear divergence in the volume of the decompactifying
manifold.
For example, when $\left(1/ \,{\rm Im}\, T^2, \,{\rm Im}\, T^3\right) \to 0$, we
observe the following leading behaviour:
\begin{equation}
{16 \, \pi^2\over g^2_{\rm grav}(\mu^{(\rm IIA)})}
\to
-2\log \,{\rm Im}\, T^2 + 2 \log \,{\rm Im}\, T^3 \, .
\label{logT}
\end{equation}
\hfil\break
However, the threshold correction blows up linearly
with respect to the modulus of the first plane in the limit $\,{\rm Im}\, T^1
\to \infty$:
\begin{equation}
{16 \, \pi^2\over g^2_{\rm grav}(\mu^{(\rm IIA)})} \to 8\pi \,{\rm Im}\, T^1
\, .
\label{linear}
\end{equation}
Finally, the $N=8$ supersymmetry is restored when
$\left(\,{\rm Im}\, T^1, 1/ \,{\rm Im}\, T^2, \,{\rm Im}\, T^3\right) \to 0$. In this limit, the
correction
behaves logarithmically in all three moduli:
\begin{equation}
{16 \, \pi^2\over g^2_{\rm grav}(\mu^{(\rm IIA)})}\to
2 \log \,{\rm Im}\, T^1- 2 \log \,{\rm Im}\, T^2 +2 \log \,{\rm Im}\, T^3
\, .
\end{equation}
\boldmath
\section{The $N=(2,0)$ type II asymmetric construction}
\unboldmath
We now consider the asymmetric type II orbifold, which is obtained
from the $N=8$ IIA superstring compactified on $T^6$
by applying the freely acting projections $Z_2^{ F_{\rm R}}$ and
$Z_2^{(1)}$. The latter is the same projection we considered
in the previous section:
it acts as a combination of rotation and translation, and it
reduces symmetrically the number of supersymmetries by one half. Instead,
$Z_2^{ F_{\rm R}}$ acts as $(-)^{F_{\rm R}}$ together
with a translation on $T^6$, and
projects out all the right-moving supersymmetries.
The properties of the $N=4$ model obtained by applying only $Z_2^{
F_{\rm R}}$
were already analysed in \cite{6auth}.
The orbifold obtained by further application of $Z_2^{(1)}$
has an $N=(2,0)$ supersymmetry realized among the left-movers only.
The partition function of the model reads:
\begin{eqnarray}
Z_{\rm II}^{2,0} & = & {1 \over \,{\rm Im}\, \tau |\eta|^{24} }
{1 \over 4} \sum_{H^1,G^1} \sum_{H^{F},G^{F}}
\Gamma_{6,6} \ar{H^1, H^F}{G^1, G^F} \nonumber \\
&& \times {1 \over 2}
\sum_{a,b}(-)^{a+b+ab}
\vartheta^2 \ar{a}{b}
\vartheta \ar{a+H^1}{b+G^1}\vartheta \ar{a-H^1}{b-G^1} \nonumber \\
&& \times {1 \over 2}
\sum_{\bar{a},\bar{b}}(-)^{\bar{a}+\bar{b}+\bar{a}\bar{b}}
(-)^{\bar{a}G^{ F}+\bar{b}H^{ F}+H^{ F}G^{F}}
\bar{\vartheta}^2 \ar{\bar{a}}{\bar{b}}
\bar{\vartheta} \ar{\bar{a}+H^1}{\bar{b}+G^1}
\bar{\vartheta} \ar{\bar{a}-H^1}{\bar{b}-G^1}\, ,
\label{z20}
\end{eqnarray}
where now
\begin{equation}
\Gamma_{6,6} \ar{H^1, H^F}{G^1, G^F} =
\Gamma_{2,2}^{(1)} \ar{0|H^1}{0|G^1}
\Gamma_{2,2}^{(2)} \ar{H^1|H^{ F}}{G^1|G^{ F}}
\Gamma_{2,2}^{(3)} \ar{H^1|H^1}{G^1|G^1}\, .
\label{g6620}
\end{equation}
The massless spectrum contains, besides the supergravity multiplet,
1 vector-tensor multiplet, dual to a vector,
2 vector multiplets and 4 hypermultiplets: it is therefore the same
as that of the type II symmetric orbifold. However, there is
an important difference in the nature of the fields:
in this case the dilaton belongs to a vector multiplet.
This is a general property of all $N=(2,0)$ string
compactifications,
where supersymmetry charges involve left-movers only.
The reason is that
the dilaton, in such cases, is uncharged under the
$SU(2)$ operators that rotate the supercharges of the $N=2$
supergravity.
The three vector moduli are in this case the dilaton $S_{\rm As}$,
the K\"{a}hler class $T_{\rm As}$, and the complex structure
$U_{\rm As}$ of the first
torus. When $(H^1,G^1)=(0,0)$, expressions (\ref{z20}) and
(\ref{g6620}) give half the partition function of an $N=(4,0)$
asymmetric orbifold with gauge group $U(1)^6$.
This model was analysed in detail in Ref. \cite{6auth}.
By using the same techniques as for the type II symmetric orbifold,
it can be shown that the model at hand possesses a spontaneously broken
$N=8$ supersymmetry, due to the free action of $Z_2^{(1)}$
and $Z_2^{ F_{\rm R}}$.
This can be seen again from the analysis of the helicity supertraces.
We find that $B_2$ is a constant also
in this asymmetric construction. There are therefore
no ``$N=2$ singularities'', i.e. lines in moduli space
with enhancement of the massless spectrum such that
$\Delta N_V \neq \Delta N_H$.
On the other hand, for finite values of the moduli,
there are no ``$N=4$ singularities'' either
(lines where $\Delta N_V= \Delta N_H$),
because the bosonic vacuum energy
is $-1/2$, and there are no points in moduli space in which
new massless states can appear.
The helicity supertrace
$B_4$ receives non-zero contributions from three $N=4$ sectors:
the $N=(4,0)$
sector with $(H^1,G^1)=(0,0)$,
the $N=(2,2)$ sector with
$(H^1,G^1)\ne (0,0)$ and $(H^{ F}, G^{ F})=(0,0)$, and
the $N=(2,2)$ sector with $(H^1,G^1)\ne (0,0)$ and $(H^{ F},
G^{ F})=(H^1,G^1)$. We obtain:
\begin{eqnarray}
B_4 & = & { 3 \over 8} {1 \over {\bar\eta}^{12}}
\sumpF (-)^{H^{ F} G^{ F}}
{\bar \vartheta}^4 \ar{1-H^F}{1-G^F} \Gamma_{6,6} \ar{0,H^{
F}}{0,G^{ F}}
\nonumber \\
&& + 12 \sumpo
\Gamma_{2,2}^{(1)} \ar{0|H^1}{0|G^1}.
\label{B420}
\end{eqnarray}
The contributions of the first line come from the $N=(4,0)$ sector,
while those of the second line are due to the $N=(2,2)$
sectors.
Expression (\ref{B420}) has to be compared with the analogous for
the type II symmetric orbifold, Eq. (\ref{B411}).
In both cases $B_4 \vert_{\rm massless}=18$, in agreement with
the expected $N=2$ supergravity result.
As in the type II symmetric orbifold, we can make $B_4$
vanish by taking appropriate limits in the
space of moduli belonging to vector multiplets and hypermultiplets.
In the asymmetric type II the vector-multiplet moduli
are the moduli of the first complex plane $T_{\rm As},U_{\rm As} $.
The moduli of second and third complex planes belong to the
hypermultiplet space.
The lattice sum in the second line of (\ref{B420}) vanishes
in some appropriate limits in $T_{\rm As}$ and $U_{\rm As}$. However,
only by taking further limits in some of the moduli
belonging to hypermultiplets can we make
also $\Gamma_{6,6}\ar{0,H^F}{0,G^F}$ vanish. This is precisely
the limit
in which the extra massive gravitinos of the asymmetric construction
become massless.
As was already pointed out in the framework of symmetric type II
constructions, the $R^2$ gravitational corrections of the asymmetric
case do not receive any contribution beyond one loop. These
corrections are related to the insertion of the operator
$2Q^2 \overline{Q}^2$. Again, this amplitude accounts for more
terms (like $H^2$) and only half of it is relevant to the $R^2$.
Therefore, the only non-zero contribution
is provided by one sixth of the $N=(2,2)$ sectors of $B_4$ (the second
term in Eq. (\ref{B420})). The part of $B_4$ associated
with the $N=(4,0)$ sector does not enter in the $R^2$ correction and
thus the
moduli dependence comes from the vector multiplets only; there is no
dependence at all on the hypermultiplet moduli, as
expected from general properties of the $N=2$ theories.
Both moduli $T_{\rm As}$ and $U_{\rm As}$
of the first plane belong to vector multiplets and appear in
the correction to the $R^2$ term.
With the specific choice of half-unit shift, $(-)^{m_1G^1}$,
induced by $Z_2^{(1)}$ acting on $\Gamma_{2,2}^{(1)}$, we obtain
the corrected gravitational coupling in terms of the moduli
$T_{\rm As}$, $U_{\rm As}$, and the appropriate string scale and infrared
cut-off:
\begin{equation}
{16 \, \pi^2\over g^2_{\rm grav}(\mu^{(\rm As)})}=
- 2 \log \,{\rm Im}\, T_{\rm As}
\left\vert \vartheta_4 \left( T_{\rm As} \right)
\right\vert^4 -
2 \log \,{\rm Im}\, U_{\rm As}
\left\vert \vartheta_2 \left( U_{\rm As} \right)
\right\vert^4 + 4\log { M^{(\rm As)} \over \mu^{(\rm As)}}
\label{thrinta}
\end{equation}
up to a constant\footnote{This constant, as well as the one appearing
later in the perturbative heterotic coupling (\ref{htr}), contains in
general $\log (M/\mu)$ terms, which account for extra massless states
that have been disregarded in the determination of the $R^2$ amplitude
from the $B_4$. Field-theoretical arguments can be advocated to fix these
terms. We will take care of them in the final expression (\ref{np}).}.
This expression deserves some comments, as was for (\ref{thrint}).
We first observe that the $Z_2^{(1)}$-shift in
$\Gamma_{2,2}^{(1)}$ breaks the
$SL(2,Z)_{T_{\rm As}} \times SL(2,Z)_{U_{\rm As}}\times
Z_2^{T_{\rm As}\leftrightarrow U_{\rm As}}$
duality
group to a subgroup that depends on the kind
of shift performed. In the limit $\,{\rm Im}\, T_{\rm As} \to \infty$, $\,{\rm Im}\,
U_{\rm As}
\to 0$
there is a restoration of an $N=(4,0)$ supersymmetry
with no linear behaviour either in $\,{\rm Im}\, T_{\rm As}$
or in $1 \big/ \,{\rm Im}\, U_{\rm As}$; the remaining contribution is
logarithmic:
\begin{equation}
{16 \, \pi^2\over g^2_{\rm grav}(\mu^{(\rm As)})}
\to
- 2 \log \,{\rm Im}\, T_{\rm As}+ 2 \log \,{\rm Im}\, U_{\rm As}\, .
\label{logTa}
\end{equation}
Finally, comparison of (\ref{thrint}) and (\ref{thrinta})
suggests that the string duality map implies the following identification
of the moduli:
\begin{equation}
T_{\rm As}\leftrightarrow T^2\ , \ \ U_{\rm As}\leftrightarrow
T^3
{\rm \ and \ \ }4 \pi S_{\rm As}\leftrightarrow -1/T^1\, .
\label{symas}
\end{equation}
The identification of the asymmetric dilaton $S_{\rm As}$ with the
$h^{1,1}$ moduli
$ T^1$ of the symmetric type II construction follows from the
behaviour in the
limit $\,{\rm Im}\, T^1\to 0$, which corresponds,
in the asymmetric construction,
to the perturbative limit
$S_{\rm As}\to \infty$.
\boldmath
\section{The $N=(2,0)$ heterotic construction}
\unboldmath
\subsection{\sl The construction of the model}
The heterotic dual to the previous type II constructions
is based on $\left(T^2 \times T^4\right)\big/ Z_2^{(\rm f)}$ freely
acting orbifold heterotic
compactification. In order to reduce the gauge group we have to
introduce a set of ``discrete Wilson lines'',
which separate the boundary conditions of the 32
right-moving fermions, $\Psi_A,A=1,\ldots,32$, in order to twist all the
currents $\Psi_A \Psi_B$ of the $c=(0,16)$ conformal block. The resulting
characters are described by those of 32 right-moving Ising's
. The $Z_2^{(\rm f)}$ projection
reduces the initial $N=4$ supersymmetry to $N=2$; it acts
as a $\pi$ rotation in $T^4$ left and right (super)coordinates and
as a translation in $T^2$. Its action on $\Psi_A$ is non-trivial
(see below), and it is chosen in such a way that neither vectors
nor hypers are produced from the $\Psi_A$'s. In this way the
heterotic massless spectrum is
identical to that of the previous type II constructions. Namely,
it consists of the $N=2$ supergravity multiplet,
1 vector-tensor multiplet that contains the dilaton $S_{\rm Het}$
and is dual to a vector, 2 vector multiplets
associated with the K\"{a}hler class $T$ and the complex structure
$U$ of the torus\footnote{For simplicity we will use systematically
$(T,U)$ for the heterotic two-torus moduli, instead of the more
natural notation, which would have been
$(T_{\rm Het},U_{\rm Het})$.}
$T^2$, and 4 hypermultiplets,
obtained by pairing the left-moving negative eigenvalues of the
projection $Z_2^{(\rm f)}$ with the 4 right-moving negative
eigenvalues in $T^4$.
The partition function of the heterotic construction has the
following expression:
\begin{eqnarray}
Z_{\rm Het} & = &
{1 \over \,{\rm Im}\, \tau | \eta|^4 } {1 \over 2}
\sum_{H^{\rm f},G^{\rm f}}
Z_{6,22} \ar{H^{\rm f}}{G^{\rm f}} \nonumber \\
&& \times {1 \over 2} \sum_{a,b}{1\over \eta^4} (-)^{a+b+ab}
\vartheta \ar{a}{b}^2
\vartheta \ar{a+H^{\rm f}}{b+G^{\rm f}}
\vartheta \ar{a-H^{\rm f}}{b-G^{\rm f}} \, ,
\label{hH}
\end{eqnarray}
where the second line stands for
the contribution of the 10 left-moving
world-sheet fermions $\psi^{\mu},\Psi^I$ and the ghosts
$\beta,\gamma$ of the super-reparametrization;
$Z_{6,22}\ar{H^{\rm f}}{G^{\rm f}}$ accounts for the $(6,6)$
compactified coordinates and the $c=(0,16 )$ conformal system, which
is described by the
32 right-moving fermions $\Psi_A$. It
takes the
following form:
\begin{equation}
Z_{6,22}\ar{H^{\rm f}}{G^{\rm f}}=
{1\over 2^{5}}\, \sum_{\vec h, \vec g}{1\over \eta^6 {\bar \eta}^6 }
\Gamma_{2,2} \ar{H^{\rm f},\vec h}{G^{\rm f},\vec g}
\,
\Gamma_{4,4} \ar{H^{\rm f}\vert \vec h}{G^{\rm f}\vert \vec g}\,
{\overline \Phi}\ar{H^{\rm f}, \vec h}{G^{\rm f}, \vec g}\, ,
\label{hH622}
\end{equation}
where $(\vec h, \vec g)$ denote the five projections that are needed
in order to separate the boundary conditions of all 32 fermions.
The function $\Phi \ar{H^{\rm f}, \vec h}{G^{\rm f}, \vec g}$
can be written explicitly using the $SO(4)$ twisted characters
(see Ref. \cite{gkp}):
\begin{equation}
{\widehat{F}}_{1}\ar{\gamma,h}{\delta, g} \equiv {1\over {\eta}^2}\,
{\vartheta}^{1/2}
\ar{\gamma-h_1-h_2-h_3}{\delta-g_1-g_2-g_3}\,
{\vartheta}^{1/2}
\ar{\gamma+h_3}{\delta+g_3}\,
{\vartheta}^{1/2}
\ar{\gamma+h_3-h_1}{\delta+g_3-g_1}\,
{\vartheta}^{1/2}
\ar{\gamma+h_2-h_3}{\delta+g_2-g_3}
\end{equation}
and
\begin{equation}
{\widehat{F}}_{2}\ar{\gamma, h}{\delta, g} \equiv {1\over {\eta}^2}\,
{\vartheta}^{1/2}
\ar{\gamma}{\delta}\,
{\vartheta}^{1/2}
\ar{\gamma+h_1-h_2}{\delta+g_1-g_2}\,
{\vartheta}^{1/2}
\ar{\gamma+h_1}{\delta+g_1}\,
{\vartheta}^{1/2}
\ar{\gamma+h_2}{\delta+g_2}\, ,
\end{equation}
where we introduced the notation
$h\equiv (h_1,h_2,h_3)$ and similarly for $g$.
Under $\tau\to \tau+1$, ${\widehat{F}}_{I}$
transform as:
\begin{eqnarray}
{\widehat{F}}_{1}\ar{\gamma, h}{\delta, g} & \to &
{\widehat{F}}_{1}\ar{\gamma, h}{\gamma+\delta +1, h + g} \nonumber \\
&& \times
\exp -{i\pi\over 4}
\left({2\over 3} +2 \gamma^2+h_1^2+h_2^2 +2 h_3^2 - 2 \gamma h_1 + h_1 h_2
- 4\gamma +2 h_1 \right) \, ;
\\
{\widehat{F}}_{2}\ar{\gamma, h}{\delta, g} & \to &
{\widehat{F}}_{2}\ar{\gamma, h}{\gamma+\delta +1, h + g} \nonumber \\
&& \times
\exp -{i\pi\over 4}
\left({2\over 3} +2 \gamma^2+h_1^2+h_2^2 + 2 \gamma h_1 - h_1 h_2
- 4 \gamma -2 h_1 \right)\, .
\end{eqnarray}
Notice that ${\widehat{F}}_{I}$ are $c=(2,0)$ conformal characters of 4
different left-moving Isings. In the fermionic language \cite{abk}
this is a system of 4 left-moving real fermions with different
boundary
conditions.
All currents ${J}^{IJ}={ \Psi}^I{ \Psi}^J$ are twisted
and therefore the initial $SO(4)$ is broken.
We have then, as in \cite{gkp}, two alternative constructions,
$\Phi$ and $\tilde{\Phi}$, that differ with respect to the embedding of
the
$Z_2^{(\rm f)}$-shift in the two-torus:
\begin{eqnarray}
\Phi \ar{H^{\rm f}, \vec h}{G^{\rm f}, \vec g} & = &
{1 \over 2} \sum_{\gamma,\delta}\,
{\widehat{F}}_{1} \ar{\gamma+H^{\rm f}, h}{\delta+G^{\rm f}, g} \,
{\widehat{F}}_{1} \ar{\gamma+h_4+H^{\rm f}, h}{\delta+g_4+G^{\rm f}, g} \,
{\widehat{F}}_{1} \ar{\gamma+h_5, h}{\delta+g_5, g} \,
{\widehat{F}}_{1} \ar{\gamma+h_4+h_5, h}{\delta+g_4+g_5, g} \nonumber \\
&& \times \, {\widehat{F}}_{2} \ar{\gamma, h}{\delta, g} \,
{\widehat{F}}_{2} \ar{\gamma+h_4, h}{\delta+g_4, g} \,
{\widehat{F}}_{2} \ar{\gamma+h_5, h}{\delta+g_5, g} \,
{\widehat{F}}_{2} \ar{\gamma+h_4+h_5, h}{\delta+g_4+g_5, g}
\end{eqnarray}
and
\begin{eqnarray}
\tilde{\Phi} \ar{H^{\rm f}, \vec h}{G^{\rm f}, \vec g} & = &
{1 \over 2} \sum_{\gamma,\delta}\,
{\widehat{F}}_{1} \ar{\gamma+H^{\rm f}, h}{\delta+G^{\rm f}, g} \,
{\widehat{F}}_{1} \ar{\gamma+h_4+H^{\rm f}, h}{\delta+g_4+G^{\rm f}, g} \,
{\widehat{F}}_{1} \ar{\gamma+h_5, h}{\delta+g_5, g} \,
{\widehat{F}}_{1} \ar{\gamma+h_4+h_5, h}{\delta+g_4+g_5, g} \nonumber \\
&& \times \,
{\widehat{F}}_{2} \ar{\gamma+H^{\rm f}, h}{\delta+G^{\rm f}, g} \,
{\widehat{F}}_{2} \ar{\gamma+h_4, h}{\delta+g_4, g} \,
{\widehat{F}}_{2} \ar{\gamma+h_5, h}{\delta+g_5, g} \,
{\widehat{F}}_{2} \ar{\gamma+h_4+h_5, h}{\delta+g_4+g_5, g} \, .
\label{88fivt}
\end{eqnarray}
The $(2,2)$ and $(4,4)$ lattice shifts are dictated by modular
invariance and are needed in order
to cancel the phases that appear under
modular transformations. These shifts are different for the two
constructions,
based on $\Phi$ or $\tilde{\Phi}$. In the case of $\Phi$, modular
invariance implies an asymmetric shift on the
$\Gamma_{2,2} \ar{H^{\rm f}}{G^{\rm f}}$, which we chose to be
$(-)^{(m_1+n_1) G^{\rm f}}$ (this projection was referred to as ``X''
in Ref. \cite{kkprn}, where the various lattice shifts were discussed
in detail); the
shift in $\Gamma_{4,4}\ar{H^{\rm f}\vert \vec h}{G^{\rm f}\vert \vec
g}$, however, has to be symmetric and is chosen to be $(-)^{M_i \,
g_i}$.
In the construction based on $\tilde{\Phi}$, on the other hand,
the $(2,2)$ lattice must be doubly shifted, $\Gamma_{2,2}\ar{H^{\rm
f}, h_1-h_2}{G^{\rm f}, g_1-g_2}$.
In this case we use the projection $(-)^{m_1G^{\rm f}+n_1(g_1-g_2)}$.
Constructions $\Phi$ and $\tilde{\Phi}$
share the same $N=4$ sector (defined by $(H^{\rm f},G^{\rm f})=(0,0)$).
The contribution of this sector to the partition function is one half
of the partition function of an $N=4$ model
in which the gauge group is $U(1)^6$, and all the vectors originating
from the torus $T^6$. In the $N=2$ sectors,
$(H^{\rm f},G^{\rm f})\neq (0,0)$, while $(h_i, g_i)$ are either
$(0,0)$ or $(H^{\rm f},G^{\rm f})$. This restriction on the values
$(h_i, g_i)$ comes from the
$(H^{\rm f},G^{\rm f})$-twisted
sector of $\Gamma_{4,4} \ar{H^{\rm f}\vert \vec h}{G^{\rm f}\vert
\vec g}$.
A quantity relevant to our purpose is the helicity
supertrace $B_2$, which receives a non-zero contribution from the $N=2$
sector, while the contribution of the $N=4$ sector to this
quantity vanishes.
For the construction based on $\Phi$ we find:
\begin{equation}
B_2 \left(\Phi \right)={1\over \bar{\eta}^{24}}\,
\sumpf
\Gamma_{2,2}^{\lambda=1} \ar{H^{\rm f}}{G^{\rm f}} \,
\overline{\Omega} \ar{H^{\rm f}}{G^{\rm f}}\, ,
\label{B288}
\end{equation}
where $\Omega \ar{H^{\rm f}}{G^{\rm f}}$ turn out to be
the same analytic functions as for the models
considered in Ref.~\cite{gkp}\footnote{The parameter $\lambda$, which
takes the
values 0 or 1, determines the phases appearing in the modular
transformations of the shifted lattice sums.
These phases are complementary of those coming
from the corresponding functions $\Omega\ar{H^{\rm f}}{G^{\rm f}}$ or
$\Omega^{(0)}\ar{H^{\rm f}}{G^{\rm f}}$ and
$\Omega^{(1)}\ar{H^{\rm f}}{G^{\rm f}}$.}, even though
the $N=2$ constructions are
different\footnote{This is not surprising since the $N=2$ models under
consideration are known to fall in a very restricted set of universality
classes with respect to their elliptic genus \cite{kkprn}.}:
\begin{eqnarray}
\Omega \ar{0}{1}&=&\hphantom{-}{1\over 16}
\left({\vartheta}_3^8 + {\vartheta}_4^8
+14\, {\vartheta}_3^4\, {\vartheta}_4^4 \right)
{\vartheta}_3^6\, {\vartheta}_4^6\nonumber \\
\Omega \ar{1}{0}&=&-{1\over 16}
\left({\vartheta}_2^8 + {\vartheta}_3^8
+14\, {\vartheta}_2^4\, {\vartheta}_3^4 \right)
{\vartheta}_2^6\, {\vartheta}_3^6\label{Om88} \\
\Omega \ar{1}{1}&=&\hphantom{-}{1\over 16}
\left({\vartheta}_2^8 + {\vartheta}_4^8
-14\, {\vartheta}_2^4\, {\vartheta}_4^4 \right)
{\vartheta}_2^6\, {\vartheta}_4 ^6\, .\nonumber
\end{eqnarray}
The construction based on $\tilde{\Phi}$ with $N_V=N_H=0$,
under study here, has the
same universality properties as the corresponding $N=2$ models of
Ref. \cite{gkp}. The
helicity supertrace $B_2$ is identical for all such models with $N_V=N_H$,
irrespectively of whether the latter
vanishes or not:
\begin{equation}
B_2 \left(\tilde{\Phi} \right)=
{1\over \bar{\eta}^{24}}\,
\sumpf {1\over 2}\left(
\Gamma_{2,2}^{\lambda=0} \ar{H^{\rm f}}{G^{\rm f}} \,
\overline{\Omega}^{(0)} \ar{H^{\rm f}}{G^{\rm f}} +
\Gamma_{2,2}^{\lambda=1} \ar{H^{\rm f}}{G^{\rm f}} \,
\overline{\Omega}^{(1)} \ar{H^{\rm f}}{G^{\rm f}}
\right) \, ,
\label{B288t}
\end{equation}
where in this case
\begin{eqnarray}
\Omega^{(0)}\ar{0}{1}&=&{\hphantom{-}}
\frac{1}{2}
\left(\vartheta_3^4+\vartheta_4^4\right)
\vartheta_3^8\, \vartheta_4^8
\nonumber\\
\Omega^{(0)}\ar{1}{0}&=&-
\frac{1}{2}
\left(\vartheta_2^4+\vartheta_3^4\right)
\vartheta_2^8\, \vartheta_3^8
\label{Om88a}\\
\Omega^{(0)}\ar{1}{1}&=&{\hphantom{-}}
\frac{1}{2}
\left(\vartheta_2^4-\vartheta_4^4\right)
\vartheta_2^8\, \vartheta_4^8\nonumber
\end{eqnarray}
and
\begin{equation}
\Omega^{(1)} \ar{H^{\rm f}}{G^{\rm f}}
= (-)^{H^{\rm f}}\left(
\omega \ar{H^{\rm f}}{G^{\rm f}}\right)^{10} \, ,
\end{equation}
with
\begin{equation}
\omega \ar{0}{1}= \vartheta_3 \, \vartheta_4 \ , \ \
\omega \ar{1}{0}= \vartheta_2 \, \vartheta_3 \ , \ \
\omega \ar{1}{1}= \vartheta_2 \, \vartheta_4 \, .
\label{Om88b}
\end{equation}
The lattice sums $\Gamma_{2,2}^{\lambda=0} \ar{H^{\rm f}}{G^{\rm f}}$
and $\Gamma_{2,2}^{\lambda=1} \ar{H^{\rm f}}{G^{\rm f}}$
correspond to simply shifted lattices with projections
$(-)^{m_1 G^{\rm f}}$ and $(-)^{(m_1+n_1) G^{\rm f}}$, respectively (the
cases ``I'' and ``X'' of \cite{kkprn}).
In both constructions $\Phi$ and $\tilde \Phi$, the massless
contribution to the $B_2$ vanishes for generic values of the moduli
$T$ and $U$, as it should for models where
$N_V = N_H$. Owing to the
$Z_2^{(\rm f)}$-translation on the two-torus,
the $N=4$ supersymmetry is spontaneously broken.
The analysis of these constructions is the same as for the
models discussed in \cite{gkp}, to which we refer for the details.
Here we simply recall that the construction based on $\Phi$
is not suitable for a comparison with the type II ground states:
the region in the space of (discrete) Wilson lines which allows for an
easy identification of the map
between the moduli of the dual models is the one that corresponds to
the construction based on $\tilde{\Phi}$.
In the latter, and for
the particular $Z_2^{(\rm f)}$-shift we have considered,
$(-)^{m_1 G^{\rm f}+n_1 g_1}$, the mass of the two extra gravitinos is
\begin{equation}
m^2_{3/2} = {1 \over 4} { \,{\rm Im}\, U \over \,{\rm Im}\, T}\, .
\label{mhet}
\end{equation}
The $N=4$ supersymmetry is restored
when $R_1= \sqrt{ \,{\rm Im}\, T / \,{\rm Im}\, U}$ is large.
For large values
of $\,{\rm Im}\, U / \,{\rm Im}\, T$ we recover instead a genuine $N=2$
non-freely-acting orbifold.
For the specific directions of the shifts in the
two-torus that we have chosen, there are lines in the $(T,U)$-plane
along which two extra hypermultiplets appear in the massless spectrum
together with two extra massless vectors leading to
an $SU(2)$ enhancement of one of the $U(1)$'s of the torus.
\subsection{\sl The gravitational corrections}
In order to obtain heterotic gravitational
corrections analogous to those of the type II constructions (see Eqs.
(\ref{thrint}) and (\ref{thrinta})), we must
proceed as follows. Instead of considering the
pure $R^2$ term, we must compute the one-loop corrections for a
special combination of gravitational and helicity operators.
The ordinary $R^2$-term correction is given by the genus-one
amplitude of \begin{equation}
Q_{\rm grav}^2\equiv Q^2\,
\overline{P}_{\rm grav}^2\, ,
\label{Qgrav}
\end{equation}
where $Q$ stands again for the left-helicity operator, and
$\overline{P}_{\rm grav}^2$ is the usual gravitational operator:
when inserted in the one-loop vacuum amplitude, it acts as
${-1\over 2 \pi i}{\partial \over
\partial{\bar \tau}}$ on ${1/ \tau_2 \, \bar{\eta}^2}$;
namely, it acts on the contribution of the two right-moving transverse
space-time coordinates $\bar X^{\mu=3,4}$, {\it including their
zero-modes.} This latter fact is responsible for the appearance of
a non-holomorphic gravity-backreaction
contribution, which ensures modular covariance but has no type II
counterpart, as was discussed in \cite {gkp}; the one-loop
amplitude of the above operator is\footnote{In general the
heterotic one-loop amplitude of an operator of the form $Q^2\, \overline{P}^2$
reads:
$
\left\langle Q^2\, \overline{P}^2
\right\rangle_{\rm genus-one}
= \overline{P}^2\,B_2\, ,
$
where, in the l.h.s., $\overline{P}^2$ acts as a differential operator on some
specific factor of $B_2$.}
\begin{equation}
F_{\rm grav}\equiv \left\langle Q_{\rm grav}^2
\right\rangle_{\rm genus-one}
= -{1\over 12}\,
\left( \overline{E}_2 -{3\over \pi\tau_2}\right) B_2\, ,
\label{Fgrav}
\end{equation}
where, for the model under consideration (constructed with
$\tilde \Phi$), $B_2$ is given in Eq. (\ref{B288t}). The massless
contribution to $F_{\rm grav}$ is precisely the gravitational anomaly,
$b_{\rm grav}= {24 - N_V +N_H\over 12}$, which in the case
at hand equals 2 ($B_2\vert_{\rm massless}$ vanishes), at generic
points of the $(T,U)$ moduli space.
The operator $\overline{P}_{\rm grav}^2$ is not suitable for comparison with
the type II result, because it is not holomorphic and its amplitude is
sensitive to the
$N=2$ singularities occurring in the $(T,U)$ plane: the corresponding
beta-function
(i.e. the gravitational
anomaly) jumps along rational lines where
$\Delta N_V\neq\Delta N_H$. We must therefore replace
$\overline{P}_{\rm grav}^2$ with an appropriate holomorphic operator
$\overline{P}_{\rm grav}^{\prime 2}$ whose amplitude
is regular everywhere in $(T,U)$, at least in the model constructed
with
$\tilde \Phi$, which is the model we will be analysing in the
following.
To this purpose, we introduce two
operators:
$\overline{H}_{\rm tw}$ and $\overline{P}^2_{2,2}$.
The operator $\overline{H}_{\rm tw}$ acts, for any $(H^{\rm f},G^{\rm f})$-twisted
sector of the orbifold, as a derivative
${ -1 \over 2 \pi i} {\partial \over \partial \bar{\tau}}$
on the factor $\bar\omega \ar{H^{\rm f}}{G^{\rm f}}
\big/ \,{\rm Im}\, \tau \, \bar{\eta}^4$, which contains the contribution of
twisted coordinates (see Eq. (\ref{Om88b})). After some
straightforward algebra we obtain the amplitude:
\begin{equation}
\overline{H}_{\rm tw}\, B_2 \left(\tilde{\Phi} \right)=
-{1\over 24}\sum_{\lambda=0,1}\sumpf
\Gamma_{2,2}^{\lambda} \ar{H^{\rm f}}{G^{\rm f}}
\left(
\overline{E}_2 -{3\over \pi\tau_2}
+{1\over 2}\overline{H}\ar{H^{\rm f}}{G^{\rm f}}
\right)
{\overline{\Omega}^{(\lambda)} \ar{H^{\rm f}}{G^{\rm f}}\over
\bar{\eta}^{24}}\, ,
\label{bHt}
\end{equation}
where we have introduced the modular-covariant functions
\begin{equation}
H{h\atopwithdelims[]g}={12 \over \pi i} \partial_\tau \log {\vartheta
{1-h\atopwithdelims[]1-g}\over
\eta}
= \cases{\hphantom{-} \vartheta_3^4 + \vartheta_4^4 \ , \ \ (h,g)=(0,1)\cr
- \vartheta_2^4 - \vartheta_3^4 \ , \ \ (h,g)=(1,0)\cr
\hphantom{-} \vartheta_2^4 - \vartheta_4^4 \ , \ \ (h,g)=(1,1)\cr }
\label{H}
\end{equation}
of weight $2$. In the model constructed with ${\Phi}$, only a
$\lambda =1$ term would
appear in (\ref{bHt}).
From expression (\ref{bHt}) we observe that the insertion of
$\overline{H}_{\rm tw}$ is covariant but not holomorphic. This latter property
will allow for cancelling the non-holomorphic terms present when
$\overline{P}_{\rm grav}^2$ and $\overline{P}^2_{2,2}$ are inserted in the vacuum
amplitude, while keeping modular covariance.
The beta-function
coefficient of this operator, i.e. the constant term of $\overline{H}_{\rm tw}\, B_2
\left(\tilde{\Phi}
\right)$, vanishes for generic $(T,U)$ while it jumps accross several
special lines:
$\Delta b\left(\overline{H}_{\rm tw}\right) = 2\, \Delta b_{\rm grav}$.
On the other hand, after insertion into the one-loop heterotic vacuum
amplitude,
$\overline{P}^2_{2,2}$ acts as ${-1 \over 2 \pi i}{\partial \over
\partial{\bar \tau}}$ on the modular-covariant factor of weight zero,
$\tau_2 \,\Gamma_{2,2}^{\lambda}\ar{H^{\rm f}}{G^{\rm f}}$.
This amounts to inserting the sum of the two right-moving
lattice momenta ${\bar p}_1^2+{\bar p}_2^2$ of $T^2$, which correspond
to the Cartan of the $U(1)$ factor. The amplitude
$\left\langle Q^2\, \overline{P}^2_{2,2}
\right\rangle$ therefore generates the corresponding gauge-coupling
correction. To be more precise, we must in fact consider the integral
$\int_{\cal F} {d^2 \tau \over \tau_2}\left\langle Q^2\, \overline{P}^2_{2,2}
\right\rangle \gamma (\tau,\bar \tau)$, where $\gamma (\tau,\bar \tau)$ is an
appropriate modular-invariant infrared-regularizing function
\cite{infra}. An integration by parts can be performed, which leads
to vanishing boundary terms {\it all over the $(T,U)$ plane},
irrespectively of the specific behaviour of the lattice sums across
rational lines. This allows us to recast the above amplitude as:
\begin{equation}
\overline{P}^2_{2,2}\, B_2 \left(\tilde{\Phi} \right)=
{1\over 2}
\sum_{\lambda=0,1}\sumpf
\left(
\Gamma_{2,2}^{\lambda} \ar{H^{\rm f}}{G^{\rm f}}
\left[
{1 \over 2 \pi i}{\partial \over
\partial{\bar \tau}}- {1\over 2\pi \tau_2}
\right]
{\overline{\Omega}^{(\lambda)} \ar{H^{\rm f}}{G^{\rm f}}\over \bar{\eta}^{24}}
\right)
\label{P22}
\end{equation}
(in the model constructed with ${\Phi}$, only a $\lambda =1$ term
appears). The differential operator inside the brackets is covariant, since
${\overline{\Omega}^{(\lambda)} \ar{H^{\rm f}}{G^{\rm f}}\Big/ \bar{\eta}^{24}}$
has modular weight $-2$, but non-holomorphic. The beta-function
coefficient $b(P_{2,2})$ (constant term in (\ref{P22})) vanishes for
generic $(T,U)$ and its discontinuity at special lines turns out to
be $\Delta b(P_{2,2})= -12\, \Delta b_{\rm grav}$.
Given the operators $\overline{P}_{\rm grav}^2$, $\overline{H}_{\rm tw}$ and $\overline{P}^2_{2,2}$,
there is a {\it unique} combination, which is holomorphic and whose
beta-function coefficient is equal to the gravitational anomaly
$b_{\rm grav}=2$ {\it everywhere} in the moduli space $(T,U)$:
\begin{equation}
\overline{P}_{\rm grav}^{\prime 2}=\overline{P}_{\rm grav}^2
-{5\over 4}
\overline{H}_{\rm tw}
-{1\over 8}
\overline{P}^2_{2,2}
\, .\label{reg}
\end{equation}
Here we want to remark that,
in some specific cases where
$N_V=N_H \neq 0$,
as for the models considered
in \cite{gkp}, the operator $\overline{H}_{\rm tw}$
can be reexpressed through an integration
by parts, as a combination of
$\overline{P}^2_{2,2}$ and
$\overline{P}^2_{\rm gauge}$, the gauge operator for the
higher-level currents (here absent).
In such cases, expression (\ref{reg}) becomes
$\overline{P}^2_{\rm grav}+{1 \over 12}\overline{P}^2_{2,2}
+{5 \over 3 N_V} \overline{P}^2_{\rm gauge}$.
The amplitude $ \left\langle Q_{\rm grav}^{\prime 2}\right\rangle=
\left\langle Q^2\, \overline{P}_{\rm grav}^{\prime 2}
\right\rangle $ at genus one reads:
\begin{equation}
\overline{P}_{\rm grav}^{\prime 2}\,B_2\left(\tilde \Phi\right)=2\sumpf
\Gamma_{2,2}^{\lambda=0} \ar{H^{\rm f}}{G^{\rm f}}
\, ,
\end{equation}
which is regular everywhere. Notice that the contribution of the
$\lambda=1$ term vanishes identically. This amplitude leads to the
thresholds
\begin{equation}
\Delta_{\rm Het}=2 \, \int_{\cal F} {d^2 \tau \over \,{\rm Im}\, \tau}
\left( \sumpf
\Gamma_{2,2}^{\lambda=0} \ar{H^{\rm f}}{G^{\rm f}}-1 \right)\, ,
\end{equation}
which is valid only for the construction based on $\tilde \Phi$.
For the specific case in which the shift
in $\Gamma_{2,2}^{\lambda = 0}$ is due to a translation of momenta,
$(-1)^{m_1 G^{\rm f}}$, we get:
\begin{equation}
\Delta_{\rm Het}= -2 \log \,{\rm Im}\, T \left\vert \vartheta_4 \left( T \right)
\right\vert^4
- 2 \log \,{\rm Im}\, U \left \vert \vartheta_2 \left( U \right)
\right\vert^4 + {\rm const.}
\label{dhet}
\end{equation}
Finally the running of the coupling is given by
\begin{eqnarray}
{16 \, \pi^2 \over g^2_{\rm grav}\left( \mu^{(\rm Het)} \right)}
& = & 16 \, \pi^2 \kappa \,{\rm Im}\, S_{\rm Het}
-2 \log \,{\rm Im}\, T \left\vert \vartheta_4 \left( T \right) \right\vert^4
- 2 \log \,{\rm Im}\, U \left \vert \vartheta_2 \left( U \right) \right\vert^4
\nonumber \\
&& +4 \log {M^{(\rm Het)} \over \mu^{(\rm Het)} }
+ {\rm const.}\, ,
\label{htr}
\end{eqnarray}
where $S_{\rm Het}$ is the heterotic dilaton--axion field and
\begin{equation}
\kappa \, \,{\rm Im}\, S_{\rm Het} ={1\over g^2_{\rm Het} }\, .
\label{dilk}
\end{equation}
In the limit where $(1/ \,{\rm Im}\, T, \,{\rm Im}\, U)\to 0$, $N=4$ supersymmetry is
restored with the following behaviour:
\begin{equation}
{16 \, \pi^2 \over g^2_{\rm grav}\left( \mu^{(\rm Het)} \right)}
=16 \, \pi^2 \kappa \,{\rm Im}\, S_{\rm Het}
-2 \log \,{\rm Im}\, T +2 \log \,{\rm Im}\, U
\, .
\label{htrli}
\end{equation}
The role of the normalization factor $\kappa$ will be
specified in the next section
where the gravitational corrections of the heterotic $\tilde{\Phi}$
construction and that of the two
type II ground states will be compared.
\section{Comparison of the three duals}
In this section we would like to test the triality relation between
all
three freely acting orbifolds we have considered, through the
analysis of the
``gravitational'' corrections.
First of all we observe that the operator
$Q_{\rm grav}^{\prime 2}= Q^2\, \overline{P}_{\rm grav}^{\prime 2}$
with $\overline{P}_{\rm grav}^{\prime 2}$
given in (\ref{reg}), coincides, on type II side, with the
operator $Q^{{\rm II}2}_{\rm grav}= 2 Q^2 \overline{Q}^2$ we considered
in Sections 2 and 3. Indeed, owing to the absence of
perturbative Ramond--Ramond charges, the contribution of the dual of
$\overline{P}^2_{\rm grav}$ vanishes; because of the symmetry
between left- and right-movers on the world-sheet, there is no
need for us to introduce an operator such as $\overline{H}_{\rm tw}$, the insertion of
$Q^{{\rm II}2}_{\rm grav}$ being automatically holomorphic.
The duality among the three orbifolds requires
the identification of
one of the three perturbative
vector multiplet moduli of the type IIA symmetric orbifold, with the
dual of the field $S_{\rm Het}$, the dilaton--axion field
of the heterotic theory, and with the inverse of
$S_{\rm As}$, the dilaton--axion field of the type II
asymmetric theory.
Modulo $SL(2,Z)$ transformations,
such modulus can be indifferently any one of
the three $T^i$, $i=1,2,3$. For definiteness, we will choose $T^1$,
as was anticipated in (\ref{symas}). In order to see what the
precise duality map is, we consider all three models in their
``$N=4$ phase''. In this limit, the heterotic amplitude
$\left\langle Q_{\rm grav}^{\prime 2}\right\rangle$ is expected
to receive contributions from genus zero only, while in the type II
$\left\langle Q_{\rm grav}^{{\rm II} 2}\right\rangle$
should vanish in the asymmetric orbifold and depend on one complex
modulus only in the type IIA symmetric one.
This behaviour can be checked by taking the appropriate
$N=4$ limits in the three models
(see Eqs. (\ref{logT}), (\ref{linear}), (\ref{logTa}) and
(\ref{mhet}), (\ref{htr})).
The other surviving contributions, with a logarithmic dependence
on the other moduli, are in fact
infrared artefacts due to an accumulation of massless
states, which can be lifted, in all the three models,
by switching on
an infrared cut-off larger than the massive gravitinos,
as we previously discussed.
By comparing Eq. (\ref{linear}) with the genus-zero contribution in
Eq. (\ref{htrli}) ($16 \, \pi^2 \kappa \, \,{\rm Im}\, S_{\rm Het} $), we obtain
\begin{equation}
T^1={\kappa \over 2} \tau_{S_{\rm Het}}=
2 \pi \kappa S_{\rm Het} \, .
\end{equation}
Here we have introduced the field $\tau_S \equiv 4 \pi S$,
the actual ``modular'' parameter of the Montonen--Olive
$S$-duality transformations. Since, after an $SL(2,Z)_T$
inversion $T \to -1/T$, the model becomes symmetric under permutations
of the fields
$\tau_{S_{\rm Het}}, T$ and $U$, the behaviour of the effective
coupling constant, for large values of the three moduli, must be
symmetric as well; this requirement forces us to fix $\kappa=2$.
This normalization of the coupling (\ref{dilk}), which differs by a
factor 2 from the usual tree-level coupling (corresponding to
$\kappa=1$), is required also for a correct interpretation in terms
of instanton contributions (see below).
In the opposite limit, $T^1 \to 0$,
the $T^1$-dependent contribution vanishes (up to an irrelevant
logarithmic term in $-1 / T^1$).
This is consistent with the identification of $-1/T^1$
with $\tau_{S_{\rm As}}= 4 \pi S_{\rm As}$, and
implies in particular that {\it the type II asymmetric orbifold
is the strong coupling limit of the heterotic}:
\begin{equation}
\tau_{S_{\rm As}}= -{1 \over \tau_{S_{\rm Het}}}\, .
\end{equation}
In order to test the above duality relations, we consider the
``$N=2$ phase'' of the various theories under consideration, where
the dependence on the other moduli, generating from genus one for all
three models, remains.
The part of the gravitational amplitude
that depends on the perturbative moduli is indeed the same
in the three models, provided we identify the moduli $(T_{\rm
As},U_{\rm As})$ with $(T^2,T^3)$ and $(T,U)$.
Through the duality map, we therefore learn that,
similarly to the symmetric type IIA orbifold, the
heterotic model possesses an $N=8$ supersymmetry that is
broken spontaneously.
This breaking on the heterotic side is non-perturbative: the Higgs
field whose vev is the order parameter for the spontaneous breaking of
the $N=8$ supersymmetry to $N=4$
is the dilaton $S_{\rm Het}$, and the $N=8$ supersymmetry can be
restored
at the strong coupling limit.
The further breaking of the supersymmetry from $N=4$ to $N=2$
is realized spontaneously in a perturbative way, by using as
Higgs fields the moduli $T$ and $U$.
The full, perturbative and non-perturbative, correction to
the effective coupling constant of the gravitational term
considered here
is given by the type IIA result, Eq. (\ref{thrint}).
In heterotic string variables, we have:
\begin{eqnarray}
{16 \, \pi^2 \over g^2_{\rm grav} \left( \mu \right)}
& = & -2 \log \,{\rm Im}\, \tau_{S_{\rm Het}} \left\vert \vartheta_2
\left( \tau_{S_{\rm Het}} \right) \right\vert^4 \nonumber \\
&& -2 \log \,{\rm Im}\, T \left\vert \vartheta_4 \left( T \right)
\right\vert^4
-2 \log \,{\rm Im}\, U\left\vert
\vartheta_2 \left( U\right)\right\vert^4 \nonumber \\
&& + 6 \log { M_{\rm Planck} \over \mu} + {\rm const.} \, ,
\label{np}
\end{eqnarray}
where we have expressed the infrared running in terms
of the duality-invariant Planck mass and the physical cut-off
$\mu$, related to the various string scales by:
\begin{equation}
{M_{\rm Planck} \over \mu}={M^{(\rm Het)} \over \mu^{(\rm Het)}}=
{M^{(\rm IIA)} \over \mu^{(\rm IIA)}}=
{M^{(\rm As)} \over \mu^{(\rm As)}}\, .
\end{equation}
From expression (\ref{np}) we can easily read off the instanton
numbers $k$,
given by the powers of $q \equiv \exp 2 \pi i \tau_{S_{\rm Het}}$ in the
expansion of the
first term. We obtain $k \in {\kappa N \over 2}$,
which for $\kappa=2$ becomes, as expected, $k \in N$.
\section{The prepotential}
\subsection{\sl The one-loop result}
The perturbative prepotential can be easily computed from the
heterotic side. Owing to the $N=2$ supersymmetry, it receives no
perturbative corrections beyond one loop.
The tree-level contribution is determined by the geometric properties
of the vector manifold, and is the same for both the constructions
based on $\Phi$ and on $\tilde{\Phi}$ \cite{wkll}:
\begin{equation}
h^{(0)}=-{ i \over 2 \pi} \tau_{S_{\rm Het}} T U\, .
\label{h0}
\end{equation}
The genus-one correction, on the other hand,
is different in the two constructions and, moreover, it
depends on the choice of shift vectors in the two-torus.
Here, we will concentrate on the choices made previously
for these shift vectors. The
one-loop corrections to the prepotential turn out to satisfy
second-order differential equations.
These equations are obtained by properly treating the universal part
of the gauge corrections in models with spontaneously broken
supersymmetries, thereby generalizing the approach of
\cite{delgrav, hmbps, agn}.
For the models based on $\Phi$ the correction
$h^{(1)}$ solves
\begin{eqnarray}
\lefteqn{{\rm Re} \Bigg( -{1 \over 2 T_2 U_2}
\left( 1-iU_2 {\partial \over \partial U} \right)
\left( 1-iT_2 {\partial \over \partial T} \right)
h^{(1)} \Bigg)}
\nonumber \\
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~ =
{1 \over 64 \pi^3} \int_{\cal F} {d^2 \tau \over \tau_2}
\sump \Gamma_{2,2}^{\lambda = 1} \ar{h}{g}
\left( i {d \over d \bar{\tau}} + {1 \over \tau_2} \right)
{ \overline{\Omega} \ar{h}{g} \over \bar{\eta}^{24} }\, ,
\label{A}
\end{eqnarray}
while for models based on $\tilde{\Phi}$ it solves
\begin{eqnarray}
\lefteqn{{\rm Re} \Bigg( -{1 \over 2 T_2 U_2}
\left( 1-iU_2 {\partial \over \partial U} \right)
\left( 1-iT_2 {\partial \over \partial T} \right)
\tilde h^{(1)} \Bigg) }
\nonumber \\
&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~ =
{1 \over 64 \pi^3} \int_{\cal F} {d^2 \tau \over \tau_2}
{1 \over 2} \sum_{\lambda=0,1 } \sump \Gamma^{\lambda}_{2,2} \ar{h}{g}
\left( i {d \over d \bar{\tau}} + {1 \over \tau_2} \right)
{ \overline{\Omega}^{(\lambda)} \ar{h}{g} \over \bar{\eta}^{24} }\, .
\label{B}
\end{eqnarray}
Notice that,
in contrast to the universal corrections \cite{kkprn},
the r.h.s. of the above equations has singularities across lines
in the moduli space. Integrals of this kind and analysis of the
singularities have been performed in several papers.
We will not present the general result here but give instead the
answer for the
prepotential. It is important to observe that the above equations
define $h^{(1)}$ up to irrelevant linear and quadratic terms as well
as cubic terms such as $T^2 U$ or $U^2 T$. These ambiguities can be
resolved by looking at the ordinary gravitational threshold
corrections \cite{kkprn, hmbps}:
\begin{equation}
\Delta_{\rm grav}
\left(\Phi\right)
= -{1 \over 12} \int_{\cal F} {d^2 \tau \over \tau_2}
\left( \sump \Gamma_{2,2}^{\lambda = 1} \ar{h}{g} \widehat{E}_2
{\overline{\Omega} \ar{h}{g} \over \bar{\eta}^{24} }
+12 \, b_{\rm grav} \right)
\label{C}
\end{equation}
or
\begin{equation}
\Delta_{\rm grav}
\left(\tilde{\Phi}\right)
= -{1 \over 12} \int_{\cal F} {d^2 \tau \over \tau_2}
\left( {1 \over 2} \sum_{\lambda=0,1}
\sump \Gamma_{2,2}^{\lambda} \ar{h}{g} \widehat{E}_2
{\overline{\Omega}^{(\lambda)} \ar{h}{g} \over \bar{\eta}^{24} }
+12 \, b_{\rm grav} \right)\, .
\label{D}
\end{equation}
We would like to stress at this point that Eqs. (\ref{A}) and (\ref{C})
(resp. (\ref{B}) and (\ref{D})) hold for heterotic constructions of the
kind $\Phi$ (resp. $\tilde{\Phi}$) with $N_V=N_H\neq 0$, as those
presented in \cite{gkp}. Therefore, our results for the perturbative
prepotential $h^{(1)}$ or $\tilde h^{(1)}$ given below (Eqs. (\ref{h})
and (\ref{htilde})) are valid for these more general models. This is again
due to the fact that the heterotic ground states under consideration fall
into the same elliptic-genus universality class, irrespectively of the
value of $N_V=N_H$. Non-perturbative contributions, however, depend on the
number of vector multiplets and hypermultiplets (through, for example, the
instanton numbers), and only the case
$N_V=N_H=0$ is analysed in the following.
After some lengthy algebra, we can solve the above equations and obtain
the one-loop prepotential
for models based on the construction $\Phi$:
\begin{eqnarray}
h^{(1)}(T,U) & = & -{1\over (2 \pi )^4 }
\left(
{\cal L}_c(T,U)+{\cal L}_a(T,U)+{\cal L}_b(T,U)
\right)
- {i \over 8 \pi} T^2 U
\ , \ \
{\rm for}
\ \
T_2 > U_2 \nonumber \\
& = & - {1 \over (2 \pi )^4 }
\left(
{\cal L}_c(U,T)+{\cal L}_a(T,U)+{\cal L}_b(U,T)
\right)
- {i \over 8 \pi} T U^2
\ , \ \
{\rm for}
\ \
T_2 < U_2\, . \nonumber \\
&& \label{h}
\end{eqnarray}
The functions ${\cal L}_{c,a,b}(T,U)$ are given in the appendix;
${\cal L}_{c,b}(T,U)$
have a branch along $T=U$, where
$\left. \Delta B_2\right\vert_{\rm massless}=
\Delta N_V - \Delta N_H= -14$. For models based on the construction
$\tilde \Phi$, we obtain:
\begin{eqnarray}
\tilde{h}^{(1)}(T,U) & = & -{1 \over (2 \pi)^4} {1 \over 2}
\Bigg(-{\cal L}^{(0)}_c(T,U)
+{\cal L}^{(0)}_a\left({T\over 2},2U\right)
+{\cal L}^{(0)}_b\left({T\over 2},U\right)
\nonumber \\
&&\ \ \ \ \ \ \ \ \ \ \ +{\cal L}^{(1)}_c(T,U)
+{\cal L}^{(1)}_a(T,U) +{\cal L}^{(1)}_b(T,U)
\Bigg)
\, ,
\label{htilde}
\end{eqnarray}
where the functions ${\cal L}^{(\lambda)}_{c,a,b}(T,U)$
are as displayed in the appendix.
In this case there is no branch at $T=U$, where now $\left. \Delta
B_2\right\vert_{\rm massless}= 0$, and (\ref{htilde}) is thus valid
for any $T$ and $U$. This makes the monodromy trivial around $T=U$.
Remember, however, that in the models at hand, where the two-torus
lattices are shifted, the target-space duality group is only a subgroup of
$SL(2,Z)_T \times SL(2,Z)_U \times Z_2^{T\leftrightarrow U}$. In
particular $T\to -1/T$ is not a symmetry, and the line $T=U$ is not
equivalent to the line $-1/T=U$. The latter, where $\left. \Delta
B_2\right\vert_{\rm massless}= 2$, is a branch for $\tilde{h}^{(1)}(T,U)$,
although this is not straightforward from expression (\ref{htilde}) -- a
Poisson resummation is needed.
Observe also the absence of cubic terms in $\tilde{h}^{(1)}$. Such terms
are present in generic models with spontaneously broken
supersymmetry, as those studied in \cite{kkprn}, for which the
corrections to the prepotential can be computed in a similar way.
Cubic terms vanish in general when the intersection form of the
underlying Calabi--Yau manifold of the type II dual becomes
trivial\footnote{This implies that the heterotic construction based
on $\Phi$, in which such cubic terms are present (see Eq. (\ref{h})),
cannot be dual to a type II,
$Z_2 \times Z_2$ symmetric orbifold, for which the intersection matrix
is trivial.},
as in our case. On the other hand, the absence of
constant term both in $h^{(1)}$ and $\tilde{h}^{(1)}$ reflects the
vanishing of the Euler characteristic $\chi=2\left(
h^{1,1}-h^{2,1}\right)$.
\subsection{\sl Non-perturbative contributions}
Let us now try to go beyond the above perturbative result. We will
concentrate on the construction based on $\tilde{\Phi}$, for which we have
been able to determine the precise duality map. For the prepotential,
however, we have no exact type II result that could be used to
obtain the heterotic non-perturbative contributions directly.
Nevertheless, the structure of the type II model is useful to infer at
least part of these contributions.
The type II $(1,1)$ ground state of
Section 2 is symmetric under permutations of the three moduli $\{T^1,
-1/T^2, T^3\}$.
The heterotic dual should therefore possess the same
property with respect to
$\{\tau_{S_{\rm Het}}, -1/T,U\}$. In fact, invariance under
$-1/T \leftrightarrow U$ is a residual target-space duality symmetry
of the shifted lattices we are
considering, and Eq. (\ref{B}) is indeed invariant owing to the
covariance property\footnote{The absence of pure power-like terms in
(\ref{cov}) reflects again the vanishing of the Euler characteristic and
intersection form of the dual symmetric type II construction.}
of the perturbative result
$\tilde{h}^{(1)}(T,U)$ given in (\ref{htilde}):
\begin{equation}
\tilde{h}^{(1)}\left(-{1\over U},-{1\over T}\right)={1 \over T^2\, U^2}
\tilde{h}^{(1)}(T,U)
\label{cov}
\end{equation}
(note, however, that $\tilde{h}^{(1)}(U,T)\neq\tilde{h}^{(1)}(T,U)$,
owing to the breakdown of the $T \leftrightarrow U$ symmetry).
In order to promote the above $-1/T \leftrightarrow U$
permutation symmetry to the level of the three
moduli, we must demand the following covariance properties for the full
prepotential $\tilde{h}\left(\tau_{S_{\rm Het}},T,U\right)$:
\begin{eqnarray}
\tilde{h}\left(\tau_{S_{\rm Het}},-{1\over U},-{1\over T}\right)&=&
{1 \over T^2\, U^2}
\tilde{h}\left(\tau_{S_{\rm Het}},T,U\right)\nonumber \\
\tilde{h}\left(-{1\over T},-{1\over \tau_{S_{\rm Het}}},U\right)&=&
{1 \over \tau_{S_{\rm Het}}^2\, T^2}
\tilde{h}\left(\tau_{S_{\rm Het}},T,U\right)\label{npcov} \\
\tilde{h}\left(U,T,\tau_{S_{\rm Het}}\right)&=&
\tilde{h}\left(\tau_{S_{\rm Het}},T,U\right)\, ,\nonumber
\end{eqnarray}
which are fulfilled by the tree-level contribution (\ref{h0}). We must
therefore add two more terms to
$\tilde{h}^{(1)}(T,U)$:
\begin{equation}
\tilde{h}\left(\tau_{S_{\rm Het}},T,U\right)= h^{(0)}+
\tilde{h}^{(1)}(T,U)+
\tilde{h}^{(1)}\left(T,\tau_{S_{\rm Het}}\right)+
T^2\, U^2 \, \tilde{h}^{(1)}\left(-{1\over U},\tau_{S_{\rm Het}}\right)\,
.
\label{htsym}
\end{equation}
These extra terms account for non-perturbative corrections and are
exponentially suppressed at large $S_{\rm Het}$.
The above covariant symmetrization, which we have been advocating in
order to determine non-perturbative corrections to the prepotential,
does not exclude the possibility of having also a
series of exponentially suppressed terms with, in the arguments,
covariant-symmetric functions of $\tau_{S_{\rm Het}},T$ and $U$,
in the sense of (\ref{npcov}).
Unfortunately, we
have no reason to rule out such non-perturbative contributions, nor
a method for computing them from the type II symmetric or asymmetric
ground states.
\section{Conclusions}
In this work we studied $N=2$ superstring ground
states obtained from the heterotic and type II ten-dimensional
superstrings through freely acting (asymmetric) orbifold
compactification.
The massless spectrum of all these models is the same. Besides
the $N=2$ supergravity multiplet, there are three vector multiplets and
four hypermultiplets. The construction presented here extends the work
of Ref. \cite{gkp},
where we analysed heterotic/type II duals with $N=2$ supersymmetries that have
$3+N_V$ vector multiplets and $4+N_H$ hypermultiplets with $N_V=N_H
\ne 0$.
Here, we presented three different $N=2$ models with $N_V=N_H = 0$
and we verified the non-perturbative duality conjecture between them.
The models we have considered are the following:
(\romannumeral1) the heterotic construction with $N=(2,0)$
supersymmetry, based on
characters $\tilde \Phi$ of the $c=(0,16)$ conformal block, Eq.
(\ref{88fivt}) (this choice is equivalent to
a particular choice of discrete Wilson lines, reducing the number of
the vectors to three and that of the hypers to four);
(\romannumeral2) the type IIA symmetric construction with $N=(1,1)$
supersymmetry, which corresponds to a self-mirror Calabi--Yau
compactification with Hodge numbers $h^{1,1}=h^{2,1}=3$;
(\romannumeral3) the type II asymmetric $N=(2,0)$
freely acting orbifold
compactification,
where the initial $N=(4,4)$ supersymmetry is spontaneously broken to
$N=(2,0)$.
The equivalence of the heterotic $N=(2,0)$, type IIA $N=(1,1)$ and
$N=(2,0)$ was verified for the
corrections of a modified gravitational and gauge combination
associated with the operator $\overline{P}_{\rm grav}^{\prime 2}$
introduced in
Section 4. This operator has the property of being regular in the
entire $(T,U)$ moduli space.
In the duality relations between the constructions described above, the
heterotic vector moduli
$(\tau_{S_{\rm Het}},T,U)$ are mapped to the three
$h^{1,1}$ moduli $(T^1,T^2,T^3)$ of the symmetric type II, as well as
to the moduli of the asymmetric type II $(\tilde{\tau}_{S_{\rm As}},
T_{\rm As}, U_{\rm As})$,
where $\tilde{\tau}_{S_{\rm As}}=-1/\tau_{S_{\rm As}}$ is the inverse of
the asymmetric type II dilaton. Thus, there is a weak--strong coupling
$S$-duality relation between the heterotic and
the asymmetric type II ground state, $\tau_{S_{\rm Het}}
=-1/\tau_{S_{\rm As}}$. In all above duals there is a
(non-)perturbative restoration of $N=8$ and $N=4$ supersymmetry in
some specific limits of the three moduli, which is in agreement
with the duality maps.
By using these duality maps,
we found that {\it the type II
corrections provide the complete, perturbative and non-perturbative,
heterotic corrections}, as was
also the case for all $N=2$ models with $N_V=N_H$ constructed in
Ref. \cite{gkp}. This remarkable property is due to the universality
of the $N=2$ sector in the heterotic orbifold and of the
corresponding $N=(2,2)$ sectors in the symmetric and asymmetric type
II orbifolds. We obtained in this way the full gravitational heterotic
corrections. These contain instanton corrections,
$n_k\,\exp {2k\pi i \tau_{S_{\rm Het}}}$, which are due
to the Euclidean five-brane wrapped around the six-dimensional internal
space; they depend only on $\tau_{S_{\rm Het}}$ and not on the other
moduli. The explicit expressions for these corrections are given in
Eq. (\ref{np}). The Olive--Montonen duality group is a $\Gamma(2)$
subgroup of $SL(2,Z)_{\tau_{S_{\rm Het}}}$.
Finally, we computed the perturbative
and part of the non-perturbative corrections to the prepotential for the
heterotic ground state.
The perturbative result is actually valid beyond
the models presented in this paper, and covers more general situations,
with $N_V=N_H\neq 0$ such as those of Ref. \cite{gkp}. The
non-perturbative piece
was reached by using the triality symmetry between the three moduli
of the type II symmetric model
$T^1\leftrightarrow -1/T^2\leftrightarrow T^3$. The requirement
of (partial) restoration
of the ($N=4$) $N=8$ supersymmetry in special limits of the
vector moduli turned to be too weak to rule out or determine
extra potential non-perturbative terms in the prepotential.
\vskip 1.cm
\centerline{\bf Acknowledgements}
\noindent
The authors thank E. Kiritsis for valuable discussions.
P.M. Petropoulos acknowledges the hospitality of the CERN Theory Division.
A. Gregori thanks the Swiss National Science Foundation and
the Swiss Office for Education and Science (ofes 95.0856).
This work was partially supported by the EEC under the contracts
TMR-ERBFMRX-CT96-0045 and TMR-ERBFMRX-CT96-0090.
\vskip 0.3cm
\setcounter{section}{0}
\setcounter{equation}{0}
\renewcommand{\oldtheequation\alph{equation}}}{A. \arabic{equation}}
\section*{\normalsize{\centerline{\bf Appendix:
The trilogarithm series}}}
\noindent
We quote here the explicit solution for the series ${\cal L}_{c,a,b}$
and ${\cal L}^{(\lambda)}_{c,a,b}(T,U)$
appearing in the expressions of the
one-loop prepotential (see
Section 6).
We have:
\begin{eqnarray}
{\cal L}_{c}(T,U) &=&
\Li_3\left(
e^{2\pi i\left(T-U\right)}
\right)
\nonumber\\
&&+c_0 \sum_{k>0}\left(2\Li_3\left(e^{4\pi i
T k }\right)-\Li_3\left(e^{2\pi i
T k}\right)\right)
\nonumber\\
&&+c_0 \sum_{\ell>0}\left(2\Li_3\left(e^{4\pi i
U \ell }\right)-\Li_3\left(e^{2\pi i
U \ell}\right)\right)
\nonumber\\
&&
+\sum_{k,\ell>0}\Big(
-c_{k\ell}\,\Lii{T k+U \ell}\nonumber\\
&&\ \ \ \ \ \ \ \ \
+2\,c_{4k\ell}\,\Lii{2 T k +2 U \ell}
\nonumber\\
&&\ \ \ \ \ \ \ \ \
+2\,c_{4k\ell-2k-2\ell+1}\Lii{T(2k-1)+U(2\ell-1)}\Big)
\label{Lc}
\end{eqnarray}
\begin{eqnarray}
{\cal L}_{a}(T,U) &=&
\sum_{k,\ell>0}\bigg(
a_{4k\ell-3k-3\ell+2}\,\Lii{\frac{T}{2}(4k-3)+
\frac{U}{2}(4\ell-3)}\nonumber\\
&&\ \ \ \ \ \
+a_{4k\ell-k-\ell}\,\Lii{\frac{T}{2}(4k-1)+
\frac{U}{2}(4\ell-1)}
\bigg)
\label{La}
\end{eqnarray}
\begin{eqnarray}
{\cal L}_{b}(T,U) &=& b_{-1}
\Li_3\left(
e^{2\pi i\left(\frac{T}{2}-\frac{U}{2}\right)}
\right)
\nonumber\\
&&+\sum_{k,\ell>0}\bigg(
b_{4k\ell-k-3\ell}\,\Lii{\frac{T}{2}(4k-3)+
\frac{U}{2}(4\ell-1)}\nonumber\\
&&\ \ \ \ \ \ \ \ \
+b_{4k\ell-3k-\ell}\,\Lii{\frac{T}{2}(4k-1)+
\frac{U}{2}(4\ell-3)}
\bigg)\, ,
\label{Lb}
\end{eqnarray}
where the coefficients in the above expansions are defined through
\begin{eqnarray}
\frac{\Omega\oao{0}{1}}{\eta^{24}}&=&
\frac{1}{q} +
\sum_{n\ge 0}c_n\,q^n\nonumber\\
\frac{\Omega\oao{1}{0}+\Omega\oao{1}{1}}{\eta^{24}}&=&
\sum_{n\ge 0}a_n\,q^{n+\frac{1}{4}}\label{Oex}\\
\frac{\Omega\oao{1}{0}-\Omega\oao{1}{1}}{\eta^{24}}&=&
\sum_{n\ge -1}b_n\,q^{n+\frac{3}{4}}\, .\nonumber
\end{eqnarray}
We also have
\begin{eqnarray}
{\cal L}^{(0)}_{c}(T,U)&=&
\Li_3
\left(e^{2\pi i
\left(T-U \right)}\right)
\nonumber \\
&&+c^{(0)}_0 \sum_{k>0}
\left(\Li_3
\left(e^{2\pi i
T k}
\right)
+ \Li_3
\left(e^{2\pi i
U k }
\right)
\right)
\nonumber \\
&&+\sum_{k,\ell>0}
c_{k\ell}^{(0)}\,\Li_3
\left(e^{2\pi i
(T k +U \ell)}\right)
\label{Lc0}
\end{eqnarray}
\begin{eqnarray}
{\cal L}_{a}^{(0)}\left(\frac{T}{2},2U\right) &=&
a^{(0)}_0 \sum_{k>0}\left(\Li_3\left(e^{\pi i
T k }\right)+\Li_3\left(e^{4\pi i
U k}\right)\right)
\nonumber\\
&&
+\sum_{k,\ell>0}
a_{k\ell}^{(0)}\,\Li_3
\left(e^{2\pi i
\left(\frac{T}{2} k +2 U \ell\right)}\right)
\label{La0}
\end{eqnarray}
\begin{eqnarray}
{\cal L}_{b}^{(0)}\left(\frac{T}{2},U\right) &=&
288 \sum_{k>0}\left(\Li_3\left(e^{2\pi i
T k }\right)+\Li_3\left(e^{4\pi i
U k}\right)\right)
\nonumber\\
&&
+\sum_{k,\ell>0}\bigg(
\left(2\, c_{2k\ell}^{(0)}-a_{2k\ell}^{(0)}\right)
\Li_3\left(e^{2\pi i(T k +2 U \ell)}\right)\nonumber\\
&&\ \ \ \ \ \ \ \ \
+b_{2k\ell-k-\ell}^{(0)}\,\Li_3
\left(e^{2\pi i
\left(\frac{T}{2} (2k-1) + U (2 \ell-1)\right)}\right)
\bigg)\, ,
\label{Lb0}
\end{eqnarray}
and
\begin{eqnarray}
{\cal L}_{c}^{(1)}(T,U) &=&
{\cal L}_{c^{(1)}}(T,U)
\label{L1c}\\
{\cal L}_{a}^{(1)}(T,U) &=&
{\cal L}_{a^{(1)}}(T,U)
\label{L1a}\\
{\cal L}_{b}^{(1)}(T,U) &=&
{\cal L}_{b^{(1)}}(T,U)\, ,
\label{L1b}\\
\end{eqnarray}
where ${\cal L}_{c^{(1)},a^{(1)},b^{(1)}}(T,U)$ are displayed in
(\ref{Lc})--(\ref{Lb}), and
$c_n^{(\lambda)}$, $a_n^{(\lambda)}$ and $b_n^{(\lambda)}$
are given by
\begin{eqnarray}
\frac{\Omega^{(\lambda)}\oao{0}{1}}{\eta^{24}}&=&
\frac{1}{q} +
\sum_{n\ge 0}c_n^{(\lambda)}\,q^n\nonumber\\
\frac{\Omega^{(\lambda)}\oao{1}{0}+\Omega^{(\lambda)}\oao{1}{1}}{\eta^{24}}&=&
\sum_{n\ge 0}a_n^{(\lambda)}\,q^{n+\frac{\lambda}{4}}\label{Oex1}\\
\frac{\Omega^{(\lambda)}\oao{1}{0}-\Omega^{(\lambda)}\oao{1}{1}}{\eta^{24}}&=&
\sum_{n\ge 0}b_n^{(\lambda)}\,q^{n+\frac{\lambda+2}{4}}\, .\nonumber
\end{eqnarray}
We finally recall that the non-trivial monodromy properties of the above
functions are due to the following connexion formula for the trilogarithm:
$$
\Li_3\left(e^{x}\right)=
\Li_3\left(e^{-x}\right)+
{\pi^2 \over 3}x-
{i\pi \over 2}x^2-
{1 \over 6}x^3 \ , \ \ {\rm for \ \ } \,{\rm Re}\, x \geq 0
\, .
$$
\clearpage
\noindent
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"redpajama_set_name": "RedPajamaArXiv"
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Huasco River is a river of Chile located in the Huasco Province, Atacama Region. Its headwaters in the upper catchments in high-altitude Andes glaciers are the Estrecho River, a tributary of El Tránsito River (catchment area 4,135 km2), and Potrerillos River, a tributary of El Carmen River (catchment area 2 890 km2). The Huasco River begins at the confluence of El Tránsito River and El Carmen River which is located in Junta del Carmen (790ma.s.l.). A small portion of its course is impounded by a small dam forming the Embalse Santa Juana.
Cities and towns along the Huasco include: Vallenar, Freirina and Huasco.
Andean glaciers and the Huasco River Basin
The inhabitants of the Huasco valley, a semi-arid region, depend on water resources from the upper catchments in high-altitude Andes glaciers which contribution to the discharge of two Huasco River headwaters: the Estrecho River and the Potrerillos River which arise from two small neighboring catchments, they actually belong to two major subcatchments of the Huasco Basin.
A collaborative study between the Centro de Estudios Avanzados en Zonas Áridas (CEAZA) and the Laboratoire de Glaciologie et Géophysique de l'Environnement (LGGE) investigated the glacier contribution to the Huasco River basins by two glaciated headwater catchments which included the monitoring of five Andean glaciers (Toro 1, Toro 2, Esperanza, Guanaco, Estrecho and Ortigas) between 2003/2004 and 2007/2008 hydrological years. The Andean "glaciers accelerated retreat" represents a "striking example of climate change impacts."
Concerns were raised by Sustainable Chile Program president, Sara Larraín, a Chilean politician and environmentalist who ran for president in 1999 presidential election, that the Andean glaciers, particularly Toro 1, Toro 2 and Esperanza, were endangered by the Pascua Lama project.
In May 2013, Chile's Superintendence of the Environment Superintendencia del Medio Ambiente (SMA) notified Barrick Gold that the company had to cease construction activities at Pascua-Lama until they complete water management system in accordance with the project's environmental permit. Barrick Gold was also fined approximately $16 million for noncompliance regarding the project's water management system.
See also
List of rivers of Chile
References
External links
Cuenca del Río Huasco, Dirección General de Aguas, Ministerio de Obras Públicas, Gobierno de Chile
Rivers of Atacama Region
Rivers of Chile
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The court was to pass the winter there, and quintanilla hoped to secure an audience for his new friend.
Your other evidences, that the new testament approves of slavery, unimportant as they are, will not be left unnoticed.
"yet, you see," she cried, appealing to her husband, and even to hortensia, who sat apart, scarce heeding this trivial matter of which so much was being made, "you see that he evades the point, avoids a direct answer to the question that is raised."
Tirant soon saw this signal which they had planned in advance, and he quickly left the camp with only a few men.
While it has never advocated 'direct action' or the avoidance of political activity, while on the contrary, it has advocated the conquest of social reforms on the fields of parliamentary and municipal government, it has not defended the state as it is, but has rather urged the need for a state which is based on democracy tempered by respect for the 'expert.
They often contribute towards the support of christian pastor or teacher, and in various other ways evince their sympathy and reveal their intellectual assent.
"rest easy on that score, evans," replies the don.
At all events, we had nothing to do but watch him.
That the exclusion of chinese labor is demanded in other countries where like conditions prevail is strongly evidenced in the dominion of canada, where chinese immigration is now regulated by laws more exclusive than our own.
There is no evidence how he came by this knowledge.
These licences were an authorized mode of evading that very prohibition which the belligerents conceived it to be for their interest to maintain.
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Aldo Montano (Livorno, 18 de noviembre de 1978) es un deportista italiano que compite en esgrima, especialista en la modalidad de sable.
Participó en cinco Juegos Olímpicos de Verano, entre los años 2004 y 2020, obteniendo en total cinco medallas: oro y plata en Atenas 2004, en las pruebas individual y por equipos (junto con Giampiero Pastore y Luigi Tarantino); bronce en Pekín 2008, en la prueba por equipos (con Diego Occhiuzzi, Luigi Tarantino y Giampiero Pastore); bronce en Londres 2012, por equipos (con Diego Occhiuzzi, Luigi Tarantino y Luigi Samele), y plata en Tokio 2020, por equipos (junto con Enrico Berrè, Luca Curatoli y Luigi Samele).
Ganó 12 medallas en el Campeonato Mundial de Esgrima entre los años 2002 y 2019, y 11 medallas en el Campeonato Europeo de Esgrima entre los años 2002 y 2019.
Proviene de una familia de exitosos esgrimidores: su padre, Mario Aldo, fue campeón olímpico y bicampeón mundial; su abuelo Aldo, dos veces subcampeón olímpico y pentacampeón mundial, y sus tíos Mario Tullio, Carlo y Tommaso, también medallistas olímpicos y mundiales.
Palmarés internacional
Referencias
Esgrimidores de Italia
Esgrimidores en los Juegos Olímpicos de Atenas 2004
Esgrimidores en los Juegos Olímpicos de Pekín 2008
Esgrimidores en los Juegos Olímpicos de Londres 2012
Esgrimidores en los Juegos Olímpicos de Río de Janeiro 2016
Esgrimidores en los Juegos Olímpicos de Tokio 2020
Medallistas olímpicos de oro de esgrima
Medallistas olímpicos de plata de esgrima
Medallistas olímpicos de bronce de esgrima
Medallistas olímpicos de oro de Italia
Medallistas olímpicos de plata de Italia
Medallistas olímpicos de bronce de Italia
Medallistas olímpicos de oro de Atenas 2004
Medallistas olímpicos de plata de Atenas 2004
Medallistas olímpicos de bronce de Pekín 2008
Medallistas olímpicos de bronce de Londres 2012
Medallistas olímpicos de plata de Tokio 2020
Deportistas de Italia en los Juegos Olímpicos de Tokio 2020
Campeones mundiales de esgrima
Nacidos en Livorno
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